TIMEOUT

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

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (50ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4iUR (151ms).
 | – Problem 3 was processed with processor SubtermCriterion (2ms).
 | – Problem 4 was processed with processor PolynomialLinearRange4iUR (189ms).
 |    | – Problem 5 was processed with processor ForwardNarrowing (6ms).
 |    |    | – Problem 6 was processed with processor ForwardNarrowing (14ms).
 |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (18ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (60ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (207ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (36ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (209ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 remains open; application of the following processors failed [ForwardNarrowing (208ms), ForwardNarrowing (202ms), ForwardNarrowing (327ms), ForwardNarrowing (359ms), ForwardNarrowing (441ms), ForwardNarrowing (371ms), ForwardNarrowing (72ms), ForwardNarrowing (352ms), ForwardNarrowing (520ms), ForwardNarrowing (683ms), ForwardNarrowing (728ms), ForwardNarrowing (360ms), ForwardNarrowing (732ms), ForwardNarrowing (380ms), ForwardNarrowing (125ms), ForwardNarrowing (688ms), ForwardNarrowing (406ms), ForwardNarrowing (314ms), ForwardNarrowing (timeout)].

The following open problems remain:

Open Dependency Pair Problem 5

Dependency Pairs

plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0)))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

quot#(s(x), s(y)) → minus#(x, y)plus#(s(x), y) → plus#(x, y)
quot#(s(x), s(y)) → quot#(minus(x, y), s(y))plus#(minus(x, s(0)), minus(y, s(s(z)))) → minus#(x, s(0))
minus#(s(x), s(y)) → minus#(x, y)plus#(minus(x, s(0)), minus(y, s(s(z)))) → minus#(y, s(s(z)))
plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0)))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The following SCCs where found

quot#(s(x), s(y)) → quot#(minus(x, y), s(y))
plus#(s(x), y) → plus#(x, y)plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0)))
minus#(s(x), s(y)) → minus#(x, y)

Problem 2: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

quot#(s(x), s(y)) → quot#(minus(x, y), s(y))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

Polynomial Interpretation

Improved Usable rules

minus(s(x), s(y)) → minus(x, y)minus(x, 0) → x

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

quot#(s(x), s(y)) → quot#(minus(x, y), s(y))

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

minus#(s(x), s(y)) → minus#(x, y)

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

minus#(s(x), s(y)) → minus#(x, y)

Problem 4: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

plus#(s(x), y) → plus#(x, y)plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0)))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

Polynomial Interpretation

Improved Usable rules

minus(s(x), s(y)) → minus(x, y)minus(x, 0) → x

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

plus#(s(x), y) → plus#(x, y)

Problem 5: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0)))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(minus(y, s(s(z))), minus(_x32, 0)) 
plus#(minus(_x22, s(z)), minus(x, s(0))) 
Thus, the rule plus#(minus(x, s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))

Problem 6: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(minus(_x22, s(z)), minus(_x32, 0)) 
plus#(minus(_x32, _x31), minus(x, s(0))) 
Thus, the rule plus#(minus(x, s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Problem 7: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(minus(_x22, _x21), minus(x, s(0))) 
plus#(minus(_x32, _x31), minus(_x32, 0)) 
plus#(_x21, minus(x, s(0))) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))

Problem 8: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x21, minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(x, s(0))) is replaced by the following rules:
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Problem 9: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(x, s(0))) 
plus#(minus(_x32, _x31), minus(x, s(0))) 
plus#(minus(_x22, _x21), minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(minus(_x32, _x31), minus(_x32, 0)) 
plus#(minus(_x22, _x21), minus(x, s(0))) 
plus#(_x21, minus(x, s(0))) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))

Problem 11: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(x, s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x21, minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(x, s(0))) is replaced by the following rules:
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(_x32, 0))

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(_x32, 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
plus#(_x21, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)

Problem 13: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(x, s(0))) is replaced by the following rules:
plus#(minus(s(_x32), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0))

Problem 14: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0))plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 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
plus#(minus(_x22, s(z)), minus(_x32, 0)) 
plus#(minus(y, s(s(z))), _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), _x41)plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Problem 15: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x41), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), _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
plus#(minus(_x22, s(z)), _x41) 
Thus, the rule plus#(minus(s(_x41), s(0)), minus(y, s(s(z)))) → plus#(minus(y, s(s(z))), _x41) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)

Problem 16: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, minus(_x32, 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
plus#(_x31, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)

Problem 17: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(x, s(0))) 
plus#(minus(_x32, _x31), minus(x, s(0))) 
plus#(minus(_x22, _x21), minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))

Problem 18: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(x, s(0))) is replaced by the following rules:
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))

Problem 19: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(_x32, 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
plus#(_x31, minus(_x32, 0)) 
plus#(minus(_x32, _x31), minus(_x32, 0)) 
plus#(minus(_x22, _x21), _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)

Problem 20: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 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
plus#(_x31, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)

