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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


app(app(app(quot, 0), app(s, y)), app(s, z)) -> 0
app(app(app(quot, app(s, x)), app(s, y)), z) -> app(app(app(quot, x), y), z)
app(app(app(quot, x), 0), app(s, z)) -> app(s, app(app(app(quot, x), app(s, z)), app(s, z)))


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 Pairs:

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


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:

QUOT(x, 0, s(z)) -> QUOT(x, s(z), s(z))
QUOT(s(x), s(y), z) -> QUOT(x, y, z)


Rule:

none


Strategy:

innermost




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

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