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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(a)) → c(n__f(g(f(a))))
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f(X) → n__f(X)
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activate(n__f(X)) → f(X)
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activate(X) → 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:
f(f(a)) → c(n__f(g(f(a))))
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f(z0) → n__f(z0)
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activate(n__f(z0)) → f(z0)
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activate(z0) → z0
Tuples:
F(f(a)) → c1(F(a))
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ACTIVATE(n__f(z0)) → c3(F(z0))
S tuples:
F(f(a)) → c1(F(a))
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ACTIVATE(n__f(z0)) → c3(F(z0))
K tuples:none
Defined Rule Symbols:
f, activate
Defined Pair Symbols:
F, ACTIVATE
Compound Symbols:
c1, c3
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(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(f(a)) → c1(F(a))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(a)) → c(n__f(g(f(a))))
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f(z0) → n__f(z0)
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activate(n__f(z0)) → f(z0)
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activate(z0) → z0
Tuples:
ACTIVATE(n__f(z0)) → c3(F(z0))
S tuples:
ACTIVATE(n__f(z0)) → c3(F(z0))
K tuples:none
Defined Rule Symbols:
f, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3
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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(a)) → c(n__f(g(f(a))))
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f(z0) → n__f(z0)
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activate(n__f(z0)) → f(z0)
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activate(z0) → z0
Tuples:
ACTIVATE(n__f(z0)) → c3
S tuples:
ACTIVATE(n__f(z0)) → c3
K tuples:none
Defined Rule Symbols:
f, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3
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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
ACTIVATE(n__f(z0)) → c3
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(a)) → c(n__f(g(f(a))))
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f(z0) → n__f(z0)
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activate(n__f(z0)) → f(z0)
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activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, activate
Defined Pair Symbols:none
Compound Symbols:none
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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(10) BOUNDS(O(1), O(1))
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