YES(O(1), O(n^1)) 11.45/3.67 YES(O(1), O(n^1)) 11.86/3.71 11.86/3.71 11.86/3.71 11.86/3.71 11.86/3.71 11.86/3.71 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 11.86/3.71 11.86/3.71 11.86/3.71
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11.86/3.71
11.86/3.71
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
11.86/3.71 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳2 CdtProblem
11.86/3.71 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳4 CdtProblem
11.86/3.71 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
11.86/3.71 
↳6 CdtProblem
11.86/3.71 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
11.86/3.71 
↳8 CdtProblem
11.86/3.71 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
11.86/3.71 
↳10 CdtProblem
11.86/3.71 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳12 CdtProblem
11.86/3.71 
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳14 CdtProblem
11.86/3.71 
↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳16 CdtProblem
11.86/3.71 
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳18 CdtProblem
11.86/3.71 
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳20 CdtProblem
11.86/3.71 
↳21 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳22 CdtProblem
11.86/3.71 
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
11.86/3.71 
↳24 CdtProblem
11.86/3.71 
↳25 SIsEmptyProof (BOTH BOUNDS(ID, ID))
11.86/3.71 
↳26 BOUNDS(O(1), O(1))
11.86/3.71
11.86/3.71 11.86/3.71
11.86/3.71
11.86/3.71

(0) Obligation:

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

a__f(0) → cons(0, f(s(0))) 11.86/3.71
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.71
a__p(s(0)) → 0 11.86/3.71
mark(f(X)) → a__f(mark(X)) 11.86/3.71
mark(p(X)) → a__p(mark(X)) 11.86/3.71
mark(0) → 0 11.86/3.71
mark(cons(X1, X2)) → cons(mark(X1), X2) 11.86/3.71
mark(s(X)) → s(mark(X)) 11.86/3.71
a__f(X) → f(X) 11.86/3.71
a__p(X) → p(X)

Rewrite Strategy: INNERMOST
11.86/3.71
11.86/3.71

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

Converted CpxTRS to CDT
11.86/3.71
11.86/3.71

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.71
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.71
a__f(z0) → f(z0) 11.86/3.71
a__p(s(0)) → 0 11.86/3.71
a__p(z0) → p(z0) 11.86/3.71
mark(f(z0)) → a__f(mark(z0)) 11.86/3.71
mark(p(z0)) → a__p(mark(z0)) 11.86/3.71
mark(0) → 0 11.86/3.71
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))), A__P(s(0))) 11.86/3.72
MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(A__P(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))), A__P(s(0))) 11.86/3.72
MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(A__P(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

A__F, MARK

Compound Symbols:

c1, c5, c6, c8, c9

11.86/3.72
11.86/3.72

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

Removed 2 trailing tuple parts
11.86/3.72
11.86/3.72

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
S tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

11.86/3.72
11.86/3.72

(5) 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.

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
We considered the (Usable) Rules:

a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0)) 11.86/3.72
a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0)
And the Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 11.86/3.72

POL(0) = 0    11.86/3.72
POL(A__F(x1)) = [2]    11.86/3.72
POL(MARK(x1)) = [4]x1    11.86/3.72
POL(a__f(x1)) = [4] + x1    11.86/3.72
POL(a__p(x1)) = 0    11.86/3.72
POL(c1(x1)) = x1    11.86/3.72
POL(c5(x1, x2)) = x1 + x2    11.86/3.72
POL(c6(x1)) = x1    11.86/3.72
POL(c8(x1)) = x1    11.86/3.72
POL(c9(x1)) = x1    11.86/3.72
POL(cons(x1, x2)) = x1    11.86/3.72
POL(f(x1)) = [2] + x1    11.86/3.72
POL(mark(x1)) = [2]x1    11.86/3.72
POL(p(x1)) = [1] + x1    11.86/3.72
POL(s(x1)) = x1   
11.86/3.72
11.86/3.72

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
S tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

11.86/3.72
11.86/3.72

(7) 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.

MARK(s(z0)) → c9(MARK(z0))
We considered the (Usable) Rules:

a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0)) 11.86/3.72
a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0)
And the Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 11.86/3.72

POL(0) = 0    11.86/3.72
POL(A__F(x1)) = [4]    11.86/3.72
POL(MARK(x1)) = [2]x1    11.86/3.72
POL(a__f(x1)) = [3]    11.86/3.72
POL(a__p(x1)) = 0    11.86/3.72
POL(c1(x1)) = x1    11.86/3.72
POL(c5(x1, x2)) = x1 + x2    11.86/3.72
POL(c6(x1)) = x1    11.86/3.72
POL(c8(x1)) = x1    11.86/3.72
POL(c9(x1)) = x1    11.86/3.72
POL(cons(x1, x2)) = x1    11.86/3.72
POL(f(x1)) = [5] + x1    11.86/3.72
POL(mark(x1)) = 0    11.86/3.72
POL(p(x1)) = [5] + x1    11.86/3.72
POL(s(x1)) = [1] + x1   
11.86/3.72
11.86/3.72

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
S tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

11.86/3.72
11.86/3.72

(9) 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.

