cryptol/docs/Syntax.md

% Cryptol version 2 Syntax
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Layout
======

Groups of declarations are organized based on indentation.
Declarations with the same indentation belong to the same group.
Lines of text that are indented more than the beginning of a
declaration belong to that declaration, while lines of text that are
indented less terminate a group of declarations.  Groups of
declarations appear at the top level of a Cryptol file, and inside
`where` blocks in expressions.  For example, consider the following
declaration group:

    f x = x + y + z
      where
      y = x * x
      z = x + y

    g y = y

This group has two declarations, one for `f` and one for `g`.  All the
lines between `f` and `g` that are indented more than `f` belong to
`f`.

This example also illustrates how groups of declarations may be nested
within each other.  For example, the `where` expression in the
definition of `f` starts another group of declarations, containing `y`
and `z`.  This group ends just before `g`, because `g` is indented
less than `y` and `z`.

Comments
========

Cryptol supports block comments, which start with `/*` and end with
`*/`, and line comments, which start with `//` and terminate at the
end of the line.  Block comments may be nested arbitrarily.

Examples:

    /* This is a block comment */
    // This is a line comment
    /* This is a /* Nested */ block comment */

Identifiers
===========

Cryptol identifiers consist of one or more characters.  The first
character must be either an English letter or underscore (`_`).  The
following characters may be an English letter, a decimal digit,
underscore (`_`), or a prime (`'`).  Some identifiers have special
meaning in the language, so they may not be used in programmer-defined
names (see [Keywords](#keywords-and-built-in-operators)).

Examples:

    name    name1    name'    longer_name
    Name    Name2    Name''   longerName



Keywords and Built-in Operators
===============================

The following identifiers have special meanings in Cryptol, and may
not be used for programmer defined names:

<!--- The table below can be generated by running `chop.hs` on this list:
else
extern
if
private
include
module
newtype
pragma
property
then
type
where
let
import
as
hiding
newtype
infixl
infixr
infix
primitive
parameter
constraint
--->

        else       include    property    let       newtype    primitive
        extern     module     then        import    infixl     parameter
        if         newtype    type        as        infixr     constraint
        private    pragma     where       hiding    infix

The following table contains Cryptol's operators and their
associativity with lowest precedence operators first, and highest
precedence last.

Operator                                   Associativity
-----------------------------------------  -------------
  `==>`                                    right
  `\/`                                     right
  `/\`                                     right
  `==` `!=` `===` `!==`                    not associative
  `>` `<` `<=` `>=` `<$` `>$` `<=$` `>=$`  not associative
  `||`                                     right
  `^`                                      left
  `&&`                                     right
  `#`                                      right
  `>>` `<<` `>>>` `<<<` `>>$`              left
  `+` `-`                                  left
  `*` `/` `%` `/$` `%$`                    left
  `^^`                                     right
  `@`  `@@`  `!` `!!`                      left
  (unary) `-` `~`                          right

Table: Operator precedences.

Built-in Type-level Operators
=============================

Cryptol includes a variety of operators that allow computations on
the numeric types used to specify the sizes of sequences.

Operator     Meaning
--------     -------------------------
  `+`        Size addition
  `-`        Size subtraction
  `*`        Size multiplication
  `/`        Size division
  `/^`       Size ceiling division (`/` rounded up)
  `%`        Size modulus
  `%^`       Size ceiling modulus (`%` rounded up)
  `^^`       Size exponentiation
  `lg2`      Size logarithm (base 2)
  `width`    Size width (`lg2` rounded up)
  `max`      Size maximum
  `min`      Size minimum

Table: Type-level operators


Numeric Literals
================

Numeric literals may be written in binary, octal, decimal, or
hexadecimal notation. The base of a literal is determined by its prefix:
`0b` for binary, `0o` for octal, no special prefix for
decimal, and `0x` for hexadecimal.

Examples:

    254                 // Decimal literal
    0254                // Decimal literal
    0b11111110          // Binary literal
    0o376               // Octal literal
    0xFE                // Hexadecimal literal
    0xfe                // Hexadecimal literal

Numeric literals in binary, octal, or hexadecimal notation result in
bit sequences of a fixed length (i.e., they have type `[n]` for
some `n`). The length is determined by the base and the number
of digits in the literal. Decimal literals are overloaded, and so the
type is inferred from context in which the literal is used. Examples:

    0b1010              // : [4],   1 * number of digits
    0o1234              // : [12],  3 * number of digits
    0x1234              // : [16],  4 * number of digits

    10                  // : {a}. (Literal 10 a) => a
                        // a = Integer or [n] where n >= width 10

Expressions
===========

This section provides an overview of the Cryptol's expression syntax.

**Calling Functions**

    f 2             // call `f` with parameter `2`
    g x y           // call `g` with two parameters: `x` and `y`
    h (x,y)         // call `h` with one parameter,  the pair `(x,y)`


**Prefix Operators**

    -2              // call unary `-` with parameter `2`
    - 2             // call unary `-` with parameter `2`
    f (-2)          // call `f` with one argument: `-2`,  parens are important
    -f 2            // call unary `-` with parameter `f 2`
    - f 2           // call unary `-` with parameter `f 2`


**Infix Functions**

    2 + 3           // call `+` with two parameters: `2` and `3`
    2 + 3 * 5       // call `+` with two parameters: `2` and `3 * 5`
    (+) 2 3         // call `+` with two parameters: `2` and `3`
    f 2 + g 3       // call `+` with two parameters: `f 2` and `g 3`
    - 2 + - 3       // call `+` with two parameters: `-2` and `-3`
    - f 2 + - g 3

**Type Annotations**

    x : Bit         // specify that `x` has type `Bit`
    f x : Bit       // specify that `f x` has type `Bit`
    - f x : [8]     // specify that `- f x` has type `[8]`
    2 + 3 : [8]     // specify that `2 + 3` has type `[8]`
    \x -> x : [8]   // type annotation is on `x`, not the function
    if x
      then y
      else z : Bit  // the type annotation is on `z`, not the whole `if`


**Local Declarations**

Local declarations have the weakest precedence of all expressions.

    2 + x : [T]
      where
      type T = 8
      x      = 2          // `T` and `x` are in scope of `2 + x : `[T]`

    if x then 1 else 2
      where x = 2         // `x` is in scope in the whole `if`

    \y -> x + y
      where x = 2         // `y` is not in scope in the defintion of `x`


**Block Arguments**

When used as the last argument to a function call,
`if` and lambda expressions do not need parens:

    f \x -> x       // call `f` with one argument `x -> x`
    2 + if x
          then y
          else z    // call `+` with two arguments: `2` and `if ...`

Bits
====

The type `Bit` has two inhabitants: `True` and `False`.  These values
may be combined using various logical operators, or constructed as
results of comparisons.

Operator                                   Associativity       Description
---------------------                      -------------       -----------
 `==>`                                     right               Short-cut implication
 `\/`                                      right               Short-cut or
 `/\`                                      right               Short-cut and
  `!=` `==`                                none                Not equals, equals
  `>` `<` `<=` `>=` `<$` `>$` `<=$` `>=$`  none                Comparisons
  `||`                                     right               Logical or
  `^`                                      left                Exclusive-or
  `&&`                                     right               Logical and
  `~`                                      right               Logical negation

Table: Bit operations.

Multi-way Conditionals
======================

The `if ... then ... else` construct can be used with
multiple branches. For example:

    x = if y % 2 == 0 then 22 else 33

    x = if y % 2 == 0 then 1
         | y % 3 == 0 then 2
         | y % 5 == 0 then 3
         else 7

Tuples and Records
==================

Tuples and records are used for packaging multiple values together.
Tuples are enclosed in parentheses, while records are enclosed in
curly braces.  The components of both tuples and records are separated by
commas.  The components of tuples are expressions, while the
components of records are a label and a value separated by an equal
sign.  Examples:

    (1,2,3)           // A tuple with 3 component
    ()                // A tuple with no components

    { x = 1, y = 2 }  // A record with two fields, `x` and `y`
    {}                // A record with no fields

The components of tuples are identified by position, while the
components of records are identified by their label, and so the
ordering of record components is not important.  Examples:

               (1,2) == (1,2)               // True
               (1,2) == (2,1)               // False

    { x = 1, y = 2 } == { x = 1, y = 2 }    // True
    { x = 1, y = 2 } == { y = 2, x = 1 }    // True


Accessing Fields
----------------

The components of a record or a tuple may be accessed in two ways: via
pattern matching or by using explicit component selectors.  Explicit
component selectors are written as follows:

    (15, 20).0           == 15
    (15, 20).1           == 20

    { x = 15, y = 20 }.x == 15

Explicit record selectors may be used only if the program contains
sufficient type information to determine the shape of the tuple or
record.  For example:

    type T = { sign : Bit, number : [15] }

    // Valid definition:
    // the type of the record is known.
    isPositive : T -> Bit
    isPositive x = x.sign

    // Invalid definition:
    // insufficient type information.
    badDef x = x.f

The components of a tuple or a record may also be accessed using
pattern matching.  Patterns for tuples and records mirror the syntax
for constructing values: tuple patterns use parentheses, while record
patterns use braces.  Examples:

    getFst (x,_) = x

    distance2 { x = xPos, y = yPos } = xPos ^^ 2 + yPos ^^ 2

    f p = x + y where
        (x, y) = p

Selectors are also lifted through sequence and function types, point-wise,
so that the following equations should hold:

    xs.l == [ x.l | x <- xs ]     // sequences
    f.l  == \x -> (f x).l         // functions

Thus, if we have a sequence of tuples, `xs`, then we can quickly obtain a
sequence with only the tuples' first components by writing `xs.0`.

Similarly, if we have a function, `f`, that computes a tuple of results,
then we can write `f.0` to get a function that computes only the first
entry in the tuple.

This behavior is quite handy when examining complex data at the REPL.




Updating Fields
---------------

The components of a record or a tuple may be updated using the following
notation:

    // Example values
    r = { x = 15, y = 20 }      // a record
    t = (True,True)             // a tuple
    n = { pt = r, size = 100 }  // nested record

    // Setting fields
    { r | x = 30 }          == { x = 30, y = 20 }
    { t | 0 = False }       == (False,True)

    // Update relative to the old value
    { r | x -> x + 5 }      == { x = 20, y = 20 }

    // Update a nested field
    { n | pt.x = 10 }       == { pt = { x = 10, y = 20 }, size = 100 }
    { n | pt.x -> x + 10 }  == { pt = { x = 25, y = 20 }, size = 100 }



Sequences
=========

A sequence is a fixed-length collection of elements of the same type.
The type of a finite sequence of length `n`, with elements of type `a`
is `[n] a`.  Often, a finite sequence of bits, `[n] Bit`, is called a
_word_.  We may abbreviate the type `[n] Bit` as `[n]`.  An infinite
sequence with elements of type `a` has type `[inf] a`, and `[inf]` is
an infinite stream of bits.

    [e1,e2,e3]                    // A sequence with three elements

    [t1 .. t3 ]                   // Sequence enumerations
    [t1, t2 .. t3 ]               // Step by t2 - t1
    [e1 ... ]                     // Infinite sequence starting at e1
    [e1, e2 ... ]                 // Infinite sequence stepping by e2-e1

    [ e | p11 <- e11, p12 <- e12  // Sequence comprehensions
        | p21 <- e21, p22 <- e22 ]

    x = generate (\i -> e)        // Sequence from generating function
    x @ i = e                     // Sequence with index binding
    arr @ i @ j = e               // Two-dimensional sequence


Note: the bounds in finite sequences (those with `..`) are type
expressions, while the bounds in infinite sequences are value
expressions.

Operator                       Description
---------------------------    -----------
  `#`                          Sequence concatenation
  `>>`  `<<`                   Shift (right, left)
  `>>>` `<<<`                  Rotate (right, left)
  `>>$`                        Arithmetic right shift (on bitvectors only)
  `@` `!`                      Access elements (front, back)
  `@@` `!!`                    Access sub-sequence (front, back)
  `update` `updateEnd`         Update the value of a sequence at a location (front, back)
  `updates` `updatesEnd`       Update multiple values of a sequence (front, back)

Table: Sequence operations.

There are also lifted pointwise operations.

    [p1, p2, p3, p4]          // Sequence pattern
    p1 # p2                   // Split sequence pattern

Functions
=========

    \p1 p2 -> e              // Lambda expression
    f p1 p2 = e              // Function definition

Local Declarations
==================

    e where ds

Note that by default, any local declarations without type signatures
are monomorphized. If you need a local declaration to be polymorphic,
use an explicit type signature.

Explicit Type Instantiation
===========================

If `f` is a polymorphic value with type:

    f : { tyParam } tyParam
    f = zero

you can evaluate `f`, passing it a type parameter:

    f `{ tyParam = 13 }


Demoting Numeric Types to Values
================================

The value corresponding to a numeric type may be accessed using the
following notation:

    `t

Here `t` should be a type expression with numeric kind.  The resulting
expression is a finite word, which is sufficiently large to accommodate
the value of the type:

    `t : {n} (fin n, n >= width t) => [n]

Explicit Type Annotations
=========================

Explicit type annotations may be added on expressions, patterns, and
in argument definitions.

    e : t

    p : t

    f (x : t) = ...

Type Signatures
===============

    f,g : {a,b} (fin a) => [a] b

Type Synonym Declarations
=========================

    type T a b = [a] b


Modules
=======

A ***module*** is used to group some related definitions.


    module M where

    type T = [8]

    f : [8]
    f = 10

Hierarchical Module Names
=========================

Module may have either simple or ***hierarchical*** names.
Hierarchical names are constructed by gluing together ordinary
identifiers using the symbol `::`.

    module Hash::SHA256 where

    sha256 = ...

The structure in the name may be used to group together related
modules.  Also, the Cryptol implementation uses the structure of the
name to locate the file containing the definition of the module.
For example, when searching for module `Hash::SHA256`, Cryptol
will look for a file named `SHA256.cry` in a directory called
`Hash`, contained in one of the directories specified by `CRYPTOLPATH`.


Module Imports
==============

To use the definitions from one module in another module, we use
`import` declarations:


    // Provide some definitions
    module M where

    f : [8]
    f = 2


    // Uses definitions from `M`
    module N where

    import M  // import all definitions from `M`

    g = f   // `f` was imported from `M`


Import Lists
============

Sometimes, we may want to import only some of the definitions
from a module.  To do so, we use an import declaration with
an ***import list***.

    module M where

    f = 0x02
    g = 0x03
    h = 0x04

    module N where

    import M(f,g)  // Imports only `f` and `g`, but not `h`

    x = f + g

Using explicit import lists helps reduce name collisions.
It also tends to make code easier to understand,  because
it makes it easy to see the source of definitions.


Hiding Imports
==============

Sometimes a module may provide many definitions, and we want to use
most of them but with a few exceptions (e.g., because those would
result to a name clash).   In such situations it is convenient
to use a ***hiding*** import:


   module M where

    f = 0x02
    g = 0x03
    h = 0x04



    module N where

    import M hiding (h) // Import everything but `h`

    x = f + g



Qualified Module Imports
========================

Another way to avoid name collisions is by using a
***qualified*** import.

    module M where

    f : [8]
    f = 2

    module N where

    import M as P

    g = P::f
    // `f` was imported from `M`
    // but when used it needs to be prefixed by the qualified `P`

Qualified imports make it possible to work with definitions
that happen to have the same name but are defined in different modules.

Qualified imports may be combined with import lists or hiding clauses:

    import A as B (f)         // introduces B::f
    import X as Y hiding (f)  // introduces everything but `f` from X
                              // using the prefix `X`

It is also possible the use the same qualifier prefix for imports
from different modules.  For example:

    import A as B
    import X as B

Such declarations will introduces all definitions from `A` and `X`
but to use them, you would have to qualify using the prefix `B:::`.


Private Blocks
==============

In some cases, definitions in a module might use helper
functions that are not intended to be used outside the module.
It is good practice to place such declarations in ***private blocks***:


    module M where

    f : [8]
    f = 0x01 + helper1 + helper2

    private

      helper1 : [8]
      helper1 = 2

      helper2 : [8]
      helper2 = 3

The keyword `private` introduce a new layout scope, and all declarations
in the block are considered to be private to the module.  A single module
may contain multiple private blocks.  For example, the following module
is equivalent to the previous one:

    module M where

    f : [8]
    f = 0x01 + helper1 + helper2

    private
      helper1 : [8]
      helper1 = 2

    private
      helper2 : [8]
      helper2 = 3






Parameterized Modules
=====================


    module M where

    parameter
      type n : #              // `n` is a numeric type parameter

      type constraint (fin n, n >= 1)
        // Assumptions about the parameter

      x : [n]                 // A value parameter

    // This definition uses the parameters.
    f : [n]
    f = 1 + x


Named Module Instantiations
===========================

One way to use a parameterized module is through a named instantiation:

    // A parameterized module
    module M where

    parameter
      type n : #
      x      : [n]
      y      : [n]

    f : [n]
    f = x + y


    // A module instantiation
    module N = M where

    type n = 32
    x      = 11
    y      = helper

    helper = 12

The second module, `N`, is computed by instantiating the parameterized
module `M`.  Module `N` will provide the exact same definitions as `M`,
except that the parameters will be replaced by the values in the body
of `N`.   In this example, `N` provides just a single definition, `f`.

Note that the only purpose of the body of `N` (the declarations
after the `where` keyword) is to define the parameters for `M`.


Parameterized Instantiations
============================

It is possible for a module instantiations to be itself parameterized.
This could be useful if we need to define some of a module's parameters
but not others.


    // A parameterized module
    module M where

    parameter
      type n : #
      x      : [n]
      y      : [n]

    f : [n]
    f = x + y


    // A parameterized instantiation
    module N = M where

    parameter
      x : [32]

    type n = 32
    y      = helper

    helper = 12

In this case `N` has a single parameter `x`.  The result of instantiating
`N` would result in instantiating `M` using the value of `x` and `12`
for the value of `y`.


Importing Parameterized Modules
===============================

It is also possible to import a parameterized module without using
a module instantiation:

    module M where

    parameter
      x : [8]
      y : [8]

    f : [8]
    f = x + y


    module N where

    import `M

    g = f { x = 2, y = 3 }

A ***backtick*** at the start of the name of an imported module indicates
that we are importing a parameterized module.  In this case, Cryptol
will import all definitions from the module as usual, however every
definition will have some additional parameters corresponding to
the parameters of a module.  All value parameters are grouped in a record.

This is why in the example `f` is applied to a record of values,
even though its definition in `M` does not look like a function.