(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
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)) → F(g(f(x), x))
F(f(x)) → G(f(x), x)
F(f(x)) → F(h(f(x), f(x)))
F(f(x)) → H(f(x), f(x))
H(x, x) → G(x, 0)
The TRS R consists of the following rules:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
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 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(x)) → F(h(f(x), f(x)))
F(f(x)) → F(g(f(x), x))
The TRS R consists of the following rules:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
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)) → F(g(f(x), x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
F(
x1)
f(
x1) =
f(
x1)
h(
x1,
x2) =
x1
g(
x1,
x2) =
x2
0 =
0
Recursive path order with status [RPO].
Quasi-Precedence:
[F1, f1] > 0
Status:
F1: multiset
f1: multiset
0: multiset
The following usable rules [FROCOS05] were oriented:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(x)) → F(h(f(x), f(x)))
The TRS R consists of the following rules:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(f(x)) → F(h(f(x), f(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
F(
x1)
f(
x1) =
f(
x1)
h(
x1,
x2) =
h
g(
x1,
x2) =
x2
0 =
0
Recursive path order with status [RPO].
Quasi-Precedence:
[F1, f1] > [h, 0]
Status:
F1: multiset
f1: multiset
h: multiset
0: multiset
The following usable rules [FROCOS05] were oriented:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE