Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	b^#
a^#	→	c^#	A^#	→	f^#(a)
U101^#(e, x)	→	T(x)	a^#	→	d^#
A^#	→	a^#	b^#	→	c^#
A^#	→	h^#(f(a), f(b))	A^#	→	f^#(b)
h^#(x, x)	→	f^#(k)	f^#(x)	→	U101^#(x, x)
g^#(d, x, x)	→	A^#	b^#	→	d^#

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Context-sensitive strategy:
μ(d) = μ(e) = μ(d#) = μ(b) = μ(c) = μ(A) = μ(a) = μ(a#) = μ(k) = μ(T) = μ(A#) = μ(b#) = μ(c#) = ∅
μ(f) = μ(f#) = μ(U101#) = μ(U101) = {1}
μ(h#) = μ(h) = {1, 2}
μ(g) = μ(g#) = {1, 2, 3}

The following SCCs where found

h^#(x, x) → g^#(x, x, f(k))	A^# → h^#(f(a), f(b))
g^#(d, x, x) → A^#

Problem 2: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))		A^#	→	h^#(f(a), f(b))
g^#(d, x, x)	→	A^#

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(a), f(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^#(U101(a, a), f(b))
h^#(f(c), f(b))
h^#(f(a), U101(b, b))
h^#(f(a), f(d))
h^#(f(d), f(b))
h^#(f(a), f(c))

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

A^# → h^#(f(c), f(b))	A^# → h^#(f(d), f(b))
A^# → h^#(U101(a, a), f(b))	A^# → h^#(f(a), f(d))
A^# → h^#(f(a), f(c))	A^# → h^#(f(a), U101(b, b))

Problem 3: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(c), f(b))
A^#	→	h^#(f(d), f(b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(a), U101(b, b))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(c), f(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(k), f(b))
h^#(U101(c, c), f(b))
h^#(f(e), f(b))
h^#(f(c), f(c))
h^#(f(c), f(d))
h^#(f(c), U101(b, b))

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

A^# → h^#(f(e), f(b))	A^# → h^#(f(c), f(d))
A^# → h^#(f(c), U101(b, b))	A^# → h^#(U101(c, c), f(b))
A^# → h^#(f(k), f(b))	A^# → h^#(f(c), f(c))

Problem 4: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(d), f(b))
A^#	→	h^#(f(e), f(b))	A^#	→	h^#(f(c), U101(b, b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(a), U101(b, b))
g^#(d, x, x)	→	A^#

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(d), f(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(d), f(c))	h^#(U101(d, d), f(b))
h^#(f(d), U101(b, b))
h^#(f(k), f(b))
h^#(f(d), f(d))

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

A^# → h^#(f(d), U101(b, b))	A^# → h^#(f(k), f(b))
A^# → h^#(f(d), f(d))	A^# → h^#(f(d), f(c))

Problem 5: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(d), U101(b, b))
A^#	→	h^#(f(e), f(b))	A^#	→	h^#(f(c), U101(b, b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(d), U101(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^#(f(k), U101(b, b))	h^#(U101(d, d), U101(b, b))
h^#(f(d), U101(c, b))	h^#(f(d), U101(d, b))

Thus, the rule A^# → h^#(f(d), U101(b, b)) is replaced by the following rules:

A^# → h^#(f(k), U101(b, b))

A^# → h^#(f(d), U101(c, b))

Problem 6: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(k), U101(b, b))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(f(e), f(b))	A^#	→	h^#(f(c), U101(b, b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(a), U101(b, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(k), U101(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^#(f(k), U101(c, b))	h^#(U101(k, k), U101(b, b))
	h^#(f(k), U101(d, b))

Thus, the rule A^# → h^#(f(k), U101(b, b)) is replaced by the following rules:

A^# → h^#(f(k), U101(c, b))

Problem 7: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(e), f(b))
A^#	→	h^#(f(c), U101(b, b))	A^#	→	h^#(f(a), f(d))
A^#	→	h^#(f(a), f(c))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(e), f(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), f(d))
h^#(f(e), f(c))
h^#(U101(e, e), f(b))
h^#(f(e), U101(b, b))

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

A^# → h^#(f(e), U101(b, b))	A^# → h^#(f(e), f(d))
A^# → h^#(U101(e, e), f(b))	A^# → h^#(f(e), f(c))

Problem 8: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(c), U101(b, b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), U101(b, b))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(c), U101(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^#(f(c), U101(c, b))	h^#(f(c), U101(d, b))
h^#(U101(c, c), U101(b, b))
h^#(f(k), U101(b, b))
h^#(f(e), U101(b, b))

Thus, the rule A^# → h^#(f(c), U101(b, b)) is replaced by the following rules:

A^# → h^#(f(k), U101(b, b))	A^# → h^#(f(e), U101(b, b))
A^# → h^#(U101(c, c), U101(b, b))	A^# → h^#(f(c), U101(c, b))

Problem 9: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(f(k), U101(b, b))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(f(c), U101(c, b))	A^#	→	h^#(f(a), f(d))
A^#	→	h^#(f(a), f(c))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(e, e), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(c), f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(f(k), U101(c, b))	h^#(U101(k, k), U101(b, b))
	h^#(f(k), U101(d, b))

Thus, the rule A^# → h^#(f(k), U101(b, b)) is replaced by the following rules:

A^# → h^#(f(k), U101(c, b))

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(c), U101(c, b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), U101(b, b))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(U101(c, c), U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(f(a), U101(b, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(c), U101(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^#(f(k), U101(c, b))	h^#(f(c), U101(k, b))
h^#(U101(c, c), U101(c, b))
h^#(f(e), U101(c, b))
h^#(f(c), U101(e, b))

Thus, the rule A^# → h^#(f(c), U101(c, b)) is replaced by the following rules:

A^# → h^#(f(e), U101(c, b))	A^# → h^#(f(c), U101(e, b))
A^# → h^#(U101(c, c), U101(c, b))	A^# → h^#(f(k), U101(c, b))

Problem 11: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(c), U101(e, b))
A^#	→	h^#(f(e), U101(c, b))	A^#	→	h^#(f(a), f(d))
A^#	→	h^#(f(a), f(c))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(e, e), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), U101(b, b))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(U101(c, c), U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(c), f(d))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(a), U101(b, b))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(c), U101(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^#(U101(c, c), U101(e, b))
h^#(f(e), U101(e, b))
h^#(f(k), U101(e, b))
h^#(f(c), b)

Thus, the rule A^# → h^#(f(c), U101(e, b)) is replaced by the following rules:

A^# → h^#(f(e), U101(e, b))	A^# → h^#(f(k), U101(e, b))
A^# → h^#(f(c), b)	A^# → h^#(U101(c, c), U101(e, b))

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(e), U101(c, b))
A^#	→	h^#(f(c), b)	A^#	→	h^#(f(k), U101(e, b))
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(e, e), f(b))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(a, a), f(b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(e), U101(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^#(U101(e, e), U101(c, b))	h^#(f(e), U101(k, b))
h^#(f(e), U101(e, b))

Thus, the rule A^# → h^#(f(e), U101(c, b)) is replaced by the following rules:

A^# → h^#(U101(e, e), U101(c, b))

A^# → h^#(f(e), U101(e, b))

Problem 13: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(k), U101(e, b))
A^#	→	h^#(f(c), b)	A^#	→	h^#(f(a), f(d))
A^#	→	h^#(f(a), f(c))	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(f(e), U101(b, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(a, a), f(b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(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^#(f(e), b)
h^#(f(c), c)
h^#(U101(c, c), b)
h^#(f(c), d)
h^#(f(k), b)

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

A^# → h^#(f(c), c)	A^# → h^#(U101(c, c), b)
A^# → h^#(f(c), d)	A^# → h^#(f(e), b)
A^# → h^#(f(k), b)

Problem 14: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(c, c), b)
A^#	→	h^#(f(k), U101(e, b))	A^#	→	h^#(f(c), d)
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(e, e), f(b))	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(f(e), U101(b, b))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(U101(c, c), U101(b, b))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), b)
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(e), b)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(c, 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^#(U101(e, c), b)	h^#(U101(k, c), b)
h^#(U101(c, c), d)
h^#(U101(c, c), c)

Thus, the rule A^# → h^#(U101(c, c), b) is replaced by the following rules:

A^# → h^#(U101(c, c), c)	A^# → h^#(U101(c, c), d)
A^# → h^#(U101(e, c), b)

Problem 15: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(k), U101(e, b))
A^#	→	h^#(f(c), d)	A^#	→	h^#(f(a), f(d))
A^#	→	h^#(f(a), f(c))	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(U101(e, c), b)	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(U101(c, c), U101(b, b))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), b)	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(f(a), U101(b, b))	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(e), b)	A^#	→	h^#(U101(c, c), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(k), U101(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(k), b)	h^#(U101(k, k), U101(e, b))

Thus, the rule A^# → h^#(f(k), U101(e, b)) is replaced by the following rules:

A^# → h^#(f(k), b)

Problem 16: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(c), d)
A^#	→	h^#(f(a), f(d))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(U101(e, e), f(b))	A^#	→	h^#(U101(e, c), b)
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), b)
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(c, c), d)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(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^#(f(e), d)
h^#(f(k), d)
h^#(f(c), k)
h^#(U101(c, c), d)

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

A^# → h^#(f(c), k)	A^# → h^#(f(e), d)
A^# → h^#(U101(c, c), d)	A^# → h^#(f(k), d)

Problem 17: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(a), f(d))
A^#	→	h^#(f(a), f(c))	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(f(k), d)	A^#	→	h^#(U101(e, c), b)
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(f(d), U101(c, b))	A^#	→	h^#(f(e), U101(b, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), b)
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(f(e), d)
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(c, c), d)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(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^#(f(a), f(k))	h^#(f(a), U101(d, d))
h^#(f(d), f(d))
h^#(f(c), f(d))
h^#(U101(a, a), f(d))

