(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

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(*(x, y), *(x, z)) → +1(y, z)
+1(+(x, y), z) → +1(x, +(y, z))
+1(+(x, y), z) → +1(y, z)
+1(*(x, y), +(*(x, z), u)) → +1(*(x, +(y, z)), u)
+1(*(x, y), +(*(x, z), u)) → +1(y, z)

The TRS R consists of the following rules:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

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 1 SCC with 1 less node.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+1(+(x, y), z) → +1(x, +(y, z))
+1(*(x, y), *(x, z)) → +1(y, z)
+1(+(x, y), z) → +1(y, z)
+1(*(x, y), +(*(x, z), u)) → +1(y, z)

The TRS R consists of the following rules:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


+1(*(x, y), *(x, z)) → +1(y, z)
+1(*(x, y), +(*(x, z), u)) → +1(y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
+1(x0, x1, x2)  =  +1(x0, x1)

Tags:
+1 has argument tags [3,1,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
+1(x1, x2)  =  +1
+(x1, x2)  =  +(x1, x2)
*(x1, x2)  =  *(x1, x2)
u  =  u

Lexicographic path order with status [LPO].
Quasi-Precedence:
[*2, u] > +^1 > +2

Status:
+^1: []
+2: [1,2]
*2: [1,2]
u: []


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

+1(+(x, y), z) → +1(x, +(y, z))
+1(+(x, y), z) → +1(y, z)

The TRS R consists of the following rules:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

Q is empty.
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(+(x, y), z) → +1(x, +(y, z))
+1(+(x, y), z) → +1(y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
+1(x0, x1, x2)  =  +1(x1)

Tags:
+1 has argument tags [0,2,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
+1(x1, x2)  =  x1
+(x1, x2)  =  +(x1, x2)
*(x1, x2)  =  x2
u  =  u

Lexicographic path order with status [LPO].
Quasi-Precedence:
u > +2

Status:
+2: [1,2]
u: []


The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

Q is empty.
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