Relevant Terms	Irrelevant Terms
h^#(f(c), f(b))
h^#(f(a), f(d))
h^#(f(a), U111(b, b))
h^#(f(d), f(b))
h^#(U111(a, a), f(b))
h^#(f(a), f(c))

Relevant Terms	Irrelevant Terms
h^#(f(c), U111(b, b))
h^#(U111(c, c), f(b))
h^#(f(l), f(b))
h^#(f(e), f(b))
h^#(f(c), f(c))
h^#(f(c), f(d))

Relevant Terms	Irrelevant Terms
h^#(f(d), f(c))	h^#(U111(d, d), f(b))
h^#(f(d), U111(b, b))
h^#(f(d), f(d))

Relevant Terms	Irrelevant Terms
h^#(f(e), f(d))
h^#(U111(e, e), f(b))
h^#(f(e), f(c))
h^#(f(e), U111(b, b))

Relevant Terms	Irrelevant Terms
h^#(U111(a, a), f(c))	h^#(U111(d, a), f(b))
h^#(U111(a, a), f(d))
h^#(U111(a, a), U111(b, b))
h^#(U111(c, a), f(b))

Relevant Terms	Irrelevant Terms
h^#(U111(a, a), U111(c, b))	h^#(U111(a, a), U111(d, b))
h^#(U111(c, a), U111(b, b))	h^#(U111(d, a), U111(b, b))

Relevant Terms	Irrelevant Terms
h^#(U111(c, a), f(c))	h^#(U111(d, a), f(c))
h^#(U111(a, a), f(l))
h^#(U111(a, a), U111(c, c))
h^#(U111(a, a), f(e))

Relevant Terms	Irrelevant Terms
h^#(U111(a, a), f(d))	h^#(f(a), U111(d, d))
h^#(f(d), f(d))
h^#(f(c), f(d))

Relevant Terms	Irrelevant Terms
h^#(f(d), f(c))
h^#(U111(a, a), f(c))
h^#(f(a), U111(c, c))
h^#(f(a), f(e))
h^#(f(c), f(c))
h^#(f(a), f(l))

Relevant Terms	Irrelevant Terms
h^#(U111(c, a), f(d))	h^#(U111(a, a), U111(d, d))
	h^#(U111(d, a), f(d))

Relevant Terms	Irrelevant Terms
h^#(f(d), f(e))	h^#(U111(d, d), f(c))
h^#(f(d), U111(c, c))
h^#(f(d), f(l))

Relevant Terms	Irrelevant Terms
h^#(U111(d, d), U111(c, c))	h^#(f(d), U111(l, c))
h^#(f(d), U111(e, c))

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(U111(e, c), f(c))
A^#	→	h^#(c, f(l))	A^#	→	h^#(l, f(c))
A^#	→	h^#(f(d), e)	A^#	→	h^#(f(e), f(l))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(a), U111(c, c))
A^#	→	h^#(U111(e, c), U111(b, b))	A^#	→	h^#(U111(a, a), U111(c, c))
A^#	→	h^#(d, f(e))	A^#	→	h^#(e, f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(c), U111(b, b))	A^#	→	h^#(f(d), U111(e, e))
A^#	→	h^#(U111(e, e), f(l))	A^#	→	h^#(e, c)
A^#	→	h^#(U111(e, e), U111(d, d))	A^#	→	h^#(l, U111(c, b))
A^#	→	h^#(U111(e, c), f(e))	A^#	→	h^#(U111(a, a), U111(b, b))
A^#	→	h^#(U111(e, c), f(l))	A^#	→	h^#(c, U111(c, c))
A^#	→	h^#(U111(e, a), e)	A^#	→	h^#(U111(c, a), c)
A^#	→	h^#(f(l), U111(b, b))	A^#	→	h^#(c, f(e))
A^#	→	h^#(e, f(c))	A^#	→	h^#(c, f(d))
A^#	→	h^#(l, f(b))	A^#	→	h^#(e, f(b))
A^#	→	h^#(U111(a, a), U111(e, e))	A^#	→	h^#(U111(l, l), U111(c, c))
A^#	→	h^#(f(d), c)	A^#	→	h^#(a, U111(e, e))
A^#	→	h^#(c, U111(c, b))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(U111(c, a), f(l))
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(f(l), c)
A^#	→	h^#(a, U111(e, c))	A^#	→	h^#(U111(e, a), f(c))
A^#	→	h^#(U111(a, a), f(l))	A^#	→	h^#(e, f(l))
A^#	→	h^#(U111(e, e), f(e))	A^#	→	h^#(U111(c, c), U111(c, b))
A^#	→	h^#(U111(a, a), e)	A^#	→	h^#(U111(e, a), f(b))
A^#	→	h^#(f(a), U111(c, b))	A^#	→	h^#(U111(c, a), f(d))
A^#	→	h^#(U111(a, a), f(c))	A^#	→	h^#(U111(c, c), U111(e, e))
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U111(e, e), U111(c, b))	A^#	→	h^#(U111(d, d), U111(e, c))
A^#	→	h^#(f(c), U111(e, e))	A^#	→	h^#(U111(e, a), c)
A^#	→	h^#(U111(e, e), U111(c, c))	A^#	→	h^#(f(e), U111(b, b))
A^#	→	h^#(f(d), U111(c, b))	A^#	→	h^#(U111(a, a), U111(c, b))
A^#	→	h^#(d, U111(e, c))	A^#	→	h^#(e, U111(e, b))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U111(c, a), U111(b, b))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Context-sensitive strategy:
μ(d) = μ(e) = μ(b) = μ(c) = μ(A) = μ(a) = μ(l) = μ(a#) = μ(k) = μ(T) = μ(A#) = μ(k#) = μ(b#) = μ(c#) = ∅
μ(f) = μ(f#) = μ(U111) = μ(U111#) = {1}
μ(h#) = μ(h) = {1, 2}
μ(g) = μ(g#) = {1, 2, 3}

The right-hand side of the rule A^# → h^#(U111(e, c), f(c)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
h^#(c, f(c))
h^#(U111(e, c), U111(c, c))
h^#(U111(e, c), f(l))
h^#(U111(e, c), f(e))

Thus, the rule A^# → h^#(U111(e, c), f(c)) is replaced by the following rules:

A^# → h^#(U111(e, c), f(e))	A^# → h^#(c, f(c))
A^# → h^#(U111(e, c), U111(c, c))	A^# → h^#(U111(e, c), f(l))

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(l), b)
A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(U111(e, c), U111(b, b))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(d), U111(e, e))	A^#	→	h^#(U111(e, e), f(l))
A^#	→	h^#(U111(e, e), U111(d, d))	A^#	→	h^#(e, c)
A^#	→	h^#(l, U111(c, b))	A^#	→	h^#(U111(e, c), f(e))
A^#	→	h^#(f(e), U111(e, c))	A^#	→	h^#(U111(a, a), U111(b, b))
A^#	→	h^#(U111(e, c), U111(c, b))	A^#	→	h^#(U111(e, c), f(l))
A^#	→	h^#(c, U111(c, c))	A^#	→	h^#(U111(e, a), e)
A^#	→	h^#(U111(c, a), c)	A^#	→	h^#(f(l), U111(b, b))
A^#	→	h^#(e, f(c))	A^#	→	h^#(c, f(e))
A^#	→	h^#(c, f(d))	A^#	→	h^#(l, f(b))
A^#	→	h^#(e, f(b))	A^#	→	h^#(f(c), U111(e, b))
A^#	→	h^#(f(e), U111(c, b))	A^#	→	h^#(U111(a, a), U111(e, e))
A^#	→	h^#(U111(l, l), U111(c, c))	A^#	→	h^#(f(d), c)
A^#	→	h^#(a, U111(e, e))	A^#	→	h^#(c, U111(c, b))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U111(c, a), f(l))	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(f(l), c)	A^#	→	h^#(a, U111(e, c))
A^#	→	h^#(U111(e, a), f(c))	A^#	→	h^#(U111(a, a), f(l))
A^#	→	h^#(e, f(l))	A^#	→	h^#(l, f(e))
A^#	→	h^#(U111(e, e), f(e))	A^#	→	h^#(U111(c, c), U111(c, b))
A^#	→	h^#(U111(a, a), e)	A^#	→	h^#(f(a), l)
A^#	→	h^#(f(a), U111(c, b))	A^#	→	h^#(U111(e, a), f(b))
A^#	→	h^#(U111(c, a), f(d))	A^#	→	h^#(l, U111(c, c))
A^#	→	h^#(U111(a, a), f(c))	A^#	→	h^#(U111(a, a), U111(e, c))
A^#	→	h^#(U111(c, c), U111(e, c))	A^#	→	h^#(U111(c, c), U111(e, e))
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U111(e, e), U111(c, b))
A^#	→	h^#(U111(d, d), U111(e, c))	A^#	→	h^#(U111(l, l), U111(e, b))
A^#	→	h^#(f(c), U111(e, e))	A^#	→	h^#(U111(e, a), c)
A^#	→	h^#(U111(e, e), U111(c, c))	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	h^#(f(e), U111(b, b))	A^#	→	h^#(f(d), U111(c, b))
A^#	→	h^#(U111(a, a), U111(c, b))	A^#	→	h^#(U111(a, a), l)
A^#	→	h^#(d, U111(e, c))	A^#	→	h^#(e, U111(e, b))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U111(c, a), U111(b, b))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule A^# → h^#(f(l), b) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
h^#(f(l), d)	h^#(U111(l, l), b)
h^#(f(l), c)

