Term Rewriting System R:
[x, ys, f, xs]
app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(consif, true), x), ys) -> APP(app(cons, x), ys)
APP(app(app(consif, true), x), ys) -> APP(cons, x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(app(consif, app(f, x)), x), app(app(filter, f), xs))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(consif, app(f, x)), x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(consif, app(f, x))
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)


Dependency Pairs:

APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)


Rules:


app(app(app(consif, true), x), ys) -> app(app(cons, x), ys)
app(app(app(consif, false), x), ys) -> ys
app(app(filter, f), nil) -> nil
app(app(filter, f), app(app(cons, x), xs)) -> app(app(app(consif, app(f, x)), x), app(app(filter, f), xs))


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Size-Change Principle


Dependency Pairs:

APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. APP(app(filter, f), app(app(cons, x), xs)) -> APP(app(filter, f), xs)
  2. APP(app(filter, f), app(app(cons, x), xs)) -> APP(f, x)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1=1
2>2
{1, 2} , {1, 2}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1
2>2
{1, 2} , {1, 2}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
app(x1, x2) -> app(x1, x2)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes