YES(O(1), O(n^2)) 51.09/19.80 YES(O(1), O(n^2)) 51.30/19.83 51.30/19.83 51.30/19.83 51.30/19.83 51.30/19.83 51.30/19.83 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 51.30/19.83 51.30/19.83 51.30/19.83
51.30/19.83 51.30/19.83 51.30/19.83
51.30/19.83
51.30/19.83

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

-(x, 0) → x 51.30/19.83
-(s(x), s(y)) → -(x, y) 51.30/19.83
p(s(x)) → x 51.30/19.83
f(s(x), y) → f(p(-(s(x), y)), p(-(y, s(x)))) 51.30/19.83
f(x, s(y)) → f(p(-(x, s(y))), p(-(s(y), x)))

Rewrite Strategy: INNERMOST
51.30/19.83
51.30/19.83

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
51.30/19.83
51.30/19.83

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.83
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.83
p(s(z0)) → z0 51.30/19.83
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.83
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1), P(-(z1, s(z0))), -'(z1, s(z0))) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)), P(-(s(z1), z0)), -'(s(z1), z0))
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), P(-(s(z0), z1)), -'(s(z0), z1), P(-(z1, s(z0))), -'(z1, s(z0))) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), P(-(z0, s(z1))), -'(z0, s(z1)), P(-(s(z1), z0)), -'(s(z1), z0))
K tuples:none
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

51.30/19.84
51.30/19.84

(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts
51.30/19.84
51.30/19.84

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0 51.30/19.84
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.84
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0))) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0))) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
K tuples:none
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

51.30/19.84
51.30/19.84

(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0))) by

F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
51.30/19.84
51.30/19.84

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0 51.30/19.84
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.84
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
K tuples:none
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3

51.30/19.84
51.30/19.84

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
51.30/19.84
51.30/19.84

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0 51.30/19.84
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.84
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
K tuples:none
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3, c3

51.30/19.84
51.30/19.84

(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

F(s(x0), 0) → c3
51.30/19.84
51.30/19.84

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0 51.30/19.84
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.84
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
K tuples:none
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3, c3

51.30/19.84
51.30/19.84

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(s(x0), 0) → c3
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.84

POL(-(x1, x2)) = 0    51.30/19.84
POL(-'(x1, x2)) = 0    51.30/19.84
POL(0) = [1]    51.30/19.84
POL(F(x1, x2)) = [1]    51.30/19.84
POL(c1(x1)) = x1    51.30/19.84
POL(c3) = 0    51.30/19.84
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.84
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.84
POL(p(x1)) = 0    51.30/19.84
POL(s(x1)) = 0   
51.30/19.84
51.30/19.84

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0 51.30/19.84
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.84
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
K tuples:

F(s(x0), 0) → c3
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3, c3

51.30/19.84
51.30/19.84

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.84
F(s(x0), 0) → c3
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.84

POL(-(x1, x2)) = x1    51.30/19.84
POL(-'(x1, x2)) = 0    51.30/19.84
POL(0) = 0    51.30/19.84
POL(F(x1, x2)) = [4]x1    51.30/19.84
POL(c1(x1)) = x1    51.30/19.84
POL(c3) = 0    51.30/19.84
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.84
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.84
POL(p(x1)) = x1    51.30/19.84
POL(s(x1)) = [2] + x1   
51.30/19.84
51.30/19.84

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.84
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.84
p(s(z0)) → z0 51.30/19.84
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.84
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.84
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.84
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.84
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.85
F(s(x0), 0) → c3
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.85
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.85
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
K tuples:

F(s(x0), 0) → c3 51.30/19.85
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3, c3

51.30/19.85
51.30/19.85

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.85
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.85
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.85
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.85
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.86

POL(-(x1, x2)) = x1    51.30/19.86
POL(-'(x1, x2)) = 0    51.30/19.86
POL(0) = 0    51.30/19.86
POL(F(x1, x2)) = [2]x2    51.30/19.86
POL(c1(x1)) = x1    51.30/19.86
POL(c3) = 0    51.30/19.86
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(p(x1)) = x1    51.30/19.86
POL(s(x1)) = [4] + x1   
51.30/19.86
51.30/19.86

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3, c3

51.30/19.86
51.30/19.86

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0)) by

F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4(F(p(-(0, s(x1))), p(s(x1))), -'(0, s(x1)), -'(s(x1), 0)) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
51.30/19.86
51.30/19.86

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4(F(p(-(0, s(x1))), p(s(x1))), -'(0, s(x1)), -'(s(x1), 0)) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4(F(p(-(0, s(x1))), p(s(x1))), -'(0, s(x1)), -'(s(x1), 0)) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4

51.30/19.86
51.30/19.86

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
51.30/19.86
51.30/19.86

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4, c4

51.30/19.86
51.30/19.86

(21) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

F(s(x0), 0) → c3 51.30/19.86
F(0, s(x1)) → c4
51.30/19.86
51.30/19.86

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4, c4

51.30/19.86
51.30/19.86

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(0, s(x1)) → c4
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.86

POL(-(x1, x2)) = 0    51.30/19.86
POL(-'(x1, x2)) = 0    51.30/19.86
POL(0) = [1]    51.30/19.86
POL(F(x1, x2)) = [1]    51.30/19.86
POL(c1(x1)) = x1    51.30/19.86
POL(c3) = 0    51.30/19.86
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(c4) = 0    51.30/19.86
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(p(x1)) = 0    51.30/19.86
POL(s(x1)) = 0   
51.30/19.86
51.30/19.86

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4, c4

51.30/19.86
51.30/19.86

(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.86

POL(-(x1, x2)) = x1    51.30/19.86
POL(-'(x1, x2)) = 0    51.30/19.86
POL(0) = 0    51.30/19.86
POL(F(x1, x2)) = x2    51.30/19.86
POL(c1(x1)) = x1    51.30/19.86
POL(c3) = 0    51.30/19.86
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(c4) = 0    51.30/19.86
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(p(x1)) = x1    51.30/19.86
POL(s(x1)) = [1] + x1   
51.30/19.86
51.30/19.86

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4, c4

51.30/19.86
51.30/19.86

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.86

POL(-(x1, x2)) = x1    51.30/19.86
POL(-'(x1, x2)) = 0    51.30/19.86
POL(0) = 0    51.30/19.86
POL(F(x1, x2)) = x1    51.30/19.86
POL(c1(x1)) = x1    51.30/19.86
POL(c3) = 0    51.30/19.86
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(c4) = 0    51.30/19.86
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(p(x1)) = x1    51.30/19.86
POL(s(x1)) = [1] + x1   
51.30/19.86
51.30/19.86

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
S tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4, c4

51.30/19.86
51.30/19.86

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

-'(s(z0), s(z1)) → c1(-'(z0, z1))
We considered the (Usable) Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0
And the Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
The order we found is given by the following interpretation:
Polynomial interpretation : 51.30/19.86

POL(-(x1, x2)) = x1    51.30/19.86
POL(-'(x1, x2)) = x2    51.30/19.86
POL(0) = [3]    51.30/19.86
POL(F(x1, x2)) = [3]x2 + x22 + [2]x1·x2 + [2]x12    51.30/19.86
POL(c1(x1)) = x1    51.30/19.86
POL(c3) = 0    51.30/19.86
POL(c3(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(c4) = 0    51.30/19.86
POL(c4(x1, x2, x3)) = x1 + x2 + x3    51.30/19.86
POL(p(x1)) = x1    51.30/19.86
POL(s(x1)) = [2] + x1   
51.30/19.86
51.30/19.86

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

-(z0, 0) → z0 51.30/19.86
-(s(z0), s(z1)) → -(z0, z1) 51.30/19.86
p(s(z0)) → z0 51.30/19.86
f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0)))) 51.30/19.86
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1)) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(x0), 0) → c3 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(0, s(x1)) → c4
S tuples:none
K tuples:

F(s(x0), 0) → c3 51.30/19.86
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(0, s(x1)) → c4 51.30/19.86
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1))) 51.30/19.86
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0))) 51.30/19.86
-'(s(z0), s(z1)) → c1(-'(z0, z1))
Defined Rule Symbols:

-, p, f

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c3, c4, c4

51.30/19.86
51.30/19.86

(31) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
51.30/19.86
51.30/19.86

(32) BOUNDS(O(1), O(1))

51.30/19.86
51.30/19.86
51.54/19.90 EOF