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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(X) -> IF(X, c, nf(true))
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(nf(X)) -> F(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(X)) -> F(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(true))


Rules:


f(X) -> if(X, c, nf(true))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
activate(nf(X)) -> f(X)
activate(X) -> X


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

IF(false, X, Y) -> ACTIVATE(Y)
one new Dependency Pair is created:

IF(false, c, nf(true)) -> ACTIVATE(nf(true))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pairs:

IF(false, c, nf(true)) -> ACTIVATE(nf(true))
F(X) -> IF(X, c, nf(true))
ACTIVATE(nf(X)) -> F(X)


Rules:


f(X) -> if(X, c, nf(true))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
activate(nf(X)) -> f(X)
activate(X) -> X


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(nf(X)) -> F(X)
one new Dependency Pair is created:

ACTIVATE(nf(true)) -> F(true)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pairs:

F(X) -> IF(X, c, nf(true))
ACTIVATE(nf(true)) -> F(true)
IF(false, c, nf(true)) -> ACTIVATE(nf(true))


Rules:


f(X) -> if(X, c, nf(true))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
activate(nf(X)) -> f(X)
activate(X) -> X


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(X) -> IF(X, c, nf(true))
one new Dependency Pair is created:

F(true) -> IF(true, c, nf(true))

The transformation is resulting in no new DP problems.


Innermost Termination of R successfully shown.
Duration:
0:01 minutes