(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) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(8) 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:
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) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(a, x0)
(10) 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:
g(a) → b
g(b) → b
The set Q consists of the following terms:
g(a)
g(b)
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
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
Knuth-Bendix order [KBO] with precedence:
trivial
and weight map:
a=3
b=1
F_2=1
g=2
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
g(a) → b
g(b) → b
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
g(a) → b
g(b) → b
The set Q consists of the following terms:
g(a)
g(b)
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) YES