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:

ack_in(0, n) → ack_out(s(n))
ack_in(s(m), 0) → u11(ack_in(m, s(0)))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACK_IN(s(m), 0) → U11(ack_in(m, s(0)))
ACK_IN(s(m), 0) → ACK_IN(m, s(0))
ACK_IN(s(m), s(n)) → U21(ack_in(s(m), n), m)
ACK_IN(s(m), s(n)) → ACK_IN(s(m), n)
U21(ack_out(n), m) → U22(ack_in(m, n))
U21(ack_out(n), m) → ACK_IN(m, n)

The TRS R consists of the following rules:

ack_in(0, n) → ack_out(s(n))
ack_in(s(m), 0) → u11(ack_in(m, s(0)))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACK_IN(s(m), s(n)) → U21(ack_in(s(m), n), m)
U21(ack_out(n), m) → ACK_IN(m, n)
ACK_IN(s(m), 0) → ACK_IN(m, s(0))
ACK_IN(s(m), s(n)) → ACK_IN(s(m), n)

The TRS R consists of the following rules:

ack_in(0, n) → ack_out(s(n))
ack_in(s(m), 0) → u11(ack_in(m, s(0)))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
0  =  0
ack_out(x1)  =  ack_out
s(x1)  =  s(x1)

From the DPs we obtained the following set of size-change graphs:

• U21(ack_out(n), m) → ACK_IN(m, n) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 2 >= 1

• ACK_IN(s(m), s(n)) → U21(ack_in(s(m), n), m) (allowed arguments on rhs = {2})
  The graph contains the following edges 1 > 2

• ACK_IN(s(m), s(n)) → ACK_IN(s(m), n) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 >= 1, 2 > 2

• ACK_IN(s(m), 0) → ACK_IN(m, s(0)) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(6) TRUE