Term Rewriting System R:
[X, Y]
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)

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:

ACKIN(s(X), 0) -> U11(ackin(X, s(0)))
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> U22(ackin(Y, X))
U21(ackout(X), Y) -> ACKIN(Y, X)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), 0) -> ACKIN(X, s(0))


Rules:


ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)


Strategy:

innermost




We number the DPs as follows:
  1. ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
  2. U21(ackout(X), Y) -> ACKIN(Y, X)
  3. ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
  4. ACKIN(s(X), 0) -> ACKIN(X, s(0))
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2
{2} , {2}
1>2
2=1
{3} , {3}
1>2
{4} , {4}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2
{3} , {2}
1>1
{1} , {4}
1>1
{4} , {1}
1>1
{2} , {3}
2>2
{4} , {2}
1>1
{3} , {1}
1>1
{1} , {1}
1>1
{1} , {2}
1>1
{3} , {4}
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)
ackout(x1) -> ackout(x1)

We obtain no new DP problems.

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