YES(O(1), O(n^1)) 0.00/0.70 YES(O(1), O(n^1)) 0.00/0.72 0.00/0.72 0.00/0.72 0.00/0.72 0.00/0.72 0.00/0.72 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.72 0.00/0.72 0.00/0.72
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0.00/0.72
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(0) Obligation:

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

and(tt, X) → activate(X) 0.00/0.72
plus(N, 0) → N 0.00/0.72
plus(N, s(M)) → s(plus(N, M)) 0.00/0.72
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:

and(tt, z0) → activate(z0) 0.00/0.72
plus(z0, 0) → z0 0.00/0.72
plus(z0, s(z1)) → s(plus(z0, z1)) 0.00/0.72
activate(z0) → z0
Tuples:

AND(tt, z0) → c(ACTIVATE(z0)) 0.00/0.72
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
S tuples:

AND(tt, z0) → c(ACTIVATE(z0)) 0.00/0.72
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

and, plus, activate

Defined Pair Symbols:

AND, PLUS

Compound Symbols:

c, c2

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

Removed 1 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(tt, z0) → activate(z0) 0.00/0.72
plus(z0, 0) → z0 0.00/0.72
plus(z0, s(z1)) → s(plus(z0, z1)) 0.00/0.72
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1)) 0.00/0.72
AND(tt, z0) → c
S tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1)) 0.00/0.72
AND(tt, z0) → c
K tuples:none
Defined Rule Symbols:

and, plus, activate

Defined Pair Symbols:

PLUS, AND

Compound Symbols:

c2, c

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

Removed 1 trailing nodes:

AND(tt, z0) → c
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0.00/0.72

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(tt, z0) → activate(z0) 0.00/0.72
plus(z0, 0) → z0 0.00/0.72
plus(z0, s(z1)) → s(plus(z0, z1)) 0.00/0.72
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
S tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

and, plus, activate

Defined Pair Symbols:

PLUS

Compound Symbols:

c2

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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.

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.72

POL(PLUS(x1, x2)) = [3]x2    0.00/0.72
POL(c2(x1)) = x1    0.00/0.72
POL(s(x1)) = [1] + x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(tt, z0) → activate(z0) 0.00/0.72
plus(z0, 0) → z0 0.00/0.72
plus(z0, s(z1)) → s(plus(z0, z1)) 0.00/0.72
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
S tuples:none
K tuples:

PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
Defined Rule Symbols:

and, plus, activate

Defined Pair Symbols:

PLUS

Compound Symbols:

c2

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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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0.00/0.73 EOF