YES(O(1), O(1)) 0.00/0.56 YES(O(1), O(1)) 0.00/0.58 0.00/0.58 0.00/0.58 0.00/0.58 0.00/0.58 0.00/0.58 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.58 0.00/0.58 0.00/0.58
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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 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳6 BOUNDS(O(1), O(1))
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(0) Obligation:

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

h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w)))))) 0.00/0.58
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w))) 0.00/0.58
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x))))) 0.00/0.58
t(t(x)) → t(c(t(x), x)) 0.00/0.58
t(x) → x 0.00/0.58
t(x) → c(0, c(0, c(0, c(0, c(0, 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:

h(c(z0, z1), c(s(z2), z2), t(z3)) → h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))) 0.00/0.58
h(z0, c(z1, z2), t(z3)) → h(c(s(z1), z0), z2, t(c(t(z3), z3))) 0.00/0.58
h(c(s(z0), c(s(0), z1)), z2, t(z0)) → h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))) 0.00/0.58
t(t(z0)) → t(c(t(z0), z0)) 0.00/0.58
t(z0) → z0 0.00/0.58
t(z0) → c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:

H(c(z0, z1), c(s(z2), z2), t(z3)) → c1(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3)) 0.00/0.58
H(z0, c(z1, z2), t(z3)) → c2(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3)) 0.00/0.58
H(c(s(z0), c(s(0), z1)), z2, t(z0)) → c3(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0)))) 0.00/0.58
T(t(z0)) → c4(T(c(t(z0), z0)), T(z0))
S tuples:

H(c(z0, z1), c(s(z2), z2), t(z3)) → c1(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3)) 0.00/0.58
H(z0, c(z1, z2), t(z3)) → c2(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3)) 0.00/0.58
H(c(s(z0), c(s(0), z1)), z2, t(z0)) → c3(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0)))) 0.00/0.58
T(t(z0)) → c4(T(c(t(z0), z0)), T(z0))
K tuples:none
Defined Rule Symbols:

h, t

Defined Pair Symbols:

H, T

Compound Symbols:

c1, c2, c3, c4

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(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

H(c(z0, z1), c(s(z2), z2), t(z3)) → c1(H(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))), T(t(c(z0, c(z1, t(z3))))), T(c(z0, c(z1, t(z3)))), T(z3)) 0.00/0.58
H(z0, c(z1, z2), t(z3)) → c2(H(c(s(z1), z0), z2, t(c(t(z3), z3))), T(c(t(z3), z3)), T(z3)) 0.00/0.58
H(c(s(z0), c(s(0), z1)), z2, t(z0)) → c3(H(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))), T(t(c(z0, s(z0)))), T(c(z0, s(z0)))) 0.00/0.58
T(t(z0)) → c4(T(c(t(z0), z0)), T(z0))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(c(z0, z1), c(s(z2), z2), t(z3)) → h(z2, c(z1, z0), t(t(c(z0, c(z1, t(z3)))))) 0.00/0.58
h(z0, c(z1, z2), t(z3)) → h(c(s(z1), z0), z2, t(c(t(z3), z3))) 0.00/0.58
h(c(s(z0), c(s(0), z1)), z2, t(z0)) → h(z1, c(s(0), c(z0, z2)), t(t(c(z0, s(z0))))) 0.00/0.58
t(t(z0)) → t(c(t(z0), z0)) 0.00/0.58
t(z0) → z0 0.00/0.58
t(z0) → c(0, c(0, c(0, c(0, c(0, z0)))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

h, t

Defined Pair Symbols:none

Compound Symbols:none

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

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

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0.00/0.63 EOF