.. Cryptol Reference Manual documentation master file, created by sphinx-quickstart on Thu Apr 29 15:31:18 2021. You can adapt this file completely to your liking, but it should at least contain the root `toctree` directive. ######################## Cryptol Reference Manual ######################## .. toctree:: :maxdepth: 3 :caption: Contents: ******************************************************************************** Basic Syntax ******************************************************************************** Declarations ============ .. code-block:: cryptol f x = x + y + z Type Signatures --------------- .. code-block:: cryptol f,g : {a,b} (fin a) => [a] b 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. Consider, for example, the following Cryptol declarations: .. code-block:: cryptol 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`. The same principle applies to the declarations in the ``where`` block of `f`, which defines two more local names, `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. .. code-block:: cryptol /* This is a block comment */ // This is a line comment /* This is a /* Nested */ block comment */ .. todo:: Document ``/** */`` 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 and Built-in Operators`_). .. code-block:: cryptol :caption: Examples of identifiers 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 infixl infixr infix primitive parameter constraint down by .. _Keywords: .. code-block:: none :caption: Keywords else include property let infixl parameter extern module then import infixr constraint if newtype type as infix down private pragma where hiding primitive by The following table contains Cryptol's operators and their associativity with lowest precedence operators first, and highest precedence last. .. table:: Operator precedences +-----------------------------------------+-----------------+ | Operator | Associativity | +=========================================+=================+ | ``==>`` | right | +-----------------------------------------+-----------------+ | ``\/`` | right | +-----------------------------------------+-----------------+ | ``/\`` | right | +-----------------------------------------+-----------------+ | ``==`` ``!=`` ``===`` ``!==`` | not associative | +-----------------------------------------+-----------------+ | ``>`` ``<`` ``<=`` ``>=`` | not associative | | ``<$`` ``>$`` ``<=$`` ``>=$`` | | +-----------------------------------------+-----------------+ | ``||`` | right | +-----------------------------------------+-----------------+ | ``^`` | left | +-----------------------------------------+-----------------+ | ``&&`` | right | +-----------------------------------------+-----------------+ | ``#`` | right | +-----------------------------------------+-----------------+ | ``>>`` ``<<`` ``>>>`` ``<<<`` ``>>$`` | left | +-----------------------------------------+-----------------+ | ``+`` ``-`` | left | +-----------------------------------------+-----------------+ | ``*`` ``/`` ``%`` ``/$`` ``%$`` | left | +-----------------------------------------+-----------------+ | ``^^`` | right | +-----------------------------------------+-----------------+ | ``@`` ``@@`` ``!`` ``!!`` | left | +-----------------------------------------+-----------------+ | (unary) ``-`` ``~`` | right | +-----------------------------------------+-----------------+ 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. .. table:: Type-level operators +------------+----------------------------------------+ | Operator | Meaning | +============+========================================+ | ``+`` | Addition | +------------+----------------------------------------+ | ``-`` | Subtraction | +------------+----------------------------------------+ | ``*`` | Multiplication | +------------+----------------------------------------+ | ``/`` | Division | +------------+----------------------------------------+ | ``/^`` | Ceiling division (``/`` rounded up) | +------------+----------------------------------------+ | ``%`` | Modulus | +------------+----------------------------------------+ | ``%^`` | Ceiling modulus (compute padding) | +------------+----------------------------------------+ | ``^^`` | Exponentiation | +------------+----------------------------------------+ | ``lg2`` | Ceiling logarithm (base 2) | +------------+----------------------------------------+ | ``width`` | Bit width (equal to ``lg2(n+1)``) | +------------+----------------------------------------+ | ``max`` | Maximum | +------------+----------------------------------------+ | ``min`` | Minimum | +------------+----------------------------------------+ 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. .. code-block:: cryptol :caption: Examples of literals 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: .. code-block:: cryptol :caption: Literals and their types 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 Numeric literals may also be written as polynomials by writing a polynomial expression in terms of `x` between an opening ``<|`` and a closing ``|>``. Numeric literals in polynomial notation result in bit sequences of length one more than the degree of the polynomial. Examples: .. code-block:: cryptol :caption: Polynomial literals <| x^^6 + x^^4 + x^^2 + x^^1 + 1 |> // : [7], equal to 0b1010111 <| x^^4 + x^^3 + x |> // : [5], equal to 0b11010 Cryptol also supports fractional literals using binary (prefix ``0b``), octal (prefix ``0o``), decimal (no prefix), and hexadecimal (prefix ``ox``) digits. A fractional literal must contain a ``.`` and may optionally have an exponent. The base of the exponent for binary, octal, and hexadecimal literals is 2 and the exponent is marked using the symbol ``p``. Decimal fractional literals use exponent base 10, and the symbol ``e``. Examples: .. code-block:: cryptol :caption: Fractional literals 10.2 10.2e3 // 10.2 * 10^3 0x30.1 // 3 * 64 + 1/16 0x30.1p4 // (3 * 64 + 1/16) * 2^4 All fractional literals are overloaded and may be used with types that support fractional numbers (e.g., ``Rational``, and the ``Float`` family of types). Some types (e.g. the ``Float`` family) cannot represent all fractional literals precisely. Such literals are rejected statically when using binary, octal, or hexadecimal notation. When using decimal notation, the literal is rounded to the closest representable even number. All numeric literals may also include ``_``, which has no effect on the literal value but may be used to improve readability. Here are some examples: .. code-block:: cryptol :caption: Using _ 0b_0000_0010 0x_FFFF_FFEA ******************************************************************************** Expressions ******************************************************************************** This section provides an overview of the Cryptol's expression syntax. Calling Functions ================= .. code-block:: cryptol 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 ================ .. code-block:: cryptol -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 =============== .. code-block:: cryptol 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 ================ Explicit type annotations may be added on expressions, patterns, and in argument definitions. .. code-block:: cryptol 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` [1..9 : [8]] // specify that elements in `[1..9]` have type `[8]` f (x : [8]) = x + 1 // type annotation on patterns .. todo:: Patterns with type variables Explicit Type Instantiation =========================== If ``f`` is a polymorphic value with type: .. code-block:: cryptol f : { tyParam } tyParam f = zero you can evaluate ``f``, passing it a type parameter: .. code-block:: cryptol f `{ tyParam = 13 } Local Declarations ================== Local declarations have the weakest precedence of all expressions. .. code-block:: cryptol 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: .. code-block:: cryptol f \x -> x // call `f` with one argument `x -> x` 2 + if x then y else z // call `+` with two arguments: `2` and `if ...` Conditionals ------------ The ``if ... then ... else`` construct can be used with multiple branches. For example: .. code-block:: cryptol 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 Demoting Numeric Types to Values -------------------------------- The value corresponding to a numeric type may be accessed using the following notation: .. code-block:: cryptol `t Here `t` should be a finite type expression with numeric kind. The resulting expression will be of a numeric base type, which is sufficiently large to accommodate the value of the type: .. code-block:: cryptol `t : {a} (Literal t a) => a This backtick notation is syntax sugar for an application of the `number` primtive, so the above may be written as: .. code-block:: cryptol number`{t} : {a} (Literal t a) => a If a type cannot be inferred from context, a suitable type will be automatically chosen if possible, usually `Integer`. ******************************************************************************** Basic Types ******************************************************************************** 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. .. table:: Bit operations. +--------------------------+---------------+------------------------+ | Operator | Associativity | Description | +--------------------------+---------------+------------------------+ | ``==>`` | right | Short-cut implication | +--------------------------+---------------+------------------------+ | ``\/`` | right | Short-cut or | +--------------------------+---------------+------------------------+ | ``/\`` | right | Short-cut and | +--------------------------+---------------+------------------------+ | `!=` `==` | none | Not equals, equals | +--------------------------+---------------+------------------------+ | ``>`` ``<`` ``<=`` ``>=``| none | Comparisons; | | ``<$`` ``>$`` | | $ indicates signed | | ``<=$`` ``>=$`` | | | +--------------------------+---------------+------------------------+ | `||` | right | Logical or | +--------------------------+---------------+------------------------+ | `^` | left | Exclusive-or | +--------------------------+---------------+------------------------+ | `&&` | right | Logical and | +--------------------------+---------------+------------------------+ | `~` | right | Logical negation | +--------------------------+---------------+------------------------+ 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: .. code-block:: cryptol (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 for most purposes. Examples: .. code-block:: cryptol (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 Ordering on tuples and records is defined lexicographically. Tuple components are compared in the order they appear, whereas record fields are compared in alphabetical order of field names. 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: .. code-block:: cryptol (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: .. code-block:: cryptol 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: .. code-block:: cryptol 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: .. code-block:: cryptol 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: .. code-block:: cryptol // 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. .. code-block:: cryptol [e1,e2,e3] // A sequence with three elements [t1 .. t2] // Enumeration [t1 .. t2 down by n] // Enumeration (downward stride, ex. bound) [t1, t2 .. t3] // Enumeration (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. .. table:: Sequence operations. +------------------------------+---------------------------------------------+ | 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) | +------------------------------+---------------------------------------------+ There are also lifted pointwise operations. .. code-block:: cryptol [p1, p2, p3, p4] // Sequence pattern p1 # p2 // Split sequence pattern Functions ========= .. code-block:: cryptol \p1 p2 -> e // Lambda expression f p1 p2 = e // Function definition ******************************************************************************** Type Declarations ******************************************************************************** Type Synonyms ============= .. code-block:: cryptol type T a b = [a] b A ``type`` declaration creates a synonym for a pre-existing type expression, which may optionally have arguments. A type synonym is transparently unfolded at use sites and is treated as though the user had instead written the body of the type synonym in line. Type synonyms may mention other synonyms, but it is not allowed to create a recursive collection of type synonyms. Newtypes ======== .. code-block:: cryptol newtype NewT a b = { seq : [a]b } A ``newtype`` declaration declares a new named type which is defined by a record body. Unlike type synonyms, each named ``newtype`` is treated as a distinct type by the type checker, even if they have the same bodies. Moreover, types created by a ``newtype`` declaration will not be members of any typeclasses, even if the record defining their body would be. For the purposes of typechecking, two newtypes are considered equal only if all their arguments are equal, even if the arguments do not appear in the body of the newtype, or are otherwise irrelevant. Just like type synonyms, newtypes are not allowed to form recursive groups. Every ``newtype`` declaration brings into scope a new function with the same name as the type which can be used to create values of the newtype. .. code-block:: cryptol x : NewT 3 Integer x = NewT { seq = [1,2,3] } Just as with records, field projections can be used directly on values of newtypes to extract the values in the body of the type. .. code-block:: none > sum x.seq 6 ******************************************************************************** Modules ******************************************************************************** A *module* is used to group some related definitions. Each file may contain at most one top-level module. .. code-block:: cryptol 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 ``::``. .. code-block:: cryptol 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: .. code-block:: cryptol :caption: module M // Provide some definitions module M where f : [8] f = 2 .. code-block:: cryptol :caption: module N // 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*. .. code-block:: cryptol module M where f = 0x02 g = 0x03 h = 0x04 .. code-block:: cryptol 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: .. code-block:: cryptol :caption: module M module M where f = 0x02 g = 0x03 h = 0x04 .. code-block:: cryptol :caption: module N 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. .. code-block:: cryptol :caption: module M module M where f : [8] f = 2 .. code-block:: cryptol :caption: module N 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 qualifier `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: .. code-block:: cryptol :caption: Example 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 to use the same qualifier prefix for imports from different modules. For example: .. code-block:: cryptol :caption: 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*: .. code-block:: cryptol :caption: Private blocks module M where f : [8] f = 0x01 + helper1 + helper2 private helper1 : [8] helper1 = 2 helper2 : [8] helper2 = 3 The private block only needs to be indented if it might be followed by additional public declarations. If all remaining declarations are to be private then no additional indentation is needed as the `private` block will extend to the end of the module. .. code-block:: cryptol :caption: Private blocks module M where f : [8] f = 0x01 + helper1 + helper2 private helper1 : [8] helper1 = 2 helper2 : [8] helper2 = 3 The keyword ``private`` introduces 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: .. code-block:: cryptol :caption: Private blocks module M where f : [8] f = 0x01 + helper1 + helper2 private helper1 : [8] helper1 = 2 private helper2 : [8] helper2 = 3 Parameterized Modules ===================== .. warning:: This section documents the current design, but we are in the process of redesigning some aspects of the parameterized modules mechanism. .. code-block:: cryptol 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: .. code-block:: cryptol // 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 instantiation to be itself parameterized. This could be useful if we need to define some of a module's parameters but not others. .. code-block:: cryptol // 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: .. code-block:: cryptol module M where parameter x : [8] y : [8] f : [8] f = x + y .. code-block:: cryptol 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.