(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))
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(s(x), y, y) → F(y, x, s(x))
The TRS R consists of the following rules:
g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))
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(s(x), y, y) → F(y, x, s(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2,
x3) =
F(
x1,
x2)
s(
x1) =
s(
x1)
g(
x1,
x2) =
g(
x1,
x2)
f(
x1,
x2,
x3) =
f(
x1,
x3)
Recursive path order with status [RPO].
Quasi-Precedence:
g2 > [F2, s1, f2]
Status:
F2: multiset
s1: multiset
g2: multiset
f2: multiset
The following usable rules [FROCOS05] were oriented:
g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) TRUE