Thus, the rule A^# → h^#(f(l), b) is replaced by the following rules:

A^# → h^#(f(l), d)

A^# → h^#(f(l), c)

Problem 69: Propagation

Dependency Pair Problem

Dependency Pairs

h^#(c, c)	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
A^#	→	h^#(e, e)	h^#(e, e)	→	g^#(e, e, f(k))
A^#	→	h^#(c, c)	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	A^#	→	h^#(l, l)
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
A^#	→	h^#(U111(e, c), U111(e, c))	A^#	→	h^#(f(e), f(e))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The dependency pairs A^# → h^#(c, c) and h^#(c, c) → g^#(c, c, f(k)) are consolidated into the rule A^# → g^#(c, c, f(k)) .

This is possible as

all subterms of h#(c, c) containing variables are rooted by a constructor symbol,
there is no variable that is replacing in h#(c, c), but non-replacing in both A# and g#(c, c, f(k))

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(e, e) are consolidated into the rule g^#(d, x, x) → h^#(e, e) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(e, e)

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(e, e) are consolidated into the rule g^#(d, x, x) → h^#(e, e) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(e, e)

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(e, e) are consolidated into the rule g^#(d, x, x) → h^#(e, e) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(e, e)

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(e, e) are consolidated into the rule g^#(d, x, x) → h^#(e, e) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(e, e)

Summary

Removed Dependency Pairs	Added Dependency Pairs
h^#(c, c) → g^#(c, c, f(k))	A^# → g^#(c, c, f(k))
A^# → h^#(e, e)	g^#(d, x, x) → h^#(e, e)
A^# → h^#(c, c)
g^#(d, x, x) → A^#

Problem 75: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(k))	h^#(e, e)	→	g^#(e, e, U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), f(c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(c), f(l), f(l))	g^#(f(c), f(c), U111(l, l))
g^#(f(e), f(c), f(l))
g^#(f(c), f(e), f(l))
g^#(U111(c, c), f(c), f(l))
g^#(f(l), f(c), f(l))
g^#(f(c), U111(c, c), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), f(c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(l))	h^#(f(c), f(c)) → g^#(f(e), f(c), f(l))
h^#(f(c), f(c)) → g^#(f(c), f(l), f(l))	h^#(f(c), f(c)) → g^#(f(c), f(e), f(l))
h^#(f(c), f(c)) → g^#(U111(c, c), f(c), f(l))	h^#(f(c), f(c)) → g^#(f(l), f(c), f(l))

Problem 76: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(k))
A^#	→	h^#(d, d)	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(k))
h^#(e, e)	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(l), U111(c, c), f(k))	g^#(f(c), U111(l, c), f(k))
g^#(f(c), U111(e, c), f(k))
g^#(f(e), U111(c, c), f(k))
g^#(f(c), U111(c, c), f(d))
g^#(U111(c, c), U111(c, c), f(k))
g^#(f(c), U111(c, c), U111(k, k))
g^#(f(c), U111(c, c), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(l))	h^#(f(c), f(c)) → g^#(f(c), U111(e, c), f(k))
h^#(f(c), f(c)) → g^#(f(e), U111(c, c), f(k))	h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c)) → g^#(f(l), U111(c, c), f(k))	h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(d))
h^#(f(c), f(c)) → g^#(f(c), U111(c, c), U111(k, k))

Problem 77: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(k))	h^#(e, e)	→	g^#(e, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	h^#(f(l), f(l))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(c, c), U111(c, c), f(d))	g^#(f(c), U111(c, c), U111(d, d))
g^#(f(c), U111(e, c), f(d))	g^#(f(c), U111(l, c), f(d))
g^#(f(e), U111(c, c), f(d))
g^#(f(l), U111(c, c), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), U111(c, c), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(c), U111(e, c), f(d))	h^#(f(c), f(c)) → g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), f(d))	h^#(f(c), f(c)) → g^#(f(l), U111(c, c), f(d))

Problem 78: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(k))
h^#(e, e)	→	g^#(e, e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	h^#(f(l), f(l))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(e, e)	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, U111(e, c), U111(k, k))
g^#(c, c, f(k))
g^#(c, U111(e, c), f(d))
g^#(e, U111(e, c), f(k))
g^#(c, U111(e, c), f(l))
g^#(l, U111(e, c), f(k))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), f(k)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(l, U111(e, c), f(k))	h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), f(l))
h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), f(d))	h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(k))
h^#(U111(e, c), U111(e, c)) → g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), U111(k, k))

Problem 79: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(e, e)	→	g^#(e, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
A^#	→	h^#(f(l), f(l))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))	A^#	→	h^#(f(d), f(d))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(e, e) → g^#(e, e, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, k))
	g^#(e, e, U111(l, k))

Thus, the rule h^#(e, e) → g^#(e, e, U111(k, k)) is deleted.

Problem 80: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	A^#	→	h^#(f(l), f(l))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(e, e)	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
A^#	→	h^#(d, d)	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(c, c), U111(c, c), f(l))	g^#(U111(c, c), U111(l, c), f(k))
g^#(U111(e, c), U111(c, c), f(k))	g^#(U111(l, c), U111(c, c), f(k))
g^#(U111(c, c), U111(c, c), f(d))
g^#(U111(c, c), U111(e, c), f(k))
g^#(U111(c, c), U111(c, c), U111(k, k))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), f(k)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), f(k))	h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), f(d))	h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(e, c), f(k))

Problem 81: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
A^#	→	h^#(f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
A^#	→	h^#(d, d)	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(c, c), f(l))	g^#(U111(l, c), U111(c, c), f(l))
g^#(U111(c, c), U111(e, c), f(l))	g^#(U111(c, c), U111(l, c), f(l))
g^#(U111(c, c), U111(c, c), U111(l, l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), f(l)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), U111(l, l))

Problem 82: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
A^#	→	h^#(f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))
h^#(e, e)	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(e), f(c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, e), f(c), f(l))	g^#(f(e), f(c), U111(l, l))
g^#(f(e), f(l), f(l))
g^#(f(e), f(e), f(l))
g^#(f(e), U111(c, c), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(f(e), f(c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(e), f(e), f(l))	h^#(f(c), f(c)) → g^#(U111(e, e), f(c), f(l))
h^#(f(c), f(c)) → g^#(f(e), f(l), f(l))	h^#(f(c), f(c)) → g^#(f(e), U111(c, c), f(l))

Problem 83: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(f(e), f(e))	→	g^#(f(e), f(e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	A^#	→	h^#(f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(f(e), f(e), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(e), U111(e, e), U111(k, k))	g^#(f(e), f(e), U111(d, k))
g^#(U111(e, e), f(e), U111(k, k))	g^#(f(e), f(e), U111(l, k))

Thus, the rule h^#(f(e), f(e)) → g^#(f(e), f(e), U111(k, k)) is replaced by the following rules:

h^#(f(e), f(e)) → g^#(U111(e, e), f(e), U111(k, k))

h^#(f(e), f(e)) → g^#(f(e), U111(e, e), U111(k, k))

Problem 84: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
A^#	→	h^#(f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))	A^#	→	h^#(f(d), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(k, k))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(c, c), U111(e, c), U111(k, k))	g^#(U111(c, c), U111(c, c), U111(d, k))
g^#(U111(e, c), U111(c, c), U111(k, k))	g^#(U111(l, c), U111(c, c), U111(k, k))
	g^#(U111(c, c), U111(c, c), U111(l, k))
	g^#(U111(c, c), U111(l, c), U111(k, k))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(c, c), U111(k, k)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), U111(k, k))

h^#(U111(c, c), U111(c, c)) → g^#(U111(c, c), U111(e, c), U111(k, k))

