YES
proof of Rubio_04_prov.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 66 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(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.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
ackin/2(YES,YES)
s/1(YES)
u21/2(YES,YES)
ackout/1(YES)
u22/1)YES(

Quasi precedence:
ackout_1 > [ackin_2, u21_2] > s_1


Status:
ackin_2: [1,2]
s_1: multiset status
u21_2: [2,1]
ackout_1: [1]

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)
   u21(ackout(X), Y) -> u22(ackin(Y, X))




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES