(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) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ACK_IN(s(m), 0) → ACK_IN(m, s(0))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACK_IN(
x0,
x1,
x2) =
ACK_IN(
x1)
U21(
x0,
x1,
x2) =
U21(
x1,
x2)
Tags:
ACK_IN has argument tags [0,3,0] and root tag 0
U21 has argument tags [4,3,3] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACK_IN(
x1,
x2) =
ACK_IN(
x1,
x2)
s(
x1) =
s(
x1)
U21(
x1,
x2) =
U21
ack_in(
x1,
x2) =
x1
ack_out(
x1) =
ack_out
0 =
0
u11(
x1) =
u11
u21(
x1,
x2) =
x2
u22(
x1) =
x1
Recursive path order with status [RPO].
Quasi-Precedence:
ACKIN2 > [s1, U21, 0, u11] > ackout
Status:
ACKIN2: [2,1]
s1: [1]
U21: multiset
ackout: []
0: multiset
u11: []
The following usable rules [FROCOS05] were oriented:
none
(6) 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), 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.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ACK_IN(s(m), s(n)) → U21(ack_in(s(m), n), m)
U21(ack_out(n), m) → ACK_IN(m, n)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACK_IN(
x0,
x1,
x2) =
ACK_IN(
x0,
x1)
U21(
x0,
x1,
x2) =
U21(
x0)
Tags:
ACK_IN has argument tags [0,0,5] and root tag 0
U21 has argument tags [0,0,0] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACK_IN(
x1,
x2) =
ACK_IN
s(
x1) =
s(
x1)
U21(
x1,
x2) =
x2
ack_in(
x1,
x2) =
ack_in
ack_out(
x1) =
ack_out
0 =
0
u11(
x1) =
u11(
x1)
u21(
x1,
x2) =
u21(
x1,
x2)
u22(
x1) =
u22
Recursive path order with status [RPO].
Quasi-Precedence:
[s1, ackin, ackout, u22] > 0 > ACKIN
[s1, ackin, ackout, u22] > u111 > ACKIN
[s1, ackin, ackout, u22] > u212 > ACKIN
Status:
ACKIN: multiset
s1: multiset
ackin: multiset
ackout: multiset
0: multiset
u111: [1]
u212: multiset
u22: []
The following usable rules [FROCOS05] were oriented:
none
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
ACK_IN(s(m), s(n)) → ACK_IN(s(m), n)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACK_IN(
x0,
x1,
x2) =
ACK_IN(
x0,
x1)
Tags:
ACK_IN has argument tags [1,2,0] and root tag 0
Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
ACK_IN(
x1,
x2) =
x2
s(
x1) =
s(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
trivial
Status:
s1: multiset
The following usable rules [FROCOS05] were oriented:
none
(10) Obligation:
Q DP problem:
P is empty.
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.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) TRUE