If we were to come up with a variable name for the given type what name would it be?

An instance for a given type Ty is simply given by:

instance Name Ty where name _ = "x"

Examples:

> name (undefined :: Int)
"x"

> name (undefined :: Bool)
"p"

> name (undefined :: [Int])
"xs"

This is then used to generate an infinite list of variable names:

> names (undefined :: Int)
["x", "y", "z", "x'", "y'", "z'", "x''", "y''", "z''", ...]

> names (undefined :: Bool)
["p", "q", "r", "p'", "q'", "r'", "p''", "q''", "r''", ...]

> names (undefined :: [Int])
["xs", "ys", "zs", "xs'", "ys'", "zs'", "xs''", "ys''", ...]

Values of type Expr represent objects or applications between objects. Each object is encapsulated together with its type and string representation. Values encoded in Exprs are always monomorphic.

An Expr can be constructed using:

• val, for values that are Show instances;
• value, for values that are not Show instances, like functions;
• :$, for applications between Exprs.

> val False
False :: Bool

> value "not" not :$ val False
not False :: Bool

An Expr can be evaluated using evaluate, eval or evl.

> evl $ val (1 :: Int) :: Int
1

> evaluate $ val (1 :: Int) :: Maybe Bool
Nothing

> eval 'a' (val 'b')
'b'

Showing a value of type Expr will return a pretty-printed representation of the expression together with its type.

> show (value "not" not :$ val False)
"not False :: Bool"

Expr is like Dynamic but has support for applications and variables (:$, var).

The var underscore convention: Functions that manipulate Exprs usually follow the convention where a value whose String representation starts with '_' represents a variable.

Express typeclass instances provide an expr function that allows values to be deeply encoded as applications of Exprs.

expr False  =  val False
expr (Just True)  =  value "Just" (Just :: Bool -> Maybe Bool) :$ val True

The function expr can be contrasted with the function val:

• val always encodes values as atomic Value Exprs -- shallow encoding.
• expr ideally encodes expressions as applications (:$) between Value Exprs -- deep encoding.

Depending on the situation, one or the other may be desirable.

Instances can be automatically derived using the TH function deriveExpress.

The following example shows a datatype and its instance:

data Stack a = Stack a (Stack a) | Empty

instance Express a => Express (Stack a) where
  expr s@(Stack x y) = value "Stack" (Stack ->>: s) :$ expr x :$ expr y
  expr s@Empty       = value "Empty" (Empty   -: s)

To declare expr it may be useful to use auxiliary type binding operators: -:, ->:, ->>:, ->>>:, ->>>>:, ->>>>>:, ...

For types with atomic values, just declare expr = val

O(n*n*m). Substitute subexpressions in an expression from the given list of substitutions. (cf. mapSubexprs).

Please consider using //- if you are replacing just terminal values as it is faster.

Given that:

> let xx = var "x" (undefined :: Int)
> let yy = var "y" (undefined :: Int)
> let zz = var "z" (undefined :: Int)
> let xx -+- yy = value "+" ((+) :: Int->Int->Int) :$ xx :$ yy

Then:

> ((xx -+- yy) -+- (yy -+- zz)) // [(xx -+- yy, yy), (yy -+- zz, yy)]
y + y :: Int

> ((xx -+- yy) -+- zz) // [(xx -+- yy, zz), (zz, xx -+- yy)]
z + (x + y) :: Int

Replacement happens only once with outer expressions having more precedence than inner expressions.

> (xx -+- yy) // [(yy,xx), (xx -+- yy,zz), (zz,xx)]
z :: Int

Given that the argument list has length m, this function is O(n*n*m). Remember that since n is the size of an expression, comparing two expressions is O(n) in the worst case, and we may need to compare with n subexpressions in the worst case.

Fill holes in an expression with the given list.

> let i_  =  hole (undefined :: Int)
> let e1 -+- e2  =  value "+" ((+) :: Int -> Int -> Int) :$ e1 :$ e2
> let xx  =  var "x" (undefined :: Int)
> let yy  =  var "y" (undefined :: Int)

> fill (i_ -+- i_) [xx, yy]
x + y :: Int

> fill (i_ -+- i_) [xx, xx]
x + x :: Int

> let one  =  val (1::Int)

> fill (i_ -+- i_) [one, one -+- one]
1 + (1 + 1) :: Int

This function silently remaining expressions:

> fill i_ [xx, yy]
x :: Int

This function silently keeps remaining holes:

> fill (i_ -+- i_ -+- i_) [xx, yy]
(x + y) + _ :: Int

This function silently skips remaining holes if one is not of the right type:

> fill (i_ -+- i_ -+- i_) [xx, val 'c', yy]
(x + _) + _ :: Int

O(n). Folds a list of Exprs into a single Expr. (cf. unfold)

This always generates an ill-typed expression.

fold [val False, val True, val (1::Int)]
[False,True,1] :: ill-typed # ExprList $ Bool #

This is useful when applying transformations on lists of Exprs, such as canonicalize, mapValues or canonicalVariations.

> let ii = var "i" (undefined::Int)
> let kk = var "k" (undefined::Int)
> let qq = var "q" (undefined::Bool)
> let notE = value "not" not
> unfold . canonicalize . fold $ [ii,kk,notE :$ qq, notE :$ val False]
[x :: Int,y :: Int,not p :: Bool,not False :: Bool]

O(n). Just the value of an expression when possible (correct type), Nothing otherwise. This does not catch errors from undefined Dynamic values.

> let one = val (1 :: Int)
> let bee = val 'b'
> let negateE = value "negate" (negate :: Int -> Int)

> evaluate one :: Maybe Int
Just 1

> evaluate one :: Maybe Char
Nothing

> evaluate bee :: Maybe Int
Nothing

> evaluate bee :: Maybe Char
Just 'b'

> evaluate $ negateE :$ one :: Maybe Int
Just (-1)

> evaluate $ negateE :$ bee :: Maybe Int
Nothing

O(n). Unfolds an Expr representing a list into a list of Exprs. This reverses the effect of fold.

> expr [1,2,3::Int]
[1,2,3] :: [Int]
> unfold $ expr [1,2,3::Int]
[1 :: Int,2 :: Int,3 :: Int]

The list constructor with Int as element type encoded as an Expr.

> cons
(:) :: Int -> [Int] -> [Int]

> cons :$ one :$ nil
[1] :: [Int]

Consider using -:- and unit when building lists of Expr.

The function div for the Int type lifted over the Expr type. (See also mod.)

> six `div'` four
6 `div` 4 :: Int

The function mod for the Int type lifted over the Expr type. (See also div.)

> six `mod'` four
6 `mod` 4 :: Int

Lists valid applications between lists of Exprs

> [notE, plus] >$$< [false, true, zero]
[not False :: Bool,not True :: Bool,(0 +) :: Int -> Int]

O(n). Returns the size of the given expression, i.e. the number of terminal values in it.

zero       =  val (0 :: Int)
one        =  val (1 :: Int)
two        =  val (2 :: Int)
xx -+- yy  =  value "+" ((+) :: Int->Int->Int) :$ xx :$ yy
abs' xx    =  value "abs" (abs :: Int->Int) :$ xx

> size zero
1

> size (one -+- two)
3

> size (abs' one)
2

O(1). It takes a string representation of a value and a value, returning an Expr with that terminal value. For instances of Show, it is preferable to use val.

> value "0" (0 :: Integer)
0 :: Integer

> value "'a'" 'a'
'a' :: Char

> value "True" True
True :: Bool

> value "id" (id :: Int -> Int)
id :: Int -> Int

> value "(+)" ((+) :: Int -> Int -> Int)
(+) :: Int -> Int -> Int

> value "sort" (sort :: [Bool] -> [Bool])
sort :: [Bool] -> [Bool]

The operator * for the Int type. (See also -*-.)

> times
(*) :: Int -> Int -> Int

> times :$ two
(2 *) :: Int -> Int

> times :$ xx :$ yy
x * y :: Int

O(n). Computes the type of an expression. This raises errors, but this should not happen if expressions are smart-constructed with $$.

> let one = val (1 :: Int)
> let bee = val 'b'
> let absE = value "abs" (abs :: Int -> Int)

> typ one
Int

> typ bee
Char

> typ absE
Int -> Int

> typ (absE :$ one)
Int

> typ (absE :$ bee)
*** Exception: type mismatch, cannot apply `Int -> Int' to `Char'

> typ ((absE :$ bee) :$ one)
*** Exception: type mismatch, cannot apply `Int -> Int' to `Char'

O(n). Return the arity of the given expression, i.e. the number of arguments that its type takes.

> arity (val (0::Int))
0

> arity (val False)
0

> arity (value "id" (id :: Int -> Int))
1

> arity (value "const" (const :: Int -> Int -> Int))
2

> arity (one -+- two)
0

Function composition encoded as an Expr:

> compose
(.) :: (Int -> Int) -> (Int -> Int) -> Int -> Int

unit constructs a list with a single element. This works for elements of type Int, Char and Bool.

