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
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Rw


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 AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> x2
app(x1, x2) -> app(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Rw


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
         ↳AFS
       →DP Problem 2
         ↳Rewriting Transformation


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




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

APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, app(p, app(s, x))))
one new Dependency Pair is created:

APP(fact, app(s, x)) -> APP(app(*, app(s, x)), app(fact, x))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Rw
           →DP Problem 4
             ↳Rewriting Transformation


Dependency Pairs:

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




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

APP(fact, app(s, x)) -> APP(fact, app(p, app(s, x)))
one new Dependency Pair is created:

APP(fact, app(s, x)) -> APP(fact, x)

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

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



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