/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 79 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) 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:
[ackin_2, u21_2] > s_1


Status:
ackin_2: [1,2]
s_1: [1]
u21_2: [2,1]

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)




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(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.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:ackin_2 > u21_2 > u22_1 > ackout_1

and weight map:

   ackout_1=1
   u22_1=1
   u21_2=0
   ackin_2=0

The variable weight is 1With 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))




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES