YES
proof of /TPDB/Der95_02.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# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty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:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

Q is empty.

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

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

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z




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

   i(1) -> 1
   i(i(x)) -> x

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:i_1 > 1

and weight map:

   1=1
   i_1=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:

   i(1) -> 1
   i(i(x)) -> 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