Term Rewriting System R:
[x, y]
plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0), y) -> s(y)
plus(0, y) -> y
ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

PLUS(s(s(x)), y) -> PLUS(x, s(y))
PLUS(x, s(s(y))) -> PLUS(s(x), y)
ACK(s(x), 0) -> ACK(x, s(0))
ACK(s(x), s(y)) -> ACK(x, plus(y, ack(s(x), y)))
ACK(s(x), s(y)) -> PLUS(y, ack(s(x), y))
ACK(s(x), s(y)) -> ACK(s(x), y)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pairs:

PLUS(x, s(s(y))) -> PLUS(s(x), y)
PLUS(s(s(x)), y) -> PLUS(x, s(y))


Rules:


plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0), y) -> s(y)
plus(0, y) -> y
ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))


Strategy:

innermost




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

PLUS(s(s(x)), y) -> PLUS(x, s(y))
two new Dependency Pairs are created:

PLUS(s(s(s(s(x'')))), y'') -> PLUS(s(s(x'')), s(y''))
PLUS(s(s(x'')), s(y'')) -> PLUS(x'', s(s(y'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

PLUS(s(s(x'')), s(y'')) -> PLUS(x'', s(s(y'')))
PLUS(s(s(s(s(x'')))), y'') -> PLUS(s(s(x'')), s(y''))
PLUS(x, s(s(y))) -> PLUS(s(x), y)


Rules:


plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0), y) -> s(y)
plus(0, y) -> y
ack(0, y) -> s(y)
ack(s(x), 0) -> ack(x, s(0))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))


Strategy:

innermost




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

PLUS(x, s(s(y))) -> PLUS(s(x), y)
three new Dependency Pairs are created:

PLUS(x'', s(s(s(s(y''))))) -> PLUS(s(x''), s(s(y'')))
PLUS(s(s(s(x''''))), s(s(y'))) -> PLUS(s(s(s(s(x'''')))), y')
PLUS(s(x''''), s(s(s(y'''')))) -> PLUS(s(s(x'''')), s(y''''))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:


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




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:01 minutes