(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(x, x, x) → g(c, d, e)
g(x, y, x) → g(c, d, e)
s(f(x, y)) → f(y, f(s(s(x)), a))
h(h(x, a), y) → h(h(a, y), h(a, x))
f(x, f(y, f(x, y))) → f(a, f(x, f(y, b)))
f(h(a, y), g(x, b, a)) → h(f(x, s(y)), s(b))
h(f(x, s(y)), b) → f(a, g(y, a, f(s(x), a)))
f(x, g(x, a, f(s(x), y))) → f(h(x, b), g(a, b, y))
s(y) → b
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(x, x, x) → G(c, d, e)
G(x, y, x) → G(c, d, e)
S(f(x, y)) → F(y, f(s(s(x)), a))
S(f(x, y)) → F(s(s(x)), a)
S(f(x, y)) → S(s(x))
S(f(x, y)) → S(x)
H(h(x, a), y) → H(h(a, y), h(a, x))
H(h(x, a), y) → H(a, y)
H(h(x, a), y) → H(a, x)
F(x, f(y, f(x, y))) → F(a, f(x, f(y, b)))
F(x, f(y, f(x, y))) → F(x, f(y, b))
F(x, f(y, f(x, y))) → F(y, b)
F(h(a, y), g(x, b, a)) → H(f(x, s(y)), s(b))
F(h(a, y), g(x, b, a)) → F(x, s(y))
F(h(a, y), g(x, b, a)) → S(y)
F(h(a, y), g(x, b, a)) → S(b)
H(f(x, s(y)), b) → F(a, g(y, a, f(s(x), a)))
H(f(x, s(y)), b) → G(y, a, f(s(x), a))
H(f(x, s(y)), b) → F(s(x), a)
H(f(x, s(y)), b) → S(x)
F(x, g(x, a, f(s(x), y))) → F(h(x, b), g(a, b, y))
F(x, g(x, a, f(s(x), y))) → H(x, b)
F(x, g(x, a, f(s(x), y))) → G(a, b, y)
The TRS R consists of the following rules:
g(x, x, x) → g(c, d, e)
g(x, y, x) → g(c, d, e)
s(f(x, y)) → f(y, f(s(s(x)), a))
h(h(x, a), y) → h(h(a, y), h(a, x))
f(x, f(y, f(x, y))) → f(a, f(x, f(y, b)))
f(h(a, y), g(x, b, a)) → h(f(x, s(y)), s(b))
h(f(x, s(y)), b) → f(a, g(y, a, f(s(x), a)))
f(x, g(x, a, f(s(x), y))) → f(h(x, b), g(a, b, y))
s(y) → b
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 15 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, f(x, y))) → F(a, f(x, f(y, b)))
The TRS R consists of the following rules:
g(x, x, x) → g(c, d, e)
g(x, y, x) → g(c, d, e)
s(f(x, y)) → f(y, f(s(s(x)), a))
h(h(x, a), y) → h(h(a, y), h(a, x))
f(x, f(y, f(x, y))) → f(a, f(x, f(y, b)))
f(h(a, y), g(x, b, a)) → h(f(x, s(y)), s(b))
h(f(x, s(y)), b) → f(a, g(y, a, f(s(x), a)))
f(x, g(x, a, f(s(x), y))) → f(h(x, b), g(a, b, y))
s(y) → b
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S(f(x, y)) → S(x)
S(f(x, y)) → S(s(x))
The TRS R consists of the following rules:
g(x, x, x) → g(c, d, e)
g(x, y, x) → g(c, d, e)
s(f(x, y)) → f(y, f(s(s(x)), a))
h(h(x, a), y) → h(h(a, y), h(a, x))
f(x, f(y, f(x, y))) → f(a, f(x, f(y, b)))
f(h(a, y), g(x, b, a)) → h(f(x, s(y)), s(b))
h(f(x, s(y)), b) → f(a, g(y, a, f(s(x), a)))
f(x, g(x, a, f(s(x), y))) → f(h(x, b), g(a, b, y))
s(y) → b
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(h(x, a), y) → H(h(a, y), h(a, x))
The TRS R consists of the following rules:
g(x, x, x) → g(c, d, e)
g(x, y, x) → g(c, d, e)
s(f(x, y)) → f(y, f(s(s(x)), a))
h(h(x, a), y) → h(h(a, y), h(a, x))
f(x, f(y, f(x, y))) → f(a, f(x, f(y, b)))
f(h(a, y), g(x, b, a)) → h(f(x, s(y)), s(b))
h(f(x, s(y)), b) → f(a, g(y, a, f(s(x), a)))
f(x, g(x, a, f(s(x), y))) → f(h(x, b), g(a, b, y))
s(y) → b
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(f(x, s(y)), b) → F(a, g(y, a, f(s(x), a)))
F(x, g(x, a, f(s(x), y))) → F(h(x, b), g(a, b, y))
F(h(a, y), g(x, b, a)) → H(f(x, s(y)), s(b))
F(x, g(x, a, f(s(x), y))) → H(x, b)
The TRS R consists of the following rules:
g(x, x, x) → g(c, d, e)
g(x, y, x) → g(c, d, e)
s(f(x, y)) → f(y, f(s(s(x)), a))
h(h(x, a), y) → h(h(a, y), h(a, x))
f(x, f(y, f(x, y))) → f(a, f(x, f(y, b)))
f(h(a, y), g(x, b, a)) → h(f(x, s(y)), s(b))
h(f(x, s(y)), b) → f(a, g(y, a, f(s(x), a)))
f(x, g(x, a, f(s(x), y))) → f(h(x, b), g(a, b, y))
s(y) → b
Q is empty.
We have to consider all minimal (P,Q,R)-chains.