YES
proof of SK90_2.30.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 64 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:

   not(x) -> xor(x, true)
   implies(x, y) -> xor(and(x, y), xor(x, true))
   or(x, y) -> xor(and(x, y), xor(x, y))
   =(x, y) -> xor(x, xor(y, true))

Q is empty.

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

   POL(=(x_1, x_2)) = 2 + x_1 + x_2
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(implies(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(not(x_1)) = 2*x_1
   POL(or(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(true) = 0
   POL(xor(x_1, x_2)) = x_1 + x_2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   implies(x, y) -> xor(and(x, y), xor(x, true))
   or(x, y) -> xor(and(x, y), xor(x, y))
   =(x, y) -> xor(x, xor(y, true))




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

   not(x) -> xor(x, true)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:true > not_1 > xor_2

and weight map:

   true=1
   not_1=1
   xor_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:

   not(x) -> xor(x, true)




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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