(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) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))
The set Q consists of the following terms:
:(:(:(:(C, x0), x1), x2), x3)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) 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))
The set Q consists of the following terms:
:(:(:(:(C, x0), x1), x2), x3)
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(:(:(:(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: Combined order from the following AFS and order.
:1(
x1,
x2) =
x1
:(
x1,
x2) =
:(
x1,
x2)
C =
C
Lexicographic path order with status [LPO].
Precedence:
trivial
Status:
:2: [1,2]
The following usable rules [FROCOS05] were oriented:
:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))
(6) 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))
The set Q consists of the following terms:
:(:(:(:(C, x0), x1), x2), x3)
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE