YES
proof of /TPDB/AotoYamada_05_023.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# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty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) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   app(id, x) -> x
   app(plus, 0) -> id
   app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

Q is empty.

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

   POL(0) = 1
   POL(app(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(id) = 1
   POL(plus) = 1
   POL(s) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   app(id, x) -> x
   app(plus, 0) -> id
   app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))




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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