The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 60 ms)
↳14 CdtProblem
↳15 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 84 ms)
↳16 CdtProblem
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳22 CdtProblem
↳23 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 11 ms)
↳24 CdtProblem
↳25 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 3387 ms)
↳26 CdtProblem
↳27 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳28 BOUNDS(O(1), O(1))

(0) Obligation:

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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
p(s(x)) → x
f(s(x), y) → f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) → f(p(-(x, s(y))), p(-(s(y), x)))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

-'(z0, 0) → c
-'(s(z0), s(z1)) → c1(-'(z0, z1))
P(s(z0)) → c2
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)))
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:

-'(z0, 0) → c
-'(s(z0), s(z1)) → c1(-'(z0, z1))
P(s(z0)) → c2
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)))
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:

-', P, F

Compound Symbols:

c, c1, c2, c3, c4

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

Removed 2 trailing nodes:

-'(z0, 0) → c
P(s(z0)) → c2

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

-'(s(z0), s(z1)) → c1(-'(z0, z1))
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)))
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))
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)))
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

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0)))
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))
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0)))
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

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(s(z0), z1) → f(p(-(s(z0), z1)), p(-(z1, s(z0))))
f(z0, s(z1)) → f(p(-(z0, s(z1))), p(-(s(z1), z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0)))
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))
F(s(z0), z1) → c3(F(p(-(s(z0), z1)), p(-(z1, s(z0)))), -'(s(z0), z1), -'(z1, s(z0)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

(9) 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)))
F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0)))
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))
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3

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

Removed 1 trailing nodes:

F(s(x0), 0) → c3(F(p(s(x0)), p(-(0, s(x0)))), -'(s(x0), 0), -'(0, s(x0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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))
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3

(13) CdtRuleRemovalProof (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:

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

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

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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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))
F(z0, s(z1)) → c4(F(p(-(z0, s(z1))), p(-(s(z1), z0))), -'(z0, s(z1)), -'(s(z1), z0))
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(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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3

(15) CdtRuleRemovalProof (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:

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

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

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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

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

F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c4, c3

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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(0, s(x1)) → c4(F(p(-(0, s(x1))), p(s(x1))), -'(0, s(x1)), -'(s(x1), 0))
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))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(0, s(x1)) → c4(F(p(-(0, s(x1))), p(s(x1))), -'(0, s(x1)), -'(s(x1), 0))
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(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

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

Removed 1 trailing nodes:

F(0, s(x1)) → c4(F(p(-(0, s(x1))), p(s(x1))), -'(0, s(x1)), -'(s(x1), 0))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

(21) CdtRuleRemovalProof (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:

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

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

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

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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))
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(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

(23) CdtRuleRemovalProof (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:

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

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

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

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

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

F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

(25) CdtRuleRemovalProof (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:

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

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

POL(-(x1, x2)) = x1   
POL(-'(x1, x2)) = x1   
POL(0) = 0   
POL(F(x1, x2)) = x2 + [2]x22 + x1·x2 + x12   
POL(c1(x1)) = x1   
POL(c3(x1, x2, x3)) = x1 + x2 + x3   
POL(c4(x1, x2, x3)) = x1 + x2 + x3   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + x1   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
-(z0, 0) → z0
-(s(z0), s(z1)) → -(z0, z1)
Tuples:

-'(s(z0), s(z1)) → c1(-'(z0, z1))
F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
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:none
K tuples:

F(s(z1), s(z0)) → c3(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c3(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
F(s(z1), s(z0)) → c4(F(p(-(s(z1), s(z0))), p(-(z0, z1))), -'(s(z1), s(z0)), -'(s(z0), s(z1)))
F(s(z0), s(z1)) → c4(F(p(-(z0, z1)), p(-(s(z1), s(z0)))), -'(s(z0), s(z1)), -'(s(z1), s(z0)))
-'(s(z0), s(z1)) → c1(-'(z0, z1))
Defined Rule Symbols:

p, -

Defined Pair Symbols:

-', F

Compound Symbols:

c1, c3, c4

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

The set S is empty

(28) BOUNDS(O(1), O(1))