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  1. (...?) :: (forall name . () => (a ~~ name) -> b ~~ f name) -> (forall name . () => p name -> Proof (q (f name))) -> (a ? p) -> b ? q

    gdp Data.Refined

    Take a simple function with one named argument and a named return, plus an implication relating a precondition to a postcondition of the function, and produce a function between refined input and output types. 

    newtype NonEmpty xs = NonEmpty Defn
type role Nonempty nominal -- disallows coercion of Nonempty's argument.

newtype Reverse  xs = Reverse  Defn
type role Reverse nominal

rev :: ([a] ~~ xs) -> ([a] ~~ Reverse xs)
rev xs = defn (reverse (the xs))

rev_nonempty_lemma :: NonEmpty xs -> Proof (NonEmpty (Reverse xs))

rev' :: ([a] ?NonEmpty) -> ([a] ?NonEmpty)
rev' = rev ...? rev_nonempty_lemma

  2. (.%) :: Hex -> Hex -> Hex

    ghci-hexcalc Data.GHex

    Integer mod 

    >>> 18 .% 16
0x0000_0000_0000_0002

  3. (.&) :: Hex -> Hex -> Hex

    ghci-hexcalc Data.GHex

    Bitwise "and" 

    >>> 0x1234 .& 0xff
0x0000_0000_0000_0034

  4. (.++) :: (Int, Hex) -> (Int, Hex) -> (Int, Hex)

    ghci-hexcalc Data.GHex

    Concatinate pairs of (length,Hex) 

    >>> (3,0b101) .++ (2,0b11)
(5,0x0000_0000_0000_0017)

>>> (4,0xa) .++ (4,0xb) .++ (8,0xcd)
(16,0x0000_0000_0000_abcd)

>>> (4,0xe) .++ (4,0xf) .@snd
0x0000_0000_0000_00ef

  5. (./) :: Hex -> Hex -> Hex

    ghci-hexcalc Data.GHex

    Integer div 

    >>> 0x1000 ./ 16
0x0000_0000_0000_0100

  6. (.<<) :: Hex -> Int -> Hex

    ghci-hexcalc Data.GHex

    Logical left shift 

    >>> 1 .<< 16
0x0000_0000_0001_0000

  7. (.>>) :: Hex -> Int -> Hex

    ghci-hexcalc Data.GHex

    Logical right shift 

    >>> 0x0f00 .>> 4
0x0000_0000_0000_00f0

  8. (.@) :: a -> (a -> b) -> b

    ghci-hexcalc Data.GHex

    Operator for postfix notation (same as Data.Function.(&)) 

    >>> 255 .@hex
"0x0000_0000_0000_00ff"

>>> 0xf1 .@bin
"0b1111_0001"

>>> 2^12 .@dec
"4096"

>>> 4 * giga .@pos1
[32]

    0x0 .@color (bits 31 24)
0b0000_0000_0000_0000_0000_0000_0000_0000_1111_1111_0000_0000_0000_0000_0000_0000
^^^^ ^^^^

  9. (.^) :: Hex -> Hex -> Hex

    ghci-hexcalc Data.GHex

    Bitwise "xor" 

    >>> 0xf .^ 0xa
0x0000_0000_0000_0005

  10. (.|) :: Hex -> Hex -> Hex

    ghci-hexcalc Data.GHex

    Bitwise "or" 

    >>> 0xf000 .| 0xa
0x0000_0000_0000_f00a

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