YES

The TRS could be proven terminating. The proof took 6133 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (16ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4iUR (1207ms).
 |    | – Problem 3 was processed with processor ForwardNarrowing (1ms).
 |    |    | – Problem 4 was processed with processor BackwardsNarrowing (6ms).
 |    |    |    | – Problem 6 was processed with processor BackwardsNarrowing (4ms).
 |    |    |    |    | – Problem 8 was processed with processor BackwardsNarrowing (5ms).
 |    |    |    |    |    | – Problem 9 was processed with processor BackwardsNarrowing (13ms).
 |    |    |    |    |    |    | – Problem 11 was processed with processor BackwardsNarrowing (5ms).
 |    |    |    |    |    |    |    | – Problem 13 was processed with processor BackwardsNarrowing (12ms).
 |    |    | – Problem 5 was processed with processor ForwardNarrowing (2ms).
 |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (23ms).
 |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (1ms).

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → a#(c(b(0, y)), 0)c#(c(c(y))) → c#(b(0, y))
c#(c(c(y))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(c(c(y))) → b#(0, y)
c#(c(c(y))) → a#(a(c(b(0, y)), 0), 0)c#(c(c(y))) → c#(a(a(c(b(0, y)), 0), 0))
a#(y, 0) → b#(y, 0)

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The following SCCs where found

c#(c(c(y))) → c#(b(0, y))c#(c(c(y))) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(c(y))) → c#(a(a(c(b(0, y)), 0), 0))

Problem 2: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → c#(b(0, y))c#(c(c(y))) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(c(y))) → c#(a(a(c(b(0, y)), 0), 0))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

Polynomial Interpretation

Improved Usable rules

b(b(0, y), x) → ya(y, 0) → b(y, 0)
c(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

c#(c(c(y))) → c#(b(0, y))c#(c(c(y))) → c#(a(a(c(b(0, y)), 0), 0))

Problem 3: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → c#(c(a(a(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

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

Problem 4: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(b(b(0, c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))
c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(b(b(0, c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is backward narrowed to the 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
c#(b(b(0, c(b(b(0, c(y)), _x72))), _x22)) 
c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) 
c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) 
c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) 
c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) 
c#(b(b(0, c(c(c(_x61)))), _x22)) 
c#(b(b(0, c(c(c(c(_x71))))), _x22)) 
c#(a(b(0, c(c(y))), 0)) 
Thus, the rule c#(b(b(0, c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is replaced by the following rules:
c#(b(b(0, c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))

Problem 6: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(b(b(0, c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))
c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(b(b(0, c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is backward narrowed to the 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
c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) 
c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) 
c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) 
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) 
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) 
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) 
c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) 
c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) 
c#(a(b(0, c(b(b(0, c(y)), _x72))), 0)) 
c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) 
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) 
Thus, the rule c#(b(b(0, c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is replaced by the following rules:
c#(a(b(0, c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x111)), 0), 0)))), 0), 0)))
c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))

Problem 8: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(a(b(0, c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x111)), 0), 0)))), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))
c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(a(b(0, c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0))) is backward narrowed to the 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
c#(a(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x32), 0)) 
c#(a(b(0, c(b(b(0, c(c(c(_x91)))), _x72))), 0)) 
c#(a(b(0, c(b(b(b(0, b(0, c(y))), _x72), _x72))), 0)) 
c#(a(b(0, c(a(b(0, c(y)), 0))), 0)) 
c#(a(b(0, c(b(b(0, c(y)), _x72))), b(b(0, 0), _x42))) 
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x92)), _x72))), 0)) 
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x62))), 0)) 
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x52)), 0)) 
c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) 
c#(a(b(0, c(b(b(b(b(0, 0), _x82), c(y)), _x72))), 0)) 
c#(a(b(b(b(0, 0), _x42), c(b(b(0, c(y)), _x72))), 0)) 
Thus, the rule c#(a(b(0, c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0))) is replaced by the following rules:
c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(b(b(0, 0), _x82), c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(b(b(0, 0), _x42), c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x32), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(b(0, b(0, c(y))), _x72), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x62))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, c(c(c(_x91)))), _x72))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x91)), 0), 0)))), 0), 0)))c#(a(b(0, c(a(b(0, c(y)), 0))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x92)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(0, c(y)), _x72))), b(b(0, 0), _x42))) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x52)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))

