(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
h(x, x) → h(a, b)
g(g(x, a), y) → g(g(a, y), g(a, x))
f(g(x, y)) → g(g(f(f(y)), h(a, a)), x)
h(h(f(f(x)), y), h(z, v)) → h(h(f(z), f(f(f(y)))), h(v, x))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(z0, z0) → c(H(a, b))
H(h(f(f(z0)), z1), h(z2, z3)) → c1(H(h(f(z2), f(f(f(z1)))), h(z3, z0)), H(f(z2), f(f(f(z1)))), F(z2), F(f(f(z1))), F(f(z1)), F(z1), H(z3, z0))
G(g(z0, a), z1) → c2(G(g(a, z1), g(a, z0)), G(a, z1), G(a, z0))
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
S tuples:
H(z0, z0) → c(H(a, b))
H(h(f(f(z0)), z1), h(z2, z3)) → c1(H(h(f(z2), f(f(f(z1)))), h(z3, z0)), H(f(z2), f(f(f(z1)))), F(z2), F(f(f(z1))), F(f(z1)), F(z1), H(z3, z0))
G(g(z0, a), z1) → c2(G(g(a, z1), g(a, z0)), G(a, z1), G(a, z0))
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
H(z0, z0) → c(H(a, b))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(z0)), z1), h(z2, z3)) → c1(H(h(f(z2), f(f(f(z1)))), h(z3, z0)), H(f(z2), f(f(f(z1)))), F(z2), F(f(f(z1))), F(f(z1)), F(z1), H(z3, z0))
G(g(z0, a), z1) → c2(G(g(a, z1), g(a, z0)), G(a, z1), G(a, z0))
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
S tuples:
H(h(f(f(z0)), z1), h(z2, z3)) → c1(H(h(f(z2), f(f(f(z1)))), h(z3, z0)), H(f(z2), f(f(f(z1)))), F(z2), F(f(f(z1))), F(f(z1)), F(z1), H(z3, z0))
G(g(z0, a), z1) → c2(G(g(a, z1), g(a, z0)), G(a, z1), G(a, z0))
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c2, c3
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
H(
h(
f(
f(
z0)),
z1),
h(
z2,
z3)) →
c1(
H(
h(
f(
z2),
f(
f(
f(
z1)))),
h(
z3,
z0)),
H(
f(
z2),
f(
f(
f(
z1)))),
F(
z2),
F(
f(
f(
z1))),
F(
f(
z1)),
F(
z1),
H(
z3,
z0)) by
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
G(g(z0, a), z1) → c2(G(g(a, z1), g(a, z0)), G(a, z1), G(a, z0))
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
S tuples:
G(g(z0, a), z1) → c2(G(g(a, z1), g(a, z0)), G(a, z1), G(a, z0))
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
G, F, H
Compound Symbols:
c2, c3, c1, c1
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
G(
g(
z0,
a),
z1) →
c2(
G(
g(
a,
z1),
g(
a,
z0)),
G(
a,
z1),
G(
a,
z0)) by
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
S tuples:
F(g(z0, z1)) → c3(G(g(f(f(z1)), h(a, a)), z0), G(f(f(z1)), h(a, a)), F(f(z1)), F(z1), H(a, a))
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
F, H, G
Compound Symbols:
c3, c1, c1, c2
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
g(
z0,
z1)) →
c3(
G(
g(
f(
f(
z1)),
h(
a,
a)),
z0),
G(
f(
f(
z1)),
h(
a,
a)),
F(
f(
z1)),
F(
z1),
H(
a,
a)) by
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3
S tuples:
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c1, c2, c3, c3
(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(g(x0, x1)) → c3
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
S tuples:
H(h(f(f(x0)), g(z0, z1)), h(x2, x3)) → c1(H(h(f(x2), f(f(f(g(z0, z1))))), h(x3, x0)), H(f(x2), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x2), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x3, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c1, c2, c3
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
H(
h(
f(
f(
x0)),
g(
z0,
z1)),
h(
x2,
x3)) →
c1(
H(
h(
f(
x2),
f(
f(
f(
g(
z0,
z1))))),
h(
x3,
x0)),
H(
f(
x2),
f(
f(
g(
g(
f(
f(
z1)),
h(
a,
a)),
z0)))),
F(
x2),
F(
f(
f(
g(
z0,
z1)))),
F(
f(
g(
z0,
z1))),
F(
g(
z0,
z1)),
H(
x3,
x0)) by
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
S tuples:
H(h(f(f(x0)), x1), h(g(z0, z1), x3)) → c1(H(h(f(g(z0, z1)), f(f(f(x1)))), h(x3, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x3, x0))
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c1, c2, c3, c1
(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
H(
h(
f(
f(
x0)),
x1),
h(
g(
z0,
z1),
x3)) →
c1(
H(
h(
f(
g(
z0,
z1)),
f(
f(
f(
x1)))),
h(
x3,
x0)),
H(
g(
g(
f(
f(
z1)),
h(
a,
a)),
z0),
f(
f(
f(
x1)))),
F(
g(
z0,
z1)),
F(
f(
f(
x1))),
F(
f(
x1)),
F(
x1),
H(
x3,
x0)) by
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
S tuples:
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, x1)) → c3(G(g(f(f(x1)), h(a, b)), x0), G(f(f(x1)), h(a, a)), F(f(x1)), F(x1), H(a, a))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c2, c3, c1, c1
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
g(
x0,
x1)) →
c3(
G(
g(
f(
f(
x1)),
h(
a,
b)),
x0),
G(
f(
f(
x1)),
h(
a,
a)),
F(
f(
x1)),
F(
x1),
H(
a,
a)) by
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, b)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3(G(f(f(x1)), h(a, a)))
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, b)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3(G(f(f(x1)), h(a, a)))
S tuples:
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, b)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3(G(f(f(x1)), h(a, a)))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c2, c3, c1, c1, c3
(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
g(
x0,
g(
z0,
z1))) →
c3(
G(
g(
f(
g(
g(
f(
f(
z1)),
h(
a,
a)),
z0)),
h(
a,
a)),
x0),
G(
f(
f(
g(
z0,
z1))),
h(
a,
a)),
F(
f(
g(
z0,
z1))),
F(
g(
z0,
z1)),
H(
a,
a)) by
F(g(x0, g(x1, x2))) → c3(G(g(f(g(g(f(f(x2)), h(a, a)), x1)), h(a, b)), x0), G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)), H(a, a))
F(g(x0, g(z1, x2))) → c3(G(g(g(g(f(f(z1)), h(a, a)), g(f(f(x2)), h(a, a))), h(a, a)), x0), G(f(f(g(z1, x2))), h(a, a)), F(f(g(z1, x2))), F(g(z1, x2)), H(a, a))
F(g(x0, g(x1, x2))) → c3(G(g(f(g(g(f(f(x2)), h(a, b)), x1)), h(a, a)), x0), G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)), H(a, a))
F(g(x0, g(x1, g(z0, z1)))) → c3(G(g(f(g(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x1)), h(a, a)), x0), G(f(f(g(x1, g(z0, z1)))), h(a, a)), F(f(g(x1, g(z0, z1)))), F(g(x1, g(z0, z1))), H(a, a))
F(g(x0, g(x1, x2))) → c3(G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)))
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, z0) → h(a, b)
h(h(f(f(z0)), z1), h(z2, z3)) → h(h(f(z2), f(f(f(z1)))), h(z3, z0))
g(g(z0, a), z1) → g(g(a, z1), g(a, z0))
f(g(z0, z1)) → g(g(f(f(z1)), h(a, a)), z0)
Tuples:
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, b)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3(G(f(f(x1)), h(a, a)))
F(g(x0, g(z1, x2))) → c3(G(g(g(g(f(f(z1)), h(a, a)), g(f(f(x2)), h(a, a))), h(a, a)), x0), G(f(f(g(z1, x2))), h(a, a)), F(f(g(z1, x2))), F(g(z1, x2)), H(a, a))
F(g(x0, g(x1, x2))) → c3(G(g(f(g(g(f(f(x2)), h(a, b)), x1)), h(a, a)), x0), G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)), H(a, a))
F(g(x0, g(x1, g(z0, z1)))) → c3(G(g(f(g(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x1)), h(a, a)), x0), G(f(f(g(x1, g(z0, z1)))), h(a, a)), F(f(g(x1, g(z0, z1)))), F(g(x1, g(z0, z1))), H(a, a))
F(g(x0, g(x1, x2))) → c3(G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)))
S tuples:
H(h(f(f(x0)), x1), h(x2, x3)) → c1(H(h(f(x2), f(f(f(x1)))), h(x3, x0)), H(x3, x0))
G(g(x0, a), x1) → c2(G(g(a, x1), g(a, x0)))
H(h(f(f(z0)), g(x1, x2)), h(x3, z0)) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(a, b)), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(z0, z0))
H(h(f(f(h(z2, z3))), g(x1, x2)), h(x3, h(f(f(z0)), z1))) → c1(H(h(f(x3), f(f(f(g(x1, x2))))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(x3), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(x3, x4)) → c1(H(h(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(f(x3), f(f(g(g(f(f(z1)), h(a, a)), z0)))), F(x3), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(g(x1, x2))))), h(x4, x0)), H(f(g(z0, z1)), f(f(g(g(f(f(x2)), h(a, a)), x1)))), F(g(z0, z1)), F(f(f(g(x1, x2)))), F(f(g(x1, x2))), F(g(x1, x2)), H(x4, x0))
H(h(f(f(x0)), g(x1, x2)), h(x3, x4)) → c1(H(f(x3), f(f(g(g(f(f(x2)), h(a, a)), x1)))))
H(h(f(f(z0)), x1), h(g(x2, x3), z0)) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(a, b)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(z0, z0))
H(h(f(f(h(z2, z3))), x1), h(g(x2, x3), h(f(f(z0)), z1))) → c1(H(h(f(g(x2, x3)), f(f(f(x1)))), h(h(f(z2), f(f(f(z1)))), h(z3, z0))), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))), F(g(x2, x3)), F(f(f(x1))), F(f(x1)), F(x1), H(h(f(f(z0)), z1), h(z2, z3)))
H(h(f(f(x0)), g(z0, z1)), h(g(x2, x3), x4)) → c1(H(h(f(g(x2, x3)), f(f(g(g(f(f(z1)), h(a, a)), z0)))), h(x4, x0)), H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(g(z0, z1))))), F(g(x2, x3)), F(f(f(g(z0, z1)))), F(f(g(z0, z1))), F(g(z0, z1)), H(x4, x0))
H(h(f(f(x0)), x1), h(g(z0, z1), x4)) → c1(H(h(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), h(x4, x0)), H(g(g(f(f(z1)), h(a, a)), z0), f(f(f(x1)))), F(g(z0, z1)), F(f(f(x1))), F(f(x1)), F(x1), H(x4, x0))
H(h(f(f(x0)), x1), h(g(x2, x3), x4)) → c1(H(g(g(f(f(x3)), h(a, a)), x2), f(f(f(x1)))))
F(g(x0, g(z0, z1))) → c3(G(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, b)), x0), G(f(f(g(z0, z1))), h(a, a)), F(f(g(z0, z1))), F(g(z0, z1)), H(a, a))
F(g(x0, x1)) → c3(G(f(f(x1)), h(a, a)))
F(g(x0, g(z1, x2))) → c3(G(g(g(g(f(f(z1)), h(a, a)), g(f(f(x2)), h(a, a))), h(a, a)), x0), G(f(f(g(z1, x2))), h(a, a)), F(f(g(z1, x2))), F(g(z1, x2)), H(a, a))
F(g(x0, g(x1, x2))) → c3(G(g(f(g(g(f(f(x2)), h(a, b)), x1)), h(a, a)), x0), G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)), H(a, a))
F(g(x0, g(x1, g(z0, z1)))) → c3(G(g(f(g(g(f(g(g(f(f(z1)), h(a, a)), z0)), h(a, a)), x1)), h(a, a)), x0), G(f(f(g(x1, g(z0, z1)))), h(a, a)), F(f(g(x1, g(z0, z1)))), F(g(x1, g(z0, z1))), H(a, a))
F(g(x0, g(x1, x2))) → c3(G(f(f(g(x1, x2))), h(a, a)), F(f(g(x1, x2))), F(g(x1, x2)))
K tuples:none
Defined Rule Symbols:
h, g, f
Defined Pair Symbols:
H, G, F
Compound Symbols:
c1, c2, c1, c1, c3, c3, c3
(21) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match(-raise)-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.
The compatible tree automaton used to show the Match(-raise)-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3]
transitions:
h0(0, 0) → 1
g0(0, 0) → 2
f0(0) → 3
a1() → 4
b1() → 5
h1(4, 5) → 1
a1() → 0
b1() → 0
a1() → 6
b1() → 7
h0(6, 0) → 1
h0(0, 6) → 1
h0(7, 0) → 1
h0(0, 7) → 1
h0(6, 6) → 1
h0(6, 7) → 1
h0(7, 6) → 1
h0(7, 7) → 1
g0(6, 0) → 2
g0(0, 6) → 2
g0(7, 0) → 2
g0(0, 7) → 2
g0(6, 6) → 2
g0(6, 7) → 2
g0(7, 6) → 2
g0(7, 7) → 2
f0(6) → 3
f0(7) → 3
h1(6, 5) → 1
h1(4, 7) → 1
h1(6, 7) → 1
(22) BOUNDS(O(1), O(n^1))