TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60045 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (60ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4iUR (696ms).
 |    | – Problem 3 was processed with processor DependencyGraph (12ms).
 |    |    | – Problem 4 was processed with processor ForwardNarrowing (3ms).
 |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (30ms).
 |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (13ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (14ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (24ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (91ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 was processed with processor ForwardNarrowing (132ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 31 was processed with processor ForwardNarrowing (123ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 32 was processed with processor ForwardNarrowing (303ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 33 was processed with processor ForwardNarrowing (149ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 34 was processed with processor ForwardNarrowing (215ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 35 remains open; application of the following processors failed [ForwardNarrowing (154ms), ForwardNarrowing (355ms), ForwardNarrowing (193ms), ForwardNarrowing (151ms), ForwardNarrowing (136ms), ForwardNarrowing (142ms), ForwardNarrowing (343ms), ForwardNarrowing (142ms), ForwardNarrowing (143ms), ForwardNarrowing (178ms), ForwardNarrowing (198ms), ForwardNarrowing (180ms), ForwardNarrowing (377ms), ForwardNarrowing (110ms), ForwardNarrowing (181ms), ForwardNarrowing (305ms), ForwardNarrowing (194ms), ForwardNarrowing (193ms), ForwardNarrowing (347ms), ForwardNarrowing (200ms), ForwardNarrowing (128ms), ForwardNarrowing (358ms), ForwardNarrowing (195ms), ForwardNarrowing (214ms), ForwardNarrowing (399ms), ForwardNarrowing (141ms), ForwardNarrowing (217ms), ForwardNarrowing (timeout)].
 |    |    | – Problem 5 was processed with processor PolynomialLinearRange4iUR (245ms).
 |    |    |    | – Problem 6 was processed with processor DependencyGraph (2ms).
 |    |    |    |    | – Problem 7 was processed with processor PolynomialLinearRange4iUR (26ms).
 |    |    |    |    |    | – Problem 8 was processed with processor PolynomialLinearRange4iUR (16ms).

The following open problems remain:



Open Dependency Pair Problem 4

Dependency Pairs

a__length#(cons(N, L))a__length#(mark(L))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2))mark#(X1)a__length#(cons(N, L))mark#(L)
mark#(and(X1, X2))a__and#(mark(X1), X2)a__and#(tt, X)mark#(X)
mark#(zeros)a__zeros#mark#(cons(X1, X2))mark#(X1)
a__length#(cons(N, L))a__length#(mark(L))mark#(length(X))mark#(X)
mark#(s(X))mark#(X)mark#(length(X))a__length#(mark(X))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The following SCCs where found

mark#(and(X1, X2)) → mark#(X1)a__length#(cons(N, L)) → mark#(L)
mark#(and(X1, X2)) → a__and#(mark(X1), X2)a__and#(tt, X) → mark#(X)
mark#(cons(X1, X2)) → mark#(X1)mark#(length(X)) → mark#(X)
a__length#(cons(N, L)) → a__length#(mark(L))mark#(s(X)) → mark#(X)
mark#(length(X)) → a__length#(mark(X))

Problem 2: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2))mark#(X1)a__length#(cons(N, L))mark#(L)
mark#(and(X1, X2))a__and#(mark(X1), X2)a__and#(tt, X)mark#(X)
mark#(cons(X1, X2))mark#(X1)mark#(length(X))mark#(X)
a__length#(cons(N, L))a__length#(mark(L))mark#(s(X))mark#(X)
mark#(length(X))a__length#(mark(X))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


Polynomial Interpretation

Improved Usable rules

mark(tt)ttmark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0a__and(X1, X2)and(X1, X2)
mark(length(X))a__length(mark(X))a__and(tt, X)mark(X)
a__length(X)length(X)a__zeroszeros
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(s(X))s(mark(X))mark(nil)nil
a__zeroscons(0, zeros)a__length(nil)0
a__length(cons(N, L))s(a__length(mark(L)))

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

mark#(length(X))mark#(X)mark#(length(X))a__length#(mark(X))

Problem 3: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2))mark#(X1)a__length#(cons(N, L))mark#(L)
mark#(and(X1, X2))a__and#(mark(X1), X2)a__and#(tt, X)mark#(X)
mark#(cons(X1, X2))mark#(X1)a__length#(cons(N, L))a__length#(mark(L))
mark#(s(X))mark#(X)

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The following SCCs where found

mark#(and(X1, X2)) → mark#(X1)mark#(and(X1, X2)) → a__and#(mark(X1), X2)
a__and#(tt, X) → mark#(X)mark#(cons(X1, X2)) → mark#(X1)
mark#(s(X)) → mark#(X)

a__length#(cons(N, L)) → a__length#(mark(L))

Problem 4: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, L))a__length#(mark(L))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, L)) → a__length#(mark(L)) 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
a__length#(a__zeros) 
a__length#(a__length(mark(_x21))) 
a__length#(cons(mark(_x21), _x22)) 
a__length#(nil) 
a__length#(a__and(mark(_x21), _x22)) 
a__length#(0) 
a__length#(s(mark(_x21))) 
a__length#(tt) 
Thus, the rule a__length#(cons(N, L)) → a__length#(mark(L)) is replaced by the following rules:
a__length#(cons(N, zeros)) → a__length#(a__zeros)a__length#(cons(N, 0)) → a__length#(0)
a__length#(cons(N, cons(_x21, _x22))) → a__length#(cons(mark(_x21), _x22))a__length#(cons(N, tt)) → a__length#(tt)
a__length#(cons(N, nil)) → a__length#(nil)a__length#(cons(N, s(_x21))) → a__length#(s(mark(_x21)))
a__length#(cons(N, and(_x21, _x22))) → a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, length(_x21))) → a__length#(a__length(mark(_x21)))

Problem 9: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, zeros))a__length#(a__zeros)a__length#(cons(N, 0))a__length#(0)
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, tt))a__length#(tt)
a__length#(cons(N, nil))a__length#(nil)a__length#(cons(N, s(_x21)))a__length#(s(mark(_x21)))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, zeros)) → a__length#(a__zeros) 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
a__length#(zeros) 
a__length#(cons(0, zeros)) 
Thus, the rule a__length#(cons(N, zeros)) → a__length#(a__zeros) is replaced by the following rules:
a__length#(cons(N, zeros)) → a__length#(cons(0, zeros))a__length#(cons(N, zeros)) → a__length#(zeros)

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, 0))a__length#(0)a__length#(cons(N, tt))a__length#(tt)
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, nil))a__length#(nil)
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(_x21)))a__length#(s(mark(_x21)))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, 0)) → a__length#(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 a__length#(cons(N, 0)) → a__length#(0) is deleted.

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, tt))a__length#(tt)
a__length#(cons(N, nil))a__length#(nil)a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(_x21)))a__length#(s(mark(_x21)))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, tt)) → a__length#(tt) 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 a__length#(cons(N, tt)) → a__length#(tt) is deleted.

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, nil))a__length#(nil)
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(_x21)))a__length#(s(mark(_x21)))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, nil)) → a__length#(nil) 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 a__length#(cons(N, nil)) → a__length#(nil) is deleted.

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(_x21)))a__length#(s(mark(_x21)))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(_x21))) → a__length#(s(mark(_x21))) 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
a__length#(s(0)) 
a__length#(s(a__length(mark(_x41)))) 
a__length#(s(cons(mark(_x41), _x42))) 
a__length#(s(nil)) 
a__length#(s(a__zeros)) 
a__length#(s(tt)) 
a__length#(s(s(mark(_x41)))) 
a__length#(s(a__and(mark(_x41), _x42))) 
Thus, the rule a__length#(cons(N, s(_x21))) → a__length#(s(mark(_x21))) is replaced by the following rules:
a__length#(cons(N, s(nil))) → a__length#(s(nil))a__length#(cons(N, s(zeros))) → a__length#(s(a__zeros))
a__length#(cons(N, s(length(_x41)))) → a__length#(s(a__length(mark(_x41))))a__length#(cons(N, s(cons(_x41, _x42)))) → a__length#(s(cons(mark(_x41), _x42)))
a__length#(cons(N, s(0))) → a__length#(s(0))a__length#(cons(N, s(s(_x41)))) → a__length#(s(s(mark(_x41))))
a__length#(cons(N, s(and(_x41, _x42)))) → a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(tt))) → a__length#(s(tt))

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(nil)))a__length#(s(nil))a__length#(cons(N, s(length(_x41))))a__length#(s(a__length(mark(_x41))))
a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(nil))) → a__length#(s(nil)) 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 a__length#(cons(N, s(nil))) → a__length#(s(nil)) is deleted.

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(_x41))))a__length#(s(a__length(mark(_x41))))
a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(_x41)))) → a__length#(s(a__length(mark(_x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__length#(s(a__length(cons(mark(_x51), _x52)))) 
a__length#(s(a__length(a__zeros))) 
a__length#(s(a__length(a__length(mark(_x51))))) 
a__length#(s(a__length(s(mark(_x51))))) 
a__length#(s(a__length(tt))) 
a__length#(s(length(mark(_x41)))) 
a__length#(s(a__length(a__and(mark(_x51), _x52)))) 
a__length#(s(a__length(nil))) 
a__length#(s(a__length(0))) 
Thus, the rule a__length#(cons(N, s(length(_x41)))) → a__length#(s(a__length(mark(_x41)))) is replaced by the following rules:
a__length#(cons(N, s(length(tt)))) → a__length#(s(a__length(tt)))a__length#(cons(N, s(length(s(_x51))))) → a__length#(s(a__length(s(mark(_x51)))))
a__length#(cons(N, s(length(_x41)))) → a__length#(s(length(mark(_x41))))a__length#(cons(N, s(length(0)))) → a__length#(s(a__length(0)))
a__length#(cons(N, s(length(nil)))) → a__length#(s(a__length(nil)))a__length#(cons(N, s(length(and(_x51, _x52))))) → a__length#(s(a__length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(cons(_x51, _x52))))) → a__length#(s(a__length(cons(mark(_x51), _x52))))a__length#(cons(N, s(length(zeros)))) → a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(length(_x51))))) → a__length#(s(a__length(a__length(mark(_x51)))))

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(_x51)))))a__length#(s(a__length(s(mark(_x51)))))a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))
a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(0)))a__length#(s(0))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(_x51))))) → a__length#(s(a__length(s(mark(_x51))))) 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
a__length#(s(a__length(s(cons(mark(_x61), _x62))))) 
a__length#(s(a__length(s(s(mark(_x61)))))) 
a__length#(s(a__length(s(a__and(mark(_x61), _x62))))) 
a__length#(s(length(s(mark(_x51))))) 
a__length#(s(a__length(s(0)))) 
a__length#(s(a__length(s(nil)))) 
a__length#(s(a__length(s(a__length(mark(_x61)))))) 
a__length#(s(a__length(s(tt)))) 
a__length#(s(a__length(s(a__zeros)))) 
Thus, the rule a__length#(cons(N, s(length(s(_x51))))) → a__length#(s(a__length(s(mark(_x51))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(_x61, _x62)))))) → a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(s(_x61)))))) → a__length#(s(a__length(s(s(mark(_x61))))))
a__length#(cons(N, s(length(s(cons(_x61, _x62)))))) → a__length#(s(a__length(s(cons(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(length(_x61)))))) → a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(length(s(tt))))) → a__length#(s(a__length(s(tt))))a__length#(cons(N, s(length(s(zeros))))) → a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(length(s(0))))) → a__length#(s(a__length(s(0))))a__length#(cons(N, s(length(s(_x51))))) → a__length#(s(length(s(mark(_x51)))))
a__length#(cons(N, s(length(s(nil))))) → a__length#(s(a__length(s(nil))))

Problem 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(a__length(s(s(mark(_x61))))))a__length#(cons(N, s(length(s(tt)))))a__length#(s(a__length(s(tt))))
a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))
a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(0)))a__length#(s(0))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))
a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(s(_x61)))))) → a__length#(s(a__length(s(s(mark(_x61)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__length#(s(a__length(s(s(0))))) 
a__length#(s(a__length(s(s(nil))))) 
a__length#(s(a__length(s(s(a__zeros))))) 
a__length#(s(a__length(s(s(a__and(mark(_x71), _x72)))))) 
a__length#(s(a__length(s(s(cons(mark(_x71), _x72)))))) 
a__length#(s(a__length(s(s(tt))))) 
a__length#(s(length(s(s(mark(_x61)))))) 
a__length#(s(a__length(s(s(a__length(mark(_x71))))))) 
a__length#(s(a__length(s(s(s(mark(_x71))))))) 
Thus, the rule a__length#(cons(N, s(length(s(s(_x61)))))) → a__length#(s(a__length(s(s(mark(_x61)))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(s(0)))))) → a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(a__length(s(s(a__zeros)))))
a__length#(cons(N, s(length(s(s(_x61)))))) → a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(length(s(s(and(_x71, _x72))))))) → a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(s(s(_x71))))))) → a__length#(s(a__length(s(s(s(mark(_x71)))))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72))))))) → a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(s(nil)))))) → a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, s(length(s(s(tt)))))) → a__length#(s(a__length(s(s(tt)))))
a__length#(cons(N, s(length(s(s(length(_x71))))))) → a__length#(s(a__length(s(s(a__length(mark(_x71)))))))

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(a__zeros)))))a__length#(cons(N, s(length(s(tt)))))a__length#(s(a__length(s(tt))))
a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))
a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(0)))a__length#(s(0))a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))
a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))
a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(a__length(s(s(a__zeros))))) 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
a__length#(s(length(s(s(a__zeros))))) 
a__length#(s(a__length(s(s(zeros))))) 
a__length#(s(a__length(s(s(cons(0, zeros)))))) 
Thus, the rule a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(a__length(s(s(a__zeros))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(a__length(s(s(zeros)))))a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(length(s(s(a__zeros)))))
a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(a__length(s(s(cons(0, zeros))))))

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(a__zeros)))))a__length#(cons(N, s(length(s(tt)))))a__length#(s(a__length(s(tt))))
a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))
a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))
a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))
a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))
a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))
a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(length(s(s(a__zeros))))) 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
a__length#(s(length(s(s(zeros))))) 
a__length#(s(length(s(s(cons(0, zeros)))))) 
Thus, the rule a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(length(s(s(a__zeros))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(s(zeros)))))) → a__length#(s(length(s(s(zeros)))))

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(tt)))))a__length#(s(a__length(s(tt))))a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))
a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))
a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(0)))a__length#(s(0))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(tt))))) → a__length#(s(a__length(s(tt)))) 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
a__length#(s(length(s(tt)))) 
Thus, the rule a__length#(cons(N, s(length(s(tt))))) → a__length#(s(a__length(s(tt)))) is replaced by the following rules:
a__length#(cons(N, s(length(s(tt))))) → a__length#(s(length(s(tt))))

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(cons(_x41, _x42))))a__length#(s(cons(mark(_x41), _x42)))a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))
a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))
a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))
a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(cons(_x41, _x42)))) → a__length#(s(cons(mark(_x41), _x42))) 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
a__length#(s(cons(0, _x42))) 
a__length#(s(cons(nil, _x42))) 
a__length#(s(cons(s(mark(_x51)), _x42))) 
a__length#(s(cons(a__and(mark(_x51), _x52), _x42))) 
a__length#(s(cons(cons(mark(_x51), _x52), _x42))) 
a__length#(s(cons(tt, _x42))) 
a__length#(s(cons(a__zeros, _x42))) 
a__length#(s(cons(a__length(mark(_x51)), _x42))) 
Thus, the rule a__length#(cons(N, s(cons(_x41, _x42)))) → a__length#(s(cons(mark(_x41), _x42))) is replaced by the following rules:
a__length#(cons(N, s(cons(0, _x42)))) → a__length#(s(cons(0, _x42)))a__length#(cons(N, s(cons(tt, _x42)))) → a__length#(s(cons(tt, _x42)))
a__length#(cons(N, s(cons(nil, _x42)))) → a__length#(s(cons(nil, _x42)))a__length#(cons(N, s(cons(zeros, _x42)))) → a__length#(s(cons(a__zeros, _x42)))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42)))) → a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(cons(length(_x51), _x42)))) → a__length#(s(cons(a__length(mark(_x51)), _x42)))
a__length#(cons(N, s(cons(s(_x51), _x42)))) → a__length#(s(cons(s(mark(_x51)), _x42)))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42)))) → a__length#(s(cons(cons(mark(_x51), _x52), _x42)))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(0))))a__length#(s(a__length(0)))a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))
a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(0)))a__length#(s(0))a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))
a__length#(cons(N, s(cons(s(_x51), _x42))))a__length#(s(cons(s(mark(_x51)), _x42)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))a__length#(cons(N, s(cons(zeros, _x42))))a__length#(s(cons(a__zeros, _x42)))
a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))
a__length#(cons(N, s(cons(0, _x42))))a__length#(s(cons(0, _x42)))a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))
a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(0)))) → a__length#(s(a__length(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
a__length#(s(length(0))) 
Thus, the rule a__length#(cons(N, s(length(0)))) → a__length#(s(a__length(0))) is replaced by the following rules:
a__length#(cons(N, s(length(0)))) → a__length#(s(length(0)))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(a__length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))
a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(cons(s(_x51), _x42))))a__length#(s(cons(s(mark(_x51)), _x42)))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))
a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))
a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(cons(zeros, _x42))))a__length#(s(cons(a__zeros, _x42)))a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))
a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(cons(0, _x42))))a__length#(s(cons(0, _x42)))
a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))
a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))
a__length#(cons(N, s(length(0))))a__length#(s(length(0)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(and(_x51, _x52))))) → a__length#(s(a__length(a__and(mark(_x51), _x52)))) 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
a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52)))) 
a__length#(s(a__length(a__and(s(mark(_x61)), _x52)))) 
a__length#(s(length(a__and(mark(_x51), _x52)))) 
a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52)))) 
a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52)))) 
a__length#(s(a__length(a__and(a__zeros, _x52)))) 
a__length#(s(a__length(a__and(0, _x52)))) 
a__length#(s(a__length(and(mark(_x51), _x42)))) 
a__length#(s(a__length(a__and(tt, _x52)))) 
a__length#(s(a__length(a__and(nil, _x52)))) 
Thus, the rule a__length#(cons(N, s(length(and(_x51, _x52))))) → a__length#(s(a__length(a__and(mark(_x51), _x52)))) is replaced by the following rules:
a__length#(cons(N, s(length(and(length(_x61), _x52))))) → a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52))))a__length#(cons(N, s(length(and(_x51, _x52))))) → a__length#(s(length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(and(nil, _x52))))) → a__length#(s(a__length(a__and(nil, _x52))))a__length#(cons(N, s(length(and(zeros, _x52))))) → a__length#(s(a__length(a__and(a__zeros, _x52))))
a__length#(cons(N, s(length(and(and(_x61, _x62), _x52))))) → a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52))))a__length#(cons(N, s(length(and(cons(_x61, _x62), _x52))))) → a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(and(0, _x52))))) → a__length#(s(a__length(a__and(0, _x52))))a__length#(cons(N, s(length(and(_x51, _x42))))) → a__length#(s(a__length(and(mark(_x51), _x42))))
a__length#(cons(N, s(length(and(tt, _x52))))) → a__length#(s(a__length(a__and(tt, _x52))))a__length#(cons(N, s(length(and(s(_x61), _x52))))) → a__length#(s(a__length(a__and(s(mark(_x61)), _x52))))

Problem 24: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(a__length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(cons(zeros, _x42))))a__length#(s(cons(a__zeros, _x42)))a__length#(cons(N, s(length(and(nil, _x52)))))a__length#(s(a__length(a__and(nil, _x52))))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))
a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))a__length#(cons(N, s(length(and(0, _x52)))))a__length#(s(a__length(a__and(0, _x52))))
a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))
a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))a__length#(cons(N, s(length(and(length(_x61), _x52)))))a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52))))
a__length#(cons(N, s(length(and(_x51, _x42)))))a__length#(s(a__length(and(mark(_x51), _x42))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(and(s(_x61), _x52)))))a__length#(s(a__length(a__and(s(mark(_x61)), _x52))))a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))
a__length#(cons(N, s(length(and(and(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))
a__length#(cons(N, s(cons(s(_x51), _x42))))a__length#(s(cons(s(mark(_x51)), _x42)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))
a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(and(cons(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52))))a__length#(cons(N, s(length(and(tt, _x52)))))a__length#(s(a__length(a__and(tt, _x52))))
a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))a__length#(cons(N, s(cons(0, _x42))))a__length#(s(cons(0, _x42)))
a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))
a__length#(cons(N, s(length(and(zeros, _x52)))))a__length#(s(a__length(a__and(a__zeros, _x52))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(0))))a__length#(s(length(0)))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(_x61, _x62)))))) → a__length#(s(a__length(s(a__and(mark(_x61), _x62))))) 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
a__length#(s(a__length(s(a__and(s(mark(_x71)), _x62))))) 
a__length#(s(a__length(s(a__and(0, _x62))))) 
a__length#(s(length(s(a__and(mark(_x61), _x62))))) 
a__length#(s(a__length(s(and(mark(_x61), _x52))))) 
a__length#(s(a__length(s(a__and(a__length(mark(_x71)), _x62))))) 
a__length#(s(a__length(s(a__and(a__zeros, _x62))))) 
a__length#(s(a__length(s(a__and(tt, _x62))))) 
a__length#(s(a__length(s(a__and(nil, _x62))))) 
a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62))))) 
a__length#(s(a__length(s(a__and(cons(mark(_x71), _x72), _x62))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(_x61, _x62)))))) → a__length#(s(a__length(s(a__and(mark(_x61), _x62))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(cons(_x71, _x72), _x62)))))) → a__length#(s(a__length(s(a__and(cons(mark(_x71), _x72), _x62)))))a__length#(cons(N, s(length(s(and(_x61, _x62)))))) → a__length#(s(length(s(a__and(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62)))))) → a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))a__length#(cons(N, s(length(s(and(0, _x62)))))) → a__length#(s(a__length(s(a__and(0, _x62)))))
a__length#(cons(N, s(length(s(and(nil, _x62)))))) → a__length#(s(a__length(s(a__and(nil, _x62)))))a__length#(cons(N, s(length(s(and(s(_x71), _x62)))))) → a__length#(s(a__length(s(a__and(s(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(and(tt, _x62)))))) → a__length#(s(a__length(s(a__and(tt, _x62)))))a__length#(cons(N, s(length(s(and(zeros, _x62)))))) → a__length#(s(a__length(s(a__and(a__zeros, _x62)))))
a__length#(cons(N, s(length(s(and(_x61, _x52)))))) → a__length#(s(a__length(s(and(mark(_x61), _x52)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(mark(_x71)), _x62)))))

Problem 25: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(_x71), _x62))))))a__length#(s(a__length(s(a__and(a__length(mark(_x71)), _x62)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))a__length#(cons(N, s(length(s(and(cons(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(cons(mark(_x71), _x72), _x62)))))
a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))a__length#(cons(N, s(cons(zeros, _x42))))a__length#(s(cons(a__zeros, _x42)))
a__length#(cons(N, s(length(and(nil, _x52)))))a__length#(s(a__length(a__and(nil, _x52))))a__length#(cons(N, s(length(s(and(tt, _x62))))))a__length#(s(a__length(s(a__and(tt, _x62)))))
a__length#(cons(N, zeros))a__length#(zeros)a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))
a__length#(cons(N, s(length(s(and(zeros, _x62))))))a__length#(s(a__length(s(a__and(a__zeros, _x62)))))a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))
a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))
a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(length(a__and(mark(_x51), _x52))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(and(0, _x62))))))a__length#(s(a__length(s(a__and(0, _x62)))))a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))
a__length#(cons(N, s(length(and(0, _x52)))))a__length#(s(a__length(a__and(0, _x52))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))
a__length#(cons(N, s(length(and(length(_x61), _x52)))))a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52))))a__length#(cons(N, s(length(and(_x51, _x42)))))a__length#(s(a__length(and(mark(_x51), _x42))))
a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))a__length#(cons(N, s(length(s(and(_x61, _x52))))))a__length#(s(a__length(s(and(mark(_x61), _x52)))))
a__length#(cons(N, s(length(and(s(_x61), _x52)))))a__length#(s(a__length(a__and(s(mark(_x61)), _x52))))a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))
a__length#(cons(N, s(length(and(and(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))
a__length#(cons(N, s(length(s(and(s(_x71), _x62))))))a__length#(s(a__length(s(a__and(s(mark(_x71)), _x62)))))a__length#(cons(N, s(cons(s(_x51), _x42))))a__length#(s(cons(s(mark(_x51)), _x42)))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))
a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))a__length#(cons(N, s(length(and(cons(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(s(and(nil, _x62))))))a__length#(s(a__length(s(a__and(nil, _x62)))))a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))
a__length#(cons(N, s(length(and(tt, _x52)))))a__length#(s(a__length(a__and(tt, _x52))))a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, s(cons(0, _x42))))a__length#(s(cons(0, _x42)))
a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))
a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(and(zeros, _x52)))))a__length#(s(a__length(a__and(a__zeros, _x52))))a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(0))))a__length#(s(length(0)))
a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(_x71), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(mark(_x71)), _x62))))) 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
a__length#(s(a__length(s(a__and(a__length(a__length(mark(_x81))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(nil), _x62))))) 
a__length#(s(length(s(a__and(a__length(mark(_x71)), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(tt), _x62))))) 
a__length#(s(a__length(s(a__and(length(mark(_x71)), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(s(mark(_x81))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(0), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__zeros), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__and(mark(_x81), _x82)), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62))))) 
a__length#(s(a__length(s(and(a__length(mark(_x71)), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(_x71), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(mark(_x71)), _x62))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(_x71), _x62)))))) → a__length#(s(length(s(a__and(a__length(mark(_x71)), _x62)))))a__length#(cons(N, s(length(s(and(length(_x71), _x52)))))) → a__length#(s(a__length(s(and(a__length(mark(_x71)), _x52)))))
a__length#(cons(N, s(length(s(and(length(tt), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(tt), _x62)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62)))))) → a__length#(s(a__length(s(a__and(length(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(and(length(zeros), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__zeros), _x62)))))a__length#(cons(N, s(length(s(and(length(and(_x81, _x82)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__and(mark(_x81), _x82)), _x62)))))
a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(s(and(length(0), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(0), _x62)))))
a__length#(cons(N, s(length(s(and(length(nil), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(nil), _x62)))))a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(s(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(mark(_x81))), _x62)))))

Problem 26: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62))))))a__length#(s(length(s(a__and(a__length(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62))))))a__length#(s(a__length(s(a__and(length(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))
a__length#(cons(N, s(length(s(and(cons(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(cons(mark(_x71), _x72), _x62)))))a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(length(and(nil, _x52)))))a__length#(s(a__length(a__and(nil, _x52))))a__length#(cons(N, s(cons(zeros, _x42))))a__length#(s(cons(a__zeros, _x42)))
a__length#(cons(N, s(length(s(and(tt, _x62))))))a__length#(s(a__length(s(a__and(tt, _x62)))))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(s(and(zeros, _x62))))))a__length#(s(a__length(s(a__and(a__zeros, _x62)))))a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))
a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))a__length#(cons(N, s(length(s(and(0, _x62))))))a__length#(s(a__length(s(a__and(0, _x62)))))
a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))a__length#(cons(N, s(length(and(0, _x52)))))a__length#(s(a__length(a__and(0, _x52))))
a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))
a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))a__length#(cons(N, s(length(and(length(_x61), _x52)))))a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52))))
a__length#(cons(N, s(length(and(_x51, _x42)))))a__length#(s(a__length(and(mark(_x51), _x42))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(and(_x61, _x52))))))a__length#(s(a__length(s(and(mark(_x61), _x52)))))a__length#(cons(N, s(length(and(s(_x61), _x52)))))a__length#(s(a__length(a__and(s(mark(_x61)), _x52))))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(a__length(s(a__and(a__length(tt), _x62)))))
a__length#(cons(N, s(length(s(and(length(and(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__and(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(and(and(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(a__length(nil), _x62)))))a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(length(s(and(s(_x71), _x62))))))a__length#(s(a__length(s(a__and(s(mark(_x71)), _x62)))))
a__length#(cons(N, s(cons(s(_x51), _x42))))a__length#(s(cons(s(mark(_x51)), _x42)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))
a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(and(length(zeros), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__zeros), _x62)))))a__length#(cons(N, s(length(and(cons(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(s(and(nil, _x62))))))a__length#(s(a__length(s(a__and(nil, _x62)))))a__length#(cons(N, s(length(s(and(length(0), _x62))))))a__length#(s(a__length(s(a__and(a__length(0), _x62)))))
a__length#(cons(N, s(length(and(tt, _x52)))))a__length#(s(a__length(a__and(tt, _x52))))a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))
a__length#(cons(N, s(tt)))a__length#(s(tt))a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))
a__length#(cons(N, s(cons(0, _x42))))a__length#(s(cons(0, _x42)))a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(a__length(mark(_x71)), _x52)))))
a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))
a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))a__length#(cons(N, s(length(and(zeros, _x52)))))a__length#(s(a__length(a__and(a__zeros, _x52))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))
a__length#(cons(N, s(length(0))))a__length#(s(length(0)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(mark(_x81))), _x62))))) 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
a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62))))) 
a__length#(s(length(s(a__and(a__length(a__length(mark(_x81))), _x62))))) 
a__length#(s(a__length(s(and(a__length(a__length(mark(_x81))), _x52))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(a__and(mark(_x91), _x92))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(s(mark(_x91)))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(a__zeros)), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(nil)), _x62))))) 
a__length#(s(a__length(s(a__and(length(a__length(mark(_x81))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(cons(mark(_x91), _x92))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(a__length(mark(_x91)))), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(a__length(0)), _x62))))) 
a__length#(s(a__length(s(a__and(a__length(length(mark(_x81))), _x62))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(mark(_x81))), _x62))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(cons(_x91, _x92))), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(cons(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62)))))) → a__length#(s(length(s(a__and(a__length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(tt)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(length(_x91))), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(a__length(mark(_x91)))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(s(_x91))), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(s(mark(_x91)))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(nil)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(nil)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(zeros)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(a__zeros)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62)))))) → a__length#(s(a__length(s(a__and(length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(_x91, _x92))), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(a__and(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(mark(_x81))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(length(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(0)), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(a__length(0)), _x62)))))

Problem 27: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(mark(_x81))), _x52)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62))))))a__length#(s(length(s(a__and(a__length(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(zeros)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62))))))a__length#(s(a__length(s(a__and(length(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(0)))))a__length#(s(a__length(s(0))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(cons(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(cons(mark(_x71), _x72), _x62)))))
a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))a__length#(cons(N, s(length(s(and(length(length(length(_x91))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(a__length(mark(_x91)))), _x62)))))
a__length#(cons(N, s(cons(zeros, _x42))))a__length#(s(cons(a__zeros, _x42)))a__length#(cons(N, s(length(and(nil, _x52)))))a__length#(s(a__length(a__and(nil, _x52))))
a__length#(cons(N, s(length(s(and(length(length(s(_x91))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(mark(_x91)))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(zeros)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(a__zeros)), _x62)))))
a__length#(cons(N, s(length(s(and(tt, _x62))))))a__length#(s(a__length(s(a__and(tt, _x62)))))a__length#(cons(N, zeros))a__length#(zeros)
a__length#(cons(N, s(length(length(_x51)))))a__length#(s(a__length(a__length(mark(_x51)))))a__length#(cons(N, s(length(s(and(zeros, _x62))))))a__length#(s(a__length(s(a__and(a__zeros, _x62)))))
a__length#(cons(N, length(_x21)))a__length#(a__length(mark(_x21)))a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))
a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(and(_x51, _x52)))))a__length#(s(length(a__and(mark(_x51), _x52))))
a__length#(cons(N, s(length(s(and(length(length(nil)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(nil)), _x62)))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(and(0, _x62))))))a__length#(s(a__length(s(a__and(0, _x62)))))a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))
a__length#(cons(N, s(length(and(0, _x52)))))a__length#(s(a__length(a__and(0, _x52))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))
a__length#(cons(N, s(length(and(length(_x61), _x52)))))a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(and(_x51, _x42)))))a__length#(s(a__length(and(mark(_x51), _x42))))a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))
a__length#(cons(N, s(length(s(and(_x61, _x52))))))a__length#(s(a__length(s(and(mark(_x61), _x52)))))a__length#(cons(N, s(length(and(s(_x61), _x52)))))a__length#(s(a__length(a__and(s(mark(_x61)), _x52))))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(a__length(s(a__and(a__length(tt), _x62)))))
a__length#(cons(N, s(length(s(and(length(and(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__and(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(and(and(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))
a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(a__length(nil), _x62)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(cons(s(_x51), _x42))))a__length#(s(cons(s(mark(_x51)), _x42)))a__length#(cons(N, s(length(s(and(s(_x71), _x62))))))a__length#(s(a__length(s(a__and(s(mark(_x71)), _x62)))))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(cons(_x51, _x52)))))a__length#(s(a__length(cons(mark(_x51), _x52))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(nil)))))a__length#(s(a__length(s(nil))))
a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(cons(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62)))))
a__length#(cons(N, s(length(s(and(length(zeros), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__zeros), _x62)))))a__length#(cons(N, s(length(and(cons(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(s(and(nil, _x62))))))a__length#(s(a__length(s(a__and(nil, _x62)))))a__length#(cons(N, s(length(s(and(length(0), _x62))))))a__length#(s(a__length(s(a__and(a__length(0), _x62)))))
a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, s(length(and(tt, _x52)))))a__length#(s(a__length(a__and(tt, _x52))))
a__length#(cons(N, s(length(s(s(length(_x71)))))))a__length#(s(a__length(s(s(a__length(mark(_x71)))))))a__length#(cons(N, s(tt)))a__length#(s(tt))
a__length#(cons(N, s(cons(0, _x42))))a__length#(s(cons(0, _x42)))a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(a__length(mark(_x71)), _x52)))))
a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))
a__length#(cons(N, s(length(s(and(_x61, _x62))))))a__length#(s(length(s(a__and(mark(_x61), _x62)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(and(zeros, _x52)))))a__length#(s(a__length(a__and(a__zeros, _x52))))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(length(s(and(length(length(and(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(a__and(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(0))))a__length#(s(length(0)))
a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))
a__length#(cons(N, s(length(s(and(length(length(0)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(0)), _x62)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(mark(_x81))), _x52))))) 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
a__length#(s(a__length(s(and(a__length(a__length(a__length(mark(_x101)))), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(a__and(mark(_x101), _x102))), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(nil)), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(0)), _x52))))) 
a__length#(s(length(s(and(a__length(a__length(mark(_x81))), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(cons(mark(_x101), _x102))), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(tt)), _x52))))) 
a__length#(s(a__length(s(and(length(a__length(mark(_x81))), _x52))))) 
a__length#(s(a__length(s(and(a__length(length(mark(_x81))), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(s(mark(_x101)))), _x52))))) 
a__length#(s(a__length(s(and(a__length(a__length(a__zeros)), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(mark(_x81))), _x52))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(nil)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(nil)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52)))))) → a__length#(s(a__length(s(and(a__length(length(mark(_x81))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(_x101))), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(s(mark(_x101)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52)))))) → a__length#(s(a__length(s(and(length(a__length(mark(_x81))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(0)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(0)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(tt)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(tt)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52)))))) → a__length#(s(length(s(and(a__length(a__length(mark(_x81))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(zeros)), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(a__zeros)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(_x101, _x102))), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(a__and(mark(_x101), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(length(_x101))), _x52)))))) → a__length#(s(a__length(s(and(a__length(a__length(a__length(mark(_x101)))), _x52)))))

