(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w))))))
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w)))
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
t(t(x)) → t(c(t(x), x))
t(x) → x
t(x) → c(0, c(0, c(0, c(0, c(0, x)))))
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:
H(c(x, y), c(s(z), z), t(w)) → H(z, c(y, x), t(t(c(x, c(y, t(w))))))
H(c(x, y), c(s(z), z), t(w)) → T(t(c(x, c(y, t(w)))))
H(c(x, y), c(s(z), z), t(w)) → T(c(x, c(y, t(w))))
H(x, c(y, z), t(w)) → H(c(s(y), x), z, t(c(t(w), w)))
H(x, c(y, z), t(w)) → T(c(t(w), w))
H(c(s(x), c(s(0), y)), z, t(x)) → H(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
H(c(s(x), c(s(0), y)), z, t(x)) → T(t(c(x, s(x))))
H(c(s(x), c(s(0), y)), z, t(x)) → T(c(x, s(x)))
T(t(x)) → T(c(t(x), x))
The TRS R consists of the following rules:
h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w))))))
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w)))
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
t(t(x)) → t(c(t(x), x))
t(x) → x
t(x) → c(0, c(0, c(0, c(0, c(0, x)))))
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 1 SCC with 6 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(x, c(y, z), t(w)) → H(c(s(y), x), z, t(c(t(w), w)))
H(c(x, y), c(s(z), z), t(w)) → H(z, c(y, x), t(t(c(x, c(y, t(w))))))
H(c(s(x), c(s(0), y)), z, t(x)) → H(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
The TRS R consists of the following rules:
h(c(x, y), c(s(z), z), t(w)) → h(z, c(y, x), t(t(c(x, c(y, t(w))))))
h(x, c(y, z), t(w)) → h(c(s(y), x), z, t(c(t(w), w)))
h(c(s(x), c(s(0), y)), z, t(x)) → h(y, c(s(0), c(x, z)), t(t(c(x, s(x)))))
t(t(x)) → t(c(t(x), x))
t(x) → x
t(x) → c(0, c(0, c(0, c(0, c(0, x)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.