Term Rewriting System R:
[x, y, z]
f(0, 1, x) -> f(x, x, x)
f(x, y, z) -> 2
0 -> 2
1 -> 2
g(x, x, y) -> y
g(x, y, y) -> 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

f(0, 1, x) -> f(x, x, x)

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

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

f(x, y, z) -> 2

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1, x2, x3)) =  x1 + x2 + x3 POL(1) =  0 POL(2) =  0 POL(f(x1, x2, x3)) =  1 + x1 + x2 + x3
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:

1 -> 2

where the Polynomial interpretation:
 POL(0) =  0 POL(g(x1, x2, x3)) =  x1 + x2 + x3 POL(1) =  1 POL(2) =  0
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:

0 -> 2

where the Polynomial interpretation:
 POL(0) =  1 POL(g(x1, x2, x3)) =  x1 + x2 + x3 POL(2) =  0
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:

g(x, y, y) -> x
g(x, x, y) -> y

where the Polynomial interpretation:
 POL(g(x1, x2, x3)) =  1 + x1 + x2 + x3
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS6`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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