Problem 28: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(s(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(cons(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(tt)), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(tt)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(_x121)), length(s(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(mark(_x121)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(0, and(0, _x112))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(and(0, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(_x101, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(mark(_x101), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(zeros)), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(a__zeros)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(mark(_x131), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(zeros)))a__length#(s(a__zeros))a__length#(cons(N, s(length(s(and(length(s(s(cons(_x101, _x102)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(cons(mark(_x101), _x102)))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(nil)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(nil)), _x62)))))a__length#(cons(N, s(length(s(and(length(s(nil)), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(nil)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52))))))a__length#(s(a__length(s(and(length(a__length(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(cons(length(_x51), _x42))))a__length#(s(cons(a__length(mark(_x51)), _x42)))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(zeros)), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__zeros)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(s(zeros))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(a__zeros))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52))))))a__length#(s(length(s(and(a__length(length(mark(_x81))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(s(nil))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(nil))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(tt)), _x62)))))
a__length#(cons(N, s(length(s(_x51)))))a__length#(s(length(s(mark(_x51)))))a__length#(cons(N, s(length(s(and(length(s(zeros)), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__zeros)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(_x91, _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__length(mark(_x121)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52))))))a__length#(s(a__length(s(and(a__length(length(mark(_x81))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(cons(nil, _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(0, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(0, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(tt), length(tt)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(tt)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, s(_x111)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(mark(_x111))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(s(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(s(mark(_x111)), _x102))), _x52)))))
a__length#(cons(N, s(length(and(s(_x61), _x52)))))a__length#(s(a__length(a__and(s(mark(_x61)), _x52))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(cons(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(0))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(0))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(nil, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, tt))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(tt))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(s(_x101))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(mark(_x101)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(length(s(a__and(length(length(mark(_x81))), _x62)))))a__length#(cons(N, cons(_x21, _x22)))a__length#(cons(mark(_x21), _x22))
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a__length#(cons(N, s(length(s(and(length(s(0)), _x52))))))a__length#(s(a__length(s(and(length(s(0)), _x52)))))a__length#(cons(N, s(length(and(0, _x52)))))a__length#(s(a__length(a__and(0, _x52))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(zeros, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(cons(0, zeros), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(tt, _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(0)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(0)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(_x121, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(length(_x101))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(mark(_x101)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(and(_x101, _x102)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(a__and(mark(_x101), _x102)))), _x62)))))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(a__length(mark(_x111))))), _x52)))))
a__length#(cons(N, s(length(and(and(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(a__and(mark(_x61), _x62), _x52))))a__length#(cons(N, s(length(s(and(length(and(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__and(mark(_x81), _x82)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(length(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(a__length(mark(_x111)), _x102))), _x52)))))a__length#(cons(N, s(length(_x41))))a__length#(s(length(mark(_x41))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, nil)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(nil)))), _x52)))))a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(a__length(nil), _x62)))))
a__length#(cons(N, s(length(s(tt)))))a__length#(s(length(s(tt))))a__length#(cons(N, s(length(s(and(s(_x71), _x62))))))a__length#(s(a__length(s(a__and(s(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(and(length(s(length(_x91))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(a__length(mark(_x91)))), _x62)))))a__length#(cons(N, s(length(s(and(length(0), _x52))))))a__length#(s(length(s(and(length(0), _x52)))))
a__length#(cons(N, and(_x21, _x22)))a__length#(a__and(mark(_x21), _x22))a__length#(cons(N, s(length(s(cons(_x61, _x62))))))a__length#(s(a__length(s(cons(mark(_x61), _x62)))))
a__length#(cons(N, s(length(s(and(length(s(s(length(_x101)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(a__length(mark(_x101))))), _x62)))))a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(length(s(a__and(length(cons(mark(_x81), _x82)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x52))))))a__length#(s(a__length(s(and(length(a__length(mark(_x81))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(zeros)), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(a__zeros)), _x52)))))
a__length#(cons(N, s(length(s(s(nil))))))a__length#(s(a__length(s(s(nil)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(s(_x121), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(s(mark(_x121)), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(cons(_x101, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(cons(nil, _x42))))a__length#(s(cons(nil, _x42)))
a__length#(cons(N, s(length(s(s(_x61))))))a__length#(s(length(s(s(mark(_x61))))))a__length#(cons(N, s(length(s(and(length(s(zeros)), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(a__zeros)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(a__and(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(s(length(_x101)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(a__length(mark(_x101))))), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, tt)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(tt)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(zeros, _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(a__zeros, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(_x91))), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(s(mark(_x91)))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(tt, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(tt))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(tt))), _x62)))))
a__length#(cons(N, s(length(s(and(length(s(s(and(_x101, _x102)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(a__and(mark(_x101), _x102)))), _x62)))))a__length#(cons(N, s(length(s(and(length(_x71), _x62))))))a__length#(s(length(s(a__and(a__length(mark(_x71)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(_x111, _x112), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(a__and(mark(_x111), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(s(and(_x71, _x72)))))))a__length#(s(a__length(s(s(a__and(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(length(s(a__and(length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(0)))a__length#(s(0))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(nil))))a__length#(s(a__length(nil)))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(a__length(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(a__and(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(nil)), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(nil)), _x52)))))a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(length(nil), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(tt)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(tt)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(mark(_x121), _x122), cons(_x111, and(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(_x111), length(_x111)))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(length(mark(_x111))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(tt))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(tt))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(length(_x91))), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(a__length(mark(_x91)))), _x62)))))
a__length#(cons(N, s(length(s(and(zeros, _x62))))))a__length#(s(a__length(s(a__and(a__zeros, _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(0, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(0, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(s(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(s(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(0))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(0))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(zeros, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__zeros, _x112)))), _x52)))))a__length#(cons(N, s(cons(tt, _x42))))a__length#(s(cons(tt, _x42)))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(s(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(0)), _x62))))))a__length#(s(length(s(a__and(a__length(s(0)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, 0)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(0)))), _x52)))))a__length#(cons(N, s(length(s(s(s(_x71)))))))a__length#(s(a__length(s(s(s(mark(_x71)))))))
a__length#(cons(N, s(length(s(and(0, _x62))))))a__length#(s(a__length(s(a__and(0, _x62)))))a__length#(cons(N, s(length(s(s(cons(_x71, _x72)))))))a__length#(s(a__length(s(s(cons(mark(_x71), _x72))))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(length(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(length(mark(_x71)), _x52)))))
a__length#(cons(N, s(and(_x41, _x42))))a__length#(s(a__and(mark(_x41), _x42)))a__length#(cons(N, s(length(s(and(length(length(s(s(_x101)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(s(mark(_x101))))), _x62)))))
a__length#(cons(N, s(length(and(length(_x61), _x52)))))a__length#(s(a__length(a__and(a__length(mark(_x61)), _x52))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, _x91)))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(mark(_x91))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(zeros)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(a__zeros)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(0)), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(0)), _x52)))))a__length#(cons(N, s(length(and(_x51, _x42)))))a__length#(s(a__length(and(mark(_x51), _x42))))
a__length#(cons(N, s(length(s(and(length(and(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(length(a__and(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(s(and(length(s(tt)), _x52))))))a__length#(s(a__length(s(and(length(s(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(_x91))), _x62))))))a__length#(s(a__length(s(a__and(length(a__length(s(mark(_x91)))), _x62)))))
a__length#(cons(N, s(length(zeros))))a__length#(s(a__length(a__zeros)))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, _x91)))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(mark(_x91))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(cons(mark(_x111), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(_x101))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(s(mark(_x101)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(zeros, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(a__zeros, _x102))), _x52)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(zeros, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__zeros, _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(nil))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(nil))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(a__length(s(and(length(length(cons(nil, _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, zeros))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__zeros))), _x52)))))a__length#(cons(N, s(length(s(s(0))))))a__length#(s(a__length(s(s(0)))))
a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62)))))a__length#(cons(N, s(length(and(cons(_x61, _x62), _x52)))))a__length#(s(a__length(a__and(cons(mark(_x61), _x62), _x52))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(zeros, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(zeros, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(0), _x62))))))a__length#(s(a__length(s(a__and(a__length(0), _x62)))))
a__length#(cons(N, s(length(and(tt, _x52)))))a__length#(s(a__length(a__and(tt, _x52))))a__length#(cons(N, s(length(s(and(length(s(s(_x91))), _x62))))))a__length#(s(a__length(s(a__and(length(s(s(mark(_x91)))), _x62)))))
a__length#(cons(N, s(length(s(and(length(zeros), _x62))))))a__length#(s(a__length(s(a__and(length(a__zeros), _x62)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(length(s(a__and(length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52))))))a__length#(s(a__length(s(and(length(length(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, 0))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(0))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(length(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__length(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(mark(_x111), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(s(_x101)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(s(mark(_x101))))), _x62)))))a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))
a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(length(s(a__and(length(tt), _x62)))))a__length#(cons(N, s(length(s(and(length(s(_x81)), _x52))))))a__length#(s(a__length(s(and(length(s(mark(_x81))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(length(_x121)), length(length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(a__length(mark(_x121)))))), _x52)))))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))
a__length#(cons(N, s(length(s(and(length(length(cons(0, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(0, _x102))), _x52)))))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(_x121))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(mark(_x121)))))), _x52))))) 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
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(tt))))), _x52))))) 
a__length#(s(length(s(and(a__length(s(s(a__length(s(mark(_x121)))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(a__zeros))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(a__and(mark(_x131), _x132)))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(nil))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(s(mark(_x131))))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(cons(mark(_x131), _x132)))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(0))))), _x52))))) 
a__length#(s(a__length(s(and(length(s(s(a__length(s(mark(_x121)))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(s(a__length(mark(_x131))))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(s(mark(_x121)))))), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(_x121))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(mark(_x121)))))), _x52))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(length(_x131)))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(a__length(mark(_x131))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(and(_x131, _x132)))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(a__and(mark(_x131), _x132)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(nil))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(nil))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(zeros))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(a__zeros))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(_x121))))), _x52)))))) → a__length#(s(length(s(and(a__length(s(s(a__length(s(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(s(_x131)))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(s(mark(_x131))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(tt))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(tt))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(0))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(0))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(_x121))))), _x52)))))) → a__length#(s(a__length(s(and(length(s(s(a__length(s(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(cons(_x131, _x132)))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(s(cons(mark(_x131), _x132)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(s(mark(_x121)), cons(_x111, s(_x121))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(s(mark(_x121)))))), _x52)))))

