(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → if(X, c, f(true))
if(true, X, Y) → X
if(false, X, Y) → Y
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:
f(X) → if(X, c, f(true))
if(true, X, Y) → X
if(false, X, Y) → Y
The set Q consists of the following terms:
f(x0)
if(true, x0, x1)
if(false, x0, x1)
(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:
F(X) → IF(X, c, f(true))
F(X) → F(true)
The TRS R consists of the following rules:
f(X) → if(X, c, f(true))
if(true, X, Y) → X
if(false, X, Y) → Y
The set Q consists of the following terms:
f(x0)
if(true, x0, x1)
if(false, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(X) → F(true)
The TRS R consists of the following rules:
f(X) → if(X, c, f(true))
if(true, X, Y) → X
if(false, X, Y) → Y
The set Q consists of the following terms:
f(x0)
if(true, x0, x1)
if(false, x0, x1)
We have to consider all minimal (P,Q,R)-chains.