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))

Innermost 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 two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Nar


Dependency Pair:

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))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(plus, x), app(s, y)) -> APP(app(plus, x), y)


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(plus)=  0  
  POL(0)=  0  
  POL(times)=  0  
  POL(s)=  0  
  POL(app(x1, x2))=  1 + x2  
  POL(APP(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Nar


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

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)


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))


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Narrowing Transformation


Dependency Pairs:

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


Strategy:

innermost




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

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



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


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))


Strategy:

innermost



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