(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(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(:(:(:(C, x), y), z), u) → :1(:(x, z), :(:(:(x, y), z), u))
:1(:(:(:(C, x), y), z), u) → :1(x, z)
:1(:(:(:(C, x), y), z), u) → :1(:(:(x, y), z), u)
:1(:(:(:(C, x), y), z), u) → :1(:(x, y), z)
:1(:(:(:(C, x), y), z), u) → :1(x, y)
The TRS R consists of the following rules:
:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))
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(:(:(:(C, x), y), z), u) → :1(:(x, z), :(:(:(x, y), z), u))
:1(:(:(:(C, x), y), z), u) → :1(x, z)
:1(:(:(:(C, x), y), z), u) → :1(:(:(x, y), z), u)
:1(:(:(:(C, x), y), z), u) → :1(:(x, y), z)
:1(:(:(:(C, x), y), z), u) → :1(x, y)
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 [1,2,3] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Lexicographic path order with status [LPO].
Quasi-Precedence:
:^12 > [:2, C]
Status:
:^12: [2,1]
:2: [1,2]
C: []
The following usable rules [FROCOS05] were oriented:
:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))
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