Relevant Terms	Irrelevant Terms
h^#(U121(a, a), f(b))
h^#(f(c), f(b))
h^#(f(a), U121(b, b))
h^#(f(a), f(d))
h^#(f(d), f(b))
h^#(f(a), f(c))

Relevant Terms	Irrelevant Terms
h^#(U121(c, c), f(b))
h^#(f(l), f(b))
h^#(f(e), f(b))
h^#(f(c), f(c))
h^#(f(c), f(d))
h^#(f(c), U121(b, b))

Relevant Terms	Irrelevant Terms
h^#(U121(c, c), f(c))	h^#(U121(l, c), f(b))
h^#(U121(c, c), f(d))
h^#(U121(e, c), f(b))
h^#(U121(c, c), U121(b, b))

Relevant Terms	Irrelevant Terms
h^#(U121(e, c), f(d))	h^#(U121(c, c), U121(d, d))
h^#(U121(c, c), f(m))	h^#(U121(l, c), f(d))

Relevant Terms	Irrelevant Terms
h^#(f(d), f(c))	h^#(U121(d, d), f(b))
h^#(f(m), f(b))
h^#(f(d), f(d))
h^#(f(d), U121(b, b))

Relevant Terms	Irrelevant Terms
h^#(f(m), U121(b, b))	h^#(U121(d, d), U121(b, b))
h^#(f(d), U121(c, b))	h^#(f(d), U121(d, b))

Relevant Terms	Irrelevant Terms
h^#(f(m), U121(c, b))	h^#(U121(d, d), U121(c, b))
h^#(f(d), U121(e, b))	h^#(f(d), U121(l, b))

Relevant Terms	Irrelevant Terms
h^#(U121(d, d), U121(e, b))
h^#(f(m), U121(e, b))
h^#(f(d), b)

Relevant Terms	Irrelevant Terms
h^#(U121(e, c), f(m))	h^#(U121(e, c), U121(d, d))
h^#(c, f(d))

Relevant Terms	Irrelevant Terms
h^#(f(m), U121(e, b))	h^#(f(m), U121(l, b))
	h^#(U121(m, m), U121(c, b))

Relevant Terms	Irrelevant Terms
h^#(f(a), f(m))	h^#(f(a), U121(d, d))
h^#(U121(a, a), f(d))
h^#(f(d), f(d))
h^#(f(c), f(d))

Relevant Terms	Irrelevant Terms
h^#(U121(a, a), f(m))	h^#(f(a), U121(m, m))
h^#(f(d), f(m))
h^#(f(c), f(m))

Problem 20: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(f(a), f(e))	A^#	→	h^#(U121(e, c), f(m))
A^#	→	h^#(c, f(d))	A^#	→	h^#(U121(a, a), f(d))
A^#	→	h^#(e, f(b))	A^#	→	h^#(l, f(b))
A^#	→	h^#(U121(e, c), f(c))	A^#	→	h^#(f(m), U121(b, b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(c, U121(b, b))
A^#	→	h^#(f(c), f(m))	A^#	→	h^#(f(e), f(b))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(a, a), f(m))	A^#	→	h^#(U121(d, d), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(l), f(b))	A^#	→	h^#(U121(e, c), U121(c, b))
A^#	→	h^#(U121(c, c), f(c))	A^#	→	h^#(f(m), f(b))
A^#	→	h^#(f(a), U121(c, c))	A^#	→	h^#(c, f(c))
A^#	→	h^#(U121(c, c), U121(b, b))	A^#	→	h^#(f(a), f(l))
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(f(c), U121(b, b))
A^#	→	h^#(U121(a, a), U121(b, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(a, a), f(c))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Context-sensitive strategy:
μ(d) = μ(e) = μ(d#) = μ(b) = μ(c) = μ(A) = μ(a) = μ(l) = μ(m) = μ(a#) = μ(k) = μ(T) = μ(A#) = μ(k#) = μ(b#) = μ(c#) = ∅
μ(f) = μ(f#) = μ(U121#) = μ(U121) = {1}
μ(h#) = μ(h) = {1, 2}
μ(g) = μ(g#) = {1, 2, 3}

The right-hand side of the rule A^# → h^#(f(d), 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^#(f(d), f(e))	h^#(U121(d, d), f(c))
h^#(f(d), U121(c, c))
h^#(f(m), f(c))
h^#(f(d), f(l))

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

A^# → h^#(f(m), f(c))	A^# → h^#(f(d), U121(c, c))
A^# → h^#(f(d), f(l))	A^# → h^#(f(d), f(e))

Problem 22: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(l, U121(b, b))
A^#	→	h^#(U121(a, a), f(d))	A^#	→	h^#(e, f(b))
A^#	→	h^#(f(d), c)	A^#	→	h^#(f(m), U121(b, b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(U121(e, c), f(c))
A^#	→	h^#(U121(c, c), f(e))	A^#	→	h^#(U121(e, c), U121(m, m))
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(c, U121(b, b))
A^#	→	h^#(l, f(m))	A^#	→	h^#(f(c), f(m))
A^#	→	h^#(f(e), f(b))	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(a, a), f(m))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(l), f(b))
A^#	→	h^#(U121(e, c), U121(c, b))	A^#	→	h^#(U121(c, c), f(c))
A^#	→	h^#(f(m), f(b))	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(m), U121(e, c))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(f(a), U121(c, c))	A^#	→	h^#(c, f(c))
A^#	→	h^#(U121(c, c), U121(b, b))	A^#	→	h^#(f(a), U121(e, e))
A^#	→	h^#(f(a), f(l))	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(l, f(c))	A^#	→	h^#(e, f(m))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(d), e)
A^#	→	h^#(f(c), U121(e, e))	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(c), U121(b, b))
A^#	→	h^#(U121(a, a), U121(b, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(e, f(d))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), f(c))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(l, U121(b, 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^#(l, U121(c, b))	h^#(l, U121(d, b))

Thus, the rule A^# → h^#(l, U121(b, b)) is replaced by the following rules:

A^# → h^#(l, U121(c, b))

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(l, U121(c, c))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(a), U121(c, c))
A^#	→	h^#(U121(c, a), f(d))	A^#	→	h^#(U121(c, c), U121(b, b))
A^#	→	h^#(f(a), f(l))	A^#	→	h^#(c, f(l))
A^#	→	h^#(U121(e, e), U121(m, m))	A^#	→	h^#(l, f(c))
A^#	→	h^#(f(d), e)	A^#	→	h^#(f(c), U121(e, e))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(m, m), U121(c, b))
A^#	→	h^#(f(c), U121(b, b))	A^#	→	h^#(U121(a, a), U121(b, b))
A^#	→	h^#(U121(e, e), U121(c, b))	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(f(l), U121(e, c))	A^#	→	h^#(e, f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U121(e, c), f(l))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(a, a), f(c))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(U121(e, e), U121(c, c))
A^#	→	h^#(e, c)	A^#	→	h^#(U121(e, c), f(e))
A^#	→	h^#(f(e), U121(e, b))	A^#	→	h^#(f(m), U121(e, e))
A^#	→	h^#(l, U121(b, b))	A^#	→	h^#(c, U121(c, b))
A^#	→	h^#(f(l), m)	A^#	→	h^#(c, f(e))
A^#	→	h^#(e, f(c))	A^#	→	h^#(e, f(b))
A^#	→	h^#(f(d), c)	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(f(l), c)	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(m))	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), U121(e, e))	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(f(m), c)
A^#	→	h^#(U121(m, m), U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, c), f(c))
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(a), U121(e, e))
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(e, f(m))
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(c, U121(e, c))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(l, U121(c, 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^#(l, U121(e, c))	h^#(l, U121(l, c))

Thus, the rule A^# → h^#(l, U121(c, c)) is replaced by the following rules:

A^# → h^#(l, U121(e, c))

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(U121(e, a), U121(d, d))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(l))
A^#	→	h^#(U121(a, a), f(l))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(c), U121(b, b))	A^#	→	h^#(U121(m, m), U121(c, b))
A^#	→	h^#(U121(a, a), U121(b, b))	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(U121(e, e), U121(c, b))	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(e, f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), f(c))	A^#	→	h^#(U121(c, a), U121(c, c))
A^#	→	h^#(U121(e, e), U121(c, c))	A^#	→	h^#(e, c)
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(f(e), l)
A^#	→	h^#(U121(e, a), f(m))	A^#	→	h^#(U121(e, c), f(e))
A^#	→	h^#(f(e), U121(e, b))	A^#	→	h^#(f(m), U121(e, e))
A^#	→	h^#(U121(a, a), U121(e, c))	A^#	→	h^#(l, U121(b, b))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(f(l), m)
A^#	→	h^#(f(m), e)	A^#	→	h^#(d, f(d))
A^#	→	h^#(c, f(e))	A^#	→	h^#(e, f(c))
A^#	→	h^#(a, f(m))	A^#	→	h^#(c, f(d))
A^#	→	h^#(f(d), U121(c, c))	A^#	→	h^#(e, f(b))
A^#	→	h^#(f(d), c)	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), c)	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(f(a), l)
A^#	→	h^#(c, c)	A^#	→	h^#(l, f(m))
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), U121(e, e))	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(f(m), c)
A^#	→	h^#(U121(m, m), U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, c), f(c))	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(a), U121(e, e))
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(e, f(m))	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(e, a), U121(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
	h^#(U121(e, a), U121(m, d))
	h^#(a, U121(d, d))

Thus, the rule A^# → h^#(U121(e, a), U121(d, d)) is deleted.

Problem 25: 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^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, e)
A^#	→	h^#(c, f(d))	A^#	→	h^#(f(d), U121(c, c))
A^#	→	h^#(e, f(b))	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(m, f(d))	A^#	→	h^#(d, f(m))
A^#	→	h^#(U121(e, a), f(l))	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(U121(c, c), U121(e, e))	A^#	→	h^#(U121(m, m), U121(e, c))
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U121(c, c), f(c))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(e, c), U121(e, e))	A^#	→	h^#(f(a), U121(e, e))
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(c, a), U121(e, c))
A^#	→	h^#(U121(a, a), l)	A^#	→	h^#(f(e), U121(b, b))
A^#	→	h^#(f(l), b)	A^#	→	h^#(f(l), U121(b, b))
A^#	→	h^#(e, f(e))	A^#	→	h^#(U121(c, c), U121(c, b))
A^#	→	h^#(U121(c, a), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(c, c), U121(b, b))	A^#	→	h^#(U121(c, a), U121(c, b))
A^#	→	h^#(U121(m, m), U121(c, b))	A^#	→	h^#(U121(e, e), U121(c, b))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(c, a), U121(c, c))
A^#	→	h^#(f(e), l)	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(f(c), b)	A^#	→	h^#(f(m), U121(e, e))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), e)
A^#	→	h^#(c, d)	A^#	→	h^#(a, f(m))
A^#	→	h^#(l, b)	A^#	→	h^#(f(d), c)
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(e, e), f(e))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(f(l), c)
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(e, f(m))	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(c, e) 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^#(l, e)
h^#(e, e)

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

A^# → h^#(e, e)

A^# → h^#(l, e)

Problem 26: 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^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(f(m), b)
A^#	→	h^#(c, c)	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(e, l)
A^#	→	h^#(m, e)	A^#	→	h^#(c, b)
A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(c, a), U121(e, c))
A^#	→	h^#(f(d), U121(e, c))	A^#	→	h^#(U121(a, a), l)
A^#	→	h^#(f(e), c)	A^#	→	h^#(f(e), U121(b, b))
A^#	→	h^#(f(l), b)	A^#	→	h^#(f(l), U121(b, b))
A^#	→	h^#(e, f(e))	A^#	→	h^#(f(c), l)
A^#	→	h^#(U121(c, a), l)	A^#	→	h^#(U121(c, c), c)
A^#	→	h^#(f(d), U121(e, e))	A^#	→	h^#(f(a), e)
A^#	→	h^#(U121(e, c), f(c))	A^#	→	h^#(U121(c, c), f(e))
A^#	→	h^#(U121(c, c), U121(c, b))	A^#	→	h^#(f(l), f(m))
A^#	→	h^#(U121(c, c), f(d))	A^#	→	h^#(U121(c, a), f(c))
A^#	→	h^#(U121(c, c), f(l))	A^#	→	h^#(U121(e, a), c)
A^#	→	h^#(U121(c, c), e)	A^#	→	h^#(f(e), f(m))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(c, c), U121(b, b))
A^#	→	h^#(U121(c, a), U121(c, b))	A^#	→	h^#(f(c), U121(e, e))
A^#	→	h^#(U121(e, e), U121(c, b))	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(U121(m, m), U121(c, b))	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), U121(e, e))	A^#	→	h^#(U121(c, a), U121(c, c))
A^#	→	h^#(U121(l, c), U121(e, b))	A^#	→	h^#(f(e), l)
A^#	→	h^#(U121(e, a), f(m))	A^#	→	h^#(f(c), b)
A^#	→	h^#(a, f(l))	A^#	→	h^#(f(m), U121(e, e))
A^#	→	h^#(c, f(m))	A^#	→	h^#(l, f(d))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), e)
A^#	→	h^#(c, d)	A^#	→	h^#(a, f(m))
A^#	→	h^#(l, b)	A^#	→	h^#(f(d), c)
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(f(l), c)
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(e, f(m))	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(m), 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(m), c)	h^#(U121(m, m), b)
h^#(f(m), d)

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

A^# → h^#(f(m), d)

A^# → h^#(f(m), c)

Problem 27: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(f(l), b)	A^#	→	h^#(U121(e, c), f(c))
A^#	→	h^#(U121(c, c), f(e))	A^#	→	h^#(U121(e, e), c)
A^#	→	h^#(f(e), U121(c, c))	A^#	→	h^#(U121(c, c), U121(c, b))
A^#	→	h^#(U121(e, e), f(c))	A^#	→	h^#(U121(e, c), c)
A^#	→	h^#(f(l), f(m))	A^#	→	h^#(e, e)
A^#	→	h^#(U121(c, c), f(d))	A^#	→	h^#(U121(c, a), f(c))
A^#	→	h^#(U121(c, c), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U121(e, a), c)	A^#	→	h^#(U121(c, c), e)
A^#	→	h^#(f(e), f(m))	A^#	→	h^#(U121(e, a), U121(e, c))
A^#	→	h^#(f(m), U121(e, c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(c, c), U121(b, b))	A^#	→	h^#(U121(c, a), U121(c, b))
A^#	→	h^#(f(d), e)	A^#	→	h^#(f(c), U121(e, e))
A^#	→	h^#(f(e), f(l))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, e), U121(c, b))	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(U121(m, m), U121(c, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(a, a), U121(e, e))
A^#	→	h^#(U121(c, a), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(U121(l, c), U121(e, b))	A^#	→	h^#(f(e), l)
A^#	→	h^#(U121(e, a), f(m))	A^#	→	h^#(f(c), b)
A^#	→	h^#(f(m), U121(e, e))	A^#	→	h^#(a, f(l))
A^#	→	h^#(c, f(m))	A^#	→	h^#(l, f(d))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), d)
A^#	→	h^#(f(l), m)	A^#	→	h^#(f(m), e)
A^#	→	h^#(c, d)	A^#	→	h^#(f(m), m)
A^#	→	h^#(a, f(m))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(U121(c, c), b)	A^#	→	h^#(U121(e, e), f(e))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(f(l), c)	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, m, 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^#(U121(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 28: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(U121(l, c), U121(c, b))	A^#	→	h^#(U121(e, e), e)
A^#	→	h^#(U121(e, e), f(c))	A^#	→	h^#(U121(e, c), c)
A^#	→	h^#(f(l), f(m))	A^#	→	h^#(U121(c, c), f(d))
A^#	→	h^#(e, e)	A^#	→	h^#(U121(c, a), f(c))
A^#	→	h^#(U121(c, c), f(l))	A^#	→	h^#(f(e), U121(e, c))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U121(e, a), c)
A^#	→	h^#(U121(c, c), e)	A^#	→	h^#(U121(e, c), U121(c, c))
A^#	→	h^#(f(e), f(m))	A^#	→	h^#(U121(e, a), U121(e, c))
A^#	→	h^#(c, f(c))	A^#	→	h^#(f(m), U121(e, c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(c, c), U121(b, b))
A^#	→	h^#(U121(c, a), U121(c, b))	A^#	→	h^#(f(d), e)
A^#	→	h^#(f(e), f(l))	A^#	→	h^#(f(c), U121(e, e))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(m, m), U121(c, b))
A^#	→	h^#(U121(e, c), U121(e, b))	A^#	→	h^#(U121(e, e), U121(c, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), U121(e, e))	A^#	→	h^#(U121(c, a), U121(c, c))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(U121(l, c), U121(e, b))
A^#	→	h^#(f(e), l)	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(f(c), b)	A^#	→	h^#(l, e)
A^#	→	h^#(f(m), U121(e, e))	A^#	→	h^#(a, f(l))
A^#	→	h^#(l, f(d))	A^#	→	h^#(c, f(m))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), d)
A^#	→	h^#(c, d)	A^#	→	h^#(f(m), e)
A^#	→	h^#(f(m), m)	A^#	→	h^#(l, b)
A^#	→	h^#(a, f(m))	A^#	→	h^#(f(d), c)
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(f(l), c)
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), l)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(f(m), c)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(f(e), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(l, c), U121(c, 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^#(U121(l, c), U121(e, b))	h^#(U121(l, c), U121(l, b))

