YES

Problem 1: 

(VAR v_NonEmpty:S m:S n:S)
(RULES
ack_in(0,n:S) -> ack_out(s(n:S))
ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
u11(ack_out(n:S)) -> ack_out(n:S)
u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
u22(ack_out(n:S)) -> ack_out(n:S)
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACK_IN(s(m:S),0) -> ACK_IN(m:S,s(0))
 ACK_IN(s(m:S),0) -> U11(ack_in(m:S,s(0)))
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
 ACK_IN(s(m:S),s(n:S)) -> U21(ack_in(s(m:S),n:S),m:S)
 U21(ack_out(n:S),m:S) -> ACK_IN(m:S,n:S)
 U21(ack_out(n:S),m:S) -> U22(ack_in(m:S,n:S))
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACK_IN(s(m:S),0) -> ACK_IN(m:S,s(0))
 ACK_IN(s(m:S),0) -> U11(ack_in(m:S,s(0)))
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
 ACK_IN(s(m:S),s(n:S)) -> U21(ack_in(s(m:S),n:S),m:S)
 U21(ack_out(n:S),m:S) -> ACK_IN(m:S,n:S)
 U21(ack_out(n:S),m:S) -> U22(ack_in(m:S,n:S))
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACK_IN(s(m:S),0) -> ACK_IN(m:S,s(0))
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
 ACK_IN(s(m:S),s(n:S)) -> U21(ack_in(s(m:S),n:S),m:S)
 U21(ack_out(n:S),m:S) -> ACK_IN(m:S,n:S)
->->-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 ACK_IN(s(m:S),0) -> ACK_IN(m:S,s(0))
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
 ACK_IN(s(m:S),s(n:S)) -> U21(ack_in(s(m:S),n:S),m:S)
 U21(ack_out(n:S),m:S) -> ACK_IN(m:S,n:S)
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)
->Projection:
 pi(ACK_IN) = 1
 pi(U21) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
 U21(ack_out(n:S),m:S) -> ACK_IN(m:S,n:S)
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
->->-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 ACK_IN(s(m:S),s(n:S)) -> ACK_IN(s(m:S),n:S)
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)
->Projection:
 pi(ACK_IN) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 ack_in(0,n:S) -> ack_out(s(n:S))
 ack_in(s(m:S),0) -> u11(ack_in(m:S,s(0)))
 ack_in(s(m:S),s(n:S)) -> u21(ack_in(s(m:S),n:S),m:S)
 u11(ack_out(n:S)) -> ack_out(n:S)
 u21(ack_out(n:S),m:S) -> u22(ack_in(m:S,n:S))
 u22(ack_out(n:S)) -> ack_out(n:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.