(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
i(x, x) → i(a, b)
g(x, x) → g(a, b)
h(s(f(x))) → h(f(x))
f(s(x)) → s(s(f(h(s(x)))))
f(g(s(x), y)) → f(g(x, s(y)))
h(g(x, s(y))) → h(g(s(x), y))
h(i(x, y)) → i(i(c, h(h(y))), x)
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
S tuples:
I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
I, G, H, F
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
G(a, g(z0, g(b, g(a, g(z0, z1))))) → c3(G(a, g(a, g(a, g(z0, g(b, g(b, z1)))))), G(a, g(a, g(z0, g(b, g(b, z1))))), G(a, g(z0, g(b, g(b, z1)))), G(z0, g(b, g(b, z1))), G(b, g(b, z1)), G(b, z1))
H(g(z0, s(z1))) → c5(H(g(s(z0), z1)), G(s(z0), z1))
H(i(z0, z1)) → c6(I(i(c, h(h(z1))), z0), I(c, h(h(z1))), H(h(z1)), H(z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
S tuples:
I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
I, G, F, H
Compound Symbols:
c1, c2, c7, c8, c4
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 2 of 5 dangling nodes:
I(z0, z0) → c1(I(a, b))
G(z0, z0) → c2(G(a, b))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
H(s(f(z0))) → c4(H(f(z0)), F(z0))
S tuples:
H(s(f(z0))) → c4(H(f(z0)), F(z0))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))), G(z0, s(z1)))
K tuples:none
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
F, H
Compound Symbols:
c7, c8, c4
(7) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
K tuples:none
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
F, H
Compound Symbols:
c7, c8, c4
(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.
H(s(f(z0))) → c4(F(z0))
We considered the (Usable) Rules:
g(z0, z0) → g(a, b)
h(s(f(z0))) → h(f(z0))
And the Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = [1] + x1
POL(H(x1)) = x1
POL(a) = 0
POL(b) = [1]
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(f(x1)) = x1
POL(g(x1, x2)) = 0
POL(h(x1)) = 0
POL(s(x1)) = [2] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
K tuples:
H(s(f(z0))) → c4(F(z0))
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
F, H
Compound Symbols:
c7, c8, c4
(11) 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.
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
We considered the (Usable) Rules:
g(z0, z0) → g(a, b)
h(s(f(z0))) → h(f(z0))
And the Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = [2]x1
POL(H(x1)) = x1
POL(a) = 0
POL(b) = 0
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(f(x1)) = [5]x1
POL(g(x1, x2)) = [4]x1 + [2]x2
POL(h(x1)) = 0
POL(s(x1)) = [1] + x1
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
K tuples:
H(s(f(z0))) → c4(F(z0))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
F, H
Compound Symbols:
c7, c8, c4
(13) 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.
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
We considered the (Usable) Rules:
g(z0, z0) → g(a, b)
h(s(f(z0))) → h(f(z0))
And the Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1)) = [3] + [2]x1
POL(H(x1)) = [1] + x1
POL(a) = 0
POL(b) = [3]
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(f(x1)) = [3] + [2]x1
POL(g(x1, x2)) = 0
POL(h(x1)) = 0
POL(s(x1)) = [4] + x1
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
i(z0, z0) → i(a, b)
g(z0, z0) → g(a, b)
g(a, g(z0, g(b, g(a, g(z0, z1))))) → g(a, g(a, g(a, g(z0, g(b, g(b, z1))))))
h(s(f(z0))) → h(f(z0))
h(g(z0, s(z1))) → h(g(s(z0), z1))
h(i(z0, z1)) → i(i(c, h(h(z1))), z0)
f(s(z0)) → s(s(f(h(s(z0)))))
f(g(s(z0), z1)) → f(g(z0, s(z1)))
Tuples:
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
H(s(f(z0))) → c4(F(z0))
S tuples:none
K tuples:
H(s(f(z0))) → c4(F(z0))
F(g(s(z0), z1)) → c8(F(g(z0, s(z1))))
F(s(z0)) → c7(F(h(s(z0))), H(s(z0)))
Defined Rule Symbols:
i, g, h, f
Defined Pair Symbols:
F, H
Compound Symbols:
c7, c8, c4
(15) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(16) BOUNDS(O(1), O(1))