PreludeWithExtras.cry

/*
 * Copyright (c) 2013-2016 Galois, Inc.
 * Distributed under the terms of the BSD3 license (see LICENSE file)
 */

module Cryptol where

/**
 * The value corresponding to a numeric type.
 */
primitive number : {val, bits} (fin val, fin bits, bits >= width val) => [bits]

infixr 10 ||
infixr 20 &&
infix  30 ==, ===, !=, !==
infix  40 >, >=, <, <=
infixl 50 ^
infixr 60 #
infixl 70 <<, <<<, >>, >>>
infixl 80 +, -
infixl 90 *, /, %
infixr 95 ^^
infixl 100 @, @@, !, !!

/**
 * Add two values.
 *  * For words, addition uses modulo arithmetic.
 *  * Structured values are added element-wise.
 */
primitive (+) : {a} (Arith a) => a -> a -> a

/**
 * For words, subtraction uses modulo arithmetic.
 * Structured values are subtracted element-wise. Defined as:
 * a - b = a + negate b
 * See also: `negate'.
 */
primitive (-) : {a} (Arith a) => a -> a -> a

/**
 * For words, multiplies two words, modulus 2^^a.
 * Structured values are multiplied element-wise.
 */
primitive (*) : {a} (Arith a) => a -> a -> a

/**
 * For words, divides two words, modulus 2^^a.
 * Structured values are divided element-wise.
 */
primitive (/) : {a} (Arith a) => a -> a -> a

/**
 * For words, takes the modulus of two words, modulus 2^^a.
 * Over structured values, operates element-wise.
 * Be careful, as this will often give unexpected results due to interaction of
 * the two moduli.
 */
primitive (%) : {a} (Arith a) => a -> a -> a

/**
 * For words, takes the exponent of two words, modulus 2^^a.
 * Over structured values, operates element-wise.
 * Be careful, due to its fast-growing nature, exponentiation is prone to
 * interacting poorly with defaulting.
 */
primitive (^^) : {a} (Arith a) => a -> a -> a

/**
 * Log base two.
 *
 * For words, computes the ceiling of log, base 2, of a number.
 * Over structured values, operates element-wise.
 */
primitive lg2 : {a} (Arith a) => a -> a


type Bool = Bit

/**
 * The constant True. Corresponds to the bit value 1.
 */
primitive True  : Bit

/**
 * The constant False. Corresponds to the bit value 0.
 */
primitive False : Bit

/**
 * Returns the twos complement of its argument.
 * Over structured values, operates element-wise.
 * negate a = ~a + 1
 */
primitive negate : {a} (Arith a) => a -> a

/**
 * Binary complement.
 */
primitive complement : {a} a -> a

/**
 * Operator form of binary complement.
 */
(~) : {a} a -> a
(~) = complement

/**
 * Less-than. Only works on comparable arguments.
 */
primitive (<) : {a} (Cmp a) => a -> a -> Bit

/**
 * Greater-than of two comparable arguments.
 */
primitive (>) : {a} (Cmp a) => a -> a -> Bit

/**
 * Less-than or equal of two comparable arguments.
 */
primitive (<=) : {a} (Cmp a) => a -> a -> Bit

/**
 * Greater-than or equal of two comparable arguments.
 */
primitive (>=) : {a} (Cmp a) => a -> a -> Bit

/**
 * Compares any two values of the same type for equality.
 */
primitive (==) : {a} (Cmp a) => a -> a -> Bit

/**
 * Compares any two values of the same type for inequality.
 */
primitive (!=) : {a} (Cmp a) => a -> a -> Bit

/**
 * Compare the outputs of two functions for equality
 */
(===) : {a,b} (Cmp b) => (a -> b) -> (a -> b) -> (a -> Bit)
f === g = \ x -> f x == g x

/**
 * Compare the outputs of two functions for inequality
 */
(!==) : {a,b} (Cmp b) => (a -> b) -> (a -> b) -> (a -> Bit)
f !== g = \x -> f x != g x

/**
 * Returns the smaller of two comparable arguments.
 */
min : {a} (Cmp a) => a -> a -> a
min x y = if x < y then x else y

/**
 * Returns the greater of two comparable arguments.
 */
max : {a} (Cmp a) => a -> a -> a
max x y = if x > y then x else y

/**
 * Logical `and' over bits. Extends element-wise over sequences, tuples.
 */
primitive (&&) : {a} a -> a -> a

/**
 * Logical `or' over bits. Extends element-wise over sequences, tuples.
 */
primitive (||) : {a} a -> a -> a

/**
 * Logical `exclusive or' over bits. Extends element-wise over sequences, tuples.
 */
primitive (^) : {a} a -> a -> a

/**
 * 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} a

/**
 * Left shift.  The first argument is the sequence to shift, the second is the
 * number of positions to shift by.
*/
primitive (<<) : {a, b, c} (fin b) => [a]c -> [b] -> [a]c

/**
 * Right shift.  The first argument is the sequence to shift, the second is the
 * number of positions to shift by.
 */
primitive (>>) : {a, b, c} (fin b) => [a]c -> [b] -> [a]c

/**
 * Left rotate.  The first argument is the sequence to rotate, the second is the
 * number of positions to rotate by.
 */
primitive (<<<) : {a, b, c} (fin a, fin b) => [a]c -> [b] -> [a]c

/**
 * Right rotate.  The first argument is the sequence to rotate, the second is
 * the number of positions to rotate by.
 */
primitive (>>>) : {a, b, c} (fin a, fin b) => [a]c -> [b] -> [a]c

primitive (#) : {front, back, a} (fin front) => [front]a -> [back]a
                                             -> [front + back] a


/**
 * Split a sequence into a tuple of sequences.
 */
primitive splitAt : {front, back, a} (fin front) => [front + back]a
                                                 -> ([front]a, [back]a)
/**
 * Joins sequences.
 */
primitive join : {parts, each, a} (fin each) => [parts][each]a
                                             -> [parts * each]a

/**
 * Splits a sequence into 'parts' groups with 'each' elements.
 */
primitive split : {parts, each, a} (fin each) => [parts * each]a
                                              -> [parts][each]a

/**
 * Reverses the elements in a sequence.
 */
primitive reverse : {a, b} (fin a) => [a]b -> [a]b

/**
 * Transposes an [a][b] matrix into a [b][a] matrix.
 */
primitive transpose : {a, b, c} [a][b]c -> [b][a]c

/**
 * 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 (@) : {a, b, c} (fin c) => [a]b -> [c] -> b

/**
 * 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.
 */
primitive (@@) : {a, b, c, d} (fin d) => [a]b -> [c][d] -> [c]b

/**
 * 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 (!) : {a, b, c} (fin a, fin c) => [a]b -> [c] -> b

/**
 * 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
 z select, starting from the end of the sequence.
 */
primitive (!!) : {a, b, c, d} (fin a, fin d) => [a]b -> [c][d] -> [c]b

primitive fromTo : {first, last, bits} (fin last, fin bits, last >= first,
                              bits >= width last) => [1 + (last - first)][bits]

primitive fromThenTo : {first, next, last, bits, len} (fin first, fin next,
                        fin last, fin bits, bits >= width first,
                        bits >= width next, bits >= width last,
                        lengthFromThenTo first next last == len) => [len][bits]

primitive infFrom : {bits} (fin bits) => [bits] -> [inf][bits]

primitive infFromThen : {bits} (fin bits) => [bits] -> [bits] -> [inf][bits]

primitive error : {at, len} (fin len) => [len][8] -> at


/**
 * Performs multiplication of polynomials over GF(2).
 */
primitive pmult : {a, b} (fin a, fin b) => [a] -> [b] -> [max 1 (a + b) - 1]

/**
 * Performs division of polynomials over GF(2).
 */
primitive pdiv : {a, b} (fin a, fin b) => [a] -> [b] -> [a]

/**
 * Performs modulus of polynomials over GF(2).
 */
primitive pmod : {a, b} (fin a, fin b) => [a] -> [1 + b] -> [b]

/**
 * Generates random values from a seed.  When called with a function, currently
 * generates a function that always returns zero.
 */
primitive random : {a} [256] -> a

type String n = [n][8]
type Word n = [n]
type Char   = [8]

take : {front,back,elem} (fin front) => [front + back] elem -> [front] elem
take (x # _) = x

drop : {front,back,elem} (fin front) => [front + back] elem -> [back] elem
drop ((_ : [front] _) # y) = y

tail : {a, b} [1 + a]b -> [a]b
tail xs = drop`{1} xs

width : {bits,len,elem} (fin len, fin bits, bits >= width len) => [len] elem -> [bits]
width _ = `len

undefined : {a} a
undefined = error "undefined"

groupBy : {each,parts,elem} (fin each) =>
  [parts * each] elem -> [parts][each]elem
groupBy = split`{parts=parts}

/**
 * Define the base 2 logarithm function in terms of width
 */
type lg2 n = width (max n 1 - 1)

/**
 * 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} [n][8] -> 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} [n][8] -> a -> a
traceVal msg x = trace msg x x

/*
 * Copyright (c) 2016 Galois, Inc.
 * Distributed under the terms of the BSD3 license (see LICENSE file)
 *
 * This module contains definitions that we wish to eventually promote
 * into the Prelude, but which currently cause typechecking of the
 * Prelude to take too long (see #299)
 */

infixr 5 ==>

/**
 * Logical implication
 */
(==>) : Bit -> Bit -> Bit
a ==> b = if a then b else True

/**
 * Logical negation
 */
not : {a} a -> a
not a = ~ a

/**
 * Conjunction
 */
and : {n} (fin n) => [n]Bit -> Bit
and xs = ~zero == xs

/**
 * Disjunction
 */
or : {n} (fin n) => [n]Bit -> Bit
or xs = zero != xs

/**
 * Conjunction after applying a predicate to all elements.
 */
all : {a,n} (fin n) => (a -> Bit) -> [n]a -> Bit
all f xs = and (map f xs)

/**
 * Disjunction after applying a predicate to all elements.
 */
any : {a,n} (fin n) => (a -> Bit) -> [n]a -> Bit
any f xs = or (map f xs)

/**
 * Map a function over an array.
 */
map : {a, b, n} (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 : {a, b, n} (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 : {a,b,n} (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 words in the array.
 */
sum : {a,n} (fin n, Arith a) => [n]a -> a
sum xs = foldl (+) zero xs

/**
 * Scan left is like a fold that emits the intermediate values.
 */
scanl : {b, a, n}  (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
 */
scanr : {a,b,n} (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]

/**
 * Zero extension
 */
extend : {total,n} (fin total, fin n, total >= n) => [n]Bit -> [total]Bit
extend n = zero # n

/**
 * Signed extension. `extendSigned 0bwxyz : [8] == 0bwwwwwxyz`.
 */
extendSigned : {total,n} (fin total, fin n, n >= 1, total >= n+1) => [n]Bit -> [total]Bit
extendSigned  xs = repeat (xs @ 0) # xs

/**
 * Repeat a value.
 */
repeat : {n, a} a -> [n]a
repeat x = [ x | _ <- zero ]

/**
 * `elem x xs` Returns true if x is equal to a value in xs.
 */
elem : {n,a} (fin n, Cmp a) => a -> [n]a -> Bit
elem a xs = any (\x -> x == a) xs

/**
 * Create a list of tuples from two lists.
 */
zip : {a,b,n} [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.
 * lists
 */
zipWith : {a,b,c,n} (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 = [x] # [ f v | v <- iterate f x ]