Term Rewriting System R:
[x, y]
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

-'(s(x), s(y)) -> -'(x, y)
MIN(s(x), s(y)) -> MIN(x, y)
TWICE(s(x)) -> TWICE(x)
F(s(x), s(y)) -> F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) -> -'(y, min(x, y))
F(s(x), s(y)) -> MIN(x, y)
F(s(x), s(y)) -> TWICE(min(x, y))
F(s(x), s(y)) -> F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) -> -'(x, min(x, y))

Furthermore, R contains four SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining`

Dependency Pair:

-'(s(x), s(y)) -> -'(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

-'(s(x), s(y)) -> -'(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-'(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining`

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, 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`
`       →DP Problem 4`
`         ↳Remaining`

Dependency Pair:

MIN(s(x), s(y)) -> MIN(x, y)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MIN(s(x), s(y)) -> MIN(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MIN(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining`

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, 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`
`       →DP Problem 4`
`         ↳Remaining`

Dependency Pair:

TWICE(s(x)) -> TWICE(x)

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TWICE(s(x)) -> TWICE(x)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(TWICE(x1)) =  x1 POL(s(x1)) =  1 + x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 7`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Remaining`

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, 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`
`         ↳AFS`
`       →DP Problem 4`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(s(x), s(y)) -> F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) -> F(-(y, min(x, y)), s(twice(min(x, y))))

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
twice(0) -> 0
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes