(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y, z) → g(<=(x, y), x, y, z)
g(true, x, y, z) → z
g(false, x, y, z) → f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
p(0) → 0
p(s(x)) → x
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, y, z) → g(<=(x, y), x, y, z)
g(true, x, y, z) → z
g(false, x, y, z) → f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(x0, x1, x2)
g(true, x0, x1, x2)
g(false, x0, x1, x2)
p(0)
p(s(x0))
(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(x, y, z) → G(<=(x, y), x, y, z)
G(false, x, y, z) → F(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
G(false, x, y, z) → F(p(x), y, z)
G(false, x, y, z) → P(x)
G(false, x, y, z) → F(p(y), z, x)
G(false, x, y, z) → P(y)
G(false, x, y, z) → F(p(z), x, y)
G(false, x, y, z) → P(z)
The TRS R consists of the following rules:
f(x, y, z) → g(<=(x, y), x, y, z)
g(true, x, y, z) → z
g(false, x, y, z) → f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
p(0) → 0
p(s(x)) → x
The set Q consists of the following terms:
f(x0, x1, x2)
g(true, x0, x1, x2)
g(false, x0, x1, x2)
p(0)
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes.
(6) TRUE