cryptol/lib/Cryptol.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 demote : {val, bits} (fin val, fin bits, bits >= width val) => [bits]

/**
 * The integer value corresponding to a numeric type.
 */
primitive integer : {val} (fin val) => Integer

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 @, @@, !, !!

/**
 * 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

/**
 * Bitwise complement. The prefix notation '~ x'
 * is syntactic sugar for 'complement x'.
 */
primitive complement : {a} (Logic a) => a -> a

/**
 * 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

/**
 * 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.
 * 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


/**
 * 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)

/**
 * 2's complement signed division.  Division rounds toward 0.
 */
primitive (/$) : {a} (Arith a) => a -> a -> a

/**
 * 2's complement signed remainder.  Division rounds toward 0.
 */
primitive (%$) : {a} (Arith a) => a -> a -> a

/**
 * Unsigned carry.  Returns true if the unsigned addition of the given
 * bitvector arguments would result in an unsigned overflow.
 */
primitive carry : {n} (fin n) => [n] -> [n] -> Bit

/**
 * Signed carry.  Returns true if the 2's complement signed addition of the
 * given bitvector arguments would result in a signed overflow.
 */
primitive scarry : {n} (fin n, n >= 1) => [n] -> [n] -> Bit

/**
 * 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 : {n, m} (fin m, m >= n) => [n] -> [m]
zext x = zero # x

/**
 * Sign extension of a bitvector.
 */
sext : {n, m} (fin m, m >= n, n >= 1) => [n] -> [m]
sext x = newbits # x
  where newbits = if x@0 then ~zero else zero

/**
 * Short-cutting boolean conjuction 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 disjuction 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

/**
 * 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

/**
 * 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

/**
 * Converts a bitvector to a non-negative integer in the range 0 to 2^^n-1.
 */
primitive toInteger : {a} (fin a) => [a] -> Integer

/**
 * Converts an unbounded integer to a finite bitvector, reducing modulo 2^^n.
 */
primitive fromInteger : {a} (fin a) => Integer -> [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, Zero c) => [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, Zero c) => [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

/**
 * 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, k} (fin n, n >= 1, fin k) => [n] -> [k] -> [n]



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

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

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

/**
 * 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 : {a,b,c,d} (fin c, fin d) => [a]b -> [d][c] -> [d]b -> [a]b
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 : {a,b,c,d} (fin a, fin c, fin d) => [a]b -> [d][c] -> [d]b -> [a]b
updatesEnd xs0 idxs vals = xss!0
 where
   xss = [ xs0 ] #
         [ updateEnd xs i b
         | xs  <- xss
         | i   <- idxs
         | b   <- vals
         ]

/**
 * A finite arithmetic sequence starting with 'first' and 'next',
 * stopping when the values would wrap around modulo '2^^bits'.
 *
 * '[a,b..]' is syntactic sugar for 'fromThen`{first=a,next=b}'.
 */
primitive fromThen : {first, next, bits, len}
                     ( fin first, fin next, fin bits
                     , bits >= width first, bits >= width next
                     , lengthFromThen first next bits == len) => [len][bits]

/**
 * A finite sequence counting up from 'first' to 'last'.
 *
 * '[a..b]' is syntactic sugar for 'fromTo`{first=a,last=b}'.
 * '[a..]' is syntactic sugar for 'fromTo`{first=a,last=(2^^bits)-1}'.
 */
primitive fromTo : {first, last, bits} (fin last, fin bits, last >= first,
                              bits >= width last) => [1 + (last - first)][bits]

/**
 * 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, 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]

/**
 * An infinite sequence counting up from the given starting value.
 * '[x...]' is syntactic sugar for 'infFrom x'.
 */
primitive infFrom : {bits} (fin bits) => [bits] -> [inf][bits]

/**
 * An infinite arithmetic sequence starting with the given two values.
 * '[x,y...]' is syntactic sugar for 'infFromThen x y'.
 */
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) => [1 + a] -> [1 + b] -> [1 + a + b]

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