(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) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) 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))
The set Q consists of the following terms:
ackin(s(x0), s(x1))
u21(ackout(x0), x1)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACKIN(s(X), s(Y)) → U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) → ACKIN(s(X), Y)
U21(ackout(X), Y) → ACKIN(Y, X)
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))
The set Q consists of the following terms:
ackin(s(x0), s(x1))
u21(ackout(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACKIN(s(X), s(Y)) → ACKIN(s(X), Y)
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))
The set Q consists of the following terms:
ackin(s(x0), s(x1))
u21(ackout(x0), x1)
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.
ACKIN(s(X), s(Y)) → ACKIN(s(X), Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACKIN(
x1,
x2) =
ACKIN(
x2)
s(
x1) =
s(
x1)
ackin(
x1,
x2) =
ackin
u21(
x1,
x2) =
x1
ackout(
x1) =
ackout
u22(
x1) =
x1
Recursive path order with status [RPO].
Quasi-Precedence:
[ACKIN1, s1] > ackin
ackout > ackin
Status:
ACKIN1: [1]
s1: multiset
ackin: []
ackout: multiset
The following usable rules [FROCOS05] were oriented:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
u21(ackout(X), Y) → u22(ackin(Y, X))
(8) Obligation:
Q DP problem:
P is empty.
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))
The set Q consists of the following terms:
ackin(s(x0), s(x1))
u21(ackout(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE