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 (1ms).
 | – Problem 2 was processed with processor ForwardNarrowing (1ms).
 |    | – Problem 3 was processed with processor ForwardNarrowing (2ms).
 |    |    | – Problem 4 was processed with processor ForwardNarrowing (6ms).
 |    |    |    | – Problem 5 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    | – Problem 6 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (13ms).
 |    |    |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (12ms).
 |    |    |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (12ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (16ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (18ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (14ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (56ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (24ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (19ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (58ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (65ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (325ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (699ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 remains open; application of the following processors failed [ForwardNarrowing (383ms), ForwardNarrowing (705ms), ForwardNarrowing (529ms), ForwardNarrowing (633ms), ForwardNarrowing (551ms), ForwardNarrowing (617ms), ForwardNarrowing (625ms), ForwardNarrowing (719ms), ForwardNarrowing (633ms), ForwardNarrowing (789ms), ForwardNarrowing (623ms), ForwardNarrowing (948ms), ForwardNarrowing (577ms), ForwardNarrowing (805ms), ForwardNarrowing (854ms), ForwardNarrowing (1101ms), ForwardNarrowing (855ms), ForwardNarrowing (1057ms), ForwardNarrowing (641ms), ForwardNarrowing (571ms), ForwardNarrowing (472ms), ForwardNarrowing (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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


Problem 1: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy


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

Problem 2: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, y)) → f#(a, f(f(f(a, a), y), x)) is narrowed to the 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, a), _x31), f(f(a, a), y)))) 
f#(a, f(f(a, f(f(f(a, a), _x41), f(a, a))), x)) 
Thus, the rule f#(x, f(a, y)) → f#(a, f(f(f(a, a), y), x)) is replaced by the following rules:
f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(f(f(a, a), _x41), f(a, a))), x))f#(f(a, _x31), f(a, y)) → f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))

Problem 3: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(x, f(a, f(a, _x41)))f#(a, f(f(a, f(f(f(a, a), _x41), f(a, a))), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, y))f#(f(a, a), y)

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(f(f(a, a), _x41), f(a, a))), x)) is narrowed to the 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, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a)))))) 
f#(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x)) 
Thus, the rule f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(f(f(a, a), _x41), f(a, a))), x)) is replaced by the following rules:
f#(f(a, _x31), f(a, f(a, _x41))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))), x))
f#(x, f(a, f(a, f(a, _x81)))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))

Problem 4: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(x, f(a, f(a, _x41)))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))), x))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))), x)) is narrowed to the 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(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41))))))) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))), x)) is replaced by the following rules:
f#(f(a, _x31), f(a, f(a, _x41))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)), x))
f#(x, f(a, f(a, f(a, _x111)))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))

Problem 5: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, _x41)))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a))))) 
Thus, the rule f#(x, f(a, f(a, _x41))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)), x)) is replaced by the following rules:
f#(f(a, _x31), f(a, f(a, _x41))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)), x))

Problem 6: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)), x))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121)))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))

Problem 7: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))

Problem 8: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)), x)) is narrowed to the 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(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x161))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121)))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))

Problem 9: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)), x)) is narrowed to the 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(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a))))) 
Thus, the rule f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121)))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x121)))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)), x)) is narrowed to the 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(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a))))) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x211)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, y))f#(f(a, a), y)

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x)) is narrowed to the 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(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x)) is narrowed to the 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(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x231)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) is narrowed to the 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(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x231)))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x)) is narrowed to the 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(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a))))) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x271)))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x)) is narrowed to the 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(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)), x))

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, _x111)))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x)) is narrowed to the 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(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x131), f(a, a))), f(a, a)))))), x)) 
f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111))))))), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))))))) 
f#(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a))))), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, _x111)))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, _x111)))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111))))))), x))f#(x, f(a, f(a, f(a, _x111)))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x111)))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))f#(x, f(a, f(a, f(a, _x111)))) → f#(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a))))), x))
f#(x, f(a, f(a, f(a, f(a, _x131))))) → f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x131), f(a, a))), f(a, a)))))), x))

Problem 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a))))), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111))))))), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, f(a, f(a, f(a, _x131)))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x131), f(a, a))), f(a, a)))))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(a, a)))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))))), a)))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x231))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))))), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)))))

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a))))), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)))))
f#(x, f(a, y))f#(f(a, a), y)f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111))))))), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x231)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))))), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, f(a, f(a, f(a, _x131)))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x131), f(a, a))), f(a, a)))))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181))))))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a))))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a)))))), f(a, a))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))), f(f(a, a), a)))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a)))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181))))))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))), f(f(a, a), a)))), a)), x))

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, y))f#(f(f(a, a), y), x)f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a))))), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181))))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)))))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111))))))), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x231)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a))))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a)))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))))), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))f#(x, f(a, f(a, f(a, f(a, _x131)))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x131), f(a, a))), f(a, a)))))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))))), a)), f(a, a))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a)))), a)), f(a, a))))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)), x)) is replaced by the following rules:
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))))), a)), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181)))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a)))), a)), f(a, a))))), a)), x))

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x121), f(a, a))))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a)))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)), f(a, a))))), a)), x))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a))))), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x141), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x191)))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x41)))))))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x41))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x121)))))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))))), a)), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x141)))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x121))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181))))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a)))))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)))))f#(x, f(a, y))f#(f(a, a), y)
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, _x121))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x121), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111))))))), x))f#(x, f(a, f(a, f(a, f(a, _x141)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, _x41)))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), _x41), f(a, a))))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x231)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))))), a)), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a))))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a)))), a)), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a)))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a)))))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a))))), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, _x211))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x211), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
f#(x, f(a, f(a, f(a, f(a, _x131)))))f#(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x131), f(a, a))), f(a, a)))))), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x)) is narrowed to the 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(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a))))) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a)))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x231)))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x231), f(a, a))), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)))))
f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191))))))), f(f(a, a), a))), f(a, a))), a)), x))

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))))), a)), f(a, a))))), a)))))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x191))), a))))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x181), f(a, a))), f(a, a)))))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x371)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x371), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x241)))), f(a, a))))), a)), f(f(a, a), a)))), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x331), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x361), f(a, a))), f(f(a, a), a)))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361))))))), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x361)))), f(f(a, a), a)))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x431)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x431), f(a, a))))))))), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x331)))), f(a, a))), f(a, a))))), a)), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))))))))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x191))), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361))))))), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a)))))), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x111)))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x201))))))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)))), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x361)))), f(f(a, a), a)))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x191), f(a, a))), f(f(a, a), a))))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x201), f(a, a))), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x231))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x231), f(a, a))))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, _x81))))f#(a, f(f(a, f(f(a, f(f(f(a, a), _x81), f(a, a))), f(a, a))), x))
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f#(x, f(a, f(a, f(a, f(a, f(a, _x271))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x271), f(a, a))), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)), x))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a)))))), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361))))))), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x141)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x141), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x311))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), _x311), f(a, a))), f(a, a))), f(a, a))))), a)), f(f(a, a), a))))))), a)), x))f#(f(a, _x31), f(a, y))f#(a, f(a, f(f(f(a, a), _x31), f(f(a, a), y))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x181), f(a, a))), f(f(a, a), a)))), a)), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x161)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x161), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x361))), a)))), a)), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, _x111))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x111), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a)))))), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a)), x)) is narrowed to the 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, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a))))), a)), f(a, a))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a)))))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a))))))))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x271), f(a, a))), f(f(a, a), a))))), f(a, a))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x291), f(a, a))), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x271))))))), f(a, a))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a)))))), f(a, a))))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a)), x)) is replaced by the following rules:
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x271))))))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x291))))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x291), f(a, a))), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a))))), a)), f(a, a))), f(a, a))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a))))))))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a)))))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x271), f(a, a))), f(f(a, a), a))))), f(a, a))), f(a, a))), f(a, a))))), a)), x))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))))), a))))))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x121))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x121)))), f(f(a, a), a)))), a)))))
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))f#(x, f(a, y))f#(f(f(a, a), y), x)
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))f#(f(a, _x31), f(a, f(a, f(a, _x111))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x111), f(a, a)))))))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x331)))), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))
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f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))), f(f(a, a), a)))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x451), f(a, a))), f(f(a, a), a))))))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))))), a)), f(a, a))))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x361), f(a, a))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a)))), a)))), a)), f(f(a, a), a))), a)))), a)), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a)))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a)))))), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))), f(a, a))))), a)))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) is narrowed to the 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, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a))))) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x531), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x461)))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x461), f(a, a)))))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a)))))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a)))))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x)) is replaced by the following rules:
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x461)))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a)))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a)))))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x461), f(a, a)))))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a)))))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461))))))))) → f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x531)))))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), _x531), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))))), a))))))), a)))))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x361), f(a, a))), f(f(a, a), a)))))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(a, f(f(f(a, a), f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a))), f(a, a))), a)))))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x331)))), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x361)))), f(f(a, a), a))), f(a, a))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)))))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))), f(f(a, a), a)))), a))))))), a)))))
f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331)))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x331), f(a, a))), f(a, a))))), a)), f(a, a))), f(a, a)))))))), a)))))
f#(f(a, _x41), f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(a, f(f(f(a, a), _x41), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201)))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))))), f(a, a))), f(f(a, a), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))))), a)), f(f(a, a), a))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x361), f(a, a))), f(f(a, a), a)))))), a)))), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x361), f(a, a))))), a)), f(a, a))))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x401)))))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x401), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x381))))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x381), f(a, a))), f(a, a)))))))), a)), f(a, a))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331)))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x331), f(a, a))))))))), f(a, a))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x371)))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x371), f(a, a)))))), f(f(a, a), a))))), f(a, a))), f(a, a)))))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x391))))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(f(a, a), _x391), f(a, a))), f(a, a))), f(a, a))), f(a, a))), f(a, a))))), a)), f(f(a, a), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x471)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x471), f(a, a)))))))))))))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))), a)), f(f(a, a), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461)))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a)))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))
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f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x191)))))), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), _x361), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))))), a)), f(f(a, a), a))), a)))), a)), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x451))))))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x331)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), _x331), f(a, a))), f(a, a)))))), f(a, a))), f(a, a)))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a)))))))))), a)))), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x271))), a))))), f(a, a))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x271))), f(a, a))), f(f(a, a), a)))), a))))), f(f(a, a), a))), a))), f(f(a, a), a))), a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))), f(a, a)))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
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f#(f(a, _x31), f(a, f(a, f(a, f(a, _x191)))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a))))), f(a, a))), f(f(a, a), a)))), a)))))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x361)))), f(f(a, a), a))))))), a)))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)), x))
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f#(x, f(a, f(a, f(a, f(a, _x191)))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x191), f(a, a)))))))), f(f(a, a), a)))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461)))))))))f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a)), f(a, a))), f(f(a, a), a)))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x241)))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x241), f(a, a))), f(a, a))), f(f(a, a), a)))), a))))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x181), f(a, a))))), a))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x451), f(a, a))), f(f(a, a), a))))))), a)), f(a, a))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x371)))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x371), f(a, a)))))), f(f(a, a), a))))), f(a, a))), f(a, a))), f(f(a, a), a))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x271), f(a, a)))))), f(a, a))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x431)))))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(a, f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x431), f(a, a)))))), f(f(a, a), a)))), a)), f(a, a))), f(f(a, a), a))))), f(f(a, a), a))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x361))), f(a, a))), f(f(a, a), a))), f(a, a))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a))), f(a, a))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x271))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(f(a, a), _x271)))), f(f(a, a), a))))), f(a, a))))), a))))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x361))), a)))), a)), f(a, a))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x361)))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, _x181))))))f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), _x181)))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), _x451), f(a, a)))))), f(f(a, a), a)))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x451)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), _x451), f(a, a))), f(f(a, a), a))))))), a))))))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x471)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), _x471), f(a, a))), f(a, a))), f(f(a, a), a)))))), a))))), f(f(a, a), a)))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x361))))))))f#(a, f(f(a, f(f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), _x361), f(a, a))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a)))), a)))), a)), f(f(a, a), a))), a)))), a)), f(f(a, a), a))), f(a, a))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x461)))))))))f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(f(a, a), f(f(a, f(f(f(a, a), f(f(f(a, a), f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(f(a, a), _x461), f(a, a))), f(a, a))), f(f(a, a), a)))), a))))), f(f(a, a), a))), f(a, a))), a)), f(f(a, a), a))), a))), f(f(a, a), a)))), a)), x))

