quox/lib/Quox/Equal.idr

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module Quox.Equal
import Quox.BoolExtra
import public Quox.Typing
import Data.Maybe
import Quox.EffExtra
%default total
public export
0 EqModeState : Type -> Type
EqModeState = State EqMode
public export
0 EqualEff : List (Type -> Type)
EqualEff = [ErrorEff, EqModeState]
public export
0 EqualE : Type -> Type
EqualE = Eff $ EqualEff
export %inline
mode : Has EqModeState fs => Eff fs EqMode
mode = get
parameters (ctx : EqContext n)
private %inline
clashT : Term 0 n -> Term 0 n -> Term 0 n -> EqualE a
clashT ty s t = throw $ ClashT ctx !mode ty s t
private %inline
clashTy : Term 0 n -> Term 0 n -> EqualE a
clashTy s t = throw $ ClashTy ctx !mode s t
private %inline
clashE : Elim 0 n -> Elim 0 n -> EqualE a
clashE e f = throw $ ClashE ctx !mode e f
private %inline
wrongType : Term 0 n -> Term 0 n -> EqualE a
wrongType ty s = throw $ WrongType ctx ty s
||| true if a term is syntactically a type, or is neutral.
|||
||| this function *doesn't* push substitutions, because its main use is as a
||| `So` argument to skip cases that are already known to be nonsense. and
||| the substitutions have already been pushed.
public export %inline
isTyCon : (t : Term {}) -> Bool
isTyCon (TYPE {}) = True
isTyCon (Pi {}) = True
isTyCon (Lam {}) = False
isTyCon (Sig {}) = True
isTyCon (Pair {}) = False
isTyCon (Enum {}) = True
isTyCon (Tag {}) = False
isTyCon (Eq {}) = True
isTyCon (DLam {}) = False
isTyCon Nat = True
isTyCon Zero = False
isTyCon (Succ {}) = False
isTyCon (BOX {}) = True
isTyCon (Box {}) = False
isTyCon (E {}) = True
isTyCon (CloT {}) = False
isTyCon (DCloT {}) = False
public export %inline
sameTyCon : (s, t : Term d n) ->
(0 ts : So (isTyCon s)) => (0 tt : So (isTyCon t)) =>
Bool
sameTyCon (TYPE {}) (TYPE {}) = True
sameTyCon (TYPE {}) _ = False
sameTyCon (Pi {}) (Pi {}) = True
sameTyCon (Pi {}) _ = False
sameTyCon (Sig {}) (Sig {}) = True
sameTyCon (Sig {}) _ = False
sameTyCon (Enum {}) (Enum {}) = True
sameTyCon (Enum {}) _ = False
sameTyCon (Eq {}) (Eq {}) = True
sameTyCon (Eq {}) _ = False
sameTyCon Nat Nat = True
sameTyCon Nat _ = False
sameTyCon (BOX {}) (BOX {}) = True
sameTyCon (BOX {}) _ = False
sameTyCon (E {}) (E {}) = True
sameTyCon (E {}) _ = False
parameters (defs : Definitions)
||| true if a type is known to be a subsingleton purely by its form.
||| a subsingleton is a type with only zero or one possible values.
||| equality/subtyping accepts immediately on values of subsingleton types.
|||
||| * a function type is a subsingleton if its codomain is.
||| * a pair type is a subsingleton if both its elements are.
||| * all equality types are subsingletons because uip is admissible by
||| boundary separation.
||| * an enum type is a subsingleton if it has zero or one tags.
public export covering
isSubSing : Has ErrorEff fs =>
{n : Nat} -> Term 0 n -> Eff fs Bool
isSubSing ty0 = do
Element ty0 nc <- whnfT defs ty0
case ty0 of
TYPE _ => pure False
Pi {res, _} => isSubSing res.term
Lam {} => pure False
Sig {fst, snd} => isSubSing fst `andM` isSubSing snd.term
Pair {} => pure False
Enum tags => pure $ length (SortedSet.toList tags) <= 1
Tag {} => pure False
Eq {} => pure True
DLam {} => pure False
Nat => pure False
Zero => pure False
Succ {} => pure False
BOX {ty, _} => isSubSing ty
Box {} => pure False
E (s :# _) => isSubSing s
E _ => pure False
export
ensureTyCon : Has ErrorEff fs =>
(ctx : EqContext n) -> (t : Term 0 n) ->
Eff fs (So (isTyCon t))
ensureTyCon ctx t = case nchoose $ isTyCon t of
Left y => pure y
Right n => throw $ NotType (toTyContext ctx) (t // shift0 ctx.dimLen)
parameters (defs : Definitions)
mutual
namespace Term
||| `compare0 ctx ty s t` compares `s` and `t` at type `ty`, according to
||| the current variance `mode`.
|||
||| ⚠ **assumes that `s`, `t` have already been checked against `ty`**. ⚠
export covering %inline
compare0 : EqContext n -> (ty, s, t : Term 0 n) -> EqualE ()
compare0 ctx ty s t =
wrapErr (WhileComparingT ctx !mode ty s t) $ do
let Val n = ctx.termLen
Element ty nty <- whnfT defs ty
Element s ns <- whnfT defs s
Element t nt <- whnfT defs t
tty <- ensureTyCon ctx ty
compare0' ctx ty s t
||| converts an elim "Γ ⊢ e" to "Γ, x ⊢ e x", for comparing with
||| a lambda "Γ ⊢ λx ⇒ t" that has been converted to "Γ, x ⊢ t".
private %inline
toLamBody : Elim d n -> Term d (S n)
toLamBody e = E $ weakE e :@ BVT 0
private covering
compare0' : EqContext n ->
(ty, s, t : Term 0 n) ->
(0 nty : NotRedex defs ty) => (0 tty : So (isTyCon ty)) =>
(0 ns : NotRedex defs s) => (0 nt : NotRedex defs t) =>
EqualE ()
compare0' ctx (TYPE _) s t = compareType ctx s t
compare0' ctx ty@(Pi {qty, arg, res}) s t {n} = local_ Equal $
case (s, t) of
-- Γ, x : A ⊢ s = t : B
-- -------------------------------------------
-- Γ ⊢ (λ x ⇒ s) = (λ x ⇒ t) : (π·x : A) → B
(Lam b1, Lam b2) => compare0 ctx' res.term b1.term b2.term
-- Γ, x : A ⊢ s = e x : B
-- -----------------------------------
-- Γ ⊢ (λ x ⇒ s) = e : (π·x : A) → B
(E e, Lam b) => eta e b
(Lam b, E e) => eta e b
(E e, E f) => Elim.compare0 ctx e f
(Lam _, t) => wrongType ctx ty t
(E _, t) => wrongType ctx ty t
(s, _) => wrongType ctx ty s
where
ctx' : EqContext (S n)
ctx' = extendTy qty res.name arg ctx
eta : Elim 0 n -> ScopeTerm 0 n -> EqualE ()
