Term Rewriting System R:
[x, y]
f(x, y) -> f(x, x)
f(s(x), y) -> f(y, x)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, y) -> F(x, x)
F(s(x), y) -> F(y, x)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Usable Rules (Innermost)`

Dependency Pairs:

F(s(x), y) -> F(y, x)
F(x, y) -> F(x, x)

Rules:

f(x, y) -> f(x, x)
f(s(x), y) -> f(y, x)

Strategy:

innermost

As we are in the innermost case, we can delete all 2 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 2`
`             ↳Non Termination`

Dependency Pairs:

F(s(x), y) -> F(y, x)
F(x, y) -> F(x, x)

Rule:

none

Strategy:

innermost

Found an infinite P-chain over R:
P =

F(s(x), y) -> F(y, x)
F(x, y) -> F(x, x)

R = none

s = F(x', x')
evaluates to t =F(x', x')

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:00 minutes