||| quantities count how many times a bound variable is used [@nuttin; @qtt].
|||
||| i tried grtt [@grtt] for a bit but i think it was more complex than
||| it's worth in a language that has other stuff going on too
module Quox.Syntax.Qty

import Quox.Pretty
import Quox.Decidable
import Data.DPair
import Derive.Prelude

%default total
%language ElabReflection


||| the possibilities we care about are:
|||
||| - 0: a variable is used only at compile time, not run time
||| - 1: a variable is used exactly once at run time
||| - ω (or #): don't care. an ω variable *can* also be used 0/1 time
public export
data Qty = Zero | One | Any
%runElab derive "Qty" [Eq, Ord, Show]
%name Qty.Qty pi, rh


export
prettyQty : {opts : _} -> Qty -> Eff Pretty (Doc opts)
prettyQty Zero = hl Qty $ text "0"
prettyQty One  = hl Qty $ text "1"
prettyQty Any  = hl Qty =<< ifUnicode (text "ω") (text "#")

||| prints in a form that can be a suffix of "case"
public export
prettySuffix : {opts : _} -> Qty -> Eff Pretty (Doc opts)
prettySuffix = prettyQty


||| e.g. if in the expression `(s, t)`, the variable `x` is
||| used π times in `s` and ρ times in `t`, then it's used
||| (π + ρ) times in the whole expression
public export
(+) : Qty -> Qty -> Qty
Zero + rh   = rh
pi   + Zero = pi
_    + _    = Any

||| e.g. if a function `f` uses its argument π times,
||| and `f x` occurs in a σ context, then `x` is used `πσ` times overall
public export
(*) : Qty -> Qty -> Qty
Zero * _    = Zero
_    * Zero = Zero
One  * rh   = rh
pi   * One  = pi
Any  * Any  = Any

||| "π ≤ ρ"
|||
||| if a variable is bound with quantity ρ, then it can be used with a total
||| quantity π as long as π ≤ ρ. for example, an ω variable can be used any
||| number of times, so π ≤ ω for any π.
public export
compat : Qty -> Qty -> Bool
compat pi Any = True
compat pi rh  = pi == rh


||| "π ∨ ρ"
|||
||| returns a quantity τ with π ≤ τ and ρ ≤ τ.
||| if π = ρ, then it's that, otherwise it's ω.
public export
lub : Qty -> Qty -> Qty
lub p q = if p == q then p else Any


||| to maintain subject reduction, only 0 or 1 can occur
||| for the subject of a typing judgment. see @qtt, §2.3 for more detail
public export
isSubj : Qty -> Bool
isSubj Zero = True
isSubj One  = True
isSubj Any  = False

public export
SQty : Type
SQty = Subset Qty $ So . isSubj

public export %inline
szero, sone : SQty
szero = Element Zero Oh
sone  = Element One  Oh

||| "σ ⨴ π"
|||
||| σ ⨭ π is 0 if either of σ or π are, otherwise it is σ.
public export
subjMult : SQty -> Qty -> SQty
subjMult _  Zero = szero
subjMult sg _    = sg


||| it doesn't make much sense for a top level declaration to have a
||| quantity of 1, so the only distinction is whether it is present
||| at runtime at all or not
public export
isGlobal : Qty -> Bool
isGlobal Zero = True
isGlobal One  = False
isGlobal Any  = True

public export
GQty : Type
GQty = Subset Qty $ So . isGlobal

public export
gzero, gany : GQty
gzero = Element Zero Oh
gany  = Element Any  Oh

||| when checking a definition, a 0 definition is checked at 0,
||| but an ω definition is checked at 1 since ω isn't a subject quantity
public export %inline
globalToSubj : GQty -> SQty
globalToSubj (Element Zero _) = szero
globalToSubj (Element Any  _) = sone