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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MIN(s(x), s(y)) -> MIN(x, y)
MAX(s(x), s(y)) -> MAX(x, y)
-'(s(x), s(y)) -> -'(x, y)
GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
GCD(s(x), s(y)) -> -'(s(max(x, y)), s(min(x, y)))
GCD(s(x), s(y)) -> MAX(x, y)
GCD(s(x), s(y)) -> MIN(x, y)

Furthermore, R contains four SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

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

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(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 implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MIN(x1, x2)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 5`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

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

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MAX(x1, x2)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 6`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

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

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(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 implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-'(x1, x2)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 7`
`             ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Nar`

Dependency Pair:

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Narrowing Transformation`

Dependency Pair:

GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GCD(s(x), s(y)) -> GCD(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
10 new Dependency Pairs are created:

GCD(s(x''), s(y'')) -> GCD(-(max(x'', y''), min(x'', y'')), s(min(x'', y'')))
GCD(s(x''), s(0)) -> GCD(-(s(x''), s(min(x'', 0))), s(min(x'', 0)))
GCD(s(0), s(y'')) -> GCD(-(s(y''), s(min(0, y''))), s(min(0, y'')))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(s(max(x'', y''))), s(min(s(x''), s(y'')))), s(min(s(x''), s(y''))))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(0)), s(min(x'', 0)))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(0)), s(min(0, y'')))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(s(min(x'', y'')))), s(min(s(x''), s(y''))))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(min(x'', 0))), s(0))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(min(0, y''))), s(0))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(min(s(x''), s(y'')))), s(s(min(x'', y''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Nar`
`           →DP Problem 8`
`             ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(min(s(x''), s(y'')))), s(s(min(x'', y''))))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(min(0, y''))), s(0))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(min(x'', 0))), s(0))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(max(s(x''), s(y''))), s(s(min(x'', y'')))), s(min(s(x''), s(y''))))
GCD(s(0), s(y'')) -> GCD(-(s(max(0, y'')), s(0)), s(min(0, y'')))
GCD(s(x''), s(0)) -> GCD(-(s(max(x'', 0)), s(0)), s(min(x'', 0)))
GCD(s(s(x'')), s(s(y''))) -> GCD(-(s(s(max(x'', y''))), s(min(s(x''), s(y'')))), s(min(s(x''), s(y''))))
GCD(s(0), s(y'')) -> GCD(-(s(y''), s(min(0, y''))), s(min(0, y'')))
GCD(s(x''), s(0)) -> GCD(-(s(x''), s(min(x'', 0))), s(min(x'', 0)))
GCD(s(x''), s(y'')) -> GCD(-(max(x'', y''), min(x'', y'')), s(min(x'', y'')))

Rules:

min(x, 0) -> 0
min(0, y) -> 0
min(s(x), s(y)) -> s(min(x, y))
max(x, 0) -> x
max(0, y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:07 minutes