Term Rewriting System R:
[x, y]
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(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)
F'(s(x), y, y) -> F'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(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))
f'(s(x), y, y) -> f'(y, x, s(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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