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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACKIN(s(X), 0) -> U11(ackin(X, s(0)))
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> U22(ackin(Y, X))
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)
ACKIN(s(X), 0) -> ACKIN(X, s(0))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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)
two new Dependency Pairs are created:

ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
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
             ↳Forward Instantiation Transformation


Dependency Pairs:

ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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), 0) -> ACKIN(X, s(0))
two new Dependency Pairs are created:

ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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)
five new Dependency Pairs are created:

ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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)
eight new Dependency Pairs are created:

U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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(0)) -> U21(u11(ackin(X'', s(0))), X'')
five new Dependency Pairs are created:

ACKIN(s(s(X'''''')), s(0)) -> U21(u11(ackin(s(X''''''), s(0))), s(X''''''))
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''')), s(0))), s(s(X'''''')))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''')), s(0))), s(s(X'''''''')))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(ackin(s(X'''''), s(0))), s(X'''''))
ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Rewriting Transformation


Dependency Pairs:

ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(ackin(s(X'''''), s(0))), s(X'''''))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''')), s(0))), s(s(X'''''''')))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''')), s(0))), s(s(X'''''')))
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> U21(u11(ackin(s(X''''''), s(0))), s(X''''''))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost




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

ACKIN(s(s(X'''''')), s(0)) -> U21(u11(ackin(s(X''''''), s(0))), s(X''''''))
one new Dependency Pair is created:

ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Rewriting Transformation


Dependency Pairs:

ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(ackin(s(X'''''), s(0))), s(X'''''))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''')), s(0))), s(s(X'''''''')))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''')), s(0))), s(s(X'''''')))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost




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

ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''')), s(0))), s(s(X'''''')))
one new Dependency Pair is created:

ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''')), 0), s(X''''''))), s(s(X'''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Rewriting Transformation


Dependency Pairs:

ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''')), 0), s(X''''''))), s(s(X'''''')))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(ackin(s(X'''''), s(0))), s(X'''''))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''')), s(0))), s(s(X'''''''')))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost




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

ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''')), s(0))), s(s(X'''''''')))
one new Dependency Pair is created:

ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''')), 0), s(X''''''''))), s(s(X'''''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Rewriting Transformation


Dependency Pairs:

ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''')), 0), s(X''''''''))), s(s(X'''''''')))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(ackin(s(X'''''), s(0))), s(X'''''))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''')), 0), s(X''''''))), s(s(X'''''')))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost




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

ACKIN(s(s(X''''')), s(0)) -> U21(u11(ackin(s(X'''''), s(0))), s(X'''''))
one new Dependency Pair is created:

ACKIN(s(s(X''''')), s(0)) -> U21(u11(u21(ackin(s(X'''''), 0), X''''')), s(X'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 10
                 ↳Rewriting Transformation


Dependency Pairs:

ACKIN(s(s(X''''')), s(0)) -> U21(u11(u21(ackin(s(X'''''), 0), X''''')), s(X'''''))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''')), 0), s(X''''''))), s(s(X'''''')))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''')), 0), s(X''''''''))), s(s(X'''''''')))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost




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

ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(ackin(s(s(X'''''''''')), s(0))), s(s(X'''''''''')))
one new Dependency Pair is created:

ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''''')), 0), s(X''''''''''))), s(s(X'''''''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 11
                 ↳Forward Instantiation Transformation


Dependency Pairs:

ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''''')), 0), s(X''''''''''))), s(s(X'''''''''')))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''')), 0), s(X''''''''))), s(s(X'''''''')))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''')), 0), s(X''''''))), s(s(X'''''')))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(u21(ackin(s(X'''''), 0), X''''')), s(X'''''))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(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''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 12
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACKIN(s(s(s(X''''''''''))), s(s(Y'''))) -> U21(u21(ackin(s(s(s(X''''''''''))), Y'''), s(s(X''''''''''))), s(s(X'''''''''')))
ACKIN(s(s(X''''')), s(s(Y'''))) -> U21(u21(ackin(s(s(X''''')), Y'''), s(X''''')), s(X'''''))
ACKIN(s(s(s(X''''''''))), s(s(Y'''))) -> U21(u21(ackin(s(s(s(X''''''''))), Y'''), s(s(X''''''''))), s(s(X'''''''')))
ACKIN(s(s(X''''')), s(0)) -> U21(u11(u21(ackin(s(X'''''), 0), X''''')), s(X'''''))
U21(ackout(s(0)), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''')), 0), s(X''''''''))), s(s(X'''''''')))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''))), s(s(Y'''))) -> U21(u21(ackin(s(s(s(X''''''))), Y'''), s(s(X''''''))), s(s(X'''''')))
ACKIN(s(X'), s(s(s(Y'''')))) -> ACKIN(s(X'), s(s(Y'''')))
U21(ackout(s(s(s(Y'''''')))), s(X''')) -> ACKIN(s(X'''), s(s(s(Y''''''))))
ACKIN(s(s(X'''''')), s(s(Y'''))) -> U21(u21(ackin(s(s(X'''''')), Y'''), s(X'''''')), s(X''''''))
U21(ackout(s(s(0))), s(X''')) -> ACKIN(s(X'''), s(s(0)))
U21(ackout(0), s(s(X''''''))) -> ACKIN(s(s(X'''''')), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(s(X''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''')), 0), s(X''''''))), s(s(X'''''')))
ACKIN(s(X'), s(s(0))) -> ACKIN(s(X'), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(X'''''')), s(0)) -> U21(u11(u21(ackin(s(X''''''), 0), X'''''')), s(X''''''))
ACKIN(s(s(X'''')), 0) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(X'''''')), s(0)) -> ACKIN(s(s(X'''''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(X'''')) -> ACKIN(s(X''''), s(0))
ACKIN(s(s(s(X''''''''''))), s(0)) -> U21(u11(u21(ackin(s(s(X'''''''''')), 0), s(X''''''''''))), s(s(X'''''''''')))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:14 minutes