MARK(cons(z0, z1)) → c8(MARK(z0))
We considered the (Usable) Rules:

a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0)) 11.86/3.72
a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0)
And the Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 11.86/3.72

POL(0) = [4]    11.86/3.72
POL(A__F(x1)) = [1]    11.86/3.72
POL(MARK(x1)) = x1    11.86/3.72
POL(a__f(x1)) = [2] + x1    11.86/3.72
POL(a__p(x1)) = 0    11.86/3.72
POL(c1(x1)) = x1    11.86/3.72
POL(c5(x1, x2)) = x1 + x2    11.86/3.72
POL(c6(x1)) = x1    11.86/3.72
POL(c8(x1)) = x1    11.86/3.72
POL(c9(x1)) = x1    11.86/3.72
POL(cons(x1, x2)) = [2] + x1    11.86/3.72
POL(f(x1)) = [2] + x1    11.86/3.72
POL(mark(x1)) = x1    11.86/3.72
POL(p(x1)) = [2] + x1    11.86/3.72
POL(s(x1)) = x1   
11.86/3.72
11.86/3.72

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

11.86/3.72
11.86/3.72

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

Use narrowing to replace MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) by

MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(0)) → c5(A__F(0), MARK(0)) 11.86/3.72
MARK(f(cons(z0, z1))) → c5(A__F(cons(mark(z0), z1)), MARK(cons(z0, z1))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
11.86/3.72
11.86/3.72

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(0)) → c5(A__F(0), MARK(0)) 11.86/3.72
MARK(f(cons(z0, z1))) → c5(A__F(cons(mark(z0), z1)), MARK(cons(z0, z1))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c1, c6, c5

11.86/3.72
11.86/3.72

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

Removed 3 trailing tuple parts
11.86/3.72
11.86/3.72

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c1, c6, c5, c5, c5

11.86/3.72
11.86/3.72

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

Removed 1 trailing nodes:

MARK(f(0)) → c5
11.86/3.72
11.86/3.72

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1(A__F(a__p(s(0)))) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c1, c6, c5, c5, c5

11.86/3.72
11.86/3.72

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

Use narrowing to replace A__F(s(0)) → c1(A__F(a__p(s(0)))) by

A__F(s(0)) → c1(A__F(0)) 11.86/3.72
A__F(s(0)) → c1(A__F(p(s(0))))
11.86/3.72
11.86/3.72

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1))) 11.86/3.72
A__F(s(0)) → c1(A__F(0)) 11.86/3.72
A__F(s(0)) → c1(A__F(p(s(0))))
S tuples:

A__F(s(0)) → c1(A__F(0)) 11.86/3.72
A__F(s(0)) → c1(A__F(p(s(0))))
K tuples:

MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c5, c1

11.86/3.72
11.86/3.72

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

Removed 2 trailing tuple parts
11.86/3.72
11.86/3.72

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1))) 11.86/3.72
A__F(s(0)) → c1
S tuples:

A__F(s(0)) → c1
K tuples:

MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c5, c1

11.86/3.72
11.86/3.72

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

Removed 1 trailing nodes:

MARK(f(0)) → c5
11.86/3.72
11.86/3.72

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1))) 11.86/3.72
A__F(s(0)) → c1
S tuples:

A__F(s(0)) → c1
K tuples:

MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c5, c1

11.86/3.72
11.86/3.72

(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.

A__F(s(0)) → c1
We considered the (Usable) Rules:

mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0)) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0)
And the Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1))) 11.86/3.72
A__F(s(0)) → c1
The order we found is given by the following interpretation:
Polynomial interpretation : 11.86/3.72

POL(0) = 0    11.86/3.72
POL(A__F(x1)) = [4]    11.86/3.72
POL(MARK(x1)) = x1    11.86/3.72
POL(a__f(x1)) = 0    11.86/3.72
POL(a__p(x1)) = 0    11.86/3.72
POL(c1) = 0    11.86/3.72
POL(c5) = 0    11.86/3.72
POL(c5(x1)) = x1    11.86/3.72
POL(c5(x1, x2)) = x1 + x2    11.86/3.72
POL(c6(x1)) = x1    11.86/3.72
POL(c8(x1)) = x1    11.86/3.72
POL(c9(x1)) = x1    11.86/3.72
POL(cons(x1, x2)) = x1    11.86/3.72
POL(f(x1)) = [4] + x1    11.86/3.72
POL(mark(x1)) = 0    11.86/3.72
POL(p(x1)) = x1    11.86/3.72
POL(s(x1)) = x1   
11.86/3.72
11.86/3.72

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0))) 11.86/3.72
a__f(s(0)) → a__f(a__p(s(0))) 11.86/3.72
a__f(z0) → f(z0) 11.86/3.72
a__p(s(0)) → 0 11.86/3.72
a__p(z0) → p(z0) 11.86/3.72
mark(f(z0)) → a__f(mark(z0)) 11.86/3.72
mark(p(z0)) → a__p(mark(z0)) 11.86/3.72
mark(0) → 0 11.86/3.72
mark(cons(z0, z1)) → cons(mark(z0), z1) 11.86/3.72
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0))) 11.86/3.72
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0))) 11.86/3.72
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0))) 11.86/3.72
MARK(f(0)) → c5 11.86/3.72
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1))) 11.86/3.72
A__F(s(0)) → c1
S tuples:none
K tuples:

MARK(p(z0)) → c6(MARK(z0)) 11.86/3.72
MARK(s(z0)) → c9(MARK(z0)) 11.86/3.72
MARK(cons(z0, z1)) → c8(MARK(z0)) 11.86/3.72
A__F(s(0)) → c1
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c5, c1

11.86/3.72
11.86/3.72

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

The set S is empty
11.86/3.72
11.86/3.72

(26) BOUNDS(O(1), O(1))

11.86/3.72
11.86/3.72
11.86/3.77 EOF