(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
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:
F(+(x, 0)) → F(x)
+1(x, +(y, z)) → +1(+(x, y), z)
+1(x, +(y, z)) → +1(x, y)
The TRS R consists of the following rules:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, +(y, z)) → +1(x, y)
+1(x, +(y, z)) → +1(+(x, y), z)
The TRS R consists of the following rules:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
+1(x, +(y, z)) → +1(x, y)
+1(x, +(y, z)) → +1(+(x, y), z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
+1(
x1,
x2) =
x2
+(
x1,
x2) =
+(
x1,
x2)
f(
x1) =
f(
x1)
0 =
0
Lexicographic path order with status [LPO].
Quasi-Precedence:
[+2, f1, 0]
Status:
+2: [2,1]
f1: [1]
0: []
The following usable rules [FROCOS05] were oriented:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(+(x, 0)) → F(x)
The TRS R consists of the following rules:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(+(x, 0)) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
x1
+(
x1,
x2) =
+(
x1,
x2)
0 =
0
f(
x1) =
f(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
+2 > f1
0 > f1
Status:
+2: [2,1]
0: []
f1: [1]
The following usable rules [FROCOS05] were oriented:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(+(x, 0)) → f(x)
+(x, +(y, z)) → +(+(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE