(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(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:
+1(a, b) → +1(b, a)
+1(a, +(b, z)) → +1(b, +(a, z))
+1(a, +(b, z)) → +1(a, z)
+1(+(x, y), z) → +1(x, +(y, z))
+1(+(x, y), z) → +1(y, z)
F(+(x, y), z) → +1(f(x, z), f(y, z))
F(+(x, y), z) → F(x, z)
F(+(x, y), z) → F(y, z)
The TRS R consists of the following rules:
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(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 3 SCCs with 3 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(a, +(b, z)) → +1(a, z)
The TRS R consists of the following rules:
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
a = a
b = b
+(x1, x2) = +(x2)
From the DPs we obtained the following set of size-change graphs:
- +1(a, +(b, z)) → +1(a, z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 >= 1, 2 > 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(7) TRUE
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(+(x, y), z) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))
The TRS R consists of the following rules:
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
+(x1, x2) = +(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- +1(+(x, y), z) → +1(y, z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- +1(+(x, y), z) → +1(x, +(y, z)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(10) TRUE
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(+(x, y), z) → F(y, z)
F(+(x, y), z) → F(x, z)
The TRS R consists of the following rules:
+(a, b) → +(b, a)
+(a, +(b, z)) → +(b, +(a, z))
+(+(x, y), z) → +(x, +(y, z))
f(a, y) → a
f(b, y) → b
f(+(x, y), z) → +(f(x, z), f(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
+(x1, x2) = +(x1, x2)
From the DPs we obtained the following set of size-change graphs:
- F(+(x, y), z) → F(y, z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
- F(+(x, y), z) → F(x, z) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(13) TRUE