Term Rewriting System R:
[y, x, f, g]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(times, app(s, x)), y) -> APP(plus, app(app(times, x), y))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(times, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(twice, f) -> APP(app(comp, f), f)
APP(twice, f) -> APP(comp, f)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(twice, f) -> APP(app(comp, f), f)
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost




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

APP(twice, f) -> APP(app(comp, f), f)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(app(comp, f), g), x) -> APP(g, x)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost




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

APP(app(times, app(s, x)), y) -> APP(app(plus, app(app(times, x), y)), y)
two new Dependency Pairs are created:

APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')
APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')
APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')
APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost




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

APP(app(times, app(s, 0)), y'') -> APP(app(plus, 0), y'')
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(comp, f), g), x) -> APP(g, x)
APP(app(app(comp, f), g), x) -> APP(f, app(g, x))
APP(app(times, app(s, x)), y) -> APP(app(times, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(times, app(s, app(s, x''))), y'') -> APP(app(plus, app(app(plus, app(app(times, x''), y'')), y'')), y'')


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(times, 0), y) -> 0
app(app(times, app(s, x)), y) -> app(app(plus, app(app(times, x), y)), y)
app(app(app(comp, f), g), x) -> app(f, app(g, x))
app(twice, f) -> app(app(comp, f), f)


Strategy:

innermost



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