Term Rewriting System R:
[X]
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(X)) -> F(f(X))
F(g(X)) -> F(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(X)) -> F(f(X))
two new Dependency Pairs are created:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(h(X''))) -> F(h(g(X'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(X)) -> F(X)


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(X)) -> F(X)
two new Dependency Pairs are created:

F(g(g(X''))) -> F(g(X''))
F(g(g(g(X'''')))) -> F(g(g(X'''')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(X''))) -> F(g(X''))
F(g(g(X''))) -> F(g(f(f(X''))))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(g(X''))) -> F(g(f(f(X''))))
two new Dependency Pairs are created:

F(g(g(g(X')))) -> F(g(f(g(f(f(X'))))))
F(g(g(h(X')))) -> F(g(f(h(g(X')))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(h(X')))) -> F(g(f(h(g(X')))))
F(g(g(g(X')))) -> F(g(f(g(f(f(X'))))))
F(g(g(X''))) -> F(g(X''))
F(g(g(g(X'''')))) -> F(g(g(X'''')))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(g(g(X')))) -> F(g(f(g(f(f(X'))))))
one new Dependency Pair is created:

F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Rewriting Transformation


Dependency Pairs:

F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(X''))) -> F(g(X''))
F(g(g(h(X')))) -> F(g(f(h(g(X')))))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(g(h(X')))) -> F(g(f(h(g(X')))))
one new Dependency Pair is created:

F(g(g(h(X')))) -> F(g(h(g(g(X')))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(X''))) -> F(g(X''))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(g(X''))) -> F(g(X''))
three new Dependency Pairs are created:

F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




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

F(g(g(g(X'''')))) -> F(g(g(X'''')))
four new Dependency Pairs are created:

F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(g(X'''))))) -> F(g(g(g(X'''))))
F(g(g(g(g(g(X'''''''')))))) -> F(g(g(g(g(X'''''''')))))
F(g(g(g(g(g(X'''''')))))) -> F(g(g(g(g(X'''''')))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(g(g(g(g(g(X'''''')))))) -> F(g(g(g(g(X'''''')))))
F(g(g(g(g(g(X'''''''')))))) -> F(g(g(g(g(X'''''''')))))
F(g(g(g(g(X'''))))) -> F(g(g(g(X'''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(g(g(g(g(g(X'''''')))))) -> F(g(g(g(g(X'''''')))))
F(g(g(g(g(g(X'''''''')))))) -> F(g(g(g(g(X'''''''')))))
F(g(g(g(g(X'''))))) -> F(g(g(g(X'''))))
F(g(g(g(g(X''''''))))) -> F(g(g(g(X''''''))))
F(g(g(g(X'''')))) -> F(g(g(X'''')))
F(g(g(g(X')))) -> F(g(g(f(f(f(f(X')))))))
F(g(g(g(g(X''''))))) -> F(g(g(g(X''''))))


Additionally, the following usable rules for innermost can be oriented:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(h(x1))=  0  
  POL(f(x1))=  x1  
  POL(F(x1))=  1 + x1  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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