cryptol-2.9.0: lib/Cryptol.cry
/*
* Copyright (c) 2013-2020 Galois, Inc.
* Distributed under the terms of the BSD3 license (see LICENSE file)
*/
module Cryptol where
infixr 5 ==>
infixr 10 \/
infixr 15 /\
infix 20 ==, ===, !=, !==
infix 30 >, >=, <, <=, <$, >$, <=$, >=$
infixr 40 ||
infixl 45 ^
infixr 50 &&
infixr 60 #
infixl 70 <<, <<<, >>, >>>, >>$
infixl 80 +, -
infixl 90 *, /, %, /$, %$, %^, /^
infixr 95 ^^
infixl 100 @, @@, !, !!
// Base types -----------------------------------------------------------------------
/** The type of boolean values. */
primitive type Bit : *
/** The type of unbounded integers. */
primitive type Integer : *
/**
* 'Z n' is the type of integers, modulo 'n'.
*
* The values of 'Z n' may be thought of as equivalance
* classes of integers according to the equivalence
* 'x ~ y' iff 'n' divides 'x - y'. 'Z n' naturally
* forms a ring, but does not support integral division
* or indexing.
*
* However, you may use the 'fromZ' operation
* to project values in 'Z n' into the integers if such operations
* are required. This will compute the reduced representative
* of the equivalance class. In other words, 'fromZ' computes
* the (unique) integer value 'i' where '0 <= i < n' and
* 'i' is in the given equivalance class.
*/
primitive type {n : #} (fin n, n >= 1) => Z n : *
/**
* 'Rational' is the type of rational numbers.
* Rational numbers form a Field (and thus a Ring).
*
* The 'ratio' operation may be used to directly create
* rational values from as a ratio of integers, or
* the 'fromInteger' method and the field operations
* can be used.
*/
primitive type Rational : *
type Bool = Bit
type Word n = [n]
type Char = [8]
type String n = [n]Char
// Numeric operators and constraints ----------------------------------------------
/** A numeric type representing infinity. */
primitive type inf : #
/** Assert that two numeric types are equal. */
primitive type (==) : # -> # -> Prop
/** Assert that two numeric types are different. */
primitive type (!=) : # -> # -> Prop
/** Assert that the first numeric type is larger than, or equal to the second.*/
primitive type (>=) : # -> # -> Prop
/** Assert that a numeric type is a proper natural number (not 'inf'). */
primitive type fin : # -> Prop
/** Add numeric types. */
primitive type (+) : # -> # -> #
/** Multiply numeric types. */
primitive type (*) : # -> # -> #
/** Subtract numeric types. */
primitive type
{m : #, n : # }
(fin n, m >= n) =>
m - n : #
/** Divide numeric types, rounding down. */
primitive type
{ m : #, n : # }
(fin m, n >= 1) =>
m / n : #
/** Remainder of numeric type division. */
primitive type
{ m : #, n : # }
(fin m, n >= 1) =>
m % n : #
/** Exponentiate numeric types. */
primitive type (^^) : # -> # -> #
/** The number of bits required to represent the value of a numeric type. */
primitive type width : # -> #
/**
* Define the base 2 logarithm function in terms of width
*/
type lg2 n = width (max n 1 - 1)
/** The smaller of two numeric types. */
primitive type min : # -> # -> #
/** The larger of two numeric types. */
primitive type max : # -> # -> #
/** Divide numeric types, rounding up. */
primitive type
{ m : #, n : # }
(fin m, fin n, n >= 1) =>
m /^ n : #
/** How much we need to add to make a proper multiple of the second argument. */
primitive type
{ m : #, n : # }
(fin m, fin n, n >= 1) =>
m %^ n : #
/** The length of an enumeration. */
primitive type
{ start : #, next : #, last : # }
(fin start, fin next, fin last, start != next) =>
lengthFromThenTo start next last : #
/**
* Assert that the first numeric type is less than or equal to the second.
*/
type constraint i <= j = (j >= i)
/**
* Assert that the first numeric type is greater than the second.
*/
type constraint i > j = i >= j + 1
/**
* Assert that the first numeric type is less than the second.
