YES(O(1), O(1)) 0.00/0.55 YES(O(1), O(1)) 0.00/0.56 0.00/0.56 0.00/0.56 0.00/0.56 0.00/0.56 0.00/0.56 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.56 0.00/0.56 0.00/0.56
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(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 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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(0) Obligation:

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

2nd(cons(X, n__cons(Y, Z))) → activate(Y) 0.00/0.56
from(X) → cons(X, n__from(s(X))) 0.00/0.56
cons(X1, X2) → n__cons(X1, X2) 0.00/0.56
from(X) → n__from(X) 0.00/0.56
activate(n__cons(X1, X2)) → cons(X1, X2) 0.00/0.56
activate(n__from(X)) → from(X) 0.00/0.56
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:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1) 0.00/0.56
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.56
from(z0) → n__from(z0) 0.00/0.56
cons(z0, z1) → n__cons(z0, z1) 0.00/0.56
activate(n__cons(z0, z1)) → cons(z0, z1) 0.00/0.56
activate(n__from(z0)) → from(z0) 0.00/0.56
activate(z0) → z0
Tuples:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1)) 0.00/0.56
FROM(z0) → c1(CONS(z0, n__from(s(z0)))) 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1)) 0.00/0.56
ACTIVATE(n__from(z0)) → c5(FROM(z0))
S tuples:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1)) 0.00/0.56
FROM(z0) → c1(CONS(z0, n__from(s(z0)))) 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1)) 0.00/0.56
ACTIVATE(n__from(z0)) → c5(FROM(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:

2ND, FROM, ACTIVATE

Compound Symbols:

c, c1, c4, c5

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1) 0.00/0.56
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.56
from(z0) → n__from(z0) 0.00/0.56
cons(z0, z1) → n__cons(z0, z1) 0.00/0.56
activate(n__cons(z0, z1)) → cons(z0, z1) 0.00/0.56
activate(n__from(z0)) → from(z0) 0.00/0.56
activate(z0) → z0
Tuples:

FROM(z0) → c1(CONS(z0, n__from(s(z0)))) 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1)) 0.00/0.56
ACTIVATE(n__from(z0)) → c5(FROM(z0))
S tuples:

FROM(z0) → c1(CONS(z0, n__from(s(z0)))) 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1)) 0.00/0.56
ACTIVATE(n__from(z0)) → c5(FROM(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:

FROM, ACTIVATE

Compound Symbols:

c1, c4, c5

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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:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1) 0.00/0.56
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.56
from(z0) → n__from(z0) 0.00/0.56
cons(z0, z1) → n__cons(z0, z1) 0.00/0.56
activate(n__cons(z0, z1)) → cons(z0, z1) 0.00/0.56
activate(n__from(z0)) → from(z0) 0.00/0.56
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(FROM(z0)) 0.00/0.56
FROM(z0) → c1 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4
S tuples:

ACTIVATE(n__from(z0)) → c5(FROM(z0)) 0.00/0.56
FROM(z0) → c1 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4
K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:

ACTIVATE, FROM

Compound Symbols:

c5, c1, c4

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

Removed 3 trailing nodes:

ACTIVATE(n__from(z0)) → c5(FROM(z0)) 0.00/0.56
FROM(z0) → c1 0.00/0.56
ACTIVATE(n__cons(z0, z1)) → c4
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1) 0.00/0.56
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.56
from(z0) → n__from(z0) 0.00/0.56
cons(z0, z1) → n__cons(z0, z1) 0.00/0.56
activate(n__cons(z0, z1)) → cons(z0, z1) 0.00/0.56
activate(n__from(z0)) → from(z0) 0.00/0.56
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:none

Compound Symbols:none

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

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

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0.00/0.58 EOF