Rewrite Rules

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

Original Signature

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

Strategy


The right-hand side of the rule f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))))), a)), x)) is narrowed to the 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(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a)))), a)), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a)))))))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a)), f(f(a, a), a))))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x301), f(a, a))))), a))))), f(a, a))), f(a, a))))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x)) 
f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a)))))))), a)), x)) 
f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))))), a))))) 
Thus, the rule f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))))), a)), x)) is replaced by the following rules:
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(a, f(f(f(a, a), f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a)), f(f(a, a), a))))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a)))))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a)))))))), f(a, a))))), a)), x))f#(f(a, _x31), f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(a, f(f(f(a, a), _x31), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))))), a)))))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, f(a, _x301)))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), _x301), f(a, a))))), a))))), f(a, a))), f(a, a))))), a)), x))f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(a, f(f(a, f(f(f(a, a), f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a)))), a)), f(a, a))), f(a, a))))), a)), x))
f#(x, f(a, f(a, f(a, f(a, f(a, f(a, _x201))))))) → f#(a, f(f(a, f(f(f(a, a), f(a, f(f(f(a, a), f(f(a, f(f(a, f(a, f(f(f(a, a), a), f(a, f(f(f(a, a), f(f(f(a, a), a), f(f(a, a), _x201))), a))))), f(a, a))), f(a, a))), f(f(a, a), a)))), a)), x))