Term Rewriting System R:
[x]
f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x)) -> F(f(p(s(x))))
F(s(x)) -> F(p(s(x)))
F(s(x)) -> P(s(x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pairs:

F(s(x)) -> F(p(s(x)))
F(s(x)) -> F(f(p(s(x))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(x)) -> F(f(p(s(x))))
one new Dependency Pair is created:

F(s(x)) -> F(f(x))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

F(s(x)) -> F(f(x))
F(s(x)) -> F(p(s(x)))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(x)) -> F(p(s(x)))
one new Dependency Pair is created:

F(s(x)) -> F(x)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(x)) -> F(x)
F(s(x)) -> F(f(x))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(x)) -> F(f(x))
two new Dependency Pairs are created:

F(s(s(x''))) -> F(s(f(f(p(s(x''))))))
F(s(0)) -> F(0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Rewriting Transformation


Dependency Pairs:

F(s(s(x''))) -> F(s(f(f(p(s(x''))))))
F(s(x)) -> F(x)


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(x''))) -> F(s(f(f(p(s(x''))))))
one new Dependency Pair is created:

F(s(s(x''))) -> F(s(f(f(x''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(x''))) -> F(s(f(f(x''))))
F(s(x)) -> F(x)


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(x)) -> F(x)
two new Dependency Pairs are created:

F(s(s(x''))) -> F(s(x''))
F(s(s(s(x'''')))) -> F(s(s(x'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 6
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(x''))) -> F(s(x''))
F(s(s(x''))) -> F(s(f(f(x''))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(x''))) -> F(s(f(f(x''))))
two new Dependency Pairs are created:

F(s(s(s(x')))) -> F(s(f(s(f(f(p(s(x'))))))))
F(s(s(0))) -> F(s(f(0)))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(0))) -> F(s(f(0)))
F(s(s(s(x')))) -> F(s(f(s(f(f(p(s(x'))))))))
F(s(s(x''))) -> F(s(x''))
F(s(s(s(x'''')))) -> F(s(s(x'''')))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(s(x')))) -> F(s(f(s(f(f(p(s(x'))))))))
one new Dependency Pair is created:

F(s(s(s(x')))) -> F(s(s(f(f(p(s(f(f(p(s(x')))))))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(s(x')))) -> F(s(s(f(f(p(s(f(f(p(s(x')))))))))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(x''))) -> F(s(x''))
F(s(s(0))) -> F(s(f(0)))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(0))) -> F(s(f(0)))
one new Dependency Pair is created:

F(s(s(0))) -> F(s(0))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(x''))) -> F(s(x''))
F(s(s(s(x')))) -> F(s(s(f(f(p(s(f(f(p(s(x')))))))))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(s(x')))) -> F(s(s(f(f(p(s(f(f(p(s(x')))))))))))
one new Dependency Pair is created:

F(s(s(s(x')))) -> F(s(s(f(f(f(f(p(s(x')))))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(s(x')))) -> F(s(s(f(f(f(f(p(s(x')))))))))
F(s(s(x''))) -> F(s(x''))
F(s(s(s(x'''')))) -> F(s(s(x'''')))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(s(x')))) -> F(s(s(f(f(f(f(p(s(x')))))))))
one new Dependency Pair is created:

F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(x''))) -> F(s(x''))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(x''))) -> F(s(x''))
three new Dependency Pairs are created:

F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 12
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(s(x'''')))) -> F(s(s(x'''')))
four new Dependency Pairs are created:

F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(s(s(s(x'''''''')))))) -> F(s(s(s(s(x'''''''')))))
F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 13
                 ↳Forward Instantiation Transformation


Dependency Pairs:

F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))
F(s(s(s(s(s(x'''''''')))))) -> F(s(s(s(s(x'''''''')))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost




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

F(s(s(s(x'''')))) -> F(s(s(x'''')))
six new Dependency Pairs are created:

F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(s(x'''''''')))))) -> F(s(s(s(s(x'''''''')))))
F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))
F(s(s(s(s(s(s(x''''''''''))))))) -> F(s(s(s(s(s(x''''''''''))))))
F(s(s(s(s(s(s(x''''''''))))))) -> F(s(s(s(s(s(x''''''''))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 14
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(s(s(s(s(s(s(x''''''''))))))) -> F(s(s(s(s(s(x''''''''))))))
F(s(s(s(s(s(s(x''''''''''))))))) -> F(s(s(s(s(s(x''''''''''))))))
F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))
F(s(s(s(s(s(x'''''''')))))) -> F(s(s(s(s(x'''''''')))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(s(s(s(x'''''''')))))) -> F(s(s(s(s(x'''''''')))))
F(s(s(s(s(x'''))))) -> F(s(s(s(x'''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))
F(s(s(s(x')))) -> F(s(s(f(f(f(f(x')))))))
F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x


Strategy:

innermost



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