Term Rewriting System R:
[X, Y, Z]
div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

DIV(X, e) -> I(X)
DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
DIV(div(X, Y), Z) -> DIV(i(X), Z)
DIV(div(X, Y), Z) -> I(X)
I(div(X, Y)) -> DIV(Y, X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

DIV(div(X, Y), Z) -> DIV(i(X), Z)
DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
I(div(X, Y)) -> DIV(Y, X)
DIV(X, e) -> I(X)

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DIV(div(X, Y), Z) -> DIV(i(X), Z)
DIV(div(X, Y), Z) -> DIV(Y, div(i(X), Z))
I(div(X, Y)) -> DIV(Y, X)
DIV(X, e) -> I(X)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

i(div(X, Y)) -> div(Y, X)
div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{i, I, div, DIV}

resulting in one new DP problem.
Used Argument Filtering System:
DIV(x1, x2) -> DIV(x1, x2)
div(x1, x2) -> div(x1, x2)
i(x1) -> i(x1)
I(x1) -> I(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

div(X, e) -> i(X)
div(div(X, Y), Z) -> div(Y, div(i(X), Z))
i(div(X, Y)) -> div(Y, X)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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