Thus, the rule A^# → h^#(U121(l, c), U121(c, b)) is replaced by the following rules:

A^# → h^#(U121(l, c), U121(e, b))

Problem 29: 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^#(f(e), f(e))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(a, l)	A^#	→	h^#(l, e)
A^#	→	h^#(m, l)	A^#	→	h^#(e, f(c))
A^#	→	h^#(U121(e, a), f(l))	A^#	→	h^#(e, f(l))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(d, c)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(e), f(m))
A^#	→	h^#(U121(c, a), f(e))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(m), U121(e, c))	A^#	→	h^#(c, f(c))
A^#	→	h^#(U121(c, c), U121(b, b))	A^#	→	h^#(U121(c, a), U121(c, b))
A^#	→	h^#(f(d), e)	A^#	→	h^#(f(e), f(l))
A^#	→	h^#(f(c), U121(e, e))	A^#	→	h^#(U121(e, e), U121(c, b))
A^#	→	h^#(U121(e, c), U121(e, b))	A^#	→	h^#(U121(m, m), U121(c, b))
A^#	→	h^#(f(l), U121(e, c))	A^#	→	h^#(U121(e, c), f(l))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(a, a), U121(e, e))
A^#	→	h^#(U121(c, a), U121(c, c))	A^#	→	h^#(U121(l, c), U121(e, b))
A^#	→	h^#(f(e), l)	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(f(c), b)	A^#	→	h^#(U121(e, c), f(d))
A^#	→	h^#(f(m), U121(e, e))	A^#	→	h^#(a, f(l))
A^#	→	h^#(c, f(m))	A^#	→	h^#(l, f(d))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), d)
A^#	→	h^#(c, d)	A^#	→	h^#(f(m), e)
A^#	→	h^#(f(m), m)	A^#	→	h^#(l, b)
A^#	→	h^#(a, f(m))	A^#	→	h^#(f(d), c)
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(f(l), c)
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(f(a), l)
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(f(m), c)	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, e) 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^#(d, e)
h^#(c, e)

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

A^# → h^#(d, e)

A^# → h^#(c, e)

Problem 30: 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^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, f(e))
A^#	→	h^#(l, f(e))	A^#	→	h^#(e, f(l))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(U121(m, m), U121(e, c))	A^#	→	h^#(l, l)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(e, f(e))
A^#	→	h^#(l, U121(c, c))	A^#	→	h^#(c, U121(c, c))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(c, c), U121(b, b))	A^#	→	h^#(U121(c, a), U121(c, b))
A^#	→	h^#(U121(e, e), U121(m, m))	A^#	→	h^#(f(d), e)
A^#	→	h^#(f(e), f(l))	A^#	→	h^#(f(c), U121(e, e))
A^#	→	h^#(U121(e, e), U121(c, b))	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(U121(m, m), U121(c, b))	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), U121(e, e))	A^#	→	h^#(U121(c, a), U121(c, c))
A^#	→	h^#(U121(l, c), U121(e, b))	A^#	→	h^#(U121(e, a), U121(e, e))
A^#	→	h^#(f(e), l)	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(f(c), b)	A^#	→	h^#(U121(e, c), f(d))
A^#	→	h^#(m, U121(e, e))	A^#	→	h^#(a, f(l))
A^#	→	h^#(f(m), U121(e, e))	A^#	→	h^#(l, f(d))
A^#	→	h^#(c, f(m))	A^#	→	h^#(c, U121(c, b))
A^#	→	h^#(U121(a, a), U121(c, b))	A^#	→	h^#(e, b)
A^#	→	h^#(f(m), d)	A^#	→	h^#(c, d)
A^#	→	h^#(f(m), e)	A^#	→	h^#(f(m), m)
A^#	→	h^#(a, f(m))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(f(l), c)	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(c, f(e)) 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^#(e, f(e))
h^#(c, U121(e, e))
h^#(l, f(e))

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

A^# → h^#(l, f(e))	A^# → h^#(e, f(e))
A^# → h^#(c, U121(e, e))

Problem 31: 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^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(e, f(l))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(U121(c, c), U121(e, e))
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(U121(m, m), U121(e, c))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(U121(c, c), U121(c, c))
A^#	→	h^#(c, b)	A^#	→	h^#(U121(c, a), l)
A^#	→	h^#(a, b)	A^#	→	h^#(U121(e, c), c)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(U121(e, a), c)
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(e, c), m)
A^#	→	h^#(U121(e, a), d)	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), U121(e, e))	A^#	→	h^#(U121(c, a), U121(c, c))
A^#	→	h^#(U121(l, c), U121(e, b))	A^#	→	h^#(f(e), l)
A^#	→	h^#(U121(e, a), U121(e, e))	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(f(c), b)	A^#	→	h^#(U121(e, c), f(d))
A^#	→	h^#(m, U121(e, e))	A^#	→	h^#(f(m), U121(e, e))
A^#	→	h^#(a, f(l))	A^#	→	h^#(l, f(d))
A^#	→	h^#(c, f(m))	A^#	→	h^#(c, U121(c, b))
A^#	→	h^#(U121(a, a), U121(c, b))	A^#	→	h^#(e, b)
A^#	→	h^#(f(m), d)	A^#	→	h^#(c, d)
A^#	→	h^#(f(m), e)	A^#	→	h^#(f(m), m)
A^#	→	h^#(a, f(m))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U121(e, e), f(e))
A^#	→	h^#(U121(c, c), b)	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), c)	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(f(m), c)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(m, f(e))	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, U121(e, 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^#(d, U121(e, b))
h^#(c, U121(e, b))
h^#(a, b)

Thus, the rule A^# → h^#(a, U121(e, b)) is replaced by the following rules:

A^# → h^#(c, U121(e, b))	A^# → h^#(a, b)
A^# → h^#(d, U121(e, b))

Problem 32: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(d, b)
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(e, c), m)
A^#	→	h^#(e, m)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, a), d)	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(f(l), U121(e, c))
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(a, a), U121(e, e))
A^#	→	h^#(U121(c, a), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(U121(l, c), U121(e, b))	A^#	→	h^#(f(e), l)
A^#	→	h^#(U121(e, a), U121(e, e))	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(a, l)	A^#	→	h^#(f(c), b)
A^#	→	h^#(U121(e, c), f(d))	A^#	→	h^#(m, U121(e, e))
A^#	→	h^#(f(m), U121(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(a, f(l))	A^#	→	h^#(m, l)
A^#	→	h^#(l, f(d))	A^#	→	h^#(c, f(m))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), d)
A^#	→	h^#(f(m), e)	A^#	→	h^#(c, d)
A^#	→	h^#(f(m), m)	A^#	→	h^#(l, b)
A^#	→	h^#(a, f(m))	A^#	→	h^#(f(d), c)
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(e, e), f(e))
A^#	→	h^#(U121(c, c), b)	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(f(l), c)
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(d, 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^#(m, b)
h^#(d, c)
h^#(d, d)

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

A^# → h^#(m, b)	A^# → h^#(d, c)
A^# → h^#(d, d)

Problem 33: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(d, e)
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(f(e), l)
A^#	→	h^#(U121(e, a), f(m))	A^#	→	h^#(a, l)
A^#	→	h^#(f(c), b)	A^#	→	h^#(U121(e, c), f(d))
A^#	→	h^#(m, U121(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(c, e)	A^#	→	h^#(a, f(l))
A^#	→	h^#(f(m), U121(e, e))	A^#	→	h^#(m, l)
A^#	→	h^#(l, f(d))	A^#	→	h^#(c, f(m))
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), d)
A^#	→	h^#(f(m), e)	A^#	→	h^#(c, d)
A^#	→	h^#(m, b)	A^#	→	h^#(f(m), m)
A^#	→	h^#(l, b)	A^#	→	h^#(a, f(m))
A^#	→	h^#(f(d), c)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(f(l), c)
A^#	→	h^#(a, d)	A^#	→	h^#(e, f(l))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), l)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(d, e) 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^#(m, e)

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

A^# → h^#(m, e)

Problem 34: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(U121(c, a), b)	A^#	→	h^#(d, d)
A^#	→	h^#(U121(e, a), b)	A^#	→	h^#(e, b)
A^#	→	h^#(f(m), d)	A^#	→	h^#(f(l), m)
A^#	→	h^#(f(m), e)	A^#	→	h^#(c, d)
A^#	→	h^#(U121(e, c), f(m))	A^#	→	h^#(f(m), m)
A^#	→	h^#(m, b)	A^#	→	h^#(a, f(m))
A^#	→	h^#(c, f(d))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(f(l), c)	A^#	→	h^#(a, d)
A^#	→	h^#(e, f(l))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), l)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(f(e), d)
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(c, a), 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^#(U121(e, a), b)	h^#(U121(l, a), b)
h^#(U121(c, a), c)
h^#(U121(c, a), d)

Thus, the rule A^# → h^#(U121(c, a), b) is replaced by the following rules:

A^# → h^#(U121(c, a), c)	A^# → h^#(U121(c, a), d)
A^# → h^#(U121(e, a), b)

Problem 35: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(U121(e, c), U121(m, m))
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(c, f(m))	A^#	→	h^#(m, b)
A^#	→	h^#(f(m), m)	A^#	→	h^#(a, f(m))
A^#	→	h^#(c, f(d))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(U121(e, e), f(e))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(f(l), c)	A^#	→	h^#(a, d)
A^#	→	h^#(e, f(l))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), l)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(f(m), c)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(f(e), d)
A^#	→	h^#(U121(c, c), f(m))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(e, c), U121(m, m)) 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, U121(m, m))

Thus, the rule A^# → h^#(U121(e, c), U121(m, m)) is deleted.

Problem 36: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(U121(c, c), d)
A^#	→	h^#(e, b)	A^#	→	h^#(c, d)
A^#	→	h^#(l, b)	A^#	→	h^#(U121(e, e), f(e))
A^#	→	h^#(a, U121(c, c))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(f(l), f(e))	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(f(l), c)	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(d, f(m))
A^#	→	h^#(e, f(l))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), l)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(f(m), c)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(c, c), 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
h^#(U121(c, c), m)	h^#(U121(l, c), d)
h^#(U121(e, c), d)

Thus, the rule A^# → h^#(U121(c, c), d) is replaced by the following rules:

A^# → h^#(U121(e, c), d)

A^# → h^#(U121(c, c), m)

Problem 37: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(a, l)
A^#	→	h^#(d, d)	A^#	→	h^#(m, l)
A^#	→	h^#(l, U121(e, b))	A^#	→	h^#(f(l), f(e))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(m, U121(c, c))
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(f(l), c)
A^#	→	h^#(a, d)	A^#	→	h^#(e, f(l))
A^#	→	h^#(d, f(m))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), l)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(f(m), c)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(d, e)

A^# → h^#(c, e)

Problem 38: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(e, b)
A^#	→	h^#(c, d)	A^#	→	h^#(m, b)
A^#	→	h^#(l, b)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(e, f(l))
A^#	→	h^#(d, f(m))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(m, f(l))	A^#	→	h^#(l, f(m))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(f(m), c)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(c, a), f(b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(e, 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^#(e, d)
h^#(e, c)

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

A^# → h^#(e, c)

A^# → h^#(e, d)

Problem 39: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(a, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(a, f(e))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(U121(e, a), U121(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(a, f(c))
A^#	→	h^#(a, U121(c, c))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(m, f(l))	A^#	→	h^#(l, f(m))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(d, m)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(m), c)	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(m, f(e))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, 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^#(d, c)
h^#(a, e)
h^#(c, c)
h^#(a, l)

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

A^# → h^#(a, l)	A^# → h^#(c, c)
A^# → h^#(d, c)	A^# → h^#(a, e)

Problem 40: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, e)	A^#	→	h^#(U121(c, c), U121(c, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(e, c)	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(m, U121(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(a, f(l))	A^#	→	h^#(c, f(e))
A^#	→	h^#(e, f(c))	A^#	→	h^#(d, f(c))
A^#	→	h^#(a, U121(c, c))	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), l)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(d, m)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(m), c)	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(m, f(e))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(m, e) 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

Thus, the rule A^# → h^#(m, e) is deleted.

Problem 41: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(d, l)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(m, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(e, c)
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(m, U121(c, c))	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(m, f(l))	A^#	→	h^#(l, f(m))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(c, c), f(m))
A^#	→	h^#(d, m)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(m), c)	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(m, f(e))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(d, 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
h^#(m, l)

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

A^# → h^#(m, l)

Problem 42: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(U121(e, e), d)	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), m)	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(c, c)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(c, a), f(b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(m, f(e))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(l, c)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(e, e), 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
h^#(U121(e, e), m)
h^#(e, d)

Thus, the rule A^# → h^#(U121(e, e), d) is replaced by the following rules:

A^# → h^#(U121(e, e), m)

A^# → h^#(e, d)

Problem 43: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(a, f(d))	A^#	→	h^#(m, m)
A^#	→	h^#(U121(e, a), U121(d, d))	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(d, f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(c, a), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(U121(e, a), U121(e, e))	A^#	→	h^#(U121(e, a), f(m))
A^#	→	h^#(U121(c, a), b)	A^#	→	h^#(U121(c, a), f(l))
A^#	→	h^#(d, d)	A^#	→	h^#(U121(e, a), b)
A^#	→	h^#(a, f(b))	A^#	→	h^#(c, f(e))
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(e, f(m))	A^#	→	h^#(m, f(e))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, 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
h^#(a, f(m))	h^#(a, U121(d, d))
h^#(d, f(d))
h^#(c, f(d))

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

A^# → h^#(d, f(d))	A^# → h^#(c, f(d))
A^# → h^#(a, f(m))

Problem 44: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(m, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U121(e, a), c)	A^#	→	h^#(d, b)
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(e, a), d)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(a, e)	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(a, l)
A^#	→	h^#(d, d)	A^#	→	h^#(m, l)
A^#	→	h^#(e, b)	A^#	→	h^#(c, d)
A^#	→	h^#(a, f(b))	A^#	→	h^#(c, f(e))
A^#	→	h^#(d, f(d))	A^#	→	h^#(c, f(d))
A^#	→	h^#(a, f(m))	A^#	→	h^#(l, b)
A^#	→	h^#(a, U121(c, c))	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(U121(e, a), f(l))	A^#	→	h^#(U121(e, a), f(c))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(m, 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^#(m, e)
h^#(m, l)

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

A^# → h^#(m, e)

A^# → h^#(m, l)

Problem 45: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(l, f(c))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(e, c)
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(m, f(c))	A^#	→	h^#(c, U121(c, b))
A^#	→	h^#(l, U121(b, b))	A^#	→	h^#(c, f(e))
A^#	→	h^#(d, f(d))	A^#	→	h^#(m, b)
A^#	→	h^#(e, f(c))	A^#	→	h^#(e, f(b))
A^#	→	h^#(d, f(c))	A^#	→	h^#(l, f(b))
A^#	→	h^#(a, f(m))	A^#	→	h^#(c, f(d))
A^#	→	h^#(l, b)	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(a, f(c))	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(a, U121(c, c))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(e, f(l))
A^#	→	h^#(m, f(d))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(m, U121(b, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(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
	h^#(l, U121(l, l))

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

Problem 46: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, e)	A^#	→	h^#(U121(c, c), U121(c, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(m, U121(e, c))	A^#	→	h^#(e, e)
A^#	→	h^#(d, l)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(m, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(d, f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(m, f(c))	A^#	→	h^#(d, f(l))
A^#	→	h^#(l, f(b))	A^#	→	h^#(e, U121(b, b))
A^#	→	h^#(c, U121(e, b))	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(a, f(c))	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(m, U121(c, c))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(l, f(e))
A^#	→	h^#(d, f(m))	A^#	→	h^#(e, f(l))
A^#	→	h^#(m, f(d))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

Thus, the rule A^# → h^#(m, e) is deleted.