> unit one
[1]

> unit false
[False]

The value 0 bound to the Int type encoded as an Expr.

> zero
0 :: Int

Given two expressions, returns a Just list of matches of subexpressions of the first expressions to variables in the second expression. Returns Nothing when there is no match.

> let zero = val (0::Int)
> let one  = val (1::Int)
> let xx   = var "x" (undefined :: Int)
> let yy   = var "y" (undefined :: Int)
> let e1 -+- e2  =  value "+" ((+)::Int->Int->Int) :$ e1 :$ e2

> (zero -+- one) `match` (xx -+- yy)
Just [(y :: Int,1 :: Int),(x :: Int,0 :: Int)]

> (zero -+- (one -+- two)) `match` (xx -+- yy)
Just [(y :: Int,1 + 2 :: Int),(x :: Int,0 :: Int)]

> (zero -+- (one -+- two)) `match` (xx -+- (yy -+- yy))
Nothing

In short:

          (zero -+- one) `match` (xx -+- yy)           =  Just [(xx,zero), (yy,one)]
(zero -+- (one -+- two)) `match` (xx -+- yy)           =  Just [(xx,zero), (yy,one-+-two)]
(zero -+- (one -+- two)) `match` (xx -+- (yy -+- yy))  =  Nothing

Canonicalizes an Expr so that variable names appear in order. Variable names are taken from the preludeNameInstances.

> canonicalize (xx -+- yy)
x + y :: Int

> canonicalize (yy -+- xx)
x + y :: Int

> canonicalize (xx -+- xx)
x + x :: Int

> canonicalize (yy -+- yy)
x + x :: Int

Constants are untouched:

> canonicalize (jj -+- (zero -+- abs' ii))
x + (y + abs y) :: Int

This also works for variable functions:

> canonicalize (gg yy -+- ff xx -+- gg xx)
(f x + g y) + f y :: Int

O(n*m). Applies a function to all terminal values in an expression. (cf. //-)

Given that:

> let zero  = val (0 :: Int)
> let one   = val (1 :: Int)
> let two   = val (2 :: Int)
> let three = val (3 :: Int)
> let xx -+- yy = value "+" ((+) :: Int->Int->Int) :$ xx :$ yy
> let intToZero e = if typ e == typ zero then zero else e

Then:

> one -+- (two -+- three)
1 + (2 + 3) :: Int

> mapValues intToZero $ one -+- (two -+- three)
0 + (0 + 0) :: Integer

Given that the argument function is O(m), this function is O(n*m).

Returns all canonical variations of an Expr by filling holes with variables. Where possible, variations are listed from most general to least general.

> canonicalVariations $ i_
[x :: Int]

> canonicalVariations $ i_ -+- i_
[ x + y :: Int
, x + x :: Int ]

> canonicalVariations $ i_ -+- i_ -+- i_
[ (x + y) + z :: Int
, (x + y) + x :: Int
, (x + y) + y :: Int
, (x + x) + y :: Int
, (x + x) + x :: Int ]

> canonicalVariations $ i_ -+- ord' c_
[x + ord c :: Int]

> canonicalVariations $ i_ -+- i_ -+- ord' c_
[ (x + y) + ord c :: Int
, (x + x) + ord c :: Int ]

> canonicalVariations $ i_ -+- i_ -+- length' (c_ -:- unit c_)
[ (x + y) + length (c:d:"") :: Int
, (x + y) + length (c:c:"") :: Int
, (x + x) + length (c:d:"") :: Int
, (x + x) + length (c:c:"") :: Int ]

In an expression without holes this functions just returns a singleton list with the expression itself:

> canonicalVariations $ val (0 :: Int)
[0 :: Int]

> canonicalVariations $ ord' bee
[ord 'b' :: Int]

When applying this to expressions already containing variables clashes are avoided and these variables are not touched:

> canonicalVariations $ i_ -+- ii -+- jj -+- i_
[ x + i + j + y :: Int
, x + i + j + y :: Int ]

> canonicalVariations $ ii -+- jj
[i + j :: Int]

> canonicalVariations $ xx -+- i_ -+- i_ -+- length' (c_ -:- unit c_) -+- yy
[ (((x + z) + x') + length (c:d:"")) + y :: Int
, (((x + z) + x') + length (c:c:"")) + y :: Int
, (((x + z) + z) + length (c:d:"")) + y :: Int
, (((x + z) + z) + length (c:c:"")) + y :: Int
]

Derives an Express instance for the given type Name.

This function needs the TemplateHaskell extension.

If -:, ->:, ->>:, ->>>:, ... are not in scope, this will derive them as well.

Returns an infinite list of variable names based on the given template.

> variableNamesFromTemplate "x"
["x", "y", "z", "x'", "y'", ...]

> variableNamesFromTemplate "p"
["p", "q", "r", "p'", "q'", ...]

> variableNamesFromTemplate "xy"
["xy", "zw", "xy'", "zw'", "xy''", ...]

Returns na infinite list of variable names from the given type: the result of variableNamesFromTemplate after name.

> names (undefined :: Int)
["x", "y", "z", "x'", "y'", "z'", "x''", "y''", "z''", ...]

> names (undefined :: Bool)
["p", "q", "r", "p'", "q'", "r'", "p''", "q''", "r''", ...]

> names (undefined :: [Int])
["xs", "ys", "zs", "xs'", "ys'", "zs'", "xs''", "ys''", ...]

Given two Exprs, checks if the first expression is an instance of the second in terms of variables. (cf. encompasses, hasInstanceOf)

> let zero = val (0::Int)
> let one  = val (1::Int)
> let xx   = var "x" (undefined :: Int)
> let yy   = var "y" (undefined :: Int)
> let e1 -+- e2  =  value "+" ((+)::Int->Int->Int) :$ e1 :$ e2

 one `isInstanceOf` one   =  True
  xx `isInstanceOf` xx    =  True
  yy `isInstanceOf` xx    =  True
zero `isInstanceOf` xx    =  True
  xx `isInstanceOf` zero  =  False
 one `isInstanceOf` zero  =  False
  (xx -+- (yy -+- xx)) `isInstanceOf`   (xx -+- yy)  =  True
  (yy -+- (yy -+- xx)) `isInstanceOf`   (xx -+- yy)  =  True
(zero -+- (yy -+- xx)) `isInstanceOf` (zero -+- yy)  =  True
 (one -+- (yy -+- xx)) `isInstanceOf` (zero -+- yy)  =  False

Derives a Name instance for the given type Name.

This function needs the TemplateHaskell extension.

Same as deriveName but does not warn when instance already exists (deriveName is preferable).

Derives a Name instance for a given type Name cascading derivation of type arguments as well.

O(1). A shorthand for value for values that are Show instances.

> val (0 :: Int)
0 :: Int

> val 'a'
'a' :: Char

> val True
True :: Bool

Example equivalences to value:

val 0     =  value "0" 0
val 'a'   =  value "'a'" 'a'
val True  =  value "True" True

O(n). Creates an Expr representing a function application. Just an Expr application if the types match, Nothing otherwise. (cf. :$)

> value "id" (id :: () -> ()) $$ val ()
Just (id () :: ())

> value "abs" (abs :: Int -> Int) $$ val (1337 :: Int)
Just (abs 1337 :: Int)

> value "abs" (abs :: Int -> Int) $$ val 'a'
Nothing

> value "abs" (abs :: Int -> Int) $$ val ()
Nothing

O(1). Creates an Expr representing a variable with the given name and argument type.

> var "x" (undefined :: Int)
x :: Int

> var "u" (undefined :: ())
u :: ()

> var "xs" (undefined :: [Int])
xs :: [Int]

This function follows the underscore convention: a variable is just a value whose string representation starts with underscore ('_').

O(n). Computes the type of an expression returning either the type of the given expression when possible or when there is a type error, the pair of types which produced the error.

> let one = val (1 :: Int)
> let bee = val 'b'
> let absE = value "abs" (abs :: Int -> Int)

> etyp one
Right Int

> etyp bee
Right Char

> etyp absE
Right (Int -> Int)

> etyp (absE :$ one)
Right Int

> etyp (absE :$ bee)
Left (Int -> Int, Char)

> etyp ((absE :$ bee) :$ one)
Left (Int -> Int, Char)

O(n). Returns Just the type of an expression or Nothing when there is an error.

> let one = val (1 :: Int)
> let bee = val 'b'
> let absE = value "abs" (abs :: Int -> Int)

> mtyp one
Just Int

> mtyp (absE :$ bee)
Nothing

O(n). Returns whether the given Expr is ill typed. (cf. isWellTyped)

> let one = val (1 :: Int)
> let bee = val 'b'
> let absE = value "abs" (abs :: Int -> Int)

> isIllTyped (absE :$ val (1 :: Int))
False

> isIllTyped (absE :$ val 'b')
True

O(n). Returns whether the given Expr is well typed. (cf. isIllTyped)

> isWellTyped (absE :$ val (1 :: Int))
True

> isWellTyped (absE :$ val 'b')
False

O(n). Returns whether the given Expr is of a functional type. This is the same as checking if the arity of the given Expr is non-zero.

> isFun (value "abs" (abs :: Int -> Int))
True

> isFun (val (1::Int))
False

> isFun (value "const" (const :: Bool -> Bool -> Bool) :$ val False)
True

O(n). Evaluates an expression when possible (correct type). Returns a default value otherwise.

> let two = val (2 :: Int)
> let three = val (3 :: Int)
> let e1 -+- e2 = value "+" ((+) :: Int->Int->Int) :$ e1 :$ e2

> eval 0 $ two -+- three :: Int
5

> eval 'z' $ two -+- three :: Char
'z'

> eval 0 $ two -+- val '3' :: Int
0

O(n). Evaluates an expression when possible (correct type). Raises an error otherwise.

> evl $ two -+- three :: Int
5

> evl $ two -+- three :: Bool
*** Exception: evl: cannot evaluate Expr `2 + 3 :: Int' at the Bool type

This may raise errors, please consider using eval or evaluate.

O(n). Evaluates an expression to a terminal Dynamic value when possible. Returns Nothing otherwise.

> toDynamic $ val (123 :: Int) :: Maybe Dynamic
Just <<Int>>

> toDynamic $ value "abs" (abs :: Int -> Int) :$ val (-1 :: Int)
Just <<Int>>

> toDynamic $ value "abs" (abs :: Int -> Int) :$ val 'a'
Nothing

O(n). Like showPrecExpr but the precedence is taken from the given operator name.

> showOpExpr "*" (two -*- three)
"(2 * 3)"

> showOpExpr "+" (two -*- three)
"2 * 3"

To imply that the surrounding environment is a function application, use " " as the given operator.

> showOpExpr " " (two -*- three)
"(2 * 3)"

O(n). Like showExpr but allows specifying the surrounding precedence.

> showPrecExpr 6 (one -+- two)
"1 + 2"

> showPrecExpr 7 (one -+- two)
"(1 + 2)"

O(n). Returns a string representation of an expression. Differently from show (:: Expr -> String) this function does not include the type in the output.

> putStrLn $ showExpr (one -+- two)
1 + 2

> putStrLn $ showExpr $ (pp -||- true) -&&- (qq -||- false)
(p || True) && (q || False)

O(n). Compares the complexity of two Exprs. An expression e1 is strictly simpler than another expression e2 if the first of the following conditions to distingish between them is:

1. e1 is smaller in size/length than e2, e.g.: x + y < x + (y + z);
2. or, e1 has more distinct variables than e2, e.g.: x + y < x + x;
3. or, e1 has more variable occurrences than e2, e.g.: x + x < 1 + x;
4. or, e1 has fewer distinct constants than e2, e.g.: 1 + 1 < 0 + 1.

They're otherwise considered of equal complexity, e.g.: x + y and y + z.

> (xx -+- yy) `compareComplexity` (xx -+- (yy -+- zz))
LT

> (xx -+- yy) `compareComplexity` (xx -+- xx)
LT

> (xx -+- xx) `compareComplexity` (one -+- xx)
LT

> (one -+- one) `compareComplexity` (zero -+- one)
LT

> (xx -+- yy) `compareComplexity` (yy -+- zz)
EQ

> (zero -+- one) `compareComplexity` (one -+- zero)
EQ

This comparison is not a total order.

O(n). Lexicographical structural comparison of Exprs where variables < constants < applications then types are compared before string representations.

> compareLexicographically one (one -+- one)
LT
> compareLexicographically one zero
GT
> compareLexicographically (xx -+- zero) (zero -+- xx)
LT
> compareLexicographically (zero -+- xx) (zero -+- xx)
EQ

(cf. compareTy)

This comparison is a total order.

O(n). A fast total order between Exprs that can be used when sorting Expr values.

This is lazier than its counterparts compareComplexity and compareLexicographically and tries to evaluate the given Exprs as least as possible.

O(n). Unfold a function application Expr into a list of function and arguments.

unfoldApp $ e0                    =  [e0]
unfoldApp $ e0 :$ e1              =  [e0,e1]
unfoldApp $ e0 :$ e1 :$ e2        =  [e0,e1,e2]
unfoldApp $ e0 :$ e1 :$ e2 :$ e3  =  [e0,e1,e2,e3]

Remember :$ is left-associative, so:

unfoldApp e0                          =  [e0]
unfoldApp (e0 :$ e1)                  =  [e0,e1]
unfoldApp ((e0 :$ e1) :$ e2)          =  [e0,e1,e2]
unfoldApp (((e0 :$ e1) :$ e2) :$ e3)  =  [e0,e1,e2,e3]

O(n). Check if an Expr has a variable. (By convention, any value whose String representation starts with '_'.)

> hasVar $ value "not" not :$ val True
False

> hasVar $ value "&&" (&&) :$ var "p" (undefined :: Bool) :$ val True
True

O(n). Returns whether a Expr has no variables. This is equivalent to "not . hasVar".

The name "ground" comes from term rewriting.

> isGround $ value "not" not :$ val True
True

> isGround $ value "&&" (&&) :$ var "p" (undefined :: Bool) :$ val True
False

O(1). Returns whether an Expr is a terminal constant. (cf. isGround).

> isConst $ var "x" (undefined :: Int)
False

> isConst $ val False
True

> isConst $ value "not" not :$ val False
False

O(1). Returns whether an Expr is a terminal variable (var). (cf. hasVar).

> isVar $ var "x" (undefined :: Int)
True

> isVar $ val False
False

> isVar $ value "not" not :$ var "p" (undefined :: Bool)
False

O(1). Returns whether an Expr is a terminal value (Value).

> isValue $ var "x" (undefined :: Int)
True

> isValue $ val False
True

> isValue $ value "not" not :$ var "p" (undefined :: Bool)
False

This is equivalent to pattern matching the Value constructor.

Properties:

•  isValue (Value e)  =  True

•  isValue (e1 :$ e2)  =  False

•  isValue  =  not . isApp

•  isValue e  =  isVar e || isConst e

O(1). Returns whether an Expr is an application (:$).

> isApp $ var "x" (undefined :: Int)
False

> isApp $ val False
False

> isApp $ value "not" not :$ var "p" (undefined :: Bool)
True

This is equivalent to pattern matching the :$ constructor.

Properties:

•  isApp (e1 :$ e2)  =  True

•  isApp (Value e)  =  False

•  isApp  =  not . isValue

•  isApp e  =  not (isVar e) && not (isConst e)

O(n) for the spine, O(n^2) for full evaluation. Lists subexpressions of a given expression in order and with repetitions. This includes the expression itself and partial function applications. (cf. nubSubexprs)

> subexprs (xx -+- yy)
[ x + y :: Int
, (x +) :: Int -> Int
, (+) :: Int -> Int -> Int
, x :: Int
, y :: Int
]

> subexprs (pp -&&- (pp -&&- true))
[ p && (p && True) :: Bool
, (p &&) :: Bool -> Bool
, (&&) :: Bool -> Bool -> Bool
, p :: Bool
, p && True :: Bool
, (p &&) :: Bool -> Bool
, (&&) :: Bool -> Bool -> Bool
, p :: Bool
, True :: Bool
]

O(n^3) for full evaluation. Lists all subexpressions of a given expression without repetitions. This includes the expression itself and partial function applications. (cf. subexprs)

> nubSubexprs (xx -+- yy)
[ x :: Int
, y :: Int
, (+) :: Int -> Int -> Int
, (x +) :: Int -> Int
, x + y :: Int
]

> nubSubexprs (pp -&&- (pp -&&- true))
[ p :: Bool
, True :: Bool
, (&&) :: Bool -> Bool -> Bool
, (p &&) :: Bool -> Bool
, p && True :: Bool
, p && (p && True) :: Bool
]

Runtime averages to O(n^2 log n) on evenly distributed expressions such as (f x + g y) + (h z + f w); and to O(n^3) on deep expressions such as f (g (h (f (g (h x))))).

