The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 77 ms)
↳12 CdtProblem
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 8 ms)
↳20 CdtProblem
↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳22 CdtProblem
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 48 ms)
↳24 CdtProblem
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 22 ms)
↳26 CdtProblem
↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳28 CdtProblem
↳29 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳30 CdtProblem
↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 31 ms)
↳32 CdtProblem
↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳34 CdtProblem
↳35 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳36 CdtProblem
↳37 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳38 CdtProblem
↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳40 CdtProblem
↳41 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳42 CdtProblem
↳43 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳44 CdtProblem
↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)
↳46 CdtProblem
↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))), 629 ms)
↳48 CdtProblem
↳49 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳50 CdtProblem
↳51 CdtKnowledgeProof (⇔, 0 ms)
↳52 BOUNDS(O(1), O(1))

(0) Obligation:

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

cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, z0) → c(COND(odd(z0), p(z0)), ODD(z0), P(z0))
ODD(s(s(z0))) → c3(ODD(z0))
S tuples:

COND(true, z0) → c(COND(odd(z0), p(z0)), ODD(z0), P(z0))
ODD(s(s(z0))) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c3

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, z0) → c(COND(odd(z0), p(z0)), ODD(z0))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, z0) → c(COND(odd(z0), p(z0)), ODD(z0))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c

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

Use narrowing to replace COND(true, z0) → c(COND(odd(z0), p(z0)), ODD(z0)) by

COND(true, 0) → c(COND(odd(0), 0), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, 0) → c(COND(false, p(0)), ODD(0))
COND(true, s(0)) → c(COND(true, p(s(0))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, x0) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, 0) → c(COND(false, p(0)), ODD(0))
COND(true, s(0)) → c(COND(true, p(s(0))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, x0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, 0) → c(COND(false, p(0)), ODD(0))
COND(true, s(0)) → c(COND(true, p(s(0))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, x0) → c
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 2 trailing nodes:

COND(true, 0) → c(COND(false, p(0)), ODD(0))
COND(true, x0) → c

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, s(0)) → c(COND(true, p(s(0))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0), ODD(0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, s(0)) → c(COND(true, p(s(0))), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c

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

Removed 2 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

(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.

COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
We considered the (Usable) Rules:

p(s(z0)) → z0
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
And the Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2)) = x2   
POL(ODD(x1)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [5]   
POL(odd(x1)) = 0   
POL(p(x1)) = x1   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Use narrowing to replace COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0))) by

COND(true, s(0)) → c(COND(true, 0), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(x0)) → c

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(0)) → c(COND(true, 0), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(x0)) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
K tuples:

COND(true, s(z0)) → c(COND(odd(s(z0)), z0), ODD(s(z0)))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

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

Removed 1 trailing nodes:

COND(true, s(x0)) → c

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(0)) → c(COND(true, 0), ODD(s(0)))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0))))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Use narrowing to replace COND(true, s(s(z0))) → c(COND(odd(z0), p(s(s(z0)))), ODD(s(s(z0)))) by

COND(true, s(s(x0))) → c(COND(odd(x0), s(x0)), ODD(s(s(x0))))
COND(true, s(s(0))) → c(COND(false, p(s(s(0)))), ODD(s(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(0))) → c(COND(false, p(s(s(0)))), ODD(s(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), s(x0)), ODD(s(s(x0))))
COND(true, s(s(0))) → c(COND(false, p(s(s(0)))), ODD(s(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Removed 1 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), s(x0)), ODD(s(s(x0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
K tuples:none
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

(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.

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
We considered the (Usable) Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(s(z0)) → z0
And the Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]   
POL(COND(x1, x2)) = [4]   
POL(ODD(x1)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(odd(x1)) = [3]   
POL(p(x1)) = [3] + [3]x1   
POL(s(x1)) = 0   
POL(true) = [3]   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(x0))) → c(COND(odd(x0), s(x0)), ODD(s(s(x0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

(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.

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
We considered the (Usable) Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(s(z0)) → z0
And the Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]   
POL(COND(x1, x2)) = [4] + [4]x2   
POL(ODD(x1)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(odd(x1)) = 0   
POL(p(x1)) = x1   
POL(s(x1)) = [4] + x1   
POL(true) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, 0) → c(COND(odd(0), 0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

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

Use narrowing to replace COND(true, 0) → c(COND(odd(0), 0)) by

COND(true, 0) → c(COND(false, 0))
COND(true, 0) → c

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c(COND(false, 0))
COND(true, 0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c(COND(false, 0))
COND(true, 0) → c
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

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

Removed 1 trailing tuple parts

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, 0) → c
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

(31) 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.

COND(true, 0) → c
We considered the (Usable) Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(s(z0)) → z0
And the Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]   
POL(COND(x1, x2)) = [4]   
POL(ODD(x1)) = 0   
POL(c) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [3]   
POL(odd(x1)) = [3] + [3]x1   
POL(p(x1)) = [2] + [4]x1   
POL(s(x1)) = x1   
POL(true) = [3]   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(0)) → c(COND(true, p(s(0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, 0) → c
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

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

Use narrowing to replace COND(true, s(0)) → c(COND(true, p(s(0)))) by

COND(true, s(0)) → c(COND(true, 0))
COND(true, s(0)) → c

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, 0) → c
COND(true, s(0)) → c
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(0)) → c
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, 0) → c
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c, c

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

Removed 3 trailing nodes:

COND(true, 0) → c
COND(true, s(0)) → c(COND(true, 0))
COND(true, s(0)) → c

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
S tuples:

ODD(s(s(z0))) → c3(ODD(z0))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

ODD, COND

Compound Symbols:

c3, c, c

(37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace ODD(s(s(z0))) → c3(ODD(z0)) by

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(0))) → c(ODD(s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing nodes:

COND(true, s(s(0))) → c(ODD(s(s(0))))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))), ODD(s(s(s(0)))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

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

Removed 1 trailing tuple parts

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
S tuples:

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

(43) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), p(s(s(s(s(z0)))))), ODD(s(s(s(s(z0)))))) by COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

(45) 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.

COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
We considered the (Usable) Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(s(z0)) → z0
And the Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2)) = [4]x2   
POL(ODD(x1)) = 0   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [2]   
POL(odd(x1)) = [3] + [2]x1   
POL(p(x1)) = x1   
POL(s(x1)) = [4] + x1   
POL(true) = [2]   

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
S tuples:

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

(47) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
We considered the (Usable) Rules:

odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(s(z0)) → z0
And the Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(COND(x1, x2)) = [1] + x1 + x12 + x13 + x1·x22   
POL(ODD(x1)) = [1] + x1   
POL(c(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(false) = [1]   
POL(odd(x1)) = [1]   
POL(p(x1)) = x1   
POL(s(x1)) = [1] + x1   
POL(true) = [1]   

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0))))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

(49) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace COND(true, s(s(s(0)))) → c(COND(true, p(s(s(s(0)))))) by COND(true, s(s(s(0)))) → c(COND(true, s(s(0))))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

cond(true, z0) → cond(odd(z0), p(z0))
odd(0) → false
odd(s(0)) → true
odd(s(s(z0))) → odd(z0)
p(0) → 0
p(s(z0)) → z0
Tuples:

COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
COND(true, s(s(s(0)))) → c(COND(true, s(s(0))))
S tuples:

COND(true, s(s(s(0)))) → c(COND(true, s(s(0))))
K tuples:

COND(true, s(s(x0))) → c(ODD(s(s(x0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(s(s(z0))))) → c(COND(odd(z0), s(s(s(z0)))), ODD(s(s(s(s(z0))))))
ODD(s(s(s(s(y0))))) → c3(ODD(s(s(y0))))
Defined Rule Symbols:

cond, odd, p

Defined Pair Symbols:

COND, ODD

Compound Symbols:

c, c, c3

(51) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

COND(true, s(s(s(0)))) → c(COND(true, s(s(0))))
COND(true, s(s(z0))) → c(COND(odd(z0), s(z0)), ODD(s(s(z0))))
COND(true, s(s(x0))) → c(COND(odd(x0), s(x0)), ODD(s(s(x0))))
COND(true, s(s(x0))) → c(ODD(s(s(x0))))
Now S is empty

(52) BOUNDS(O(1), O(1))