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

A^# → h^#(U101(a, a), f(d))	A^# → h^#(f(a), f(k))
A^# → h^#(f(c), f(d))	A^# → h^#(f(d), f(d))

Problem 18: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(a), f(c))
A^#	→	h^#(f(k), d)	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(U101(e, c), b)	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(U101(a, a), f(d))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(f(a), f(k))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), b)	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(f(a), U101(b, b))	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U101(c, c), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(a), 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(c))
h^#(f(a), f(k))
h^#(f(a), U101(c, c))
h^#(f(a), f(e))
h^#(f(c), f(c))
h^#(U101(a, a), f(c))

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

A^# → h^#(f(a), f(e))	A^# → h^#(f(a), U101(c, c))
A^# → h^#(f(a), f(k))	A^# → h^#(U101(a, a), f(c))
A^# → h^#(f(d), f(c))	A^# → h^#(f(c), f(c))

Problem 19: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(a, a), f(c))
A^#	→	h^#(U101(e, e), f(b))	A^#	→	h^#(f(k), d)
A^#	→	h^#(U101(e, c), b)	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(U101(a, a), f(d))
A^#	→	h^#(f(a), f(e))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(U101(c, c), U101(b, b))
A^#	→	h^#(f(a), f(k))	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), b)
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U101(c, c), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(a, a), 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^#(U101(c, a), f(c))	h^#(U101(d, a), f(c))
h^#(U101(a, a), f(k))
h^#(U101(a, a), f(e))
h^#(U101(a, a), U101(c, c))

Thus, the rule A^# → h^#(U101(a, a), f(c)) is replaced by the following rules:

A^# → h^#(U101(a, a), f(e))	A^# → h^#(U101(c, a), f(c))
A^# → h^#(U101(a, a), U101(c, c))	A^# → h^#(U101(a, a), f(k))

Problem 20: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(a, a), U101(c, c))
A^#	→	h^#(f(k), d)	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(U101(e, c), b)	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(U101(a, a), f(d))
A^#	→	h^#(f(a), f(e))	A^#	→	h^#(U101(a, a), f(e))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(U101(c, c), U101(b, b))	A^#	→	h^#(f(a), f(k))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), b)	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(f(a), U101(b, b))	A^#	→	h^#(U101(a, a), f(k))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(f(e), d)
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(c, c), d)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(a, a), U101(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^#(U101(a, a), U101(e, c))	h^#(U101(d, a), U101(c, c))
h^#(U101(c, a), U101(c, c))	h^#(U101(a, a), U101(k, c))

Thus, the rule A^# → h^#(U101(a, a), U101(c, c)) is replaced by the following rules:

A^# → h^#(U101(a, a), U101(e, c))

A^# → h^#(U101(c, a), U101(c, c))

Problem 21: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, e), f(b))
A^#	→	h^#(f(k), d)	A^#	→	h^#(U101(e, c), b)
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(U101(a, a), f(d))	A^#	→	h^#(U101(a, a), f(e))
A^#	→	h^#(f(a), f(e))	A^#	→	h^#(f(d), U101(c, b))
A^#	→	h^#(f(e), U101(b, b))	A^#	→	h^#(U101(c, c), U101(b, b))
A^#	→	h^#(f(a), f(k))	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), b)
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(f(a), U101(b, b))
A^#	→	h^#(U101(a, a), f(k))	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(f(e), d)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(k), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U101(c, c), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(k), 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^#(f(k), k)	h^#(U101(k, k), d)

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

A^# → h^#(f(k), k)

Problem 22: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(k), b)
A^#	→	h^#(U101(e, e), c)	A^#	→	h^#(e, U101(c, c))
A^#	→	h^#(f(e), U101(e, e))	A^#	→	h^#(f(k), f(b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(e, c), c)
A^#	→	h^#(U101(c, a), f(d))	A^#	→	h^#(U101(e, c), U101(e, e))
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(k, b)
A^#	→	h^#(f(d), e)	A^#	→	h^#(U101(c, c), d)
A^#	→	h^#(U101(e, c), d)	A^#	→	h^#(e, f(d))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(c, a), U101(e, e))	A^#	→	h^#(e, c)
A^#	→	h^#(f(d), f(k))	A^#	→	h^#(U101(e, e), f(e))
A^#	→	h^#(f(e), U101(c, b))	A^#	→	h^#(f(k), U101(e, b))
A^#	→	h^#(U101(e, c), U101(b, b))	A^#	→	h^#(f(c), U101(c, b))
A^#	→	h^#(f(a), U101(e, b))	A^#	→	h^#(f(d), U101(b, b))
A^#	→	h^#(f(c), U101(b, b))	A^#	→	h^#(f(c), f(e))
A^#	→	h^#(e, f(k))	A^#	→	h^#(f(c), f(k))
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(c, d)	A^#	→	h^#(U101(a, a), f(e))
A^#	→	h^#(e, f(c))	A^#	→	h^#(e, f(b))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(e))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(U101(a, a), f(k))	A^#	→	h^#(U101(a, a), U101(c, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(f(e), d)
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(k), 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(k), c)	h^#(U101(k, k), b)
h^#(f(k), d)

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

A^# → h^#(f(k), c)

A^# → h^#(f(k), d)

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(f(d), U101(b, b))	A^#	→	h^#(f(c), U101(c, b))
A^#	→	h^#(f(a), U101(e, b))	A^#	→	h^#(f(c), f(e))
A^#	→	h^#(f(c), U101(b, b))	A^#	→	h^#(f(k), c)
A^#	→	h^#(c, e)	A^#	→	h^#(e, f(k))
A^#	→	h^#(a, f(k))	A^#	→	h^#(f(c), f(k))
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(k, d)	A^#	→	h^#(c, d)
A^#	→	h^#(f(k), U101(c, c))	A^#	→	h^#(U101(a, a), f(e))
A^#	→	h^#(d, f(d))	A^#	→	h^#(e, f(c))
A^#	→	h^#(k, k)	A^#	→	h^#(c, f(d))
A^#	→	h^#(e, f(b))	A^#	→	h^#(f(k), e)
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(e))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(U101(c, a), f(k))
A^#	→	h^#(U101(a, a), f(k))	A^#	→	h^#(U101(a, a), U101(c, b))
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(f(k), U101(b, b))	h^#(U101(d, d), U101(b, b))
h^#(f(d), U101(c, b))	h^#(f(d), U101(d, b))

Thus, the rule A^# → h^#(f(d), U101(b, b)) is replaced by the following rules:

A^# → h^#(f(k), U101(b, b))

A^# → h^#(f(d), U101(c, b))

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(f(a), e)
A^#	→	h^#(U101(a, a), c)	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(d, b)
A^#	→	h^#(a, k)	A^#	→	h^#(U101(c, a), e)
A^#	→	h^#(k, b)	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(a, a), k)	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(f(c), U101(e, b))
A^#	→	h^#(f(c), b)	A^#	→	h^#(f(k), U101(e, b))
A^#	→	h^#(f(c), d)	A^#	→	h^#(f(c), U101(b, b))
A^#	→	h^#(f(c), f(e))	A^#	→	h^#(f(k), c)
A^#	→	h^#(c, e)	A^#	→	h^#(U101(c, a), c)
A^#	→	h^#(e, f(k))	A^#	→	h^#(U101(e, c), b)
A^#	→	h^#(f(c), f(k))	A^#	→	h^#(a, f(k))
A^#	→	h^#(e, b)	A^#	→	h^#(f(d), f(c))
A^#	→	h^#(k, d)	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(f(k), U101(c, c))	A^#	→	h^#(U101(a, a), f(e))
A^#	→	h^#(c, d)	A^#	→	h^#(d, f(d))
A^#	→	h^#(e, f(c))	A^#	→	h^#(c, f(d))
A^#	→	h^#(k, k)	A^#	→	h^#(e, f(b))
A^#	→	h^#(f(d), c)	A^#	→	h^#(f(k), e)
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(e))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(U101(c, a), f(k))
A^#	→	h^#(U101(a, a), f(k))	A^#	→	h^#(U101(a, a), U101(c, b))
A^#	→	h^#(a, d)	A^#	→	h^#(f(d), k)
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(f(c), e)
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(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^#(U101(a, a), e)
h^#(f(d), e)
h^#(f(c), e)

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

A^# → h^#(f(c), e)	A^# → h^#(f(d), e)
A^# → h^#(U101(a, a), e)

Problem 25: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(e, c), d)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(f(k), U101(e, b))
A^#	→	h^#(f(c), d)	A^#	→	h^#(f(c), U101(b, b))
A^#	→	h^#(f(c), f(e))	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(f(k), c)
A^#	→	h^#(U101(c, a), c)	A^#	→	h^#(e, f(k))
A^#	→	h^#(U101(e, c), b)	A^#	→	h^#(f(c), f(k))
A^#	→	h^#(a, f(k))	A^#	→	h^#(e, b)
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(k, d)
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(f(k), U101(c, c))
A^#	→	h^#(U101(a, a), f(e))	A^#	→	h^#(c, d)
A^#	→	h^#(d, f(d))	A^#	→	h^#(e, f(c))
A^#	→	h^#(c, f(d))	A^#	→	h^#(k, k)
A^#	→	h^#(e, f(b))	A^#	→	h^#(f(d), c)
A^#	→	h^#(f(k), e)	A^#	→	h^#(f(e), f(d))
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(k), f(e))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(U101(a, a), f(k))
A^#	→	h^#(U101(a, a), U101(c, b))	A^#	→	h^#(a, d)
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(a, a), U101(b, b))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(f(c), e)	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(e, 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^#(U101(e, c), k)
h^#(c, d)

Thus, the rule A^# → h^#(U101(e, c), d) is replaced by the following rules:

A^# → h^#(c, d)

A^# → h^#(U101(e, c), k)

