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
Narrowing Transformation


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




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

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
one new Dependency Pair is created:

ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Narrowing Transformation


Dependency Pairs:

U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
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




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

ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
one new Dependency Pair is created:

ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Forward Instantiation Transformation


Dependency Pairs:

ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
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




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

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')


Rules:


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


Strategy:

innermost




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

U21(ackout(X), Y) -> ACKIN(Y, X)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 5
Forward Instantiation Transformation


Dependency Pairs:

U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))


Rules:


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


Strategy:

innermost




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

ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 6
Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 7
Polynomial Ordering


Dependency Pairs:

ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(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 pairs can be strictly oriented:

ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(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(s(x1))=  1 + x1  
  POL(ACKIN(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 8
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