YES(O(1), O(n^1)) 0.00/0.73 YES(O(1), O(n^1)) 0.00/0.74 0.00/0.74 0.00/0.74 0.00/0.74 0.00/0.74 0.00/0.74 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.74 0.00/0.74 0.00/0.74
0.00/0.74 0.00/0.74 0.00/0.74
0.00/0.74
0.00/0.74

(0) Obligation:

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

f(0) → cons(0, n__f(n__s(n__0))) 0.00/0.74
f(s(0)) → f(p(s(0))) 0.00/0.74
p(s(X)) → X 0.00/0.74
f(X) → n__f(X) 0.00/0.74
s(X) → n__s(X) 0.00/0.74
0n__0 0.00/0.74
activate(n__f(X)) → f(activate(X)) 0.00/0.74
activate(n__s(X)) → s(activate(X)) 0.00/0.74
activate(n__0) → 0 0.00/0.74
activate(X) → X

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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0.00/0.74

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0))) 0.00/0.74
f(s(0)) → f(p(s(0))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
p(s(z0)) → z0 0.00/0.74
s(z0) → n__s(z0) 0.00/0.74
0n__0 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__s(z0)) → s(activate(z0)) 0.00/0.74
activate(n__0) → 0 0.00/0.74
activate(z0) → z0
Tuples:

F(0) → c(0') 0.00/0.74
F(s(0)) → c1(F(p(s(0))), P(s(0)), S(0), 0') 0.00/0.74
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8(0')
S tuples:

F(0) → c(0') 0.00/0.74
F(s(0)) → c1(F(p(s(0))), P(s(0)), S(0), 0') 0.00/0.74
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8(0')
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c, c1, c6, c7, c8

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(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

F(0) → c(0') 0.00/0.74
F(s(0)) → c1(F(p(s(0))), P(s(0)), S(0), 0')
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0))) 0.00/0.74
f(s(0)) → f(p(s(0))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
p(s(z0)) → z0 0.00/0.74
s(z0) → n__s(z0) 0.00/0.74
0n__0 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__s(z0)) → s(activate(z0)) 0.00/0.74
activate(n__0) → 0 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8(0')
S tuples:

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8(0')
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

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

Removed 3 trailing tuple parts
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0))) 0.00/0.74
f(s(0)) → f(p(s(0))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
p(s(z0)) → z0 0.00/0.74
s(z0) → n__s(z0) 0.00/0.74
0n__0 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__s(z0)) → s(activate(z0)) 0.00/0.74
activate(n__0) → 0 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
S tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

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

Removed 1 trailing nodes:

ACTIVATE(n__0) → c8
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0))) 0.00/0.74
f(s(0)) → f(p(s(0))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
p(s(z0)) → z0 0.00/0.74
s(z0) → n__s(z0) 0.00/0.74
0n__0 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__s(z0)) → s(activate(z0)) 0.00/0.74
activate(n__0) → 0 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
S tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

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(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.

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.74

POL(ACTIVATE(x1)) = [5] + [3]x1    0.00/0.74
POL(c6(x1)) = x1    0.00/0.74
POL(c7(x1)) = x1    0.00/0.74
POL(c8) = 0    0.00/0.74
POL(n__0) = [5]    0.00/0.74
POL(n__f(x1)) = [5] + x1    0.00/0.74
POL(n__s(x1)) = [5] + x1   
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0))) 0.00/0.74
f(s(0)) → f(p(s(0))) 0.00/0.74
f(z0) → n__f(z0) 0.00/0.74
p(s(z0)) → z0 0.00/0.74
s(z0) → n__s(z0) 0.00/0.74
0n__0 0.00/0.74
activate(n__f(z0)) → f(activate(z0)) 0.00/0.74
activate(n__s(z0)) → s(activate(z0)) 0.00/0.74
activate(n__0) → 0 0.00/0.74
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0)) 0.00/0.74
ACTIVATE(n__0) → c8
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

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

The set S is empty
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(12) BOUNDS(O(1), O(1))

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0.00/0.78 EOF