Problem 26: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(a, k)
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), c)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(e, f(k))
A^#	→	h^#(U101(e, c), b)	A^#	→	h^#(a, f(k))
A^#	→	h^#(f(c), f(k))	A^#	→	h^#(e, b)
A^#	→	h^#(f(d), f(c))	A^#	→	h^#(k, d)
A^#	→	h^#(U101(e, e), U101(c, b))	A^#	→	h^#(f(k), U101(c, c))
A^#	→	h^#(U101(a, a), f(e))	A^#	→	h^#(c, d)
A^#	→	h^#(e, f(c))	A^#	→	h^#(d, f(d))
A^#	→	h^#(c, f(e))	A^#	→	h^#(k, k)
A^#	→	h^#(c, f(d))	A^#	→	h^#(e, f(b))
A^#	→	h^#(f(d), c)	A^#	→	h^#(f(k), e)
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(e))
A^#	→	h^#(U101(c, c), U101(e, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(U101(c, a), f(k))
A^#	→	h^#(U101(a, a), f(k))	A^#	→	h^#(U101(a, a), U101(c, b))
A^#	→	h^#(a, d)	A^#	→	h^#(f(d), k)
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(f(c), e)
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(d, k)

A^# → h^#(c, k)

Problem 27: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(e, b)	A^#	→	h^#(U101(e, e), U101(c, b))
A^#	→	h^#(k, d)	A^#	→	h^#(f(k), U101(c, c))
A^#	→	h^#(c, d)	A^#	→	h^#(U101(a, a), f(e))
A^#	→	h^#(c, f(e))	A^#	→	h^#(d, f(d))
A^#	→	h^#(e, f(c))	A^#	→	h^#(e, f(b))
A^#	→	h^#(k, k)	A^#	→	h^#(c, f(d))
A^#	→	h^#(c, f(k))	A^#	→	h^#(f(d), c)
A^#	→	h^#(f(k), e)	A^#	→	h^#(f(c), f(d))
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(k), f(e))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(U101(a, a), f(k))
A^#	→	h^#(U101(a, a), U101(c, b))	A^#	→	h^#(a, d)
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(a, a), U101(b, b))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, 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 28: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(d, f(d))	A^#	→	h^#(c, f(e))
A^#	→	h^#(e, f(c))	A^#	→	h^#(e, f(b))
A^#	→	h^#(k, k)	A^#	→	h^#(c, f(k))
A^#	→	h^#(c, f(d))	A^#	→	h^#(f(d), c)
A^#	→	h^#(e, U101(e, b))	A^#	→	h^#(f(k), e)
A^#	→	h^#(f(e), f(d))	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(f(k), f(e))
A^#	→	h^#(k, U101(e, e))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(U101(a, a), f(k))
A^#	→	h^#(U101(a, a), U101(c, b))	A^#	→	h^#(a, d)
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(a, a), U101(b, b))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(d, 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^#(k, f(d))	h^#(d, U101(d, d))
h^#(d, f(k))

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

A^# → h^#(k, f(d))

A^# → h^#(d, f(k))

Problem 29: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(k, k)	A^#	→	h^#(U101(c, c), c)
A^#	→	h^#(f(c), f(d))	A^#	→	h^#(k, U101(e, e))
A^#	→	h^#(f(k), f(e))	A^#	→	h^#(U101(c, c), U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(k), U101(c, b))
A^#	→	h^#(U101(a, a), f(k))	A^#	→	h^#(U101(c, a), f(k))
A^#	→	h^#(a, d)	A^#	→	h^#(U101(a, a), U101(c, b))
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(a, a), U101(b, b))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(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^#(U101(c, c), e)	h^#(U101(k, c), c)
h^#(U101(c, c), k)
h^#(U101(e, c), c)

Thus, the rule A^# → h^#(U101(c, c), c) is replaced by the following rules:

A^# → h^#(U101(e, c), c)	A^# → h^#(U101(c, c), k)
A^# → h^#(U101(c, 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(c), f(c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(e, b)
A^#	→	h^#(k, d)	A^#	→	h^#(U101(e, c), U101(e, b))
A^#	→	h^#(c, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(k), U101(c, b))	A^#	→	h^#(U101(a, a), f(k))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(a, d)
A^#	→	h^#(U101(a, a), U101(c, b))	A^#	→	h^#(f(d), k)
A^#	→	h^#(f(e), U101(e, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

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

A^# → h^#(e, e)

A^# → h^#(k, e)

Problem 31: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(k, d)	A^#	→	h^#(c, d)
A^#	→	h^#(k, k)	A^#	→	h^#(e, U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(U101(c, a), f(k))
A^#	→	h^#(a, d)	A^#	→	h^#(U101(a, a), U101(c, b))
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(U101(a, a), U101(b, b))	A^#	→	h^#(c, c)
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

Problem 32: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(k, c)
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(e, k)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(e, U101(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(k, d)
A^#	→	h^#(c, d)	A^#	→	h^#(k, k)
A^#	→	h^#(d, U101(c, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, U101(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(k, e)

Problem 33: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(e, k)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(k, d)
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(d), k)	A^#	→	h^#(f(e), U101(e, b))
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

Problem 34: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, k)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(k, b)	A^#	→	h^#(e, k)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(U101(e, a), U101(c, b))
A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, a), U101(b, b))
A^#	→	h^#(e, U101(c, b))	A^#	→	h^#(e, b)
A^#	→	h^#(c, d)	A^#	→	h^#(k, k)
A^#	→	h^#(d, U101(c, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, U101(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(e, k)

Problem 35: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, k)
A^#	→	h^#(e, e)	A^#	→	h^#(k, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(d, b)
A^#	→	h^#(c, U101(c, b))	A^#	→	h^#(a, k)
A^#	→	h^#(k, b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(e, U101(c, b))
A^#	→	h^#(e, b)	A^#	→	h^#(k, d)
A^#	→	h^#(c, d)	A^#	→	h^#(k, k)
A^#	→	h^#(d, U101(c, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, U101(e, b))	A^#	→	h^#(a, U101(b, b))
A^#	→	h^#(a, d)	A^#	→	h^#(k, U101(c, b))
A^#	→	h^#(c, c)	A^#	→	h^#(f(a), U101(c, c))
A^#	→	h^#(U101(c, a), f(c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

Problem 36: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(e, k)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(k, d)
A^#	→	h^#(k, k)	A^#	→	h^#(e, U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(a, U101(b, b))
A^#	→	h^#(a, d)	A^#	→	h^#(k, U101(c, b))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

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

Problem 37: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(e, k)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(d, U101(c, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, U101(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(c, c)
A^#	→	h^#(f(a), U101(c, c))	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

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

Problem 38: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(a, a), k)
A^#	→	h^#(f(d), U101(e, c))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(U101(a, a), U101(c, c))	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(U101(c, a), c)
A^#	→	h^#(k, d)	A^#	→	h^#(f(k), U101(c, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(c), U101(e, c))
A^#	→	h^#(f(e), U101(c, c))	A^#	→	h^#(f(d), c)
A^#	→	h^#(f(k), e)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(a, a), 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
h^#(U101(c, a), k)	h^#(U101(d, a), k)

Thus, the rule A^# → h^#(U101(a, a), k) is replaced by the following rules:

A^# → h^#(U101(c, a), k)

Problem 39: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(f(e), U101(e, c))
A^#	→	h^#(f(k), c)	A^#	→	h^#(d, d)
A^#	→	h^#(k, k)	A^#	→	h^#(f(d), c)
A^#	→	h^#(f(e), U101(c, c))	A^#	→	h^#(f(k), e)
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(f(e), U101(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^#(U101(e, e), U101(e, c))
h^#(f(e), c)

Thus, the rule A^# → h^#(f(e), U101(e, c)) is replaced by the following rules:

A^# → h^#(f(e), c)

A^# → h^#(U101(e, e), U101(e, c))

Problem 40: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, e)	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(c, a), U101(e, e))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(k, f(e))
A^#	→	h^#(c, f(e))	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, a), f(c))	A^#	→	h^#(k, U101(e, e))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(U101(c, a), f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

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

A^# → h^#(k, e)

Problem 41: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(a, U101(c, c))	A^#	→	h^#(a, f(k))
A^#	→	h^#(e, f(c))	A^#	→	h^#(c, f(e))
A^#	→	h^#(k, f(c))	A^#	→	h^#(c, f(k))
A^#	→	h^#(d, f(c))	A^#	→	h^#(k, k)
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(k, U101(e, e))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(U101(e, a), U101(c, c))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(c, c)
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, U101(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^#(c, U101(c, c))	h^#(a, U101(k, c))
h^#(a, U101(e, c))
h^#(d, U101(c, c))

Thus, the rule A^# → h^#(a, U101(c, c)) is replaced by the following rules:

A^# → h^#(a, U101(e, c))	A^# → h^#(c, U101(c, c))
A^# → h^#(d, U101(c, c))

Problem 42: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(a, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(a, k)	A^#	→	h^#(U101(e, a), e)
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(a, U101(c, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(a, U101(e, c))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(f(e), d)	A^#	→	h^#(c, U101(e, c))
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(U101(c, c), f(b))	A^#	→	h^#(f(a), k)
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

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

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

Problem 43: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(e, e)
A^#	→	h^#(k, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(a, k)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(k, U101(e, c))	A^#	→	h^#(c, c)
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(f(e), d)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(c, U101(e, c))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(c, c), f(b))
A^#	→	h^#(f(a), k)	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(k, k)

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

A^# → h^#(k, k)