Problem 47: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(d, e)
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(a, l)
A^#	→	h^#(d, d)	A^#	→	h^#(c, e)
A^#	→	h^#(m, l)	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(a, f(c))	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(m, U121(c, c))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(e, f(l))
A^#	→	h^#(d, f(m))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(m, f(d))	A^#	→	h^#(l, f(e))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(m, e)

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

A^# → h^#(m, e)

Problem 48: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(d, l)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(m, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(d, f(e))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(m, f(c))	A^#	→	h^#(d, f(l))
A^#	→	h^#(a, U121(c, c))	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(m, U121(c, c))	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(m, f(d))
A^#	→	h^#(d, f(m))	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(m, l)

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

A^# → h^#(m, l)

Problem 49: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, e)	A^#	→	h^#(U121(c, c), U121(c, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(d, l)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(m, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(a, l)	A^#	→	h^#(d, d)
A^#	→	h^#(m, l)	A^#	→	h^#(e, b)
A^#	→	h^#(c, d)	A^#	→	h^#(m, b)
A^#	→	h^#(l, b)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(e, f(l))
A^#	→	h^#(m, f(d))	A^#	→	h^#(d, f(m))
A^#	→	h^#(l, f(e))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

Thus, the rule A^# → h^#(m, e) is deleted.

Problem 50: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(c, m)	A^#	→	h^#(l, d)
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(m, f(d))
A^#	→	h^#(e, f(l))	A^#	→	h^#(d, f(m))
A^#	→	h^#(U121(e, a), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(d, m)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(m, U121(b, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(c, m) 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^#(l, m)
h^#(e, m)

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

A^# → h^#(l, m)

A^# → h^#(e, m)

Problem 51: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(l, e)	A^#	→	h^#(d, d)
A^#	→	h^#(m, l)	A^#	→	h^#(c, f(e))
A^#	→	h^#(a, f(c))	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(U121(e, a), f(l))	A^#	→	h^#(c, c)
A^#	→	h^#(l, f(m))	A^#	→	h^#(m, f(l))
A^#	→	h^#(U121(c, a), f(m))	A^#	→	h^#(m, d)
A^#	→	h^#(d, m)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(l, l)	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), f(d))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(f(e), b)
A^#	→	h^#(l, c)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(l, e) 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

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

Problem 52: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(m, U121(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(d, l)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(m, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(d, f(e))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(d, f(l))
A^#	→	h^#(m, f(c))	A^#	→	h^#(a, U121(c, c))
A^#	→	h^#(m, U121(c, c))	A^#	→	h^#(e, f(l))
A^#	→	h^#(l, f(e))	A^#	→	h^#(U121(e, a), f(l))
A^#	→	h^#(c, c)	A^#	→	h^#(l, f(m))
A^#	→	h^#(m, f(l))	A^#	→	h^#(U121(c, a), f(m))
A^#	→	h^#(m, d)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(d, m)	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(a, U121(e, c))	A^#	→	h^#(l, l)
A^#	→	h^#(m, U121(b, b))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(f(d), f(m))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(m, f(e))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(m, U121(e, 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^#(m, c)

Thus, the rule A^# → h^#(m, U121(e, c)) is replaced by the following rules:

A^# → h^#(m, c)

Problem 53: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(d, f(m))
A^#	→	h^#(c, c)	A^#	→	h^#(l, f(m))
A^#	→	h^#(m, d)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(d, m)	A^#	→	h^#(a, U121(e, c))
A^#	→	h^#(e, U121(e, e))	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(d), f(m))
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(a, a), f(e))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(e, d)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(l), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(m, f(e))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(f(e), b)
A^#	→	h^#(l, c)	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(d, f(m)) 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^#(m, f(m))	h^#(d, U121(m, m))

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

A^# → h^#(m, f(m))

Problem 54: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(l, l)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U121(c, c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(U121(e, e), b)
A^#	→	h^#(f(c), c)	A^#	→	h^#(f(l), f(d))
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(a, a), f(e))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(m), U121(c, c))	A^#	→	h^#(U121(e, e), f(d))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(c, c), e) 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^#(U121(e, c), e)	h^#(U121(l, c), e)

Thus, the rule A^# → h^#(U121(c, c), e) is replaced by the following rules:

A^# → h^#(U121(e, c), e)

Problem 55: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(e, U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(l, l)	A^#	→	h^#(l, U121(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(e, e), f(d))	A^#	→	h^#(f(m), U121(c, c))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(e, f(m))
A^#	→	h^#(m, f(e))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(a, e)
A^#	→	h^#(d, f(e))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(U121(a, a), U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(e, U121(e, e)) 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^#(e, e)

Thus, the rule A^# → h^#(e, U121(e, e)) is replaced by the following rules:

A^# → h^#(e, e)

Problem 56: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(l, d)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(e, U121(e, b))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(e, U121(c, b))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(e, d)	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(e, f(m))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, m)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(e, f(d))	g^#(d, x, x)	→	A^#
A^#	→	h^#(d, f(e))	A^#	→	h^#(a, e)
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(e, l)
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(a, a), U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(l, 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
h^#(l, m)

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

A^# → h^#(l, m)

Problem 57: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(c, d)
A^#	→	h^#(a, m)	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(m, d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(m, U121(b, b))
A^#	→	h^#(l, l)	A^#	→	h^#(m, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(d, U121(e, e))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(l), l)
A^#	→	h^#(e, f(m))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(m, f(e))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(e, m)	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(a, a), U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(c, 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
h^#(c, m)
h^#(e, d)
h^#(l, d)

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

A^# → h^#(c, m)	A^# → h^#(e, d)
A^# → h^#(l, d)

Problem 58: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(a, b)
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(a, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U121(e, a), c)	A^#	→	h^#(d, b)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(e, a), U121(e, b))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(a, a), c)	A^#	→	h^#(U121(m, a), U121(e, b))
A^#	→	h^#(m, f(e))	A^#	→	h^#(e, f(m))
A^#	→	h^#(l, c)	A^#	→	h^#(f(a), U121(b, b))
A^#	→	h^#(e, m)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U121(c, a), d)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(e, a), d)	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(a, a), U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, 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^#(a, c)
h^#(c, b)
h^#(a, d)
h^#(d, b)

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

A^# → h^#(c, b)	A^# → h^#(a, d)
A^# → h^#(d, b)	A^# → h^#(a, c)

Problem 59: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(l, d)	A^#	→	h^#(m, m)
A^#	→	h^#(l, m)	A^#	→	h^#(a, b)
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U121(e, a), c)	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(e, m)	A^#	→	h^#(U121(e, a), d)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(m, b)
A^#	→	h^#(l, b)	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(U121(a, a), d)	A^#	→	h^#(c, c)
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(U121(m, a), U121(e, b))	A^#	→	h^#(e, f(m))
A^#	→	h^#(m, f(e))	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(l, c)
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(e, l)	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(U121(e, c), l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(l, m)

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

A^# → h^#(l, m)

Problem 60: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(l, b)
A^#	→	h^#(l, m)	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(m, d)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(l, l)
A^#	→	h^#(m, c)	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(U121(m, a), U121(e, b))	A^#	→	h^#(m, f(e))
A^#	→	h^#(e, f(m))	A^#	→	h^#(l, c)
A^#	→	h^#(f(a), U121(b, b))	A^#	→	h^#(f(e), b)
A^#	→	h^#(e, m)	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(e, l)
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(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^#(l, c)
h^#(l, d)

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

A^# → h^#(l, c)

A^# → h^#(l, d)

Problem 63: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(f(c), U121(c, b))
A^#	→	h^#(f(d), U121(e, b))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(l, l)
A^#	→	h^#(f(c), U121(e, b))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(c, a), U121(c, b))
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(U121(c, a), d)
A^#	→	h^#(U121(a, a), U121(b, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(f(a), b)	A^#	→	h^#(f(c), U121(b, b))
A^#	→	h^#(U121(m, m), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(a, e)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(c), U121(c, 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^#(U121(c, c), U121(c, b))	h^#(f(c), U121(l, b))
h^#(f(l), U121(c, b))
h^#(f(c), U121(e, b))
h^#(f(e), U121(c, b))

Thus, the rule A^# → h^#(f(c), U121(c, b)) is replaced by the following rules:

A^# → h^#(f(l), U121(c, b))	A^# → h^#(f(c), U121(e, b))
A^# → h^#(f(e), U121(c, b))	A^# → h^#(U121(c, c), U121(c, b))

Problem 64: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), U121(e, b))	A^#	→	h^#(f(e), U121(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U121(e, c), U121(c, b))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(c, c), U121(b, b))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, b))
A^#	→	h^#(U121(m, m), U121(c, b))	A^#	→	h^#(U121(e, e), U121(c, b))
A^#	→	h^#(U121(a, a), U121(b, b))	A^#	→	h^#(f(a), b)
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(U121(c, a), b)	A^#	→	h^#(f(e), U121(e, b))
A^#	→	h^#(f(c), b)	A^#	→	h^#(d, d)
A^#	→	h^#(U121(c, c), d)	A^#	→	h^#(c, U121(c, b))
A^#	→	h^#(U121(e, a), b)	A^#	→	h^#(e, b)
A^#	→	h^#(f(l), m)	A^#	→	h^#(c, d)
A^#	→	h^#(l, b)	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(f(l), c)
A^#	→	h^#(a, d)	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(d, d), U121(e, b))
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(c), U121(e, 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(e), U121(e, b))
h^#(f(l), U121(e, b))
h^#(f(c), b)
h^#(U121(c, c), U121(e, b))

Thus, the rule A^# → h^#(f(c), U121(e, b)) is replaced by the following rules:

A^# → h^#(f(e), U121(e, b))	A^# → h^#(f(c), b)
A^# → h^#(U121(c, c), U121(e, b))	A^# → h^#(f(l), U121(e, b))

Problem 65: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(c, a), d)	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(f(c), b)
A^#	→	h^#(f(c), d)	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(U121(c, c), d)
A^#	→	h^#(c, U121(c, b))	A^#	→	h^#(l, U121(b, b))
A^#	→	h^#(U121(e, a), b)	A^#	→	h^#(U121(a, a), U121(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(f(m), d)
A^#	→	h^#(f(l), m)	A^#	→	h^#(f(m), e)
A^#	→	h^#(c, d)	A^#	→	h^#(f(m), m)
A^#	→	h^#(U121(m, m), U121(e, b))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(c, a), c)
A^#	→	h^#(f(a), m)	A^#	→	h^#(U121(c, c), l)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(f(l), c)
A^#	→	h^#(U121(a, a), b)	A^#	→	h^#(a, d)
A^#	→	h^#(U121(a, a), d)	A^#	→	h^#(f(a), l)
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(U121(d, d), U121(e, b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(f(l), l)
A^#	→	h^#(f(d), b)	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(f(e), b)
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(c, a), 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
h^#(U121(e, a), d)	h^#(U121(l, a), d)
h^#(U121(c, a), m)

Thus, the rule A^# → h^#(U121(c, a), d) is replaced by the following rules:

A^# → h^#(U121(c, a), m)

A^# → h^#(U121(e, a), d)

Problem 66: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, e)	A^#	→	h^#(U121(c, c), U121(c, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(d, l)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(m, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(e, a), d)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(a, e)	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(a, l)
A^#	→	h^#(d, d)	A^#	→	h^#(m, l)
A^#	→	h^#(U121(a, a), U121(c, b))	A^#	→	h^#(e, b)
A^#	→	h^#(f(m), d)	A^#	→	h^#(f(l), m)
A^#	→	h^#(f(m), e)	A^#	→	h^#(c, d)
A^#	→	h^#(m, b)	A^#	→	h^#(f(m), m)
A^#	→	h^#(U121(m, m), U121(e, b))	A^#	→	h^#(l, b)
A^#	→	h^#(f(d), c)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(f(a), m)
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(f(l), c)	A^#	→	h^#(U121(a, a), b)
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(l, l)
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

Thus, the rule A^# → h^#(m, e) is deleted.

Problem 67: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(e, a), d)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(a, l)	A^#	→	h^#(d, d)
A^#	→	h^#(m, l)	A^#	→	h^#(e, b)
A^#	→	h^#(f(m), d)	A^#	→	h^#(f(l), m)
A^#	→	h^#(f(m), e)	A^#	→	h^#(c, d)
A^#	→	h^#(f(m), m)	A^#	→	h^#(m, b)
A^#	→	h^#(l, b)	A^#	→	h^#(U121(m, m), U121(e, b))
A^#	→	h^#(f(d), c)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(e, U121(b, b))	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(f(a), m)
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(f(l), c)	A^#	→	h^#(U121(a, a), b)
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(e, a), 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
h^#(U121(e, a), m)
h^#(a, d)

Thus, the rule A^# → h^#(U121(e, a), d) is replaced by the following rules:

A^# → h^#(a, d)

A^# → h^#(U121(e, a), m)

Problem 68: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(e, b)
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(U121(c, c), b)
A^#	→	h^#(f(a), m)	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(f(l), c)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(U121(a, a), b)
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(m, d)
A^#	→	h^#(f(m), c)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(e, c)

A^# → h^#(e, d)

Problem 69: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(c, m)	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(e, b)	A^#	→	h^#(c, d)
A^#	→	h^#(l, b)	A^#	→	h^#(l, U121(e, b))
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(f(l), c)
A^#	→	h^#(a, U121(e, b))	A^#	→	h^#(U121(a, a), b)
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(f(a), l)	A^#	→	h^#(c, c)
A^#	→	h^#(U121(c, c), U121(e, b))	A^#	→	h^#(m, d)
A^#	→	h^#(f(m), c)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(f(l), l)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(l, m)
h^#(e, m)

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

A^# → h^#(l, m)

A^# → h^#(e, m)

Problem 70: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(e, m)	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	h^#(m, b)	A^#	→	h^#(l, b)
A^#	→	h^#(c, U121(e, b))	A^#	→	h^#(f(l), c)
A^#	→	h^#(U121(a, a), b)	A^#	→	h^#(a, d)
A^#	→	h^#(U121(a, a), d)	A^#	→	h^#(f(a), l)
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(m, d)	A^#	→	h^#(f(m), c)
A^#	→	h^#(f(e), d)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(U121(e, e), U121(e, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(U121(d, d), U121(e, b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(e, m) 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

Thus, the rule A^# → h^#(e, m) is deleted.

Problem 71: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, e)	A^#	→	h^#(U121(c, c), U121(c, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(d, l)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(m, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(d, b)	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(e, a), d)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(a, l)	A^#	→	h^#(d, d)
A^#	→	h^#(m, l)	A^#	→	h^#(e, b)
A^#	→	h^#(c, d)	A^#	→	h^#(l, b)
A^#	→	h^#(U121(c, a), c)	A^#	→	h^#(a, d)
A^#	→	h^#(U121(a, a), d)	A^#	→	h^#(f(a), l)
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(m, d)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(f(m), c)
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(U121(d, d), U121(e, b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(l, c)	A^#	→	h^#(f(e), b)
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

Thus, the rule A^# → h^#(m, e) is deleted.

