(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
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:
IF(if(x, y, z), u, v) → IF(x, if(y, u, v), if(z, u, v))
IF(if(x, y, z), u, v) → IF(y, u, v)
IF(if(x, y, z), u, v) → IF(z, u, v)
The TRS R consists of the following rules:
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
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.
IF(if(x, y, z), u, v) → IF(x, if(y, u, v), if(z, u, v))
IF(if(x, y, z), u, v) → IF(y, u, v)
IF(if(x, y, z), u, v) → IF(z, u, v)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:
IF3 > if3
Status:
if3: [1,2,3]
IF3: [1,3,2]
The following usable rules [FROCOS05] were oriented:
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
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