(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(c) = 0
POL(f(x1)) = x1
POL(false) = 2
POL(if(x1, x2, x3)) = x1 + 2·x2 + 2·x3
POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(false, X, Y) → Y
(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
Q is empty.
(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
Q is empty.
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
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0,
σ' = [ ], and μ' = [x0 / true] on the rule
F(true)[ ]n[ ] → F(true)[ ]n[x0 / true]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Simplify mu) - Instantiate mu - Instantiation
F(X)[ ]n[ ] → F(true)[ ]n[ ]
by OriginalRule from TRS P
(8) NO