Problem 72: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(c, l)
A^#	→	h^#(e, e)	A^#	→	h^#(d, l)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, e)	A^#	→	h^#(d, d)
A^#	→	h^#(m, l)	A^#	→	h^#(a, d)
A^#	→	h^#(U121(a, a), d)	A^#	→	h^#(f(a), l)
A^#	→	h^#(c, c)	A^#	→	h^#(U121(c, c), U121(e, b))
A^#	→	h^#(m, d)	A^#	→	h^#(f(e), d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(f(m), c)
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(U121(d, d), U121(e, b))
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(e, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(a, U121(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, a), m)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(c, 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
h^#(l, l)
h^#(e, l)

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

A^# → h^#(l, l)

A^# → h^#(e, l)

Problem 73: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(c, d)
A^#	→	h^#(l, b)	A^#	→	h^#(c, U121(e, b))
A^#	→	h^#(U121(c, c), b)	A^#	→	h^#(c, c)
A^#	→	h^#(m, d)	A^#	→	h^#(f(e), d)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(d, m)
A^#	→	h^#(f(m), c)	A^#	→	h^#(U121(l, l), U121(c, b))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(c, m)	A^# → h^#(e, d)
A^# → h^#(l, d)

Problem 74: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	h^#(l, b)
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(c, c)
A^#	→	h^#(m, d)	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(f(m), c)
A^#	→	h^#(d, m)	A^#	→	h^#(U121(e, e), U121(e, b))
A^#	→	h^#(U121(l, l), U121(c, b))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(d, d), U121(e, b))	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U121(e, c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(l, c)

A^# → h^#(l, d)

Problem 75: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(U121(c, c), c)
A^#	→	h^#(U121(c, c), l)	A^#	→	h^#(U121(e, e), c)
A^#	→	h^#(f(l), c)	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(c), e)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(e, e), m)	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(U121(c, a), m)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(a, U121(b, b))
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(l, l), U121(e, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U121(c, c), 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^#(U121(c, c), e)	h^#(U121(l, c), c)
h^#(U121(e, c), c)
h^#(U121(c, c), l)

Thus, the rule A^# → h^#(U121(c, c), c) is replaced by the following rules:

A^# → h^#(U121(c, c), e)	A^# → h^#(U121(c, c), l)
A^# → h^#(U121(e, c), c)

Problem 76: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(l, d)
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(l, U121(c, b))	A^#	→	h^#(d, U121(b, b))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(e, U121(e, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(e, U121(c, b))
A^#	→	h^#(a, U121(c, b))	A^#	→	h^#(f(m), f(m))
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(f(m), U121(e, b))	A^#	→	h^#(f(l), l)
A^#	→	h^#(f(d), b)	A^#	→	h^#(U121(e, e), U121(b, b))
A^#	→	h^#(U121(a, a), U121(e, b))	A^#	→	h^#(f(e), b)
A^#	→	h^#(l, c)	A^#	→	h^#(e, m)
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(U121(e, a), U121(c, b))
A^#	→	h^#(f(e), U121(e, e))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(U121(e, e), U121(e, c))	A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(l, m)

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

A^# → h^#(l, m)

Problem 77: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(m, m)	A^#	→	h^#(a, b)
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(d, U121(c, b))	A^#	→	h^#(l, U121(c, b))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(e, U121(e, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(e, U121(c, b))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(e, d)
A^#	→	h^#(U121(e, e), l)	A^#	→	h^#(f(m), U121(e, b))
A^#	→	h^#(f(l), l)	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(U121(a, a), U121(e, b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(e, m)	A^#	→	h^#(l, U121(e, c))
A^#	→	h^#(m, U121(e, b))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(c, a), U121(e, e))	A^#	→	h^#(U121(l, l), U121(e, b))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(a, U121(e, e))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(c, b)	A^# → h^#(a, d)
A^# → h^#(d, b)	A^# → h^#(a, c)

Problem 79: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(m, m)	A^#	→	h^#(e, e)
A^#	→	h^#(a, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(d, b)	A^#	→	h^#(U121(e, a), U121(e, b))
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(U121(a, a), c)
A^#	→	h^#(e, m)	A^#	→	h^#(U121(c, a), d)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, a), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(a, l)	A^#	→	h^#(d, d)
A^#	→	h^#(m, l)	A^#	→	h^#(e, b)
A^#	→	h^#(c, d)	A^#	→	h^#(l, b)
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(c, c)	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, a), l)	A^#	→	h^#(U121(c, a), e)
A^#	→	h^#(e, d)	A^#	→	h^#(U121(m, a), U121(e, b))
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(l, c)
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(d, U121(e, b))	A^#	→	h^#(U121(e, c), l)
A^#	→	h^#(a, U121(e, e))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(U121(e, e), U121(e, c))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(a, l)	A^# → h^#(c, c)
A^# → h^#(d, c)	A^# → h^#(a, e)

Problem 80: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, e)	A^#	→	h^#(U121(c, c), U121(c, c))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(m, m)
A^#	→	h^#(e, e)	A^#	→	h^#(d, l)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(m, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U121(e, a), c)	A^#	→	h^#(d, b)
A^#	→	h^#(f(m), f(m))	A^#	→	h^#(e, m)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, a), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(a, l)	A^#	→	h^#(d, d)
A^#	→	h^#(e, b)	A^#	→	h^#(c, d)
A^#	→	h^#(l, b)	A^#	→	h^#(a, U121(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(U121(a, a), d)
A^#	→	h^#(c, c)	A^#	→	h^#(m, d)
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, a), l)
A^#	→	h^#(U121(c, a), e)	A^#	→	h^#(U121(c, a), m)
A^#	→	h^#(e, d)	A^#	→	h^#(U121(e, a), m)
A^#	→	h^#(U121(m, a), U121(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(U121(e, e), U121(b, b))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), b)	A^#	→	h^#(m, U121(e, b))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(U121(c, a), U121(e, e))
A^#	→	h^#(U121(l, l), U121(e, b))	A^#	→	h^#(f(e), U121(e, e))
A^#	→	h^#(U121(e, a), U121(c, b))	A^#	→	h^#(d, U121(e, b))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(c, U121(e, c))	A^#	→	h^#(e, l)
A^#	→	h^#(a, U121(e, e))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

Thus, the rule A^# → h^#(m, e) is deleted.

Problem 91: Propagation

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(c, c), U121(c, c))	h^#(c, c)	→	g^#(c, c, f(k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(e, e)	→	g^#(e, e, f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(m, m)	→	g^#(m, m, f(k))
A^#	→	h^#(m, m)	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	h^#(U121(c, c), U121(c, c))	→	g^#(U121(c, c), U121(c, c), f(k))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
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))
A^#	→	h^#(f(m), f(m))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Context-sensitive strategy:
μ(d) = μ(e) = μ(d#) = μ(b) = μ(c) = μ(A) = μ(a) = μ(l) = μ(a#) = μ(m) = μ(k) = μ(T) = μ(A#) = μ(k#) = μ(b#) = μ(c#) = ∅
μ(f) = μ(f#) = μ(U121#) = μ(U121) = {1}
μ(h#) = μ(h) = {1, 2}
μ(g) = μ(g#) = {1, 2, 3}

The dependency pairs A^# → h^#(U121(c, c), U121(c, c)) and h^#(U121(c, c), U121(c, c)) → g^#(U121(c, c), U121(c, c), f(k)) are consolidated into the rule A^# → g^#(U121(c, c), U121(c, c), f(k)) .

This is possible as

all subterms of h#(U121(c, c), U121(c, c)) containing variables are rooted by a constructor symbol,
there is no variable that is replacing in h#(U121(c, c), U121(c, c)), but non-replacing in both A# and g#(U121(c, c), U121(c, c), f(k))

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(U121(c, c), U121(c, c)) are consolidated into the rule g^#(d, x, x) → h^#(U121(c, c), U121(c, c)) .

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#(U121(c, c), U121(c, c))

Summary

Removed Dependency Pairs	Added Dependency Pairs
A^# → h^#(U121(c, c), U121(c, c))	g^#(d, x, x) → h^#(U121(c, c), U121(c, c))
h^#(U121(c, c), U121(c, c)) → g^#(U121(c, c), U121(c, c), f(k))	A^# → g^#(U121(c, c), U121(c, c), f(k))
g^#(d, x, x) → A^#

Problem 92: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(c, c)	→	g^#(c, c, f(k))
h^#(e, e)	→	g^#(e, e, f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(k))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(m, m)	→	g^#(m, m, f(k))
A^#	→	h^#(m, m)	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	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))	A^#	→	h^#(f(m), f(m))
h^#(l, l)	→	g^#(l, l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, 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, c, f(m))
g^#(c, e, f(k))
g^#(c, c, U121(k, k))
g^#(e, c, f(k))
g^#(l, c, f(k))
g^#(c, c, f(l))
g^#(c, l, f(k))

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

h^#(c, c) → g^#(c, c, U121(k, k))	h^#(c, c) → g^#(c, l, f(k))
h^#(c, c) → g^#(c, c, f(l))	h^#(c, c) → g^#(e, c, f(k))
h^#(c, c) → g^#(l, c, f(k))	h^#(c, c) → g^#(c, c, f(m))
h^#(c, c) → g^#(c, e, f(k))

Problem 93: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(e, e)	→	g^#(e, e, f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(k))	A^#	→	h^#(d, d)
A^#	→	g^#(U121(c, c), U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))
h^#(m, m)	→	g^#(m, m, f(k))	A^#	→	h^#(m, m)
h^#(c, c)	→	g^#(c, c, U121(k, k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(c, c)	→	g^#(c, e, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(e, e)
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
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^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(e, e) → g^#(e, 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^#(e, e, f(l))
g^#(e, e, U121(k, k))
g^#(e, e, f(m))

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

h^#(e, e) → g^#(e, e, U121(k, k))	h^#(e, e) → g^#(e, e, f(m))
h^#(e, e) → g^#(e, e, f(l))

Problem 94: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(k))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(m, m)	→	g^#(m, m, f(k))
A^#	→	h^#(m, m)	h^#(c, c)	→	g^#(c, c, U121(k, k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(c, e, f(k))
h^#(c, c)	→	g^#(c, c, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(e, e)	→	g^#(e, e, U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
A^#	→	h^#(e, e)	h^#(c, c)	→	g^#(c, l, f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	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^#(c, c)	→	g^#(c, c, f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(l, l)	→	g^#(l, l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(m), f(m)) → g^#(f(m), f(m), 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(m), f(m), f(m))	g^#(f(m), U121(m, m), f(k))
g^#(f(m), f(m), f(l))	g^#(U121(m, m), f(m), f(k))
g^#(f(m), f(m), U121(k, k))

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

h^#(f(m), f(m)) → g^#(f(m), f(m), U121(k, k))	h^#(f(m), f(m)) → g^#(f(m), f(m), f(m))
h^#(f(m), f(m)) → g^#(f(m), f(m), f(l))

Problem 95: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(m), f(m))	→	g^#(f(m), f(m), U121(k, k))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	A^#	→	h^#(m, m)
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(c, c, U121(k, k))
h^#(c, c)	→	g^#(c, c, f(m))	h^#(c, c)	→	g^#(c, e, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(e, e)	→	g^#(e, e, U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(e, e)
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(c, c)	→	g^#(c, l, f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	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))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(m), f(m)) → g^#(f(m), f(m), U121(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(m), f(m), U121(l, k))
	g^#(f(m), U121(m, m), U121(k, k))
	g^#(f(m), f(m), U121(m, k))
	g^#(U121(m, m), f(m), U121(k, k))

Thus, the rule h^#(f(m), f(m)) → g^#(f(m), f(m), U121(k, k)) is deleted.

Problem 96: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	g^#(U121(c, c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(m, m)	→	g^#(m, m, f(k))
A^#	→	h^#(m, m)	h^#(c, c)	→	g^#(c, c, U121(k, k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(c, e, f(k))
h^#(c, c)	→	g^#(c, c, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(e, e)	→	g^#(e, e, U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
A^#	→	h^#(e, e)	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
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^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(U121(c, c), U121(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^#(U121(c, c), U121(c, c), U121(k, k))	g^#(U121(l, c), U121(c, c), f(k))
g^#(U121(c, c), U121(c, c), f(m))	g^#(U121(c, c), U121(l, c), f(k))
g^#(U121(c, c), U121(e, c), f(k))
g^#(U121(e, c), U121(c, c), f(k))
g^#(U121(c, c), U121(c, c), f(l))

Thus, the rule A^# → g^#(U121(c, c), U121(c, c), f(k)) is replaced by the following rules:

A^# → g^#(U121(c, c), U121(c, c), f(m))	A^# → g^#(U121(e, c), U121(c, c), f(k))
A^# → g^#(U121(c, c), U121(c, c), U121(k, k))	A^# → g^#(U121(c, c), U121(e, c), f(k))
A^# → g^#(U121(c, c), U121(c, c), f(l))

Problem 97: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	h^#(m, m)	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(c, c, U121(k, k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(c, c)	→	g^#(c, e, f(k))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(e, e)	→	g^#(e, e, U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(e, e)
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
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))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(m), f(m)) → g^#(f(m), f(m), 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^#(U121(m, m), f(m), f(l))
	g^#(f(m), f(m), U121(l, l))
	g^#(f(m), U121(m, m), f(l))

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

Problem 98: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	A^#	→	h^#(m, m)
h^#(c, c)	→	g^#(c, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(c, e, f(k))	h^#(c, c)	→	g^#(c, c, f(m))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(e, e)	→	g^#(e, e, U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
A^#	→	h^#(e, e)	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(c, c)	→	g^#(c, l, f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(l), f(l))	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))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), f(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(c), f(e), f(k))
g^#(f(c), U121(c, c), f(k))
g^#(U121(c, c), f(c), f(k))
g^#(f(c), f(l), f(k))
g^#(f(c), f(c), U121(k, k))
g^#(f(l), f(c), f(k))
g^#(f(c), f(c), f(m))
g^#(f(c), f(c), f(l))
g^#(f(e), f(c), f(k))

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

h^#(f(c), f(c)) → g^#(U121(c, c), f(c), f(k))	h^#(f(c), f(c)) → g^#(f(c), U121(c, c), f(k))
h^#(f(c), f(c)) → g^#(f(e), f(c), f(k))	h^#(f(c), f(c)) → g^#(f(c), f(c), f(l))
h^#(f(c), f(c)) → g^#(f(l), f(c), f(k))	h^#(f(c), f(c)) → g^#(f(c), f(c), f(m))
h^#(f(c), f(c)) → g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c)) → g^#(f(c), f(e), f(k))
h^#(f(c), f(c)) → g^#(f(c), f(l), f(k))

Problem 99: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(m, m)	→	g^#(m, m, f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(c, c, U121(k, k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(c, c)	→	g^#(c, e, f(k))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(e, e)	→	g^#(e, e, U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, m, 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), U121(l, l))
g^#(f(e), f(c), f(l))
g^#(f(c), U121(c, c), f(l))
g^#(f(c), f(e), f(l))
g^#(U121(c, c), f(c), f(l))
g^#(f(l), f(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(e), f(c), f(l))	h^#(f(c), f(c)) → g^#(f(c), U121(c, 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^#(f(l), f(c), f(l))	h^#(f(c), f(c)) → g^#(U121(c, c), f(c), f(l))

Problem 100: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, m)	h^#(c, c)	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))	h^#(c, c)	→	g^#(c, e, f(k))
h^#(c, c)	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(e, e)	→	g^#(e, e, U121(k, k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(m))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	A^#	→	h^#(c, c)
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
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^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → g^#(c, c, U121(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, e, U121(k, k))	g^#(c, c, U121(l, k))
g^#(e, c, U121(k, k))	g^#(c, c, U121(m, k))
g^#(c, l, U121(k, k))
g^#(l, c, U121(k, k))

Thus, the rule h^#(c, c) → g^#(c, c, U121(k, k)) is replaced by the following rules:

h^#(c, c) → g^#(c, e, U121(k, k))	h^#(c, c) → g^#(e, c, U121(k, k))
h^#(c, c) → g^#(l, c, U121(k, k))	h^#(c, c) → g^#(c, l, U121(k, k))

Problem 101: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(c, c)	→	g^#(c, e, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))	h^#(c, c)	→	g^#(c, e, f(k))
h^#(c, c)	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(e, e)	→	g^#(e, e, U121(k, k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
A^#	→	h^#(c, c)	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))	h^#(c, c)	→	g^#(c, l, U121(k, k))
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^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → g^#(c, e, U121(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, U121(k, k))	g^#(c, e, U121(l, k))
g^#(l, e, U121(k, k))	g^#(c, e, U121(m, k))

Thus, the rule h^#(c, c) → g^#(c, e, U121(k, k)) is replaced by the following rules:

h^#(c, c) → g^#(l, e, U121(k, k))

h^#(c, c) → g^#(e, e, U121(k, k))

Problem 102: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(l))
h^#(c, c)	→	g^#(c, e, f(k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(d, d)
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
A^#	→	h^#(c, c)	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))	h^#(c, c)	→	g^#(c, l, U121(k, k))
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^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, l, m, 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^#(f(e), f(l), f(l))	g^#(f(e), f(c), U121(l, l))
g^#(U121(e, e), f(c), f(l))
g^#(f(e), U121(c, c), f(l))
g^#(f(e), f(e), 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^#(f(e), U121(c, c), f(l))
h^#(f(c), f(c)) → g^#(f(e), f(l), f(l))	h^#(f(c), f(c)) → g^#(U121(e, e), f(c), f(l))

Problem 103: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), f(l))	A^#	→	h^#(m, m)
h^#(c, c)	→	g^#(c, e, f(k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))	A^#	→	h^#(d, d)
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
A^#	→	h^#(c, c)	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), 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(c), f(m))
h^#(c, c)	→	g^#(c, l, U121(k, k))	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^#(U121(c, c), f(c), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, 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^#(U121(e, e), f(l), f(l))	g^#(U121(e, e), f(c), U121(l, l))
g^#(e, f(c), f(l))
g^#(U121(e, e), U121(c, c), f(l))
g^#(U121(e, e), f(e), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), f(c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c)) → g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c)) → g^#(e, f(c), f(l))	h^#(f(c), f(c)) → g^#(U121(e, e), f(l), f(l))

Problem 104: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	A^#	→	h^#(m, m)
h^#(c, c)	→	g^#(c, e, f(k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))	A^#	→	h^#(d, d)
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
A^#	→	h^#(c, c)	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(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^#(U121(e, e), U121(e, e), f(l))	g^#(U121(e, e), f(e), U121(l, l))
g^#(e, f(e), f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), f(e), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, f(e), f(l))

h^#(f(c), f(c)) → g^#(U121(e, e), U121(e, e), f(l))

Problem 105: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(c, c)	→	g^#(c, e, f(k))	h^#(c, c)	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(c, c)	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	A^#	→	h^#(c, c)
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))
h^#(c, c)	→	g^#(c, l, U121(k, k))	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^#(U121(c, c), f(c), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, e), U121(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, U121(e, e), f(l))	g^#(U121(e, e), U121(e, e), U121(l, l))
g^#(U121(e, e), e, f(l))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), U121(e, e), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), e, f(l))

h^#(f(c), f(c)) → g^#(e, U121(e, e), f(l))

