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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rule:


g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(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)) -> G(x)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rule:


g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(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)) -> G(y)
five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rule:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes