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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
b(x, y) → c(a(c(y), a(0, x)))
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a(y, x) → y
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a(y, c(b(a(0, x), 0))) → b(a(c(b(0, y)), x), 0)
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:
b(z0, z1) → c(a(c(z1), a(0, z0)))
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a(z0, z1) → z0
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a(z0, c(b(a(0, z1), 0))) → b(a(c(b(0, z0)), z1), 0)
Tuples:
B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
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A(z0, c(b(a(0, z1), 0))) → c3(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0))
S tuples:
B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
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A(z0, c(b(a(0, z1), 0))) → c3(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0))
K tuples:none
Defined Rule Symbols:
b, a
Defined Pair Symbols:
B, A
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:
A(z0, c(b(a(0, z1), 0))) → c3(B(a(c(b(0, z0)), z1), 0), A(c(b(0, z0)), z1), B(0, z0))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(z0, z1) → c(a(c(z1), a(0, z0)))
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a(z0, z1) → z0
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a(z0, c(b(a(0, z1), 0))) → b(a(c(b(0, z0)), z1), 0)
Tuples:
B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
S tuples:
B(z0, z1) → c1(A(c(z1), a(0, z0)), A(0, z0))
K tuples:none
Defined Rule Symbols:
b, a
Defined Pair Symbols:
B
Compound Symbols:
c1
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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(z0, z1) → c(a(c(z1), a(0, z0)))
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a(z0, z1) → z0
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a(z0, c(b(a(0, z1), 0))) → b(a(c(b(0, z0)), z1), 0)
Tuples:
B(z0, z1) → c1
S tuples:
B(z0, z1) → c1
K tuples:none
Defined Rule Symbols:
b, a
Defined Pair Symbols:
B
Compound Symbols:
c1
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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
B(z0, z1) → c1
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(z0, z1) → c(a(c(z1), a(0, z0)))
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a(z0, z1) → z0
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a(z0, c(b(a(0, z1), 0))) → b(a(c(b(0, z0)), z1), 0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
b, a
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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