Problem 106: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, m)	h^#(c, c)	→	g^#(c, e, f(k))
h^#(c, c)	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(e, e)	→	g^#(e, e, U121(k, k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))	A^#	→	h^#(d, d)
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
A^#	→	h^#(c, c)	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, 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, U121(k, k))
g^#(l, e, f(k))
g^#(c, e, f(m))
g^#(c, e, f(l))
g^#(e, e, f(k))

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

h^#(c, c) → g^#(c, e, f(l))	h^#(c, c) → g^#(c, e, U121(k, k))
h^#(c, c) → g^#(l, e, f(k))	h^#(c, c) → g^#(e, e, f(k))
h^#(c, c) → g^#(c, e, f(m))

Problem 107: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(c, c)	→	g^#(l, e, f(k))	h^#(c, c)	→	g^#(c, e, U121(k, k))
A^#	→	h^#(m, m)	h^#(c, c)	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(c, c)	→	g^#(c, e, f(l))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
h^#(c, c)	→	g^#(e, e, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(c, c)	→	g^#(c, c, f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(l, l)	→	g^#(l, l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))
A^#	→	h^#(d, d)	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(c, c)	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(c, c)	→	g^#(c, e, f(m))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))	h^#(c, c)	→	g^#(c, l, U121(k, k))
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^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(c, c)	→	g^#(e, e, U121(k, k))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → 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, f(m))
g^#(l, e, U121(k, k))
g^#(l, e, f(l))

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

h^#(c, c) → g^#(l, e, U121(k, k))	h^#(c, c) → g^#(l, e, f(l))
h^#(c, c) → g^#(l, e, f(m))

Problem 108: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(c, c)	→	g^#(c, e, U121(k, k))	A^#	→	h^#(m, m)
h^#(c, c)	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(e, e)	→	g^#(e, e, U121(k, k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(c, c)	→	g^#(c, e, f(l))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(c, c)	→	g^#(e, e, f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(c, c)	→	g^#(l, e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))
A^#	→	h^#(d, d)	h^#(c, c)	→	g^#(l, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(c, c)	→	g^#(c, e, f(m))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	A^#	→	h^#(c, c)
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(c, c)	→	g^#(e, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
g^#(e, e, U121(k, k))	g^#(c, e, U121(l, k))
g^#(l, e, U121(k, k))	g^#(c, e, U121(m, k))

Thus, the rule h^#(c, c) → g^#(c, e, U121(k, k)) is replaced by the following rules:

h^#(c, c) → g^#(l, e, U121(k, k))

h^#(c, c) → g^#(e, e, U121(k, k))

Problem 109: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(m, m)	h^#(c, c)	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(c, c)	→	g^#(c, e, f(l))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(c, c)	→	g^#(e, e, f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(c, c)	→	g^#(c, c, f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(l, l)	→	g^#(l, l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))	A^#	→	h^#(d, d)
h^#(c, c)	→	g^#(l, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(c, c)	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(c, c)	→	g^#(c, e, f(m))
h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))	h^#(c, c)	→	g^#(c, l, U121(k, k))
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^#(U121(c, c), f(c), f(l))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(c, c)	→	g^#(e, e, U121(k, k))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → g^#(c, c, f(m)) 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(m))	g^#(c, c, U121(m, m))
g^#(c, e, f(m))
g^#(c, l, f(m))
g^#(e, c, f(m))

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

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

Problem 110: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(c, c)	→	g^#(l, c, f(m))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(e, e)	→	g^#(e, e, U121(k, k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(c, c)	→	g^#(c, e, f(l))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(d, d)	→	g^#(d, d, f(k))
h^#(c, c)	→	g^#(e, c, f(m))	h^#(c, c)	→	g^#(e, e, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
A^#	→	h^#(f(m), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(l, l)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(c, c)	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(k))	h^#(c, c)	→	g^#(l, e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))
A^#	→	h^#(d, d)	h^#(c, c)	→	g^#(l, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(c, c)	→	g^#(c, e, f(m))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	A^#	→	h^#(c, c)
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), 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(c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(c, c)	→	g^#(e, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))
h^#(c, c)	→	g^#(c, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → g^#(l, c, f(m)) 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(m))	g^#(l, c, U121(m, m))
g^#(l, l, f(m))

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

h^#(c, c) → g^#(l, l, f(m))

h^#(c, c) → g^#(l, e, f(m))

Problem 111: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(d, d)	→	g^#(d, m, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	h^#(c, c)	→	g^#(e, e, f(k))
h^#(c, c)	→	g^#(e, c, f(m))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(l))	h^#(c, c)	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), 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^#(e, U121(e, e), f(l))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(c, c)	→	g^#(l, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(k))
h^#(e, e)	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(c, c)	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(c, e, f(m))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(k))	A^#	→	h^#(c, c)
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(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^#(U121(c, c), f(c), f(l))	h^#(d, d)	→	g^#(m, d, 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^#(f(e), f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(c, c)	→	g^#(e, e, U121(k, k))	h^#(d, d)	→	g^#(d, d, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(c, c)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
A^#	→	g^#(U121(c, c), U121(e, c), f(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(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(c, c)	→	g^#(c, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(d, d) → g^#(d, m, 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^#(d, m, f(l))
g^#(m, m, f(k))
g^#(d, m, f(m))
g^#(d, m, U121(k, k))

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

h^#(d, d) → g^#(d, m, f(l))	h^#(d, d) → g^#(m, m, f(k))
h^#(d, d) → g^#(d, m, U121(k, k))	h^#(d, d) → g^#(d, m, f(m))

Problem 112: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, f(l), U121(k, k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(e, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(k))
h^#(d, d)	→	g^#(d, m, f(l))	h^#(e, e)	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(c, c)	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(c, e, f(m))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(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^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(l))
h^#(d, d)	→	g^#(m, d, f(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^#(f(l), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(m))
h^#(d, d)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
h^#(d, d)	→	g^#(d, d, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), f(m))
h^#(d, d)	→	g^#(d, m, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(m))
h^#(c, c)	→	g^#(l, e, f(k))	h^#(c, c)	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(l), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(d, d)	→	g^#(d, m, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(l, l)	→	g^#(l, l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(c, c)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
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^#(f(l), c, f(l))	A^#	→	g^#(U121(c, c), U121(e, c), f(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(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(d, d)	→	g^#(d, d, f(m))	h^#(c, c)	→	g^#(e, e, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(c, c)	→	g^#(c, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, f(l), U121(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, f(l), U121(k, k))	g^#(c, f(l), U121(m, k))
g^#(l, f(l), U121(k, k))	g^#(c, U121(l, l), U121(k, k))
	g^#(c, f(l), U121(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, f(l), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, f(l), U121(k, k))

h^#(f(c), f(c)) → g^#(e, f(l), U121(k, k))

Problem 113: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(c, c)	→	g^#(c, e, f(m))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(c, c)	h^#(c, c)	→	g^#(c, l, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(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^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(l))
h^#(d, d)	→	g^#(m, d, f(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^#(f(l), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(m))
h^#(d, d)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(c, c)	→	g^#(e, e, U121(k, k))
h^#(d, d)	→	g^#(d, d, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(m))	h^#(d, d)	→	g^#(d, m, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(m))
h^#(c, c)	→	g^#(l, e, f(k))	h^#(c, c)	→	g^#(l, c, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(l), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(l), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(d, d)	→	g^#(d, m, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, f(l), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(c, c, f(l))	A^#	→	h^#(f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(c, c)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	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^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(d, d)	→	g^#(d, d, f(m))	h^#(c, c)	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), 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^#(U121(c, c), U121(e, c), f(m))	h^#(c, c)	→	g^#(c, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, c), U121(c, c), f(m)) 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^#(U121(e, c), U121(c, c), U121(m, m))	g^#(U121(e, c), U121(l, c), f(m))
g^#(U121(e, c), U121(e, c), f(m))
g^#(c, U121(c, c), f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, c), U121(c, c), f(m)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, c), U121(c, c), U121(m, m))	h^#(f(c), f(c)) → g^#(c, U121(c, c), f(m))
h^#(f(c), f(c)) → g^#(U121(e, c), U121(e, c), f(m))

Problem 114: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(c, c)	→	g^#(e, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(m))
h^#(d, d)	→	g^#(d, m, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, e), U121(k, k))
h^#(c, c)	→	g^#(l, e, f(k))	h^#(c, c)	→	g^#(l, c, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(l), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(m))
h^#(f(c), f(c))	→	g^#(f(l), c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(c, c, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(d, d)	→	g^#(d, m, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(k))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(c, c)	→	g^#(e, e, f(l))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	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^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(m))	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^#(U121(c, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(c, c)	→	g^#(e, e, f(m))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(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^#(c, c)	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(l))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
h^#(d, d)	→	g^#(m, d, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))
A^#	→	g^#(c, U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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(l), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, l, f(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(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → g^#(e, e, U121(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, U121(l, k))
	g^#(e, e, U121(m, k))

Thus, the rule h^#(c, c) → g^#(e, e, U121(k, k)) is deleted.

Problem 115: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(d, d)	→	g^#(d, m, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), f(m))	h^#(c, c)	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))	h^#(l, l)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(k))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(c, c)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
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^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	A^#	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(m))	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^#(U121(c, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(c, c)	→	g^#(e, e, f(m))
h^#(d, d)	→	g^#(d, d, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(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^#(c, c)	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))	h^#(c, c)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	h^#(d, d)	→	g^#(m, d, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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(l), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, 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^#(U121(e, e), e, U121(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), e, f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, f(l))

Problem 116: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(k))
h^#(m, m)	→	g^#(m, m, f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(d, d)	→	g^#(d, d, f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(c, c)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), l, f(k))	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^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(m))	A^#	→	g^#(U121(c, c), U121(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^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(c, c)	→	g^#(e, e, f(m))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(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^#(c, c)	→	g^#(c, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(c, c)	→	g^#(l, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(c, f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(l), c, f(m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	h^#(e, e)
h^#(c, c)	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
h^#(d, d)	→	g^#(m, d, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, e, U121(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, U121(k, k))	g^#(c, e, U121(l, k))
g^#(l, e, U121(k, k))	g^#(c, e, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U121(k, k))

h^#(f(c), f(c)) → g^#(l, e, U121(k, k))

Problem 117: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(k))	h^#(m, m)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(d, d)	→	g^#(d, d, f(l))	h^#(c, c)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
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^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(m))	A^#	→	g^#(U121(c, c), U121(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^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(d, d)	→	g^#(d, d, f(m))	h^#(c, c)	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(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^#(c, c)	→	g^#(c, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(c, c)	→	g^#(l, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	h^#(e, e)	h^#(c, c)	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	h^#(d, d)	→	g^#(m, d, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, c, U121(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, U121(k, k))	g^#(e, c, U121(m, k))
g^#(e, e, U121(k, k))	g^#(e, c, U121(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, c, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U121(k, k))

h^#(f(c), f(c)) → g^#(e, l, U121(k, k))

Problem 118: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(k))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))
A^#	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), f(m))
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^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(c, c)	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(c, c)	→	g^#(c, l, f(l))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(c, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))
h^#(c, c)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	h^#(e, e)
h^#(c, c)	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), c, f(l))	h^#(d, d)	→	g^#(m, d, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U121(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^#(e, U121(c, c), f(l))	g^#(e, U121(l, c), f(k))
g^#(e, U121(c, c), U121(k, k))
g^#(e, U121(e, c), f(k))
g^#(e, U121(c, c), f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U121(c, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U121(c, c), f(m))	h^#(f(c), f(c)) → g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c)) → g^#(e, U121(c, c), f(l))	h^#(f(c), f(c)) → g^#(e, U121(e, c), f(k))

Problem 119: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	g^#(U121(e, c), c, f(k))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(e, c), f(l))
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^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(m), f(m))	→	g^#(f(m), f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(c, c)	→	g^#(e, e, f(m))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(c, c)	→	g^#(c, l, f(l))
h^#(c, c)	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))
h^#(c, c)	→	g^#(l, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(c, f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	h^#(e, e)	h^#(c, c)	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	A^#	→	g^#(U121(c, c), c, f(l))
h^#(d, d)	→	g^#(m, d, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(c, c)	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	A^#	→	g^#(U121(c, c), e, f(k))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(U121(e, 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^#(U121(e, c), l, f(k))
g^#(c, c, f(k))
g^#(U121(e, c), c, f(l))
g^#(U121(e, c), c, f(m))
g^#(U121(e, c), e, f(k))
g^#(U121(e, c), c, U121(k, k))

Thus, the rule A^# → g^#(U121(e, c), c, f(k)) is replaced by the following rules:

A^# → g^#(U121(e, c), l, f(k))	A^# → g^#(c, c, f(k))
A^# → g^#(U121(e, c), c, f(l))	A^# → g^#(U121(e, c), c, U121(k, k))
A^# → g^#(U121(e, c), c, f(m))	A^# → g^#(U121(e, c), e, f(k))

Problem 120: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	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, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(c, c)	→	g^#(e, e, f(m))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(c, c)	→	g^#(c, l, f(l))
h^#(c, c)	→	g^#(e, l, f(l))	A^#	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))
A^#	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))
h^#(c, c)	→	g^#(l, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(c, f(l), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), c, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	h^#(e, e)
h^#(c, c)	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), c, f(l))	h^#(d, d)	→	g^#(m, d, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))
A^#	→	g^#(U121(e, c), l, f(k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(c, c)
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	A^#	→	g^#(U121(c, c), c, f(m))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U121(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, U121(e, c), U121(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U121(e, c), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, c, f(l))

Problem 121: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(d, d)	→	g^#(d, d, f(m))
h^#(c, c)	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(c, c)	→	g^#(e, l, f(l))	h^#(c, c)	→	g^#(c, l, f(l))
A^#	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	h^#(c, c)	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(c, f(l), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
h^#(f(e), f(e))	→	g^#(f(e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	h^#(e, e)	h^#(c, c)	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	A^#	→	g^#(U121(c, c), c, f(l))
h^#(d, d)	→	g^#(m, d, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	g^#(U121(e, c), l, f(k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(c, c)	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, e), f(e), f(m)) 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^#(U121(e, e), U121(e, e), f(m))
g^#(U121(e, e), f(e), U121(m, m))
g^#(e, f(e), f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), f(e), f(m)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c)) → g^#(e, f(e), f(m))
h^#(f(c), f(c)) → g^#(U121(e, e), U121(e, e), f(m))

Problem 122: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, e, f(m))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(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^#(l, U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(c, c)	→	g^#(l, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), c, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
h^#(f(e), f(e))	→	g^#(f(e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	h^#(e, e)
h^#(c, c)	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
h^#(f(e), f(e))	→	g^#(e, e, U121(k, k))	A^#	→	g^#(U121(c, c), c, f(l))
h^#(d, d)	→	g^#(m, d, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	A^#	→	g^#(U121(e, c), e, f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	g^#(U121(e, c), l, f(k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(c, c)
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	A^#	→	g^#(U121(e, c), e, U121(k, 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^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(l, e, f(m)) 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, U121(m, m))

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

Problem 123: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, e), U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	g^#(U121(e, c), c, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(c, f(l), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	h^#(e, e)	h^#(c, c)	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), c, f(l))	h^#(d, d)	→	g^#(m, d, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
A^#	→	g^#(U121(e, c), e, f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(c, c)	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
g^#(e, e, U121(k, k))	g^#(c, e, U121(l, k))
g^#(l, e, U121(k, k))	g^#(c, e, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U121(k, k))

h^#(f(c), f(c)) → g^#(l, e, U121(k, k))

Problem 124: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(l), c, f(m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
h^#(f(e), f(e))	→	g^#(f(e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	h^#(e, e)
h^#(c, c)	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, U121(k, k))
h^#(d, d)	→	g^#(m, d, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(l))	A^#	→	g^#(U121(e, c), e, f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))	A^#	→	g^#(c, U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(c, c)
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U121(e, c), U121(m, m)) 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, U121(m, m))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U121(e, c), U121(m, m)) is deleted.

Problem 125: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, l, U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(k))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))
A^#	→	g^#(U121(e, c), l, f(k))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	A^#	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	A^#	→	h^#(c, c)
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, l, U121(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, U121(l, k))
	g^#(e, l, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, l, U121(k, k)) is deleted.

