Term Rewriting System R:
[n, m]
ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACKIN(s(m), 0) -> U11(ackin(m, s(0)))
ACKIN(s(m), 0) -> ACKIN(m, s(0))
ACKIN(s(m), s(n)) -> U21(ackin(s(m), n), m)
ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
U21(ackout(n), m) -> U22(ackin(m, n))
U21(ackout(n), m) -> ACKIN(m, n)

Furthermore, R contains one SCC.

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

Dependency Pairs:

ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
U21(ackout(n), m) -> ACKIN(m, n)
ACKIN(s(m), s(n)) -> U21(ackin(s(m), n), m)
ACKIN(s(m), 0) -> ACKIN(m, s(0))

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

The following dependency pairs can be strictly oriented:

U21(ackout(n), m) -> ACKIN(m, n)
ACKIN(s(m), 0) -> ACKIN(m, s(0))

Additionally, the following rules can be oriented:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ack_in(x1, x2)) =  0 POL(0) =  0 POL(u11(x1)) =  0 POL(u22(x1)) =  0 POL(ACK_IN(x1, x2)) =  x1 POL(U21(x1, x2)) =  1 + x2 POL(ack_out(x1)) =  0 POL(u21(x1, x2)) =  0 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pairs:

ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
ACKIN(s(m), s(n)) -> U21(ackin(s(m), n), m)

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

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(m), s(n)) -> ACKIN(s(m), n)

Rules:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

The following dependency pair can be strictly oriented:

ACKIN(s(m), s(n)) -> ACKIN(s(m), n)

Additionally, the following rules can be oriented:

ackin(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ack_in(x1, x2)) =  0 POL(0) =  0 POL(u11(x1)) =  0 POL(u22(x1)) =  0 POL(ACK_IN(x1, x2)) =  x2 POL(ack_out(x1)) =  0 POL(u21(x1, x2)) =  0 POL(s(x1)) =  1 + x1

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(0, n) -> ackout(s(n))
ackin(s(m), 0) -> u11(ackin(m, s(0)))
ackin(s(m), s(n)) -> u21(ackin(s(m), n), m)
u11(ackout(n)) -> ackout(n)
u21(ackout(n), m) -> u22(ackin(m, n))
u22(ackout(n)) -> ackout(n)

Using the Dependency Graph resulted in no new DP problems.

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