YES(O(1), O(n^1)) 0.00/0.80 YES(O(1), O(n^1)) 0.00/0.82 0.00/0.82 0.00/0.82 0.00/0.82 0.00/0.82 0.00/0.82 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.82 0.00/0.82 0.00/0.82
0.00/0.82 0.00/0.82 0.00/0.82
0.00/0.82
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(0) Obligation:

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

a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
mark(f(X)) → a__f(mark(X)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(X)) → g(X) 0.00/0.82
a__f(X) → f(X)

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

Converted CpxTRS to CDT
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0.00/0.82

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
Tuples:

A__F(f(a)) → c(A__F(g(f(a)))) 0.00/0.82
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
S tuples:

A__F(f(a)) → c(A__F(g(f(a)))) 0.00/0.82
MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

A__F, MARK

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:

a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
S tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c2, c

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

Removed 1 trailing nodes:

A__F(f(a)) → c
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
S tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
K tuples:none
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c2, c

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0.00/0.82

(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.

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
We considered the (Usable) Rules:

mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0) 0.00/0.82
a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0)
And the Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.82

POL(A__F(x1)) = 0    0.00/0.82
POL(MARK(x1)) = [4]x1    0.00/0.82
POL(a) = 0    0.00/0.82
POL(a__f(x1)) = [3]    0.00/0.82
POL(c) = 0    0.00/0.82
POL(c2(x1, x2)) = x1 + x2    0.00/0.82
POL(f(x1)) = [4] + x1    0.00/0.82
POL(g(x1)) = [3]    0.00/0.82
POL(mark(x1)) = 0   
0.00/0.82
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(f(a)) → a__f(g(f(a))) 0.00/0.82
a__f(z0) → f(z0) 0.00/0.82
mark(f(z0)) → a__f(mark(z0)) 0.00/0.82
mark(a) → a 0.00/0.82
mark(g(z0)) → g(z0)
Tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0)) 0.00/0.82
A__F(f(a)) → c
S tuples:

A__F(f(a)) → c
K tuples:

MARK(f(z0)) → c2(A__F(mark(z0)), MARK(z0))
Defined Rule Symbols:

a__f, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c2, c

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(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

A__F(f(a)) → c
Now S is empty
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(10) BOUNDS(O(1), O(1))

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0.00/0.84 EOF