(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 0)))
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:
+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 0)))
The set Q consists of the following terms:
+(0, x0)
+(s(x0), 0)
+(s(x0), 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:
+1(s(x), s(y)) → +1(s(x), +(y, 0))
+1(s(x), s(y)) → +1(y, 0)
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 0)))
The set Q consists of the following terms:
+(0, x0)
+(s(x0), 0)
+(s(x0), s(x1))
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 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(s(x), s(y)) → +1(s(x), +(y, 0))
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 0)))
The set Q consists of the following terms:
+(0, x0)
+(s(x0), 0)
+(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
+1(s(x), s(y)) → +1(s(x), +(y, 0))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
+1(
x1,
x2) =
x2
s(
x1) =
s(
x1)
+(
x1,
x2) =
+(
x1,
x2)
0 =
0
Lexicographic path order with status [LPO].
Precedence:
s1 > 0 > +2
Status:
s1: [1]
0: []
+2: [1,2]
The following usable rules [FROCOS05] were oriented:
+(0, y) → y
+(s(x), 0) → s(x)
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 0)))
The set Q consists of the following terms:
+(0, x0)
+(s(x0), 0)
+(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE