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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
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b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
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b(y, z) → z
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(c(a, z0, z1)) → b(a, z0)
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b(z0, b(z1, z2)) → f(b(f(f(z1)), c(z0, z1, z2)))
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b(z0, z1) → z1
Tuples:
F(c(a, z0, z1)) → c1(B(a, z0))
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B(z0, b(z1, z2)) → c2(F(b(f(f(z1)), c(z0, z1, z2))), B(f(f(z1)), c(z0, z1, z2)), F(f(z1)), F(z1))
S tuples:
F(c(a, z0, z1)) → c1(B(a, z0))
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B(z0, b(z1, z2)) → c2(F(b(f(f(z1)), c(z0, z1, z2))), B(f(f(z1)), c(z0, z1, z2)), F(f(z1)), F(z1))
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F, B
Compound Symbols:
c1, c2
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(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
B(z0, b(z1, z2)) → c2(F(b(f(f(z1)), c(z0, z1, z2))), B(f(f(z1)), c(z0, z1, z2)), F(f(z1)), F(z1))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(a, z0, z1)) → b(a, z0)
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b(z0, b(z1, z2)) → f(b(f(f(z1)), c(z0, z1, z2)))
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b(z0, z1) → z1
Tuples:
F(c(a, z0, z1)) → c1(B(a, z0))
S tuples:
F(c(a, z0, z1)) → c1(B(a, z0))
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F
Compound Symbols:
c1
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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(c(a, z0, z1)) → b(a, z0)
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b(z0, b(z1, z2)) → f(b(f(f(z1)), c(z0, z1, z2)))
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b(z0, z1) → z1
Tuples:
F(c(a, z0, z1)) → c1
S tuples:
F(c(a, z0, z1)) → c1
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F
Compound Symbols:
c1
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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(c(a, z0, z1)) → c1
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(a, z0, z1)) → b(a, z0)
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b(z0, b(z1, z2)) → f(b(f(f(z1)), c(z0, z1, z2)))
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b(z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, b
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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