YES
proof of /TPDB/AG01_#3.15.ari
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 0 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:

   average(s(x), y) -> average(x, s(y))
   average(x, s(s(s(y)))) -> s(average(s(x), y))
   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(average(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   average(s(x), y) -> average(x, s(y))
   average(x, s(s(s(y)))) -> s(average(s(x), y))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:average_2 > s_1

and weight map:

   s_1=1
   average_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:

   average(s(x), y) -> average(x, s(y))
   average(x, s(s(s(y)))) -> s(average(s(x), y))




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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