mirror of
https://github.com/GaloisInc/cryptol.git
synced 2024-12-14 08:43:06 +03:00
93 lines
2.2 KiB
Plaintext
93 lines
2.2 KiB
Plaintext
// == Q is a commutative ring ==
|
|
|
|
property QaddUnit (x : Rational) =
|
|
x + 0 == x
|
|
|
|
property QaddComm (x:Rational) (y:Rational) =
|
|
x + y == y + x
|
|
|
|
property QaddAssoc (x:Rational) (y:Rational) (z:Rational) =
|
|
x + (y + z) == (x + y) + z
|
|
|
|
property Qneg (x: Rational) = x + negate x == 0
|
|
|
|
property QmulUnit (x : Rational) =
|
|
x * 1 == x
|
|
|
|
property QmulComm (x:Rational) (y:Rational) =
|
|
x * y == y * x
|
|
|
|
property QmulAssoc (x:Rational) (y:Rational) (z:Rational) =
|
|
x * (y * z) == (x * y) * z
|
|
|
|
// Distributivity in Q
|
|
property Qdistrib (x:Rational) (y:Rational) (z:Rational) =
|
|
x * (y + z) == x*y + x*z
|
|
|
|
// == Q is a field ==
|
|
|
|
property Qrecip (x: Rational) = x == 0 \/ x * recip x == 1
|
|
|
|
property QdivisionEquiv (x : Rational) (y : Rational) =
|
|
y == 0 \/ x /. y == x * (recip y)
|
|
|
|
// == Q is a total order ==
|
|
|
|
property QordEquiv1 (x : Rational) (y : Rational) =
|
|
(x <= y) == ~(y < x)
|
|
|
|
property QordEquiv2 (x : Rational) (y : Rational) =
|
|
(x <= y) == (x == y \/ x < y)
|
|
|
|
property QordTrans (x : Rational) (y : Rational) (z : Rational) =
|
|
x < y ==> y < z ==> x < z
|
|
|
|
property QordIrreflexive (x : Rational) =
|
|
~(x < x)
|
|
|
|
property QordExclusive (x : Rational) (y:Rational) =
|
|
~(x < y) || ~(y < x)
|
|
|
|
property Qtrichotomy (x : Rational) (y : Rational) =
|
|
x < y \/ x == y \/ x > y
|
|
|
|
|
|
// == Q is an ordered field
|
|
|
|
property QordCompatible (x : Rational) (y : Rational) (z:Rational) =
|
|
x < y ==> x+z < y+z
|
|
|
|
property QordPositive (x : Rational) (y : Rational) =
|
|
0 < x ==> 0 < y ==> 0 < x*y
|
|
|
|
// == Q is a dense total order ==
|
|
property Qdense (x : Rational) (y : Rational) = x < y ==> x < mid /\ mid < y
|
|
where
|
|
mid = (x + y) /. 2
|
|
|
|
|
|
// == Integer division rounds down ==
|
|
property intDivDown (x : Integer) (y:Integer) =
|
|
y == 0 \/ fromInteger (x / y) <= ratio x y
|
|
|
|
|
|
// == correctness of floor and ceiling
|
|
|
|
// floor(x) is an integer below x
|
|
property floorCorrect1 (x:Rational) =
|
|
fromInteger (floor x) <= x
|
|
|
|
// floor(x) is the largest integer below x
|
|
property floorCorrect2 (x:Rational) (y:Integer) =
|
|
fromInteger y <= x ==>
|
|
y <= floor x
|
|
|
|
// ceiling(x) is an integer above x
|
|
property ceilingCorrect1 (x:Rational) =
|
|
fromInteger (ceiling x) >= x
|
|
|
|
// ceiling(x) is the smallest integer above x
|
|
property ceilingCorrect2 (x:Rational) (y:Integer) =
|
|
fromInteger y >= x ==>
|
|
y >= ceiling x
|