Problem 29: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(s(s(0))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(0))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(nil, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(and(_x111, _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__and(mark(_x111), _x112), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(s(_x101))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(mark(_x101)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(s(zeros))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(s(a__zeros))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(length(s(a__and(length(length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(tt))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(s(tt))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(tt, _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(tt, _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(nil)), length(s(nil))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(nil))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(_x131), _x132), and(cons(_x131, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(mark(_x131), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(0, _x122), cons(_x111, and(0, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(length(a__and(0, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(cons(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x81))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(mark(_x81)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(0)), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(0)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(0, _x132))), length(s(and(0, _x132)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(a__and(0, _x132)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(_x111, _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(a__and(mark(_x111), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(tt)), _x62)))))
a__length#(cons(N, s(length(s(and(length(zeros), _x62))))))a__length#(s(length(s(a__and(length(a__zeros), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(cons(_x141, _x142), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(cons(mark(_x141), _x142), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(_x131))), length(s(length(_x131)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(mark(_x131))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x52))))))a__length#(s(length(s(and(a__length(s(mark(_x81))), _x52)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(and(length(length(cons(s(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(cons(s(mark(_x111)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(_x111, _x112), _x82))), _x52))))))a__length#(s(length(s(and(length(length(and(a__and(mark(_x111), _x112), _x82))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(0, _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(cons(0, _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(cons(_x131, _x132))), length(s(cons(_x131, _x132)))))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(length(s(cons(mark(_x131), _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(0, and(0, _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(a__length(and(0, _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(tt), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__length(mark(_x111))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(0)), _x52))))))a__length#(s(a__length(s(and(a__length(s(0)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(nil, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(_x131, _x132))), length(s(and(_x131, _x132)))))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(length(s(a__and(mark(_x131), _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(cons(0, zeros), _x92))), _x62)))))
a__length#(cons(N, s(zeros)))a__length#(s(cons(0, zeros)))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(and(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__and(mark(_x131), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(nil, _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(nil, _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(length(s(and(a__length(length(cons(nil, _x102))), _x52)))))
a__length#(cons(N, s(cons(length(s(_x61)), _x42))))a__length#(s(cons(a__length(s(mark(_x61))), _x42)))a__length#(cons(N, s(length(s(and(length(s(and(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(a__and(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(0), length(0)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(0)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(_x101, _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(mark(_x101), _x92))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(a__length(a__zeros), _x92))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(0, _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(0, _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(0)), _x62))))))a__length#(s(length(s(a__and(length(s(0)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(length(_x121), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(a__length(mark(_x121)), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(zeros))), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(s(zeros))), _x62)))))
a__length#(cons(N, s(length(and(nil, _x52)))))a__length#(s(a__length(a__and(nil, _x52))))a__length#(cons(N, s(length(s(and(length(s(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(cons(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(a__length(s(a__and(length(tt), _x62)))))a__length#(cons(N, s(length(s(and(length(s(and(length(0), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(0), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(s(nil))), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(s(nil))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(nil)), _x52))))))a__length#(s(a__length(s(and(a__length(s(nil)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(_x121), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(mark(_x121)), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(length(s(a__and(length(nil), _x62)))))
a__length#(cons(N, s(length(s(and(length(and(_x81, _x82)), _x62))))))a__length#(s(length(s(a__and(length(a__and(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))
a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(cons(a__length(mark(_x111)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(cons(_x111, _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(cons(mark(_x111), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(nil, _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(length(_x101))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(a__length(mark(_x101)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(cons(0, zeros)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(_x111), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(mark(_x111)), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(and(cons(_x151, _x152), _x142), _x132))), length(s(and(and(cons(_x151, _x152), _x142), _x132)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(a__and(a__and(cons(mark(_x151), _x152), _x142), _x132)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(length(cons(cons(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(_x91, _x92))), _x62))))))a__length#(s(length(s(a__and(length(length(a__and(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(s(_x131))), length(s(s(_x131)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(s(mark(_x131))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(zeros, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(cons(cons(0, zeros), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(_x61, _x52))))))a__length#(s(a__length(s(and(mark(_x61), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(_x121), and(_x121, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(mark(_x121), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(a__length(s(a__and(a__length(tt), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(length(s(and(length(s(a__length(length(s(0))))), _x52)))))
a__length#(cons(N, s(cons(length(tt), _x42))))a__length#(s(cons(a__length(tt), _x42)))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(a__length(mark(_x111))))), _x52)))))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(and(_x131, _x132), _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(cons(cons(a__and(mark(_x131), _x132), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(s(_x121), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(s(mark(_x121)), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(0))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(s(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(s(mark(_x111)), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(nil))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(s(nil))), _x52)))))
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a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(_x131, _x122))), length(s(and(_x131, _x122)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(and(mark(_x131), _x122)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(mark(_x141)), _x132), _x122), cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(s(nil), cons(_x111, s(nil))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(length(s(nil))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(length(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__length(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(and(_x121, _x122)), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(a__and(mark(_x121), _x122)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(and(length(length(cons(0, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(0, _x102))), _x52)))))
a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(s(s(0))), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(s(s(0))), _x62))))) 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
a__length#(s(length(s(a__and(a__length(s(s(0))), _x62))))) 
a__length#(s(a__length(s(and(a__length(s(s(0))), _x52))))) 
a__length#(s(a__length(s(a__and(length(s(s(0))), _x62))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(s(s(0))), _x62)))))) → a__length#(s(a__length(s(a__and(a__length(s(s(0))), _x62))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(s(s(0))), _x62)))))) → a__length#(s(length(s(a__and(a__length(s(s(0))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(s(0))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(0))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(0))), _x62)))))) → a__length#(s(a__length(s(a__and(length(s(s(0))), _x62)))))

Problem 30: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(a__and(tt, _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(length(tt)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(_x131))), length(s(length(_x131)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(mark(_x131))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(cons(nil, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(cons(nil, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(nil), _x132), and(cons(nil, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(nil, _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(0, _x112), cons(0, and(0, _x112))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(length(and(0, _x112))))), _x52)))))
a__length#(cons(N, s(cons(and(_x51, _x52), _x42))))a__length#(s(cons(a__and(mark(_x51), _x52), _x42)))a__length#(cons(N, s(length(s(and(length(length(cons(s(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(cons(s(mark(_x111)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(_x111, _x112), _x82))), _x52))))))a__length#(s(length(s(and(length(length(and(a__and(mark(_x111), _x112), _x82))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(and(_x111, _x102), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(and(mark(_x111), _x102), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(cons(_x131, _x132))), length(s(cons(_x131, _x132)))))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(length(s(cons(mark(_x131), _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(zeros)), _x132), _x112), cons(cons(mark(s(zeros)), _x132), and(cons(s(zeros), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__zeros), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(tt), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x101))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(mark(_x101)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__length(mark(_x111))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(cons(0, zeros), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(nil, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(0)), _x52))))))a__length#(s(a__length(s(and(a__length(s(0)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(s(s(_x181)))), _x132), _x112), cons(cons(mark(s(s(s(_x181)))), _x132), and(cons(s(s(s(_x181))), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(and(cons(s(s(s(mark(_x181)))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(zeros)))a__length#(s(cons(0, zeros)))
a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(length(s(and(a__length(length(cons(nil, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(0)), _x62))))))a__length#(s(a__length(s(a__and(length(length(0)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(nil))), _x132), _x112), cons(cons(mark(s(length(nil))), _x132), and(cons(s(length(nil)), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(nil)), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(s(_x141), _x122), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(and(s(mark(_x141)), _x122), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(s(cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(a__length(a__zeros), _x92))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(0)), _x62))))))a__length#(s(length(s(a__and(length(s(0)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(s(zeros))), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(s(zeros))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(cons(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(and(nil, _x52)))))a__length#(s(a__length(a__and(nil, _x52))))a__length#(cons(N, s(length(s(and(length(s(nil)), _x52))))))a__length#(s(a__length(s(and(a__length(s(nil)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(s(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(s(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(_x121), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(mark(_x121)), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(length(s(a__and(length(nil), _x62)))))a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(length(s(and(length(s(and(cons(_x111, _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(cons(mark(_x111), _x112), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(cons(0, zeros)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(_x111), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(mark(_x111)), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(s(_x131))), length(s(s(_x131)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(s(mark(_x131))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x112), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(and(mark(_x121), _x112), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(0)), _x132), and(cons(s(0), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(0), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(length(_x181)))), _x132), _x112), cons(cons(mark(s(length(length(_x181)))), _x132), and(cons(s(length(length(_x181))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(a__length(mark(_x181)))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(_x121), and(_x121, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(mark(_x121), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(a__length(s(a__and(a__length(tt), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(length(s(and(length(s(a__length(length(s(0))))), _x52)))))
a__length#(cons(N, s(cons(length(tt), _x42))))a__length#(s(cons(a__length(tt), _x42)))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(nil)), _x132), _x112), cons(cons(mark(s(nil)), _x132), and(cons(s(nil), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(nil), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(a__length(mark(_x111))))), _x52)))))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(0))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x91)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(mark(_x91))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(s(nil))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(s(nil))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(length(s(a__and(a__length(s(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(length(_x141), _x132))), length(s(and(length(_x141), _x132)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(a__and(a__length(mark(_x141)), _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(zeros), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__zeros), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(0, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(length(cons(cons(0, _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(cons(_x181, _x182)))), _x132), _x112), cons(cons(mark(s(length(cons(_x181, _x182)))), _x132), and(cons(s(length(cons(_x181, _x182))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(cons(mark(_x181), _x182))), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(zeros), _x132), and(cons(zeros, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__zeros, _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(length(cons(mark(_x81), _x82)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(and(_x141, _x142), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(a__and(mark(_x141), _x142), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(zeros, _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(a__and(a__zeros, _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(mark(_x121), _x122), cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(nil, _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(nil, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(mark(_x131), _x132), _x122), cons(_x111, and(cons(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(mark(_x131), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(s(_x131)))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(s(s(mark(_x131))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(0, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(length(cons(0, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(nil))), _x52)))))a__length#(cons(N, s(length(0))))a__length#(s(length(0)))
a__length#(cons(N, s(length(s(and(length(length(and(tt, _x92))), _x52))))))a__length#(s(length(s(and(a__length(length(a__and(tt, _x92))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x81))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(mark(_x81)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(nil, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(_x101, _x102), _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(a__and(mark(_x101), _x102), _x92))), _x62)))))
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a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(s(_x171))), _x132), and(cons(length(s(_x171)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(s(mark(_x171))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(length(_x101)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(a__length(mark(_x101))))), _x62)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(tt, _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(zeros, _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(a__zeros, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(s(cons(0, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(cons(0, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(tt))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(s(tt))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(length(a__length(cons(mark(_x91), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(and(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(a__and(mark(_x131), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(_x111, _x112), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(a__and(mark(_x111), _x112), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(length(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(0, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(cons(0, _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(_x151)), _x132), _x112), cons(cons(mark(s(_x151)), _x132), and(cons(s(_x151), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(and(cons(s(mark(_x151)), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(zeros, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(a__zeros, _x122))))), _x52)))))
a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(a__and(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(a__length(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(tt, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(tt, _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(length(nil), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(and(_x151, _x152)), _x132), and(cons(and(_x151, _x152), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__and(mark(_x151), _x152), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(s(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(s(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(length(s(a__length(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(cons(_x131, _x132), _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__and(cons(mark(_x131), _x132), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x112), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(cons(and(mark(_x121), _x112), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(zeros), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(length(a__zeros), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(tt), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(length(mark(_x71)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(s(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(s(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(length(a__zeros), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(zeros), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x52))))))a__length#(s(length(s(and(a__length(a__length(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(nil, _x122), cons(_x111, cons(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(cons(_x171, _x172))), _x132), and(cons(s(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(a__length(cons(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(and(length(_x121), _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__and(a__length(mark(_x121)), _x112), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(tt)), _x132), _x112), cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(tt), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(tt)), _x52))))))a__length#(s(a__length(s(and(length(s(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(nil))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(s(nil))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(zeros, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(a__zeros, _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(tt, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(tt, _x112)))), _x52)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(s(and(length(s(s(nil))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(nil))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(0)), _x132), and(cons(length(0), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(0), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(0), _x62))))))a__length#(s(a__length(s(a__and(a__length(0), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(_x131, _x122))), length(s(and(_x131, _x122)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(and(mark(_x131), _x122)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(mark(_x141)), _x132), _x122), cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(length(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__length(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(and(_x121, _x122)), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(a__and(mark(_x121), _x122)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(a__length(length(and(a__and(tt, _x112), _x82))), _x62))))) 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
a__length#(s(length(s(a__and(a__length(length(and(mark(_x101), _x82))), _x62))))) 
a__length#(s(length(s(a__and(a__length(length(and(and(tt, _x102), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(tt, _x112), _x82))), _x62))))) 
a__length#(s(length(s(and(a__length(length(and(a__and(tt, _x112), _x82))), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(a__length(length(and(a__and(tt, _x112), _x82))), _x62))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x102), _x82))), _x62)))))) → a__length#(s(length(s(a__and(a__length(length(and(and(tt, _x102), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(tt, _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x101), _x82))), _x62)))))) → a__length#(s(length(s(a__and(a__length(length(and(mark(_x101), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x112), _x82))), _x52)))))) → a__length#(s(length(s(and(a__length(length(and(a__and(tt, _x112), _x82))), _x52)))))

Problem 31: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(s(and(and(_x111, _x102), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(and(mark(_x111), _x102), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(_x141))), _x122), cons(_x111, cons(s(length(_x141)), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(s(_x141), _x122), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(and(s(mark(_x141)), _x122), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(nil)), _x52))))))a__length#(s(a__length(s(and(a__length(s(nil)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(_x121), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(mark(_x121)), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(length(s(a__and(length(nil), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(tt, _x102), _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(and(tt, _x102), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(s(cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(s(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(tt))))a__length#(s(a__length(tt)))a__length#(cons(N, s(length(s(and(and(_x71, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(mark(_x71), _x72), _x62)))))
a__length#(cons(N, s(length(s(and(length(s(and(cons(_x111, _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(cons(mark(_x111), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(s(s(zeros)))), _x132), _x112), cons(cons(mark(s(s(s(zeros)))), _x132), and(cons(s(s(s(zeros))), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(and(cons(s(s(s(a__zeros))), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(cons(0, zeros)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(_x111), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(mark(_x111)), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(length(s(and(a__length(a__zeros), _x92))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x112), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(and(mark(_x121), _x112), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(s(_x131))), length(s(s(_x131)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(s(mark(_x131))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(0)), _x132), and(cons(s(0), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(0), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(tt), _x132), _x122), cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(length(_x181)))), _x132), _x112), cons(cons(mark(s(length(length(_x181)))), _x132), and(cons(s(length(length(_x181))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(a__length(mark(_x181)))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(a__length(zeros), _x92))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(length(a__zeros), _x92))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(_x121), and(_x121, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(mark(_x121), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(cons(nil, _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(s(cons(nil, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(length(s(and(length(s(a__length(length(s(0))))), _x52)))))
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a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(zeros)), _x132), and(cons(s(zeros), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(s(a__zeros), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(s(_x171))), _x132), and(cons(length(s(_x171)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(s(mark(_x171))), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(tt, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(tt, _x122))))), _x52)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(length(s(s(zeros)))))
a__length#(cons(N, s(length(s(and(length(length(s(length(_x101)))), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(s(a__length(mark(_x101))))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(tt, _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(zeros, _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(a__zeros, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(s(tt)), length(cons(_x121, length(s(tt))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(length(a__length(s(tt))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x101))))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(s(a__length(mark(_x101)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(s(cons(0, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(cons(0, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, and(tt, _x111))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(mark(_x111))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(tt))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(s(tt))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(length(a__length(cons(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(_x111, _x112), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(a__and(mark(_x111), _x112), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(and(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(a__and(mark(_x131), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(length(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(0, _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(length(cons(cons(0, _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(_x151)), _x132), _x112), cons(cons(mark(s(_x151)), _x132), and(cons(s(_x151), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(and(cons(s(mark(_x151)), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(a__and(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(zeros, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(a__zeros, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(a__length(s(and(length(s(a__and(length(tt), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(a__length(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(s(and(_x151, _x152))), length(cons(_x121, length(s(and(_x151, _x152)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(s(a__and(mark(_x151), _x152)))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(tt, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(tt, _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(s(zeros)), length(cons(_x121, length(s(zeros))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(s(zeros))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(length(nil), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(and(_x151, _x152)), _x132), and(cons(and(_x151, _x152), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__and(mark(_x151), _x152), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(cons(and(_x51, _x42), _x42))))a__length#(s(cons(and(mark(_x51), _x42), _x42)))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(tt, _x122), cons(nil, and(tt, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(a__and(tt, _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(a__length(tt), _x132), and(cons(length(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(tt), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(s(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(s(mark(_x131)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(length(s(a__length(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(cons(_x131, _x132), _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__and(cons(mark(_x131), _x132), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(tt), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x112), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(cons(and(mark(_x121), _x112), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(tt), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(length(a__zeros), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(zeros), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x92))), _x52))))))a__length#(s(length(s(and(a__length(length(a__and(cons(0, zeros), _x92))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(tt), _x132), _x122), cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(s(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(s(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(length(mark(_x71)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(zeros), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(length(a__zeros), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x52))))))a__length#(s(length(s(and(a__length(a__length(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(nil, _x122), cons(_x111, cons(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(length(_x141)), length(cons(_x121, length(length(_x141))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(a__length(mark(_x141)))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(cons(_x171, _x172))), _x132), and(cons(s(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(a__length(cons(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(and(length(_x121), _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__and(a__length(mark(_x121)), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(tt)), _x132), _x112), cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(tt), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(mark(_x101), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(tt)), _x52))))))a__length#(s(a__length(s(and(length(s(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(nil))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(s(nil))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(zeros, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(a__zeros, _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(a__length(cons(0, zeros)), _x92))), _x52)))))
a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(nil, _x122), length(cons(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(cons(nil, _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(tt, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(tt, _x112)))), _x52)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(s(and(length(s(s(nil))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(nil))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(s(_x141)))), length(s(length(s(_x141))))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(s(mark(_x141)))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(0)), _x132), and(cons(length(0), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(0), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(s(length(_x141)), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(s(a__length(mark(_x141))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(0), _x62))))))a__length#(s(a__length(s(a__and(a__length(0), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(_x131, _x122))), length(s(and(_x131, _x122)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(and(mark(_x131), _x122)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(s(length(zeros)), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(s(a__length(a__zeros)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(cons(0, zeros), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(mark(_x141)), _x132), _x122), cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(length(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__length(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(and(_x121, _x122)), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(a__and(mark(_x121), _x122)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(and(length(length(cons(length(0), length(cons(0, length(0)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(length(0)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(zeros))), length(s(length(zeros)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(a__zeros)))))), _x52)))))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(_x141))), _x122), cons(_x111, cons(s(length(_x141)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52))))) 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
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(s(mark(_x151)))), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(length(mark(_x141))), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(a__and(mark(_x151), _x152))), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(nil)), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(a__length(mark(_x151)))), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(tt)), _x122))))), _x52))))) 
a__length#(s(length(s(and(a__length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(a__zeros)), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(0)), _x122))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(cons(mark(_x151), _x152))), _x122))))), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(_x141))), _x122), cons(_x111, cons(s(length(_x141)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(and(_x151, _x152)))), _x122), cons(_x111, cons(s(length(and(_x151, _x152))), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(a__and(mark(_x151), _x152))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(_x141))), _x122), cons(_x111, cons(s(length(_x141)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(length(mark(_x141))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(_x141))), _x122), cons(_x111, cons(s(length(_x141)), _x122))))), _x52)))))) → a__length#(s(length(s(and(a__length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(s(_x151)))), _x122), cons(_x111, cons(s(length(s(_x151))), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(s(mark(_x151)))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(nil))), _x122), cons(_x111, cons(s(length(nil)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(nil)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(tt))), _x122), cons(_x111, cons(s(length(tt)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(tt)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(cons(_x151, _x152)))), _x122), cons(_x111, cons(s(length(cons(_x151, _x152))), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(cons(mark(_x151), _x152))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(_x141))), _x122), cons(_x111, cons(s(length(_x141)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(length(s(s(length(cons(s(a__length(mark(_x141))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(_x151)))), _x122), cons(_x111, cons(s(length(length(_x151))), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(a__length(mark(_x151)))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(zeros))), _x122), cons(_x111, cons(s(length(zeros)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(a__zeros)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(0))), _x122), cons(_x111, cons(s(length(0)), _x122))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(0)), _x122))))), _x52)))))

