Term Rewriting System R:
[x, y]
app(app(app(f, app(g, x)), app(s, 0)), y) -> app(app(app(f, y), y), app(g, x))
app(g, app(s, x)) -> app(s, app(g, x))
app(g, 0) -> 0

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(app(f, y), y), app(g, x))
APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(app(f, y), y)
APP(app(app(f, app(g, x)), app(s, 0)), y) -> APP(f, y)
APP(g, app(s, x)) -> APP(s, app(g, x))
APP(g, app(s, x)) -> APP(g, x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rules:


app(app(app(f, app(g, x)), app(s, 0)), y) -> app(app(app(f, y), y), app(g, x))
app(g, app(s, x)) -> app(s, app(g, x))
app(g, 0) -> 0


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳A-Transformation


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳ATrans
             ...
               →DP Problem 3
                 ↳Size-Change Principle


Dependency Pair:

G(s(x)) -> G(x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. G(s(x)) -> G(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

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