TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60000 ms.

The following DP Processors were used


Problem 1 was processed with processor ForwardNarrowing (2ms).
 | – Problem 2 was processed with processor ForwardNarrowing (1ms).
 |    | – Problem 3 was processed with processor ForwardNarrowing (4ms).
 |    |    | – Problem 4 was processed with processor ForwardNarrowing (3ms).
 |    |    |    | – Problem 5 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    | – Problem 6 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (14ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (14ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (12ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (81ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (218ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (211ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (604ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 remains open; application of the following processors failed [ForwardNarrowing (406ms), ForwardNarrowing (376ms), ForwardNarrowing (318ms), ForwardNarrowing (332ms), ForwardNarrowing (353ms), ForwardNarrowing (347ms), ForwardNarrowing (370ms), ForwardNarrowing (366ms), ForwardNarrowing (299ms), ForwardNarrowing (313ms), ForwardNarrowing (182ms), ForwardNarrowing (359ms), ForwardNarrowing (584ms), ForwardNarrowing (370ms), ForwardNarrowing (597ms), ForwardNarrowing (320ms), ForwardNarrowing (534ms), ForwardNarrowing (482ms), ForwardNarrowing (704ms), ForwardNarrowing (466ms), ForwardNarrowing (563ms), ForwardNarrowing (371ms), ForwardNarrowing (592ms), ForwardNarrowing (228ms), ForwardNarrowing (713ms), ForwardNarrowing (261ms), ForwardNarrowing (481ms), ForwardNarrowing (226ms), ForwardNarrowing (565ms), ForwardNarrowing (487ms)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, x), y)f#(a, f(f(x, h(y)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h


Problem 1: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, x), y)f#(a, f(f(x, h(y)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, x), y) → f#(a, f(f(x, h(y)), a)) is narrowed to the 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#(a, f(f(a, f(f(_x42, h(h(y))), a)), a)) 
f#(a, f(a, f(f(h(y), h(a)), a))) 
Thus, the rule f#(f(a, x), y) → f#(a, f(f(x, h(y)), a)) is replaced by the following rules:
f#(f(a, a), y) → f#(a, f(a, f(f(h(y), h(a)), a)))f#(f(a, f(a, _x42)), y) → f#(a, f(f(a, f(f(_x42, h(h(y))), a)), a))

Problem 2: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, a), y)f#(a, f(a, f(f(h(y), h(a)), a)))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, _x42)), y)f#(a, f(f(a, f(f(_x42, h(h(y))), a)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, a), y) → f#(a, f(a, f(f(h(y), h(a)), a))) is narrowed to the 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#(f(a, a), y) → f#(a, f(a, f(f(h(y), h(a)), a))) is deleted.

Problem 3: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, _x42)), y)f#(a, f(f(a, f(f(_x42, h(h(y))), a)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, _x42)), y) → f#(a, f(f(a, f(f(_x42, h(h(y))), a)), a)) is narrowed to the 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#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a))) 
f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a)) 
f#(a, f(f(a, f(a, f(f(h(h(y)), h(a)), a))), a)) 
Thus, the rule f#(f(a, f(a, _x42)), y) → f#(a, f(f(a, f(f(_x42, h(h(y))), a)), a)) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(f(a, f(a, f(f(h(h(y)), h(a)), a))), a))f#(f(a, f(a, _x42)), y) → f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y) → f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))

Problem 4: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, a)), y)f#(a, f(f(a, f(a, f(f(h(h(y)), h(a)), a))), a))f#(f(a, x), y)f#(x, h(y))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, _x42)), y)f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, a)), y) → f#(a, f(f(a, f(a, f(f(h(h(y)), h(a)), a))), a)) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a))) 
Thus, the rule f#(f(a, f(a, a)), y) → f#(a, f(f(a, f(a, f(f(h(h(y)), h(a)), a))), a)) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a)))

Problem 5: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, a)), y)f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a)))f#(f(a, f(a, _x42)), y)f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a))) 
Thus, the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a)))

Problem 6: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, a)), y)f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, _x42)), y)f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a))) is narrowed to the 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#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a)))) 
Thus, the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a))) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Problem 7: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, _x42)), y)f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, _x42)), y) → f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, _x42)), y) → f#(a, f(a, f(f(f(f(_x42, h(h(y))), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y) → f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a)))

