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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(fix, f), x) -> APP(f, app(fix, f))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))


Rules:


app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)


Strategy:

innermost




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

APP(app(app(subst, f), g), x) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(app(subst, app(app(subst, f''), g'')), g), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(app(subst, app(fix, f'')), g), x'') -> APP(app(fix, f''), x'')

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(subst, app(fix, f'')), g), x'') -> APP(app(fix, f''), x'')
APP(app(app(subst, app(app(subst, f''), g'')), g), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(fix, f), x) -> APP(f, app(fix, f))


Rules:


app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)


Strategy:

innermost




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

APP(app(app(subst, f), g), x) -> APP(g, x)
four new Dependency Pairs are created:

APP(app(app(subst, f), app(app(subst, f''), g'')), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(app(subst, f), app(fix, f'')), x'') -> APP(app(fix, f''), x'')
APP(app(app(subst, f), app(app(subst, app(app(subst, f''''), g'''')), g'')), x') -> APP(app(app(subst, app(app(subst, f''''), g'''')), g''), x')
APP(app(app(subst, f), app(app(subst, app(fix, f'''')), g'')), x') -> APP(app(app(subst, app(fix, f'''')), g''), x')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(subst, f), app(app(subst, app(fix, f'''')), g'')), x') -> APP(app(app(subst, app(fix, f'''')), g''), x')
APP(app(app(subst, f), app(app(subst, app(app(subst, f''''), g'''')), g'')), x') -> APP(app(app(subst, app(app(subst, f''''), g'''')), g''), x')
APP(app(app(subst, f), app(fix, f'')), x'') -> APP(app(fix, f''), x'')
APP(app(app(subst, f), app(app(subst, f''), g'')), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(app(subst, app(app(subst, f''), g'')), g), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, app(fix, f'')), g), x'') -> APP(app(fix, f''), x'')


Rules:


app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes