cryptol/docs/CryptolPrims.markdown

Comparisons and Ordering

Old types:

(==)   : {a}   (fin a) => (a,a) -> Bit
(!=)   : {a}   (fin a) => (a,a) -> Bit
(===)  : {a b} (fin b) => (a -> b,a -> b) -> a -> Bit
(!===) : {a b} (fin b) => (a -> b,a -> b) -> a -> Bit

(<)    : {n} (fin n) => ([n],[n]) -> Bit
(>)    : {n} (fin n) => ([n],[n]) -> Bit
(<=)   : {n} (fin n) => ([n],[n]) -> Bit
(>=)   : {n} (fin n) => ([n],[n]) -> Bit

min    : {n} (fin n) => ([n],[n]) -> [n]
max    : {n} (fin n) => ([n],[n]) -> [n]

New types:

(==)   :  {a}   (Cmp a) => a -> a -> Bit
(!=)   :  {a}   (Cmp a) => a -> a -> Bit
(===)  :  {a,b} (Cmp b) => (a -> b) -> (a -> b) -> a -> Bit
(!==)  :  {a,b} (Cmp b) => (a -> b) -> (a -> b) -> a -> Bit

(<)    :  {a} (Cmp a) => a -> a -> Bit
(>)    :  {a} (Cmp a) => a -> a -> Bit
(<=)   :  {a} (Cmp a) => a -> a -> Bit
(>=)   :  {a} (Cmp a) => a -> a -> Bit

min    :  {a} (Cmp a) => a -> a -> a
max    :  {a} (Cmp a) => a -> a -> a

instance Cmp Bit
// No instance for functions.
instance (Cmp a, fin n) => Cmp [n] a
instance (Cmp a, Cmp b) => Cmp (a,b)
instance (Cmp a, Cmp b) => Cmp { x : a, y : b }

Arithmetic

Old types:

negate : {n a} (n >= 1) => [n]a -> [n]a
(+)    : {n a} ([n]a,[n]a) -> [n]a
(-)    : {n a} ([n]a,[n]a) -> [n]a
(*)    : {n a} ([n]a,[n]a) -> [n]a
(/)    : {n a} ([n]a,[n]a) -> [n]a
(%)    : {n a} ([n]a,[n]a) -> [n]a
(**)   : {n a} ([n]a,[n]a) -> [n]a

New types:

negate : {a} (Arith a) => a -> a
(+)    : {a} (Arith a) => a -> a -> a
(-)    : {a} (Arith a) => a -> a -> a
(*)    : {a} (Arith a) => a -> a -> a
(/)    : {a} (Arith a) => a -> a -> a
(%)    : {a} (Arith a) => a -> a -> a
(^^)   : {a} (Arith a) => a -> a -> a

// No instance for `Bit`.
instance (fin n)            => Arith ( [n] Bit)
instance (Arith a)          => Arith ( [n] a)
instance (Arith b)          => Arith (a -> b)
instance (Arith a, Arith b) => Arith (a,b)
instance (Arith a, Arith b) => Arith { x : a, y : b }

Note that because there is no instances for Arith Bit the top two instances do not actually overlap.

A corner case: unlike the old system, we'd also have to define negate at type 0. This makes sense, there is only one element of type [0], so it is naturally its own inverse, thus negate should behave as the identity function.

Boolean

Old types:

False : Bit
True  : Bit

zero  : {a} a
(&)   : {a} (a,a) -> a
(|)   : {a} (a,a) -> a
(^)   : {a} (a,a) -> a
(~)   : {a} a -> a

New types:

False : Bit
True  : Bit

zero  : (Logic a) => a
(&&)  : (Logic a) => a -> a -> a
(||)  : (Logic a) => a -> a -> a
(^)   : (Logic a) => a -> a -> a
(~)   : (Logic a) => a -> a

// There are instances for all types,
// so we could potentially omit the constraints.
instance Logic Bit
instance Logic a            => Logic ([n] a)
instance Logic b            => Logic (a -> b)
instance (Logic a, Logic b) => Logic (a,b)
instance (Logic a, Logic b) => Logic { x : a, y : b }

Sequences

Old types:

width     : {n a m} (m >= width n) => [n]a -> [m]

join      : {n m a} [n][m]a -> [n*m]a
split     : {n m a} [n*m]a -> [n][m]a
splitBy   : {n m a} (n,[n*m]a) -> [n][m]a
groupBy   : {n m a} (m,[n*m]a) -> [n][m]a

(#)       : {n a m}   (fin n) => ([n]a,[m]a) -> [n+m]a
tail      : {n a} [n+1]a -> [n]a
take      : {n m a} (fin n,m >= n) => (n,[n+m]a) -> [n]a
drop      : {n m a} (fin n,n >= n) => (n,[n+m]a) -> [m]a

reverse   : {n a} (fin n) => [n]a -> [n]a
transpose : {n m a} [n][m]a -> [m][n]a

(@)        : {n a m}              ([n]a,[m]) -> a
(@@)       : {n a m i}            ([n]a,[m][i]) -> [m]a
(!)        : {n a m}   (fin n) => ([n]a,[m]) -> a
(!!)       : {n a m i} (fin n) => ([n]a,[m][i]) -> [m]a

New types:

length    : {n,a,m} (m >= width n) => [n]a -> [m]

join      : {parts,ench,a} (fin each) => [parts][each]a -> [parts * each]a
split     : {parts,each,a} (fin each) => [parts * each]a -> [parts][each]a

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

reverse   : {n,a} (fin n) => [n]a -> [n]a
transpose : {n,m,a} [n][m]a -> [m][n]a

(@)        : {n a m}              [n]a -> [m]    -> a
(@@)       : {n a m i}            [n]a -> [m][i] -> [m]a
(!)        : {n a m}   (fin n) => [n]a -> [m]    -> a
(!!)       : {n a m i} (fin n) => [n]a -> [m][i] -> [m]a


// Abbreviations
splitBy n = split`{parts = n}
groupBy n = split`{each  = n}
tail n    = splitAt`{front = 1}.2
take n    = splitAt`{front = n}.1
drop n    = splitAt`{front = n}.2

/* Also, `length` is not really needed:
   length : {n,a,m} (m >= width n) => [n]a -> [m]
   length _ = `n
 */

Shift And Rotate

Old types:

(<<)   : {n a m} (m >= lg2 n,fin n) => ([n]a,[m]) -> [n]a
(>>)   : {n a m} (m >= lg2 n,fin n) => ([n]a,[m]) -> [n]a
(<<<)  : {n a m} (m >= lg2 n,fin n) => ([n]a,[m]) -> [n]a
(>>>)  : {n a m} (m >= lg2 n,fin n) => ([n]a,[m]) -> [n]a

New types:

(<<)   : {n,a,m} (fin n, Logic a) => [n]a -> [m] -> [n]a
(>>)   : {n,a,m} (fin n, Logic a) => [n]a -> [m] -> [n]a
(<<<)  : {n,a,m} (fin n, Logic a) => [n]a -> [m] -> [n]a
(>>>)  : {n,a,m} (fin n, Logic a) => [n]a -> [m] -> [n]a

Random Values

Old types:

random    : {a b} => [a] -> b

New types:

random    : {a} => [32] -> a

Debugging

 ASSERT      : {n a} (Bit,[n][8],a) -> a
 undefined   : {a} a
 error       : {n a} [n][8] -> a
 trace       : {n a j} ([n][8],a,j) -> j

Hints for Hardware Generation?

pipeline_stop : {a} a -> a
pipeline      : {n a} (fin n) => ([n],a) -> a

seq           : {n a} [n]a -> [n]a
par           : {n a} [n]a -> [n]a
reg           : {a} a -> a
const         : {a} a -> a