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