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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
d(x) → e(u(x))
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d(u(x)) → c(x)
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c(u(x)) → b(x)
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v(e(x)) → x
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b(u(x)) → a(e(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:
d(z0) → e(u(z0))
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d(u(z0)) → c(z0)
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c(u(z0)) → b(z0)
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v(e(z0)) → z0
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b(u(z0)) → a(e(z0))
Tuples:
D(u(z0)) → c2(C(z0))
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C(u(z0)) → c3(B(z0))
S tuples:
D(u(z0)) → c2(C(z0))
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C(u(z0)) → c3(B(z0))
K tuples:none
Defined Rule Symbols:
d, c, v, b
Defined Pair Symbols:
D, C
Compound Symbols:
c2, c3
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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:
d(z0) → e(u(z0))
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d(u(z0)) → c(z0)
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c(u(z0)) → b(z0)
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v(e(z0)) → z0
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b(u(z0)) → a(e(z0))
Tuples:
D(u(z0)) → c2(C(z0))
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C(u(z0)) → c3
S tuples:
D(u(z0)) → c2(C(z0))
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C(u(z0)) → c3
K tuples:none
Defined Rule Symbols:
d, c, v, b
Defined Pair Symbols:
D, C
Compound Symbols:
c2, c3
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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
C(u(z0)) → c3
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D(u(z0)) → c2(C(z0))
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
d(z0) → e(u(z0))
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d(u(z0)) → c(z0)
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c(u(z0)) → b(z0)
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v(e(z0)) → z0
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b(u(z0)) → a(e(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
d, c, v, b
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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