(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
+(minus(1), 1) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
+(minus(+(x, 1)), 1) → minus(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(+(x1, x2)) = x1 + x2
POL(0) = 0
POL(1) = 1
POL(minus(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
+(minus(1), 1) → 0
+(minus(+(x, 1)), 1) → minus(x)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(0) → 0
+(x, 0) → x
+(0, y) → y
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(+(x1, x2)) = x1 + x2
POL(0) = 1
POL(minus(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
+(x, 0) → x
+(0, y) → y
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
minus(0) → 0
minus(minus(x)) → x
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(+(x1, x2)) = x1 + 2·x2
POL(0) = 0
POL(minus(x1)) = 2 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
minus(0) → 0
minus(minus(x)) → x
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(x, minus(y)) → minus(+(minus(x), y))
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(+(x1, x2)) = 1 + x1 + 2·x2
POL(minus(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
+(x, +(y, z)) → +(+(x, y), z)
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(x, minus(y)) → minus(+(minus(x), y))
Q is empty.
(9) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(x, minus(y)) → minus(+(minus(x), y))
The set Q consists of the following terms:
+(x0, minus(x1))
(11) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, minus(y)) → +1(minus(x), y)
The TRS R consists of the following rules:
+(x, minus(y)) → minus(+(minus(x), y))
The set Q consists of the following terms:
+(x0, minus(x1))
We have to consider all minimal (P,Q,R)-chains.
(13) 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.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, minus(y)) → +1(minus(x), y)
R is empty.
The set Q consists of the following terms:
+(x0, minus(x1))
We have to consider all minimal (P,Q,R)-chains.
(15) 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].
+(x0, minus(x1))
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, minus(y)) → +1(minus(x), y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) 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:
- +1(x, minus(y)) → +1(minus(x), y)
The graph contains the following edges 2 > 2
(18) TRUE