(0) Obligation:

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

.(.(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:

.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), z) → .(x, .(y, z))

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(.(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(x0, x1)

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

Comparison: MS
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)

Recursive path order with status [RPO].
Quasi-Precedence:
[.^1, .2]

Status:
.^1: multiset
.2: multiset


The following usable rules [FROCOS05] were oriented: none

(4) Obligation:

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

.(.(x, y), z) → .(x, .(y, z))

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