TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60001 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (44ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 | – Problem 4 was processed with processor ForwardNarrowing (2ms).
 |    | – Problem 5 was processed with processor ForwardNarrowing (2ms).
 |    |    | – Problem 6 was processed with processor ForwardNarrowing (2ms).
 |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (9ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (68ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (56ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (35ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (48ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (45ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 was processed with processor ForwardNarrowing (35ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 31 was processed with processor ForwardNarrowing (51ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 32 was processed with processor ForwardNarrowing (48ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 33 was processed with processor ForwardNarrowing (149ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 34 was processed with processor ForwardNarrowing (205ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 35 was processed with processor ForwardNarrowing (23ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 36 was processed with processor ForwardNarrowing (191ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 37 remains open; application of the following processors failed [ForwardNarrowing (130ms), ForwardNarrowing (130ms), ForwardNarrowing (206ms), ForwardNarrowing (213ms), ForwardNarrowing (312ms), ForwardNarrowing (timeout)].

The following open problems remain:

Open Dependency Pair Problem 4

Dependency Pairs

f#(s(x), y) → f#(half(s(x)), double(y))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

g#(d, s(x)) → g#(d, x)g#(h, s(s(x))) → g#(h, x)
double#(x) → g#(d, x)f#(s(x), y) → f#(half(s(x)), double(y))
id#(x) → f#(x, s(0))half#(x) → g#(h, x)
f#(s(x), y) → half#(s(x))f#(s(x), y) → double#(y)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The following SCCs where found

g#(d, s(x)) → g#(d, x)
g#(h, s(s(x))) → g#(h, x)
f#(s(x), y) → f#(half(s(x)), double(y))

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

g#(h, s(s(x))) → g#(h, x)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

g#(h, s(s(x))) → g#(h, x)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

g#(d, s(x)) → g#(d, x)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

g#(d, s(x)) → g#(d, x)

Problem 4: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), y) → f#(half(s(x)), double(y))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), y) → f#(half(s(x)), double(y)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(half(s(x)), g(d, _x31)) 
f#(g(h, s(x)), double(y)) 
Thus, the rule f#(s(x), y) → f#(half(s(x)), double(y)) is replaced by the following rules:
f#(s(x), _x31) → f#(half(s(x)), g(d, _x31))f#(s(x), y) → f#(g(h, s(x)), double(y))

Problem 5: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), _x31) → f#(half(s(x)), g(d, _x31))f#(s(x), y) → f#(g(h, s(x)), double(y))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), y) → f#(g(h, s(x)), double(y)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(g(h, _x21)), double(y)) 
f#(g(h, s(x)), g(d, _x31)) 
f#(0, double(y)) 
Thus, the rule f#(s(x), y) → f#(g(h, s(x)), double(y)) is replaced by the following rules:
f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))
f#(s(0), y) → f#(0, double(y))

Problem 6: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), _x31) → f#(half(s(x)), g(d, _x31))f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(0), y) → f#(0, double(y))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), _x31) → f#(half(s(x)), g(d, _x31)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(g(h, s(x)), g(d, _x31)) 
f#(half(s(x)), s(s(g(d, _x41)))) 
f#(half(s(x)), 0) 
Thus, the rule f#(s(x), _x31) → f#(half(s(x)), g(d, _x31)) is replaced by the following rules:
f#(s(x), s(_x41)) → f#(half(s(x)), s(s(g(d, _x41))))f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))
f#(s(x), 0) → f#(half(s(x)), 0)

Problem 7: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(x), s(_x41)) → f#(half(s(x)), s(s(g(d, _x41))))
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(x), 0) → f#(half(s(x)), 0)
f#(s(0), y) → f#(0, double(y))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(_x41)) → f#(half(s(x)), s(s(g(d, _x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(half(s(x)), s(s(s(s(g(d, _x61)))))) 
f#(half(s(x)), s(s(0))) 
f#(g(h, s(x)), s(s(g(d, _x41)))) 
Thus, the rule f#(s(x), s(_x41)) → f#(half(s(x)), s(s(g(d, _x41)))) is replaced by the following rules:
f#(s(x), s(s(_x61))) → f#(half(s(x)), s(s(s(s(g(d, _x61))))))f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))

Problem 8: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), s(s(_x61))) → f#(half(s(x)), s(s(s(s(g(d, _x61))))))f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))
f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))f#(s(x), 0) → f#(half(s(x)), 0)
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(_x61))) → f#(half(s(x)), s(s(s(s(g(d, _x61)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(half(s(x)), s(s(s(s(s(s(g(d, _x81)))))))) 
f#(half(s(x)), s(s(s(s(0))))) 
f#(g(h, s(x)), s(s(s(s(g(d, _x61)))))) 
Thus, the rule f#(s(x), s(s(_x61))) → f#(half(s(x)), s(s(s(s(g(d, _x61)))))) is replaced by the following rules:
f#(s(x), s(s(_x61))) → f#(g(h, s(x)), s(s(s(s(g(d, _x61))))))f#(s(x), s(s(0))) → f#(half(s(x)), s(s(s(s(0)))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))

Problem 9: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), s(s(_x61))) → f#(g(h, s(x)), s(s(s(s(g(d, _x61))))))f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))
f#(s(x), s(s(0))) → f#(half(s(x)), s(s(s(s(0)))))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))
f#(s(x), 0) → f#(half(s(x)), 0)f#(s(0), y) → f#(0, double(y))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(_x61))) → f#(g(h, s(x)), s(s(s(s(g(d, _x61)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(g(h, _x21)), s(s(s(s(g(d, _x61)))))) 
f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81)))))))) 
f#(g(h, s(x)), s(s(s(s(0))))) 
f#(0, s(s(s(s(g(d, _x61)))))) 
Thus, the rule f#(s(x), s(s(_x61))) → f#(g(h, s(x)), s(s(s(s(g(d, _x61)))))) is replaced by the following rules:
f#(s(0), s(s(_x61))) → f#(0, s(s(s(s(g(d, _x61))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0)))))f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(0), s(s(_x61))) → f#(0, s(s(s(s(g(d, _x61))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(x), s(s(0))) → f#(half(s(x)), s(s(s(s(0)))))
f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0)))))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))f#(s(x), 0) → f#(half(s(x)), 0)
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(_x61))) → f#(0, s(s(s(s(g(d, _x61)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(0))))) 
f#(0, s(s(s(s(s(s(g(d, _x81)))))))) 
Thus, the rule f#(s(0), s(s(_x61))) → f#(0, s(s(s(s(g(d, _x61)))))) is replaced by the following rules:
f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))

Problem 11: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(x), s(s(0))) → f#(half(s(x)), s(s(s(s(0)))))
f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0)))))f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(x), 0) → f#(half(s(x)), 0)
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))
f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(0))) → f#(half(s(x)), s(s(s(s(0))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(g(h, s(x)), s(s(s(s(0))))) 
Thus, the rule f#(s(x), s(s(0))) → f#(half(s(x)), s(s(s(s(0))))) is replaced by the following rules:
f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0)))))

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0)))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))
f#(s(x), 0) → f#(half(s(x)), 0)f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))
f#(s(0), y) → f#(0, double(y))f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(0))))) 
f#(s(g(h, _x21)), s(s(s(s(0))))) 
Thus, the rule f#(s(x), s(s(0))) → f#(g(h, s(x)), s(s(s(s(0))))) is replaced by the following rules:
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))

Problem 13: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(x), 0) → f#(half(s(x)), 0)
f#(s(0), y) → f#(0, double(y))f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81)))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(0))))))) 
f#(0, s(s(s(s(s(s(s(s(g(d, _x101)))))))))) 
Thus, the rule f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81)))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))

Problem 14: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(x), s(0)) → f#(half(s(x)), s(s(0)))f#(s(x), 0) → f#(half(s(x)), 0)
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))
f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(0)) → f#(half(s(x)), s(s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(g(h, s(x)), s(s(0))) 
Thus, the rule f#(s(x), s(0)) → f#(half(s(x)), s(s(0))) is replaced by the following rules:
f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))