Problem 85: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(k, k))
A^#	→	h^#(f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))	A^#	→	h^#(f(d), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(k, k))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(c, c), U111(e, c), f(d))	g^#(U111(c, c), U111(l, c), f(d))
g^#(U111(c, c), U111(c, c), U111(d, d))	g^#(U111(l, c), U111(c, c), f(d))
g^#(U111(e, c), U111(c, c), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c)) → g^#(U111(e, c), U111(c, c), f(d))
h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), U111(d, d))

Problem 86: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(k, k))	A^#	→	h^#(f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))
h^#(e, e)	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(k, k))
A^#	→	h^#(d, d)	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, c, f(k))
g^#(e, U111(e, c), f(l))
g^#(e, U111(e, c), U111(k, k))
g^#(e, U111(e, c), f(d))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(k)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(d))	h^#(U111(e, c), U111(e, c)) → g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(l))

Problem 87: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	A^#	→	h^#(f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(k, k))
A^#	→	h^#(d, d)	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, c), U111(d, d))
g^#(e, c, f(d))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(d)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(e, c, f(d))

h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), U111(d, d))

Problem 88: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(e, e)	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(k, k))	A^#	→	h^#(f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), f(d))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))
h^#(e, e)	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))	A^#	→	h^#(f(d), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(k, k))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(e, c, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, l, f(k))
g^#(e, c, f(l))
g^#(e, c, U111(k, k))
g^#(e, c, f(d))
g^#(e, e, f(k))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(e, c, f(k)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c)) → g^#(e, c, f(l))
h^#(U111(e, c), U111(e, c)) → g^#(e, l, f(k))	h^#(U111(e, c), U111(e, c)) → g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c)) → g^#(e, e, f(k))

Problem 89: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(k))	h^#(e, e)	→	g^#(e, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(d))
h^#(f(e), f(e))	→	g^#(e, f(e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(d, d))	A^#	→	h^#(l, l)
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))
g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(d, d)	→	g^#(d, d, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(c, c), f(l))
g^#(U111(e, c), U111(l, c), f(k))
g^#(U111(e, c), U111(c, c), f(d))
g^#(c, U111(c, c), f(k))
g^#(U111(e, c), U111(e, c), f(k))
g^#(U111(e, c), U111(c, c), U111(k, k))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), f(k)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c)) → g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), f(d))	h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), f(l))

Problem 90: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(k))
A^#	→	h^#(l, l)	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(d, d)	→	g^#(d, d, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(e, e, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(l, l))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(e, e, f(l)) is deleted.

Problem 91: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))	h^#(d, d)	→	g^#(d, d, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(f(e), f(e))	→	g^#(f(e), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(d))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(U111(e, e), f(e), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, f(e), f(d))
g^#(U111(e, e), f(e), U111(d, d))
g^#(U111(e, e), U111(e, e), f(d))

Thus, the rule h^#(f(e), f(e)) → g^#(U111(e, e), f(e), f(d)) is replaced by the following rules:

h^#(f(e), f(e)) → g^#(U111(e, e), f(e), U111(d, d))	h^#(f(e), f(e)) → g^#(e, f(e), f(d))
h^#(f(e), f(e)) → g^#(U111(e, e), U111(e, e), f(d))

Problem 92: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))	h^#(d, d)	→	g^#(d, d, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(f(e), f(e))	→	g^#(f(e), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(c), f(c))	→	g^#(f(c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(l), U111(c, c), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(l), U111(e, c), f(d))	g^#(U111(l, l), U111(c, c), f(d))
g^#(f(l), U111(c, c), U111(d, d))	g^#(f(l), U111(l, c), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(f(l), U111(c, c), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(l), U111(c, c), U111(d, d))

h^#(f(c), f(c)) → g^#(f(l), U111(e, c), f(d))

Problem 93: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(l, l))	h^#(d, d)	→	g^#(d, d, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(f(e), f(e))	→	g^#(f(e), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(c), f(c))	→	g^#(f(c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, e), l, f(d))
g^#(U111(e, e), c, U111(d, d))
g^#(U111(e, e), e, f(d))
g^#(e, c, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, e), l, f(d))	h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(d, d))
h^#(f(c), f(c)) → g^#(U111(e, e), e, f(d))	h^#(f(c), f(c)) → g^#(e, c, f(d))

Problem 94: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(f(e), f(e))	→	g^#(f(e), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(c), f(c))	→	g^#(f(c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(c, l, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, l, f(d))	g^#(c, l, U111(d, d))
g^#(e, l, f(d))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(c, l, f(d)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(e, l, f(d))

h^#(U111(c, c), U111(c, c)) → g^#(l, l, f(d))

Problem 95: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(k))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(f(e), f(e))	→	g^#(f(e), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), f(e), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, f(e), f(l))
g^#(U111(e, e), f(e), U111(l, l))
g^#(U111(e, e), U111(e, e), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), f(e), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c)) → g^#(U111(e, e), U111(e, e), f(l))
h^#(f(c), f(c)) → g^#(e, f(e), f(l))

Problem 96: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(d, d))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))
h^#(f(e), f(e))	→	g^#(f(e), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, e), e, U111(d, d))	g^#(e, c, U111(d, d))
g^#(U111(e, e), l, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(d, d))

h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(d, d))

Problem 97: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(e), f(e))	→	g^#(e, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(k))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, e), U111(e, e)) → g^#(U111(e, e), e, U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(l, l))

Thus, the rule h^#(U111(e, e), U111(e, e)) → g^#(U111(e, e), e, U111(l, l)) is deleted.

Problem 98: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(k, k))	A^#	→	h^#(l, l)
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(f(e), f(e))	→	g^#(f(e), e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(U111(e, c), e, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), e, U111(l, k))	g^#(U111(e, c), e, U111(d, k))
g^#(c, e, U111(k, k))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(U111(e, c), e, U111(k, k)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(c, e, U111(k, k))

h^#(U111(e, c), U111(e, c)) → g^#(U111(e, c), e, U111(l, k))

Problem 99: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(l), U111(e, c), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(l, l), U111(e, c), U111(k, k))	g^#(f(l), U111(e, c), U111(d, k))
g^#(f(l), c, U111(k, k))
g^#(f(l), U111(e, c), U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(f(l), U111(e, c), U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(l), U111(e, c), U111(l, k))	h^#(f(c), f(c)) → g^#(f(l), c, U111(k, k))
h^#(f(c), f(c)) → g^#(U111(l, l), U111(e, c), U111(k, k))

Problem 100: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(f(e), U111(e, e), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(e), e, U111(k, k))	g^#(f(e), U111(e, e), U111(l, k))
g^#(U111(e, e), U111(e, e), U111(k, k))	g^#(f(e), U111(e, e), U111(d, k))

Thus, the rule h^#(f(e), f(e)) → g^#(f(e), U111(e, e), U111(k, k)) is replaced by the following rules:

h^#(f(e), f(e)) → g^#(f(e), e, U111(k, k))

h^#(f(e), f(e)) → g^#(U111(e, e), U111(e, e), U111(k, k))

Problem 101: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(d))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), l, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(c), l, U111(k, k))
g^#(f(e), l, f(k))
g^#(U111(c, c), l, f(k))
g^#(f(l), l, f(k))
g^#(f(c), l, f(l))
g^#(f(c), l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), l, f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(c), l, U111(k, k))	h^#(f(c), f(c)) → g^#(U111(c, c), l, f(k))
h^#(f(c), f(c)) → g^#(f(c), l, f(d))	h^#(f(c), f(c)) → g^#(f(l), l, f(k))
h^#(f(c), f(c)) → g^#(f(e), l, f(k))	h^#(f(c), f(c)) → g^#(f(c), l, f(l))

Problem 102: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(l, l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, l, f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), l, f(d))
h^#(f(c), f(c))	→	g^#(f(e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, l, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(l, l)) is deleted.

Problem 103: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, e, f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(d))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), l, f(d))	h^#(f(c), f(c))	→	g^#(f(e), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, e, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, e, f(l)) is deleted.

Problem 104: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	A^#	→	h^#(f(c), f(c))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, e, f(d))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), l, f(d))
h^#(f(c), f(c))	→	g^#(f(e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(c, U111(l, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), U111(l, l)) is deleted.

