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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(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)) -> G(x)
one new Dependency Pair is created:

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

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


Rules:


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


Strategy:

innermost




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

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

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

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


Rules:


g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(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'''')))) -> G(s(s(x'''')))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
s > F
s > G

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
F(x1) -> F(x1)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes