(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(y, z), f(x, f(a, x))) → f(f(f(a, z), f(x, a)), f(a, y))
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:
F(f(y, z), f(x, f(a, x))) → F(f(f(a, z), f(x, a)), f(a, y))
F(f(y, z), f(x, f(a, x))) → F(f(a, z), f(x, a))
F(f(y, z), f(x, f(a, x))) → F(a, z)
F(f(y, z), f(x, f(a, x))) → F(x, a)
F(f(y, z), f(x, f(a, x))) → F(a, y)
The TRS R consists of the following rules:
f(f(y, z), f(x, f(a, x))) → f(f(f(a, z), f(x, a)), f(a, y))
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 4 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(y, z), f(x, f(a, x))) → F(f(f(a, z), f(x, a)), f(a, y))
The TRS R consists of the following rules:
f(f(y, z), f(x, f(a, x))) → f(f(f(a, z), f(x, a)), f(a, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(y, z), f(x, f(a, x))) → F(f(f(a, z), f(x, a)), f(a, y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
f(
y,
z),
f(
x,
f(
a,
x))) →
F(
f(
f(
a,
z),
f(
x,
a)),
f(
a,
y)) we obtained the following new rules [LPAR04]:
F(f(f(a, z1), f(z2, a)), f(a, f(a, a))) → F(f(f(a, f(z2, a)), f(a, a)), f(a, f(a, z1)))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, z1), f(z2, a)), f(a, f(a, a))) → F(f(f(a, f(z2, a)), f(a, a)), f(a, f(a, z1)))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
f(
f(
a,
z1),
f(
z2,
a)),
f(
a,
f(
a,
a))) →
F(
f(
f(
a,
f(
z2,
a)),
f(
a,
a)),
f(
a,
f(
a,
z1))) we obtained the following new rules [LPAR04]:
F(f(f(a, f(z1, a)), f(a, a)), f(a, f(a, a))) → F(f(f(a, f(a, a)), f(a, a)), f(a, f(a, f(z1, a))))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, f(z1, a)), f(a, a)), f(a, f(a, a))) → F(f(f(a, f(a, a)), f(a, a)), f(a, f(a, f(z1, a))))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(12) TRUE