Term Rewriting System R:
[X, Y]
*(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
*(X, 1) -> X
*(X, 0) -> X
*(X, 0) -> 0

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))
*'(X, +(Y, 1)) -> *'(1, 0)

Furthermore, R contains one SCC.

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

Dependency Pair:

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))

Rules:

*(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
*(X, 1) -> X
*(X, 0) -> X
*(X, 0) -> 0

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 Pair:

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))

Rules:

*(X, 0) -> 0
*(X, 0) -> X

Strategy:

innermost

Found an infinite P-chain over R:
P =

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))

R =

*(X, 0) -> 0
*(X, 0) -> X

s = *'(X', +(Y', *(1, 0)))
evaluates to t =*'(X', +(Y', *(1, 0)))

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:03 minutes