YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.85 0.00/0.85 0.00/0.85
0.00/0.85 0.00/0.85 0.00/0.85
0.00/0.85
0.00/0.85

(0) Obligation:

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

U11(tt, M, N) → U12(tt, activate(M), activate(N)) 0.00/0.85
U12(tt, M, N) → s(plus(activate(N), activate(M))) 0.00/0.85
plus(N, 0) → N 0.00/0.85
plus(N, s(M)) → U11(tt, M, N) 0.00/0.85
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:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 0.00/0.85
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 0.00/0.85
plus(z0, 0) → z0 0.00/0.85
plus(z0, s(z1)) → U11(tt, z1, z0) 0.00/0.85
activate(z0) → z0
Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 0.00/0.85
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 0.00/0.85
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
K tuples:none
Defined Rule Symbols:

U11, U12, plus, activate

Defined Pair Symbols:

U11', U12', PLUS

Compound Symbols:

c, c1, c3

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

Removed 4 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 0.00/0.85
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 0.00/0.85
plus(z0, 0) → z0 0.00/0.85
plus(z0, s(z1)) → U11(tt, z1, z0) 0.00/0.85
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:none
Defined Rule Symbols:

U11, U12, plus, activate

Defined Pair Symbols:

PLUS, U11', U12'

Compound Symbols:

c3, c, c1

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(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.

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
We considered the (Usable) Rules:

activate(z0) → z0
And the Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(PLUS(x1, x2)) = x2    0.00/0.85
POL(U11'(x1, x2, x3)) = x1 + x2    0.00/0.85
POL(U12'(x1, x2, x3)) = x1 + x2    0.00/0.85
POL(activate(x1)) = x1    0.00/0.85
POL(c(x1)) = x1    0.00/0.85
POL(c1(x1)) = x1    0.00/0.85
POL(c3(x1)) = x1    0.00/0.85
POL(s(x1)) = [2] + x1    0.00/0.85
POL(tt) = 0   
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 0.00/0.85
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 0.00/0.85
plus(z0, 0) → z0 0.00/0.85
plus(z0, s(z1)) → U11(tt, z1, z0) 0.00/0.85
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0)) 0.00/0.85
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:

PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
Defined Rule Symbols:

U11, U12, plus, activate

Defined Pair Symbols:

PLUS, U11', U12'

Compound Symbols:

c3, c, c1

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(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 0.00/0.85
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 0.00/0.85
PLUS(z0, s(z1)) → c3(U11'(tt, z1, z0))
Now S is empty
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(8) BOUNDS(O(1), O(1))

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0.00/0.90 EOF