Term Rewriting System R:
[X]
f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))
F(s(0)) -> P(s(0))
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(n0) -> 0'

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> ACTIVATE(X)


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(X)) -> ACTIVATE(X)
two new Dependency Pairs are created:

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(ns(X)) -> ACTIVATE(X)


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(nf(X''))) -> ACTIVATE(nf(X''))
two new Dependency Pairs are created:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nf(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
two new Dependency Pairs are created:

ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 7
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(ns(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nf(ns(ns(X''''''))))) -> ACTIVATE(nf(ns(ns(X''''''))))
ACTIVATE(ns(nf(ns(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ns(nf(nf(X'''''''')))))
ACTIVATE(ns(nf(ns(nf(ns(X'''''''')))))) -> ACTIVATE(nf(ns(nf(ns(X'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Polynomial Ordering


Dependency Pairs:

ACTIVATE(ns(nf(ns(nf(ns(X'''''''')))))) -> ACTIVATE(nf(ns(nf(ns(X'''''''')))))
ACTIVATE(ns(nf(ns(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ns(nf(nf(X'''''''')))))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))
ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(ns(X''''''))))) -> ACTIVATE(nf(ns(ns(X''''''))))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVATE(ns(nf(ns(nf(ns(X'''''''')))))) -> ACTIVATE(nf(ns(nf(ns(X'''''''')))))
ACTIVATE(ns(nf(ns(nf(nf(X'''''''')))))) -> ACTIVATE(nf(ns(nf(nf(X'''''''')))))
ACTIVATE(ns(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(ns(X''''''))))) -> ACTIVATE(nf(nf(ns(X''''''))))
ACTIVATE(ns(nf(nf(nf(X''''''))))) -> ACTIVATE(nf(nf(nf(X''''''))))
ACTIVATE(ns(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nf(ns(ns(X''''''))))) -> ACTIVATE(nf(ns(ns(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__f(x1))=  x1  
  POL(n__s(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 9
Dependency Graph


Dependency Pairs:

ACTIVATE(nf(ns(nf(nf(X''''''))))) -> ACTIVATE(ns(nf(nf(X''''''))))
ACTIVATE(nf(nf(ns(X'''')))) -> ACTIVATE(nf(ns(X'''')))
ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))
ACTIVATE(nf(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nf(ns(nf(ns(X''''''))))) -> ACTIVATE(ns(nf(ns(X''''''))))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 10
Polynomial Ordering


Dependency Pair:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(X'''')))


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVATE(nf(nf(nf(X'''')))) -> ACTIVATE(nf(nf(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__f(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 11
Dependency Graph


Dependency Pair:


Rules:


f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes