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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

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:

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


Rules:


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


Strategy:

innermost




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

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

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

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


Rules:


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


Strategy:

innermost




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

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

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

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:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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:
{H, G}

resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> G(x1, x2)
H(x1, x2) -> H(x1, x2)
f(x1) -> f(x1)


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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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