(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → 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(x, y, z), u, f(x, y, v)) → F(x, y, f(z, u, v))
F(f(x, y, z), u, f(x, y, v)) → F(z, u, v)
The TRS R consists of the following rules:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(f(x, y, z), u, f(x, y, v)) → F(x, y, f(z, u, v))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x0,
x1,
x2,
x3) =
F(
x1,
x2)
Tags:
F has argument tags [1,1,2,2] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(F(x1, x2, x3)) = x2 + x3
POL(f(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(g(x1)) = 0
The following usable rules [FROCOS05] were oriented:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(x, y, z), u, f(x, y, v)) → F(z, u, v)
The TRS R consists of the following rules:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(f(x, y, z), u, f(x, y, v)) → F(z, u, v)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x0,
x1,
x2,
x3) =
F(
x1)
Tags:
F has argument tags [0,0,0,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(F(x1, x2, x3)) = x2
POL(f(x1, x2, x3)) = 1 + x3
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE