Term Rewriting System R:
[y, x]
app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(ack, 0), y) -> APP(succ, y)
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))
APP(app(ack, app(succ, x)), y) -> APP(ack, x)
APP(app(ack, app(succ, x)), y) -> APP(succ, 0)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(ack, x)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳A-Transformation


Dependency Pairs:

APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, app(succ, x)), y)
APP(app(ack, app(succ, x)), app(succ, y)) -> APP(app(ack, x), app(app(ack, app(succ, x)), y))
APP(app(ack, app(succ, x)), y) -> APP(app(ack, x), app(succ, 0))


Rules:


app(app(ack, 0), y) -> app(succ, y)
app(app(ack, app(succ, x)), y) -> app(app(ack, x), app(succ, 0))
app(app(ack, app(succ, x)), app(succ, y)) -> app(app(ack, x), app(app(ack, app(succ, x)), y))


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
         ↳ATrans
           →DP Problem 2
             ↳Size-Change Principle


Dependency Pairs:

ACK(succ(x), succ(y)) -> ACK(succ(x), y)
ACK(succ(x), succ(y)) -> ACK(x, ack(succ(x), y))
ACK(succ(x), y) -> ACK(x, succ(0))


Rules:


ack(0, y) -> succ(y)
ack(succ(x), y) -> ack(x, succ(0))
ack(succ(x), succ(y)) -> ack(x, ack(succ(x), y))


Strategy:

innermost




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

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

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems.

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