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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(f(x), y) -> H(x, y)
H(x, y) -> G(x, f(y))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pairs:

H(x, y) -> G(x, f(y))
G(f(x), y) -> H(x, y)


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))


Strategy:

innermost




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

G(f(x), y) -> H(x, y)
one new Dependency Pair is created:

G(f(x''), f(y'')) -> H(x'', f(y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pairs:

G(f(x''), f(y'')) -> H(x'', f(y''))
H(x, y) -> G(x, f(y))


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))


Strategy:

innermost




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

H(x, y) -> G(x, f(y))
one new Dependency Pair is created:

H(x', f(y'''')) -> G(x', f(f(y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pairs:

H(x', f(y'''')) -> G(x', f(f(y'''')))
G(f(x''), f(y'')) -> H(x'', f(y''))


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))


Strategy:

innermost




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

G(f(x''), f(y'')) -> H(x'', f(y''))
one new Dependency Pair is created:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 4
                 ↳Instantiation Transformation


Dependency Pairs:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))
H(x', f(y'''')) -> G(x', f(f(y'''')))


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))


Strategy:

innermost




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

H(x', f(y'''')) -> G(x', f(f(y'''')))
one new Dependency Pair is created:

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 5
                 ↳Polynomial Ordering


Dependency Pairs:

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))
G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))


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(G(x1, x2))=  x1  
  POL(H(x1, x2))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem. 



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


Dependency Pair:

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))


Rules:


g(f(x), y) -> f(h(x, y))
h(x, y) -> g(x, f(y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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