*/
type constraint i < j = j >= i + 1
// The Literal class ----------------------------------------------------
/** 'Literal n a' asserts that type 'a' contains the number 'n'. */
primitive type Literal : # -> * -> Prop
/**
* The value corresponding to a numeric type.
*/
primitive number : {val, rep} Literal val rep => rep
/**
* An alternative name for 'number', present for backward compatibility.
*/
demote : {val, rep} Literal val rep => rep
demote = number`{val}
/**
* Return the length of a sequence. Note that the result depends only
* on the type of the argument, not its value.
*/
length : {n, a, b} (fin n, Literal n b) => [n]a -> b
length _ = `n
/**
* A finite sequence counting up from 'first' to 'last'.
*
* '[a..b]' is syntactic sugar for 'fromTo`{first=a,last=b}'.
*/
primitive fromTo : {first, last, a} (fin last, last >= first,
Literal first a, Literal last a) =>
[1 + (last - first)]a
/**
* A finite arithmetic sequence starting with 'first' and 'next',
* stopping when the values reach or would skip over 'last'.
*
* '[a,b..c]' is syntactic sugar for 'fromThenTo`{first=a,next=b,last=c}'.
*/
primitive fromThenTo : {first, next, last, a, len}
( fin first, fin next, fin last
, Literal first a, Literal next a, Literal last a
, first != next
, lengthFromThenTo first next last == len) => [len]a
// Fractional Literals ---------------------
/** 'FLiteral m n r a' asserts that the type `a' contains the
fraction `m/n`. The flag `r` indicates if we should round (`r >= 1`)
or report an error if the number can't be represented exactly. */
primitive type FLiteral : # -> # -> # -> * -> Prop
/** A fractional literal corresponding to `m/n` */
primitive
fraction : { m, n, r, a } FLiteral m n r a => a
// The Zero class -------------------------------------------------------
/** Value types that have a notion of 'zero'. */
primitive type Zero : * -> Prop
/**
* Gives an arbitrary shaped value whose bits are all False.
* ~zero likewise gives an arbitrary shaped value whose bits are all True.
*/
primitive zero : {a} (Zero a) => a
// The Logic class ------------------------------------------------------
/** Value types that support logical operations. */
primitive type Logic : * -> Prop
/**
* Logical 'and' over bits. Extends element-wise over sequences, tuples.
*/
primitive (&&) : {a} (Logic a) => a -> a -> a
/**
* Logical 'or' over bits. Extends element-wise over sequences, tuples.
*/
primitive (||) : {a} (Logic a) => a -> a -> a
/**
* Logical 'exclusive or' over bits. Extends element-wise over sequences, tuples.
*/
primitive (^) : {a} (Logic a) => a -> a -> a
/**
* Bitwise complement. The prefix notation '~ x'
* is syntactic sugar for 'complement x'.
*/
primitive complement : {a} (Logic a) => a -> a
// The Ring class -------------------------------------------------------
/**
* Value types that support ring addition and multiplication.
*
* Floating-point values are only approximately a ring, but
* nonetheless inhabit this class.
*/
primitive type Ring : * -> Prop
/**
* Converts an unbounded integer to a value in a Ring. When converting
* to the bitvector type [n], the value is reduced modulo 2^^n. Likewise,
* when converting to Z n, the value is reduced modulo n. When converting
* to a floating-point value, the value is rounded to the nearest
* representable value.
*/
primitive fromInteger : {a} (Ring a) => Integer -> a
/**
* Add two values.
* * For type [n], addition is modulo 2^^n.
* * Structured values are added element-wise.
*/
primitive (+) : {a} (Ring a) => a -> a -> a
/**
* Subtract two values.
* * For type [n], subtraction is modulo 2^^n.
* * Structured values are subtracted element-wise.
* * Satisfies 'a - b = a + negate b'.
* See also: 'negate'.
*/
primitive (-) : {a} (Ring a) => a -> a -> a
/**
* Multiply two values.
* * For type [n], multiplication is modulo 2^^n.
* * Structured values are multiplied element-wise.