O(n). Lists all terminal values in an expression in order and with repetitions. (cf. nubValues)

> values (xx -+- yy)
[ (+) :: Int -> Int -> Int
, x :: Int
, y :: Int
]

> values (xx -+- (yy -+- zz))
[ (+) :: Int -> Int -> Int
, x :: Int
, (+) :: Int -> Int -> Int
, y :: Int
, z :: Int
]

> values (zero -+- (one -*- two))
[ (+) :: Int -> Int -> Int
, 0 :: Int
, (*) :: Int -> Int -> Int
, 1 :: Int
, 2 :: Int
]

> values (pp -&&- true)
[ (&&) :: Bool -> Bool -> Bool
, p :: Bool
, True :: Bool
]

O(n^2). Lists all terminal values in an expression without repetitions. (cf. values)

> nubValues (xx -+- yy)
[ x :: Int
, y :: Int
, (+) :: Int -> Int -> Int

]

> nubValues (xx -+- (yy -+- zz))
[ x :: Int
, y :: Int
, z :: Int
, (+) :: Int -> Int -> Int
]

> nubValues (zero -+- (one -*- two))
[ 0 :: Int
, 1 :: Int
, 2 :: Int
, (*) :: Int -> Int -> Int
, (+) :: Int -> Int -> Int
]

> nubValues (pp -&&- pp)
[ p :: Bool
, (&&) :: Bool -> Bool -> Bool
]

Runtime averages to O(n log n) on evenly distributed expressions such as (f x + g y) + (h z + f w); and to O(n^2) on deep expressions such as f (g (h (f (g (h x))))).

O(n). List terminal constants in an expression in order and with repetitions. (cf. nubConsts)

> consts (xx -+- yy)
[ (+) :: Int -> Int -> Int ]

> consts (xx -+- (yy -+- zz))
[ (+) :: Int -> Int -> Int
, (+) :: Int -> Int -> Int
]

> consts (zero -+- (one -*- two))
[ (+) :: Int -> Int -> Int
, 0 :: Int
, (*) :: Int -> Int -> Int
, 1 :: Int
, 2 :: Int
]

> consts (pp -&&- true)
[ (&&) :: Bool -> Bool -> Bool
, True :: Bool
]

O(n^2). List terminal constants in an expression without repetitions. (cf. consts)

> nubConsts (xx -+- yy)
[ (+) :: Int -> Int -> Int ]

> nubConsts (xx -+- (yy -+- zz))
[ (+) :: Int -> Int -> Int ]

> nubConsts (pp -&&- true)
[ True :: Bool
, (&&) :: Bool -> Bool -> Bool
]

Runtime averages to O(n log n) on evenly distributed expressions such as (f x + g y) + (h z + f w); and to O(n^2) on deep expressions such as f (g (h (f (g (h x))))).

O(n). Lists all variables in an expression in order and with repetitions. (cf. nubVars)

> vars (xx -+- yy)
[ x :: Int
, y :: Int
]

> vars (xx -+- (yy -+- xx))
[ x :: Int
, y :: Int
, x :: Int
]

> vars (zero -+- (one -*- two))
[]

> vars (pp -&&- true)
[p :: Bool]

O(n^2). Lists all variables in an expression without repetitions. (cf. vars)

> nubVars (yy -+- xx)
[ x :: Int
, y :: Int
]

> nubVars (xx -+- (yy -+- xx))
[ x :: Int
, y :: Int
]

> nubVars (zero -+- (one -*- two))
[]

> nubVars (pp -&&- true)
[p :: Bool]

Runtime averages to O(n log n) on evenly distributed expressions such as (f x + g y) + (h z + f w); and to O(n^2) on deep expressions such as f (g (h (f (g (h x))))).

O(n). Returns the maximum depth of a given expression given by the maximum number of nested function applications. Curried function application is counted only once, i.e. the application of a two argument function increases the depth of both its arguments by one. (cf. height)

With

zero       =  val (0 :: Int)
one        =  val (1 :: Int)
two        =  val (2 :: Int)
xx -+- yy  =  value "+" ((+) :: Int->Int->Int) :$ xx :$ yy
abs' xx    =  value "abs" (abs :: Int->Int) :$ xx

> depth zero
1

> depth (one -+- two)
2

> depth (abs' one -+- two)
3

Flipping arguments of applications in any of the subterms does not affect the result.

O(n). Returns the maximum height of a given expression given by the maximum number of nested function applications. Curried function application is counted each time, i.e. the application of a two argument function increases the depth of its first argument by two and of its second argument by one. (cf. depth)

With:

zero          =  val (0 :: Int)
one           =  val (1 :: Int)
two           =  val (2 :: Int)
const' xx yy  =  value "const" (const :: Int->Int->Int) :$ xx :$ yy
abs' xx       =  value "abs" (abs :: Int->Int) :$ xx

Then:

> height zero
1

> height (abs' one)
2

> height ((const' one) two)
3

> height ((const' (abs' one)) two)
4

> height ((const' one) (abs' two))
3

Flipping arguments of applications in subterms may change the result of the function.

Like match but allowing predefined bindings.

matchWith [(xx,zero)] (zero -+- one) (xx -+- yy)  =  Just [(xx,zero), (yy,one)]
matchWith [(xx,one)]  (zero -+- one) (xx -+- yy)  =  Nothing

Given two Exprs, checks if the first expression encompasses the second expression in terms of variables.

This is equivalent to flipping the arguments of isInstanceOf.

zero `encompasses` xx    =  False
  xx `encompasses` zero  =  True

Checks if any of the subexpressions of the first argument Expr is an instance of the second argument Expr.

O(n^2). Checks if an Expr is a subexpression of another.

> (xx -+- yy) `isSubexprOf` (zz -+- (xx -+- yy))
True

> (xx -+- yy) `isSubexprOf` abs' (yy -+- xx)
False

> xx `isSubexprOf` yy
False

O(n*m). Applies a function to all variables in an expression.

Given that:

> let primeify e = if isVar e
|                  then case e of (Value n d) -> Value (n ++ "'") d
|                  else e
> let xx = var "x" (undefined :: Int)
> let yy = var "y" (undefined :: Int)
> let xx -+- yy = value "+" ((+) :: Int->Int->Int) :$ xx :$ yy

Then:

> xx -+- yy
x + y :: Int

> primeify xx
x' :: Int

> mapVars primeify $ xx -+- yy
x' + y' :: Int

> mapVars (primeify . primeify) $ xx -+- yy
x'' + y'' :: Int

Given that the argument function is O(m), this function is O(n*m).

O(n*m). Applies a function to all terminal constants in an expression.

Given that:

> let one   = val (1 :: Int)
> let two   = val (2 :: Int)
> let xx -+- yy = value "+" ((+) :: Int->Int->Int) :$ xx :$ yy
> let intToZero e = if typ e == typ zero then zero else e

Then:

> one -+- (two -+- xx)
1 + (2 + x) :: Int

> mapConsts intToZero (one -+- (two -+- xx))
0 + (0 + x) :: Integer

Given that the argument function is O(m), this function is O(n*m).

O(n*m). Substitute subexpressions of an expression using the given function. Outer expressions have more precedence than inner expressions. (cf. //)

With:

> let xx = var "x" (undefined :: Int)
> let yy = var "y" (undefined :: Int)
> let zz = var "z" (undefined :: Int)
> let plus = value "+" ((+) :: Int->Int->Int)
> let times = value "*" ((*) :: Int->Int->Int)
> let xx -+- yy = plus :$ xx :$ yy
> let xx -*- yy = times :$ xx :$ yy

> let pluswap (o :$ xx :$ yy) | o == plus = Just $ o :$ yy :$ xx
|     pluswap _                           = Nothing

Then:

> mapSubexprs pluswap $ (xx -*- yy) -+- (yy -*- zz)
y * z + x * y :: Int

> mapSubexprs pluswap $ (xx -+- yy) -*- (yy -+- zz)
(y + x) * (z + y) :: Int

Substitutions do not stack, in other words a replaced expression or its subexpressions are not further replaced:

> mapSubexprs pluswap $ (xx -+- yy) -+- (yy -+- zz)
(y + z) + (x + y) :: Int

Given that the argument function is O(m), this function is O(n*m).

O(n*m). Substitute occurrences of values in an expression from the given list of substitutions. (cf. mapValues)

Given that:

> let xx = var "x" (undefined :: Int)
> let yy = var "y" (undefined :: Int)
> let zz = var "z" (undefined :: Int)
> let xx -+- yy = value "+" ((+) :: Int->Int->Int) :$ xx :$ yy

Then:

> ((xx -+- yy) -+- (yy -+- zz)) //- [(xx, yy), (zz, yy)]
(y + y) + (y + y) :: Int

> ((xx -+- yy) -+- (yy -+- zz)) //- [(yy, yy -+- zz)]
(x + (y + z)) + ((y + z) + z) :: Int

This function does not work for substituting non-terminal subexpressions:

> (xx -+- yy) //- [(xx -+- yy, zz)]
x + y :: Int

Please use the slower // if you want the above replacement to work.

Replacement happens only once:

> xx //- [(xx,yy), (yy,zz)]
y :: Int

Given that the argument list has length m, this function is O(n*m).

Rename variables in an Expr.

> renameVarsBy (++ "'") (xx -+- yy)
x' + y' :: Int

> renameVarsBy (++ "'") (yy -+- (zz -+- xx))
(y' + (z' + x')) :: Int

> renameVarsBy (++ "1") (abs' xx)
abs x1 :: Int

> renameVarsBy (++ "2") $ abs' (xx -+- yy)
abs (x2 + y2) :: Int

NOTE: this will affect holes!

O(1). Creates a variable with the same type as the given Expr.

> let one = val (1::Int)
> "x" `varAsTypeOf` one
x :: Int

> "p" `varAsTypeOf` val False
p :: Bool

O(1). Creates an Expr representing a typed hole with the type of the given Expr. (cf. hole)

> val (1::Int)
1 :: Int
> holeAsTypeOf $ val (1::Int)
_ :: Int

O(1). Creates an Expr representing a typed hole of the given argument type.

> hole (undefined :: Int)
_ :: Int

> hole (undefined :: Maybe String)
_ :: Maybe [Char]

A hole is represented as a variable with no name or a value named "_":

hole x = var "" x
hole x = value "_" x

O(1). Checks if an Expr represents a typed hole. (cf. hole)

> isHole $ hole (undefined :: Int)
True

> isHole $ value "not" not :$ val True
False

> isHole $ val 'a'
False

O(n). Lists all holes in an expression, in order and with repetitions. (cf. nubHoles)

> holes $ hole (undefined :: Bool)
[_ :: Bool]

> holes $ value "&&" (&&) :$ hole (undefined :: Bool) :$ hole (undefined :: Bool)
[_ :: Bool,_ :: Bool]

> holes $ hole (undefined :: Bool->Bool) :$ hole (undefined::Bool)
[_ :: Bool -> Bool,_ :: Bool]

O(n^2). Lists all holes in an expression without repetitions. (cf. holes)

> nubHoles $ hole (undefined :: Bool)
[_ :: Bool]

> nubHoles $ value "&&" (&&) :$ hole (undefined :: Bool) :$ hole (undefined :: Bool)
[_ :: Bool]

> nubHoles $ hole (undefined :: Bool->Bool) :$ hole (undefined::Bool)
[_ :: Bool,_ :: Bool -> Bool]

Runtime averages to O(n log n) on evenly distributed expressions such as (f x + g y) + (h z + f w); and to O(n^2) on deep expressions such as f (g (h (f (g (h x))))).

O(n). Returns whether an expression contains a hole

> hasHole $ hole (undefined :: Bool)
True

> hasHole $ value "not" not :$ val True
False

> hasHole $ value "not" not :$ hole (undefined :: Bool)
True

O(n). Returns whether an expression is complete. A complete expression is one without holes.

> isComplete $ hole (undefined :: Bool)
False

> isComplete $ value "not" not :$ val True
True

> isComplete $ value "not" not :$ hole (undefined :: Bool)
False

isComplete is the negation of hasHole.

isComplete  =  not . hasHole

isComplete is to hasHole what isGround is to hasVar.

Generate an infinite list of variables based on a template and a given type. (cf. listVarsAsTypeOf)

> putL 10 $ listVars "x" (undefined :: Int)
[ x :: Int
, y :: Int
, z :: Int
, x' :: Int
, y' :: Int
, z' :: Int
, x'' :: Int
, ...
]

> putL 10 $ listVars "p" (undefined :: Bool)
[ p :: Bool
, q :: Bool
, r :: Bool
, p' :: Bool
, q' :: Bool
, r' :: Bool
, p'' :: Bool
, ...
]

Generate an infinite list of variables based on a template and the type of a given Expr. (cf. listVars)

> let one = val (1::Int)
> putL 10 $ "x" `listVarsAsTypeOf` one
[ x :: Int
, y :: Int
, z :: Int
, x' :: Int
, ...
]

> let false = val False
> putL 10 $ "p" `listVarsAsTypeOf` false
[ p :: Bool
, q :: Bool
, r :: Bool
, p' :: Bool
, ...
]

O(n). Folds a list of Expr with function application (:$). This reverses the effect of unfoldApp.

foldApp [e0]           =  e0
foldApp [e0,e1]        =  e0 :$ e1
foldApp [e0,e1,e2]     =  e0 :$ e1 :$ e2
foldApp [e0,e1,e2,e3]  =  e0 :$ e1 :$ e2 :$ e3

Remember :$ is left-associative, so:

foldApp [e0]           =    e0
foldApp [e0,e1]        =   (e0 :$ e1)
foldApp [e0,e1,e2]     =  ((e0 :$ e1) :$ e2)
foldApp [e0,e1,e2,e3]  = (((e0 :$ e1) :$ e2) :$ e3)

This function may produce an ill-typed expression.

O(1). Folds a pair of Expr values into a single Expr. (cf. unfoldPair)

This always generates an ill-typed expression, as it uses a fake pair constructor.

> foldPair (val False, val (1::Int))
(False,1) :: ill-typed # ExprPair $ Bool #

> foldPair (val (0::Int), val True)
(0,True) :: ill-typed # ExprPair $ Int #

This is useful when applying transformations on pairs of Exprs, such as canonicalize, mapValues or canonicalVariations.

> let ii = var "i" (undefined::Int)
> let kk = var "k" (undefined::Int)
> unfoldPair $ canonicalize $ foldPair (ii,kk)
(x :: Int,y :: Int)

O(1). Unfolds an Expr representing a pair. This reverses the effect of foldPair.

> value "," ((,) :: Bool->Bool->(Bool,Bool)) :$ val True :$ val False
(True,False) :: (Bool,Bool)
> unfoldPair $ value "," ((,) :: Bool->Bool->(Bool,Bool)) :$ val True :$ val False
(True :: Bool,False :: Bool)

O(1). Folds a trio/triple of Expr values into a single Expr. (cf. unfoldTrio)

This always generates an ill-typed expression as it uses a fake trio/triple constructor.

> foldTrio (val False, val (1::Int), val 'a')
(False,1,'a') :: ill-typed # ExprTrio $ Bool #

> foldTrio (val (0::Int), val True, val 'b')
(0,True,'b') :: ill-typed # ExprTrio $ Int #

This is useful when applying transformations on pairs of Exprs, such as canonicalize, mapValues or canonicalVariations.

> let ii = var "i" (undefined::Int)
> let kk = var "k" (undefined::Int)
> let zz = var "z" (undefined::Int)
> unfoldPair $ canonicalize $ foldPair (ii,kk,zz)
(x :: Int,y :: Int,z :: Int)

O(1). Unfolds an Expr representing a trio/triple. This reverses the effect of foldTrio.

> value ",," ((,,) :: Bool->Bool->Bool->(Bool,Bool,Bool)) :$ val True :$ val False :$ val True
(True,False,True) :: (Bool,Bool,Bool)
> unfoldTrio $ value ",," ((,,) :: Bool->Bool->Bool->(Bool,Bool,Bool)) :$ val True :$ val False :$ val True
(True :: Bool,False :: Bool,True :: Bool)

(cf. unfoldPair)

Same as deriveExpress but does not warn when instance already exists (deriveExpress is preferable).

Derives a Express instance for a given type Name cascading derivation of type arguments as well.

Lists valid applications between a list of Exprs and an Expr.

> [plus, times] >$$ zero
[(0 +) :: Int -> Int,(0 *) :: Int -> Int]

Lists valid applications between an Expr and a list of Exprs.

> notE >$$< [false, true, zero]
[not False :: Bool,not True :: Bool]

O(1). Reifies an Eq instance into a list of Exprs. The list will contain == and /= for the given type. (cf. mkEq, mkEquation)

> reifyEq (undefined :: Int)
[ (==) :: Int -> Int -> Bool
, (/=) :: Int -> Int -> Bool ]

> reifyEq (undefined :: Bool)
[ (==) :: Bool -> Bool -> Bool
, (/=) :: Bool -> Bool -> Bool ]

> reifyEq (undefined :: String)
[ (==) :: [Char] -> [Char] -> Bool
, (/=) :: [Char] -> [Char] -> Bool ]

O(1). Reifies an Ord instance into a list of Exprs. The list will contain compare, <= and < for the given type. (cf. mkOrd, mkOrdLessEqual, mkComparisonLE, mkComparisonLT)

> reifyOrd (undefined :: Int)
[ (<=) :: Int -> Int -> Bool
, (<) :: Int -> Int -> Bool ]

> reifyOrd (undefined :: Bool)
[ (<=) :: Bool -> Bool -> Bool
, (<) :: Bool -> Bool -> Bool ]

> reifyOrd (undefined :: [Bool])
[ (<=) :: [Bool] -> [Bool] -> Bool
, (<) :: [Bool] -> [Bool] -> Bool ]

O(1). Reifies Eq and Ord instances into a list of Expr.

O(1). Reifies a Name instance into a list of Exprs. The list will contain name for the given type. (cf. mkName, lookupName, lookupNames)

> reifyName (undefined :: Int)
[name :: Int -> [Char]]

> reifyName (undefined :: Bool)
[name :: Bool -> [Char]]

O(1). Builds a reified Eq instance from the given == function. (cf. reifyEq)

> mkEq ((==) :: Int -> Int -> Bool)
[ (==) :: Int -> Int -> Bool
, (/=) :: Int -> Int -> Bool ]

O(1). Builds a reified Ord instance from the given compare function. (cf. reifyOrd, mkOrdLessEqual)

O(1). Builds a reified Ord instance from the given <= function. (cf. reifyOrd, mkOrd)

O(1). Builds a reified Name instance from the given name function. (cf. reifyName, mkNameWith)

O(1). Builds a reified Name instance from the given String and type. (cf. reifyName, mkName)

O(n). Lookups for a comparison function (:: a -> a -> Bool) with the given name and argument type.

O(n). Returns whether an Eq instance exists in the given instances list for the given TypeRep.

> isEqT (reifyEqOrd (undefined :: Int)) (typeOf (undefined :: Int))
True

> isEqT (reifyEqOrd (undefined :: Int)) (typeOf (undefined :: [[[Int]]]))
False

Given that the instances list has length n, this function is O(n).

O(n). Returns whether an Ord instance exists in the given instances list for the given TypeRep.

> isOrdT (reifyEqOrd (undefined :: Int)) (typeOf (undefined :: Int))
True

> isOrdT (reifyEqOrd (undefined :: Int)) (typeOf (undefined :: [[[Int]]]))
False

Given that the instances list has length n, this function is O(n).

O(n). Returns whether both Eq and Ord instance exist in the given list for the given TypeRep.

Given that the instances list has length n, this function is O(n).

O(n+m). Returns whether an Eq instance exists in the given instances list for the given Expr.

> isEq (reifyEqOrd (undefined :: Int)) (val (0::Int))
True

> isEq (reifyEqOrd (undefined :: Int)) (val ([[[0::Int]]]))
False

Given that the instances list has length m and that the given Expr has size n, this function is O(n+m).

O(n+m). Returns whether an Ord instance exists in the given instances list for the given Expr.

> isOrd (reifyEqOrd (undefined :: Int)) (val (0::Int))
True

> isOrd (reifyEqOrd (undefined :: Int)) (val ([[[0::Int]]]))
False

Given that the instances list has length m and that the given Expr has size n, this function is O(n+m).

O(n+m). Returns whether both Eq and Ord instance exist in the given list for the given Expr.

Given that the instances list has length m and that the given Expr has size n, this function is O(n+m).

O(n+m). Like mkEquation, mkComparisonLE and mkComparisonLT but allows providing the binary operator name.

When not possible, this function returns False encoded as an Expr.

O(n+m). Returns an equation between two expressions given that it is possible to do so from == operators given in the argument instances list.

When not possible, this function returns False encoded as an Expr.

O(n+m). Returns a less-than inequation between two expressions given that it is possible to do so from < operators given in the argument instances list.

When not possible, this function returns False encoded as an Expr.

O(n+m). Returns a less-than-or-equal-to inequation between two expressions given that it is possible to do so from <= operators given in the argument instances list.

When not possible, this function returns False encoded as an Expr.

O(n+m). Like name but lifted over an instance list and an Expr.

> lookupName preludeNameInstances (val False)
"p"

> lookupName preludeNameInstances (val (0::Int))
"x"

This function defaults to "x" when no appropriate name is found.

> lookupName [] (val False)
"x"

O(n+m). A mix between lookupName and names: this returns an infinite list of names based on an instances list and an Expr.

O(n+m). Like lookupNames but returns a list of variables encoded as Exprs.

Given a list of functional expressions and another expression, returns a list of valid applications.

Like validApps but returns a Maybe value.

A list of reified Name instances for an arbitrary selection of types from the Haskell Prelude.

Like canonicalize but allows customization of the list of variable names. (cf. lookupNames, variableNamesFromTemplate)

> canonicalizeWith (const ["i","j","k","l",...]) (xx -+- yy)
i + j :: Int

The argument Expr of the argument function allows to provide a different list of names for different types:

> let namesFor e
>   | typ e == typeOf (undefined::Char) = variableNamesFromTemplate "c1"
>   | typ e == typeOf (undefined::Int)  = variableNamesFromTemplate "i"
>   | otherwise                         = variableNamesFromTemplate "x"

> canonicalizeWith namesFor ((xx -+- ord' dd) -+- (ord' cc -+- yy))
(i + ord c1) + (ord c2 + j) :: Int

Like canonicalization but allows customization of the list of variable names. (cf. lookupNames, variableNamesFromTemplate)

Like isCanonical but allows specifying the list of variable names.

Return a canonicalization of an Expr that makes variable names appear in order using names as provided by preludeNameInstances. By using //- it can canonicalize Exprs.

> canonicalization (gg yy -+- ff xx -+- gg xx)
[ (x :: Int,        y :: Int)
, (f :: Int -> Int, g :: Int -> Int)
, (y :: Int,        x :: Int)
, (g :: Int -> Int, f :: Int -> Int) ]

> canonicalization (yy -+- xx -+- yy)
[ (x :: Int, y :: Int)
, (y :: Int, x :: Int) ]

Returns whether an Expr is canonical: if applying canonicalize is an identity using names as provided by preludeNameInstances.

Returns the most general canonical variation of an Expr by filling holes with variables.

> mostGeneralCanonicalVariation $ i_
x :: Int

> mostGeneralCanonicalVariation $ i_ -+- i_
x + y :: Int

> mostGeneralCanonicalVariation $ i_ -+- i_ -+- i_
(x + y) + z :: Int

> mostGeneralCanonicalVariation $ i_ -+- ord' c_
x + ord c :: Int

> mostGeneralCanonicalVariation $ i_ -+- i_ -+- ord' c_
(x + y) + ord c :: Int

> mostGeneralCanonicalVariation $ i_ -+- i_ -+- length' (c_ -:- unit c_)
(x + y) + length (c:d:"") :: Int

In an expression without holes this functions just returns the given expression itself:

> mostGeneralCanonicalVariation $ val (0 :: Int)
0 :: Int

> mostGeneralCanonicalVariation $ ord' bee
ord 'b' :: Int

This function is the same as taking the head of canonicalVariations but a bit faster.

Returns the most specific canonical variation of an Expr by filling holes with variables.

> mostSpecificCanonicalVariation $ i_
x :: Int

> mostSpecificCanonicalVariation $ i_ -+- i_
x + x :: Int

> mostSpecificCanonicalVariation $ i_ -+- i_ -+- i_
(x + x) + x :: Int

> mostSpecificCanonicalVariation $ i_ -+- ord' c_
x + ord c :: Int

> mostSpecificCanonicalVariation $ i_ -+- i_ -+- ord' c_
(x + x) + ord c :: Int

> mostSpecificCanonicalVariation $ i_ -+- i_ -+- length' (c_ -:- unit c_)
(x + x) + length (c:c:"") :: Int

In an expression without holes this functions just returns the given expression itself:

> mostSpecificCanonicalVariation $ val (0 :: Int)
0 :: Int

> mostSpecificCanonicalVariation $ ord' bee
ord 'b' :: Int

This function is the same as taking the last of canonicalVariations but a bit faster.

A faster version of canonicalVariations that disregards name clashes across different types. Results are confusing to the user but fine for Express which differentiates between variables with the same name but different types.

Without applying canonicalize, the following Expr may seem to have only one variable:

> fastCanonicalVariations $ i_ -+- ord' c_
[x + ord x :: Int]

Where in fact it has two, as the second x has a different type. Applying canonicalize disambiguates:

> map canonicalize . fastCanonicalVariations $ i_ -+- ord' c_
[x + ord c :: Int]

This function is useful when resulting Exprs are not intended to be presented to the user but instead to be used by another function. It is simply faster to skip the step where clashes are resolved.

A faster version of mostGeneralCanonicalVariation that disregards name clashes across different types. Consider using mostGeneralCanonicalVariation instead.

The same caveats of fastCanonicalVariations do apply here.

A faster version of mostSpecificCanonicalVariation that disregards name clashes across different types. Consider using mostSpecificCanonicalVariation instead.

The same caveats of fastCanonicalVariations do apply here.

Expr representing a hole of Bool type.

> b_
_ :: Bool

Expr representing a variable p :: Bool.

> pp
p :: Bool

Expr representing a variable q :: Bool.

> qq
q :: Bool

Expr representing a variable r :: Bool.

> rr
r :: Bool

Expr representing a variable p' :: Bool.

> pp'
p' :: Bool

False encoded as an Expr.

> false
False :: Bool

True encoded as an Expr.

> true
True :: Bool

The function not encoded as an Expr.

> notE
not :: Bool -> Bool

The function and encoded as an Expr.

> andE
(&&) :: Bool -> Bool -> Bool

The function or encoded as an Expr.

> orE
(||) :: Bool -> Bool -> Bool

The function ==> lifted over Exprs.

> false -==>- true
False ==> True :: Bool

> evl $ false -==>- true :: Bool
True

The ==> operator encoded as an Expr

The function not lifted over the Expr type.

> not' false
not False :: Bool

> evalBool $ not' false
True

> not' pp
not p :: Bool

The function && lifted over the Expr type.

