mirror of
https://github.com/GaloisInc/cryptol.git
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93 lines
2.2 KiB
Plaintext
93 lines
2.2 KiB
Plaintext
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// == Q is a commutative ring ==
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property QaddUnit (x : Rational) =
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x + 0 == x
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property QaddComm (x:Rational) (y:Rational) =
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x + y == y + x
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property QaddAssoc (x:Rational) (y:Rational) (z:Rational) =
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x + (y + z) == (x + y) + z
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property Qneg (x: Rational) = x + negate x == 0
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property QmulUnit (x : Rational) =
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x * 1 == x
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property QmulComm (x:Rational) (y:Rational) =
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x * y == y * x
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property QmulAssoc (x:Rational) (y:Rational) (z:Rational) =
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x * (y * z) == (x * y) * z
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// Distributivity in Q
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property Qdistrib (x:Rational) (y:Rational) (z:Rational) =
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x * (y + z) == x*y + x*z
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// == Q is a field ==
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property Qrecip (x: Rational) = x == 0 \/ x * recip x == 1
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property QdivisionEquiv (x : Rational) (y : Rational) =
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y == 0 \/ x /. y == x * (recip y)
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// == Q is a total order ==
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property QordEquiv1 (x : Rational) (y : Rational) =
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(x <= y) == ~(y < x)
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property QordEquiv2 (x : Rational) (y : Rational) =
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(x <= y) == (x == y \/ x < y)
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property QordTrans (x : Rational) (y : Rational) (z : Rational) =
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x < y ==> y < z ==> x < z
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property QordIrreflexive (x : Rational) =
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~(x < x)
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property QordExclusive (x : Rational) (y:Rational) =
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~(x < y) || ~(y < x)
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property Qtrichotomy (x : Rational) (y : Rational) =
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x < y \/ x == y \/ x > y
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// == Q is an ordered field
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property QordCompatible (x : Rational) (y : Rational) (z:Rational) =
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x < y ==> x+z < y+z
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property QordPositive (x : Rational) (y : Rational) =
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0 < x ==> 0 < y ==> 0 < x*y
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// == Q is a dense total order ==
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property Qdense (x : Rational) (y : Rational) = x < y ==> x < mid /\ mid < y
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where
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mid = (x + y) /. 2
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// == Integer division rounds down ==
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property intDivDown (x : Integer) (y:Integer) =
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y == 0 \/ fromInteger (x / y) <= ratio x y
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// == correctness of floor and ceiling
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// floor(x) is an integer below x
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property floorCorrect1 (x:Rational) =
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fromInteger (floor x) <= x
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// floor(x) is the largest integer below x
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property floorCorrect2 (x:Rational) (y:Integer) =
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fromInteger y <= x ==>
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y <= floor x
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// ceiling(x) is an integer above x
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property ceilingCorrect1 (x:Rational) =
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fromInteger (ceiling x) >= x
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// ceiling(x) is the smallest integer above x
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property ceilingCorrect2 (x:Rational) (y:Integer) =
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fromInteger y >= x ==>
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y >= ceiling x
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