Problem 32: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(s(and(and(_x111, _x102), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(and(mark(_x111), _x102), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(s(_x141), _x122), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(and(s(mark(_x141)), _x122), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(a__zeros, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x112), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(and(mark(_x121), _x112), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(tt), _x62))))))a__length#(s(a__length(s(a__and(a__length(tt), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(length(s(and(length(s(a__length(length(s(0))))), _x52)))))
a__length#(cons(N, s(cons(length(tt), _x42))))a__length#(s(cons(a__length(tt), _x42)))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(nil)), _x132), _x112), cons(cons(mark(s(nil)), _x132), and(cons(s(nil), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(nil), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(a__length(mark(_x111))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(0)))), _x122), cons(_x111, cons(s(length(length(0))), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(cons(s(length(length(0))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(s(_x111), _x102))), _x52))))))a__length#(s(a__length(s(and(length(a__length(cons(s(mark(_x111)), _x102))), _x52)))))a__length#(cons(N, zeros))a__length#(cons(0, zeros))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(0)), length(s(0))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(0))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(length(s(a__and(a__length(s(mark(_x81))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x91)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(mark(_x91))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(s(nil))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(s(nil))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(_x151)))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(s(mark(_x151))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(a__and(0, _x142), _x112), cons(mark(and(0, _x142)), and(and(0, _x142), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(and(a__and(0, _x142), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(length(_x141), _x132))), length(s(and(length(_x141), _x132)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(a__and(a__length(mark(_x141)), _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(_x131), length(cons(_x121, length(_x131)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(s(a__length(a__length(mark(_x131))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, _x91)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(mark(_x91))))), _x52)))))a__length#(cons(N, s(length(s(and(length(zeros), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__zeros), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(nil), _x132), and(cons(nil, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(nil, _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(length(length(tt)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(0, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(length(cons(cons(0, _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(tt))), _x122), cons(_x111, cons(s(length(tt)), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(s(a__length(tt)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(cons(_x181, _x182)))), _x132), _x112), cons(cons(mark(s(length(cons(_x181, _x182)))), _x132), and(cons(s(length(cons(_x181, _x182))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(cons(mark(_x181), _x182))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(s(length(_x151))), length(cons(_x121, length(s(length(_x151)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(s(a__length(mark(_x151))))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(cons(_x161, _x162))))), _x122), cons(_x111, cons(s(length(length(cons(_x161, _x162)))), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(cons(s(length(a__length(cons(mark(_x161), _x162)))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(zeros), _x132), and(cons(zeros, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__zeros, _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(length(cons(mark(_x81), _x82)), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(and(_x141, _x142), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(a__and(mark(_x141), _x142), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(_x81)), _x62))))))a__length#(s(length(s(a__and(a__length(a__length(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(zeros, _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(a__and(a__zeros, _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(cons(zeros, _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(cons(a__zeros, _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(mark(_x121), _x122), cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(nil, _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(nil, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(mark(_x131), _x132), _x122), cons(_x111, and(cons(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(mark(_x131), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(s(_x131)))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(s(s(mark(_x131))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(0, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(length(cons(0, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(length(s(and(length(s(a__and(a__length(tt), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(nil))), _x52)))))
a__length#(cons(N, s(length(s(and(length(nil), _x52))))))a__length#(s(length(s(and(length(nil), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(s(length(_x171)))))), _x122), cons(_x111, cons(s(length(length(s(length(_x171))))), _x122))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(cons(s(length(length(s(a__length(mark(_x171)))))), _x122))))), _x52)))))
a__length#(cons(N, s(length(0))))a__length#(s(length(0)))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(length(cons(0, zeros)), _x132), and(cons(length(zeros), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(length(cons(0, zeros)), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(tt, _x92))), _x52))))))a__length#(s(length(s(and(a__length(length(a__and(tt, _x92))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x81))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(mark(_x81)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(nil, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(_x101, _x102), _x92))), _x62))))))a__length#(s(length(s(a__and(a__length(length(a__and(a__and(mark(_x101), _x102), _x92))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(nil)), length(s(nil))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(length(s(nil))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(and(_x151, _x152)))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(a__and(mark(_x151), _x152)))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(zeros)), _x102), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(and(s(a__length(a__zeros)), _x102), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(cons(_x171, _x172))), _x132), _x112), cons(cons(mark(s(cons(_x171, _x172))), _x132), and(cons(s(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(and(cons(s(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, and(zeros, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__zeros, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(_x121), and(_x121, _x112))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(and(mark(_x121), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(s(_x171))), _x132), and(cons(s(s(_x171)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(s(s(mark(_x171))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, and(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(a__and(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(s(0), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(s(0), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(s(length(_x121)), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(a__length(cons(s(a__length(mark(_x121))), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(cons(s(_x121), _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(cons(s(mark(_x121)), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(length(length(_x151)))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(a__length(a__length(mark(_x151))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(cons(_x141, _x142), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(a__and(cons(mark(_x141), _x142), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(length(_x61))))))a__length#(s(a__length(s(a__length(mark(_x61))))))
a__length#(cons(N, s(length(s(and(length(s(s(s(_x111)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(s(mark(_x111))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(0), _x132), and(cons(0, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(0, _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(_x111, _x112), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(cons(mark(_x111), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(length(nil))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(a__length(nil))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(and(and(_x81, _x82), _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(a__and(mark(_x81), _x82), _x72), _x62)))))a__length#(cons(N, s(length(s(and(tt, _x62))))))a__length#(s(a__length(s(a__and(tt, _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(s(zeros))), length(s(s(zeros)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(s(a__zeros)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(length(cons(mark(_x111), _x112)))), _x52)))))
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a__length#(cons(N, s(length(s(and(length(length(cons(length(s(and(_x151, _x152))), length(cons(_x121, length(s(and(_x151, _x152)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(s(a__and(mark(_x151), _x152)))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(tt, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(tt, _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(length(s(zeros)), length(cons(_x121, length(s(zeros))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(s(zeros))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(nil), _x62))))))a__length#(s(a__length(s(a__and(length(nil), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(and(_x151, _x152)), _x132), and(cons(and(_x151, _x152), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__and(mark(_x151), _x152), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(cons(_x161, _x162))))), _x122), cons(_x111, cons(s(length(length(cons(_x161, _x162)))), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(cons(s(a__length(a__length(cons(mark(_x161), _x162)))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(cons(and(_x51, _x42), _x42))))a__length#(s(cons(and(mark(_x51), _x42), _x42)))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(a__and(nil, _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(tt, _x122), cons(nil, and(tt, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(a__and(tt, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(a__length(tt), _x132), and(cons(length(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(tt), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(s(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(s(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(length(s(a__length(cons(a__and(mark(_x121), _x122), _x112)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(cons(_x131, _x132), _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__and(cons(mark(_x131), _x132), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(tt), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(_x121, _x112), _x112)))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(cons(and(mark(_x121), _x112), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(tt), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(length(a__zeros), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(zeros), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x92))), _x52))))))a__length#(s(length(s(and(a__length(length(a__and(cons(0, zeros), _x92))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(tt), _x132), _x122), cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(nil)), _x52))))))a__length#(s(a__length(s(and(length(s(nil)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(s(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(s(mark(_x141)), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(length(mark(_x71)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(a__length(zeros), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(zeros), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(length(a__zeros), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x52))))))a__length#(s(length(s(and(a__length(a__length(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(nil, _x122), cons(_x111, cons(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(cons(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(length(_x141)), length(cons(_x121, length(length(_x141))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(a__length(mark(_x141)))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(cons(_x171, _x172))), _x132), and(cons(s(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(a__length(cons(mark(_x121), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(and(length(_x121), _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__and(a__length(mark(_x121)), _x112), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(tt)), _x132), _x112), cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(tt), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(nil), and(nil, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(nil, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, _x102))), _x52))))))a__length#(s(length(s(and(a__length(a__length(cons(mark(_x101), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(tt)), _x52))))))a__length#(s(a__length(s(and(length(s(tt)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(nil))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(s(nil))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(zeros, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(a__zeros, _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x92))), _x52))))))a__length#(s(a__length(s(and(a__length(s(and(a__length(cons(0, zeros)), _x92))), _x52)))))a__length#(cons(N, s(length(s(and(length(cons(_x81, _x82)), _x62))))))a__length#(s(a__length(s(a__and(a__length(cons(mark(_x81), _x82)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(nil, _x122), length(cons(nil, _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(cons(nil, _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(and(_x161, _x162))))), _x122), cons(_x111, cons(s(length(length(and(_x161, _x162)))), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(cons(s(length(a__length(a__and(mark(_x161), _x162)))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(tt, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(tt, _x112)))), _x52)))))a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))
a__length#(cons(N, s(length(s(and(length(s(s(nil))), _x62))))))a__length#(s(a__length(s(a__and(a__length(s(s(nil))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(s(_x141)))), length(s(length(s(_x141))))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(s(mark(_x141)))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(length(s(_x131))), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(a__length(s(mark(_x131)))), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x62))))))a__length#(s(a__length(s(a__and(a__length(a__length(tt)), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(0)), _x132), and(cons(length(0), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(0), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(s(length(_x141)), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(s(a__length(mark(_x141))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, and(nil, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(nil, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(and(zeros, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(a__zeros, _x72), _x62)))))
a__length#(cons(N, s(length(s(and(length(0), _x62))))))a__length#(s(a__length(s(a__and(a__length(0), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(_x131, _x122))), length(s(and(_x131, _x122)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(and(mark(_x131), _x122)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(length(length(_x131))), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(a__length(a__length(mark(_x131)))), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(s(length(zeros)), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(s(a__length(a__zeros)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(s(s(_x171)))))), _x122), cons(_x111, cons(s(length(length(s(s(_x171))))), _x122))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(cons(s(length(length(s(s(mark(_x171)))))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(cons(0, zeros), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(mark(_x141)), _x132), _x122), cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(length(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__length(mark(_x131)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(length(_x121)), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(length(a__length(mark(_x121))), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(and(_x121, _x122)), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(a__and(mark(_x121), _x122)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(and(length(length(cons(length(0), length(cons(0, length(0)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(length(0)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(zeros))), length(s(length(zeros)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(a__zeros)))))), _x52)))))a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(and(a__zeros, _x112))))), _x52))))) 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
a__length#(s(a__length(s(and(a__length(s(s(a__length(and(zeros, _x112))))), _x52))))) 
a__length#(s(length(s(and(a__length(s(s(a__length(and(a__zeros, _x112))))), _x52))))) 
a__length#(s(a__length(s(and(length(s(s(a__length(and(a__zeros, _x112))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(length(and(a__zeros, _x112))))), _x52))))) 
a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(0, zeros), _x112))))), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(and(a__zeros, _x112))))), _x52))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(and(zeros, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(0, zeros), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(length(s(and(a__length(s(s(a__length(and(a__zeros, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(a__zeros, _x112), cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(a__length(s(and(a__length(s(s(length(and(a__zeros, _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(zeros), and(zeros, _x112))))), _x52)))))) → a__length#(s(a__length(s(and(length(s(s(a__length(and(a__zeros, _x112))))), _x52)))))

Problem 33: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x172)))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(s(and(and(_x111, _x102), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(and(mark(_x111), _x102), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x151)))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(s(a__length(mark(_x151))))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(cons(_x161, _x162))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(s(cons(mark(_x161), _x162))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(0))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(s(a__length(0))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(s(_x141), _x122), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(and(s(mark(_x141)), _x122), _x122))))), _x52)))))
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a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(tt)))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(tt)))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(s(cons(and(length(_x141), _x132), _x122)), _x102))), _x52))))))a__length#(s(a__length(s(and(length(a__length(cons(s(cons(a__and(a__length(mark(_x141)), _x132), _x122)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(s(s(_x171)))))), _x122), cons(_x111, cons(s(length(length(s(s(_x171))))), _x122))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(cons(s(length(length(s(s(mark(_x171)))))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(s(cons(_x121, _x122)), _x81))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(mark(_x81)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(tt)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(a__length(tt)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(cons(0, zeros), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(mark(_x141)), _x132), _x122), cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(length(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(a__length(mark(_x131)), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(length(_x121)), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(length(a__length(mark(_x121))), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(and(_x121, _x122)), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(a__and(mark(_x121), _x122)), _x102))), _x52)))))
a__length#(cons(N, s(length(s(zeros)))))a__length#(s(a__length(s(a__zeros))))a__length#(cons(N, s(length(s(and(length(length(cons(length(0), length(cons(0, length(0)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(length(0)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(zeros))), length(s(length(zeros)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(a__zeros)))))), _x52)))))a__length#(cons(N, s(cons(length(tt), _x42))))a__length#(s(cons(length(tt), _x42)))
a__length#(cons(N, s(cons(cons(_x51, _x52), _x42))))a__length#(s(cons(cons(mark(_x51), _x52), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x62))))) 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
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(a__length(mark(_x181)), _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(tt, _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(s(a__length(mark(_x151))))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(0, _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(cons(mark(_x181), _x182), _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(a__zeros, _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(a__and(mark(_x181), _x182), _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(nil, _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(s(mark(_x181)), _x172)))))), _x112), _x82))), _x62))))) 
a__length#(s(length(s(and(length(length(and(a__and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x52))))) 
a__length#(s(length(s(a__and(length(length(and(and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x102), _x82))), _x62))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x62))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(nil, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(nil, _x172)))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x151)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(s(a__length(mark(_x151))))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(zeros, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(a__zeros, _x172)))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(and(_x181, _x182), _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(a__and(mark(_x181), _x182), _x172)))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(0, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(0, _x172)))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x172)))))), _x102), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x102), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(cons(_x181, _x182), _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(cons(mark(_x181), _x182), _x172)))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(tt, _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(tt, _x172)))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(s(_x181), _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(s(mark(_x181)), _x172)))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x172)))))), _x112), _x82))), _x52)))))) → a__length#(s(length(s(and(length(length(and(a__and(s(length(s(s(a__length(cons(mark(_x171), _x172)))))), _x112), _x82))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(length(_x181), _x172)))))), _x112), _x82))), _x62)))))) → a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(a__length(cons(a__length(mark(_x181)), _x172)))))), _x112), _x82))), _x62)))))

Problem 34: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(s(_x141), _x122), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(and(s(mark(_x141)), _x122), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52))))))a__length#(s(length(s(and(length(s(a__length(nil))), _x52)))))
a__length#(cons(N, zeros))a__length#(cons(0, zeros))a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(length(s(and(length(s(a__and(a__length(tt), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(nil, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(nil, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(and(_x151, _x152)))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(a__and(mark(_x151), _x152)))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(zeros)), _x102), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(and(s(a__length(a__zeros)), _x102), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, _x151)))))), _x112), _x82))), _x52))))))a__length#(s(length(s(and(length(length(and(a__and(s(length(s(s(s(a__length(mark(_x151))))))), _x112), _x82))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(cons(_x171, _x172))), _x132), _x112), cons(cons(mark(s(cons(_x171, _x172))), _x132), and(cons(s(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(and(cons(s(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(s(_x171))), _x132), and(cons(s(s(_x171)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(s(s(mark(_x171))), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(s(0)))))), _x52))))))a__length#(s(a__length(s(and(length(s(length(a__length(s(s(0)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(cons(_x181, _x161)))), _x132), _x112), cons(cons(mark(s(length(cons(_x181, _x161)))), _x132), and(cons(s(length(cons(_x181, _x161))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(s(a__length(mark(_x161)))), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(length(cons(_x171, and(0, _x182))))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(length(s(s(s(a__length(a__and(0, _x182))))))), _x112), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(length(s(s(cons(tt, _x162))))))))), _x52))))))a__length#(s(a__length(s(and(length(s(length(a__length(s(a__length(s(s(cons(tt, _x162))))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(length(0), _x132))), length(s(and(length(0), _x132)))))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(length(s(a__and(a__length(0), _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(cons(_x141, _x142), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(cons(a__and(cons(mark(_x141), _x142), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(0), _x132), and(cons(0, _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(0, _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(length(s(s(length(cons(_x171, _x172)))))))))), _x52))))))a__length#(s(a__length(s(and(length(s(length(a__length(s(length(s(s(a__length(cons(mark(_x171), _x172)))))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(cons(nil, _x132)))))), _x52))))))a__length#(s(a__length(s(and(length(s(length(a__length(s(cons(nil, _x132)))))), _x52)))))a__length#(cons(N, s(length(s(and(and(and(_x81, _x82), _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(a__and(mark(_x81), _x82), _x72), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(length(s(s(length(cons(_x171, _x172)))))))))), _x52))))))a__length#(s(length(s(and(length(s(length(a__length(s(a__length(s(s(a__length(cons(mark(_x171), _x172)))))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(zeros, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__zeros, _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(s(_x131), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(s(mark(_x131)), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(cons(cons(_x171, _x172), _x162))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(s(cons(cons(mark(_x171), _x172), _x162))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(s(_x131)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(a__length(s(s(mark(_x131))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(cons(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(cons(mark(_x131), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(s(_x171))), _x132), _x112), cons(cons(mark(s(s(_x171))), _x132), and(cons(s(s(_x171)), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(s(mark(_x171))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(and(0, _x52)))))a__length#(s(a__length(a__and(0, _x52))))
a__length#(cons(N, s(length(s(and(length(s(0)), _x52))))))a__length#(s(a__length(s(and(length(s(0)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(cons(_x171, _x172))), _x132), and(cons(length(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(tt, _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(length(cons(tt, _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(0)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(0)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(_x121, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(cons(and(zeros, _x52), _x42))))a__length#(s(cons(a__and(a__zeros, _x52), _x42)))
a__length#(cons(N, s(length(s(s(tt))))))a__length#(s(a__length(s(s(tt)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(length(length(_x191))))), _x132), _x112), cons(cons(mark(s(length(length(length(_x191))))), _x132), and(cons(s(length(length(length(_x191)))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(a__length(a__length(mark(_x191))))), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(length(cons(length(s(_x201)), _x182)))), _x132), _x112), cons(cons(mark(s(length(cons(length(s(_x201)), _x182)))), _x132), and(cons(s(length(cons(length(s(_x201)), _x182))), _x132), _x112))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(and(cons(s(a__length(cons(a__length(s(mark(_x201))), _x182))), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(0)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__length(0)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(_x111)))), _x52))))))a__length#(s(a__length(s(and(length(s(a__length(a__length(mark(_x111))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(0, _x132), _x122), cons(_x111, and(cons(0, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(0, _x132), _x122))))), _x52)))))
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a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(and(_x161, _x162))))), _x102), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(and(s(a__length(s(s(a__and(mark(_x161), _x162))))), _x102), _x82))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(_x121, _x101))))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(s(a__length(mark(_x101)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(s(cons(0, _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(cons(0, _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, and(tt, _x111))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(mark(_x111))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(s(a__length(s(s(mark(0))))), length(s(length(s(s(0)))))))), _x52))))))a__length#(s(a__length(s(and(length(s(length(length(s(length(s(s(0)))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(length(length(_x151))), length(cons(_x121, length(length(length(_x151)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(s(length(a__length(a__length(a__length(mark(_x151))))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(length(_x131)))))), _x52))))))a__length#(s(length(s(and(length(s(length(a__length(s(a__length(mark(_x131))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(s(mark(_x121)), length(s(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(length(s(mark(_x121)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x91, _x92))), _x62))))))a__length#(s(a__length(s(a__and(length(a__length(cons(mark(_x91), _x92))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(cons(and(_x131, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(cons(a__and(mark(_x131), _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(a__and(cons(length(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(and(cons(s(mark(_x151)), _x132), _x112), cons(cons(mark(s(_x151)), _x132), and(cons(s(_x151), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(and(cons(s(mark(_x151)), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(and(_x121, _x122), _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(a__and(mark(_x121), _x122), _x112)))), _x52)))))a__length#(cons(N, s(s(_x41))))a__length#(s(s(mark(_x41))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(a__length(a__length(mark(_x121)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(cons(s(nil), _x162))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(s(cons(s(nil), _x162))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(length(s(and(_x151, _x152))), length(cons(_x121, length(s(and(_x151, _x152)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(s(a__and(mark(_x151), _x152)))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(cons(_x161, _x162))))), _x112), _x82))), _x52))))))a__length#(s(length(s(and(length(length(and(a__and(s(a__length(s(s(cons(mark(_x161), _x162))))), _x112), _x82))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(cons(length(_x171), _x162))))), _x122), cons(_x111, cons(s(length(length(cons(length(_x171), _x162)))), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(cons(s(length(a__length(cons(a__length(mark(_x171)), _x162)))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(0)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(a__length(0)))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(and(tt, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(cons(a__and(tt, _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(cons(_x161, _x162))))), _x122), cons(_x111, cons(s(length(length(cons(_x161, _x162)))), _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(cons(s(a__length(a__length(cons(mark(_x161), _x162)))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(_x81)), _x62))))))a__length#(s(a__length(s(a__and(length(s(mark(_x81))), _x62)))))a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(0)))), _x122), cons(_x111, cons(s(length(length(0))), _x122))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(cons(s(length(length(0))), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, cons(_x121, _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(cons(mark(_x121), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(tt, _x122), cons(nil, and(tt, _x122))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(length(a__and(tt, _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(a__length(tt), _x132), and(cons(length(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(tt), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(tt), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__length(tt), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(and(cons(_x131, _x132), _x122), _x112)))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(cons(a__and(cons(mark(_x131), _x132), _x122), _x112)))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(a__length(zeros), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(zeros), _x102))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__and(length(a__zeros), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(tt)), _x132), and(cons(s(tt), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(tt), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(zeros), _x52))))))a__length#(s(length(s(and(a__length(a__zeros), _x52)))))a__length#(cons(N, s(length(s(and(length(_x71), _x52))))))a__length#(s(a__length(s(and(length(mark(_x71)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(length(length(cons(_x151, _x152))), length(cons(_x121, length(length(cons(_x151, _x152)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(s(length(a__length(a__length(cons(mark(_x151), _x152)))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(tt)), _x52))))))a__length#(s(length(s(and(a__length(a__length(tt)), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(length(length(_x141)), length(cons(_x121, length(length(_x141))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(s(a__length(a__length(a__length(mark(_x141)))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(and(s(length(s(s(cons(s(and(_x181, _x182)), _x162))))), _x112), _x82))), _x62))))))a__length#(s(length(s(a__and(length(length(and(a__and(s(a__length(s(s(cons(s(a__and(mark(_x181), _x182)), _x162))))), _x112), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(s(cons(_x171, _x172))), _x132), and(cons(s(cons(_x171, _x172)), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(and(cons(s(cons(mark(_x171), _x172)), _x132), _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(length(s(and(and(length(_x121), _x112), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(a__and(a__length(mark(_x121)), _x112), _x102))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(length(length(and(_x151, _x152))), length(cons(_x121, length(length(and(_x151, _x152)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(s(length(a__length(a__length(a__and(mark(_x151), _x152)))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(mark(nil), and(nil, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(nil, _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(tt)), _x52))))))a__length#(s(a__length(s(and(length(s(tt)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, s(nil))))), _x52))))))a__length#(s(a__length(s(and(length(s(s(a__length(s(nil))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, length(s(length(s(nil))))))), _x52))))))a__length#(s(a__length(s(and(length(s(length(a__length(s(a__length(s(nil))))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(length(s(length(0))), length(cons(_x121, length(s(length(0)))))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(length(s(a__length(a__length(s(a__length(0)))))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(_x111, and(cons(length(tt), _x132), _x122))))), _x52))))))a__length#(s(length(s(and(a__length(s(s(a__length(a__and(cons(a__length(tt), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, and(tt, _x112)))), _x52))))))a__length#(s(length(s(and(a__length(s(a__length(a__and(tt, _x112)))), _x52)))))
a__length#(cons(N, s(length(s(s(zeros))))))a__length#(s(a__length(s(s(cons(0, zeros))))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(cons(mark(length(0)), _x132), and(cons(length(0), _x132), _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(cons(a__length(0), _x132), _x112))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, and(nil, _x112))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(a__length(and(nil, _x112))))), _x52)))))a__length#(cons(N, s(length(s(and(and(zeros, _x72), _x62))))))a__length#(s(a__length(s(a__and(a__and(a__zeros, _x72), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(and(_x131, _x122))), length(s(and(_x131, _x122)))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(a__length(length(s(and(mark(_x131), _x122)))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, cons(nil, length(_x121))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__length(mark(_x121)))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(length(cons(cons(s(a__length(mark(length(s(s(_x171)))))), _x122), cons(_x111, cons(s(length(length(s(s(_x171))))), _x122))))), _x52))))))a__length#(s(length(s(and(length(s(s(length(cons(s(length(length(s(s(mark(_x171)))))), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(and(zeros, _x82))), _x62))))))a__length#(s(length(s(a__and(a__length(length(and(cons(0, zeros), _x82))), _x62)))))
a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(a__length(mark(_x141)), _x132), _x122), cons(_x111, and(cons(length(_x141), _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(a__length(mark(_x141)), _x132), _x122))))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(a__and(cons(mark(0), _x132), _x122), cons(_x111, and(cons(0, _x132), _x122))))), _x52))))))a__length#(s(a__length(s(and(a__length(s(s(length(a__and(cons(0, _x132), _x122))))), _x52)))))
a__length#(cons(N, s(length(s(and(length(s(and(length(length(_x121)), _x102))), _x52))))))a__length#(s(length(s(and(a__length(s(a__and(length(a__length(mark(_x121))), _x102))), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(mark(s(length(zeros))), length(s(length(zeros)))))), _x52))))))a__length#(s(length(s(and(a__length(s(length(length(s(a__length(a__zeros)))))), _x52)))))
a__length#(cons(N, s(cons(length(tt), _x42))))a__length#(s(cons(length(tt), _x42)))

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


The right-hand side of the rule a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52)))))) → a__length#(s(length(s(and(length(s(a__length(nil))), _x52))))) 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
a__length#(s(length(s(and(length(s(0)), _x52))))) 
a__length#(s(length(s(and(length(s(length(nil))), _x52))))) 
Thus, the rule a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52)))))) → a__length#(s(length(s(and(length(s(a__length(nil))), _x52))))) is replaced by the following rules:
a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52)))))) → a__length#(s(length(s(and(length(s(0)), _x52)))))a__length#(cons(N, s(length(s(and(length(length(cons(_x101, nil))), _x52)))))) → a__length#(s(length(s(and(length(s(length(nil))), _x52)))))

Problem 5: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(and(X1, X2))mark#(X1)mark#(and(X1, X2))a__and#(mark(X1), X2)
a__and#(tt, X)mark#(X)mark#(cons(X1, X2))mark#(X1)
mark#(s(X))mark#(X)

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


Polynomial Interpretation

Improved Usable rules

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

mark#(and(X1, X2))mark#(X1)mark#(and(X1, X2))a__and#(mark(X1), X2)

Problem 6: DependencyGraph



Dependency Pair Problem

Dependency Pairs

a__and#(tt, X)mark#(X)mark#(cons(X1, X2))mark#(X1)
mark#(s(X))mark#(X)

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


The following SCCs where found

mark#(cons(X1, X2)) → mark#(X1)mark#(s(X)) → mark#(X)

Problem 7: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2))mark#(X1)mark#(s(X))mark#(X)

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, zeros, tt, a__length, length, mark, cons, a__and, nil, and

Strategy


Polynomial Interpretation

There are no usable rules

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

mark#(cons(X1, X2))mark#(X1)

Problem 8: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(s(X))mark#(X)

Rewrite Rules

a__zeroscons(0, zeros)a__and(tt, X)mark(X)
a__length(nil)0a__length(cons(N, L))s(a__length(mark(L)))
mark(zeros)a__zerosmark(and(X1, X2))a__and(mark(X1), X2)
mark(length(X))a__length(mark(X))mark(cons(X1, X2))cons(mark(X1), X2)
mark(0)0mark(tt)tt
mark(nil)nilmark(s(X))s(mark(X))
a__zeroszerosa__and(X1, X2)and(X1, X2)
a__length(X)length(X)

Original Signature

Termination of terms over the following signature is verified: a__zeros, 0, s, tt, zeros, a__length, length, mark, and, nil, a__and, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

mark#(s(X))mark#(X)