Problem 105: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, l, f(d))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, e), U111(e, e))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(c, e, f(d))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, l, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, l, f(d))	g^#(c, l, U111(d, d))
g^#(e, l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(c, l, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, l, f(d))

h^#(f(c), f(c)) → g^#(e, l, f(d))

Problem 106: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	A^#	→	h^#(l, l)
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), U111(l, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(e, c), U111(l, k))	g^#(c, U111(c, c), U111(l, k))
g^#(U111(e, c), U111(l, c), U111(l, k))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(c, c), U111(l, k)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(e, c), U111(l, k))

h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), U111(l, c), U111(l, k))

Problem 107: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	A^#	→	h^#(f(e), f(e))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U111(c, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, c), U111(l, l))	g^#(e, U111(l, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U111(c, c), U111(l, l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(l, l))

Problem 108: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	A^#	→	h^#(f(c), f(c))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), e, U111(l, l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, U111(e, c), f(l))
g^#(c, c, f(l))
g^#(c, U111(e, c), U111(l, l))
g^#(e, U111(e, c), f(l))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), f(l)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(c, U111(e, c), U111(l, l))	h^#(U111(e, c), U111(e, c)) → g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c)) → g^#(e, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c)) → g^#(l, U111(e, c), f(l))

Problem 109: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(e), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(e, e)	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, e), l, f(l))
g^#(U111(e, e), e, f(l))
g^#(U111(e, e), c, U111(l, l))
g^#(e, c, f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, e), e, f(l))	h^#(f(c), f(c)) → g^#(U111(e, e), l, f(l))
h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c)) → g^#(e, c, f(l))

Problem 110: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(l, e, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(l, l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(l, e, f(l)) is deleted.

Problem 111: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, c, f(k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, c, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, l, f(k))
g^#(e, c, f(l))
g^#(e, c, U111(k, k))
g^#(e, c, f(d))
g^#(e, e, f(k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, c, f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, l, f(k))	h^#(f(c), f(c)) → g^#(e, c, f(d))
h^#(f(c), f(c)) → g^#(e, c, U111(k, k))	h^#(f(c), f(c)) → g^#(e, e, f(k))
h^#(f(c), f(c)) → g^#(e, c, f(l))

Problem 112: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(d))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, U111(k, k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	g^#(d, x, x)	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(c, c), U111(e, e), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(c, c), e, f(d))	g^#(U111(l, c), U111(e, e), f(d))
g^#(U111(e, c), U111(e, e), f(d))
g^#(U111(c, c), U111(e, e), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(c, c), U111(e, e), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(c, c), e, f(d))	h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, e), f(d))
h^#(f(c), f(c)) → g^#(U111(c, c), U111(e, e), U111(d, d))

Problem 113: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, U111(k, k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, e, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, e, f(l))	g^#(c, e, U111(l, l))
g^#(l, e, f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, f(l))

h^#(f(c), f(c)) → g^#(l, e, f(l))

Problem 114: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(d, d))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, e), U111(e, e))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))
h^#(f(c), f(c))	→	g^#(c, e, f(d))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(l), U111(e, e), U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(l, l), U111(e, e), U111(d, d))	g^#(f(l), e, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(f(l), U111(e, e), U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(l, l), U111(e, e), U111(d, d))

Problem 115: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(k, k))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, e), U111(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))	g^#(d, x, x)	→	h^#(e, e)
h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), l, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, l, U111(k, k))	g^#(U111(e, c), l, U111(d, k))
g^#(U111(e, c), l, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), l, U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), l, U111(l, k))

h^#(f(c), f(c)) → g^#(c, l, U111(k, k))

Problem 116: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	A^#	→	h^#(U111(e, e), U111(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
g^#(d, x, x)	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), c, U111(l, l))
g^#(c, U111(e, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), U111(l, l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(l, l))

h^#(f(c), f(c)) → g^#(U111(e, c), c, U111(l, l))

Problem 117: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, c, U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, U111(k, k))
g^#(d, x, x)	→	h^#(e, e)	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(e, e, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, d))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, e, f(d)) is deleted.

Problem 118: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), f(k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, c, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, e, f(k))
g^#(l, c, f(k))
g^#(e, c, f(k))
g^#(c, c, f(l))
g^#(c, l, f(k))
g^#(c, c, f(d))
g^#(c, c, U111(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, c, f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, c, f(k))	h^#(f(c), f(c)) → g^#(c, e, f(k))
h^#(f(c), f(c)) → g^#(e, c, f(k))	h^#(f(c), f(c)) → g^#(c, c, f(l))
h^#(f(c), f(c)) → g^#(c, c, U111(k, k))	h^#(f(c), f(c)) → g^#(c, c, f(d))
h^#(f(c), f(c)) → g^#(c, l, f(k))

Problem 119: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, f(e), U111(k, k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), U111(k, k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, f(e), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, e), U111(k, k))	g^#(e, f(e), U111(d, k))
	g^#(e, f(e), U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, f(e), U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(k, k))

Problem 120: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(k, k))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(l, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, U111(e, c), U111(d, d))
g^#(l, U111(e, c), f(d))
g^#(c, c, f(d))
g^#(e, U111(e, c), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, c), f(d))	h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(d, d))
h^#(f(c), f(c)) → g^#(c, c, f(d))	h^#(f(c), f(c)) → g^#(l, U111(e, c), f(d))

Problem 121: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, e, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, f(l)) is deleted.

Problem 122: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(d, d))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), l, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), l, U111(d, d))
g^#(c, l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), l, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, l, f(d))

h^#(f(c), f(c)) → g^#(U111(e, c), l, U111(d, d))

Problem 123: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, e, f(k))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, U111(e, e), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, U111(e, e), U111(d, d))
g^#(c, e, f(d))
g^#(l, U111(e, e), f(d))
g^#(e, U111(e, e), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U111(e, e), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U111(e, e), f(d))	h^#(f(c), f(c)) → g^#(c, e, f(d))
h^#(f(c), f(c)) → g^#(e, U111(e, e), f(d))	h^#(f(c), f(c)) → g^#(c, U111(e, e), U111(d, d))

Problem 124: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, e, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U111(e, e), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, e, f(l))
g^#(e, U111(e, e), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U111(e, e), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(l, l))

h^#(f(c), f(c)) → g^#(e, e, f(l))

Problem 125: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), e, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(e), e, f(l))	g^#(f(c), e, U111(l, l))
g^#(U111(c, c), e, f(l))
g^#(f(l), e, f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), e, f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(e), e, f(l))	h^#(f(c), f(c)) → g^#(f(l), e, f(l))
h^#(f(c), f(c)) → g^#(U111(c, c), e, f(l))

Problem 126: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(d, d))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(d))	h^#(f(c), f(c))	→	g^#(f(c), c, f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
g^#(U111(e, e), e, U111(d, d))	g^#(e, c, U111(d, d))
g^#(U111(e, e), l, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(d, d))

h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(d, d))

Problem 127: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(d, d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), U111(e, e), U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(l), U111(e, e), U111(d, d))	g^#(f(c), e, U111(d, d))
g^#(f(e), U111(e, e), U111(d, d))
g^#(U111(c, c), U111(e, e), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), U111(e, e), U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(e), U111(e, e), U111(d, d))	h^#(f(c), f(c)) → g^#(f(l), U111(e, e), U111(d, d))
h^#(f(c), f(c)) → g^#(U111(c, c), U111(e, e), U111(d, d))

Problem 128: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(d, d)) is deleted.

Problem 129: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), l, U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(c, l, U111(l, l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), l, U111(l, l)) is deleted.

Problem 130: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(e, l, U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), f(l))
A^#	→	h^#(l, l)	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), f(k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(e, l, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, l, U111(d, k))
	g^#(e, l, U111(l, k))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, l, U111(k, k)) is deleted.

Problem 131: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(e, U111(e, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, c, U111(l, l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, U111(e, c), U111(l, l)) is deleted.

Problem 132: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
A^#	→	h^#(f(d), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(l, c, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, l, f(k))
g^#(l, c, f(l))
g^#(l, c, f(d))
g^#(l, e, f(k))
g^#(l, c, U111(k, k))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(l, c, f(k)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(l, c, U111(k, k))	h^#(U111(c, c), U111(c, c)) → g^#(l, e, f(k))
h^#(U111(c, c), U111(c, c)) → g^#(l, l, f(k))	h^#(U111(c, c), U111(c, c)) → g^#(l, c, f(d))
h^#(U111(c, c), U111(c, c)) → g^#(l, c, f(l))

Problem 133: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, e, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, k))
	g^#(e, e, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, e, U111(k, k)) is deleted.

Problem 134: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	A^#	→	h^#(f(d), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, e, U111(k, k))	g^#(U111(e, e), e, U111(l, k))
	g^#(U111(e, e), e, U111(d, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U111(k, k))

Problem 135: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(k))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(c, c), U111(l, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(l, c), U111(l, k))	g^#(c, U111(c, c), U111(l, k))
g^#(U111(e, c), U111(e, c), U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(c, c), U111(l, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), U111(l, k))

h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), U111(l, k))

Problem 136: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, f(l)) is deleted.

Problem 137: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(c, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(c), f(c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(e, l, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, l, U111(d, d))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, l, f(d)) is deleted.

Problem 138: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, e, f(d))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))	h^#(f(c), f(c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, c, U111(l, l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, U111(e, c), U111(l, l)) is deleted.

Problem 139: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(f(c), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, c), U111(d, d))	g^#(c, c, U111(d, d))
g^#(l, U111(e, c), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(d, d))

h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(d, d))

Problem 140: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(d))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(l, c), U111(e, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(U111(l, c), c, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(l, c), U111(e, c), U111(l, l)) is deleted.

Problem 141: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, l, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, l, f(l))	g^#(c, l, U111(l, l))
g^#(e, l, f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(c, l, f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, l, f(l))

h^#(f(c), f(c)) → g^#(l, l, f(l))

Problem 142: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(c, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(l, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), l, U111(d, d))
g^#(c, l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), l, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, l, f(d))

h^#(f(c), f(c)) → g^#(U111(e, c), l, U111(d, d))

Problem 143: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, l, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, l, U111(k, k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(d))
h^#(f(c), f(c))	→	g^#(c, e, f(k))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, l, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, l, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(e, l, f(d)) is deleted.

Problem 144: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U111(c, c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, U111(e, c), f(l))	g^#(l, U111(c, c), U111(l, l))
	g^#(l, U111(l, c), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U111(c, c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U111(e, c), f(l))

Problem 145: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, e, U111(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, e, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, e, U111(k, k))	g^#(c, e, U111(l, k))
g^#(e, e, U111(k, k))	g^#(c, e, U111(d, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, e, U111(k, k))

h^#(f(c), f(c)) → g^#(e, e, U111(k, k))

Problem 146: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(l, l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(c, c), U111(e, c), U111(l, l))	g^#(U111(l, c), U111(c, c), U111(l, l))
g^#(U111(e, c), U111(c, c), U111(l, l))	g^#(U111(c, c), U111(l, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(c, c), U111(c, c), U111(l, l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(c, c), U111(e, c), U111(l, l))

h^#(f(c), f(c)) → g^#(U111(e, c), U111(c, c), U111(l, l))

Problem 147: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(c, U111(l, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), U111(l, l)) is deleted.

Problem 148: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, l, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, l, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, l, f(l)) is deleted.

Problem 149: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, l, U111(k, k))	g^#(U111(e, e), l, U111(d, k))
	g^#(U111(e, e), l, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, l, U111(k, k))

Problem 150: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, l, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, l, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, l, f(l)) is deleted.

Problem 151: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(e, c), U111(k, k))
g^#(U111(e, c), U111(e, c), f(l))
g^#(c, U111(e, c), f(k))
g^#(U111(e, c), c, f(k))
g^#(U111(e, c), U111(e, c), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), U111(k, k))	h^#(f(c), f(c)) → g^#(U111(e, c), c, f(k))
h^#(f(c), f(c)) → g^#(c, U111(e, c), f(k))	h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), f(l))

Problem 152: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, l, U111(k, k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(c, e, f(k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(c, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, l, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, l, U111(d, k))
	g^#(e, l, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, l, U111(k, k)) is deleted.

Problem 153: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, e, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(d, k))
	g^#(l, e, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, U111(k, k)) is deleted.

Problem 154: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(l, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, c, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, e, f(d))	g^#(l, c, U111(d, d))
g^#(l, l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(l, c, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, e, f(d))

h^#(f(c), f(c)) → g^#(l, l, f(d))

Problem 155: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U111(e, c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, c, f(l))
g^#(l, U111(e, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U111(e, c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(l, l))

h^#(f(c), f(c)) → g^#(l, c, f(l))

Problem 156: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(l, l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, e, U111(k, k))	g^#(e, U111(e, e), U111(d, k))
	g^#(e, U111(e, e), U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U111(k, k))

Problem 157: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), U111(l, l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), U111(e, e), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, e), U111(l, l))
g^#(U111(e, e), e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), U111(e, e), U111(l, l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(l, l))

h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(l, l))

Problem 158: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(d, k))
	g^#(l, e, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, U111(k, k)) is deleted.

Problem 159: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U111(c, c), U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, U111(e, c), U111(k, k))	g^#(l, U111(c, c), U111(d, k))
	g^#(l, U111(c, c), U111(l, k))
	g^#(l, U111(l, c), U111(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U111(c, c), U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(k, k))

Problem 160: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, e, U111(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(l, k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(l, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, c), U111(l, k))	g^#(c, c, U111(l, k))
g^#(l, U111(e, c), U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(l, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(l, k))

h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(l, k))

Problem 161: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, e, f(k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	A^#	→	h^#(d, d)
h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, e, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, e, f(d))
g^#(l, e, f(k))
g^#(c, e, U111(k, k))
g^#(c, e, f(l))
g^#(e, e, f(k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, e, U111(k, k))	h^#(f(c), f(c)) → g^#(l, e, f(k))
h^#(f(c), f(c)) → g^#(c, e, f(d))	h^#(f(c), f(c)) → g^#(c, e, f(l))
h^#(f(c), f(c)) → g^#(e, e, f(k))

Problem 162: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), e, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))	A^#	→	h^#(d, d)
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, e, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, f(d)) is deleted.

Problem 163: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(e), f(e))	→	g^#(e, U111(e, e), f(k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, k))
	g^#(e, e, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, e, U111(k, k)) is deleted.

Problem 164: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, f(c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(l, c), U111(d, d))	g^#(c, U111(l, c), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), U111(d, d))

Problem 165: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(e, e), U111(e, e))	→	g^#(e, e, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
A^#	→	g^#(c, c, f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, e), U111(e, e)) → g^#(e, e, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, d))

Thus, the rule h^#(U111(e, e), U111(e, e)) → g^#(e, e, f(d)) is deleted.

Problem 166: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	A^#	→	g^#(c, c, f(k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(c, l, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, l, f(l))	g^#(c, l, U111(l, l))
g^#(e, l, f(l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(c, l, f(l)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(e, l, f(l))

h^#(U111(c, c), U111(c, c)) → g^#(l, l, f(l))

Problem 167: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, e, f(k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	A^#	→	g^#(l, c, U111(k, k))
A^#	→	g^#(c, c, f(d))	A^#	→	g^#(l, c, f(d))
A^#	→	g^#(e, c, f(l))	A^#	→	g^#(e, e, U111(k, k))
A^#	→	g^#(c, e, f(k))	A^#	→	g^#(c, l, f(d))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	g^#(c, l, U111(k, k))
A^#	→	g^#(e, e, f(d))	A^#	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule A^# → g^#(l, e, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, e, U111(k, k))
g^#(l, e, f(d))
g^#(l, e, f(l))

Thus, the rule A^# → g^#(l, e, f(k)) is replaced by the following rules:

A^# → g^#(l, e, U111(k, k))	A^# → g^#(l, e, f(l))
A^# → g^#(l, e, f(d))

Problem 168: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, c, f(l))
A^#	→	h^#(f(e), f(e))	h^#(U111(e, c), U111(e, c))	→	g^#(e, e, f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(e, c), U111(e, c))	→	g^#(e, l, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, l, f(l))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(e), f(e))	→	g^#(U111(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(e, c), U111(e, c)) → g^#(l, c, f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, l, f(l))	g^#(l, c, U111(l, l))
g^#(l, e, f(l))

Thus, the rule h^#(U111(e, c), U111(e, c)) → g^#(l, c, f(l)) is replaced by the following rules:

h^#(U111(e, c), U111(e, c)) → g^#(l, e, f(l))

h^#(U111(e, c), U111(e, c)) → g^#(l, l, f(l))

Problem 169: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(d))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, e), f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, e, f(l)) is deleted.

Problem 170: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), f(l))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	A^#	→	g^#(e, c, U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, c, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(l, l)) is deleted.

Problem 171: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, U111(e, c), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), U111(e, c), U111(l, l))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(c, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
A^#	→	g^#(e, c, U111(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U111(e, c), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, c, f(l))
g^#(e, U111(e, c), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U111(e, c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(l, l))

h^#(f(c), f(c)) → g^#(e, c, f(l))

Problem 172: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(d))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(l, e, f(d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(k, k))	h^#(U111(e, e), U111(e, e))	→	g^#(e, U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
A^#	→	g^#(e, c, U111(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U111(e, e), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, e, f(d))
g^#(l, U111(e, e), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U111(e, e), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, e, f(d))

h^#(f(c), f(c)) → g^#(l, U111(e, e), U111(d, d))

Problem 173: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, e), U111(d, d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(c, c), f(c), U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), f(c), U111(d, d))	g^#(U111(c, c), f(l), U111(d, d))
	g^#(U111(c, c), U111(c, c), U111(d, d))
	g^#(U111(c, c), f(e), U111(d, d))
	g^#(U111(l, c), f(c), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(c, c), f(c), U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), f(c), U111(d, d))

Problem 174: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), c, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), e, f(d))
g^#(U111(e, c), c, U111(d, d))
g^#(c, c, f(d))
g^#(U111(e, c), l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), c, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), l, f(d))	h^#(f(c), f(c)) → g^#(c, c, f(d))
h^#(f(c), f(c)) → g^#(U111(e, c), e, f(d))	h^#(f(c), f(c)) → g^#(U111(e, c), c, U111(d, d))

Problem 175: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(l, U111(e, c), U111(d, d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(e, U111(c, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(l, l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	A^#	→	h^#(l, l)
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, c, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(d, d)) is deleted.