Problem 15: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(x), 0) → f#(half(s(x)), 0)f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))
f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))
f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), 0) → f#(half(s(x)), 0) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(g(h, s(x)), 0) 
Thus, the rule f#(s(x), 0) → f#(half(s(x)), 0) is replaced by the following rules:
f#(s(x), 0) → f#(g(h, s(x)), 0)

Problem 16: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), 0) → f#(g(h, s(x)), 0)f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))
f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))
f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), 0) → f#(g(h, s(x)), 0) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(g(h, _x21)), 0) 
f#(0, 0) 
Thus, the rule f#(s(x), 0) → f#(g(h, s(x)), 0) is replaced by the following rules:
f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)f#(s(0), 0) → f#(0, 0)

Problem 17: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31))f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))
f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, g(d, _x31)) 
f#(s(g(h, _x21)), g(d, _x31)) 
f#(g(h, s(x)), s(s(g(d, _x41)))) 
f#(g(h, s(x)), 0) 
Thus, the rule f#(s(x), _x31) → f#(g(h, s(x)), g(d, _x31)) is replaced by the following rules:
f#(s(0), _x31) → f#(0, g(d, _x31))f#(s(x), 0) → f#(g(h, s(x)), 0)
f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))

Problem 18: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(x), 0) → f#(g(h, s(x)), 0)f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))
f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)f#(s(0), _x31) → f#(0, g(d, _x31))
f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), 0) → f#(g(h, s(x)), 0) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(g(h, _x21)), 0) 
f#(0, 0) 
Thus, the rule f#(s(x), 0) → f#(g(h, s(x)), 0) is replaced by the following rules:
f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)f#(s(0), 0) → f#(0, 0)

Problem 19: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))
f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)
f#(s(0), _x31) → f#(0, g(d, _x31))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))
f#(s(0), y) → f#(0, double(y))f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(0), double(y)) 
f#(s(g(h, _x21)), g(d, _x31)) 
f#(s(s(g(h, _x41))), double(y)) 
Thus, the rule f#(s(s(_x21)), y) → f#(s(g(h, _x21)), double(y)) is replaced by the following rules:
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(_x41)))), y) → f#(s(s(g(h, _x41))), double(y))
f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))

Problem 20: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(_x41)))), y) → f#(s(s(g(h, _x41))), double(y))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0)))))))f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))
f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)
f#(s(0), _x31) → f#(0, g(d, _x31))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))
f#(s(0), y) → f#(0, double(y))f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule f#(s(0), s(s(s(0)))) → f#(0, s(s(s(s(s(s(0))))))) is deleted.

Problem 21: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(_x41)))), y) → f#(s(s(g(h, _x41))), double(y))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)f#(s(0), _x31) → f#(0, g(d, _x31))
f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(g(h, _x21)), s(s(0))) 
f#(0, s(s(0))) 
Thus, the rule f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0))) is replaced by the following rules:
f#(s(0), s(0)) → f#(0, s(s(0)))f#(s(s(_x21)), s(0)) → f#(s(g(h, _x21)), s(s(0)))

Problem 22: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(_x41)))), y) → f#(s(s(g(h, _x41))), double(y))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(_x21)), s(0)) → f#(s(g(h, _x21)), s(s(0)))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))
f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)f#(s(0), _x31) → f#(0, g(d, _x31))
f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))f#(s(0), s(0)) → f#(0, s(s(0)))
f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))
f#(s(0), y) → f#(0, double(y))f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(0))) → f#(0, s(s(s(s(0))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule f#(s(0), s(s(0))) → f#(0, s(s(s(s(0))))) is deleted.

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(_x41)))), y) → f#(s(s(g(h, _x41))), double(y))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(_x21)), s(0)) → f#(s(g(h, _x21)), s(s(0)))f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), 0) → f#(s(g(h, _x21)), 0)
f#(s(0), _x31) → f#(0, g(d, _x31))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(0), s(0)) → f#(0, s(s(0)))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), _x31) → f#(s(g(h, _x21)), g(d, _x31))f#(s(0), y) → f#(0, double(y))
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: id, f, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(g(d, _x41)))) 
f#(s(g(h, _x21)), s(s(g(d, _x41)))) 
f#(g(h, s(x)), s(s(0))) 
f#(g(h, s(x)), s(s(s(s(g(d, _x61)))))) 
Thus, the rule f#(s(x), s(_x41)) → f#(g(h, s(x)), s(s(g(d, _x41)))) is replaced by the following rules:
f#(s(x), s(s(_x61))) → f#(g(h, s(x)), s(s(s(s(g(d, _x61))))))f#(s(0), s(_x41)) → f#(0, s(s(g(d, _x41))))
f#(s(x), s(0)) → f#(g(h, s(x)), s(s(0)))f#(s(s(_x21)), s(_x41)) → f#(s(g(h, _x21)), s(s(g(d, _x41))))

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(0))), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), _x31) → f#(s(s(s(s(s(s(g(h, _x81))))))), g(d, _x31))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(_x51)))))), _x31) → f#(s(s(s(g(h, _x51)))), g(d, _x31))f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(_x61)))))))), y) → f#(s(s(s(s(g(h, _x61))))), double(y))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(_x21)), s(_x41)) → f#(s(g(h, _x21)), s(s(g(d, _x41))))f#(s(0), s(s(0))) → f#(0, s(s(s(s(0)))))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(0), s(s(s(_x81)))) → f#(0, s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(0), s(0)) → f#(0, s(s(0)))f#(s(x), s(s(s(_x81)))) → f#(half(s(x)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(0)), 0) → f#(s(0), 0)
f#(s(x), s(s(s(_x81)))) → f#(g(h, s(x)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(0), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(_x51)))))), s(_x41)) → f#(s(s(s(g(h, _x51)))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(g(h, _x41))), 0)
f#(s(s(s(s(s(s(s(s(_x61)))))))), 0) → f#(s(s(s(s(g(h, _x61))))), 0)f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), 0) → f#(s(s(s(s(s(g(h, _x71)))))), 0)f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(0), s(_x41)) → f#(0, s(s(g(d, _x41))))
f#(s(0), y) → f#(0, double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(0), 0) → f#(0, 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(s(s(s(s(s(_x51)))))), _x31) → f#(s(s(s(g(h, _x51)))), g(d, _x31)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(s(s(g(h, _x51)))), 0) 
f#(s(s(s(g(h, _x51)))), s(s(g(d, _x41)))) 
f#(s(s(s(s(g(h, _x61))))), g(d, _x31)) 
f#(s(s(s(0))), g(d, _x31)) 
Thus, the rule f#(s(s(s(s(s(s(_x51)))))), _x31) → f#(s(s(s(g(h, _x51)))), g(d, _x31)) is replaced by the following rules:
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(_x61)))))))), _x31) → f#(s(s(s(s(g(h, _x61))))), g(d, _x31))f#(s(s(s(s(s(s(_x51)))))), s(_x41)) → f#(s(s(s(g(h, _x51)))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), 0) → f#(s(s(s(g(h, _x51)))), 0)

Problem 25: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(0)), y) → f#(s(0), double(y))f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))
f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(0)), 0) → f#(s(0), 0)
f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)
f#(s(x), s(s(s(0)))) → f#(g(h, s(x)), s(s(s(s(s(s(0)))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(_x51)))))), s(_x41)) → f#(s(s(s(g(h, _x51)))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(g(h, _x41))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(_x61)))))))), 0) → f#(s(s(s(s(g(h, _x61))))), 0)f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))f#(s(0), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(x), s(s(s(0)))) → f#(half(s(x)), s(s(s(s(s(s(0)))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), 0) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), 0)f#(s(0), s(s(s(s(_x101))))) → f#(0, s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), 0) → f#(s(s(s(s(s(g(h, _x71)))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(0)))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), g(d, _x31))f#(s(0), s(_x41)) → f#(0, s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(x), s(s(s(s(_x101))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221)))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241)))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221)))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))