Problem 126: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(k))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	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, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(k))	A^#	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	g^#(U121(e, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
A^#	→	g^#(c, e, f(k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
g^#(e, U121(c, c), f(l))	g^#(e, U121(l, c), f(k))
g^#(e, U121(c, c), U121(k, k))
g^#(e, U121(e, c), f(k))
g^#(e, U121(c, c), f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U121(c, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, U121(c, c), f(m))	h^#(f(c), f(c)) → g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c)) → g^#(e, U121(e, c), f(k))	h^#(f(c), f(c)) → g^#(e, U121(c, c), f(l))

Problem 127: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	A^#	→	h^#(e, e)
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	g^#(c, U121(c, c), f(k))
A^#	→	g^#(e, l, f(l))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	g^#(U121(e, c), l, f(k))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(c, c)	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
A^#	→	g^#(c, e, f(k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, U121(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, U121(e, c), f(k))
g^#(c, U121(e, c), U121(k, k))
g^#(c, U121(e, c), f(m))
g^#(c, c, f(k))
g^#(c, U121(e, c), f(l))
g^#(l, U121(e, c), f(k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U121(e, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, c, f(k))	h^#(f(c), f(c)) → g^#(c, U121(e, c), f(l))
h^#(f(c), f(c)) → g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c)) → g^#(l, U121(e, c), f(k))
h^#(f(c), f(c)) → g^#(c, U121(e, c), f(m))	h^#(f(c), f(c)) → g^#(e, U121(e, c), f(k))

Problem 128: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, e, f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	g^#(c, U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))	A^#	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	g^#(U121(e, c), l, f(k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	A^#	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, e, f(m)) 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, U121(m, m))

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

Problem 129: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	g^#(l, U121(e, c), U121(m, m))
A^#	→	h^#(l, l)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(e, e)	A^#	→	g^#(l, e, f(l))
A^#	→	g^#(l, c, f(m))	A^#	→	g^#(c, U121(c, c), f(m))
A^#	→	g^#(l, e, f(k))	A^#	→	g^#(e, l, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	A^#	→	g^#(e, l, f(l))
A^#	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
A^#	→	g^#(e, c, f(m))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
A^#	→	g^#(e, e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	A^#	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(l, U121(e, c), U121(m, m)) 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, U121(m, m))

Thus, the rule A^# → g^#(l, U121(e, c), U121(m, m)) is deleted.

Problem 130: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	g^#(l, l, f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	g^#(e, l, f(m))	A^#	→	g^#(l, l, U121(k, k))
A^#	→	g^#(U121(e, c), l, f(l))	A^#	→	h^#(e, e)
A^#	→	g^#(e, l, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	g^#(c, l, f(m))	A^#	→	h^#(f(e), f(e))
A^#	→	g^#(e, l, f(l))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(k))
A^#	→	g^#(e, e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	A^#	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(l, l)	→	g^#(l, l, U121(k, k))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → 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, U121(l, l))

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

Problem 131: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(l), l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
A^#	→	g^#(e, e, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
A^#	→	g^#(c, e, f(k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(e, c, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
g^#(e, e, U121(k, k))	g^#(c, e, U121(l, k))
g^#(l, e, U121(k, k))	g^#(c, e, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, e, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U121(k, k))

h^#(f(c), f(c)) → g^#(l, e, U121(k, k))

Problem 132: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
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))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	A^#	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(c, c)	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), l, U121(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(l), l, U121(k, k))	g^#(f(c), l, U121(m, k))
g^#(U121(c, c), l, U121(k, k))	g^#(f(c), l, U121(l, k))
g^#(f(e), l, U121(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(f(c), l, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(e), l, U121(k, k))	h^#(f(c), f(c)) → g^#(U121(c, c), l, U121(k, k))
h^#(f(c), f(c)) → g^#(f(l), l, U121(k, k))

Problem 133: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(k))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
A^#	→	g^#(e, e, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(d, d)	→	g^#(m, m, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(e), c, f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(c, l, f(k))	A^#	→	h^#(c, c)
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, 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^#(e, l, U121(k, k))
g^#(e, l, f(l))
g^#(e, l, f(m))

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

h^#(f(c), f(c)) → g^#(e, l, f(m))	h^#(f(c), f(c)) → g^#(e, l, U121(k, k))
h^#(f(c), f(c)) → g^#(e, l, f(l))

Problem 134: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(l))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
A^#	→	g^#(e, e, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(l, l)	→	g^#(l, l, U121(k, k))	A^#	→	h^#(c, c)
h^#(c, c)	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	A^#	→	g^#(U121(c, c), c, f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(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^#(f(l), l, f(l))	g^#(f(l), c, U121(l, l))
g^#(f(l), e, f(l))	g^#(U121(l, l), c, f(l))

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

h^#(f(c), f(c)) → g^#(f(l), e, f(l))

h^#(f(c), f(c)) → g^#(f(l), l, f(l))

Problem 135: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
A^#	→	h^#(e, e)	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))	A^#	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))	h^#(c, c)	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(l, l)	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))
A^#	→	h^#(c, c)	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
A^#	→	g^#(U121(c, c), c, f(m))	A^#	→	g^#(U121(c, c), e, f(k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(d, d)	→	g^#(m, d, f(k))
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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, e), e, U121(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, U121(k, k))	g^#(U121(e, e), e, U121(l, k))
	g^#(U121(e, e), e, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), e, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U121(k, k))

Problem 136: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, e, U121(k, k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(m, m))
A^#	→	g^#(e, e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, U121(k, k))
h^#(c, c)	→	g^#(l, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(d, d)	→	g^#(m, m, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))	A^#	→	g^#(U121(c, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	h^#(c, c)	→	g^#(e, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	A^#	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(c, c)
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
A^#	→	g^#(U121(c, c), c, f(m))	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(d, d)	→	g^#(m, d, f(k))	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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, c), U121(e, c)) → g^#(e, U121(e, c), f(m)) 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, U121(e, c), U121(m, m))
g^#(e, c, f(m))

Thus, the rule h^#(U121(e, c), U121(e, c)) → g^#(e, U121(e, c), f(m)) is replaced by the following rules:

h^#(U121(e, c), U121(e, c)) → g^#(e, U121(e, c), U121(m, m))

h^#(U121(e, c), U121(e, c)) → g^#(e, c, f(m))

Problem 137: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
h^#(c, c)	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(c, l, f(k))	h^#(l, l)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(l))	A^#	→	h^#(c, c)
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	A^#	→	g^#(U121(c, c), c, f(m))
A^#	→	g^#(U121(c, c), e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
A^#	→	h^#(l, l)	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(m, d, f(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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	h^#(l, l)	→	g^#(l, l, f(m))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, e), U121(e, e), f(m)) 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, U121(e, e), f(m))
g^#(U121(e, e), U121(e, e), U121(m, m))
g^#(U121(e, e), e, f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), U121(e, e), f(m)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), e, f(m))	h^#(f(c), f(c)) → g^#(e, U121(e, e), f(m))
h^#(f(c), f(c)) → g^#(U121(e, e), U121(e, e), U121(m, m))

Problem 138: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	h^#(l, l)	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(c, c), l, f(m))	h^#(d, d)	→	g^#(m, d, f(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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
A^#	→	g^#(U121(e, c), e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, e, f(k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	g^#(e, e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(c, l, f(l))	h^#(c, c)	→	g^#(e, c, U121(k, k))
A^#	→	g^#(U121(e, c), e, f(k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(c, c)	→	g^#(c, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, f(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, f(c), f(l))
g^#(c, f(c), U121(k, k))
g^#(c, f(e), f(k))
g^#(e, f(c), f(k))
g^#(c, f(c), f(m))
g^#(c, U121(c, c), f(k))
g^#(c, f(l), f(k))
g^#(l, f(c), f(k))

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

h^#(f(c), f(c)) → g^#(c, f(e), f(k))	h^#(f(c), f(c)) → g^#(c, f(c), U121(k, k))
h^#(f(c), f(c)) → g^#(c, f(c), f(l))	h^#(f(c), f(c)) → g^#(e, f(c), f(k))
h^#(f(c), f(c)) → g^#(c, U121(c, c), f(k))	h^#(f(c), f(c)) → g^#(c, f(c), f(m))
h^#(f(c), f(c)) → g^#(c, f(l), f(k))	h^#(f(c), f(c)) → g^#(l, f(c), f(k))

Problem 139: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(e, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, c, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, f(l))	h^#(c, c)	→	g^#(e, l, f(l))
A^#	→	g^#(U121(e, c), c, f(m))	A^#	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(c, c)	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), U121(k, k))	A^#	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	A^#	→	h^#(e, e)
A^#	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(c), f(k))	h^#(f(c), f(c))	→	g^#(c, f(l), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(c, f(c), f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	g^#(U121(c, c), l, f(m))
h^#(d, d)	→	g^#(m, d, f(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))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	A^#	→	g^#(c, c, f(m))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, m, 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, U121(k, k))
g^#(l, e, f(k))
g^#(c, e, f(m))
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^#(l, e, f(k))	h^#(f(c), f(c)) → g^#(c, e, f(m))
h^#(f(c), f(c)) → g^#(c, e, f(l))	h^#(f(c), f(c)) → g^#(e, e, f(k))
h^#(f(c), f(c)) → g^#(c, e, U121(k, k))

Problem 140: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(c, l, f(m))
A^#	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), U121(k, k))	A^#	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	A^#	→	h^#(e, e)
A^#	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(c, f(c), f(l))
A^#	→	g^#(c, l, f(m))	A^#	→	h^#(f(e), f(e))
A^#	→	g^#(U121(e, c), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(m))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(l, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	A^#	→	g^#(U121(c, c), l, f(m))
h^#(d, d)	→	g^#(m, d, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
A^#	→	g^#(U121(e, c), e, U121(k, 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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	A^#	→	g^#(c, c, f(m))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(e), e, f(m)) 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^#(U121(e, e), e, f(m))	g^#(f(e), e, U121(m, m))

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

h^#(f(c), f(c)) → g^#(U121(e, e), e, f(m))

Problem 141: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), U121(k, k))
A^#	→	h^#(l, l)	h^#(f(c), f(c))	→	g^#(e, f(l), f(l))
h^#(f(c), f(c))	→	g^#(l, f(c), f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, f(c), f(l))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(c, f(c), f(l))	A^#	→	g^#(c, l, f(m))
A^#	→	g^#(U121(e, c), e, f(l))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), U121(k, k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))
h^#(m, m)	→	g^#(m, m, f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	A^#	→	g^#(U121(c, c), l, f(m))
h^#(d, d)	→	g^#(m, d, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
A^#	→	g^#(U121(e, c), e, U121(k, 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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	h^#(l, l)	→	g^#(l, l, f(m))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
A^#	→	g^#(c, c, f(m))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, f(e), U121(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, U121(e, e), U121(k, k))	g^#(l, f(e), U121(m, k))
	g^#(l, f(e), U121(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(l, f(e), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, U121(e, e), U121(k, k))

Problem 142: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, f(e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))
A^#	→	h^#(l, l)	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	A^#	→	g^#(U121(c, c), l, f(m))
h^#(d, d)	→	g^#(m, d, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(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))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(l, l)	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(l, f(e), f(l))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	g^#(e, e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, m, 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, U121(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 143: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(l, f(e), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, 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^#(e, f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))
A^#	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	A^#	→	h^#(e, e)
A^#	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
A^#	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(e, f(l), f(m))
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(l))	h^#(m, m)	→	g^#(m, m, f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(c, c), l, f(m))	h^#(d, d)	→	g^#(m, d, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
h^#(l, l)	→	g^#(l, l, f(m))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	g^#(e, e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, 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^#(l, U121(e, e), f(l))	g^#(l, f(e), U121(l, l))

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

h^#(f(c), f(c)) → g^#(l, U121(e, e), f(l))

Problem 144: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), U121(m, m))	A^#	→	h^#(l, l)
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(c, c), l, f(m))	h^#(d, d)	→	g^#(m, d, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
h^#(l, l)	→	g^#(l, l, f(m))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, e, f(m))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, e), U121(e, e)) → g^#(U121(e, e), e, U121(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, U121(k, k))	g^#(U121(e, e), e, U121(l, k))
	g^#(U121(e, e), e, U121(m, k))

Thus, the rule h^#(U121(e, e), U121(e, e)) → g^#(U121(e, e), e, U121(k, k)) is replaced by the following rules:

h^#(U121(e, e), U121(e, e)) → g^#(e, e, U121(k, k))

Problem 145: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(k, k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(c, f(e), f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(l, l)
A^#	→	g^#(U121(e, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(m))
A^#	→	g^#(U121(c, c), l, f(m))	h^#(d, d)	→	g^#(m, d, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	g^#(e, e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, c), U121(e, e), U121(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^#(U121(e, c), e, U121(k, k))	g^#(U121(e, c), U121(e, e), U121(l, k))
g^#(c, U121(e, e), U121(k, k))	g^#(U121(e, c), U121(e, e), U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, c), U121(e, e), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, c), e, U121(k, k))

h^#(f(c), f(c)) → g^#(c, U121(e, e), U121(k, k))

Problem 146: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(f(l), e, U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(c, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(U121(c, c), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), c, f(m))	A^#	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(l, f(e), f(k))	A^#	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))	A^#	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))	A^#	→	h^#(e, e)
A^#	→	g^#(l, e, f(k))	A^#	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
A^#	→	g^#(e, c, f(m))	A^#	→	g^#(U121(e, c), l, f(k))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
A^#	→	g^#(U121(c, c), l, f(m))	h^#(d, d)	→	g^#(m, d, f(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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(l), e, U121(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^#(U121(l, l), e, U121(k, k))
	g^#(f(l), e, U121(l, k))
	g^#(f(l), e, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(f(l), e, U121(k, k)) is deleted.

Problem 147: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(l, l)
h^#(f(c), f(c))	→	g^#(l, f(e), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(k, k))
A^#	→	g^#(e, c, f(k))	A^#	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(k))
A^#	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(l))
A^#	→	h^#(e, e)	A^#	→	g^#(l, c, f(m))
A^#	→	g^#(l, e, f(k))	A^#	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(l))
A^#	→	h^#(f(e), f(e))	A^#	→	g^#(U121(e, c), e, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))	A^#	→	g^#(e, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))	A^#	→	g^#(U121(e, c), l, f(k))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
A^#	→	g^#(U121(c, c), l, f(m))	h^#(d, d)	→	g^#(m, d, f(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^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	A^#	→	g^#(U121(e, c), e, U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))
h^#(l, l)	→	g^#(l, l, f(m))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(e, e)	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
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
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
g^#(l, U121(e, e), f(l))	g^#(l, f(e), U121(l, l))

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

h^#(f(c), f(c)) → g^#(l, U121(e, e), f(l))

Problem 148: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	g^#(e, e, f(k))	A^#	→	g^#(e, c, f(m))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), f(m))
A^#	→	g^#(U121(e, c), l, f(k))	A^#	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(m))
A^#	→	g^#(l, l, f(m))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
A^#	→	h^#(l, l)	A^#	→	g^#(U121(c, c), l, f(m))
h^#(d, d)	→	g^#(m, d, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, U121(k, k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	g^#(U121(e, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	h^#(l, l)	→	g^#(l, l, f(m))
A^#	→	g^#(U121(c, c), l, f(k))	A^#	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(e, e)	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(e, c, f(m)) 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(m))	g^#(e, c, U121(m, m))
g^#(e, e, f(m))

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

A^# → g^#(e, e, f(m))

A^# → g^#(e, l, f(m))

Problem 149: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(d), f(d))	→	g^#(f(m), f(m), f(l))	A^#	→	h^#(e, e)
A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(d), f(d))	→	g^#(f(d), f(m), U121(k, k))	A^#	→	h^#(c, c)
A^#	→	h^#(l, l)	h^#(l, l)	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), f(l))	A^#	→	g^#(U121(c, c), l, f(k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(d, d)	→	g^#(m, m, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(e, e)	→	g^#(e, e, f(l))	A^#	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))	A^#	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))	h^#(f(d), f(d))	→	g^#(f(m), f(m), f(m))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
A^#	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(m, m))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(d), f(d))	→	g^#(f(m), f(m), U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(d), f(d)) → g^#(f(m), f(m), 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^#(U121(m, m), f(m), f(l))
	g^#(f(m), f(m), U121(l, l))
	g^#(f(m), U121(m, m), f(l))

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

Problem 150: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(k))
A^#	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
h^#(f(d), f(d))	→	g^#(f(m), f(m), f(m))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	A^#	→	g^#(U121(e, c), l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(m, m))	A^#	→	g^#(e, c, f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(d), f(d))	→	g^#(f(m), f(m), U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(m))
A^#	→	g^#(U121(c, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
A^#	→	g^#(l, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, c), U121(e, c)) → g^#(l, U121(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^#(l, c, f(k))
g^#(l, U121(e, c), f(m))
g^#(l, U121(e, c), f(l))
g^#(l, U121(e, c), U121(k, k))

Thus, the rule h^#(U121(e, c), U121(e, c)) → g^#(l, U121(e, c), f(k)) is replaced by the following rules:

h^#(U121(e, c), U121(e, c)) → g^#(l, U121(e, c), f(l))	h^#(U121(e, c), U121(e, c)) → g^#(l, c, f(k))
h^#(U121(e, c), U121(e, c)) → g^#(l, U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c)) → g^#(l, U121(e, c), f(m))

Problem 151: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(k))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, f(m))
h^#(f(d), f(d))	→	g^#(f(m), f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(m))	A^#	→	g^#(U121(e, c), l, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(m, m))	A^#	→	g^#(e, c, f(l))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(d), f(d))	→	g^#(f(m), f(m), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(l))
A^#	→	g^#(U121(c, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(l, e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))
h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	A^#	→	g^#(l, c, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, c, f(m)) 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(m))	g^#(c, c, U121(m, m))
g^#(c, e, f(m))
g^#(c, l, f(m))
g^#(e, c, f(m))

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

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

Problem 152: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(l))	A^#	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(e, c, f(m))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, e, U121(k, k))	A^#	→	h^#(f(l), f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	g^#(c, l, f(m))
A^#	→	h^#(f(e), f(e))	A^#	→	g^#(e, l, f(l))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	A^#	→	g^#(e, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))
A^#	→	h^#(m, m)	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(l))
A^#	→	g^#(U121(c, c), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
A^#	→	g^#(c, c, f(m))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	g^#(e, e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(U121(c, c), l, U121(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^#(U121(e, c), l, U121(k, k))	g^#(U121(c, c), l, U121(l, k))
	g^#(U121(l, c), l, U121(k, k))
	g^#(U121(c, c), l, U121(m, k))

Thus, the rule A^# → g^#(U121(c, c), l, U121(k, k)) is replaced by the following rules:

A^# → g^#(U121(e, c), l, U121(k, k))

Problem 153: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(l), f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))
A^#	→	g^#(e, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, U121(k, k))	A^#	→	h^#(m, m)
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(m))	h^#(f(l), f(l))	→	g^#(f(l), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, f(m))	h^#(f(c), f(c))	→	g^#(l, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(m, m))	A^#	→	g^#(U121(e, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	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))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U121(m, m))

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

Problem 154: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(m, m))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), c, f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
A^#	→	g^#(c, e, f(k))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
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))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, c), l, U121(m, m)) 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, U121(m, m))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, c), l, U121(m, m)) is deleted.

