Term Rewriting System R:
[x, y]
and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(false, y) -> not(false)
implies(x, false) -> not(x)
implies(not(x), not(y)) -> implies(y, and(x, y))

Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

and(x, false) -> false
and(x, not(false)) -> x
not(not(x)) -> x
implies(not(x), not(y)) -> implies(y, and(x, y))

where the Polynomial interpretation:
 POL(and(x1, x2)) =  1 + x1 + x2 POL(false) =  0 POL(implies(x1, x2)) =  1 + 2·x1 + 2·x2 POL(not(x1)) =  1 + 2·x1
was used.

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

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

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

implies(x, false) -> not(x)
implies(false, y) -> not(false)

where the Polynomial interpretation:
 POL(false) =  0 POL(implies(x1, x2)) =  1 + x1 + x2 POL(not(x1)) =  x1
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Overlay and local confluence Check`

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳OC`
`             ...`
`               →TRS4`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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