(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
The set Q consists of the following terms:
f(x0, 0, 0)
f(0, x0, 0)
f(0, 0, x0)
f(s(0), x0, x1)
f(s(x0), s(x1), 0)
f(s(x0), 0, s(x1))
f(0, s(0), s(0))
f(s(x0), s(x1), s(x2))
f(0, s(s(x0)), s(0))
f(0, s(0), s(s(x0)))
f(0, s(s(x0)), s(s(x1)))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(0), y, z) → F(0, s(y), s(z))
F(s(x), s(y), 0) → F(x, y, s(0))
F(s(x), 0, s(z)) → F(x, s(0), z)
F(s(x), s(y), s(z)) → F(x, y, f(s(x), s(y), z))
F(s(x), s(y), s(z)) → F(s(x), s(y), z)
F(0, s(s(y)), s(0)) → F(0, y, s(0))
F(0, s(0), s(s(z))) → F(0, s(0), z)
F(0, s(s(y)), s(s(z))) → F(0, y, f(0, s(s(y)), s(z)))
F(0, s(s(y)), s(s(z))) → F(0, s(s(y)), s(z))
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
The set Q consists of the following terms:
f(x0, 0, 0)
f(0, x0, 0)
f(0, 0, x0)
f(s(0), x0, x1)
f(s(x0), s(x1), 0)
f(s(x0), 0, s(x1))
f(0, s(0), s(0))
f(s(x0), s(x1), s(x2))
f(0, s(s(x0)), s(0))
f(0, s(0), s(s(x0)))
f(0, s(s(x0)), s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 1 less node.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(0, s(0), s(s(z))) → F(0, s(0), z)
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
The set Q consists of the following terms:
f(x0, 0, 0)
f(0, x0, 0)
f(0, 0, x0)
f(s(0), x0, x1)
f(s(x0), s(x1), 0)
f(s(x0), 0, s(x1))
f(0, s(0), s(0))
f(s(x0), s(x1), s(x2))
f(0, s(s(x0)), s(0))
f(0, s(0), s(s(x0)))
f(0, s(s(x0)), s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(0, s(s(y)), s(0)) → F(0, y, s(0))
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
The set Q consists of the following terms:
f(x0, 0, 0)
f(0, x0, 0)
f(0, 0, x0)
f(s(0), x0, x1)
f(s(x0), s(x1), 0)
f(s(x0), 0, s(x1))
f(0, s(0), s(0))
f(s(x0), s(x1), s(x2))
f(0, s(s(x0)), s(0))
f(0, s(0), s(s(x0)))
f(0, s(s(x0)), s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(0, s(s(y)), s(s(z))) → F(0, s(s(y)), s(z))
F(0, s(s(y)), s(s(z))) → F(0, y, f(0, s(s(y)), s(z)))
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
The set Q consists of the following terms:
f(x0, 0, 0)
f(0, x0, 0)
f(0, 0, x0)
f(s(0), x0, x1)
f(s(x0), s(x1), 0)
f(s(x0), 0, s(x1))
f(0, s(0), s(0))
f(s(x0), s(x1), s(x2))
f(0, s(s(x0)), s(0))
f(0, s(0), s(s(x0)))
f(0, s(s(x0)), s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), 0, s(z)) → F(x, s(0), z)
F(s(x), s(y), 0) → F(x, y, s(0))
F(s(x), s(y), s(z)) → F(x, y, f(s(x), s(y), z))
F(s(x), s(y), s(z)) → F(s(x), s(y), z)
The TRS R consists of the following rules:
f(x, 0, 0) → s(x)
f(0, y, 0) → s(y)
f(0, 0, z) → s(z)
f(s(0), y, z) → f(0, s(y), s(z))
f(s(x), s(y), 0) → f(x, y, s(0))
f(s(x), 0, s(z)) → f(x, s(0), z)
f(0, s(0), s(0)) → s(s(0))
f(s(x), s(y), s(z)) → f(x, y, f(s(x), s(y), z))
f(0, s(s(y)), s(0)) → f(0, y, s(0))
f(0, s(0), s(s(z))) → f(0, s(0), z)
f(0, s(s(y)), s(s(z))) → f(0, y, f(0, s(s(y)), s(z)))
The set Q consists of the following terms:
f(x0, 0, 0)
f(0, x0, 0)
f(0, 0, x0)
f(s(0), x0, x1)
f(s(x0), s(x1), 0)
f(s(x0), 0, s(x1))
f(0, s(0), s(0))
f(s(x0), s(x1), s(x2))
f(0, s(s(x0)), s(0))
f(0, s(0), s(s(x0)))
f(0, s(s(x0)), s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.