(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
The TRS R 2 is
g(0, x) → g(f(x, x), x)
The signature Sigma is {
g}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
(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(s(x), s(y)) → F(x, y)
G(0, x) → G(f(x, x), x)
G(0, x) → F(x, x)
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
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 2 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), s(y)) → F(x, y)
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
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(s(x), s(y)) → F(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x2)
s(
x1) =
s(
x1)
f(
x1,
x2) =
f(
x1)
0 =
0
g(
x1,
x2) =
x2
Recursive Path Order [RPO].
Precedence:
[F1, s1, f1] > 0
The following usable rules [FROCOS05] were oriented:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x, 0) → s(0)
f(s(x), s(y)) → s(f(x, y))
g(0, x) → g(f(x, x), x)
The set Q consists of the following terms:
f(x0, 0)
f(s(x0), s(x1))
g(0, x0)
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