Term Rewriting System R:
[x, y, z]
app(app(times, x), app(app(plus, y), app(s, z))) -> app(app(plus, app(app(times, x), app(app(plus, y), app(app(times, app(s, z)), 0)))), app(app(times, x), app(s, z)))
app(app(times, x), 0) -> 0
app(app(times, x), app(s, y)) -> app(app(plus, app(app(times, x), y)), x)
app(app(plus, x), 0) -> x
app(app(plus, x), app(s, y)) -> app(s, app(app(plus, x), y))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

APP(app(times, x), app(app(plus, y), app(s, z))) -> APP(app(times, x), app(app(plus, y), app(app(times, app(s, z)), 0)))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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





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

APP(app(times, x), app(app(plus, y), app(s, z))) -> APP(app(plus, y), app(app(times, app(s, z)), 0))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

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


Rules:


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




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