Problem 155: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(c, e, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	A^#	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(m))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
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))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(c, e, U121(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, U121(k, k))	g^#(c, e, U121(l, k))
g^#(l, e, U121(k, k))	g^#(c, e, U121(m, k))

Thus, the rule A^# → g^#(c, e, U121(k, k)) is replaced by the following rules:

A^# → g^#(l, e, U121(k, k))

A^# → g^#(e, e, U121(k, k))

Problem 156: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, c, f(m))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))
A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
A^#	→	h^#(f(m), f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(m))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
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))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U121(m, m))

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

Problem 157: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), f(l), U121(k, k))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(f(e), f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, e), f(l))	A^#	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	h^#(f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
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))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(c, c), f(l), U121(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^#(U121(e, c), f(l), U121(k, k))	g^#(U121(c, c), f(l), U121(m, k))
	g^#(U121(c, c), f(l), U121(l, k))
	g^#(U121(c, c), U121(l, l), U121(k, k))
	g^#(U121(l, c), f(l), U121(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(c, c), f(l), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, c), f(l), U121(k, k))

Problem 158: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(e, e, f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(e, e, U121(k, k))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(m, m))
A^#	→	g^#(c, c, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), f(e), f(l))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
A^#	→	h^#(f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(m))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	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))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → 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, U121(l, l))

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

Problem 159: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, e, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, e, f(l))
A^#	→	h^#(c, c)	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	h^#(f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))	A^#	→	g^#(U121(e, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))	g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(e), f(e))	→	g^#(e, e, f(m))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), e, f(m))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	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^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, c), U121(e, c)) → g^#(e, c, f(m)) 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(m))	g^#(e, c, U121(m, m))
g^#(e, e, f(m))

Thus, the rule h^#(U121(e, c), U121(e, c)) → g^#(e, c, f(m)) is replaced by the following rules:

h^#(U121(e, c), U121(e, c)) → g^#(e, l, f(m))

h^#(U121(e, c), U121(e, c)) → g^#(e, e, f(m))

Problem 160: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(U121(e, c), U121(e, c))	→	g^#(c, e, f(l))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))	A^#	→	h^#(f(m), f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(c), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	A^#	→	h^#(f(d), f(d))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	g^#(e, e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, c), U121(e, 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, U121(l, l))
g^#(e, l, f(l))

Thus, the rule h^#(U121(e, c), U121(e, c)) → g^#(c, l, f(l)) is replaced by the following rules:

h^#(U121(e, c), U121(e, c)) → g^#(e, l, f(l))

h^#(U121(e, c), U121(e, c)) → g^#(l, l, f(l))

Problem 161: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, e, f(l))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(l))
g^#(d, x, x)	→	h^#(U121(c, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(e), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	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^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, m, 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, U121(l, l))

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

Problem 162: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
g^#(d, x, x)	→	h^#(c, l)	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(e), f(e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(f(c), e, f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	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^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U121(e, c), f(m)) 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, U121(e, c), U121(m, m))
g^#(l, c, f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U121(e, c), f(m)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, c, f(m))

h^#(f(c), f(c)) → g^#(l, U121(e, c), U121(m, m))

Problem 163: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, e, f(l))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(U121(e, c), U121(e, c))	→	g^#(e, l, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(c), f(m))	h^#(c, c)	→	g^#(l, e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), U121(k, k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	g^#(d, x, x)	→	h^#(c, l)
h^#(f(c), f(c))	→	g^#(f(e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, U121(e, c), f(m))	h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	A^#	→	g^#(e, e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	A^#	→	h^#(d, d)
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, c), U121(e, 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, U121(l, l))

Thus, the rule h^#(U121(e, c), U121(e, c)) → g^#(l, l, f(l)) is deleted.

Problem 164: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, e, f(l))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(m, m))
h^#(f(d), f(d))	→	g^#(f(d), f(m), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, U121(k, k))	A^#	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), l, f(m))	A^#	→	g^#(e, e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	A^#	→	h^#(d, d)
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	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^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U121(l, l))

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

Problem 165: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(e), f(e))	→	g^#(e, e, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(m))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(e), f(e))	→	g^#(e, f(e), f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(m))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(d), f(d))	→	g^#(f(m), f(d), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))
h^#(f(e), f(e))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
A^#	→	g^#(e, e, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))	A^#	→	h^#(d, d)
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(e, 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^#(e, e, f(l))
g^#(e, e, U121(k, k))
g^#(e, e, f(m))

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

h^#(f(e), f(e)) → g^#(e, e, f(l))	h^#(f(e), f(e)) → g^#(e, e, f(m))
h^#(f(e), f(e)) → g^#(e, e, U121(k, k))

Problem 166: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(e, f(e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), f(l))
A^#	→	g^#(c, e, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(c, c)	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	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^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
A^#	→	h^#(f(d), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, U121(e, e), U121(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, U121(k, k))	g^#(l, U121(e, e), U121(l, k))
	g^#(l, U121(e, e), U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(l, U121(e, e), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(l, e, U121(k, k))

Problem 167: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(e, e, f(m))
A^#	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(f(c), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
A^#	→	h^#(d, d)	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, e), e, f(m)) 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^#(U121(e, e), e, U121(m, m))
g^#(e, e, f(m))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, e), e, f(m)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), e, U121(m, m))

h^#(f(c), f(c)) → g^#(e, e, f(m))

Problem 168: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), c, U121(k, k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(f(e), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), U121(c, c), U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
g^#(d, x, x)	→	h^#(U121(e, c), c)	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))
h^#(f(e), f(e))	→	g^#(f(e), e, f(m))	A^#	→	g^#(U121(e, c), U121(c, c), f(m))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
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^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(c, c), U121(e, c), U121(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^#(U121(c, c), c, U121(k, k))	g^#(U121(c, c), U121(e, c), U121(l, k))
g^#(U121(e, c), U121(e, c), U121(k, k))	g^#(U121(l, c), U121(e, c), U121(k, k))
	g^#(U121(c, c), U121(e, c), U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(c, c), U121(e, c), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, c), U121(e, c), U121(k, k))

h^#(f(c), f(c)) → g^#(U121(c, c), c, U121(k, k))

Problem 169: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(f(e), l, U121(k, k))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(U121(c, c), l, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	g^#(d, x, x)	→	h^#(U121(e, c), c)
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(k, k))	h^#(c, c)	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	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^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	A^#	→	h^#(m, d)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
A^#	→	h^#(f(d), f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(e), l, U121(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^#(U121(e, e), l, U121(k, k))	g^#(f(e), l, U121(m, k))
	g^#(f(e), l, U121(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(f(e), l, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), l, U121(k, k))

Problem 170: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(e, e, f(l))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(e, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), c, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(k))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(c, c, f(m))	g^#(d, x, x)	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(c, c, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(c, c, f(l))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(c), f(l))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	A^#	→	h^#(m, d)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U121(l, l))

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

Problem 171: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, e, f(k))	h^#(f(c), f(c))	→	g^#(l, c, f(m))
h^#(f(c), f(c))	→	g^#(c, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), c, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(l))	h^#(f(c), f(c))	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, e, U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(l, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(k))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(l))	h^#(f(e), f(e))	→	g^#(f(e), e, f(m))
A^#	→	g^#(U121(e, c), U121(c, c), f(m))	A^#	→	g^#(U121(c, c), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	A^#	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(c, c)	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))
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^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(c, c)	→	g^#(e, c, f(l))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	A^#	→	h^#(m, d)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
A^#	→	h^#(f(d), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
A^#	→	g^#(c, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, l, f(m)) 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, U121(m, m))

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

Problem 172: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, U121(e, c), f(m))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), U121(c, c), U121(m, m))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	A^#	→	g^#(e, U121(c, c), f(m))
A^#	→	g^#(c, U121(e, c), f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	g^#(U121(c, c), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(c, c)	→	g^#(l, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
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^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	h^#(f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(l, U121(e, c), f(m)) 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, U121(e, c), U121(m, m))
g^#(l, c, f(m))

Thus, the rule A^# → g^#(l, U121(e, c), f(m)) is replaced by the following rules:

A^# → g^#(l, c, f(m))

A^# → g^#(l, U121(e, c), U121(m, m))

Problem 173: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	A^#	→	h^#(U121(e, e), U121(e, e))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(m))
A^#	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, f(e), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))
h^#(c, c)	→	g^#(l, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
A^#	→	g^#(e, U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))
A^#	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(c, c)	→	g^#(l, l, f(k))
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^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	h^#(f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, 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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U121(e, e), U121(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, U121(k, k))	g^#(e, U121(e, e), U121(m, k))
	g^#(e, U121(e, e), U121(l, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U121(e, e), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, U121(k, k))

Problem 174: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, e, f(l))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(l, l, f(m))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	A^#	→	g^#(l, c, U121(k, k))
A^#	→	g^#(e, c, f(l))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	g^#(c, c, f(m))	A^#	→	g^#(c, e, f(k))
A^#	→	g^#(c, e, f(l))	A^#	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(e, f(c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(e), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))	h^#(c, c)	→	g^#(l, e, U121(k, k))
A^#	→	g^#(e, U121(e, c), U121(k, k))	A^#	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))	h^#(f(c), f(c))	→	g^#(f(e), c, f(l))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(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^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	h^#(f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	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^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → 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, U121(l, l))

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

Problem 175: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, U121(k, k))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, c, U121(k, k))
A^#	→	h^#(U121(e, e), U121(e, e))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(m, m))
h^#(c, c)	→	g^#(l, e, U121(k, k))	A^#	→	g^#(e, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, l, f(m))	A^#	→	g^#(c, U121(e, c), U121(m, m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(c, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(c, c), f(m))
A^#	→	g^#(e, U121(e, c), U121(m, m))	h^#(c, c)	→	g^#(l, l, f(k))
A^#	→	g^#(l, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), c, f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(c, e, f(l))	h^#(c, c)	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	A^#	→	h^#(m, d)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
A^#	→	h^#(f(d), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
A^#	→	g^#(c, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(e, c), l, U121(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, U121(k, k))	g^#(U121(e, c), l, U121(l, k))
	g^#(U121(e, c), l, U121(m, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(e, c), l, U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, l, U121(k, k))

Problem 176: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(c, c)	→	g^#(l, l, f(k))	A^#	→	g^#(l, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(c, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, f(e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))
h^#(c, c)	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
A^#	→	g^#(l, U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))
h^#(f(c), f(c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(l, e, f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
A^#	→	h^#(c, c)	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
g^#(d, x, x)	→	h^#(c, c)	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	h^#(f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(c, c) → g^#(l, 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^#(l, l, f(l))
g^#(l, l, U121(k, k))
g^#(l, l, f(m))

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

h^#(c, c) → g^#(l, l, f(m))	h^#(c, c) → g^#(l, l, U121(k, k))
h^#(c, c) → g^#(l, l, f(l))

Problem 177: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), f(k))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(k, k))	A^#	→	g^#(l, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), l, f(m))	h^#(f(c), f(c))	→	g^#(c, l, f(k))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(l))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, e, f(m))	h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))
A^#	→	g^#(U121(e, c), c, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(c, c)	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	A^#	→	g^#(c, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U121(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, U121(e, e), U121(l, l))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U121(e, e), f(l)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, e, f(l))

Problem 178: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), U121(e, c), f(m))
h^#(f(e), f(e))	→	g^#(e, e, f(l))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	A^#	→	g^#(U121(e, c), c, f(l))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	g^#(d, x, x)	→	h^#(c, c)
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	A^#	→	g^#(l, l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
A^#	→	h^#(f(d), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(l, c, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(e), U121(c, c), f(m)) 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), U121(e, c), f(m))	g^#(f(e), U121(l, c), f(m))
g^#(U121(e, e), U121(c, c), f(m))	g^#(f(e), U121(c, c), U121(m, m))

Thus, the rule h^#(f(c), f(c)) → g^#(f(e), U121(c, c), f(m)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, e), U121(c, c), f(m))

h^#(f(c), f(c)) → g^#(f(e), U121(e, c), f(m))

Problem 179: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(c), l, f(m))	h^#(f(c), f(c))	→	g^#(f(c), e, f(m))
h^#(f(c), f(c))	→	g^#(e, U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(c, c), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(e), f(e))	→	g^#(e, e, f(l))
A^#	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(c, c)
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	A^#	→	h^#(m, d)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	g^#(l, l, f(k))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(l, c, f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), l, f(m)) 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), l, f(m))	g^#(f(c), l, U121(m, m))
g^#(f(e), l, f(m))
g^#(U121(c, c), l, f(m))

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

h^#(f(c), f(c)) → g^#(f(e), l, f(m))	h^#(f(c), f(c)) → g^#(U121(c, c), l, f(m))
h^#(f(c), f(c)) → g^#(f(l), l, f(m))

Problem 180: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(c, c), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(m))
g^#(d, x, x)	→	h^#(c, c)	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))
A^#	→	h^#(m, d)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(l), e, f(m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))	h^#(c, c)	→	g^#(l, c, f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
A^#	→	h^#(f(d), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	A^#	→	g^#(c, l, f(l))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, l, f(l))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	A^#	→	g^#(c, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U121(c, c), U121(c, c), U121(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^#(U121(e, c), U121(c, c), U121(k, k))	g^#(U121(c, c), U121(l, c), U121(k, k))
g^#(U121(c, c), U121(e, c), U121(k, k))	g^#(U121(c, c), U121(c, c), U121(m, k))
	g^#(U121(c, c), U121(c, c), U121(l, k))
	g^#(U121(l, c), U121(c, c), U121(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U121(c, c), U121(c, c), U121(k, k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U121(e, c), U121(c, c), U121(k, k))

h^#(f(c), f(c)) → g^#(U121(c, c), U121(e, c), U121(k, k))

Problem 181: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, c, f(m))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	g^#(c, l, f(m))
A^#	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	h^#(f(e), f(e))
A^#	→	g^#(e, c, f(m))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	g^#(e, e, f(m))
A^#	→	g^#(c, U121(e, c), U121(m, m))	A^#	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(l, l, f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), f(m))	A^#	→	h^#(m, d)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	A^#	→	g^#(c, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(l, c, f(m)) 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(m))	g^#(l, c, U121(m, m))
g^#(l, l, f(m))

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

A^# → g^#(l, l, f(m))

A^# → g^#(l, e, f(m))

Problem 182: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(l, e, f(l))	h^#(f(c), f(c))	→	g^#(c, e, f(l))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	A^#	→	h^#(f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
A^#	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, l, m, 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, U121(l, l))

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

Problem 183: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(l, l, f(k))	h^#(f(c), f(c))	→	g^#(l, e, f(m))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(l, c, f(k))	A^#	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(c, c, f(m))
h^#(f(c), f(c))	→	g^#(l, c, U121(k, k))	g^#(d, x, x)	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(m))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(c, l, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))
A^#	→	h^#(f(d), f(m))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(U121(c, c), U121(c, c), f(m))	h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	A^#	→	g^#(c, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
g^#(l, l, f(l))	g^#(c, l, U121(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 184: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(l, e, f(m))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(c, e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(e, f(e), f(m))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))	A^#	→	g^#(l, l, f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
A^#	→	h^#(f(d), f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(l, e, f(m)) 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, U121(m, m))

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

Problem 185: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, f(l))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(f(c), f(c))	→	g^#(f(e), l, f(l))	h^#(f(c), f(c))	→	g^#(e, c, f(k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(f(l), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), l, f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(m))
A^#	→	g^#(l, l, f(k))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(c, c)	→	g^#(c, l, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	A^#	→	h^#(f(d), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))	h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(e, e, U121(l, l))

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

Problem 186: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, l, U121(k, k))	A^#	→	h^#(U121(e, e), U121(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	g^#(l, l, f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(f(e), l, f(k))
h^#(c, c)	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(e, U121(e, e), f(l))
h^#(U121(e, c), U121(e, c))	→	g^#(l, l, U121(k, k))	h^#(c, c)	→	g^#(c, l, U121(k, k))
A^#	→	h^#(f(d), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(U121(e, c), U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
A^#	→	g^#(U121(e, c), U121(e, c), f(l))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, l, f(l))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(l, l, U121(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, l, U121(m, k))
	g^#(l, l, U121(l, k))

Thus, the rule A^# → g^#(l, l, U121(k, k)) is deleted.

