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

Termination of R to be shown. 



   R
     ↳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)
ACK(s(x), y) -> F(x, x)
F(s(x), y) -> F(x, s(x))
F(x, s(y)) -> F(y, x)
F(x, y) -> ACK(x, y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

F(x, y) -> ACK(x, y)
F(x, s(y)) -> F(y, x)
F(s(x), y) -> F(x, s(x))
ACK(s(x), y) -> F(x, x)
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))
ack(s(x), y) -> f(x, x)
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)





We number the DPs as follows:
  1. F(x, y) -> ACK(x, y)
  2. F(x, s(y)) -> F(y, x)
  3. F(s(x), y) -> F(x, s(x))
  4. ACK(s(x), y) -> F(x, x)
  5. ACK(s(x), s(y)) -> ACK(s(x), y)
  6. ACK(s(x), s(y)) -> ACK(x, ack(s(x), y))
  7. ACK(s(x), 0) -> ACK(x, s(0))
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2=2
{3, 2} , {3, 2}
1=2
2>1
{3, 2} , {3, 2}
1>1
1=2
{4} , {4}
1>1
1>2
{5} , {5}
1=1
2>2
{6} , {6}
1>1
{7} , {7}
1>1

which lead(s) to this/these maximal multigraph(s):
{5} , {5}
1=1
2>2
{6} , {6}
1>1
{3, 2} , {3, 2}
1>1
2>2
{7} , {5}
1>1
{6} , {5}
1>1
{3, 2} , {3, 2}
1>1
1>2
{4} , {1}
1>1
1>2
{3, 2} , {3, 2}
2>1
2>2
{5} , {6}
1>1
{1} , {4}
1>1
1>2
{5} , {7}
1>1
{3, 2} , {4}
2>1
2>2
{4} , {7}
1>1
{6} , {1}
1>1
1>2
{7} , {1}
1>1
1>2
{4} , {6}
1>1
{6} , {7}
1>1
{5} , {1}
1>1
1>2
{5} , {5}
1>1
{3, 2} , {4}
1>1
1>2
{4} , {5}
1>1
1>2
{1} , {3, 2}
1>1
1>2
{4} , {5}
1>1
{7} , {5}
1>1
1>2
{6} , {5}
1>1
1>2
{5} , {5}
1>1
1>2

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