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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


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


Dependency Pairs:

APP(app(app(f, x), y), app(s, z)) -> APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) -> APP(app(app(f, app(s, x)), x), 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 Pairs:

F(x, y, s(z)) -> F(0, 1, z)
F(0, 1, x) -> F(s(x), x, x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(x, y, s(z)) -> F(0, 1, z)
  2. F(0, 1, x) -> F(s(x), x, x)
and get the following Size-Change Graph(s):
{1} , {1}
3>3
{2} , {2}
3=2
3=3

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
3>3
{2} , {1}
3>3
{1} , {2}
3>2
3>3

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