(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y, w, w, a) → g1(x, x, y, w)
f(x, y, w, a, a) → g1(y, x, x, w)
f(x, y, a, a, w) → g2(x, y, y, w)
f(x, y, a, w, w) → g2(y, y, x, w)
g1(x, x, y, a) → h(x, y)
g1(y, x, x, a) → h(x, y)
g2(x, y, y, a) → h(x, y)
g2(y, y, x, a) → h(x, y)
h(x, 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, w, w, a) → g1(x, x, y, w)
f(x, y, w, a, a) → g1(y, x, x, w)
f(x, y, a, a, w) → g2(x, y, y, w)
f(x, y, a, w, w) → g2(y, y, x, w)
g1(x, x, y, a) → h(x, y)
g1(y, x, x, a) → h(x, y)
g2(x, y, y, a) → h(x, y)
g2(y, y, x, a) → h(x, y)
h(x, x) → x
The set Q consists of the following terms:
f(x0, x1, x2, x2, a)
f(x0, x1, x2, a, a)
f(x0, x1, a, a, x2)
f(x0, x1, a, x2, x2)
g1(x0, x0, x1, a)
g1(x0, x1, x1, a)
g2(x0, x1, x1, a)
g2(x0, x0, x1, a)
h(x0, 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, w, w, a) → G1(x, x, y, w)
F(x, y, w, a, a) → G1(y, x, x, w)
F(x, y, a, a, w) → G2(x, y, y, w)
F(x, y, a, w, w) → G2(y, y, x, w)
G1(x, x, y, a) → H(x, y)
G1(y, x, x, a) → H(x, y)
G2(x, y, y, a) → H(x, y)
G2(y, y, x, a) → H(x, y)
The TRS R consists of the following rules:
f(x, y, w, w, a) → g1(x, x, y, w)
f(x, y, w, a, a) → g1(y, x, x, w)
f(x, y, a, a, w) → g2(x, y, y, w)
f(x, y, a, w, w) → g2(y, y, x, w)
g1(x, x, y, a) → h(x, y)
g1(y, x, x, a) → h(x, y)
g2(x, y, y, a) → h(x, y)
g2(y, y, x, a) → h(x, y)
h(x, x) → x
The set Q consists of the following terms:
f(x0, x1, x2, x2, a)
f(x0, x1, x2, a, a)
f(x0, x1, a, a, x2)
f(x0, x1, a, x2, x2)
g1(x0, x0, x1, a)
g1(x0, x1, x1, a)
g2(x0, x1, x1, a)
g2(x0, x0, x1, a)
h(x0, 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