Term Rewriting System R:
[X, Y]
minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MINUS(s(X), s(Y)) -> P(minus(X, Y))
MINUS(s(X), s(Y)) -> MINUS(X, Y)
DIV(s(X), s(Y)) -> DIV(minus(X, Y), s(Y))
DIV(s(X), s(Y)) -> MINUS(X, Y)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

MINUS(s(X), s(Y)) -> MINUS(X, Y)

Rules:

minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

The following dependency pair can be strictly oriented:

MINUS(s(X), s(Y)) -> MINUS(X, Y)

The following rules can be oriented:

minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
div > s > p

resulting in one new DP problem.
Used Argument Filtering System:
MINUS(x1, x2) -> MINUS(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
p(x1) -> p(x1)
div(x1, x2) -> div(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

Rules:

minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

DIV(s(X), s(Y)) -> DIV(minus(X, Y), s(Y))

Rules:

minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

The following dependency pair can be strictly oriented:

DIV(s(X), s(Y)) -> DIV(minus(X, Y), s(Y))

The following rules can be oriented:

minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
div > s > p

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

minus(X, 0) -> X
minus(s(X), s(Y)) -> p(minus(X, Y))
p(s(X)) -> X
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

Using the Dependency Graph resulted in no new DP problems.

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