Problem 187: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(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(l), f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), f(l), f(m))
A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(l, f(l), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), f(l), f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), f(l), f(k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, U121(e, e), f(l))	A^#	→	h^#(f(d), f(m))
h^#(f(c), f(c))	→	g^#(f(l), f(l), f(k))	h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), U121(e, e), U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(e, U121(e, c), U121(m, m))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(l, c, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	A^#	→	g^#(e, U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	A^#	→	g^#(c, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(c), 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^#(f(e), f(l), f(l))	g^#(f(c), U121(l, l), f(l))
g^#(U121(c, c), f(l), f(l))	g^#(f(c), f(l), U121(l, l))
g^#(f(l), f(l), f(l))

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

h^#(f(c), f(c)) → g^#(U121(c, c), f(l), f(l))	h^#(f(c), f(c)) → g^#(f(l), f(l), f(l))
h^#(f(c), f(c)) → g^#(f(e), f(l), f(l))

Problem 188: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
h^#(U121(e, e), U121(e, e))	→	g^#(e, U121(e, e), U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(U121(e, e), U121(e, e))	→	g^#(U121(e, e), e, U121(k, k))	A^#	→	g^#(U121(e, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(l, l, U121(k, k))	A^#	→	g^#(U121(c, c), U121(c, c), f(m))
h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))	A^#	→	g^#(l, c, f(l))
A^#	→	g^#(e, U121(e, c), f(l))	h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	A^#	→	g^#(c, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U121(e, e), U121(e, e)) → g^#(e, U121(e, e), U121(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, U121(k, k))	g^#(e, U121(e, e), U121(m, k))
	g^#(e, U121(e, e), U121(l, k))

Thus, the rule h^#(U121(e, e), U121(e, e)) → g^#(e, U121(e, e), U121(k, k)) is replaced by the following rules:

h^#(U121(e, e), U121(e, e)) → g^#(e, e, U121(k, k))

Problem 189: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, e), U121(e, e))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U121(e, c), U121(e, c))
A^#	→	g^#(U121(e, c), U121(e, c), U121(m, m))	A^#	→	g^#(l, l, f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
A^#	→	g^#(U121(e, c), U121(e, c), f(m))	g^#(d, x, x)	→	h^#(c, c)
A^#	→	g^#(U121(c, c), c, f(m))	h^#(U121(e, e), U121(e, e))	→	g^#(e, e, f(l))
A^#	→	g^#(l, c, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	A^#	→	g^#(e, U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(c, c)	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
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^#(c, e, U121(k, k))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	h^#(f(c), f(c))	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(U121(e, c), U121(e, c), U121(m, m)) 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, U121(e, c), U121(m, m))
g^#(U121(e, c), c, U121(m, m))

Thus, the rule A^# → g^#(U121(e, c), U121(e, c), U121(m, m)) is replaced by the following rules:

A^# → g^#(c, U121(e, c), U121(m, m))

A^# → g^#(U121(e, c), c, U121(m, m))

Problem 190: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(l, l, f(l))	A^#	→	g^#(l, e, f(l))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U121(e, c), U121(e, c))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
A^#	→	g^#(e, U121(e, c), f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, U121(k, k))
h^#(U121(e, c), U121(e, c))	→	g^#(c, l, f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), U121(e, e), f(m))
A^#	→	g^#(c, U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, e, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	A^#	→	g^#(c, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	g^#(d, x, x)	→	h^#(U121(c, c), e)
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(l, l, U121(l, l))

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

Problem 191: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(e, U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, c), e, U121(m, m))	h^#(f(c), f(c))	→	g^#(l, e, f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	h^#(c, c)
g^#(d, x, x)	→	h^#(c, c)	h^#(f(c), f(c))	→	g^#(c, e, f(m))
A^#	→	h^#(f(c), f(c))	A^#	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), c, f(m))
A^#	→	g^#(c, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))	A^#	→	g^#(c, U121(c, c), f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(c, c)	→	g^#(e, l, f(l))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))
h^#(f(c), f(c))	→	g^#(f(e), e, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
h^#(f(c), f(c))	→	g^#(l, l, f(m))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	g^#(d, x, x)	→	h^#(U121(c, c), e)

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U121(e, e), U121(m, m)) 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, U121(m, m))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U121(e, e), U121(m, m)) is deleted.

Problem 192: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(l, e, f(l))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), f(l))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(f(c), f(c))	→	g^#(l, l, f(l))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(e, e, f(l))	h^#(f(c), f(c))	→	g^#(l, c, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, e), U121(k, k))
A^#	→	g^#(c, U121(c, c), f(l))	h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))
h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(e, e, f(m))	A^#	→	g^#(U121(e, c), c, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))	h^#(c, c)	→	g^#(e, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(m, m))	A^#	→	h^#(f(e), f(e))
h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
	g^#(l, e, U121(l, l))

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

Problem 193: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
h^#(f(e), f(e))	→	g^#(e, e, U121(k, k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(c, U121(e, c), U121(k, k))	h^#(c, c)	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, c), U121(k, k))	h^#(f(c), f(c))	→	g^#(e, e, f(m))
A^#	→	g^#(U121(e, c), c, f(m))	h^#(f(c), f(c))	→	g^#(f(e), f(l), f(k))
h^#(c, c)	→	g^#(e, c, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))	h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))
A^#	→	g^#(c, c, U121(k, k))	g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))
h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))
A^#	→	g^#(c, c, f(l))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))
h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(f(c), l, f(l))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(e, e, U121(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, U121(l, k))
	g^#(e, e, U121(m, k))

Thus, the rule h^#(f(e), f(e)) → g^#(e, e, U121(k, k)) is deleted.

Problem 194: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U121(e, c), U121(e, c))	A^#	→	g^#(e, e, f(m))
g^#(d, x, x)	→	h^#(U121(e, c), U121(e, c))	A^#	→	g^#(c, e, f(m))
A^#	→	h^#(c, c)	A^#	→	g^#(l, c, f(m))
g^#(d, x, x)	→	h^#(c, c)	A^#	→	h^#(f(c), f(c))
A^#	→	g^#(c, l, f(m))	h^#(f(c), f(c))	→	g^#(U121(e, c), l, f(m))
h^#(f(c), f(c))	→	g^#(l, U121(e, c), f(k))	h^#(f(e), f(e))	→	g^#(U121(e, e), e, f(m))
h^#(f(c), f(c))	→	g^#(f(e), f(l), f(m))	h^#(U121(e, c), U121(e, c))	→	g^#(l, c, U121(k, k))
g^#(d, x, x)	→	h^#(U121(e, c), U121(c, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(m, m))
h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(e, l, f(l))	h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
A^#	→	g^#(c, c, f(l))	h^#(c, c)	→	g^#(e, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	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^#(c, e, U121(k, k))
h^#(f(c), f(c))	→	g^#(e, f(l), f(k))	g^#(d, x, x)	→	h^#(U121(c, c), e)
h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))	A^#	→	g^#(e, l, f(m))
h^#(f(c), f(c))	→	g^#(l, l, f(m))	h^#(f(c), f(c))	→	g^#(e, c, f(l))
A^#	→	g^#(e, l, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(e, e, f(m)) 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, U121(m, m))

Thus, the rule A^# → g^#(e, e, f(m)) is deleted.

Problem 195: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U121(e, c), c)	g^#(d, x, x)	→	h^#(c, U121(e, c))
A^#	→	h^#(c, c)	g^#(d, x, x)	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U121(e, c), e)
g^#(d, x, x)	→	h^#(e, U121(e, c))	A^#	→	g^#(c, c, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(m, m))	h^#(f(c), f(c))	→	g^#(U121(e, e), U121(e, e), U121(m, m))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(f(l), l, U121(k, k))
h^#(f(c), f(c))	→	g^#(U121(e, e), f(l), f(k))	h^#(f(c), f(c))	→	g^#(e, l, f(l))
h^#(f(c), f(c))	→	g^#(U121(e, e), l, U121(k, k))	h^#(f(e), f(e))	→	g^#(U121(e, e), U121(e, e), f(l))
h^#(f(c), f(c))	→	g^#(U121(c, c), e, U121(k, k))	h^#(f(c), f(c))	→	g^#(c, e, U121(k, k))
h^#(c, c)	→	g^#(e, e, U121(k, k))	h^#(f(c), f(c))	→	g^#(e, f(l), f(k))
h^#(f(c), f(c))	→	g^#(U121(c, c), U121(e, c), f(m))	h^#(f(c), f(c))	→	g^#(f(c), l, f(l))
A^#	→	h^#(U121(e, c), l)	h^#(f(c), f(c))	→	g^#(f(e), e, f(m))
A^#	→	h^#(c, U121(e, c))	g^#(d, x, x)	→	h^#(l, U121(e, c))
h^#(f(c), f(c))	→	g^#(f(l), f(c), f(l))	A^#	→	g^#(c, c, f(l))
g^#(d, x, x)	→	h^#(U121(c, c), e)	h^#(f(c), f(c))	→	g^#(l, U121(e, c), U121(m, m))
A^#	→	g^#(e, l, f(m))	h^#(f(c), f(c))	→	g^#(l, l, f(m))
h^#(f(c), f(c))	→	g^#(e, c, f(l))	A^#	→	g^#(e, l, f(l))

Rewrite Rules

a	→	c	b	→	c
c	→	e	k	→	l
d	→	m	a	→	d
b	→	d	c	→	l
k	→	m	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U121(x, x)	U121(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule g^#(d, x, x) → h^#(U121(e, c), 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^#(U121(e, c), l)
h^#(c, c)
h^#(U121(e, c), e)

Thus, the rule g^#(d, x, x) → h^#(U121(e, c), c) is replaced by the following rules:

g^#(d, x, x) → h^#(U121(e, c), e)	g^#(d, x, x) → h^#(U121(e, c), l)
g^#(d, x, x) → h^#(c, c)

A^# → h^#(f(c), f(b))	A^# → h^#(f(d), f(b))
A^# → h^#(f(a), f(d))	A^# → h^#(f(a), U121(b, b))
A^# → h^#(f(a), f(c))	A^# → h^#(U121(a, a), f(b))

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	b^#
a^#	→	c^#	A^#	→	f^#(a)
a^#	→	d^#	A^#	→	a^#
b^#	→	c^#	h^#(x, x)	→	k^#
A^#	→	h^#(f(a), f(b))	A^#	→	f^#(b)
h^#(x, x)	→	f^#(k)	U121^#(e, x)	→	T(x)
g^#(d, x, x)	→	A^#	b^#	→	d^#
f^#(x)	→	U121^#(x, x)

h^#(x, x) → g^#(x, x, f(k))	A^# → h^#(f(a), f(b))
g^#(d, x, x) → A^#

A^# → h^#(f(l), f(b))	A^# → h^#(U121(c, c), f(b))
A^# → h^#(f(e), f(b))	A^# → h^#(f(c), f(d))
A^# → h^#(f(c), U121(b, b))	A^# → h^#(f(c), f(c))

A^# → h^#(U121(c, c), f(c))	A^# → h^#(U121(c, c), f(d))
A^# → h^#(U121(c, c), U121(b, b))	A^# → h^#(U121(e, c), f(b))

A^# → h^#(f(d), U121(b, b))	A^# → h^#(f(m), f(b))
A^# → h^#(f(d), f(d))	A^# → h^#(f(d), f(c))

A^# → h^#(f(m), U121(e, b))	A^# → h^#(f(d), b)
A^# → h^#(U121(d, d), U121(e, b))

A^# → h^#(U121(a, a), f(d))	A^# → h^#(f(a), f(m))
A^# → h^#(f(c), f(d))	A^# → h^#(f(d), f(d))

A^# → h^#(f(c), f(m))	A^# → h^#(U121(a, a), f(m))
A^# → h^#(f(d), f(m))

A^# → h^#(f(a), f(e))	A^# → h^#(f(a), U121(c, c))
A^# → h^#(f(a), f(l))	A^# → h^#(f(c), f(c))
A^# → h^#(U121(a, a), f(c))	A^# → h^#(f(d), f(c))

Relevant Terms	Irrelevant Terms
h^#(U121(c, a), f(b))	h^#(U121(d, a), f(b))
h^#(U121(a, a), f(d))
h^#(U121(a, a), U121(b, b))
h^#(U121(a, a), f(c))

A^# → h^#(U121(a, a), f(d))	A^# → h^#(U121(c, a), f(b))
A^# → h^#(U121(a, a), U121(b, b))	A^# → h^#(U121(a, a), f(c))

Relevant Terms	Irrelevant Terms
h^#(U121(e, c), f(d))
h^#(c, f(b))
h^#(U121(e, c), U121(b, b))
h^#(U121(e, c), f(c))

A^# → h^#(U121(e, c), U121(b, b))	A^# → h^#(U121(e, c), f(d))
A^# → h^#(U121(e, c), f(c))	A^# → h^#(c, f(b))

Relevant Terms	Irrelevant Terms
h^#(U121(e, c), U121(c, b))	h^#(U121(e, c), U121(d, b))
h^#(c, U121(b, b))

A^# → h^#(c, U121(b, b))	A^# → h^#(c, f(c))
A^# → h^#(c, f(d))	A^# → h^#(e, f(b))
A^# → h^#(l, f(b))

h^#(c, c) → g^#(c, c, f(k))	h^#(f(e), f(e)) → g^#(f(e), f(e), f(k))
h^#(U121(c, c), U121(c, c)) → g^#(U121(c, c), U121(c, c), f(k))	h^#(e, e) → g^#(e, e, f(k))
h^#(f(m), f(m)) → g^#(f(m), f(m), f(k))	h^#(f(c), f(c)) → g^#(f(c), f(c), f(k))
h^#(d, d) → g^#(d, d, f(k))	h^#(m, m) → g^#(m, m, f(k))
h^#(f(d), f(d)) → g^#(f(d), f(d), f(k))	h^#(l, l) → g^#(l, l, f(k))
h^#(U121(e, c), U121(e, c)) → g^#(U121(e, c), U121(e, c), f(k))	h^#(U121(e, e), U121(e, e)) → g^#(U121(e, e), U121(e, e), f(k))
h^#(f(l), f(l)) → g^#(f(l), f(l), f(k))

Relevant Terms	Irrelevant Terms
g^#(f(l), e, f(l))	g^#(U121(l, l), U121(e, e), f(l))
	g^#(f(l), U121(e, e), U121(l, l))

A^# → h^#(c, e)	A^# → h^#(e, U121(e, e))
A^# → h^#(l, U121(e, e))

A^#	→	h^#(U121(c, c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(f(l), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(U121(e, c), U121(e, e))	A^#	→	h^#(U121(e, e), l)
A^#	→	h^#(c, f(l))	A^#	→	h^#(l, f(c))
A^#	→	h^#(f(c), U121(e, e))	A^#	→	h^#(l, c)
A^#	→	h^#(f(e), f(l))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(l, U121(e, c))	A^#	→	h^#(f(l), U121(e, c))
A^#	→	h^#(U121(e, c), f(l))	A^#	→	h^#(c, U121(e, c))
A^#	→	h^#(U121(e, c), l)	A^#	→	h^#(U121(e, e), U121(e, c))
A^#	→	h^#(e, l)	A^#	→	h^#(f(e), f(c))
g^#(d, x, x)	→	h^#(U121(c, c), e)	A^#	→	g^#(e, l, f(l))

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