(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(y))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(s(x), y) → +1(x, y)
+1(s(x), y) → +1(x, s(y))
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
+1(s(x), y) → +1(x, y)
+1(s(x), y) → +1(x, s(y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
+1(
x1,
x2) =
+1(
x1)
s(
x1) =
s(
x1)
+(
x1,
x2) =
+(
x1,
x2)
0 =
0
Lexicographic path order with status [LPO].
Precedence:
+2 > s1 > +^11
0 > +^11
Status:
+^11: [1]
s1: [1]
+2: [1,2]
0: []
The following usable rules [FROCOS05] were oriented:
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(y))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) TRUE