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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(x)) → f(c(f(x)))
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f(f(x)) → f(d(f(x)))
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g(c(x)) → x
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g(d(x)) → x
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g(c(h(0))) → g(d(1))
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g(c(1)) → g(d(h(0)))
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g(h(x)) → g(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(z0)) → f(c(f(z0)))
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f(f(z0)) → f(d(f(z0)))
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g(c(z0)) → z0
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g(d(z0)) → z0
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g(c(h(0))) → g(d(1))
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g(c(1)) → g(d(h(0)))
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g(h(z0)) → g(z0)
Tuples:
F(f(z0)) → c1(F(c(f(z0))), F(z0))
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F(f(z0)) → c2(F(d(f(z0))), F(z0))
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G(c(h(0))) → c5(G(d(1)))
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G(c(1)) → c6(G(d(h(0))))
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G(h(z0)) → c7(G(z0))
S tuples:
F(f(z0)) → c1(F(c(f(z0))), F(z0))
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F(f(z0)) → c2(F(d(f(z0))), F(z0))
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G(c(h(0))) → c5(G(d(1)))
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G(c(1)) → c6(G(d(h(0))))
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G(h(z0)) → c7(G(z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c1, c2, c5, c6, c7
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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(z0)) → c1(F(c(f(z0))), F(z0))
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F(f(z0)) → c2(F(d(f(z0))), F(z0))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(c(f(z0)))
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f(f(z0)) → f(d(f(z0)))
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g(c(z0)) → z0
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g(d(z0)) → z0
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g(c(h(0))) → g(d(1))
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g(c(1)) → g(d(h(0)))
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g(h(z0)) → g(z0)
Tuples:
G(c(h(0))) → c5(G(d(1)))
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G(c(1)) → c6(G(d(h(0))))
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G(h(z0)) → c7(G(z0))
S tuples:
G(c(h(0))) → c5(G(d(1)))
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G(c(1)) → c6(G(d(h(0))))
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G(h(z0)) → c7(G(z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
G
Compound Symbols:
c5, c6, c7
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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:
f(f(z0)) → f(c(f(z0)))
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f(f(z0)) → f(d(f(z0)))
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g(c(z0)) → z0
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g(d(z0)) → z0
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g(c(h(0))) → g(d(1))
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g(c(1)) → g(d(h(0)))
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g(h(z0)) → g(z0)
Tuples:
G(h(z0)) → c7(G(z0))
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G(c(h(0))) → c5
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G(c(1)) → c6
S tuples:
G(h(z0)) → c7(G(z0))
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G(c(h(0))) → c5
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G(c(1)) → c6
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
G
Compound Symbols:
c7, c5, c6
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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
G(c(h(0))) → c5
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G(c(1)) → c6
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(c(f(z0)))
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f(f(z0)) → f(d(f(z0)))
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g(c(z0)) → z0
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g(d(z0)) → z0
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g(c(h(0))) → g(d(1))
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g(c(1)) → g(d(h(0)))
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g(h(z0)) → g(z0)
Tuples:
G(h(z0)) → c7(G(z0))
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G(c(h(0))) → c5
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G(c(1)) → c6
S tuples:
G(h(z0)) → c7(G(z0))
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G(c(h(0))) → c5
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G(c(1)) → c6
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
G
Compound Symbols:
c7, c5, c6
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(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
G(h(z0)) → c7(G(z0))
We considered the (Usable) Rules:none
And the Tuples:
G(h(z0)) → c7(G(z0))
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G(c(h(0))) → c5
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G(c(1)) → c6
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(0) = [1]
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POL(1) = [1]
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POL(G(x1)) = [2]x1
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POL(c(x1)) = 0
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POL(c5) = 0
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POL(c6) = 0
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POL(c7(x1)) = x1
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POL(h(x1)) = [1] + x1
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(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0)) → f(c(f(z0)))
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f(f(z0)) → f(d(f(z0)))
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g(c(z0)) → z0
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g(d(z0)) → z0
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g(c(h(0))) → g(d(1))
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g(c(1)) → g(d(h(0)))
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g(h(z0)) → g(z0)
Tuples:
G(h(z0)) → c7(G(z0))
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G(c(h(0))) → c5
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G(c(1)) → c6
S tuples:
G(c(h(0))) → c5
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G(c(1)) → c6
K tuples:
G(h(z0)) → c7(G(z0))
Defined Rule Symbols:
f, g
Defined Pair Symbols:
G
Compound Symbols:
c7, c5, c6
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(11) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
G(c(h(0))) → c5
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G(c(1)) → c6
Now S is empty
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(12) BOUNDS(O(1), O(1))
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