*/
primitive (*) : {a} (Ring a) => a -> a -> a
/**
* Returns the additive inverse of its argument.
* Over structured values, operates element-wise.
* The prefix notation '- x' is syntactic sugar
* for 'negate x'.
*
* Satisfies 'a + negate a = 0'.
* Satisfies 'negate a = ~a + 1' for bitvector values.
*/
primitive negate : {a} (Ring a) => a -> a
// The Integral class -------------------------------------------------
/**
* Value types that correspond to a segment of the
* integers. These types support integer division and
* modulus, indexing into sequences, and enumeration.
*/
primitive type Integral : * -> Prop
/**
* Divide two values, rounding down (toward negative infinity).
* * For type [n], the arguments are treated as unsigned.
* * Division by zero is undefined.
*/
primitive (/) : {a} (Integral a) => a -> a -> a
/**
* Compute the remainder from dividing two values.
* * For type [n], the arguments are treated as unsigned.
* * Remainder of division by zero is undefined.
* * Satisfies 'x % y == x - (x / y) * y'.
*/
primitive (%) : {a} (Integral a) => a -> a -> a
/**
* Converts a value of an integral type to an integer.
*/
primitive toInteger : {a} (Integral a) => a -> Integer
/**
* Compute the exponentiation of a value in a ring.
* * For type [n], the exponent is treated as unsigned.
* * It is an error to raise a value to a negative integer exponent.
* * Satisfies: 'x ^^ 0 == fromInteger 1'
* * Satisfies: 'x ^^ e == x * x ^^ (e-1)' when 'e > 0'.
*/
primitive (^^) : {a, e} (Ring a, Integral e) => a -> e -> a
/**
* An infinite sequence counting up from the given starting value.
* '[x...]' is syntactic sugar for 'infFrom x'.
*/
primitive infFrom : {a} (Integral a) => a -> [inf]a
/**
* An infinite arithmetic sequence starting with the given two values.
* '[x,y...]' is syntactic sugar for 'infFromThen x y'.
*/
primitive infFromThen : {a} (Integral a) => a -> a -> [inf]a
// The Field class -------------------------------------------------
/**
* Value types that correspond to a field; that is,
* a ring also posessing multiplicative inverses for
* non-zero elements.
*
* Floating-point values are only approximately a field,
* but nonetheless inhabit this class.
*/
primitive type Field : * -> Prop
/**
* Reciprocal
*
* Compute the multiplicative inverse of an element of a field.
* The reciprocal of 0 is undefined.
*/
primitive recip : {a} (Field a) => a -> a
/**
* Field division
*
* The division operation in a field.
* Satisfies 'x /. y == x * (recip y)'
*
* Field division by 0 is undefined.
*/
primitive (/.) : {a} (Field a) => a -> a -> a
// The Round class -------------------------------------------------
/** Value types that can be rounded to integer values. */
primitive type Round : * -> Prop
/**
* Ceiling function.
*
* Given 'x', compute the smallest integer 'i'
* such that 'x <= i'.
*/
primitive ceiling : {a} (Round a) => a -> Integer
/**
* Floor function.
*
* Given 'x', compute the largest integer 'i'
* such that 'i <= x'.
*/
primitive floor : {a} (Round a) => a -> Integer
/**
* Truncate the value toward 0.
*
* Given 'x' compute the nearest integer between
* 'x' and 0. For nonnegative 'x', this is floor,
* and for negative 'x' this is ceiling.
*/
primitive trunc : {a} (Round a) => a -> Integer
/**
* Round to the nearest integer, ties away from 0.
*
* Ties are broken away from 0. For nonnegative 'x'
* this is 'floor (x + 0.5)'. For negative 'x' this
* is 'ceiling (x - 0.5)'.
*/
primitive roundAway : {a} (Round a) => a -> Integer
/**
* Round to the nearest integer, ties to even.
*
* Ties are broken to the nearest even integer.
*/
primitive roundToEven : {a} (Round a) => a -> Integer
// The Eq class ----------------------------------------------------
/** Value types that support equality comparisons. */
primitive type Eq : * -> Prop
/**
* Compares any two values of the same type for equality.
