(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, y) → f(y, g(y))
g(a) → b
g(b) → b
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
g(a) → b
g(b) → b
The TRS R 2 is
f(a, y) → f(y, g(y))
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, y) → f(y, g(y))
g(a) → b
g(b) → b
The set Q consists of the following terms:
f(a, x0)
g(a)
g(b)
(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(a, y) → F(y, g(y))
F(a, y) → G(y)
The TRS R consists of the following rules:
f(a, y) → f(y, g(y))
g(a) → b
g(b) → b
The set Q consists of the following terms:
f(a, x0)
g(a)
g(b)
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(a, y) → F(y, g(y))
The TRS R consists of the following rules:
f(a, y) → f(y, g(y))
g(a) → b
g(b) → b
The set Q consists of the following terms:
f(a, x0)
g(a)
g(b)
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(a, y) → F(y, g(y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
F(
x1,
x2)
a =
a
g(
x1) =
g
b =
b
Recursive path order with status [RPO].
Precedence:
a > g > b > F2
Status:
F2: multiset
a: multiset
g: multiset
b: multiset
The following usable rules [FROCOS05] were oriented:
g(a) → b
g(b) → b
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, y) → f(y, g(y))
g(a) → b
g(b) → b
The set Q consists of the following terms:
f(a, x0)
g(a)
g(b)
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