Problem 8: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, a)), y)f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a))) 
Thus, the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(f(a, f(f(h(h(y)), h(a)), a)), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a)))

Problem 9: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, a)), y)f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a))) is narrowed to the 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#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a)))) 
Thus, the rule f#(f(a, f(a, a)), y) → f#(a, f(a, f(f(a, f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a)), a))) is replaced by the following rules:
f#(f(a, f(a, a)), y) → f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, _x82))), y) → f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a)), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, _x82))), y) → f#(a, f(a, f(f(f(f(a, f(f(_x82, h(h(h(y)))), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, _x122)))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, a))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a)), a), h(a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, _x122)))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, _x122)))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, _x122)))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, a)))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, _x122)))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a)), a), h(a)), a)))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a))), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a))), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a))), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a))), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, x), y)f#(x, h(y))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, a))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, a))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(y))), h(a)), a))), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, a))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a)))

Problem 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, x), y)f#(x, h(y))f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a))), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a)), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a), h(a)), a)), h(a)), a)))

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, x), y)f#(x, h(y))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a)), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a), h(a)), a))), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a)), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a)), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a)), a)), a), h(a)), a)))

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a)), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a))), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a), h(a)), a))), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x262, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a)), a)), a)), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a), h(a)), a))), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x262))))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x262, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a)), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a)), a)), a)), a), h(a)), a)))

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, a)), y)f#(a, f(a, f(a, f(f(f(f(f(f(h(h(y)), h(a)), a), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(a, f(f(f(a, f(f(f(f(_x82, h(h(h(y)))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, a)))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(y)))), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, x), y)f#(f(x, h(y)), a)f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x262))))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x262, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a)), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, _x82))), y)f#(a, f(f(a, f(f(a, f(f(_x82, h(h(h(y)))), a)), a)), a))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x202)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a), h(a)), a))), a)), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a)), a)), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a)), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a), h(a)), a))), a)), a), h(a)), a)))

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a)), h(a)), a))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x252, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x252, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x192, h(h(h(h(h(h(y))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a)), a)), a)), a), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x122, h(h(h(h(y))))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x252, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a)), a)), a), h(a)), a)))
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f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(y))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a)), h(a)), a))), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a), h(a)), a))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x142, h(h(h(h(h(y)))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x282))))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x282, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a)), h(a)), a))), a)), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, a))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(y))), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a)))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, a))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(y))))), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a), h(h(a))), a)), a))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x282))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x282, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a))), a)), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x182, h(h(h(h(h(h(y))))))), a)), a), h(a)), a))), a)), a), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x222, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a), h(a)), a))), a), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x222))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x222, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))f#(f(a, x), y)f#(x, h(y))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(a, f(f(f(f(f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(h(y)))))))))), h(a)), a))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y)f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x352)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x352, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))
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f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, _x122)))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x122, h(h(h(h(y))))), a)), a), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292)))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a)), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x422)))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(_x422, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x252, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x442)))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(_x442, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x182)))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x182, h(h(h(h(h(h(y))))))), a), h(a)), a))), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x392)))))))))), y)f#(a, f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x392, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x192, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(_x252, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a)), a)), a)), a), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x262)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x262, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a)))f#(f(a, x), y)f#(f(x, h(y)), a)
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a), h(h(a))), a)), a))), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x282))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(_x282, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a))), a)), a)), a)), a), h(a)), a)), h(a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) is narrowed to the 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#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) 
f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) 
f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) 
f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a)))) 
f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) 
f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) 
f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a))) 
f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x302))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x302, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a))), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, a)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(y)))))), h(a)), a))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a)), h(a)), a))), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a), h(h(a))), a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x352)))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x352, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x252, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292)))))))), y)f#(a, f(a, f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, _x142))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x142, h(h(h(h(h(y)))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a), h(a)), a))), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(a, f(f(f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(f(f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a), h(a)), a)), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x352)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x352, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(a, f(f(f(f(f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x212))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x212, h(h(h(h(h(h(h(y)))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, _x212)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x212, h(h(h(h(h(h(y))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(y))))))), h(a)), a))), a), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
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f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x192, h(h(h(h(h(h(y))))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a)), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x202, h(h(h(h(h(h(h(y)))))))), a)), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292)))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402))))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(h(y)))))))))))), a)), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a), h(h(h(a)))), a)), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x392)))))))))), y)f#(a, f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x392, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a)), a), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(f(a, f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a)), a)), a), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a))), a)), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x252)))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x252, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a), h(h(a))), a)), a))), a)), a)), a)), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a), h(h(h(a)))), a)), a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a))) is narrowed to the 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#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a))), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a))) 
f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)), h(a)), a)), h(a)), a))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a))), a)), a), h(a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a))), a), h(a)), a)))

Problem 24: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, x), y)f#(x, h(y))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(a, f(f(f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(f(f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a), h(a)), a)), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(a, f(f(f(f(f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a)), a)))
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f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(y)))))))), h(a)), a))), a)), a)), a), h(a)), a))), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, _x192)))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(_x192, h(h(h(h(h(h(y))))))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292)))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(f(a, f(f(_x292, h(h(h(h(h(h(h(h(y))))))))), a)), a), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a)), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a))), a)), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y)f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x292))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(f(_x292, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(f(f(a, f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(a, f(a, f(f(f(f(f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a), h(h(a))), a), h(a)), a)))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a)), h(a)), a)), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x372)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x372, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a))), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x202))))))), y)f#(a, f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(a, f(f(a, f(a, f(f(f(f(_x202, h(h(h(h(h(h(h(y)))))))), a), h(a)), a))), a)), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a)), h(a)), a)), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332)))))))))), y)f#(a, f(a, f(f(f(f(a, f(a, f(f(f(f(a, f(a, f(a, f(f(f(f(f(f(f(f(a, f(f(a, f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a)), a), h(a)), a), h(h(a))), a), h(a)), a)))), a), h(a)), a))), a), h(a)), a)))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x402)))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(f(a, f(f(f(f(a, f(a, f(f(f(f(_x402, h(h(h(h(h(h(h(h(h(h(y))))))))))), a), h(a)), a))), a), h(a)), a)), h(a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a)))))))))), y)f#(a, f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(h(y)))))))))), h(a)), a))), a)), a)), a), h(a)), a), h(h(a))), a)), a))), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x242)))))))), y)f#(a, f(a, f(f(f(f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(a, f(f(f(f(f(f(_x242, h(h(h(h(h(h(h(h(y))))))))), a), h(a)), a), h(h(a))), a)), a))), a)), a)), a)), a), h(a)), a)))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y)f#(a, f(a, f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(f(f(f(f(a, f(f(f(a, f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a))), h(a)), a)), h(a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a), h(h(h(a)))), a)), a)), a)))

Rewrite Rules

f(f(a, x), y)f(a, f(f(x, h(y)), a))

Original Signature

Termination of terms over the following signature is verified: f, a, h

Strategy


The right-hand side of the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) is narrowed to the 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#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(_x452, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(a, f(f(f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a), h(a)), a)), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(a, f(f(f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) 
Thus, the rule f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a)))) is replaced by the following rules:
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(a, f(f(f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a), h(a)), a)), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(a, f(f(f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a), h(a)), a)), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(a, f(f(f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(a, f(f(f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a), h(a)), a))), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x332))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(a, f(f(f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(_x332, h(h(h(h(h(h(h(h(h(y)))))))))), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a)), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))
f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x452)))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(f(a, f(f(_x452, h(h(h(h(h(h(h(h(h(h(y))))))))))), a)), a)), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))f#(f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, a))))))))), y) → f#(a, f(a, f(a, f(f(f(f(a, f(f(f(f(f(f(f(f(a, f(f(f(f(a, f(f(f(f(f(a, f(f(a, f(f(f(f(f(f(a, f(a, f(f(h(h(h(h(h(h(h(h(h(y))))))))), h(a)), a))), a), h(a)), a), h(h(a))), a)), a)), h(a)), a), h(h(a))), a)), a), h(h(a))), a)), a), h(h(a))), a), h(a)), a), h(h(h(a)))), a)), a), h(a)), a))))