*/
primitive (==) : {a} (Eq a) => a -> a -> Bit
/**
* Compares any two values of the same type for inequality.
*/
primitive (!=) : {a} (Eq a) => a -> a -> Bit
/**
* Compare the outputs of two functions for equality.
*/
(===) : {a, b} (Eq b) => (a -> b) -> (a -> b) -> (a -> Bit)
f === g = \ x -> f x == g x
/**
* Compare the outputs of two functions for inequality.
*/
(!==) : {a, b} (Eq b) => (a -> b) -> (a -> b) -> (a -> Bit)
f !== g = \x -> f x != g x
// The Cmp class ---------------------------------------------------
/** Value types that support equality and ordering comparisons. */
primitive type Cmp : * -> Prop
/**
* Less-than. Only works on comparable arguments.
*
* Bitvectors are compared using unsigned arithmetic.
*/
primitive (<) : {a} (Cmp a) => a -> a -> Bit
/**
* Greater-than of two comparable arguments.
*
* Bitvectors are compared using unsigned arithmetic.
*/
primitive (>) : {a} (Cmp a) => a -> a -> Bit
/**
* Less-than or equal of two comparable arguments.
*
* Bitvectors are compared using unsigned arithmetic.
*/
primitive (<=) : {a} (Cmp a) => a -> a -> Bit
/**
* Greater-than or equal of two comparable arguments.
*
* Bitvectors are compared using unsigned arithmetic.
*/
primitive (>=) : {a} (Cmp a) => a -> a -> Bit
/**
* Returns the smaller of two comparable arguments.
* Bitvectors are compared using unsigned arithmetic.
*/
min : {a} (Cmp a) => a -> a -> a
min x y = if x < y then x else y
/**
* Returns the greater of two comparable arguments.
* Bitvectors are compared using unsigned arithmetic.
*/
max : {a} (Cmp a) => a -> a -> a
max x y = if x > y then x else y
/**
* Compute the absolute value of a value from an ordered ring.
* Bitvector values are considered unsigned, so this is
* the identity function on [n].
*/
abs : {a} (Cmp a, Ring a) => a -> a
abs x = if x < fromInteger 0 then negate x else x
// The SignedCmp class ----------------------------------------------
/** Value types that support signed comparisons. */
primitive type SignedCmp : * -> Prop
/**
* 2's complement signed less-than.
*/
primitive (<$) : {a} (SignedCmp a) => a -> a -> Bit
/**
* 2's complement signed greater-than.
*/
(>$) : {a} (SignedCmp a) => a -> a -> Bit
x >$ y = y <$ x
/**
* 2's complement signed less-than-or-equal.
*/
(<=$) : {a} (SignedCmp a) => a -> a -> Bit
x <=$ y = ~(y <$ x)
/**
* 2's complement signed greater-than-or-equal.
*/
(>=$) : {a} (SignedCmp a) => a -> a -> Bit
x >=$ y = ~(x <$ y)
// Bit specific operations ----------------------------------------
/**
* The constant True. Corresponds to the bit value 1.
*/
primitive True : Bit
/**
* The constant False. Corresponds to the bit value 0.
*/
primitive False : Bit
/**
* Short-cutting boolean conjunction function.
* If the first argument is False, the second argument
* is not evaluated.
*/
(/\) : Bit -> Bit -> Bit
x /\ y = if x then y else False
/**
* Short-cutting boolean disjunction function.
* If the first argument is True, the second argument
* is not evaluated.
*/
(\/) : Bit -> Bit -> Bit
x \/ y = if x then True else y
/**
* Short-cutting logical implication.
* If the first argument is False, the second argument is
* not evaluated.
*/
(==>) : Bit -> Bit -> Bit
a ==> b = if a then b else True
// Bitvector specific operations ----------------------------------
/**
* 2's complement signed division. Division rounds toward 0.
* Division by 0 is undefined.
*
* * Satisfies 'x == x %$ y + (x /$ y) * y' for 'y != 0'.
*/
primitive (/$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n]
/**
* 2's complement signed remainder. Division rounds toward 0.
* Division by 0 is undefined. Satisfies the following for 'y != 0'
*
* * 'x %$ y == x - (x /$ y) * y'.
* * 'x >=$ 0 ==> x %$ y >=$ 0'
* * 'x <=$ 0 ==> x %$ y <=$ 0'
*/
primitive (%$) : {n} (fin n, n >= 1) => [n] -> [n] -> [n]
/**
* Unsigned carry. Returns true if the unsigned addition of the given
* bitvector arguments would result in an unsigned overflow.
*/
carry : {n} (fin n) => [n] -> [n] -> Bit
carry x y = (x + y) < x
/**
* Signed carry. Returns true if the 2's complement signed addition of the
* given bitvector arguments would result in a signed overflow.
*/
scarry : {n} (fin n, n >= 1) => [n] -> [n] -> Bit
scarry x y = (sx == sy) && (sx != sz)
where
z = x + y
sx = x@0
sy = y@0
sz = z@0
/**
* Signed borrow. Returns true if the 2's complement signed subtraction of the
* given bitvector arguments would result in a signed overflow.
*/
sborrow : {n} (fin n, n >= 1) => [n] -> [n] -> Bit
sborrow x y = ( x <$ (x-y) ) ^ y@0
/**
* Zero extension of a bitvector.
*/
zext : {m, n} (fin m, m >= n) => [n] -> [m]
zext x = zero # x
/**
* Sign extension of a bitvector.
*/
sext : {m, n} (fin m, m >= n, n >= 1) => [n] -> [m]
sext x = newbits # x
where newbits = if x@0 then ~zero else zero
/**
* 2's complement signed (arithmetic) right shift. The first argument
* is the sequence to shift (considered as a signed value),
* the second argument is the number of positions to shift
* by (considered as an unsigned value).
*/
primitive (>>$) : {n, ix} (fin n, n >= 1, Integral ix) => [n] -> ix -> [n]
/**
* Log base two.
*
* For words, computes the ceiling of log, base 2, of a number.
* We set 'lg2 0 = 0'
*/
primitive lg2 : {n} (fin n) => [n] -> [n]
// Rational specific operations ----------------------------------------------
/**
* Compute the ratio of two integers as a rational.
* Ratio is undefined if the denominator is 0.
*
* 'ratio x y = (fromInteger x /. fromInteger y) : Rational'
*/
primitive ratio : Integer -> Integer -> Rational
// Zn specific operations ----------------------------------------------------
/**
* Converts an integer modulo n to an unbounded integer in the range 0 to n-1.
*/
primitive fromZ : {n} (fin n, n >= 1) => Z n -> Integer
// Sequence operations -------------------------------------------------------
/**
* Concatenates two sequences. On bitvectors, the most-significant bits
* are in the left argument, and the least-significant bits are in the right.
*/
primitive (#) : {front, back, a} (fin front) => [front]a -> [back]a
-> [front + back] a
/**
* Splits a sequence into a pair of sequences.
* 'splitAt z = (x, y)' iff 'x # y = z'.
*/
primitive splitAt : {front, back, a} (fin front) => [front + back]a
-> ([front]a, [back]a)
/**
* Concatenates a list of sequences.
* 'join' is the inverse function to 'split'.
*/
primitive join : {parts, each, a} (fin each) => [parts][each]a
-> [parts * each]a
/**
* Splits a sequence into 'parts' groups with 'each' elements.
* 'split' is the inverse function to 'join'.
*/
primitive split : {parts, each, a} (fin each) => [parts * each]a
-> [parts][each]a
/**
* Reverses the elements in a sequence.
*/
primitive reverse : {n, a} (fin n) => [n]a -> [n]a
/**
* Transposes a matrix.
* Satisfies the property 'transpose m @ i @ j == m @ j @ i'.
*/
primitive transpose : {rows, cols, a} [rows][cols]a -> [cols][rows]a
/**
* Select the first (left-most) 'front' elements of a sequence.
*/
take : {front, back, a} (fin front) => [front + back]a -> [front]a
take (x # _) = x
/**
* Select all the elements after (to the right of) the 'front' elements of a sequence.
*/
drop : {front, back, a} (fin front) => [front + back]a -> [back]a
drop ((_ : [front] _) # y) = y
/**
* Drop the first (left-most) element of a sequence.
*/
tail : {n, a} [1 + n]a -> [n]a
tail xs = drop`{1} xs
/**
* Return the first (left-most) element of a sequence.
*/
head : {n, a} [1 + n]a -> a
head xs = xs @ 0
/**
* Return the right-most element of a sequence.
*/
last : {n, a} (fin n) => [1 + n]a -> a
last xs = xs ! 0
/**
* Same as 'split', but with a different type argument order.
* Take a sequence of elements and break it into 'parts' sequences
* of 'each' elements.
*/
groupBy : {each, parts, a} (fin each) => [parts * each]a -> [parts][each]a
groupBy = split`{parts=parts}
/**
* Left shift. The first argument is the sequence to shift, the second is the
* number of positions to shift by.
*/
primitive (<<) : {n, ix, a} (Integral ix, Zero a) => [n]a -> ix -> [n]a
/**
* Right shift. The first argument is the sequence to shift, the second is the
* number of positions to shift by.
*/
primitive (>>) : {n, ix, a} (Integral ix, Zero a) => [n]a -> ix -> [n]a
/**
* Left rotate. The first argument is the sequence to rotate, the second is the
* number of positions to rotate by.
*/
primitive (<<<) : {n, ix, a} (fin n, Integral ix) => [n]a -> ix -> [n]a
/**
* Right rotate. The first argument is the sequence to rotate, the second is
* the number of positions to rotate by.
*/
primitive (>>>) : {n, ix, a} (fin n, Integral ix) => [n]a -> ix -> [n]a
/**
* Index operator. The first argument is a sequence. The second argument is
* the zero-based index of the element to select from the sequence.
*/
primitive (@) : {n, a, ix} (Integral ix) => [n]a -> ix -> a
/**
* Bulk index operator. The first argument is a sequence. The second argument
* is a sequence of the zero-based indices of the elements to select.
*/
(@@) : {n, k, ix, a} (Integral ix) => [n]a -> [k]ix -> [k]a
xs @@ is = [ xs @ i | i <- is ]
/**
* Reverse index operator. The first argument is a finite sequence. The second
* argument is the zero-based index of the element to select, starting from the
* end of the sequence.
*/
primitive (!) : {n, a, ix} (fin n, Integral ix) => [n]a -> ix -> a
/**
* Bulk reverse index operator. The first argument is a finite sequence. The
* second argument is a sequence of the zero-based indices of the elements to
* select, starting from the end of the sequence.
*/
(!!) : {n, k, ix, a} (fin n, Integral ix) => [n]a -> [k]ix -> [k]a
xs !! is = [ xs ! i | i <- is ]
/**
* Update the given sequence with new value at the given index position.
* The first argument is a sequence. The second argument is the zero-based
* index of the element to update, starting from the front of the sequence.
* The third argument is the new element. The return value is the
* initial sequence updated so that the indicated index has the given value.
*/
primitive update : {n, a, ix} (Integral ix) => [n]a -> ix -> a -> [n]a
/**
* Update the given sequence with new value at the given index position.
* The first argument is a sequence. The second argument is the zero-based
* index of the element to update, starting from the end of the sequence.
* The third argument is the new element. The return value is the
* initial sequence updated so that the indicated index has the given value.
*/
primitive updateEnd : {n, a, ix} (fin n, Integral ix) => [n]a -> ix -> a -> [n]a
/**
* Perform a series of updates to a sequence. The first argument is
* the initial sequence to update. The second argument is a sequence
* of indices, and the third argument is a sequence of values.
* This function applies the 'update' function in sequence with the
* given update pairs.
*/
updates : {n, k, ix, a} (Integral ix, fin k) => [n]a -> [k]ix -> [k]a -> [n]a
updates xs0 idxs vals = xss!0
where
xss = [ xs0 ] #
[ update xs i b
| xs <- xss
| i <- idxs
| b <- vals
]
/**
* Perform a series of updates to a sequence. The first argument is
* the initial sequence to update. The second argument is a sequence
* of indices, and the third argument is a sequence of values.
* This function applies the 'updateEnd' function in sequence with the
* given update pairs.
*/
updatesEnd : {n, k, ix, a} (fin n, Integral ix, fin k) => [n]a -> [k]ix -> [k]a -> [n]a
updatesEnd xs0 idxs vals = xss!0
where
xss = [ xs0 ] #
[ updateEnd xs i b
| xs <- xss
| i <- idxs
| b <- vals
]
/**
* Produce a sequence using a generating function.
* Satisfies 'generate f @ i == f i' for all 'i' between '0' and 'n-1'.
*
* Declarations of the form 'x @ i = e' are syntactic sugar for
* 'x = generate (\i -> e)'.
*/
generate : {n, a} (fin n, n >= 1) => (Integer -> a) -> [n]a
generate f = [ f i | i <- [0 .. n-1] ]
// GF_2^n polynomial computations -------------------------------------------
/**
* Performs multiplication of polynomials over GF(2).
*/
pmult : {u, v} (fin u, fin v) => [1 + u] -> [1 + v] -> [1 + u + v]
pmult x y = last zs
where
zs = [0] # [ (z << 1) ^ (if yi then 0 # x else 0) | yi <- y | z <- zs ]
/**
* Performs division of polynomials over GF(2).
*/
pdiv : {u, v} (fin u, fin v) => [u] -> [v] -> [u]
pdiv x y = [ z ! degree | z <- zs ]
where
degree : [width v]
degree = last (ds : [1 + v]_)
where ds = [0/0] # [if yi then i else d | yi <- reverse y | i <- [0..v] | d <- ds ]
reduce : [v] -> [v]
reduce u = if u ! degree then u ^ y else u
zs : [u][v]
zs = [ tail (reduce z # [xi]) | z <- [0] # zs | xi <- x ]
/**
* Performs modulus of polynomials over GF(2).
*/
pmod : {u, v} (fin u, fin v) => [u] -> [1 + v] -> [v]
pmod x y = if y == 0 then 0/0 else last zs
where
degree : [width v]
degree = last (ds : [2 + v]_)
where ds = [0/0] # [if yi then i else d | yi <- reverse y | i <- [0..v] | d <- ds ]
reduce : [1 + v] -> [1 + v]
reduce u = if u ! degree then u ^ y else u
powers : [inf][1 + v]
powers = [reduce 1] # [ reduce (p << 1) | p <- powers ]
zs = [0] # [ z ^ (if xi then tail p else 0) | xi <- reverse x | p <- powers | z <- zs ]
// Experimental primitives ------------------------------------------------------------
/**
* Parallel map. The given function is applied to each element in the
* given finite seqeuence, and the results are computed in parallel.
*
* This function is experimental.
*/
primitive parmap : {a, b, n} (fin n) => (a -> b) -> [n]a -> [n]b
// Utility operations -----------------------------------------------------------------
/**
* Raise a run-time error with the given message.
* This function can be called at any type.
*/
primitive error : {a, n} (fin n) => String n -> a
/**
* Raise a run-time error with a generic message.
* This function can be called at any type.
*/
undefined : {a} a
undefined = error "undefined"
/**
* Assert that the given condition holds, and raise an error
* with the given message if it does not. If the condition
* holds, return the third argument unchanged.
*/
assert : {a, n} (fin n) => Bit -> String n -> a -> a
assert pred msg x = if pred then x else error msg
/**
* Generates random values from a seed. When called with a function, currently
* generates a function that always returns zero.
*/
primitive random : {a} [256] -> a
/**
* Debugging function for tracing. The first argument is a string,
* which is prepended to the printed value of the second argument.
* This combined string is then printed when the trace function is
* evaluated. The return value is equal to the third argument.
*
* The exact timing and number of times the trace message is printed
* depend on the internal details of the Cryptol evaluation order,
* which are unspecified. Thus, the output produced by this
* operation may be difficult to predict.
*/
primitive trace : {n, a, b} (fin n) => String n -> a -> b -> b
/**
* Debugging function for tracing values. The first argument is a string,
* which is prepended to the printed value of the second argument.
* This combined string is then printed when the trace function is
* evaluated. The return value is equal to the second argument.
*
* The exact timing and number of times the trace message is printed
* depend on the internal details of the Cryptol evaluation order,
* which are unspecified. Thus, the output produced by this
* operation may be difficult to predict.
*/
traceVal : {n, a} (fin n) => String n -> a -> a
traceVal msg x = trace msg x x
/* Functions previously in Cryptol::Extras */
/**
* Conjunction of all bits in a sequence.
*/
and : {n} (fin n) => [n]Bit -> Bit
and xs = ~zero == xs
/**
* Disjunction of all bits in a sequence.
*/
or : {n} (fin n) => [n]Bit -> Bit
or xs = zero != xs
/**
* Conjunction after applying a predicate to all elements.
*/
all : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit
all f xs = and (map f xs)
/**
* Disjunction after applying a predicate to all elements.
*/
any : {n, a} (fin n) => (a -> Bit) -> [n]a -> Bit
any f xs = or (map f xs)
/**
* Map a function over a sequence.
*/
map : {n, a, b} (a -> b) -> [n]a -> [n]b
map f xs = [f x | x <- xs]
/**
* Functional left fold.
*
* foldl (+) 0 [1,2,3] = ((0 + 1) + 2) + 3
*/
foldl : {n, a, b} (fin n) => (a -> b -> a) -> a -> [n]b -> a
foldl f acc xs = ys ! 0
where ys = [acc] # [f a x | a <- ys | x <- xs]
/**
* Functional right fold.
*
* foldr (-) 0 [1,2,3] = 0 - (1 - (2 - 3))
*/
foldr : {n, a, b} (fin n) => (a -> b -> b) -> b -> [n]a -> b
foldr f acc xs = ys ! 0
where ys = [acc] # [f x a | a <- ys | x <- reverse xs]
/**
* Compute the sum of the values in the sequence.
*/
sum : {n, a} (fin n, Ring a) => [n]a -> a
sum xs = foldl (+) (fromInteger 0) xs
/**
* Scan left is like a foldl that also emits the intermediate values.
*/
scanl : {n, b, a} (b -> a -> b) -> b -> [n]a -> [n+1]b
scanl f acc xs = ys
where ys = [acc] # [f a x | a <- ys | x <- xs]
/**
* Scan right is like a foldr that also emits the intermediate values.
*/
scanr : {n, a, b} (fin n) => (a -> b -> b) -> b -> [n]a -> [n+1]b
scanr f acc xs = reverse ys
where ys = [acc] # [f x a | a <- ys | x <- reverse xs]
/**
* Repeat a value.
*/
repeat : {n, a} a -> [n]a
repeat x = [ x | _ <- zero : [n] ]
/**
* 'elem x xs' returns true if x is equal to a value in xs.
*/
elem : {n, a} (fin n, Eq a) => a -> [n]a -> Bit
elem a xs = any (\x -> x == a) xs
/**
* Create a list of tuples from two lists.
*/
zip : {n, a, b} [n]a -> [n]b -> [n](a, b)
zip xs ys = [(x,y) | x <- xs | y <- ys]
/**
* Create a list by applying the function to each pair of elements in the input.
*/
zipWith : {n, a, b, c} (a -> b -> c) -> [n]a -> [n]b -> [n]c
zipWith f xs ys = [f x y | x <- xs | y <- ys]
/**
* Transform a function into uncurried form.
*/
uncurry : {a, b, c} (a -> b -> c) -> (a, b) -> c
uncurry f = \(a, b) -> f a b
/**
* Transform a function into curried form.
*/
curry : {a, b, c} ((a, b) -> c) -> a -> b -> c
curry f = \a b -> f (a, b)
/**
* Map a function iteratively over a seed value, producing an infinite
* list of successive function applications.
*/
iterate : {a} (a -> a) -> a -> [inf]a
iterate f x = xs
where xs = [x] # [ f v | v <- xs ]