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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

We obtain no new DP problems.

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