Termination Proof using AProVE

Term Rewriting System R:
[X, Z, N, Y]
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
PI(X) -> 2NDSPOS(X, from(0))
PI(X) -> FROM(0)
PLUS(s(X), Y) -> PLUS(X, Y)
TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
TIMES(s(X), Y) -> TIMES(X, Y)
SQUARE(X) -> TIMES(X, X)
ACTIVATE(n_{_from}(X)) -> FROM(X)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))

two new Dependency Pairs are created:

2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))

two new Dependency Pairs are created:

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))

two new Dependency Pairs are created:

2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))

two new Dependency Pairs are created:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSPOS(s(N), cons(X, n_{_from}(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))

two new Dependency Pairs are created:

2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, n_{_from}(X''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSNEG(N, from(X''))

two new Dependency Pairs are created:

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, n_{_from}(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSNEG(s(N), cons(X, n_{_from}(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))

two new Dependency Pairs are created:

2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, n_{_from}(X''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X'')))) -> 2NDSPOS(N, from(X''))

two new Dependency Pairs are created:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, n_{_from}(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))

two new Dependency Pairs are created:

2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N''), cons(X'', cons(Y'', n_{_from}(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N''), cons(X'', cons(Y'', n_{_from}(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')

four new Dependency Pairs are created:

2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', n_{_from}(X'''''))))) -> 2NDSNEG(s(N''), cons(X'', n_{_from}(X''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', n_{_from}(X''''')))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 13

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', n_{_from}(X''''')))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', n_{_from}(X'''''))))) -> 2NDSNEG(s(N''), cons(X'', n_{_from}(X''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))
2NDSPOS(s(N''), cons(X'', cons(Y'', n_{_from}(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))

two new Dependency Pairs are created:

2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N''), cons(X'', cons(Y'', n_{_from}(X''''')))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 14

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N''), cons(X'', cons(Y'', n_{_from}(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', n_{_from}(X'''''))))) -> 2NDSNEG(s(N''), cons(X'', n_{_from}(X''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N''), cons(X'', cons(Y'', n_{_from}(X''''')))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))
2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', n_{_from}(X''''')))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')

eight new Dependency Pairs are created:

2NDSNEG(s(s(N'')), cons2(X, cons(Y, cons(X'', n_{_from}(X'''''))))) -> 2NDSPOS(s(N''), cons(X'', n_{_from}(X''''')))
2NDSNEG(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', n_{_from}(X''''')))))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', n_{_from}(X'''''))))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', Z'''''))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', Z''''')))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', n_{_from}(X''''''')))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', n_{_from}(X'''''''))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', Z''''')))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', Z'''''))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z'''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', n_{_from}(X'''''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', n_{_from}(X''''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', n_{_from}(X''''''')))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', n_{_from}(X'''''''))))))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 15

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

2NDSPOS(s(N), cons(X, n_{_from}(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSNEG(N, cons(X''', n_{_from}(s(X'''))))

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(n__from(x₁)) = 0

POL(cons(x₁, x₂)) = 1

POL(2NDSNEG(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

POL(cons2(x₁, x₂)) = 0

POL(2NDSPOS(x₁, x₂)) = x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 17

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSNEG(s(N), cons2(X, cons(Y, n_{_from}(X''')))) -> 2NDSPOS(N, cons(X''', n_{_from}(s(X'''))))
2NDSNEG(s(N), cons(X, n_{_from}(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', n_{_from}(s(X''')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 16

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pairs:

2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', n_{_from}(X''''''')))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', n_{_from}(X'''''''))))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', n_{_from}(X'''''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', n_{_from}(X''''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z'''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', Z''''')))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', Z'''''))))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', n_{_from}(X''''''')))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', n_{_from}(X'''''''))))
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', Z'''''))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', Z''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', n_{_from}(X''''''')))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', n_{_from}(X'''''''))))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', n_{_from}(X'''''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', n_{_from}(X''''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z'''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', Z''''')))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', Z'''''))))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', n_{_from}(X''''''')))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', n_{_from}(X'''''''))))
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', Z'''''))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', Z''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))

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(n__from(x₁)) = 0

POL(cons(x₁, x₂)) = 1 + x₂

POL(2NDSNEG(x₁, x₂)) = x₂

POL(s(x₁)) = 0

POL(cons2(x₁, x₂)) = x₂

POL(2NDSPOS(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 4

             ↳Nar

...

               →DP Problem 18

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳FwdInst

Dependency Pair:

PLUS(s(X), Y) -> PLUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

PLUS(s(X), Y) -> PLUS(X, Y)

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

           →DP Problem 19

             ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳FwdInst

Dependency Pair:

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

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

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(s(X''), Y'')

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

           →DP Problem 19

             ↳FwdInst

...

               →DP Problem 20

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳FwdInst

Dependency Pair:

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

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

           →DP Problem 19

             ↳FwdInst

...

               →DP Problem 21

                 ↳Dependency Graph

       →DP Problem 3

         ↳FwdInst

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Forward Instantiation Transformation

Dependency Pair:

TIMES(s(X), Y) -> TIMES(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

TIMES(s(X), Y) -> TIMES(X, Y)

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 22

             ↳Forward Instantiation Transformation

Dependency Pair:

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

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 22

             ↳FwdInst

...

               →DP Problem 23

                 ↳Polynomial Ordering

Dependency Pair:

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

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳FwdInst

           →DP Problem 22

             ↳FwdInst

...

               →DP Problem 24

                 ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes

POL(n__from(x₁))	= 0
POL(cons(x₁, x₂))	= 1
POL(2NDSNEG(x₁, x₂))	= x₁
POL(s(x₁))	= 1 + x₁
POL(cons2(x₁, x₂))	= 0
POL(2NDSPOS(x₁, x₂))	= x₁ + x₂

POL(n__from(x₁))	= 0
POL(cons(x₁, x₂))	= 1 + x₂
POL(2NDSNEG(x₁, x₂))	= x₂
POL(s(x₁))	= 0
POL(cons2(x₁, x₂))	= x₂
POL(2NDSPOS(x₁, x₂))	= x₂

POL(PLUS(x₁, x₂))	= 1 + x₁
POL(s(x₁))	= 1 + x₁

POL(TIMES(x₁, x₂))	= 1 + x₁
POL(s(x₁))	= 1 + x₁