Problem 26: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(0)), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(s(0))), 0) → f#(s(0), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(_x51)))))), s(_x41)) → f#(s(s(s(g(h, _x51)))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(0), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), 0) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), 0) → f#(s(s(s(s(s(g(h, _x71)))))), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(g(h, _x81))))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(0)))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), g(d, _x31))f#(s(0), s(_x41)) → f#(0, s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x451))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(x), s(s(s(s(_x101))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351)))))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371)))))))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351)))))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))

Problem 27: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))
f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(_x61))) → f#(s(g(h, _x21)), s(s(s(s(g(d, _x61))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(0))), 0) → f#(s(0), 0)f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(_x51)))))), s(_x41)) → f#(s(s(s(g(h, _x51)))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x451))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(x), s(s(s(s(_x101))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))f#(s(0), s(s(s(s(s(s(s(s(0))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(0), y) → f#(0, double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)
f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))f#(s(0), 0) → f#(0, 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331)))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351)))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331)))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))

Problem 28: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(0))), y) → f#(s(0), double(y))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(0), s(s(s(s(s(s(s(s(0))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(0), s(s(s(s(s(s(0))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351)))))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371)))))))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351)))))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))

Problem 29: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(s(0))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))f#(s(0), s(s(s(s(s(s(s(s(0))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(x), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(0))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))
f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(x), s(s(s(s(s(s(s(_x161)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)
f#(s(0), y) → f#(0, double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)
f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))f#(s(0), 0) → f#(0, 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) is deleted.

Problem 30: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))f#(s(s(0)), 0) → f#(s(0), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(x), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(0))), y) → f#(s(0), double(y))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(0))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))
f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(x), s(s(s(s(s(s(s(_x161)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261)))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281)))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261)))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))

Problem 31: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))f#(s(s(0)), 0) → f#(s(0), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(0))), y) → f#(s(0), double(y))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(0))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))
f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(x), s(s(s(s(s(s(s(_x161)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) 
Thus, the rule f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))

Problem 32: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))
f#(s(s(0)), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))f#(s(s(0)), 0) → f#(s(0), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(x), s(s(s(s(s(s(0))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(x), s(s(s(s(s(s(s(_x161)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411)))))))))))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411)))))))))))))))))))))))))))))))))))))))) 
f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411)))))))))))))))))))))))))))))))))))))))) 
f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x431)))))))))))))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411)))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x431))))))))))))))))))))))))))))))))))))))))))

Problem 33: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))
f#(s(s(0)), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), 0) → f#(s(0), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x451))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x431))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x471))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x471))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(x), s(s(s(s(s(s(0))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(x), s(s(s(s(s(s(s(_x161)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x491)))))))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x491))))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)
f#(s(0), y) → f#(0, double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)
f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))f#(s(0), 0) → f#(0, 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371)))))))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391)))))))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371)))))))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))

Problem 34: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))
f#(s(s(0)), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x511))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x511))))))))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(0))), y) → f#(s(0), double(y))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x451))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)
f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x431))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x471))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x471))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(x), s(s(s(s(s(s(0))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(x), s(s(s(s(s(s(s(_x161)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x491)))))))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x491))))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)
f#(s(0), y) → f#(0, double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)
f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))f#(s(0), 0) → f#(0, 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x511))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x511)))))))))))))))))))))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531)))))))))))))))))))))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x511))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x511)))))))))))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x531)))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x531))))))))))))))))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))

Problem 35: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(s(_x181))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(0)), y) → f#(s(0), double(y))
f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))
f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(0), s(s(s(s(s(s(s(_x161)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(s(s(0))), 0) → f#(s(0), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(_x161)))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x451))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(0))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x431))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x471))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x471))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x811)))))))))))))))))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x811))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))
f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x491)))))))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x491))))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))
f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)
f#(s(0), y) → f#(0, double(y))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)
f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))f#(s(0), 0) → f#(0, 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(x), s(s(s(s(s(s(s(s(_x181))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181)))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181)))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181)))))))))))))))))) 
f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201)))))))))))))))))))) 
f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) 
Thus, the rule f#(s(x), s(s(s(s(s(s(s(s(_x181))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181)))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(_x181))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(0))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))

Problem 36: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), g(d, _x31))
f#(s(s(0)), y) → f#(s(0), double(y))f#(s(s(s(s(s(s(s(s(0)))))))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(0)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))
f#(s(s(s(0))), s(_x41)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x91)))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(g(h, _x91)))))))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(0))))))), y) → f#(s(s(s(0))), double(y))f#(s(s(_x21)), s(s(s(_x81)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(g(d, _x81))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(_x261))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x261))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)
f#(s(s(0)), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x101)))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(g(h, _x101))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), 0) → f#(s(s(s(s(0)))), 0)f#(s(s(s(s(s(0))))), 0) → f#(s(s(0)), 0)
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(0))))))))), y) → f#(s(s(s(s(0)))), double(y))f#(s(s(s(s(s(s(_x51)))))), s(0)) → f#(s(s(s(g(h, _x51)))), s(s(0)))
f#(s(s(0)), 0) → f#(s(0), 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(0)))))), 0)
f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(0))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(0))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(_x71)))))))))), s(_x41)) → f#(s(s(s(s(s(g(h, _x71)))))), s(s(g(d, _x41))))
f#(s(s(s(s(s(s(_x51)))))), s(s(_x61))) → f#(s(s(s(g(h, _x51)))), s(s(s(s(g(d, _x61))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))
f#(s(s(s(0))), 0) → f#(s(0), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x351))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371))))))))))))))))))))))))))))))))))))f#(s(s(s(0))), _x31) → f#(s(0), g(d, _x31))
f#(s(s(s(s(s(0))))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0) → f#(s(s(s(s(s(0))))), 0)
f#(s(s(0)), s(h)) → f#(s(0), s(s(g(d, _x41))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(0))), 0)
f#(s(s(_x21)), s(s(s(s(_x101))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(g(d, _x101))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(s(g(h, _x131)))))))))))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x201)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x201))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), 0) → f#(s(s(s(0))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(0)))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(_x181))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x181))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), s(s(g(d, _x41))))
f#(s(s(s(s(d)))), 0) → f#(s(s(0)), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(_x161)))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x161))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(0))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(0))))))))), g(d, _x31))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))
f#(s(s(0)), _x31) → f#(s(0), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(0))), y) → f#(s(0), double(y))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)
f#(s(s(s(s(s(s(0)))))), y) → f#(s(s(s(0))), double(y))f#(s(0), s(s(s(s(s(s(s(s(0))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(_x221))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x221))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x451))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(_x121)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(g(d, _x121))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(_x21)), s(s(s(0)))) → f#(s(g(h, _x21)), s(s(s(s(s(s(0)))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), _x31) → f#(s(s(s(s(s(s(0)))))), g(d, _x31))f#(s(s(0)), s(0)) → f#(s(0), s(s(0)))
f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), _x31) → f#(s(s(s(s(s(0))))), g(d, _x31))
f#(s(s(_x21)), s(s(s(s(s(s(0))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))f#(s(s(_x21)), s(s(s(s(s(s(s(s(0))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))
f#(s(s(_x21)), s(s(s(s(0))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(0)))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(0)))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(0))))))))), _x31) → f#(s(s(s(s(0)))), g(d, _x31))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(s(g(h, _x141))))))))))))), 0)f#(s(s(s(s(s(s(s(s(0)))))))), y) → f#(s(s(s(s(0)))), double(y))
