YES
(VAR X Y)
(RULES 
ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
u21(ackout(X),Y) -> u22(ackin(Y,X))
)

The TRS is an overlay system and all critical pairs are trivial, thus termination of innermost rewriting is equivalent to termination of rewriting.

Proving termination of innermost rewriting for prov:

-> Dependency pairs:
nF_ackin(s(X),s(Y)) -> nF_u21(ackin(s(X),Y),X)
nF_ackin(s(X),s(Y)) -> nF_ackin(s(X),Y)
nF_u21(ackout(X),Y) -> nF_ackin(Y,X)

-> Proof of termination for prov_1:
-> -> Dependency pairs in cycle:
nF_ackin(s(X),s(Y)) -> nF_ackin(s(X),Y)
nF_u21(ackout(X),Y) -> nF_ackin(Y,X)
nF_ackin(s(X),s(s(Y))) -> nF_u21(u21(ackin(s(X),Y),X),X)

Dependency pairs oriented using subterm criterion.

-> -> Dependency pairs in cycle:
nF_ackin(s(X),s(Y)) -> nF_ackin(s(X),Y)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in DG + base ordering
Upper bound for coeffs: 1
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Linear
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.