Term Rewriting System R:
[X, Y]
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pairs:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

The following dependency pair can be strictly oriented:

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)

Additionally, the following rules can be oriented:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(u22(x1)) =  0 POL(U21(x1, x2)) =  x2 POL(ackin(x1, x2)) =  0 POL(u21(x1, x2)) =  0 POL(s(x1)) =  1 + x1 POL(ACKIN(x1, x2)) =  x1 POL(ackout(x1)) =  0

resulting in one new DP problem.

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

Dependency Pairs:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(u22(x1)) =  0 POL(ackin(x1, x2)) =  0 POL(u21(x1, x2)) =  0 POL(s(x1)) =  1 + x1 POL(ACKIN(x1, x2)) =  x2 POL(ackout(x1)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Using the Dependency Graph resulted in no new DP problems.

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