YES(O(1), O(1)) 0.00/0.54 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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
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↳4 CdtProblem
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↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
0.00/0.56 
↳6 CdtProblem
0.00/0.56 
↳7 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳8 BOUNDS(O(1), O(1))
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0.00/0.56

(0) Obligation:

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

rewrite(Op(Val(n), y)) → rewrite[Ite][True][Let][Ite](False, Op(Val(n), y), Val(n)) 0.00/0.56
rewrite(Op(Op(x, y), y')) → rewrite[Ite][True][Let][Ite](True, Op(Op(x, y), y'), Op(x, y)) 0.00/0.56
rewrite(Val(n)) → Val(n) 0.00/0.56
second(Op(x, y)) → y 0.00/0.56
isOp(Val(n)) → False 0.00/0.56
isOp(Op(x, y)) → True 0.00/0.56
first(Val(n)) → Val(n) 0.00/0.56
first(Op(x, y)) → x 0.00/0.56
assrewrite(exp) → rewrite(exp)

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:

rewrite(Op(Val(z0), z1)) → rewrite[Ite][True][Let][Ite](False, Op(Val(z0), z1), Val(z0)) 0.00/0.56
rewrite(Op(Op(z0, z1), z2)) → rewrite[Ite][True][Let][Ite](True, Op(Op(z0, z1), z2), Op(z0, z1)) 0.00/0.56
rewrite(Val(z0)) → Val(z0) 0.00/0.56
second(Op(z0, z1)) → z1 0.00/0.56
isOp(Val(z0)) → False 0.00/0.56
isOp(Op(z0, z1)) → True 0.00/0.56
first(Val(z0)) → Val(z0) 0.00/0.56
first(Op(z0, z1)) → z0 0.00/0.56
assrewrite(z0) → rewrite(z0)
Tuples:

ASSREWRITE(z0) → c8(REWRITE(z0))
S tuples:

ASSREWRITE(z0) → c8(REWRITE(z0))
K tuples:none
Defined Rule Symbols:

rewrite, second, isOp, first, assrewrite

Defined Pair Symbols:

ASSREWRITE

Compound Symbols:

c8

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

Removed 1 trailing tuple parts
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0.00/0.56

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

rewrite(Op(Val(z0), z1)) → rewrite[Ite][True][Let][Ite](False, Op(Val(z0), z1), Val(z0)) 0.00/0.56
rewrite(Op(Op(z0, z1), z2)) → rewrite[Ite][True][Let][Ite](True, Op(Op(z0, z1), z2), Op(z0, z1)) 0.00/0.56
rewrite(Val(z0)) → Val(z0) 0.00/0.56
second(Op(z0, z1)) → z1 0.00/0.56
isOp(Val(z0)) → False 0.00/0.56
isOp(Op(z0, z1)) → True 0.00/0.56
first(Val(z0)) → Val(z0) 0.00/0.56
first(Op(z0, z1)) → z0 0.00/0.56
assrewrite(z0) → rewrite(z0)
Tuples:

ASSREWRITE(z0) → c8
S tuples:

ASSREWRITE(z0) → c8
K tuples:none
Defined Rule Symbols:

rewrite, second, isOp, first, assrewrite

Defined Pair Symbols:

ASSREWRITE

Compound Symbols:

c8

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

Removed 1 trailing nodes:

ASSREWRITE(z0) → c8
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

rewrite(Op(Val(z0), z1)) → rewrite[Ite][True][Let][Ite](False, Op(Val(z0), z1), Val(z0)) 0.00/0.56
rewrite(Op(Op(z0, z1), z2)) → rewrite[Ite][True][Let][Ite](True, Op(Op(z0, z1), z2), Op(z0, z1)) 0.00/0.56
rewrite(Val(z0)) → Val(z0) 0.00/0.56
second(Op(z0, z1)) → z1 0.00/0.56
isOp(Val(z0)) → False 0.00/0.56
isOp(Op(z0, z1)) → True 0.00/0.56
first(Val(z0)) → Val(z0) 0.00/0.56
first(Op(z0, z1)) → z0 0.00/0.56
assrewrite(z0) → rewrite(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

rewrite, second, isOp, first, assrewrite

Defined Pair Symbols:none

Compound Symbols:none

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

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

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0.00/0.58 EOF