YES(O(1), O(n^1)) 0.00/0.76 YES(O(1), O(n^1)) 0.00/0.77 0.00/0.77 0.00/0.77 0.00/0.77 0.00/0.77 0.00/0.77 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.77 0.00/0.77 0.00/0.77
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳8 CdtProblem
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↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳10 CdtProblem
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↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳12 BOUNDS(O(1), O(1))
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(0) Obligation:

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

from(X) → cons(X, n__from(n__s(X))) 0.00/0.77
after(0, XS) → XS 0.00/0.77
after(s(N), cons(X, XS)) → after(N, activate(XS)) 0.00/0.77
from(X) → n__from(X) 0.00/0.77
s(X) → n__s(X) 0.00/0.77
activate(n__from(X)) → from(activate(X)) 0.00/0.77
activate(n__s(X)) → s(activate(X)) 0.00/0.77
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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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.77
from(z0) → n__from(z0) 0.00/0.77
after(0, z0) → z0 0.00/0.77
after(s(z0), cons(z1, z2)) → after(z0, activate(z2)) 0.00/0.77
s(z0) → n__s(z0) 0.00/0.77
activate(n__from(z0)) → from(activate(z0)) 0.00/0.77
activate(n__s(z0)) → s(activate(z0)) 0.00/0.77
activate(z0) → z0
Tuples:

AFTER(s(z0), cons(z1, z2)) → c3(AFTER(z0, activate(z2)), ACTIVATE(z2)) 0.00/0.77
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
S tuples:

AFTER(s(z0), cons(z1, z2)) → c3(AFTER(z0, activate(z2)), ACTIVATE(z2)) 0.00/0.78
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, after, s, activate

Defined Pair Symbols:

AFTER, ACTIVATE

Compound Symbols:

c3, c5, c6

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

AFTER(s(z0), cons(z1, z2)) → c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.78
from(z0) → n__from(z0) 0.00/0.78
after(0, z0) → z0 0.00/0.78
after(s(z0), cons(z1, z2)) → after(z0, activate(z2)) 0.00/0.78
s(z0) → n__s(z0) 0.00/0.78
activate(n__from(z0)) → from(activate(z0)) 0.00/0.78
activate(n__s(z0)) → s(activate(z0)) 0.00/0.78
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, after, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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

Removed 2 trailing tuple parts
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.78
from(z0) → n__from(z0) 0.00/0.78
after(0, z0) → z0 0.00/0.78
after(s(z0), cons(z1, z2)) → after(z0, activate(z2)) 0.00/0.78
s(z0) → n__s(z0) 0.00/0.78
activate(n__from(z0)) → from(activate(z0)) 0.00/0.78
activate(n__s(z0)) → s(activate(z0)) 0.00/0.78
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, after, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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(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.

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.78

POL(ACTIVATE(x1)) = [2]x1    0.00/0.78
POL(c5(x1)) = x1    0.00/0.78
POL(c6(x1)) = x1    0.00/0.78
POL(n__from(x1)) = [1] + x1    0.00/0.78
POL(n__s(x1)) = x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.78
from(z0) → n__from(z0) 0.00/0.78
after(0, z0) → z0 0.00/0.78
after(s(z0), cons(z1, z2)) → after(z0, activate(z2)) 0.00/0.78
s(z0) → n__s(z0) 0.00/0.78
activate(n__from(z0)) → from(activate(z0)) 0.00/0.78
activate(n__s(z0)) → s(activate(z0)) 0.00/0.78
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
K tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
Defined Rule Symbols:

from, after, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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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__s(z0)) → c6(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.78

POL(ACTIVATE(x1)) = [3]x1    0.00/0.78
POL(c5(x1)) = x1    0.00/0.78
POL(c6(x1)) = x1    0.00/0.78
POL(n__from(x1)) = x1    0.00/0.78
POL(n__s(x1)) = [1] + x1   
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.78
from(z0) → n__from(z0) 0.00/0.78
after(0, z0) → z0 0.00/0.78
after(s(z0), cons(z1, z2)) → after(z0, activate(z2)) 0.00/0.78
s(z0) → n__s(z0) 0.00/0.78
activate(n__from(z0)) → from(activate(z0)) 0.00/0.78
activate(n__s(z0)) → s(activate(z0)) 0.00/0.78
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.78
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
Defined Rule Symbols:

from, after, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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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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0.00/0.82 EOF