> pp -&&- qq
p && q :: Bool

> false -&&- true
False && True :: Bool

> evalBool $ false -&&- true
False

The function || lifted over the Expr type.

> pp -||- qq
p || q :: Bool

> false -||- true
False || True :: Bool

> evalBool $ false -||- true
True

A typed hole of Int type.

> i_
_ :: Int

A variable x of Int type.

> xx
x :: Int

A variable y of Int type.

> yy
y :: Int

A variable z of Int type.

> zz
z :: Int

A variable x' of Int type.

> xx'
x' :: Int

A variable i of Int type.

> ii
i :: Int

A variable j of Int type.

> jj
j :: Int

A variable k of Int type.

> kk
k :: Int

A variable i' of Int type.

> ii'
i' :: Int

A variable l of Int type.

> ll
l :: Int

A variable m of Int type.

> mm
m :: Int

A variable n of Int type.

> nn
n :: Int

The value 1 bound to the Int type encoded as an Expr.

> one
1 :: Int

The value 2 bound to the Int type encoded as an Expr.

> two
2 :: Int

The value 3 bound to the Int type encoded as an Expr.

> three
3 :: Int

The value 4 bound to the Int type encoded as an Expr.

> four
4 :: Int

The value 5 bound to the Int type encoded as an Expr.

> five
5 :: Int

The value 6 bound to the Int type encoded as an Expr.

> six
6 :: Int

The value 7 bound to the Int type encoded as an Expr.

> seven
7 :: Int

The value 8 bound to the Int type encoded as an Expr.

> eight
8 :: Int

The value 9 bound to the Int type encoded as an Expr.

> nine
9 :: Int

The value 10 bound to the Int type encoded as an Expr.

> ten
10 :: Int

The value 11 bound to the Int type encoded as an Expr.

> eleven
11 :: Int

The value 12 bound to the Int type encoded as an Expr.

> twelve
12 :: Int

The value -1 bound to the Int type encoded as an Expr.

> minusOne
-1 :: Int

The value -2 bound to the Int type encoded as an Expr.

> minusOne
-2 :: Int

A variable function f of 'Int -> Int' type lifted over the Expr type.

> ff xx
f x :: Int

> ff one
f 1 :: Int

A variable f of 'Int -> Int' type encoded as an Expr.

> ffE
f :: Int -> Int

A variable function g of 'Int -> Int' type lifted over the Expr type.

> gg yy
g y :: Int

> gg minusTwo
gg (-2) :: Int

A variable g of 'Int -> Int' type encoded as an Expr.

> ggE
g :: Int -> Int

A variable function h of 'Int -> Int' type lifted over the Expr type.

> hh zz
h z :: Int

A variable h of 'Int -> Int' type encoded as an Expr.

> hhE
h :: Int -> Int

A variable binary operator ? lifted over the Expr type. Works for Int, Bool, Char, [Int] and String.

> xx -?- yy
x ? y :: Int

> pp -?- qq
p ? q :: Bool

> xx -?- qq
*** Exception: (-?-): unhandled type: 1 :: Int, False :: Bool
    accepted types are:
    (?) :: Int -> Int -> Int
    (?) :: Bool -> Bool -> Bool
    (?) :: Char -> Char -> Char
    (?) :: [Int] -> [Int] -> [Int]
    (?) :: [Char] -> [Char] -> [Char]
    (?) :: Int -> [Int] -> [Int]
    (?) :: Char -> [Char] -> [Char]

A variable binary operator ? encoded as an Expr (cf. -?-)

> question :$ xx :$ yy
x ? y :: Int

> question :$ pp :$ qq
p ? q :: Bool

A variable binary operator o lifted over the Expr type. Works for Int, Bool, Char, [Int] and String.

> xx `oo` yy
x `o` y :: Int

> pp `oo` qq
p `o` q :: Bool

> xx `oo` qq
*** Exception: oo: unhandled type: 1 :: Int, False :: Bool
    accepted types are:
    o :: Int -> Int -> Int
    o :: Bool -> Bool -> Bool
    o :: Char -> Char -> Char
    o :: [Int] -> [Int] -> [Int]
    o :: [Char] -> [Char] -> [Char]

A variable binary function o encoded as an Expr (cf. oo)

> ooE :$ xx :$ yy
x `o` y :: Int

> ooE :$ pp :$ qq
p `o` q :: Bool

The operator + for the Int type for use on Exprs. (See also plus.)

> two -+- three
2 + 3 :: Int

> minusOne -+- minusTwo -+- zero
((-1) + (-2)) + 0 :: Int

> xx -+- (yy -+- zz)
x + (y + z) :: Int

The operator + for the Int type. (See also -+-.)

> plus
(+) :: Int -> Int -> Int

> plus :$ one
(1 +) :: Int -> Int

> plus :$ xx :$ yy
x + y :: Int

The operator * for the Int type lifted over the Expr type. (See also times.)

> three -*- three
3 * 3 :: Int

> one -*- two -*- three
(1 * 2) * 3 :: Int

> two -*- xx
2 * x :: Int

The subtraction - operator encoded as an Expr.

> minus :$ one
(1 -) :: Int -> Int

> minus :$ one :$ zero
1 - 0 :: Int

Integer division div encoded as an Expr.

> divE :$ two
(2 `div`) :: Int -> Int

> divE :$ two :$ three
2 `div` 3 :: Int

Integer modulo mod encoded as an Expr.

> modE :$ two
(2 `mod`) :: Int -> Int

> modE :$ two :$ three
2 `mod` 3 :: Int

The function quot for the Int type lifted over the Expr type. (See also rem.)

> six `quot'` four
6 `quot` 4 :: Int

Integer quotient quot encoded as an Expr.

> quotE :$ two
(2 `quot`) :: Int -> Int

> quotE :$ two :$ three
2 `quot` 3 :: Int

The function rem for the Int type lifted over the Expr type. (See also quot.)

> six `rem'` four
6 `rem` 4 :: Int

Integer remainder rem encoded as an Expr.

> remE :$ two
(2 `rem`) :: Int -> Int

> remE :$ two :$ three
2 `rem` 3 :: Int

Constructs an application of id as an Expr. Only works for Int, Bool, Char, String, [Int], [Bool].

> id' yy
id yy :: Int

> id' one
id 1 :: Int

> evl (id' one) :: Int
1

> id' pp
id p :: Bool

> id' false
id' False :: Bool

> evl (id' true) :: Bool
True :: Bool

The function id for the Int type encoded as an Expr. (See also id'.)

> idE :$ xx
id x :: Int

> idE :$ zero
id 0 :: Int

> evaluate $ idE :$ zero :: Maybe Int
Just 0

The function id encoded as an Expr. (cf. id')

The function id encoded as an Expr. (cf. id')

The function id encoded as an Expr. (cf. id')

The function id encoded as an Expr. (cf. id')

The function id encoded as an Expr. (cf. id')

The function id encoded as an Expr. (cf. id')

The const function lifted over the Expr type.

> const' zero one
const 0 1 :: Int

This works for the argument types Int, Char, Bool and their lists.

negate over the Int type lifted over the Expr type.

> negate' xx
negate x :: Int

> evl (negate' one) :: Int
-1

negate over the Int type encoded as an Expr

> negateE
negate :: Int -> Int

abs over the Int type lifted over the Expr type.

> abs' xx'
abs x' :: Int

> evl (abs' minusTwo) :: Int
2

abs over the Int type encoded as an Expr.

> absE
abs :: Int -> Int

signum over the Int type lifted over the Expr type.

> signum' xx'
signum x' :: Int

> evl (signum' minusTwo) :: Int
-1

signum over the Int type encoded as an Expr.

> signumE
signum :: Int -> Int

odd with an Int argument lifted over the Expr type.

> odd' (xx -+- one)
odd (x + 1) :: Bool

> evl (odd' two) :: Bool
False

even with an Int argument lifted over the Expr type.

> even' (xx -+- two)
even (x + 2) :: Bool

> evl (even' two) :: Bool
True

A hole of Char type encoded as an Expr.

> c_
_ :: Char

A hole of String type encoded as an Expr.

> cs_
_ :: [Char]

A variable named c of type Char encoded as an Expr.

> cc
c :: Char

A variable named c of type Char encoded as an Expr.

> dd
d :: Char

A variable named cs of type String encoded as an Expr.

> ccs
cs :: [Char]

The character 'a' encoded as an Expr.

> ae
'a' :: Char

> evl ae :: Char
'a'

The character 'b' encoded as an Expr

> bee
'b' :: Char

> evl bee :: Char
'b'

The character 'c' encoded as an Expr

> cee
'c' :: Char

> evl cee :: Char
'c'

The character 'd' encoded as an Expr

> dee
'd' :: Char

> evl dee :: Char
'd'

The character 'z' encoded as an Expr

> zed
'z' :: Char

> evl zed :: Char
'z'

(cf. zee)

The character 'z' encoded as an Expr

> zee
'z' :: Char

> evl zee :: Char
'z'

(cf. zed)

The space character encoded as an Expr

> space
' ' :: Char

The line break character encoded as an Expr

> lineBreak
'\n' :: Char

The ord function lifted over Expr

> ord' bee
ord 'b' :: Int

> evl (ord' bee)
98

The ord function encoded as an Expr

A typed hole of [Int] type encoded as an Expr.

