Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, f(true))
if(true, X, Y) -> X
if(false, X, Y) -> Y

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

if(false, X, Y) -> Y

where the Polynomial interpretation:
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(c)=  0  
  POL(false)=  1  
  POL(true)=  0  
  POL(f(x1))=  x1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(X) -> IF(X, c, f(true))
F(X) -> F(true)

Furthermore, R contains one SCC. 


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pair:

F(X) -> F(true)


Rules:


f(X) -> if(X, c, f(true))
if(true, X, Y) -> X


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Non Termination


Dependency Pair:

F(X) -> F(true)


Rule:

none


Strategy:

innermost




Found an infinite P-chain over R:
P =

F(X) -> F(true)

R = none

s = F(true)
evaluates to t =F(true)

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:01 minutes