YES(O(1), O(n^1)) 0.00/0.71 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) Obligation:

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

rev(ls) → r1(ls, empty) 0.00/0.72
r1(empty, a) → a 0.00/0.72
r1(cons(x, k), a) → r1(k, cons(x, a))

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:

rev(z0) → r1(z0, empty) 0.00/0.72
r1(empty, z0) → z0 0.00/0.72
r1(cons(z0, z1), z2) → r1(z1, cons(z0, z2))
Tuples:

REV(z0) → c(R1(z0, empty)) 0.00/0.72
R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
S tuples:

REV(z0) → c(R1(z0, empty)) 0.00/0.72
R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
K tuples:none
Defined Rule Symbols:

rev, r1

Defined Pair Symbols:

REV, R1

Compound Symbols:

c, c2

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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

REV(z0) → c(R1(z0, empty))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(z0) → r1(z0, empty) 0.00/0.72
r1(empty, z0) → z0 0.00/0.72
r1(cons(z0, z1), z2) → r1(z1, cons(z0, z2))
Tuples:

R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
S tuples:

R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
K tuples:none
Defined Rule Symbols:

rev, r1

Defined Pair Symbols:

R1

Compound Symbols:

c2

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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.

R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
We considered the (Usable) Rules:none
And the Tuples:

R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.72

POL(R1(x1, x2)) = [4]x1    0.00/0.72
POL(c2(x1)) = x1    0.00/0.72
POL(cons(x1, x2)) = [4] + x1 + x2   
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(z0) → r1(z0, empty) 0.00/0.72
r1(empty, z0) → z0 0.00/0.72
r1(cons(z0, z1), z2) → r1(z1, cons(z0, z2))
Tuples:

R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
S tuples:none
K tuples:

R1(cons(z0, z1), z2) → c2(R1(z1, cons(z0, z2)))
Defined Rule Symbols:

rev, r1

Defined Pair Symbols:

R1

Compound Symbols:

c2

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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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0.00/0.73 EOF