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:

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

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

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)) =  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)
U21(ackout(n), m) -> ACKIN(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)

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
Forward Instantiation Transformation

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)

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

ACKIN(s(m), s(n)) -> ACKIN(s(m), n)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Polo
→DP Problem 2
DGraph
...
→DP Problem 4
Forward Instantiation Transformation

Dependency Pair:

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

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

ACKIN(s(m''), s(s(n''))) -> ACKIN(s(m''), s(n''))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Polo
→DP Problem 2
DGraph
...
→DP Problem 5
Polynomial Ordering

Dependency Pair:

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

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

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
DGraph
...
→DP Problem 6
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