eta e (S _ (Y b)) = compare0 ctx' res.term (toLamBody e) b
eta e (S _ (N _)) = clashT ctx ty s t
compare0' ctx ty@(Sig {fst, snd, _}) s t = local_ Equal $
case (s, t) of
-- Γ ⊢ s₁ = t₁ : A Γ ⊢ s₂ = t₂ : B{s₁/x}
-- --------------------------------------------
-- Γ ⊢ (s₁, t₁) = (s₂,t₂) : (x : A) × B
--
-- [todo] η for π ≥ 0 maybe
(Pair sFst sSnd, Pair tFst tSnd) => do
compare0 ctx fst sFst tFst
compare0 ctx (sub1 snd (sFst :# fst)) sSnd tSnd
(E e, E f) => Elim.compare0 ctx e f
(Pair {}, E _) => clashT ctx ty s t
(E _, Pair {}) => clashT ctx ty s t
(Pair {}, t) => wrongType ctx ty t
(E _, t) => wrongType ctx ty t
(s, _) => wrongType ctx ty s
compare0' ctx ty@(Enum tags) s t = local_ Equal $
case (s, t) of
-- --------------------
-- Γ ⊢ `t = `t : {ts}
--
-- t ∈ ts is in the typechecker, not here, ofc
(Tag t1, Tag t2) => unless (t1 == t2) $ clashT ctx ty s t
(E e, E f) => Elim.compare0 ctx e f
(Tag _, E _) => clashT ctx ty s t
(E _, Tag _) => clashT ctx ty s t
(Tag _, t) => wrongType ctx ty t
(E _, t) => wrongType ctx ty t
(s, _) => wrongType ctx ty s
compare0' _ (Eq {}) _ _ =
-- ✨ uip ✨
--
-- Γ ⊢ e = f : Eq [i ⇒ A] s t
pure ()
compare0' ctx Nat s t = local_ Equal $
case (s, t) of
-- ---------------
-- Γ ⊢ 0 = 0 :
(Zero, Zero) => pure ()
-- Γ ⊢ m = n :
-- -------------------------
-- Γ ⊢ succ m = succ n :
(Succ m, Succ n) => compare0 ctx Nat m n
(E e, E f) => Elim.compare0 ctx e f
(Zero, Succ _) => clashT ctx Nat s t
(Zero, E _) => clashT ctx Nat s t
(Succ _, Zero) => clashT ctx Nat s t
(Succ _, E _) => clashT ctx Nat s t
(E _, Zero) => clashT ctx Nat s t
(E _, Succ _) => clashT ctx Nat s t
(Zero, t) => wrongType ctx Nat t
(Succ _, t) => wrongType ctx Nat t
(E _, t) => wrongType ctx Nat t
(s, _) => wrongType ctx Nat s
compare0' ctx ty@(BOX q ty') s t = local_ Equal $
case (s, t) of
-- Γ ⊢ s = t : A
-- -----------------------
-- Γ ⊢ [s] = [t] : [π.A]
(Box s, Box t) => compare0 ctx ty' s t
(E e, E f) => Elim.compare0 ctx e f
(Box _, t) => wrongType ctx ty t
(E _, t) => wrongType ctx ty t
(s, _) => wrongType ctx ty s
compare0' ctx ty@(E _) s t = do
-- a neutral type can only be inhabited by neutral values
-- e.g. an abstract value in an abstract type, bound variables, …
E e <- pure s | _ => wrongType ctx ty s
E f <- pure t | _ => wrongType ctx ty t
Elim.compare0 ctx e f
||| compares two types, using the current variance `mode` for universes.
||| fails if they are not types, even if they would happen to be equal.
export covering %inline
compareType : EqContext n -> (s, t : Term 0 n) -> EqualE ()
compareType ctx s t = do
let Val n = ctx.termLen
Element s ns <- whnfT defs s
Element t nt <- whnfT defs t
ts <- ensureTyCon ctx s
tt <- ensureTyCon ctx t
st <- either pure (const $ clashTy ctx s t) $
nchoose $ sameTyCon s t
compareType' ctx s t
private covering
compareType' : EqContext n -> (s, t : Term 0 n) ->
(0 ns : NotRedex defs s) => (0 ts : So (isTyCon s)) =>
(0 nt : NotRedex defs t) => (0 tt : So (isTyCon t)) =>
(0 st : So (sameTyCon s t)) =>
EqualE ()
-- equality is the same as subtyping, except with the
-- "≤" in the TYPE rule being replaced with "="
compareType' ctx (TYPE k) (TYPE l) =
-- 𝓀
-- ----------------------
-- Γ ⊢ Type 𝓀 <: Type
expectModeU !mode k l
compareType' ctx (Pi {qty = sQty, arg = sArg, res = sRes, _})
(Pi {qty = tQty, arg = tArg, res = tRes, _}) = do
-- Γ ⊢ A₁ :> A₂ Γ, x : A₁ ⊢ B₁ <: B₂
-- ----------------------------------------
-- Γ ⊢ (π·x : A₁) → B₁ <: (π·x : A₂) → B₂
expectEqualQ sQty tQty
local flip $ compareType ctx sArg tArg -- contra
compareType (extendTy Zero sRes.name sArg ctx) sRes.term tRes.term
compareType' ctx (Sig {fst = sFst, snd = sSnd, _})
(Sig {fst = tFst, snd = tSnd, _}) = do
-- Γ ⊢ A₁ <: A₂ Γ, x : A₁ ⊢ B₁ <: B₂
-- --------------------------------------
-- Γ ⊢ (x : A₁) × B₁ <: (x : A₂) × B₂
compareType ctx sFst tFst
compareType (extendTy Zero sSnd.name sFst ctx) sSnd.term tSnd.term
compareType' ctx (Eq {ty = sTy, l = sl, r = sr, _})
(Eq {ty = tTy, l = tl, r = tr, _}) = do
-- Γ ⊢ A₁ε/i <: A₂ε/i
-- Γ ⊢ l₁ = l₂ : A₁𝟎/i Γ ⊢ r₁ = r₂ : A₁𝟏/i
-- ------------------------------------------------
-- Γ ⊢ Eq [i ⇒ A₁] l₁ r₂ <: Eq [i ⇒ A₂] l₂ r₂
compareType (extendDim sTy.name Zero ctx) sTy.zero tTy.zero
compareType (extendDim sTy.name One ctx) sTy.one tTy.one
local_ Equal $ do
Term.compare0 ctx sTy.zero sl tl
Term.compare0 ctx sTy.one sr tr
compareType' ctx s@(Enum tags1) t@(Enum tags2) = do
-- ------------------
-- Γ ⊢ {ts} <: {ts}
--
-- no subtyping based on tag subsets, since that would need
-- a runtime coercion
unless (tags1 == tags2) $ clashTy ctx s t
compareType' ctx Nat Nat =
-- ------------
-- Γ ⊢ <:
pure ()
compareType' ctx (BOX pi a) (BOX rh b) = do
expectEqualQ pi rh
compareType ctx a b
compareType' ctx (E e) (E f) = do
-- no fanciness needed here cos anything other than a neutral
-- has been inlined by whnf
Elim.compare0 ctx e f
||| performs the minimum work required to recompute the type of an elim.
|||
||| ⚠ **assumes the elim is already typechecked.** ⚠
private covering
computeElimType : EqContext n -> (e : Elim 0 n) ->
(0 ne : NotRedex defs e) ->
EqualE (Term 0 n)
computeElimType ctx (F x) _ = do
defs <- lookupFree' defs x
pure $ injectT ctx defs.type
computeElimType ctx (B i) _ = pure $ ctx.tctx !! i
computeElimType ctx (f :@ s) ne = do
(_, arg, res) <- expectPiE defs ctx !(computeElimType ctx f (noOr1 ne))
pure $ sub1 res (s :# arg)
computeElimType ctx (CasePair {pair, ret, _}) _ = pure $ sub1 ret pair
computeElimType ctx (CaseEnum {tag, ret, _}) _ = pure $ sub1 ret tag
computeElimType ctx (CaseNat {nat, ret, _}) _ = pure $ sub1 ret nat
computeElimType ctx (CaseBox {box, ret, _}) _ = pure $ sub1 ret box
computeElimType ctx (f :% p) ne = do
(ty, _, _) <- expectEqE defs ctx !(computeElimType ctx f (noOr1 ne))
pure $ dsub1 ty p
computeElimType ctx (_ :# ty) _ = pure ty
private covering
replaceEnd : EqContext n ->
(e : Elim 0 n) -> DimConst -> (0 ne : NotRedex defs e) ->
EqualE (Elim 0 n)
replaceEnd ctx e p ne = do
(ty, l, r) <- expectEqE defs ctx !(computeElimType ctx e ne)
pure $ ends l r p :# dsub1 ty (K p)
namespace Elim
-- [fixme] the following code ends up repeating a lot of work in the
-- computeElimType calls. the results should be shared better
||| compare two eliminations according to the given variance `mode`.
|||
||| ⚠ **assumes that they have both been typechecked, and have
||| equal types.** ⚠
export covering %inline
compare0 : EqContext n -> (e, f : Elim 0 n) -> EqualE ()
compare0 ctx e f =
wrapErr (WhileComparingE ctx !mode e f) $ do
let Val n = ctx.termLen
Element e ne <- whnfT defs e
Element f nf <- whnfT defs f
-- [fixme] there is a better way to do this "isSubSing" stuff for sure
unless !(isSubSing defs !(computeElimType ctx e ne)) $
compare0' ctx e f ne nf
private covering
compare0' : EqContext n ->
(e, f : Elim 0 n) ->
(0 ne : NotRedex defs e) -> (0 nf : NotRedex defs f) ->
EqualE ()
-- replace applied equalities with the appropriate end first
-- e.g. e : Eq [i ⇒ A] s t ⊢ e 𝟎 = s : A𝟎/i
--
-- [todo] maybe have typed whnf and do this (and η???) there instead
compare0' ctx (e :% K p) f ne nf =
compare0 ctx !(replaceEnd ctx e p $ noOr1 ne) f
compare0' ctx e (f :% K q) ne nf =
compare0 ctx e !(replaceEnd ctx f q $ noOr1 nf)
compare0' ctx e@(F x) f@(F y) _ _ = unless (x == y) $ clashE ctx e f
compare0' ctx e@(F _) f _ _ = clashE ctx e f
compare0' ctx e@(B i) f@(B j) _ _ = unless (i == j) $ clashE ctx e f
compare0' ctx e@(B _) f _ _ = clashE ctx e f
compare0' ctx (e :@ s) (f :@ t) ne nf =
local_ Equal $ do
compare0 ctx e f
(_, arg, _) <- expectPiE defs ctx !(computeElimType ctx e (noOr1 ne))
Term.compare0 ctx arg s t
compare0' ctx e@(_ :@ _) f _ _ = clashE ctx e f
compare0' ctx (CasePair epi e eret ebody)
(CasePair fpi f fret fbody) ne nf =
local_ Equal $ do
compare0 ctx e f
ety <- computeElimType ctx e (noOr1 ne)
compareType (extendTy Zero eret.name ety ctx) eret.term fret.term
(fst, snd) <- expectSigE defs ctx ety
let [< x, y] = ebody.names
Term.compare0 (extendTyN [< (epi, x, fst), (epi, y, snd.term)] ctx)
(substCasePairRet ety eret)
ebody.term fbody.term
expectEqualQ epi fpi
compare0' ctx e@(CasePair {}) f _ _ = clashE ctx e f
compare0' ctx (CaseEnum epi e eret earms)
(CaseEnum fpi f fret farms) ne nf =
local_ Equal $ do
compare0 ctx e f
ety <- computeElimType ctx e (noOr1 ne)
compareType (extendTy Zero eret.name ety ctx) eret.term fret.term
for_ !(expectEnumE defs ctx ety) $ \t =>
compare0 ctx (sub1 eret $ Tag t :# ety)
!(lookupArm t earms) !(lookupArm t farms)
expectEqualQ epi fpi
where
lookupArm : TagVal -> CaseEnumArms d n -> EqualE (Term d n)
lookupArm t arms = case lookup t arms of
Just arm => pure arm
Nothing => throw $ TagNotIn t (fromList $ keys arms)
compare0' ctx e@(CaseEnum {}) f _ _ = clashE ctx e f
compare0' ctx (CaseNat epi epi' e eret ezer esuc)
(CaseNat fpi fpi' f fret fzer fsuc) ne nf =
local_ Equal $ do
compare0 ctx e f
ety <- computeElimType ctx e (noOr1 ne)
compareType (extendTy Zero eret.name ety ctx) eret.term fret.term
compare0 ctx (sub1 eret (Zero :# Nat)) ezer fzer
let [< p, ih] = esuc.names
compare0 (extendTyN [< (epi, p, Nat), (epi', ih, eret.term)] ctx)
(substCaseSuccRet eret)
esuc.term fsuc.term
expectEqualQ epi fpi
expectEqualQ epi' fpi'
compare0' ctx e@(CaseNat {}) f _ _ = clashE ctx e f
compare0' ctx (CaseBox epi e eret ebody)
(CaseBox fpi f fret fbody) ne nf =
local_ Equal $ do
compare0 ctx e f
ety <- computeElimType ctx e (noOr1 ne)
compareType (extendTy Zero eret.name ety ctx) eret.term fret.term
(q, ty) <- expectBOXE defs ctx ety
compare0 (extendTy (epi * q) ebody.name ty ctx)
(substCaseBoxRet ety eret)
ebody.term fbody.term
expectEqualQ epi fpi
compare0' ctx e@(CaseBox {}) f _ _ = clashE ctx e f
compare0' ctx (s :# a) (t :# b) _ _ =
Term.compare0 ctx !(bigger a b) s t
where
bigger : forall a. a -> a -> EqualE a
bigger l r = mode <&> \case Super => l; _ => r
compare0' ctx (s :# a) f _ _ = Term.compare0 ctx a s (E f)
compare0' ctx e (t :# b) _ _ = Term.compare0 ctx b (E e) t
compare0' ctx e@(_ :# _) f _ _ = clashE ctx e f
parameters {auto _ : (Has DefsReader fs, Has ErrorEff fs)} (ctx : TyContext d n)
-- [todo] only split on the dvars that are actually used anywhere in
-- the calls to `splits`
parameters (mode : EqMode)
namespace Term
export covering
compare : (ty, s, t : Term d n) -> Eff fs ()
compare ty s t = do
defs <- ask
map fst $ runState @{Z} mode $
for_ (splits ctx.dctx) $ \th =>
let ectx = makeEqContext ctx th in
lift $ compare0 defs ectx (ty // th) (s // th) (t // th)
export covering
compareType : (s, t : Term d n) -> Eff fs ()
compareType s t = do
defs <- ask
map fst $ runState @{Z} mode $
for_ (splits ctx.dctx) $ \th =>
let ectx = makeEqContext ctx th in
lift $ compareType defs ectx (s // th) (t // th)
namespace Elim
||| you don't have to pass the type in but the arguments must still be
||| of the same type!!
export covering %inline
compare : (e, f : Elim d n) -> Eff fs ()
compare e f = do
defs <- ask
map fst $ runState @{Z} mode $
for_ (splits ctx.dctx) $ \th =>
let ectx = makeEqContext ctx th in
lift $ compare0 defs ectx (e // th) (f // th)
namespace Term
export covering %inline
equal, sub, super : (ty, s, t : Term d n) -> Eff fs ()
equal = compare Equal
sub = compare Sub
super = compare Super
export covering %inline
equalType, subtype, supertype : (s, t : Term d n) -> Eff fs ()
equalType = compareType Equal
subtype = compareType Sub
supertype = compareType Super
namespace Elim
export covering %inline
equal, sub, super : (e, f : Elim d n) -> Eff fs ()
equal = compare Equal
sub = compare Sub
super = compare Super