(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x), y) → f(x, s(s(x)))
f(x, s(s(y))) → f(y, x)
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
f(s(x), y) → f(x, s(s(x)))
f(x, s(s(y))) → f(y, x)
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(s(x), y) → f(x, s(s(x)))
f(x, s(s(y))) → f(y, x)
The set Q consists of the following terms:
f(s(x0), x1)
f(x0, s(s(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(s(x), y) → F(x, s(s(x)))
F(x, s(s(y))) → F(y, x)
The TRS R consists of the following rules:
f(s(x), y) → f(x, s(s(x)))
f(x, s(s(y))) → f(y, x)
The set Q consists of the following terms:
f(s(x0), x1)
f(x0, s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
(5) 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.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(s(x)))
F(x, s(s(y))) → F(y, x)
R is empty.
The set Q consists of the following terms:
f(s(x0), x1)
f(x0, s(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
(7) 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(s(x0), x1)
f(x0, s(s(x1)))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(s(x)))
F(x, s(s(y))) → F(y, x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
F(x, s(s(y))) → F(y, 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 | + | 4A | · | x1 | + | 0A | · | x2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(s(x), y) → F(x, s(s(x)))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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(s(x), y) → F(x, s(s(x)))
The graph contains the following edges 1 > 1
(12) YES