Problem 176: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, e, f(l))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(l, c), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
A^#	→	h^#(l, l)	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	A^#	→	g^#(e, c, U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, e, f(l)) is deleted.

Problem 177: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(d, d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	A^#	→	h^#(U111(e, e), U111(e, e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
A^#	→	h^#(l, l)	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, c), U111(d, d))	g^#(c, c, U111(d, d))
g^#(l, U111(e, c), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U111(e, c), U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U111(e, c), U111(d, d))

h^#(f(c), f(c)) → g^#(e, U111(e, c), U111(d, d))

Problem 178: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
A^#	→	h^#(U111(e, e), U111(e, e))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(c, c), f(d))
h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))
h^#(U111(e, e), U111(e, e))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, e), U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, d))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, e, f(d)) is deleted.

Problem 179: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, U111(k, k))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(l, l)	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), l, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
A^#	→	h^#(l, l)	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(k, k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, e), e, U111(k, k))	g^#(U111(e, e), c, U111(d, k))
g^#(e, c, U111(k, k))	g^#(U111(e, e), c, U111(l, k))
g^#(U111(e, e), l, U111(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), c, U111(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c)) → g^#(U111(e, e), l, U111(k, k))
h^#(f(c), f(c)) → g^#(e, c, U111(k, k))

Problem 180: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(l, l)	→	g^#(l, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	A^#	→	h^#(l, l)
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	A^#	→	h^#(U111(e, e), e)
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	A^#	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), e, U111(l, l)) is deleted.

Problem 181: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(c, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(d, d))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), e, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	A^#	→	g^#(l, l, f(k))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
A^#	→	g^#(e, c, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
g^#(l, l, f(l))	g^#(c, l, U111(l, l))
g^#(e, l, f(l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(c, l, f(l)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(e, l, f(l))

h^#(U111(c, c), U111(c, c)) → g^#(l, l, f(l))

Problem 182: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(U111(c, c), U111(c, c))	→	g^#(e, e, U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(c, l, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(e), l, f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
A^#	→	g^#(l, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, d))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(e, e, f(d)) is deleted.

Problem 183: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, U111(l, l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(e, l, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(e, e, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), l, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), l, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	A^#	→	h^#(U111(e, e), e)
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(c, l, U111(l, l))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), l, U111(l, l)) is deleted.

Problem 184: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(e, c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(d, d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
h^#(f(c), f(c))	→	g^#(f(e), l, f(k))	A^#	→	h^#(l, l)
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	A^#	→	h^#(U111(e, e), e)
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), e, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, e, f(d))
g^#(U111(e, c), e, U111(d, d))

Thus, the rule h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), e, f(d)) is replaced by the following rules:

h^#(U111(c, c), U111(c, c)) → g^#(U111(e, c), e, U111(d, d))

h^#(U111(c, c), U111(c, c)) → g^#(c, e, f(d))

Problem 185: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(c), f(d))
A^#	→	g^#(l, l, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	A^#	→	h^#(U111(e, e), e)
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	A^#	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	A^#	→	g^#(l, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), U111(l, c), U111(l, k))	g^#(c, U111(c, c), U111(l, k))
g^#(U111(e, c), U111(e, c), U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), U111(c, c), U111(l, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, c), U111(l, k))

h^#(f(c), f(c)) → g^#(U111(e, c), U111(l, c), U111(l, k))

Problem 186: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(c, c), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(l, f(c), f(d))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(e, f(c), f(d))
A^#	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, c, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), f(e), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), f(e), U111(d, d))
g^#(U111(e, c), U111(e, e), f(d))
g^#(c, f(e), f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), f(e), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), f(e), U111(d, d))	h^#(f(c), f(c)) → g^#(c, f(e), f(d))
h^#(f(c), f(c)) → g^#(U111(e, c), U111(e, e), f(d))

Problem 187: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(l, c, f(d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(l, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(e, U111(e, c), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, l, f(d))
h^#(f(c), f(c))	→	g^#(l, f(c), f(d))	h^#(f(c), f(c))	→	g^#(c, f(l), f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, e, f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	A^#	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	A^#	→	h^#(U111(e, e), e)
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
g^#(l, e, f(d))	g^#(l, c, U111(d, d))
g^#(l, l, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(l, c, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, e, f(d))

h^#(f(c), f(c)) → g^#(l, l, f(d))

Problem 188: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, f(e), f(d))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, U111(e, e), f(d))	h^#(f(c), f(c))	→	g^#(l, U111(e, e), f(d))
h^#(f(c), f(c))	→	g^#(e, f(c), f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(c, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	A^#	→	g^#(e, c, U111(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	A^#	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, f(e), f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, U111(e, e), f(d))	g^#(e, f(e), U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(e, f(e), f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U111(e, e), f(d))

Problem 189: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(l, f(l), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), U111(k, k))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), f(l), f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(d))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	A^#	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, f(l), f(l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(l, U111(l, l), f(l))
	g^#(l, f(l), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, f(l), f(l)) is deleted.

Problem 190: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, e), U111(l, l), U111(l, l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), U111(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), f(e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	A^#	→	h^#(U111(e, e), e)
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, e), U111(l, l), U111(l, l)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, U111(l, l), U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, e), U111(l, l), U111(l, l)) is deleted.

Problem 191: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(c, e, f(d))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))
A^#	→	h^#(U111(e, e), e)	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	A^#	→	g^#(l, c, f(l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
A^#	→	g^#(l, l, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, e, f(d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(l, e, f(d))	g^#(c, e, U111(d, d))
g^#(e, e, f(d))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, f(d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, e, f(d))

h^#(f(c), f(c)) → g^#(e, e, f(d))

Problem 192: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(l, e, f(d))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(l, c, f(l))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(l, e, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(d))
h^#(f(c), f(c))	→	g^#(l, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, e, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, l))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, e, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	A^#	→	h^#(U111(e, e), e)
h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))	A^#	→	g^#(e, c, U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), c, f(d))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, f(d)) is deleted.

Problem 193: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(e, U111(e, e), U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, c), U111(d, d))
A^#	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(l, e, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))
A^#	→	g^#(l, l, U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U111(e, e), U111(d, d)) is deleted.

Problem 194: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, e, U111(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	h^#(f(c), f(c))	→	g^#(e, l, U111(k, k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(e), l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), e, U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(f(e), e, U111(k, k))	h^#(U111(e, c), U111(e, c))	→	g^#(U111(e, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))	h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))
A^#	→	g^#(e, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	g^#(l, l, U111(k, k))	A^#	→	h^#(U111(e, c), U111(e, c))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U111(d, k))
	g^#(e, e, U111(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, e, U111(k, k)) is deleted.

Problem 195: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(l, l))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(e, c), U111(l, k))
h^#(U111(c, c), U111(c, c))	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(e, c), U111(l, k))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(k, k))	A^#	→	g^#(c, l, f(l))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	g^#(e, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U111(k, k))	h^#(f(c), f(c))	→	g^#(c, U111(e, c), U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), f(e), U111(l, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(l, l), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
h^#(f(c), f(c))	→	g^#(U111(e, c), c, f(d))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	A^#	→	g^#(l, l, U111(k, k))
h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

Relevant Terms	Irrelevant Terms
	g^#(l, e, U111(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(l, e, f(l)) is deleted.

