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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0) → 1
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f(s(x)) → g(x, s(x))
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g(0, y) → y
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g(s(x), y) → g(x, +(y, s(x)))
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+(x, 0) → x
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+(x, s(y)) → s(+(x, y))
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g(s(x), y) → g(x, s(+(y, 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(0) → 1
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f(s(z0)) → g(z0, s(z0))
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g(0, z0) → z0
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g(s(z0), z1) → g(z0, +(z1, s(z0)))
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g(s(z0), z1) → g(z0, s(+(z1, z0)))
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+(z0, 0) → z0
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+(z0, s(z1)) → s(+(z0, z1))
Tuples:
F(s(z0)) → c1(G(z0, s(z0)))
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G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
S tuples:
F(s(z0)) → c1(G(z0, s(z0)))
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G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:
f, g, +
Defined Pair Symbols:
F, G, +'
Compound Symbols:
c1, c3, c4, c6
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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
F(s(z0)) → c1(G(z0, s(z0)))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → 1
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f(s(z0)) → g(z0, s(z0))
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g(0, z0) → z0
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g(s(z0), z1) → g(z0, +(z1, s(z0)))
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g(s(z0), z1) → g(z0, s(+(z1, z0)))
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+(z0, 0) → z0
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+(z0, s(z1)) → s(+(z0, z1))
Tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
S tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:
f, g, +
Defined Pair Symbols:
G, +'
Compound Symbols:
c3, c4, c6
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(5) 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(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
We considered the (Usable) Rules:
+(z0, 0) → z0
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+(z0, s(z1)) → s(+(z0, z1))
And the Tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(+(x1, x2)) = [5] + [3]x2
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POL(+'(x1, x2)) = [3]
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POL(0) = 0
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POL(G(x1, x2)) = [2]x1
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POL(c3(x1, x2)) = x1 + x2
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POL(c4(x1, x2)) = x1 + x2
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POL(c6(x1)) = x1
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POL(s(x1)) = [3] + x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → 1
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f(s(z0)) → g(z0, s(z0))
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g(0, z0) → z0
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g(s(z0), z1) → g(z0, +(z1, s(z0)))
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g(s(z0), z1) → g(z0, s(+(z1, z0)))
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+(z0, 0) → z0
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+(z0, s(z1)) → s(+(z0, z1))
Tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
S tuples:
+'(z0, s(z1)) → c6(+'(z0, z1))
K tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
Defined Rule Symbols:
f, g, +
Defined Pair Symbols:
G, +'
Compound Symbols:
c3, c4, c6
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(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
+'(z0, s(z1)) → c6(+'(z0, z1))
We considered the (Usable) Rules:
+(z0, 0) → z0
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+(z0, s(z1)) → s(+(z0, z1))
And the Tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(+(x1, x2)) = [2] + [2]x22
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POL(+'(x1, x2)) = x2
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POL(0) = [3]
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POL(G(x1, x2)) = x12
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POL(c3(x1, x2)) = x1 + x2
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POL(c4(x1, x2)) = x1 + x2
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POL(c6(x1)) = x1
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POL(s(x1)) = [2] + x1
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0) → 1
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f(s(z0)) → g(z0, s(z0))
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g(0, z0) → z0
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g(s(z0), z1) → g(z0, +(z1, s(z0)))
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g(s(z0), z1) → g(z0, s(+(z1, z0)))
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+(z0, 0) → z0
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+(z0, s(z1)) → s(+(z0, z1))
Tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
S tuples:none
K tuples:
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
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G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
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+'(z0, s(z1)) → c6(+'(z0, z1))
Defined Rule Symbols:
f, g, +
Defined Pair Symbols:
G, +'
Compound Symbols:
c3, c4, c6
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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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