f#(s(s(s(s(0)))), _x31) → f#(s(s(0)), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x301))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(0)))))))), 0)f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x431))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(0))))))), _x31) → f#(s(s(s(0))), g(d, _x31))f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(0)))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x471))))))))))))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x471))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(_x21)), s(s(s(s(s(0)))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(0)))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(s(_x61))) → f#(s(s(g(h, _x41))), s(s(s(s(g(d, _x61))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), 0) → f#(s(s(s(s(s(s(s(0))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), y) → f#(s(s(s(s(s(0))))), double(y))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x281))))))))))))))))))))))))))))f#(s(s(s(s(s(s(0)))))), 0) → f#(s(s(s(0))), 0)
f#(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))), 0) → f#(s(s(s(s(s(s(0)))))), 0)f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x411))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(0)))))))))), 0)f#(s(0), s(s(s(s(s(s(s(0)))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x811)))))))))))))))))))))))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x811))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(0)))))), _x31) → f#(s(s(s(0))), g(d, _x31))
f#(s(s(s(0))), s(s(_x61))) → f#(s(0), s(s(s(s(g(d, _x61))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), g(d, _x31))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x121)))))))))))))))))))), s(_x41)) → f#(s(s(s(s(s(s(s(s(s(s(g(h, _x121))))))))))), s(s(g(d, _x41))))f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), _x31) → f#(s(s(s(s(s(s(s(0))))))), g(d, _x31))
f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(s(s(s(s(s(0)))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(s(_x21)), s(s(0))) → f#(s(g(h, _x21)), s(s(s(s(0)))))f#(s(s(s(s(s(0))))), y) → f#(s(s(0)), double(y))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x331))))))))))))))))))))))))))))))))f#(s(s(s(s(s(s(s(0))))))), s(_x41)) → f#(s(s(s(0))), s(s(g(d, _x41))))
f#(s(0), s(s(s(s(s(0)))))) → f#(0, s(s(s(s(s(s(s(s(s(s(0)))))))))))f#(s(s(s(s(_x41)))), s(_x41)) → f#(s(s(g(h, _x41))), s(s(g(d, _x41))))
f#(s(x), s(s(s(s(s(s(s(0)))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), s(_x41)) → f#(s(s(s(s(s(s(g(h, _x81))))))), s(s(g(d, _x41))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x241))))))))))))))))))))))))f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(0)))), 0)
f#(s(s(s(s(s(s(s(s(_x61)))))))), s(_x41)) → f#(s(s(s(s(g(h, _x61))))), s(s(g(d, _x41))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))
f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → f#(half(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))f#(s(s(s(s(_x41)))), s(0)) → f#(s(s(g(h, _x41))), s(s(0)))
f#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → f#(s(g(h, _x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))f#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x491)))))))))))))))))))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x491))))))))))))))))))))))))))))))))))))))))))))))))
f#(s(s(s(s(s(s(s(s(s(s(s(s(_x81)))))))))))), y) → f#(s(s(s(s(s(s(g(h, _x81))))))), double(y))f#(s(s(s(s(0)))), y) → f#(s(s(0)), double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x111)))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)f#(s(0), y) → f#(0, double(y))
f#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))), 0) → f#(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), 0)f#(s(x), s(s(s(s(s(s(_x141))))))) → f#(g(h, s(x)), s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x141))))))))))))))
f#(s(0), 0) → f#(0, 0)f#(s(s(s(s(d)))), 0) → f#(s(s(s(s(s(s(s(s(s(g(h, _x111)))))))))), 0)

Rewrite Rules

g(x, 0) → 0g(d, s(x)) → s(s(g(d, x)))
g(h, s(0)) → 0g(h, s(s(x))) → s(g(h, x))
double(x) → g(d, x)half(x) → g(h, x)
f(s(x), y) → f(half(s(x)), double(y))f(s(0), y) → y
id(x) → f(x, s(0))

Original Signature

Termination of terms over the following signature is verified: f, id, g, d, 0, s, half, double, h

Strategy

The right-hand side of the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371)))))))))))))))))))))))))))))))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391)))))))))))))))))))))))))))))))))))))) 
f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) 
Thus, the rule f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x371)))))))))))))))))))))))))))))))))))) is replaced by the following rules:
f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391))))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(g(d, _x391))))))))))))))))))))))))))))))))))))))f#(s(0), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → f#(0, s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))