Problem 9: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(b(b(0, 0), _x82), c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x111)), 0), 0)))), 0), 0)))c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x62))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(a(b(0, c(y)), 0))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(b(b(0, 0), _x42), c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(0, c(c(c(_x91)))), _x72))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x91)), 0), 0)))), 0), 0)))
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x92)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x52)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, c(y)), _x72))), b(b(0, 0), _x42))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x32), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(b(0, b(0, c(y))), _x72), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is backward narrowed to the 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
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)), _x32)) 
c#(b(b(0, b(b(0, c(b(b(b(b(0, 0), _x142), c(y)), _x72))), _x62)), _x22)) 
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x82)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x92), _x62)), _x22)) 
c#(b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) 
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x112)), _x62)), _x22)) 
c#(b(b(0, b(b(b(b(0, 0), _x102), c(b(b(0, c(y)), _x72))), _x62)), _x22)) 
c#(b(b(b(0, b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62))), _x42), _x22)) 
c#(b(b(0, b(b(0, c(b(b(b(0, b(0, c(y))), _x132), _x72))), _x62)), _x22)) 
c#(b(b(b(b(0, 0), _x52), b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) 
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), 0)) 
c#(b(b(0, b(b(0, c(b(b(0, c(c(c(_x151)))), _x72))), _x62)), _x22)) 
c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x152)), _x72))), _x62)), _x22)) 
c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x122))), _x62)), _x22)) 
c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) 
Thus, the rule c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is replaced by the following rules:
c#(b(b(0, b(b(0, c(b(b(b(b(0, 0), _x142), c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x112)), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x152)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, c(c(c(_x151)))), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x151)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x92), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x122))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x82)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(b(0, b(0, c(y))), _x132), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(b(0, 0), _x102), c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(b(0, 0), _x52), b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))

Problem 11: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(b(b(0, 0), _x82), c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x82)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(b(0, b(0, c(y))), _x132), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(b(0, 0), _x102), c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x111)), 0), 0)))), 0), 0)))c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x62))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(a(b(0, c(y)), 0))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(b(b(0, 0), _x142), c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x112)), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(b(b(0, 0), _x42), c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x152)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, c(c(c(_x151)))), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x151)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x92), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, c(c(c(_x91)))), _x72))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x91)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x122))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x92)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(0, c(y)), _x72))), b(b(0, 0), _x42))) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x52)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x32), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(b(0, b(0, c(y))), _x72), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0))) is backward narrowed to the 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
c#(b(b(b(b(0, 0), _x52), c(c(c(c(_x71))))), _x22)) 
c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x62)), _x22)) 
c#(b(b(0, c(c(b(b(0, c(c(_x71))), _x92)))), _x22)) 
c#(b(b(0, c(c(c(c(c(c(_x101))))))), _x22)) 
c#(a(b(0, c(c(c(c(_x71))))), 0)) 
c#(b(b(0, c(c(c(c(c(_x91)))))), _x22)) 
c#(b(b(0, c(c(c(b(b(0, c(_x71)), _x102))))), _x22)) 
c#(b(b(b(0, b(0, c(c(c(c(_x71)))))), _x42), _x22)) 
c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x22)), _x32)) 
c#(b(b(0, c(b(b(0, c(c(c(_x71)))), _x82))), _x22)) 
Thus, the rule c#(b(b(0, c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0))) is replaced by the following rules:
c#(a(b(0, c(c(c(c(_x71))))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(b(b(0, c(_x71)), _x102))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(b(0, b(0, c(c(c(c(_x71)))))), _x42), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, c(c(b(b(0, c(c(_x71))), _x92)))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(c(c(_x91)))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, a(a(c(b(0, _x91)), 0), 0))), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x22)), _x32)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(c(c(c(_x101))))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, c(a(a(c(b(0, _x101)), 0), 0)))), 0), 0)))), 0), 0)))c#(b(b(0, c(b(b(0, c(c(c(_x71)))), _x82))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))

Problem 13: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))
c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x22)), _x32)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(b(b(b(0, b(0, c(y))), _x132), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(b(b(0, 0), _x82), c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(b(0, 0), _x102), c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(a(b(0, c(y)), 0))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(b(b(0, 0), _x142), c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(c(c(_x151)))), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x151)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x92), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, c(c(c(_x91)))), _x72))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x91)), 0), 0)))), 0), 0)))c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, c(y)), _x72))), b(b(0, 0), _x42))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x32), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(b(0, b(0, c(y))), _x72), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(c(c(c(_x71)))))), _x42), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))
c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x82)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x62))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x111)), 0), 0)))), 0), 0)))
c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(b(b(0, 0), _x42), c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x112)), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x152)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(c(c(c(_x71))))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, c(c(c(b(b(0, c(_x71)), _x102))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(b(0, b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x122))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x92)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x52)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(b(b(0, c(c(_x71))), _x92)))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, c(c(c(c(c(_x91)))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, a(a(c(b(0, _x91)), 0), 0))), 0), 0)))), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(c(c(c(_x101))))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, c(a(a(c(b(0, _x101)), 0), 0)))), 0), 0)))), 0), 0)))
c#(b(b(0, c(b(b(0, c(c(c(_x71)))), _x82))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is backward narrowed to the 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
c#(b(b(0, b(b(0, c(a(b(0, b(b(0, c(y)), _x142)), 0))), _x62)), _x22)) 
c#(b(b(0, b(b(0, c(a(b(0, c(c(c(_x141)))), 0))), _x62)), _x22)) 
c#(b(b(0, b(b(0, c(a(b(b(b(0, 0), _x132), c(y)), 0))), _x62)), _x22)) 
c#(b(b(b(b(0, 0), _x52), b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(0, c(b(b(0, a(b(0, c(y)), 0)), _x112))), _x62)), _x22)) 
c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)), _x32)) 
c#(a(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), 0)) 
c#(b(b(0, b(b(0, c(a(b(0, c(y)), b(b(0, 0), _x132)))), _x62)), _x22)) 
c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x72)), _x22)) 
c#(b(b(0, a(b(0, c(a(b(0, c(y)), 0))), 0)), _x22)) 
c#(b(b(0, b(b(0, c(a(b(b(0, b(0, c(y))), _x122), 0))), _x62)), _x22)) 
c#(b(b(0, b(b(b(b(0, 0), _x92), c(a(b(0, c(y)), 0))), _x62)), _x22)) 
c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x102)), _x62)), _x22)) 
c#(b(b(b(0, b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62))), _x42), _x22)) 
Thus, the rule c#(b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is replaced by the following rules:
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, a(b(0, c(a(b(0, c(y)), 0))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(a(b(0, c(y)), b(b(0, 0), _x132)))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(a(b(b(0, b(0, c(y))), _x122), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(b(0, 0), _x92), c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, a(b(0, c(y)), 0)), _x112))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(a(b(0, b(b(0, c(y)), _x142)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x72)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(a(b(b(b(0, 0), _x132), c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x102)), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(a(b(0, c(c(c(_x141)))), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x141)), 0), 0)))), 0), 0)))

Problem 14: BackwardsNarrowing

Dependency Pair Problem

Dependency Pairs

c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(c(c(_x61)))), _x22)) → c#(c(a(a(c(b(0, a(a(c(b(0, _x61)), 0), 0))), 0), 0)))c#(b(b(0, c(b(b(b(b(0, 0), _x102), c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(a(b(0, c(y)), 0))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x22)), _x32)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(b(0, b(0, c(y))), _x132), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(b(b(0, 0), _x82), c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(0, b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(b(0, 0), _x102), c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(a(b(0, c(y)), 0))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(a(b(0, c(c(c(_x141)))), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x141)), 0), 0)))), 0), 0)))
c#(b(b(b(b(0, 0), _x52), b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(b(b(0, 0), _x142), c(y)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, a(b(0, c(a(b(0, c(y)), 0))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(c(c(_x151)))), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x151)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(a(b(0, c(y)), b(b(0, 0), _x132)))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x92), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(0, c(c(c(_x91)))), _x72))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x91)), 0), 0)))), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(b(b(0, c(y)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, c(y)), _x72))), b(b(0, 0), _x42))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x32), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(b(0, b(0, c(y))), _x72), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x82))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(c(c(c(_x71)))))), _x42), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(c(c(c(_x21)))) → c#(c(a(a(c(b(0, a(a(c(b(0, _x21)), 0), 0))), 0), 0)))
c#(c(c(c(c(_x31))))) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x31)), 0), 0)))), 0), 0)))c#(b(b(0, c(b(b(0, b(b(0, c(y)), _x112)), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, c(b(b(b(0, b(0, c(y))), _x92), _x72))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(a(b(b(0, b(0, c(y))), _x122), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, a(b(0, c(y)), 0)), _x112))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62)), _x82)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(a(b(0, b(b(0, c(y)), _x142)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, c(c(y))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x72)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x62))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(0, c(c(c(_x111)))), _x72))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x111)), 0), 0)))), 0), 0)))
c#(a(b(0, c(c(y))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(a(b(b(b(0, 0), _x132), c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x102)), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(c(y)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(c(b(b(0, c(y)), _x32))) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(b(b(0, 0), _x42), c(b(b(0, c(y)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x112)), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(c(c(c(_x71))))), 0)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x152)), _x72))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(c(c(b(b(0, c(_x71)), _x102))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(b(0, b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(0, c(b(b(0, b(b(0, c(y)), _x72)), _x122))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(0, c(a(b(0, c(y)), 0))), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, c(b(b(0, b(b(0, c(y)), _x92)), _x72))), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(b(0, 0), _x92), c(a(b(0, c(y)), 0))), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(0, b(0, c(b(b(0, c(y)), _x72)))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(a(b(0, b(b(0, c(b(b(0, c(y)), _x72))), _x52)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, a(b(0, c(b(b(0, c(y)), _x72))), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(c(c(_x71))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, b(b(0, c(c(c(c(_x71))))), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(b(b(0, c(c(_x71))), _x92)))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))c#(b(b(0, c(c(c(c(c(_x91)))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, a(a(c(b(0, _x91)), 0), 0))), 0), 0)))), 0), 0)))
c#(b(b(b(b(0, 0), _x52), c(c(y))), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, c(b(b(0, c(c(c(_x71)))), _x82))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x71)), 0), 0)))), 0), 0)))
c#(b(b(0, c(c(c(c(c(c(_x101))))))), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, c(a(a(c(b(0, _x101)), 0), 0)))), 0), 0)))), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

The left-hand side of the rule c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is backward narrowed to the 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
c#(b(b(0, b(b(b(b(0, b(0, b(0, c(a(b(0, c(y)), 0))))), _x102), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(b(b(0, a(b(0, c(y)), 0)), _x152)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(c(c(_x181)))), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(b(b(0, 0), _x112), b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, a(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), 0)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), b(b(0, 0), _x172))))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, b(b(0, c(a(b(0, c(y)), 0))), _x142))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x72)), _x22)) 
c#(a(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), 0)) 
c#(b(b(b(0, b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62))), _x42), _x22)) 
c#(b(b(0, b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)), _x32)) 
c#(b(b(b(b(0, 0), _x52), b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(a(b(0, b(0, c(a(b(0, c(y)), 0)))), 0), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82)), _x92), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x122)), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(a(b(b(b(0, 0), _x172), c(y)), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(b(b(0, 0), _x132), c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(a(b(b(0, b(0, c(y))), _x162), 0)))), _x82), _x62)), _x22)) 
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, b(b(0, c(y)), _x182)), 0)))), _x82), _x62)), _x22)) 
Thus, the rule c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0))) is replaced by the following rules:
c#(b(b(b(0, b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62))), _x42), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(0, b(0, c(a(b(b(0, b(0, c(y))), _x162), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, a(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), 0)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(0, b(b(b(0, 0), _x132), c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x72)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(0, b(0, b(b(0, c(a(b(0, c(y)), 0))), _x142))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(b(0, b(0, b(0, c(a(b(0, c(y)), 0))))), _x102), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(b(b(0, 0), _x52), b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(a(b(b(b(0, 0), _x172), c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(a(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), 0)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, b(b(0, c(y)), _x182)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(b(b(0, 0), _x112), b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(0, b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x122)), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(0, b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82)), _x92), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(b(b(0, a(b(0, c(y)), 0)), _x152)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(a(b(0, b(0, c(a(b(0, c(y)), 0)))), 0), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))
c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), b(b(0, 0), _x172))))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, y)), 0), 0)))c#(b(b(0, b(b(b(0, b(0, c(a(b(0, c(c(c(_x181)))), 0)))), _x82), _x62)), _x22)) → c#(c(a(a(c(b(0, c(a(a(c(b(0, _x181)), 0), 0)))), 0), 0)))
c#(b(b(0, b(b(0, b(b(b(0, b(0, c(a(b(0, c(y)), 0)))), _x82), _x62)), _x22)), _x32)) → c#(c(a(a(c(b(0, y)), 0), 0)))

Problem 5: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → c#(c(b(a(c(b(0, y)), 0), 0)))c#(c(c(y))) → c#(c(a(b(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

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

Problem 7: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → c#(c(b(b(c(b(0, y)), 0), 0)))c#(c(c(y))) → c#(c(a(b(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

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

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → c#(c(a(b(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

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

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

c#(c(c(y))) → c#(c(b(b(c(b(0, y)), 0), 0)))

Rewrite Rules

b(b(0, y), x) → yc(c(c(y))) → c(c(a(a(c(b(0, y)), 0), 0)))
a(y, 0) → b(y, 0)

Original Signature

Termination of terms over the following signature is verified: 0, b, c, a

Strategy

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