Term Rewriting System R:
[x, y]
app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(fact, 0) -> APP(s, 0)
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
APP(fact, app(s, x)) -> APP(*, app(s, x))
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(p, app(s, x))
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(app(*, app(s, x)), y) -> APP(+, app(app(*, x), y))
APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(*, x)
APP(app(+, x), app(s, y)) -> APP(s, app(app(+, x), y))
APP(app(+, x), app(s, y)) -> APP(app(+, x), y)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(+, x), app(s, y)) -> APP(app(+, 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(0)=  0  
  POL(fact)=  0  
  POL(*)=  0  
  POL(s)=  0  
  POL(app(x1, x2))=  1 + x2  
  POL(+)=  0  
  POL(APP(x1, x2))=  x2  
  POL(p)=  0  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(*, app(s, x)), y) -> APP(app(*, x), y)
APP(app(*, app(s, x)), y) -> APP(app(+, app(app(*, x), y)), y)
APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))


Rules:


app(p, app(s, x)) -> x
app(fact, 0) -> app(s, 0)
app(fact, app(s, x)) -> app(app(*, app(s, x)), app(fact, app(p, app(s, x))))
app(app(*, 0), y) -> 0
app(app(*, app(s, x)), y) -> app(app(+, app(app(*, x), y)), y)
app(app(+, x), 0) -> x
app(app(+, x), app(s, y)) -> app(s, app(app(+, x), y))


Strategy:

innermost



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