Problem 21: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 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
plus#(_x31, minus(_x32, 0)) 
plus#(minus(_x32, _x31), minus(_x32, 0)) 
plus#(minus(_x22, _x21), _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x32), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))

Problem 22: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(_x32, 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
plus#(_x31, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 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
plus#(_x21, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 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
plus#(_x31, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)

Problem 25: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(0))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(_x21)))))), s(s(s(s(s(s(0)))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(0)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x21)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x22))))))), s(s(s(s(s(s(s(_x21))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(_x21)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(_x31)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x31))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x31))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x22))))))))), s(s(s(s(s(s(s(s(s(_x21))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(_x31)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))), s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, minus(_x32, 0))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, minus(_x32, 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
plus#(_x31, _x41) 
Thus, the rule plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, minus(_x32, 0)) is replaced by the following rules:
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, _x41)

Problem 26: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(_x21)))))), s(s(s(s(s(s(0)))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x21)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x31))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(0))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(0)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x22))))))), s(s(s(s(s(s(s(_x21))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(_x21)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(_x31)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x31))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(_x31)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x22))))))))), s(s(s(s(s(s(s(s(s(_x21))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))), s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(_x21, _x41)

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(x, s(0))) 
plus#(minus(_x32, _x31), minus(x, s(0))) 
plus#(minus(_x22, _x21), minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))

Problem 27: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x21)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(0))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(0)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x22))))))), s(s(s(s(s(s(s(_x21))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(_x21)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(_x31)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x31))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x22))))))))), s(s(s(s(s(s(s(s(s(_x21))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(_x21)))))), s(s(s(s(s(s(0)))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))), s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x31))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(_x31)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(_x21, _x41)

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(x, s(0))) 
plus#(minus(_x32, _x31), minus(x, s(0))) 
plus#(minus(_x22, _x21), minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))

Problem 28: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x21)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(0))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(0)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x22))))))), s(s(s(s(s(s(s(_x21))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(_x21)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(_x31)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x31))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x22))))))))), s(s(s(s(s(s(s(s(s(_x21))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(_x21)))))), s(s(s(s(s(s(0)))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))), s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, minus(x, s(0)))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x31))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(_x21, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(_x31)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(_x21, _x41)

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(_x31, minus(x, s(0))) 
plus#(minus(_x32, _x31), minus(x, s(0))) 
plus#(minus(_x22, _x21), minus(_x32, 0)) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))

Problem 29: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x22))), s(s(s(_x21))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(_x31))), s(s(s(0))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(0))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(_x21)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x31))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(_x31)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(_x21)))))), s(s(s(s(s(s(0)))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))), s(s(s(s(s(s(s(s(s(s(s(0))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x31))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(0))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(0))), s(s(s(0))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(0))))))), s(s(s(s(s(s(s(0))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(_x31)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x31))))), s(s(s(s(s(0))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x21)))))))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(0))))))))), s(s(s(s(s(s(s(s(s(0))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(_x21)))), s(s(s(s(0)))))) → plus#(_x21, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(_x32)))))), s(s(s(s(s(s(0)))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(_x22))))))), s(s(s(s(s(s(s(_x21))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(_x32)))), s(s(s(s(0)))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(_x31)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(_x22), s(s(z)))) → plus#(minus(_x22, s(z)), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(_x22))))))))), s(s(s(s(s(s(s(s(s(_x21))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(_x21)), s(s(0)))) → plus#(_x21, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))), s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(_x22))))), s(s(s(s(s(_x21))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(0)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x32), s(0)), minus(s(s(_x32)), s(s(_x31)))) → plus#(minus(_x32, _x31), minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(_x32)))))))), s(s(s(s(s(s(s(s(0)))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, _x41)plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, minus(_x32, 0))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(_x32)))))))))), s(s(s(s(s(s(s(s(s(s(_x31)))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))) → plus#(_x31, _x41)
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))) → plus#(minus(_x22, _x21), minus(_x32, 0))
plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))) → plus#(minus(_x22, _x21), _x41)plus#(minus(s(_x41), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))) → plus#(_x21, _x41)
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x31, minus(x, s(0)))

Rewrite Rules

minus(x, 0) → xminus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → yplus(s(x), y) → s(plus(x, y))
plus(minus(x, s(0)), minus(y, s(s(z)))) → plus(minus(y, s(s(z))), minus(x, s(0)))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, quot

Strategy

The right-hand side of the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(x, s(0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
plus#(minus(_x32, _x31), minus(_x32, 0)) 
plus#(minus(_x22, _x21), minus(x, s(0))) 
plus#(_x21, minus(x, s(0))) 
Thus, the rule plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x31)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(x, s(0))) is replaced by the following rules:
plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x22))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x22, _x21), minus(x, s(0)))plus#(minus(x, s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x21)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(_x21, minus(x, s(0)))
plus#(minus(s(_x32), s(0)), minus(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x32)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → plus#(minus(_x32, _x31), minus(_x32, 0))