(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