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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳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))





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):
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 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:
succ(x1) -> succ(x1)

We obtain no new DP problems.

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