Problem 196: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U111(e, c), c, U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), l, f(d))
h^#(f(c), f(c))	→	g^#(c, l, f(d))	h^#(f(c), f(c))	→	g^#(e, c, f(d))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(l, k))
h^#(f(c), f(c))	→	g^#(U111(c, c), l, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, c), e, U111(l, l))
A^#	→	h^#(U111(e, c), U111(e, c))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), f(l))
h^#(f(c), f(c))	→	g^#(U111(e, c), U111(l, l), U111(d, d))	h^#(U111(e, c), U111(e, c))	→	g^#(l, U111(e, c), U111(l, k))
A^#	→	g^#(l, l, U111(k, k))	h^#(f(c), f(c))	→	g^#(U111(e, e), l, U111(d, d))
h^#(f(c), f(c))	→	g^#(f(c), U111(e, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(c, e, f(d))
h^#(f(c), f(c))	→	g^#(f(e), U111(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U111(c, c), U111(d, d))
h^#(f(c), f(c))	→	g^#(c, U111(c, c), U111(d, d))	h^#(f(c), f(c))	→	g^#(U111(e, c), f(l), f(l))
h^#(f(c), f(c))	→	g^#(e, U111(e, c), U111(l, k))	A^#	→	h^#(f(e), f(e))
h^#(U111(c, c), U111(c, c))	→	g^#(U111(c, c), c, f(d))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(l, l))	h^#(U111(c, c), U111(c, c))	→	g^#(l, U111(c, c), f(l))
h^#(f(e), f(e))	→	g^#(U111(e, e), U111(e, e), U111(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(d))
h^#(f(e), f(e))	→	g^#(f(e), U111(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U111(c, c), U111(l, l), U111(d, d))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(U111(c, c), U111(c, c))	→	g^#(e, c, f(d))	h^#(f(c), f(c))	→	g^#(U111(e, e), c, f(d))
h^#(U111(e, c), U111(e, c))	→	g^#(e, c, U111(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	A^#	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	l
k	→	l	k	→	d
A	→	h(f(a), f(b))	h(x, x)	→	g(x, x, f(k))
g(d, x, x)	→	A	f(x)	→	U111(x, x)
U111(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, k, h

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U111(e, c), c, U111(d, d)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(U111(e, c), l, U111(d, d))	g^#(c, c, U111(d, d))
g^#(U111(e, c), e, U111(d, d))

Thus, the rule h^#(f(c), f(c)) → g^#(U111(e, c), c, U111(d, d)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U111(e, c), e, U111(d, d))

h^#(f(c), f(c)) → g^#(U111(e, c), l, U111(d, d))

A^# → h^#(f(c), f(b))	A^# → h^#(f(d), f(b))
A^# → h^#(U111(a, a), f(b))	A^# → h^#(f(a), f(d))
A^# → h^#(f(a), U111(b, b))	A^# → h^#(f(a), f(c))

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	b^#
a^#	→	c^#	A^#	→	f^#(a)
A^#	→	a^#	f^#(x)	→	U111^#(x, x)
b^#	→	c^#	h^#(x, x)	→	k^#
A^#	→	h^#(f(a), f(b))	A^#	→	f^#(b)
h^#(x, x)	→	f^#(k)	U111^#(e, x)	→	T(x)
g^#(d, x, x)	→	A^#

h^#(x, x) → g^#(x, x, f(k))	A^# → h^#(f(a), f(b))
g^#(d, x, x) → A^#

A^# → h^#(U111(c, c), f(b))	A^# → h^#(f(l), f(b))
A^# → h^#(f(e), f(b))	A^# → h^#(f(c), f(d))
A^# → h^#(f(c), U111(b, b))	A^# → h^#(f(c), f(c))

A^# → h^#(f(d), U111(b, b))	A^# → h^#(f(d), f(d))
A^# → h^#(f(d), f(c))

A^# → h^#(U111(e, e), f(b))	A^# → h^#(f(e), f(d))
A^# → h^#(f(e), U111(b, b))	A^# → h^#(f(e), f(c))

A^# → h^#(U111(c, a), f(b))	A^# → h^#(U111(a, a), U111(b, b))
A^# → h^#(U111(a, a), f(c))	A^# → h^#(U111(a, a), f(d))

A^# → h^#(U111(a, a), f(l))	A^# → h^#(U111(a, a), f(e))
A^# → h^#(U111(c, a), f(c))	A^# → h^#(U111(a, a), U111(c, c))

A^# → h^#(f(a), f(e))	A^# → h^#(f(a), f(l))
A^# → h^#(U111(a, a), f(c))	A^# → h^#(f(a), U111(c, c))
A^# → h^#(f(c), f(c))	A^# → h^#(f(d), f(c))

A^# → h^#(f(d), U111(c, c))	A^# → h^#(f(d), f(l))
A^# → h^#(f(d), f(e))

Relevant Terms	Irrelevant Terms
h^#(U111(e, e), f(d))
h^#(U111(e, e), f(c))
h^#(e, f(b))
h^#(U111(e, e), U111(b, b))

A^# → h^#(U111(e, e), U111(b, b))	A^# → h^#(e, f(b))
A^# → h^#(U111(e, e), f(c))	A^# → h^#(U111(e, e), f(d))

Relevant Terms	Irrelevant Terms
h^#(U111(c, c), f(d))	h^#(U111(l, c), f(b))
h^#(U111(e, c), f(b))
h^#(U111(c, c), f(c))
h^#(U111(c, c), U111(b, b))

A^# → h^#(U111(c, c), U111(b, b))	A^# → h^#(U111(e, c), f(b))
A^# → h^#(U111(c, c), f(d))	A^# → h^#(U111(c, c), f(c))

Relevant Terms	Irrelevant Terms
h^#(U111(e, c), f(d))	h^#(U111(c, c), U111(d, d))
	h^#(U111(l, c), f(d))

Relevant Terms	Irrelevant Terms
h^#(U111(c, c), f(l))	h^#(U111(l, c), f(c))
h^#(U111(c, c), U111(c, c))
h^#(U111(e, c), f(c))
h^#(U111(c, c), f(e))

A^# → h^#(U111(c, c), f(l))	A^# → h^#(U111(c, c), U111(c, c))
A^# → h^#(U111(e, c), f(c))	A^# → h^#(U111(c, c), f(e))

Relevant Terms	Irrelevant Terms
h^#(f(a), U111(e, e))
h^#(f(d), f(e))
h^#(U111(a, a), f(e))
h^#(f(c), f(e))

A^# → h^#(U111(a, a), f(e))	A^# → h^#(f(a), U111(e, e))
A^# → h^#(f(c), f(e))	A^# → h^#(f(d), f(e))

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U111(e, a), U111(e, e))
A^#	→	h^#(U111(c, a), e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(e, U111(c, c))	A^#	→	h^#(f(l), f(b))
A^#	→	h^#(a, U111(c, c))	A^#	→	h^#(U111(c, c), U111(b, b))
A^#	→	h^#(U111(c, a), f(b))	A^#	→	h^#(U111(e, c), f(b))
A^#	→	h^#(U111(e, c), U111(d, d))	A^#	→	h^#(U111(a, a), f(e))
A^#	→	h^#(e, U111(e, e))	A^#	→	h^#(U111(c, c), f(l))
A^#	→	h^#(a, f(e))	A^#	→	h^#(U111(c, a), U111(e, c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(U111(e, e), f(c))
A^#	→	h^#(f(a), f(l))	A^#	→	h^#(U111(e, c), f(c))
A^#	→	h^#(f(d), e)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(a), U111(c, c))	A^#	→	h^#(U111(a, a), U111(c, c))
A^#	→	h^#(e, f(d))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(c), U111(b, b))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(d), U111(e, e))	A^#	→	h^#(U111(e, e), U111(d, d))
A^#	→	h^#(U111(e, c), f(e))	A^#	→	h^#(U111(a, a), U111(b, b))
A^#	→	h^#(U111(e, a), e)	A^#	→	h^#(f(d), c)
A^#	→	h^#(a, U111(e, e))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(U111(c, a), f(l))
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(a, U111(e, c))
A^#	→	h^#(U111(e, a), f(c))	A^#	→	h^#(U111(a, a), f(l))
A^#	→	h^#(e, f(l))	A^#	→	h^#(U111(a, a), e)
A^#	→	h^#(f(a), U111(c, b))	A^#	→	h^#(U111(c, a), f(d))
A^#	→	h^#(U111(a, a), f(c))	A^#	→	h^#(U111(c, c), U111(e, e))
A^#	→	h^#(f(c), e)	A^#	→	h^#(U111(e, e), U111(c, b))
A^#	→	h^#(U111(d, d), U111(e, c))	A^#	→	h^#(f(c), U111(e, e))
A^#	→	h^#(U111(e, a), c)	A^#	→	h^#(f(e), U111(b, b))
A^#	→	h^#(f(d), U111(c, b))	A^#	→	h^#(U111(a, a), U111(c, b))
A^#	→	h^#(e, U111(e, b))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U111(c, a), U111(b, b))

A^# → h^#(c, U111(e, c))	A^# → h^#(a, c)
A^# → h^#(d, U111(e, c))

The following DP Processors were used

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

The following SCCs where found

Problem 2: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 3: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 4: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 5: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 6: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 7: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 8: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 9: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 11: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Problem 13: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

Rewrite Rules

Original Signature

Strategy

Relevant Terms	Irrelevant Terms
h^#(U111(c, c), U111(l, b))	h^#(U111(l, c), U111(c, b))
h^#(U111(e, c), U111(c, b))
h^#(U111(c, c), U111(e, b))

A^# → h^#(U111(c, c), U111(l, b))	A^# → h^#(U111(e, c), U111(c, b))
A^# → h^#(U111(c, c), U111(e, b))

A^# → h^#(f(d), l)	A^# → h^#(f(c), l)
A^# → h^#(U111(a, a), l)

Relevant Terms	Irrelevant Terms
h^#(U111(e, a), U111(e, b))	h^#(U111(l, a), U111(e, b))
h^#(U111(c, a), b)

A^# → h^#(e, U111(c, c))	A^# → h^#(e, f(l))
A^# → h^#(e, f(e))

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(l, U111(b, b))
A^#	→	h^#(d, U111(b, b))	A^#	→	h^#(e, e)
A^#	→	h^#(e, U111(b, b))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(U111(e, a), U111(c, b))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(d, d)	A^#	→	h^#(a, U111(c, b))
A^#	→	h^#(c, U111(c, b))	A^#	→	h^#(a, U111(e, c))
A^#	→	h^#(l, U111(e, b))	A^#	→	h^#(U111(e, e), U111(e, e))
A^#	→	h^#(U111(e, a), f(c))	A^#	→	h^#(a, U111(e, b))
A^#	→	h^#(e, f(l))	A^#	→	h^#(l, f(e))
A^#	→	h^#(c, c)	A^#	→	h^#(U111(e, a), f(d))
A^#	→	h^#(U111(c, a), f(d))	A^#	→	h^#(l, U111(c, c))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U111(a, a), f(c))
A^#	→	h^#(c, U111(e, b))	A^#	→	h^#(U111(a, a), U111(e, c))
A^#	→	h^#(U111(c, c), d)	A^#	→	h^#(U111(c, c), U111(e, c))
A^#	→	h^#(l, l)	A^#	→	h^#(U111(c, c), U111(e, e))
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U111(e, c), U111(e, b))
A^#	→	h^#(U111(c, a), l)	A^#	→	h^#(U111(e, e), e)
A^#	→	h^#(U111(e, e), U111(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(U111(d, d), U111(e, c))	A^#	→	h^#(U111(e, e), l)
A^#	→	h^#(e, d)	A^#	→	h^#(U111(l, l), U111(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U111(e, a), c)
A^#	→	h^#(f(c), U111(e, e))	A^#	→	h^#(U111(e, e), U111(c, c))
A^#	→	h^#(f(d), b)	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U111(c, b))	A^#	→	h^#(f(e), U111(b, b))
A^#	→	h^#(U111(a, a), U111(c, b))	A^#	→	h^#(e, l)
A^#	→	h^#(d, U111(e, c))	A^#	→	h^#(U111(a, a), l)
A^#	→	h^#(e, U111(e, b))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U111(c, a), U111(b, b))

A^#	→	h^#(d, e)	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(c, U111(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(d, d)
A^#	→	h^#(a, f(c))	A^#	→	h^#(l, U111(e, c))
A^#	→	h^#(a, U111(e, c))	A^#	→	h^#(U111(e, e), U111(e, e))
A^#	→	h^#(a, U111(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(l, f(e))	A^#	→	h^#(e, f(l))
A^#	→	h^#(U111(e, a), f(d))	A^#	→	h^#(c, c)
A^#	→	h^#(U111(c, a), f(d))	A^#	→	h^#(l, U111(c, c))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U111(a, a), f(c))
A^#	→	h^#(c, U111(e, b))	A^#	→	h^#(U111(a, a), U111(e, c))
A^#	→	h^#(U111(c, c), U111(e, c))	A^#	→	h^#(U111(c, c), d)
A^#	→	h^#(l, l)	A^#	→	h^#(U111(c, c), U111(e, e))
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U111(e, c), U111(e, b))
A^#	→	h^#(U111(c, a), l)	A^#	→	h^#(U111(e, e), e)
A^#	→	h^#(U111(e, e), U111(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(U111(d, d), U111(e, c))	A^#	→	h^#(U111(e, e), l)
A^#	→	h^#(U111(l, l), U111(e, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U111(e, a), c)	A^#	→	h^#(f(l), l)
A^#	→	h^#(f(c), U111(e, e))	A^#	→	h^#(U111(e, c), U111(e, c))
A^#	→	h^#(U111(e, e), U111(c, c))	A^#	→	h^#(f(d), b)
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(e), U111(b, b))	A^#	→	h^#(f(d), U111(c, b))
A^#	→	h^#(U111(a, a), U111(c, b))	A^#	→	h^#(e, l)
A^#	→	h^#(d, U111(e, c))	A^#	→	h^#(U111(a, a), l)
A^#	→	h^#(e, U111(e, b))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U111(c, a), U111(b, b))

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U111(e, a), U111(e, c))
A^#	→	h^#(U111(e, a), U111(e, e))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(a, U111(c, c))	A^#	→	h^#(a, f(e))
A^#	→	h^#(U111(c, c), U111(c, c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(d, d)	A^#	→	h^#(a, f(l))
A^#	→	h^#(a, f(c))	A^#	→	h^#(U111(e, e), U111(e, e))
A^#	→	h^#(l, f(e))	A^#	→	h^#(e, f(l))
A^#	→	h^#(U111(a, a), e)	A^#	→	h^#(c, c)
A^#	→	h^#(l, U111(c, c))	A^#	→	h^#(c, U111(e, b))
A^#	→	h^#(U111(a, a), U111(e, c))	A^#	→	h^#(U111(c, c), U111(e, c))
A^#	→	h^#(U111(c, c), d)	A^#	→	h^#(l, l)
A^#	→	h^#(U111(c, c), U111(e, e))	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U111(e, e), e)	A^#	→	h^#(U111(c, a), l)
A^#	→	h^#(U111(e, c), U111(e, b))	A^#	→	h^#(U111(e, e), U111(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(U111(d, d), U111(e, c))
A^#	→	h^#(U111(e, e), l)	A^#	→	h^#(U111(l, l), U111(e, b))
A^#	→	h^#(e, d)	A^#	→	h^#(f(l), l)
A^#	→	h^#(f(c), U111(e, e))	A^#	→	h^#(U111(e, a), c)
A^#	→	h^#(U111(e, c), U111(e, c))	A^#	→	h^#(U111(e, e), U111(c, c))
A^#	→	h^#(f(d), b)	A^#	→	h^#(U111(d, a), U111(e, c))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(e), U111(b, b))	A^#	→	h^#(f(d), U111(c, b))
A^#	→	h^#(U111(a, a), U111(c, b))	A^#	→	h^#(e, U111(e, b))
A^#	→	h^#(e, l)	A^#	→	h^#(U111(a, a), l)
A^#	→	h^#(d, U111(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U111(c, a), U111(b, b))

A^# → h^#(l, f(e))	A^# → h^#(l, U111(c, c))
A^# → h^#(l, f(l))

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, c)
A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U111(c, c), U111(c, c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(a, l)	A^#	→	h^#(d, d)
A^#	→	h^#(c, U111(e, c))	A^#	→	h^#(U111(e, e), U111(e, e))
A^#	→	h^#(e, f(l))	A^#	→	h^#(l, f(e))
A^#	→	h^#(U111(a, a), e)	A^#	→	h^#(c, c)
A^#	→	h^#(l, U111(c, c))	A^#	→	h^#(c, U111(e, b))
A^#	→	h^#(U111(a, a), U111(e, c))	A^#	→	h^#(l, l)
A^#	→	h^#(U111(c, c), d)	A^#	→	h^#(U111(c, c), U111(e, c))
A^#	→	h^#(U111(c, c), U111(e, e))	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U111(e, e), e)	A^#	→	h^#(U111(c, a), l)
A^#	→	h^#(U111(e, c), U111(e, b))	A^#	→	h^#(U111(e, e), U111(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(U111(d, d), U111(e, c))
A^#	→	h^#(U111(e, e), l)	A^#	→	h^#(U111(l, l), U111(e, b))
A^#	→	h^#(e, d)	A^#	→	h^#(f(l), l)
A^#	→	h^#(f(c), U111(e, e))	A^#	→	h^#(U111(e, a), c)
A^#	→	h^#(U111(e, e), U111(c, c))	A^#	→	h^#(f(d), b)
A^#	→	h^#(U111(e, c), U111(e, c))	A^#	→	h^#(U111(d, a), U111(e, c))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(e), U111(b, b))	A^#	→	h^#(f(d), U111(c, b))
A^#	→	h^#(U111(a, a), U111(c, b))	A^#	→	h^#(e, U111(e, b))
A^#	→	h^#(e, l)	A^#	→	h^#(U111(a, a), l)
A^#	→	h^#(U111(c, a), U111(b, b))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(d, U111(e, c))