(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
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(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
(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(f(0, x), 1) → F(g(f(x, x)), x)
F(f(0, x), 1) → F(x, x)
F(g(x), y) → F(x, y)
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
F(f(0, x), 1) → F(x, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(F(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
| POL(f(x1, x2)) = | -I | + | 0A | · | x1 | + | 5A | · | x2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
f(g(x), y) → g(f(x, y))
f(f(0, x), 1) → f(g(f(x, x)), x)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(0, x), 1) → F(g(f(x, x)), x)
F(g(x), y) → F(x, y)
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(7) TransformationProof (EQUIVALENT transformation)
By forward instantiating [JAR06] the rule
F(
g(
x),
y) →
F(
x,
y) we obtained the following new rules [LPAR04]:
F(g(f(0, y_0)), 1) → F(f(0, y_0), 1) → F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1) → F(g(g(y_0)), x1) → F(g(y_0), x1)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(0, x), 1) → F(g(f(x, x)), x)
F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(9) TransformationProof (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
F(
f(
0,
x),
1) →
F(
g(
f(
x,
x)),
x) at position [] we obtained the following new rules [LPAR04]:
F(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0)) → F(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)
F(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0))
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
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(f(0, g(x0)), 1) → F(g(g(f(x0, g(x0)))), g(x0))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x2
1 =
1
g(
x1) =
g
Knuth-Bendix order [KBO] with precedence:
1 > g
and weight map:
1=1
g=1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(f(0, y_0)), 1) → F(f(0, y_0), 1)
F(g(g(y_0)), x1) → F(g(y_0), x1)
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
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(g(g(y_0)), x1) → F(g(y_0), x1)
The TRS R consists of the following rules:
f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(x, y))
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(15) 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.
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(g(y_0)), x1) → F(g(y_0), x1)
R is empty.
The set Q consists of the following terms:
f(f(0, x0), 1)
f(g(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(17) 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(f(0, x0), 1)
f(g(x0), x1)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(g(y_0)), x1) → F(g(y_0), x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) 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(g(g(y_0)), x1) → F(g(y_0), x1)
The graph contains the following edges 1 > 1, 2 >= 2
(20) YES