Term Rewriting System R:
[Y, X]
minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MINUS(s(X), s(Y)) -> MINUS(X, Y)
GEQ(s(X), s(Y)) -> GEQ(X, Y)
DIV(s(X), s(Y)) -> IF(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
DIV(s(X), s(Y)) -> GEQ(X, Y)
DIV(s(X), s(Y)) -> DIV(minus(X, Y), s(Y))
DIV(s(X), s(Y)) -> MINUS(X, Y)

Furthermore, R contains three SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

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

Rules:

minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

Rules:

minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

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

Rules:

minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

Rules:

minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rules:

minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

minus(0, Y) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
geq(X, 0) -> true
geq(0, s(Y)) -> false
geq(s(X), s(Y)) -> geq(X, Y)
div(0, s(Y)) -> 0
div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) -> X
if(false, X, Y) -> Y

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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