Term Rewriting System R:
[x, y]
f(x, y) -> g1(x, x, y)
f(x, y) -> g1(y, x, x)
f(x, y) -> g2(x, y, y)
f(x, y) -> g2(y, y, x)
g1(x, x, y) -> h(x, y)
g1(y, x, x) -> h(x, y)
g2(x, y, y) -> h(x, y)
g2(y, y, x) -> h(x, y)
h(x, x) -> x

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

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

f(x, y) -> g1(x, x, y)
f(x, y) -> g1(y, x, x)
f(x, y) -> g2(x, y, y)
f(x, y) -> g2(y, y, x)

where the Polynomial interpretation:
 POL(g2(x1, x2, x3)) =  x1 + x2 + x3 POL(h(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  1 + 2·x1 + 2·x2 POL(g1(x1, x2, x3)) =  x1 + x2 + x3
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:

h(x, x) -> x

where the Polynomial interpretation:
 POL(g2(x1, x2, x3)) =  1 + x1 + x2 + x3 POL(h(x1, x2)) =  1 + x1 + x2 POL(g1(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`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Removing Redundant Rules`

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

g1(y, x, x) -> h(x, y)
g1(x, x, y) -> h(x, y)

where the Polynomial interpretation:
 POL(g2(x1, x2, x3)) =  x1 + x2 + x3 POL(h(x1, x2)) =  x1 + x2 POL(g1(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`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

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

g2(x, y, y) -> h(x, y)
g2(y, y, x) -> h(x, y)

where the Polynomial interpretation:
 POL(g2(x1, x2, x3)) =  1 + x1 + x2 + x3 POL(h(x1, x2)) =  x1 + x2
was used.

All Rules of R can be deleted.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Dependency Pair Analysis`

R contains no Dependency Pairs and therefore no SCCs.

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