(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
if(true, x, y) → x
if(false, x, y) → y
if(x, y, y) → y
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
if(x, if(x, y, z), z) → if(x, y, z)
if(x, y, if(x, y, z)) → if(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:
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(true, x, y) → x
if(false, x, y) → y
if(x, y, y) → y
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
if(x, if(x, y, z), z) → if(x, y, z)
if(x, y, if(x, y, z)) → if(x, y, z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
if(x1, x2, x3) = if(x1, x2, x3)
From the DPs we obtained the following set of size-change graphs:
- IF(if(x, y, z), u, v) → IF(x, if(y, u, v), if(z, u, v)) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- IF(if(x, y, z), u, v) → IF(y, u, v) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
- IF(if(x, y, z), u, v) → IF(z, u, v) (allowed arguments on rhs = {1, 2, 3})
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(4) TRUE