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

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules for Innermost Termination`

Removing the following rules from R which left hand sides contain non normal subterms

if(true, X, Y) -> X

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳Removing Redundant Rules`

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

f(X) -> nf(X)

where the Polynomial interpretation:
 POL(n__f(x1)) =  1 + x1 POL(n__true) =  0 POL(activate(x1)) =  2·x1 POL(if(x1, x2, x3)) =  x1 + x2 + 2·x3 POL(c) =  0 POL(false) =  0 POL(true) =  0 POL(f(x1)) =  2 + x1
was used.

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

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Removing Redundant Rules`

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

if(false, X, Y) -> activate(Y)

where the Polynomial interpretation:
 POL(activate(x1)) =  x1 POL(n__true) =  0 POL(n__f(x1)) =  x1 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`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

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

true -> ntrue
activate(X) -> X

where the Polynomial interpretation:
 POL(n__f(x1)) =  x1 POL(n__true) =  0 POL(activate(x1)) =  1 + x1 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(c) =  0 POL(true) =  1 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`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Removing Redundant Rules`

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

f(X) -> if(X, c, nf(ntrue))

where the Polynomial interpretation:
 POL(activate(x1)) =  2·x1 POL(n__true) =  0 POL(n__f(x1)) =  1 + x1 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(c) =  0 POL(true) =  0 POL(f(x1)) =  2 + x1
was used.

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

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS6`
`                 ↳Removing Redundant Rules`

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

activate(ntrue) -> true

where the Polynomial interpretation:
 POL(activate(x1)) =  x1 POL(n__f(x1)) =  x1 POL(n__true) =  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`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS7`
`                 ↳Removing Redundant Rules`

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

activate(nf(X)) -> f(activate(X))

where the Polynomial interpretation:
 POL(activate(x1)) =  x1 POL(n__f(x1)) =  1 + x1 POL(f(x1)) =  x1
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS8`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes