(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
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, a), y), h(a)), x)
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, a), y), h(a)), x)
F(x, f(y, a)) → F(f(f(a, a), y), h(a))
F(x, f(y, a)) → F(f(a, a), y)
F(x, f(y, a)) → F(a, a)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
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(f(a, a), y)
F(x, f(y, a)) → F(f(f(f(a, a), y), h(a)), x)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(7) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
x,
f(
y,
a)) →
F(
f(
a,
a),
y) we obtained the following new rules [LPAR04]:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(f(f(f(a, a), y), h(a)), x)
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(9) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
x,
f(
y,
a)) →
F(
f(
a,
a),
y) we obtained the following new rules [LPAR04]:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(f(f(f(a, a), y), h(a)), x)
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(11) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
x,
f(
y,
a)) →
F(
f(
f(
f(
a,
a),
y),
h(
a)),
x) we obtained the following new rules [LPAR04]:
F(f(y_0, h(a)), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(y_0, h(a)))
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
F(f(y_0, h(a)), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(y_0, h(a)))
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
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 1 less node.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))
F(f(y_0, h(a)), f(x1, a)) → F(f(a, a), x1)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(15) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
f(
y_0,
h(
a)),
f(
x1,
a)) →
F(
f(
a,
a),
x1) we obtained the following new rules [LPAR04]:
F(f(y_0, h(a)), f(a, a)) → F(f(a, a), a)
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
F(f(a, a), f(x1, a)) → F(f(f(f(a, a), x1), h(a)), f(a, a))
F(f(y_0, h(a)), f(a, a)) → F(f(a, a), a)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(17) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(f(f(a, a), y), h(a)), x)
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(19) 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.
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(x1, a)) → F(f(a, a), x1)
R is empty.
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(21) ForwardInstantiation (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
F(
f(
a,
a),
f(
x1,
a)) →
F(
f(
a,
a),
x1) we obtained the following new rules [LPAR04]:
F(f(a, a), f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
R is empty.
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
(23) 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(f(a, a), f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
The graph contains the following edges 1 >= 1, 2 > 2
(24) TRUE