(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
(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(x, f(y, a)) → F(f(f(f(a, x), y), a), h(a))
F(x, f(y, a)) → F(f(f(a, x), y), a)
F(x, f(y, a)) → F(f(a, x), y)
F(x, f(y, a)) → F(a, x)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
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(x, f(y, a)) → F(a, x)
F(x, f(y, a)) → F(f(a, x), y)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(7) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
F(
x,
f(
y,
a)) →
F(
a,
x) we obtained the following new rules [LPAR04]:
F(f(y_1, a), f(x1, a)) → F(a, f(y_1, a))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(f(a, x), y)
F(f(y_1, a), f(x1, a)) → F(a, f(y_1, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(9) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
F(
x,
f(
y,
a)) →
F(
f(
a,
x),
y) we obtained the following new rules [LPAR04]:
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))
F(a, f(f(y_1, a), a)) → F(f(a, a), f(y_1, a))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(y_1, a), f(x1, a)) → F(a, f(y_1, a))
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))
F(a, f(f(y_1, a), a)) → F(f(a, a), f(y_1, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(11) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
F(
f(
y_1,
a),
f(
x1,
a)) →
F(
a,
f(
y_1,
a)) we obtained the following new rules [LPAR04]:
F(f(f(y_1, a), a), f(x1, a)) → F(a, f(f(y_1, a), a))
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))
F(a, f(f(y_1, a), a)) → F(f(a, a), f(y_1, a))
F(f(f(y_1, a), a), f(x1, a)) → F(a, f(f(y_1, a), a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(13) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, x), y), a), h(a))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(15) 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:
- F(x0, f(f(y_1, a), a)) → F(f(a, x0), f(y_1, a))
The graph contains the following edges 2 > 2
(16) TRUE