Term Rewriting System R:
[X, Y]
f(X, g(X), Y) -> f(Y, Y, Y)
g(b) -> c
b -> c

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

g(b) -> c

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(X, g(X), Y) -> F(Y, Y, Y)

Furthermore, R contains one SCC.

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Non-Overlappingness Check`

Dependency Pair:

F(X, g(X), Y) -> F(Y, Y, Y)

Rules:

f(X, g(X), Y) -> f(Y, Y, Y)
b -> c

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

`   R`
`     ↳RRRI`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳NOC`
`             ...`
`               →DP Problem 2`
`                 ↳Dependency Graph`

Dependency Pair:

F(X, g(X), Y) -> F(Y, Y, Y)

Rules:

f(X, g(X), Y) -> f(Y, Y, Y)
b -> c

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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