Term Rewriting System R:
[x, y, z]
app(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> app(app(app(app(f, app(app(g, x), y)), app(app(g, x), y)), app(app(g, x), y)), app(h, x))
app(app(g, 0), 1) -> 0
app(app(g, 0), 1) -> 1
app(h, app(app(g, x), y)) -> app(h, x)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> APP(app(app(app(f, app(app(g, x), y)), app(app(g, x), y)), app(app(g, x), y)), app(h, x))
APP(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> APP(app(app(f, app(app(g, x), y)), app(app(g, x), y)), app(app(g, x), y))
APP(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> APP(app(f, app(app(g, x), y)), app(app(g, x), y))
APP(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> APP(f, app(app(g, x), y))
APP(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> APP(h, x)
APP(h, app(app(g, x), y)) -> APP(h, x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

APP(h, app(app(g, x), y)) -> APP(h, x)


Rules:


app(app(app(app(f, 0), 1), app(app(g, x), y)), z) -> app(app(app(app(f, app(app(g, x), y)), app(app(g, x), y)), app(app(g, x), y)), app(h, x))
app(app(g, 0), 1) -> 0
app(app(g, 0), 1) -> 1
app(h, app(app(g, x), y)) -> app(h, x)


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
             ↳A-Transformation


Dependency Pair:

APP(h, app(app(g, x), y)) -> APP(h, 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:

H(g(x, y)) -> H(x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. H(g(x, y)) -> H(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:
g(x1, x2) -> g(x1, x2)

We obtain no new DP problems.

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