Problem 44: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, e), k)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(U101(c, c), f(d))
A^#	→	h^#(k, b)	A^#	→	h^#(e, k)
A^#	→	h^#(U101(e, c), f(d))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(a, a), k)
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(U101(e, c), U101(b, b))
A^#	→	h^#(c, e)	A^#	→	h^#(k, f(b))
A^#	→	h^#(d, d)	A^#	→	h^#(c, f(e))
A^#	→	h^#(e, f(c))	A^#	→	h^#(k, k)
A^#	→	h^#(k, f(c))	A^#	→	h^#(c, f(k))
A^#	→	h^#(e, f(b))	A^#	→	h^#(c, f(d))
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(U101(e, c), f(c))	A^#	→	h^#(c, c)
A^#	→	h^#(U101(c, c), U101(c, b))	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(e, e), e)	A^#	→	h^#(f(c), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(f(e), e)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(e, e), 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
h^#(e, k)

Thus, the rule A^# → h^#(U101(e, e), k) is replaced by the following rules:

A^# → h^#(e, k)

Problem 45: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, f(e))
A^#	→	h^#(c, f(d))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(c, f(k))	A^#	→	h^#(k, k)
A^#	→	h^#(e, U101(e, b))	A^#	→	h^#(c, U101(c, c))
A^#	→	h^#(k, U101(e, e))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(k, U101(c, b))
A^#	→	h^#(U101(e, c), f(c))	A^#	→	h^#(k, U101(c, c))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, c), U101(c, b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(c), e)	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(U101(a, a), U101(e, c))	g^#(d, x, x)	→	A^#
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(k, U101(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^#(k, U101(e, c))	h^#(k, U101(k, c))

Thus, the rule A^# → h^#(k, U101(c, c)) is replaced by the following rules:

A^# → h^#(k, U101(e, c))

Problem 46: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(k, c)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(e, k)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, c), U101(c, c))
A^#	→	h^#(U101(e, c), d)	A^#	→	h^#(U101(c, c), d)
A^#	→	h^#(e, f(d))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(c, e)
A^#	→	h^#(e, b)	A^#	→	h^#(U101(e, c), U101(e, b))
A^#	→	h^#(k, d)	A^#	→	h^#(c, d)
A^#	→	h^#(e, f(c))	A^#	→	h^#(c, f(e))
A^#	→	h^#(k, f(c))	A^#	→	h^#(c, f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(c, U101(c, c))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(k, e)

Problem 47: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	A^#	→	h^#(c, U101(c, c))
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(k, f(k))
A^#	→	h^#(k, U101(e, e))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, U101(e, b))	A^#	→	h^#(e, e)
A^#	→	h^#(k, U101(c, c))	A^#	→	h^#(c, c)
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(c, k)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(f(c), e)	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), c)	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(U101(c, c), e)	h^#(U101(k, c), c)
h^#(U101(c, c), k)
h^#(U101(e, c), c)

Thus, the rule A^# → h^#(U101(c, c), c) is replaced by the following rules:

A^# → h^#(U101(e, c), c)	A^# → h^#(U101(c, c), k)
A^# → h^#(U101(c, c), e)

Problem 48: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(a, k)
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(k, b)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(e, k)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, a), c)
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(c, e)
A^#	→	h^#(e, b)	A^#	→	h^#(c, d)
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(a, d)	A^#	→	h^#(c, c)
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(f(c), e)	A^#	→	h^#(f(c), c)
A^#	→	h^#(U101(e, c), U101(c, b))	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(d, k)

A^# → h^#(c, k)

Problem 49: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(e, c)
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(c, d)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(a, b)	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(a, d)	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(d, b)
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, c), U101(c, b))
A^#	→	h^#(U101(c, c), e)	A^#	→	h^#(f(e), e)
A^#	→	h^#(a, k)	A^#	→	h^#(f(k), U101(e, e))
A^#	→	h^#(U101(e, e), k)	A^#	→	h^#(k, U101(e, b))
A^#	→	h^#(f(k), k)	A^#	→	h^#(k, b)
A^#	→	h^#(U101(e, a), e)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(e, k)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(e, e)

A^# → h^#(e, k)

Problem 50: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(e, b)
A^#	→	h^#(c, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(e, U101(e, b))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(c, k)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(e, c), c)
A^#	→	h^#(c, U101(c, b))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(a, k)	A^#	→	h^#(f(e), e)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(U101(e, e), k)
A^#	→	h^#(U101(e, a), e)	A^#	→	h^#(k, b)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(f(k), k)
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, k)	A^#	→	h^#(U101(a, a), f(b))
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, c), d)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), c)	A^#	→	h^#(f(d), U101(c, c))
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, 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 51: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, k)
A^#	→	h^#(U101(a, a), f(b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(U101(e, c), d)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, a), c)	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

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

Problem 52: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(a, f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(U101(c, a), U101(b, b))
A^#	→	h^#(e, e)	A^#	→	h^#(U101(a, a), U101(b, b))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), f(c))
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(U101(e, a), f(b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, a), f(d))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(c, a), e)
A^#	→	h^#(a, f(e))	A^#	→	h^#(U101(e, a), e)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, c), d)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), c)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(a, e)
A^#	→	h^#(U101(e, e), U101(b, b))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(a, 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
h^#(d, f(k))	h^#(a, U101(k, k))
h^#(c, f(k))

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

A^# → h^#(c, f(k))

A^# → h^#(d, f(k))

Problem 53: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(k, c)
A^#	→	h^#(d, k)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(k, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(d, b)
A^#	→	h^#(c, U101(c, b))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(c, a), f(d))	A^#	→	h^#(a, k)
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(a, f(e))
A^#	→	h^#(c, f(c))	A^#	→	h^#(k, b)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(U101(e, a), e)
A^#	→	h^#(d, f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, c), d)	A^#	→	h^#(U101(e, a), c)
g^#(d, x, x)	→	A^#	A^#	→	h^#(a, e)
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(c, a), U101(e, e))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(a, U101(c, c))	A^#	→	h^#(a, f(k))
A^#	→	h^#(e, b)	A^#	→	h^#(c, d)
A^#	→	h^#(k, k)	A^#	→	h^#(d, f(c))
A^#	→	h^#(d, U101(c, b))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(U101(e, a), U101(c, c))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(a, d)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(c, c)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, a), f(b))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(k, e)

Problem 54: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(U101(e, a), U101(e, e))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(k, f(e))
A^#	→	h^#(e, U101(c, b))	A^#	→	h^#(a, U101(c, c))
A^#	→	h^#(a, f(k))	A^#	→	h^#(e, b)
A^#	→	h^#(k, d)	A^#	→	h^#(c, d)
A^#	→	h^#(c, f(e))	A^#	→	h^#(d, f(d))
A^#	→	h^#(e, f(c))	A^#	→	h^#(k, f(c))
A^#	→	h^#(c, f(d))	A^#	→	h^#(c, f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(d, f(c))
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(d, U101(c, b))
A^#	→	h^#(k, U101(e, e))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(U101(e, a), U101(c, c))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(a, d)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(c, c)
A^#	→	h^#(U101(e, c), k)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(U101(e, a), f(b))	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(e, a), U101(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^#(U101(e, a), e)
h^#(a, U101(e, e))

Thus, the rule A^# → h^#(U101(e, a), U101(e, e)) is replaced by the following rules:

A^# → h^#(U101(e, a), e)

A^# → h^#(a, U101(e, e))

Problem 55: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(k, f(e))	A^#	→	h^#(d, U101(c, c))
A^#	→	h^#(c, f(k))	A^#	→	h^#(k, k)
A^#	→	h^#(k, f(c))	A^#	→	h^#(e, U101(e, b))
A^#	→	h^#(d, U101(e, c))	A^#	→	h^#(d, U101(c, b))
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(k, U101(e, e))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(k, U101(e, c))
A^#	→	h^#(U101(e, a), U101(c, c))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(U101(c, a), f(k))	A^#	→	h^#(a, U101(e, c))
A^#	→	h^#(a, d)	A^#	→	h^#(k, U101(c, b))
A^#	→	h^#(c, c)	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, a), f(b))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(e, a), d)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(k, 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^#(k, U101(e, e))

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

A^# → h^#(k, U101(e, e))

Problem 56: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, d)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(a, U101(e, c))
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(e, e)
A^#	→	h^#(k, U101(c, c))	A^#	→	h^#(c, c)
A^#	→	h^#(e, U101(c, c))	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(k, e)	A^#	→	h^#(c, U101(e, c))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(U101(c, a), U101(e, b))	A^#	→	h^#(U101(e, a), f(b))
A^#	→	h^#(c, k)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(a, k)	A^#	→	h^#(k, b)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(e, k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(d, f(k))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(k, k)

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

A^# → h^#(k, k)

Problem 57: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(U101(e, a), f(b))	A^#	→	h^#(d, U101(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(c, k)
A^#	→	h^#(d, b)	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(c, a), e)
A^#	→	h^#(a, k)	A^#	→	h^#(k, U101(e, b))
A^#	→	h^#(k, b)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(e, k)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(d, f(k))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(a, e)
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(d, U101(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^#(d, e)
h^#(k, U101(e, e))

Thus, the rule A^# → h^#(d, U101(e, e)) is replaced by the following rules:

A^# → h^#(d, e)

A^# → h^#(k, U101(e, e))

Problem 58: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, a), U101(e, c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(k, f(b))	A^#	→	h^#(d, d)
A^#	→	h^#(e, U101(c, b))	A^#	→	h^#(a, f(k))
A^#	→	h^#(e, b)	A^#	→	h^#(k, d)
A^#	→	h^#(c, d)	A^#	→	h^#(e, f(c))
A^#	→	h^#(c, f(e))	A^#	→	h^#(d, f(d))
A^#	→	h^#(e, f(b))	A^#	→	h^#(k, f(c))
A^#	→	h^#(d, f(c))	A^#	→	h^#(c, f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(c, f(d))
A^#	→	h^#(U101(e, a), f(c))	A^#	→	h^#(c, U101(c, c))
A^#	→	h^#(d, U101(c, b))	A^#	→	h^#(a, f(c))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(a, U101(b, b))	A^#	→	h^#(a, d)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(c, c)
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(e, a), U101(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^#(U101(e, a), c)
h^#(a, U101(e, c))

Thus, the rule A^# → h^#(U101(e, a), U101(e, c)) is replaced by the following rules:

A^# → h^#(a, U101(e, c))

A^# → h^#(U101(e, a), c)

Problem 59: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(d, f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(d, f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(e, f(d))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
A^#	→	h^#(k, f(c))	A^#	→	h^#(c, f(k))
A^#	→	h^#(e, f(b))	A^#	→	h^#(k, k)
A^#	→	h^#(d, U101(e, c))	A^#	→	h^#(U101(e, a), f(c))
A^#	→	h^#(e, U101(e, b))	A^#	→	h^#(d, U101(c, b))
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(a, f(c))
A^#	→	h^#(k, U101(e, e))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(k, U101(e, c))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(a, U101(b, b))	A^#	→	h^#(a, U101(e, c))
A^#	→	h^#(a, d)	A^#	→	h^#(k, U101(c, b))
A^#	→	h^#(c, c)	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(k, 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
	h^#(k, U101(k, k))

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

Problem 60: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(e, e)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(a, e)	g^#(d, x, x)	→	A^#
A^#	→	h^#(d, f(e))	A^#	→	h^#(e, f(d))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(d, d)	A^#	→	h^#(a, f(k))
A^#	→	h^#(a, U101(c, c))	A^#	→	h^#(e, f(c))
A^#	→	h^#(c, f(e))	A^#	→	h^#(d, f(c))
A^#	→	h^#(k, k)	A^#	→	h^#(k, f(c))
A^#	→	h^#(c, f(k))	A^#	→	h^#(d, U101(c, b))
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(k, U101(e, e))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(k, U101(e, c))
A^#	→	h^#(U101(e, a), U101(c, c))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(a, U101(b, b))	A^#	→	h^#(a, U101(e, c))
A^#	→	h^#(a, d)	A^#	→	h^#(k, U101(c, b))
A^#	→	h^#(c, c)	A^#	→	h^#(a, U101(e, e))
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(e, d)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, 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 61: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(k, U101(e, c))	A^#	→	h^#(c, U101(e, b))
A^#	→	h^#(a, U101(b, b))	A^#	→	h^#(a, U101(e, c))
A^#	→	h^#(a, d)	A^#	→	h^#(e, e)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(k, U101(c, c))
A^#	→	h^#(c, c)	A^#	→	h^#(e, U101(c, c))
A^#	→	h^#(k, e)	A^#	→	h^#(c, U101(e, c))
A^#	→	h^#(a, U101(e, e))	A^#	→	h^#(d, U101(e, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(e, d)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(d, f(k))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, a), U101(e, c))
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(d, f(e))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(U101(a, a), U101(e, c))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(k, U101(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^#(k, c)

Thus, the rule A^# → h^#(k, U101(e, c)) is replaced by the following rules:

A^# → h^#(k, c)

Problem 62: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(e, c)
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(d, d)
A^#	→	h^#(c, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(c, U101(e, b))	A^#	→	h^#(a, d)
A^#	→	h^#(k, U101(c, b))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(a, c)
A^#	→	h^#(d, U101(e, b))	A^#	→	h^#(d, U101(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(c, k)
A^#	→	h^#(c, U101(c, b))	A^#	→	h^#(d, b)
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, a), U101(e, b))
A^#	→	h^#(a, k)	A^#	→	h^#(k, U101(e, b))
A^#	→	h^#(k, b)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, k)	A^#	→	h^#(e, d)
A^#	→	h^#(f(c), k)	A^#	→	h^#(d, f(k))
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(a, e)	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(d, f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(e, e)

A^# → h^#(e, k)

Problem 63: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(a, k)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(k, b)
A^#	→	h^#(e, k)	A^#	→	h^#(c, U101(b, b))
A^#	→	h^#(e, d)	A^#	→	h^#(d, f(k))
A^#	→	h^#(U101(a, a), U101(e, b))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(a, e)
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(d, f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(e, a), U101(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^#(U101(e, a), b)
h^#(a, U101(e, b))

Thus, the rule A^# → h^#(U101(e, a), U101(e, b)) is replaced by the following rules:

A^# → h^#(a, U101(e, b))

A^# → h^#(U101(e, a), b)

Problem 64: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(a, k)	A^#	→	h^#(k, b)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(e, k)
A^#	→	h^#(c, U101(b, b))	A^#	→	h^#(e, d)
A^#	→	h^#(d, f(k))	A^#	→	h^#(U101(a, a), U101(e, b))
A^#	→	h^#(f(c), k)	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(a, e)
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(d, f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(d, k)

A^# → h^#(c, k)

Problem 65: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(U101(a, a), c)	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(U101(c, a), k)
A^#	→	h^#(U101(e, a), b)	A^#	→	h^#(U101(c, a), U101(e, b))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(f(c), k)
A^#	→	h^#(d, f(k))	A^#	→	h^#(f(d), b)
A^#	→	h^#(f(e), b)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(f(d), U101(c, c))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(d, f(e))
A^#	→	h^#(a, e)	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(U101(a, a), k)	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(a, 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^#(U101(a, a), e)	h^#(U101(d, a), c)
h^#(U101(a, a), k)
h^#(U101(c, a), c)

Thus, the rule A^# → h^#(U101(a, a), c) is replaced by the following rules:

A^# → h^#(U101(c, a), c)	A^# → h^#(U101(a, a), e)
A^# → h^#(U101(a, a), k)

Problem 66: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(e, a), U101(e, b))	A^#	→	h^#(U101(c, a), e)
A^#	→	h^#(a, k)	A^#	→	h^#(k, b)
A^#	→	h^#(e, k)	A^#	→	h^#(e, d)
A^#	→	h^#(f(c), k)	A^#	→	h^#(d, f(k))
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), U101(c, c))	A^#	→	h^#(U101(e, a), c)
A^#	→	h^#(a, e)	A^#	→	h^#(U101(e, e), U101(b, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(d, f(e))	A^#	→	h^#(U101(a, a), k)
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(U101(e, a), b)
h^#(a, U101(e, b))

Thus, the rule A^# → h^#(U101(e, a), U101(e, b)) is replaced by the following rules:

A^# → h^#(a, U101(e, b))

A^# → h^#(U101(e, a), b)

Problem 67: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(c, a), e)	A^#	→	h^#(a, k)
A^#	→	h^#(k, U101(e, b))	A^#	→	h^#(k, b)
A^#	→	h^#(e, k)	A^#	→	h^#(e, d)
A^#	→	h^#(d, f(k))	A^#	→	h^#(f(c), k)
A^#	→	h^#(f(d), b)	A^#	→	h^#(f(e), b)
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, a), U101(e, c))
A^#	→	h^#(U101(e, a), c)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(a, e)	A^#	→	h^#(U101(e, e), U101(b, b))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(a, a), U101(e, c))
A^#	→	h^#(d, f(e))	A^#	→	h^#(U101(a, a), k)
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(U101(c, a), U101(c, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(d, k)

A^# → h^#(c, k)

Problem 68: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(e, e), e)
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, e), k)
A^#	→	h^#(e, k)	A^#	→	h^#(e, d)
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), c)	A^#	→	h^#(f(d), U101(c, c))
A^#	→	h^#(d, f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(a, e)
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(a, a), k)	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(e, 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^#(U101(e, e), e) is replaced by the following rules:

A^# → h^#(e, e)

Problem 69: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(k, e)	A^#	→	h^#(d, U101(e, e))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(e, e), U101(b, b))	A^#	→	h^#(a, e)
A^#	→	h^#(U101(a, a), U101(e, c))	A^#	→	h^#(f(d), U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(a, a), k)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(k, 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^#(k, e) is deleted.

Problem 70: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, e)
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(k, c)
A^#	→	h^#(d, k)	A^#	→	h^#(d, d)
A^#	→	h^#(k, k)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(a, k)
A^#	→	h^#(U101(e, a), e)	A^#	→	h^#(f(e), f(e))
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(e, e), U101(b, b))
A^#	→	h^#(a, e)	A^#	→	h^#(f(d), U101(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(U101(a, a), k)
A^#	→	h^#(U101(c, a), U101(c, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, e)

Problem 71: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(f(e), k)
A^#	→	h^#(c, c)	A^#	→	h^#(e, U101(c, c))
A^#	→	h^#(d, d)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(f(e), e)
A^#	→	h^#(k, k)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(k), f(k))	g^#(d, x, x)	→	A^#
A^#	→	h^#(U101(c, a), U101(c, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(U101(a, a), k)	A^#	→	h^#(f(d), U101(e, c))
A^#	→	h^#(f(d), f(d))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(U101(e, a), d)	A^#	→	h^#(U101(e, e), c)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(U101(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
h^#(e, f(k))	h^#(U101(e, e), U101(k, k))

Thus, the rule A^# → h^#(U101(e, e), f(k)) is replaced by the following rules:

A^# → h^#(e, f(k))

Problem 72: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(k, c)	A^#	→	h^#(d, d)
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(e, U101(c, c))
A^#	→	h^#(a, c)	A^#	→	h^#(k, e)
A^#	→	h^#(c, U101(e, c))	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(a, k)
A^#	→	h^#(U101(e, a), U101(e, c))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(U101(e, a), c)	A^#	→	h^#(a, e)
g^#(d, x, x)	→	A^#	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(U101(a, a), k)	A^#	→	h^#(f(d), U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(e, U101(e, c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(k, e)

Problem 73: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, e)	A^#	→	h^#(U101(e, e), U101(e, e))
h^#(x, x)	→	g^#(x, x, f(k))	A^#	→	h^#(e, e)
A^#	→	h^#(c, c)	A^#	→	h^#(c, e)
A^#	→	h^#(d, d)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(k), f(k))	g^#(d, x, x)	→	A^#
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(d), f(d))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, a), d)
A^#	→	h^#(U101(a, a), k)	A^#	→	h^#(f(d), U101(e, c))
A^#	→	h^#(U101(e, a), k)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, e)

Problem 74: BackwardInstantiation

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, e), U101(e, e))	h^#(x, x)	→	g^#(x, x, f(k))
A^#	→	h^#(e, e)	A^#	→	h^#(c, c)
A^#	→	h^#(d, d)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(e), f(e))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Instantiation

For all potential predecessors l → r of the rule h^#(x, x) → g^#(x, x, f(k)) on dependency pair chains it holds that:

h#(x, x) matches r,
all variables of h#(x, x) are embedded in constructor contexts, i.e., each subterm of h#(x, x), containing a variable is rooted by a constructor symbol.

Thus, h^#(x, x) → g^#(x, x, f(k)) is replaced by instances determined through the above matching. These instances are:

h^#(f(d), f(d)) → g^#(f(d), f(d), f(k))	h^#(f(e), f(e)) → g^#(f(e), f(e), f(k))
h^#(c, c) → g^#(c, c, f(k))	h^#(U101(c, c), U101(c, c)) → g^#(U101(c, c), U101(c, c), f(k))
h^#(e, e) → g^#(e, e, f(k))	h^#(U101(e, c), U101(e, c)) → g^#(U101(e, c), U101(e, c), f(k))
h^#(k, k) → g^#(k, k, f(k))	h^#(f(k), f(k)) → g^#(f(k), f(k), f(k))
h^#(U101(e, e), U101(e, e)) → g^#(U101(e, e), U101(e, e), f(k))	h^#(f(c), f(c)) → g^#(f(c), f(c), f(k))
h^#(d, d) → g^#(d, d, f(k))

Problem 75: Propagation

Dependency Pair Problem

Dependency Pairs

h^#(c, c)	→	g^#(c, c, f(k))	h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))
A^#	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(e, e)
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(c, c), f(k))	h^#(e, e)	→	g^#(e, e, f(k))
A^#	→	h^#(c, c)	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	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^#(U101(c, c), U101(c, c))	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))
h^#(k, k)	→	g^#(k, k, f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(k), f(k))	g^#(d, x, x)	→	A^#
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

This is possible as

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

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(U101(e, e), U101(e, e)) are consolidated into the rule g^#(d, x, x) → h^#(U101(e, e), U101(e, e)) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(U101(e, e), U101(e, e))

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(U101(e, e), U101(e, e)) are consolidated into the rule g^#(d, x, x) → h^#(U101(e, e), U101(e, e)) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(U101(e, e), U101(e, e))

The dependency pairs g^#(d, x, x) → A^# and A^# → h^#(U101(e, e), U101(e, e)) are consolidated into the rule g^#(d, x, x) → h^#(U101(e, e), U101(e, e)) .

This is possible as

all subterms of A# containing variables are rooted by a constructor symbol,
there is no variable that is replacing in A#, but non-replacing in both g#(d, x, x) and h#(U101(e, e), U101(e, e))

Summary

Removed Dependency Pairs	Added Dependency Pairs
A^# → h^#(U101(e, e), U101(e, e))	A^# → g^#(c, c, f(k))
h^#(c, c) → g^#(c, c, f(k))	g^#(d, x, x) → h^#(U101(e, e), U101(e, e))
A^# → h^#(c, c)
g^#(d, x, x) → A^#

Problem 76: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(e), f(e))	→	g^#(f(e), f(e), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(e, e)	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(c, c), f(k))
h^#(e, e)	→	g^#(e, e, f(k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	A^#	→	g^#(c, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(k, k)	→	g^#(k, k, f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(f(e), f(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^#(f(e), U101(e, e), f(k))	g^#(f(e), f(e), U101(k, k))
g^#(U101(e, e), f(e), f(k))

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

h^#(f(e), f(e)) → g^#(f(e), U101(e, e), f(k))

h^#(f(e), f(e)) → g^#(U101(e, e), f(e), f(k))

Problem 77: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(e, e)
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(c, c), f(k))	h^#(e, e)	→	g^#(e, e, f(k))
h^#(f(e), f(e))	→	g^#(U101(e, e), f(e), f(k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
A^#	→	g^#(c, c, f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))
h^#(k, k)	→	g^#(k, k, f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(d), f(d))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U101(c, c), U101(c, c)) → g^#(U101(c, c), U101(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^#(U101(e, c), U101(c, c), f(k))	g^#(U101(c, c), U101(c, c), U101(k, k))
g^#(U101(c, c), U101(e, c), f(k))	g^#(U101(c, c), U101(k, c), f(k))
	g^#(U101(k, c), U101(c, c), f(k))

Thus, the rule h^#(U101(c, c), U101(c, c)) → g^#(U101(c, c), U101(c, c), f(k)) is replaced by the following rules:

h^#(U101(c, c), U101(c, c)) → g^#(U101(e, c), U101(c, c), f(k))

h^#(U101(c, c), U101(c, c)) → g^#(U101(c, c), U101(e, c), f(k))

Problem 78: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(e, e)
h^#(e, e)	→	g^#(e, e, f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), f(e), f(k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	g^#(c, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(k, k)	→	g^#(k, k, f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, 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, U101(k, k))

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

Problem 79: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(e, e)
h^#(f(e), f(e))	→	g^#(U101(e, e), f(e), f(k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
A^#	→	g^#(c, c, f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(k, k)	→	g^#(k, k, f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → h^#(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

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

Problem 80: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(e), f(e))	→	g^#(U101(e, e), f(e), f(k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	g^#(c, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	A^#	→	h^#(k, k)
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(U101(e, e), f(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^#(U101(e, e), U101(e, e), f(k))	g^#(U101(e, e), f(e), U101(k, k))
g^#(e, f(e), f(k))

Thus, the rule h^#(f(e), f(e)) → g^#(U101(e, e), f(e), f(k)) is replaced by the following rules:

h^#(f(e), f(e)) → g^#(e, f(e), f(k))

h^#(f(e), f(e)) → g^#(U101(e, e), U101(e, e), f(k))

Problem 81: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(e), f(e))	→	g^#(e, f(e), f(k))
A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	g^#(c, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(e, f(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, U101(e, e), f(k))	g^#(e, f(e), U101(k, k))

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

h^#(f(e), f(e)) → g^#(e, U101(e, e), f(k))

Problem 82: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
h^#(f(e), f(e))	→	g^#(e, U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	g^#(c, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(e), f(e)) → g^#(e, U101(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(k))	g^#(e, U101(e, e), U101(k, k))

Thus, the rule h^#(f(e), f(e)) → g^#(e, U101(e, e), f(k)) is replaced by the following rules:

h^#(f(e), f(e)) → g^#(e, e, f(k))

Problem 83: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
h^#(f(e), f(e))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	g^#(c, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(k, k)
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, c, A, a, 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, U101(k, k))

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

Problem 84: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(c), f(c), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
A^#	→	g^#(c, c, f(k))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(k, k)
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(k), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

Termination of terms over the following signature is verified: f, g, d, e, b, A, c, a, 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(k), f(k))	g^#(f(c), f(c), U101(k, k))
g^#(f(c), f(e), f(k))
g^#(f(k), f(c), f(k))
g^#(f(c), U101(c, c), f(k))
g^#(U101(c, c), f(c), f(k))
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^#(f(e), f(c), f(k))	h^#(f(c), f(c)) → g^#(f(c), U101(c, c), f(k))
h^#(f(c), f(c)) → g^#(f(c), f(e), f(k))	h^#(f(c), f(c)) → g^#(f(k), f(c), f(k))
h^#(f(c), f(c)) → g^#(U101(c, c), f(c), f(k))	h^#(f(c), f(c)) → g^#(f(c), f(k), f(k))

Problem 85: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	h^#(f(c), f(c))	→	g^#(f(k), f(c), f(k))
A^#	→	g^#(c, c, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(k, k)
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(k), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))
g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(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(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(k), 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(k), f(k), f(k))	g^#(f(k), f(c), U101(k, k))
g^#(f(k), f(e), f(k))	g^#(U101(k, k), f(c), f(k))
g^#(f(k), U101(c, c), f(k))

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

h^#(f(c), f(c)) → g^#(f(k), U101(c, c), f(k))	h^#(f(c), f(c)) → g^#(f(k), f(e), f(k))
h^#(f(c), f(c)) → g^#(f(k), f(k), f(k))

Problem 86: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	A^#	→	g^#(c, c, f(k))
A^#	→	h^#(k, k)	h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(f(c), f(c))	→	g^#(f(k), U101(c, c), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	A^#	→	h^#(f(k), f(k))
h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(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(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → 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^#(k, c, f(k))	g^#(c, c, U101(k, k))
g^#(c, e, f(k))
g^#(e, c, f(k))
g^#(c, k, f(k))

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

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

Problem 87: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(e, c, f(k))	A^#	→	h^#(d, d)
h^#(f(c), f(c))	→	g^#(f(k), U101(c, c), f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(k, k)
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(k), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))
A^#	→	g^#(c, k, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

Relevant Terms	Irrelevant Terms
g^#(e, k, f(k))	g^#(e, c, U101(k, k))
g^#(e, e, f(k))

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

A^# → g^#(e, e, f(k))

A^# → g^#(e, k, f(k))

Problem 88: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	g^#(e, e, f(k))	A^#	→	h^#(d, d)
A^#	→	g^#(e, k, f(k))	A^#	→	h^#(k, k)
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(f(c), f(c))	→	g^#(f(k), U101(c, c), f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
A^#	→	h^#(f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))
g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))	A^#	→	g^#(c, k, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
A^#	→	g^#(k, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → 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, U101(k, k))

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

Problem 89: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	A^#	→	g^#(e, k, f(k))
h^#(f(c), f(c))	→	g^#(f(k), U101(c, c), f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(k, k)
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(k), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))
A^#	→	g^#(c, k, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))
h^#(k, k)	→	g^#(k, k, f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule A^# → g^#(e, k, 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, k, U101(k, k))

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

Problem 90: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	A^#	→	h^#(k, k)
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(f(c), f(c))	→	g^#(f(k), U101(c, c), f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
A^#	→	h^#(f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))
g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))	A^#	→	g^#(c, k, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
A^#	→	g^#(k, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(f(k), U101(c, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(f(k), U101(e, c), f(k))	g^#(U101(k, k), U101(c, c), f(k))
	g^#(f(k), U101(k, c), f(k))
	g^#(f(k), U101(c, c), U101(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(f(k), U101(c, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(f(k), U101(e, c), f(k))

Problem 91: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(k, k)
h^#(f(c), f(c))	→	g^#(f(k), U101(e, c), f(k))	h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))
h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(k), f(k))
h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))
g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))
h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))	A^#	→	g^#(c, k, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	h^#(k, k)	→	g^#(k, k, f(k))
A^#	→	g^#(k, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U101(e, c), U101(e, c)) → g^#(U101(e, c), U101(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^#(U101(e, c), c, f(k))	g^#(U101(e, c), U101(e, c), U101(k, k))
g^#(c, U101(e, c), f(k))

Thus, the rule h^#(U101(e, c), U101(e, c)) → g^#(U101(e, c), U101(e, c), f(k)) is replaced by the following rules:

h^#(U101(e, c), U101(e, c)) → g^#(U101(e, c), c, f(k))

h^#(U101(e, c), U101(e, c)) → g^#(c, U101(e, c), f(k))

Problem 92: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	h^#(U101(e, c), U101(e, c))	→	g^#(c, U101(e, c), f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(k), U101(e, c), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	A^#	→	h^#(f(k), f(k))
h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))
A^#	→	g^#(c, k, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))
h^#(k, k)	→	g^#(k, k, f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U101(e, c), U101(e, c)) → g^#(c, U101(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, U101(e, c), f(k))	g^#(c, U101(e, c), U101(k, k))
g^#(k, U101(e, c), f(k))
g^#(c, c, f(k))

Thus, the rule h^#(U101(e, c), U101(e, c)) → g^#(c, U101(e, c), f(k)) is replaced by the following rules:

h^#(U101(e, c), U101(e, c)) → g^#(e, U101(e, c), f(k))	h^#(U101(e, c), U101(e, c)) → g^#(c, c, f(k))
h^#(U101(e, c), U101(e, c)) → g^#(k, U101(e, c), f(k))

Problem 93: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	h^#(U101(e, c), U101(e, c))	→	g^#(e, U101(e, c), f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	A^#	→	h^#(k, k)
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(k), U101(e, c), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))
A^#	→	h^#(f(k), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))
A^#	→	g^#(c, k, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))
h^#(k, k)	→	g^#(k, k, f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))	A^#	→	h^#(f(e), f(e))
h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U101(e, c), U101(e, c)) → g^#(e, U101(e, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, c, f(k))	g^#(e, U101(e, c), U101(k, k))

Thus, the rule h^#(U101(e, c), U101(e, c)) → g^#(e, U101(e, c), f(k)) is replaced by the following rules:

h^#(U101(e, c), U101(e, c)) → g^#(e, c, f(k))

Problem 94: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	h^#(U101(e, c), U101(e, c))	→	g^#(e, c, f(k))
A^#	→	h^#(k, k)	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(f(k), U101(e, c), f(k))
h^#(f(k), f(k))	→	g^#(f(k), f(k), f(k))	A^#	→	h^#(f(k), f(k))
h^#(f(e), f(e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(k), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), U101(e, c), f(k))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), f(c), f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))
A^#	→	g^#(c, k, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))
h^#(d, d)	→	g^#(d, d, f(k))	A^#	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))
h^#(k, k)	→	g^#(k, k, f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	A^#	→	h^#(f(e), f(e))
h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U101(e, c), U101(e, c)) → g^#(e, c, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, k, f(k))	g^#(e, c, U101(k, k))
g^#(e, e, f(k))

Thus, the rule h^#(U101(e, c), U101(e, c)) → g^#(e, c, f(k)) is replaced by the following rules:

h^#(U101(e, c), U101(e, c)) → g^#(e, e, f(k))

h^#(U101(e, c), U101(e, c)) → g^#(e, k, f(k))

Problem 95: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(e), c, f(k))	h^#(f(c), f(c))	→	g^#(e, U101(e, c), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(U101(e, c), k, f(k))	h^#(f(c), f(c))	→	g^#(f(k), f(e), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(c, c), c, f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(e, k, f(k))
h^#(f(c), f(c))	→	g^#(f(c), f(e), f(k))	A^#	→	g^#(c, k, f(k))
h^#(f(c), f(c))	→	g^#(e, k, f(k))	h^#(d, d)	→	g^#(d, d, f(k))
A^#	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(k), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(k, e, f(k))	h^#(k, k)	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(c), f(c))	→	g^#(f(k), e, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(e, c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(e, U101(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U101(e, e), f(c), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))	A^#	→	h^#(f(d), f(d))
h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(e, U101(e, c), f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(e, c, f(k))	g^#(e, U101(e, c), U101(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(e, U101(e, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, c, f(k))

Problem 96: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(d, d)	A^#	→	h^#(U101(e, c), U101(e, c))
A^#	→	h^#(k, k)	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
h^#(f(c), f(c))	→	g^#(f(c), f(k), f(k))	A^#	→	h^#(f(c), f(c))
A^#	→	g^#(c, e, f(k))	h^#(f(e), f(e))	→	g^#(f(e), U101(e, e), f(k))
h^#(f(c), f(c))	→	g^#(f(e), f(e), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(d), f(d))	→	g^#(f(d), f(d), f(k))	h^#(f(c), f(c))	→	g^#(f(e), f(k), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U101(e, e), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(k, e, f(k))
h^#(k, k)	→	g^#(k, k, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), U101(c, c), f(k))
h^#(f(c), f(c))	→	g^#(U101(e, c), f(e), f(k))	h^#(f(c), f(c))	→	g^#(f(k), e, f(k))
A^#	→	g^#(k, c, f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(e, c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(e, U101(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U101(e, e), f(c), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))	A^#	→	h^#(f(d), f(d))
h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, d)

A^# → h^#(d, k)

Problem 97: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(e, c), U101(e, c))	g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))
A^#	→	h^#(f(c), f(c))	h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(c, c), f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(f(c), f(c))	→	g^#(U101(e, c), f(e), f(k))
h^#(f(c), f(c))	→	g^#(f(k), e, f(k))	A^#	→	g^#(k, c, f(k))
h^#(f(d), f(d))	→	g^#(f(d), f(k), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(c, c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(e, c), f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(e, U101(e, c), f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U101(e, e), f(c), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(U101(c, c), U101(c, c)) → g^#(k, U101(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^#(k, U101(e, c), f(k))	g^#(k, U101(c, c), U101(k, k))
	g^#(k, U101(k, c), f(k))

Thus, the rule h^#(U101(c, c), U101(c, c)) → g^#(k, U101(c, c), f(k)) is replaced by the following rules:

h^#(U101(c, c), U101(c, c)) → g^#(k, U101(e, c), f(k))

Problem 98: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, c, f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), e, f(k))
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(f(c), f(c))	→	g^#(f(k), U101(e, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U101(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(e, U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(f(c), U101(e, c), f(k))
h^#(f(c), f(c))	→	g^#(f(e), U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(k, e, f(k))
g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U101(e, c), e, f(k))	h^#(f(c), f(c))	→	g^#(U101(e, c), k, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), k, f(k))	h^#(f(c), f(c))	→	g^#(e, k, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(k, k, f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(U101(c, c), U101(c, c))	→	g^#(e, U101(e, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(e, c), f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U101(e, e), f(c), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

Relevant Terms	Irrelevant Terms
g^#(k, c, f(k))	g^#(c, c, U101(k, k))
g^#(c, e, f(k))
g^#(e, c, f(k))
g^#(c, k, f(k))

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

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

Problem 99: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(k, c, f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(f(c), f(c))	→	g^#(c, k, f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), c, f(k))
h^#(f(c), f(c))	→	g^#(c, U101(e, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(k))
g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(U101(e, e), c, f(k))	h^#(f(c), f(c))	→	g^#(f(e), c, f(k))
h^#(f(c), f(c))	→	g^#(e, U101(e, c), f(k))	h^#(f(c), f(c))	→	g^#(U101(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(f(c), e, f(k))	h^#(f(c), f(c))	→	g^#(U101(e, c), k, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), k, f(k))	h^#(f(c), f(c))	→	g^#(e, k, f(k))
A^#	→	h^#(f(c), f(c))	h^#(f(c), f(c))	→	g^#(k, k, f(k))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(f(c), f(c))	→	g^#(f(k), e, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(e, U101(e, c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(e, c), f(k))
h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))
A^#	→	h^#(f(e), f(e))	h^#(f(c), f(c))	→	g^#(U101(e, e), f(c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(k, 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^#(k, e, f(k))	g^#(k, c, U101(k, k))
g^#(k, k, f(k))

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

h^#(f(c), f(c)) → g^#(k, k, f(k))

h^#(f(c), f(c)) → g^#(k, e, f(k))

Problem 100: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

g^#(d, x, x)	→	h^#(U101(e, e), U101(e, e))	h^#(f(c), f(c))	→	g^#(U101(c, c), k, f(k))
h^#(f(c), f(c))	→	g^#(e, k, f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(k, k, f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
A^#	→	h^#(U101(e, c), U101(e, c))	h^#(f(c), f(c))	→	g^#(c, k, f(k))
h^#(f(c), f(c))	→	g^#(f(k), e, f(k))	h^#(U101(e, e), U101(e, e))	→	g^#(U101(e, e), U101(e, e), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(e, U101(e, c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(k, U101(e, c), f(k))
h^#(U101(e, c), U101(e, c))	→	g^#(c, c, f(k))	A^#	→	h^#(f(e), f(e))
h^#(f(c), f(c))	→	g^#(U101(e, e), f(c), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U101(c, c), k, 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^#(U101(e, c), k, f(k))	g^#(U101(c, c), k, U101(k, k))
	g^#(U101(k, c), k, f(k))

Thus, the rule h^#(f(c), f(c)) → g^#(U101(c, c), k, f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(U101(e, c), k, f(k))

Problem 101: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(U101(e, e), f(k), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U101(e, e), U101(c, c), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(c), f(c))	→	g^#(U101(e, e), f(e), f(k))	A^#	→	h^#(U101(e, c), U101(e, c))
h^#(f(c), f(c))	→	g^#(e, f(c), f(k))	h^#(U101(e, c), U101(e, c))	→	g^#(k, U101(e, c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))	A^#	→	h^#(f(d), f(d))
h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))
h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U101(e, e), f(k), 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, f(k), f(k))	g^#(U101(e, e), U101(k, k), f(k))
	g^#(U101(e, e), f(k), U101(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U101(e, e), f(k), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(e, f(k), f(k))

Problem 102: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(k, c)	A^#	→	h^#(f(c), f(c))
A^#	→	h^#(U101(c, c), U101(c, c))	h^#(U101(c, c), U101(c, c))	→	g^#(k, k, f(k))
A^#	→	h^#(f(d), f(d))	h^#(U101(c, c), U101(c, c))	→	g^#(c, c, f(k))
A^#	→	h^#(e, U101(e, c))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), f(c), f(k))	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(k, e)

Problem 103: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(k, U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, f(k), f(k))
h^#(f(c), f(c))	→	g^#(k, f(k), f(k))	h^#(f(c), f(c))	→	g^#(k, f(e), f(k))
h^#(f(c), f(c))	→	g^#(U101(e, c), U101(e, e), f(k))	A^#	→	h^#(f(c), f(c))
h^#(f(c), f(c))	→	g^#(U101(e, c), U101(e, c), f(k))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(c), f(c))	→	g^#(U101(e, c), f(k), f(k))	h^#(f(c), f(c))	→	g^#(U101(e, c), U101(c, c), f(k))
h^#(f(c), f(c))	→	g^#(c, U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(U101(e, 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^#(f(c), f(c))	→	g^#(U101(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(k, e, f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(k, U101(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^#(k, U101(e, c), f(k))	g^#(k, U101(c, c), U101(k, k))
	g^#(k, U101(k, c), f(k))

Thus, the rule h^#(f(c), f(c)) → g^#(k, U101(c, c), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(k, U101(e, c), f(k))

Problem 104: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

h^#(f(c), f(c))	→	g^#(c, U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(k, f(e), f(k))
A^#	→	h^#(f(c), f(c))	A^#	→	h^#(U101(c, c), U101(c, c))
h^#(f(c), f(c))	→	g^#(U101(c, c), c, f(k))	h^#(f(c), f(c))	→	g^#(e, f(c), f(k))
h^#(f(c), f(c))	→	g^#(e, U101(c, c), f(k))	h^#(f(c), f(c))	→	g^#(e, c, f(k))
h^#(f(c), f(c))	→	g^#(c, U101(e, c), f(k))	h^#(f(c), f(c))	→	g^#(k, e, f(k))
h^#(f(c), f(c))	→	g^#(e, e, f(k))	h^#(f(c), f(c))	→	g^#(e, f(e), f(k))
h^#(f(c), f(c))	→	g^#(U101(c, c), U101(e, e), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))
h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(c, U101(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^#(c, e, f(k))	g^#(c, U101(e, e), U101(k, k))
g^#(e, U101(e, e), f(k))
g^#(k, U101(e, e), f(k))

Thus, the rule h^#(f(c), f(c)) → g^#(c, U101(e, e), f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, e, f(k))	h^#(f(c), f(c)) → g^#(k, U101(e, e), f(k))
h^#(f(c), f(c)) → g^#(e, U101(e, e), f(k))

Problem 105: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(U101(c, c), U101(c, c))	h^#(f(c), f(c))	→	g^#(U101(e, c), e, f(k))
h^#(f(c), f(c))	→	g^#(k, U101(e, e), f(k))	h^#(f(c), f(c))	→	g^#(k, e, f(k))
h^#(f(c), f(c))	→	g^#(e, k, f(k))	h^#(f(c), f(c))	→	g^#(e, e, f(k))
h^#(f(c), f(c))	→	g^#(e, f(e), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(c, e, f(k))
h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))	h^#(U101(c, c), U101(c, c))	→	g^#(U101(e, c), c, f(k))
A^#	→	h^#(f(c), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

The right-hand side of the rule h^#(f(c), f(c)) → g^#(U101(e, c), e, f(k)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
g^#(c, e, f(k))	g^#(U101(e, c), e, U101(k, k))

Thus, the rule h^#(f(c), f(c)) → g^#(U101(e, c), e, f(k)) is replaced by the following rules:

h^#(f(c), f(c)) → g^#(c, e, f(k))

Problem 106: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, k)	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(k, e)	A^#	→	h^#(c, U101(e, c))
A^#	→	h^#(c, e)	A^#	→	h^#(U101(e, c), U101(c, c))
A^#	→	h^#(c, k)	h^#(f(c), f(c))	→	g^#(e, U101(e, e), f(k))
A^#	→	h^#(f(c), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

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

Problem 107: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#

→

h^#(c, k)

A^#

→

h^#(f(c), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

Relevant Terms	Irrelevant Terms
h^#(U101(c, c), f(c))
h^#(f(c), U101(c, c))
h^#(f(c), f(k))
h^#(f(e), f(c))
h^#(f(k), f(c))
h^#(f(c), f(e))

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

A^# → h^#(U101(c, c), f(c))	A^# → h^#(f(c), f(e))
A^# → h^#(f(c), U101(c, c))	A^# → h^#(f(k), f(c))
A^# → h^#(f(c), f(k))	A^# → h^#(f(e), f(c))

Problem 108: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(k, f(e))	A^#	→	h^#(f(k), k)
A^#	→	h^#(k, k)	A^#	→	h^#(c, f(k))
A^#	→	h^#(e, k)	A^#	→	h^#(f(c), U101(e, c))
A^#	→	h^#(f(e), U101(c, c))	A^#	→	h^#(f(k), e)
A^#	→	h^#(c, U101(c, c))	A^#	→	h^#(k, f(k))
A^#	→	h^#(k, U101(e, e))	A^#	→	h^#(f(k), f(e))
A^#	→	h^#(U101(e, e), f(k))	A^#	→	h^#(U101(c, c), U101(e, c))
A^#	→	h^#(U101(e, c), U101(c, c))	A^#	→	h^#(f(c), U101(e, e))
A^#	→	h^#(f(k), f(k))	A^#	→	h^#(f(e), f(e))
A^#	→	h^#(k, U101(e, c))	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(f(e), f(c))	A^#	→	h^#(c, U101(e, e))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms
h^#(k, U101(e, e))

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

A^# → h^#(k, U101(e, e))

Problem 109: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(k, c)	A^#	→	h^#(U101(e, c), k)
A^#	→	h^#(c, e)	A^#	→	h^#(c, k)
A^#	→	h^#(f(c), e)	A^#	→	h^#(U101(c, c), e)
A^#	→	h^#(f(e), e)	A^#	→	h^#(U101(e, e), k)
A^#	→	h^#(f(k), U101(e, e))	A^#	→	h^#(k, k)
A^#	→	h^#(f(k), k)	A^#	→	h^#(e, k)
A^#	→	h^#(f(c), k)	A^#	→	h^#(U101(c, c), U101(e, c))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, c), U101(c, c))
A^#	→	h^#(f(c), U101(e, e))	A^#	→	h^#(k, U101(e, c))
A^#	→	h^#(U101(e, e), c)	A^#	→	h^#(e, U101(e, c))
A^#	→	h^#(c, U101(e, e))	A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(k, k)

A^# → h^#(k, e)

Problem 110: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(k, e)	A^#	→	h^#(c, k)
A^#	→	h^#(k, k)	A^#	→	h^#(e, k)
A^#	→	h^#(U101(c, c), c)	A^#	→	h^#(f(c), U101(e, e))
A^#	→	h^#(f(e), f(e))	A^#	→	h^#(U101(e, c), U101(c, c))
A^#	→	h^#(k, U101(e, c))	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(U101(e, e), c)
A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

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

Problem 111: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, e)	A^#	→	h^#(f(e), U101(e, e))
A^#	→	h^#(U101(e, c), U101(c, c))	A^#	→	h^#(e, f(e))
A^#	→	h^#(k, U101(e, c))	A^#	→	h^#(e, U101(e, e))
A^#	→	h^#(U101(e, e), c)	A^#	→	h^#(c, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

Relevant Terms	Irrelevant Terms

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

Problem 112: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, c)	A^#	→	h^#(e, e)
A^#	→	h^#(U101(e, e), k)	A^#	→	h^#(e, U101(e, e))
A^#	→	h^#(e, U101(e, c))	A^#	→	h^#(f(e), f(c))
A^#	→	h^#(c, k)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(e, e)

A^# → h^#(e, k)

Problem 113: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

A^#	→	h^#(e, c)		A^#	→	h^#(e, e)
A^#	→	h^#(U101(e, e), k)		A^#	→	h^#(c, k)

Rewrite Rules

a	→	c	b	→	c
a	→	d	b	→	d
c	→	e	c	→	k
d	→	k	A	→	h(f(a), f(b))
h(x, x)	→	g(x, x, f(k))	g(d, x, x)	→	A
f(x)	→	U101(x, x)	U101(e, x)	→	x

Original Signature

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

Strategy

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

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

A^# → h^#(e, e)

A^# → h^#(e, k)

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