Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
ackin(x1, x2)  =  ackin(x1, x2)
s(x1)  =  s(x1)
u21(x1, x2)  =  u21(x1, x2)
ackout(x1)  =  x1
u22(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[ackin₂, u21₂] > s₁

Status:

ackin₂: [1,2]
u21₂: [2,1]
s₁: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

u21(ackout(X), Y) → u22(ackin(Y, X))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(ackin(x₁, x₂)) = x₁ + x₂   
POL(ackout(x₁)) = x₁   
POL(u21(x₁, x₂)) = 1 + x₁ + x₂   
POL(u22(x₁)) = x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

u21(ackout(X), Y) → u22(ackin(Y, X))

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE