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))

Innermost 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`
`         ↳Argument Filtering and 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
ACKIN(x1, x2) -> x1
s(x1) -> s(x1)
U21(x1, x2) -> x2

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →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))

Strategy:

innermost

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 3`
`                 ↳Argument Filtering and 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
ACKIN(x1, x2) -> ACKIN(x1, x2)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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