(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, x) → f(g(x), x)
h(g(x)) → h(a)
g(h(x)) → g(x)
h(h(x)) → 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(a, x) → F(g(x), x)
F(a, x) → G(x)
H(g(x)) → H(a)
G(h(x)) → G(x)
The TRS R consists of the following rules:
f(a, x) → f(g(x), x)
h(g(x)) → h(a)
g(h(x)) → g(x)
h(h(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(h(x)) → G(x)
The TRS R consists of the following rules:
f(a, x) → f(g(x), x)
h(g(x)) → h(a)
g(h(x)) → g(x)
h(h(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(h(x)) → G(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- G(h(x)) → G(x)
The graph contains the following edges 1 > 1
(9) YES
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, x) → F(g(x), x)
The TRS R consists of the following rules:
f(a, x) → f(g(x), x)
h(g(x)) → h(a)
g(h(x)) → g(x)
h(h(x)) → x
Q is empty.
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, x) → F(g(x), x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x1
a =
a
g(
x1) =
g
Knuth-Bendix order [KBO] with precedence:
a > g
and weight map:
a=1
g=1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
g(h(x)) → g(x)
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, x) → f(g(x), x)
h(g(x)) → h(a)
g(h(x)) → g(x)
h(h(x)) → x
Q is empty.
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