> is_
_ :: [Int]

A variable named xs of type [Int] encoded as an Expr.

> xxs
xs :: [Int]

A variable named ys of type [Int] encoded as an Expr.

> yys
ys :: [Int]

A variable named zs of type [Int] encoded as an Expr.

> yys
ys :: [Int]

An empty list of type [Int] encoded as an Expr.

> nil
[] :: [Int]

An empty String encoded as an Expr.

> emptyString
"" :: String

The empty list '[]' encoded as an Expr.

The empty list '[]' encoded as an Expr.

The empty list '[]' encoded as an Expr.

The list constructor : encoded as an Expr.

The list constructor : encoded as an Expr.

The list constructor : encoded as an Expr.

The list constructor lifted over the Expr type. Works for the element types Int, Char and Bool.

> zero -:- one -:- unit two
[0,1,2] :: [Int]

> zero -:- one -:- two -:- nil
[0,1,2] :: [Int]

> bee -:- unit cee
"bc" :: [Char]

Append for list of Ints encoded as an Expr.

List concatenation lifted over the Expr type. Works for the element types Int, Char and Bool.

> (zero -:- one -:- nil) -:- (two -:- three -:- nil)
[0,1] -++- [2,3] :: [Int]

> (bee -:- unit cee) -:- unit dee
"bc" -++- "c" :: [Char]

List head lifted over the Expr type. Works for the element types Int, Char and Bool.

> head' $ unit one
head [1] :: Int

> head' $ unit bee
head "b" :: Char

> head' $ zero -:- unit two
head [0,2] :: Int

> evl $ head' $ unit one :: Int
1

List tail lifted over the Expr type. Works for the element types Int, Char and Bool.

> tail' $ unit one
tail [1] :: [Int]

> tail' $ unit bee
tail "b" :: [Char]

> tail' $ zero -:- unit two
tail [0,2] :: [Int]

> evl $ tail' $ zero -:- unit two :: [Int]
[2]

List null lifted over the Expr type. Works for the element types Int, Char and Bool.

> null' $ unit one
null [1] :: Bool

> null' $ nil
null [] :: Bool

> evl $ null' nil :: Bool
True

List length lifted over the Expr type. Works for the element types Int, Char and Bool.

> length' $ unit one
length [1] :: Int

> length' $ unit bee
length "b" :: Int

> length' $ zero -:- unit two
length [0,2] :: Int

> evl $ length' $ unit one :: Int
1

List init lifted over the Expr type. Works for the element types Int, Char and Bool.

> init' $ unit one
init [1] :: [Int]

> init' $ unit bee
init "b" :: [Char]

> init' $ zero -:- unit two
init [0,2] :: [Int]

> evl $ init' $ zero -:- unit two :: [Int]
[0]

List sort lifted over the Expr type. Works for the element types Int, Char and Bool.

> sort' $ unit one
sort [1] :: Int

> sort' $ unit bee
sort "b" :: Int

> sort' $ zero -:- unit two
sort [0,2] :: Int

> evl $ sort' $ two -:- unit one :: [Int]
[1,2]

List insert lifted over the Expr type. Works for the element types Int, Char and Bool.

> insert' zero nilInt
insert 0 [] :: [Int]

> insert' false (false -:- unit true)
insert False [False,True] :: [Bool]

List elem lifted over the Expr type. Works for the element types Int, Char and Bool.

> elem' false (false -:- unit true)
elem False [False,True] :: Bool

> evl $ elem' false (false -:- unit true) :: Bool
True

$ lifted over Exprs

> absE -$- one
abs $ 1 :: Int

Works for Int, Bool, Char argument types and their lists.

Constructs an inequation between two Exprs.

> xx -/=- zero
x /= 0 :: Bool

> cc -/=- ae
c /= 'a' :: Bool

Constructs a less-than-or-equal inequation between two Exprs.

> xx -<=- zero
x <= 0 :: Bool

> cc -<=- ae
c <= 'a' :: Bool

Constructs a less-than inequation between two Exprs.

> xx -<- zero
x < 0 :: Bool

> cc -<- bee
c < 'b' :: Bool

A function if :: Bool -> a -> a -> a lifted over the Expr type that encodes if-then-else functionality. This is properly displayed as an if-then-else.

> if' pp zero xx
(if p then 0 else x) :: Int

> zz -*- if' pp xx yy
z * (if p then x else y) :: Int

> if' pp false true -||- if' qq true false
(if p then False else True) || (if q then True else False) :: Bool

> evl $ if' true (val 't') (val 'f') :: Char
't'

A function case :: Bool -> a -> a -> a lifted over the Expr type that encodes case-of-False-True functionality. This is properly displayed as a case-of-False-True expression.

> caseBool pp zero xx
(case p of False -> 0; True -> x) :: Int

> zz -*- caseBool pp xx yy
z * (case p of False -> x; True -> y) :: Int

> caseBool pp false true -||- caseBool qq true false
(caseBool p of False -> False; True -> True) || (caseBool q of False -> True; True -> False) :: Bool

> evl $ caseBool true (val 'f') (val 't') :: Char
't'

By convention, the False case comes before True as False < True and data Bool = False | True.

When evaluating, this is equivalent to if with arguments reversed. Instead of using this, you are perhaps better of using if encoded as an expression. This is just here to be consistent with caseOrdering.

A function case :: Ordering -> a -> a -> a -> a lifted over the Expr type that encodes case-of-LT-EQ-GT functionality. This is properly displayed as a case-of-LT-EQ-GT expression. (cf. caseBool)

> caseOrdering (xx `compare'` yy) zero one two
(case compare x y of LT -> 0; EQ -> 1; GT -> 2) :: Int

> evl $ caseOrdering (val EQ) (val 'l') (val 'e') (val 'g') :: Char
'e'

By convention cases are given in LT, EQ and GT order as LT < EQ < GT and data Ordering = LT | EQ | GT.

Constructs an Expr-encoded compare operation between two Exprs.

> xx `compare'` zero
compare x 0 :: Ordering

> compare' ae bee
compare 'a' 'b' :: Ordering

Nothing bound to the Maybe Int type encoded as an Expr.

This is an alias to nothingInt.

Nothing bound to the Maybe Int type encoded as an Expr.

Nothing bound to the Maybe Bool type encoded as an Expr.

The Just constructor of the Int element type encoded as an Expr.

The Just constructor of the Bool element type encoded as an Expr.

The Just constructor lifted over the Expr type.

This works for the Bool and Int argument types.

> just zero
Just 0 :: Maybe Int
> just false
Just False :: Maybe Bool

An infix synonym of pair.

The pair constructor lifted over Exprs.

This works for the Int and Bool element types by differently from foldPair by returning a well-typed expression.

The pair constructor ( :: ... -> (Int,Int) ) encoded as an Expr.

The triple/trio constructor lifted over Exprs.

This only works for the Int element type.

The quadruple constructor lifted over Exprs.

This only works for the Int element type.

The quintuple constructor lifted over Exprs.

This only works for the Int element type.

The sixtuple constructor lifted over Exprs.

This only works for the Int element type.

A typed hole of [Bool] type encoded as an Expr.

> bs_
_ :: [Bool]

Expr representing a variable p' :: `[Bool]`.

> pps
ps :: [Bool]

A typed hole of '[Bool]' type

> qqs
qs :: [Bool]

and lifted over the Expr type.

> and' pps
and ps :: Bool

> evl (and' $ expr [False,True]) :: Bool
False

or lifted over the Expr type.

> or' pps
or ps :: Bool

> evl (or' $ expr [False,True]) :: Bool
True

sum of Int elements lifted over the Expr type.

> sum' xxs
sum xs :: Int

> evl (sum' $ expr [1,2,3::Int]) :: Int
6

product of Int elements lifted over the Expr type.

> product' xxs
product xs :: Int

> evl (product' $ expr [1,2,3::Int]) :: Int
6

The % constructor lifted over Exprs.

> val (2 :: Integer) -%- val (3 :: Integer)
2 % 3 :: Ratio Integer

This only accepts Exprs bound to the Integer type.

Function composition . lifted over Expr.

> absE -.- negateE
abs . negate :: Int -> Int

> absE -.- negateE :$ one
(abs . negate) 1 :: Int

This works for Int, Bool, Char and their lists.

map over the Int element type encoded as an Expr

> mapE
map :: (Int -> Int) -> [Int] -> [Int]

map lifted over Exprs.

> map' absE (unit one)
map abs [1] :: [Int]