(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
f(h(x)) → h(g(x))
f(g(x)) → g(f(f(x)))
The TRS R 2 is
f'(s(x), y, y) → f'(y, x, s(x))
The signature Sigma is {
f'}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))
The set Q consists of the following terms:
f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)
(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(g(x)) → F(f(x))
F(g(x)) → F(x)
F'(s(x), y, y) → F'(y, x, s(x))
The TRS R consists of the following rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))
The set Q consists of the following terms:
f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(x)) → F(x)
F(g(x)) → F(f(x))
The TRS R consists of the following rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))
The set Q consists of the following terms:
f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)
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(g(x)) → F(x)
F(g(x)) → F(f(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
x1
g(
x1) =
g(
x1)
f(
x1) =
x1
h(
x1) =
h
f'(
x1,
x2,
x3) =
f'(
x3)
s(
x1) =
s
Recursive Path Order [RPO].
Precedence:
g1 > s
h > s
f'1 > s
The following usable rules [FROCOS05] were oriented:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))
The set Q consists of the following terms:
f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)
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