YES(O(1), O(1)) 0.00/0.54 YES(O(1), O(1)) 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 0.00/0.55 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.55 0.00/0.55 0.00/0.55
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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:

a(c(d(x))) → c(x) 0.00/0.55
u(b(d(d(x)))) → b(x) 0.00/0.55
v(a(a(x))) → u(v(x)) 0.00/0.55
v(a(c(x))) → u(b(d(x))) 0.00/0.55
v(c(x)) → b(x) 0.00/0.55
w(a(a(x))) → u(w(x)) 0.00/0.55
w(a(c(x))) → u(b(d(x))) 0.00/0.55
w(c(x)) → b(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:

a(c(d(z0))) → c(z0) 0.00/0.55
u(b(d(d(z0)))) → b(z0) 0.00/0.55
v(a(a(z0))) → u(v(z0)) 0.00/0.55
v(a(c(z0))) → u(b(d(z0))) 0.00/0.55
v(c(z0)) → b(z0) 0.00/0.55
w(a(a(z0))) → u(w(z0)) 0.00/0.55
w(a(c(z0))) → u(b(d(z0))) 0.00/0.55
w(c(z0)) → b(z0)
Tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0)) 0.00/0.55
V(a(c(z0))) → c4(U(b(d(z0)))) 0.00/0.55
W(a(a(z0))) → c6(U(w(z0)), W(z0)) 0.00/0.55
W(a(c(z0))) → c7(U(b(d(z0))))
S tuples:

V(a(a(z0))) → c3(U(v(z0)), V(z0)) 0.00/0.55
V(a(c(z0))) → c4(U(b(d(z0)))) 0.00/0.55
W(a(a(z0))) → c6(U(w(z0)), W(z0)) 0.00/0.55
W(a(c(z0))) → c7(U(b(d(z0))))
K tuples:none
Defined Rule Symbols:

a, u, v, w

Defined Pair Symbols:

V, W

Compound Symbols:

c3, c4, c6, c7

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

V(a(a(z0))) → c3(U(v(z0)), V(z0)) 0.00/0.55
V(a(c(z0))) → c4(U(b(d(z0)))) 0.00/0.55
W(a(a(z0))) → c6(U(w(z0)), W(z0)) 0.00/0.55
W(a(c(z0))) → c7(U(b(d(z0))))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(c(d(z0))) → c(z0) 0.00/0.55
u(b(d(d(z0)))) → b(z0) 0.00/0.55
v(a(a(z0))) → u(v(z0)) 0.00/0.55
v(a(c(z0))) → u(b(d(z0))) 0.00/0.55
v(c(z0)) → b(z0) 0.00/0.55
w(a(a(z0))) → u(w(z0)) 0.00/0.55
w(a(c(z0))) → u(b(d(z0))) 0.00/0.55
w(c(z0)) → b(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, u, v, w

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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0.00/0.58 EOF