module Quox.Equal import public Quox.Syntax import public Quox.Definition import public Quox.Typing import public Control.Monad.Either import public Control.Monad.Reader import Data.Maybe public export record CmpContext where constructor MkCmpContext mode : EqMode public export 0 HasCmpContext : (Type -> Type) -> Type HasCmpContext = MonadReader CmpContext public export 0 CanEqual : (q : Type) -> (Type -> Type) -> Type CanEqual q m = (HasErr q m, HasCmpContext m) private %inline mode : HasCmpContext m => m EqMode mode = asks mode private %inline clashT : CanEqual q m => Term q d n -> Term q d n -> Term q d n -> m a clashT ty s t = throwError $ ClashT !mode ty s t private %inline clashE : CanEqual q m => Elim q d n -> Elim q d n -> m a clashE e f = throwError $ ClashE !mode e f ||| 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 (Eq {}) = True isTyCon (DLam {}) = False isTyCon (E {}) = True isTyCon (CloT {}) = False isTyCon (DCloT {}) = False public export %inline sameTyCon : (s, t : Term q 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 (Eq {}) (Eq {}) = True sameTyCon (Eq {}) _ = False sameTyCon (E {}) (E {}) = True sameTyCon (E {}) _ = False parameters (defs : Definitions' q g) ||| 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. private isSubSing : Term q 0 n -> Bool isSubSing ty = let Element ty nc = whnfD defs ty in case ty of TYPE _ => False Pi {res, _} => isSubSing res.term Lam {} => False Sig {fst, snd, _} => isSubSing fst && isSubSing snd.term Pair {} => False Eq {} => True DLam {} => False E (s :# _) => isSubSing s E _ => False parameters {auto _ : HasErr q m} export %inline ensure : (a -> Error q) -> (p : a -> Bool) -> (t : a) -> m (So (p t)) ensure e p t = case nchoose $ p t of Left y => pure y Right _ => throwError $ e t export %inline ensureTyCon : HasErr q m => (t : Term q d n) -> m (So (isTyCon t)) ensureTyCon = ensure NotType isTyCon parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)} 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 : TContext q 0 n -> (ty, s, t : Term q 0 n) -> m () compare0 ctx ty s t = do let Element ty nty = whnfD defs ty Element s ns = whnfD defs s Element t nt = whnfD defs t tty <- ensureTyCon 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 q d n -> Term q d (S n) toLamBody e = E $ weakE e :@ BVT 0 private covering compare0' : TContext q 0 n -> (ty, s, t : Term q 0 n) -> (0 nty : NotRedex defs ty) => (0 tty : So (isTyCon ty)) => (0 ns : NotRedex defs s) => (0 nt : NotRedex defs t) => m () compare0' ctx (TYPE _) s t = compareType ctx s t compare0' ctx ty@(Pi {arg, res, _}) s t {n} = local {mode := Equal} $ case (s, t) of -- Γ, x : A ⊢ s = t : B -- ----------------------------------------- -- Γ ⊢ (λx ⇒ s) = (λx ⇒ 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 ⊢ (λx ⇒ 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 _ => throwError $ WrongType ty s t where ctx' : TContext q 0 (S n) ctx' = ctx :< arg eta : Elim q 0 n -> ScopeTerm q 0 n -> m () eta e (TUsed b) = compare0 ctx' res.term (toLamBody e) b eta e (TUnused _) = clashT ty s t compare0' ctx ty@(Sig {fst, snd, _}) s t = local {mode := 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 _ => throwError $ WrongType ty s t compare0' _ (Eq {}) _ _ = -- ✨ uip ✨ -- -- Γ ⊢ e = f : Eq [i ⇒ A] s t pure () 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 | _ => throwError $ WrongType ty s t E f <- pure t | _ => throwError $ WrongType ty s 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 : TContext q 0 n -> (s, t : Term q 0 n) -> m () compareType ctx s t = do let Element s ns = whnfD defs s Element t nt = whnfD defs t ts <- ensureTyCon s tt <- ensureTyCon t st <- either pure (const $ clashT (TYPE UAny) s t) $ nchoose $ sameTyCon s t compareType' ctx s t private covering compareType' : TContext q 0 n -> (s, t : Term q 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)) => m () -- equality is the same as subtyping, except with the -- "≤" in the TYPE rule being replaced with "=" compareType' ctx (TYPE k) (TYPE l) = -- 𝓀 ≤ ℓ -- ---------------------- -- Γ ⊢ Type 𝓀 <: Type ℓ 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 {mode $= flip} $ compareType ctx sArg tArg -- contra compareType (ctx :< sArg) 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 (ctx :< sFst) sSnd.term tSnd.term compareType' ctx (Eq {ty = sTy, l = sl, r = sr, _}) (Eq {ty = tTy, l = tl, r = tr, _}) = do -- Γ ⊢ A₁‹ε/i› <: A₂‹ε/i› -- Γ ⊢ l₁ = l₂ : A₁‹𝟎/i› Γ ⊢ r₁ = r₂ : A₁‹𝟏/i› -- ------------------------------------------------ -- Γ ⊢ Eq [i ⇒ A₁] l₁ r₂ <: Eq [i ⇒ A₂] l₂ r₂ compareType ctx sTy.zero tTy.zero compareType ctx sTy.one tTy.one local {mode := Equal} $ do Term.compare0 ctx sTy.zero sl tl Term.compare0 ctx sTy.one sr tr compareType' ctx (E e) (E f) = do -- no fanciness needed here cos anything other than a neutral -- has been inlined by whnfD 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 : TContext q 0 n -> (e : Elim q 0 n) -> (0 ne : NotRedex defs e) -> m (Term q 0 n) computeElimType ctx (F x) _ = do defs <- lookupFree' defs x pure $ defs.type.get computeElimType ctx (B i) _ = do pure $ ctx !! i computeElimType ctx (f :@ s) ne = do (_, arg, res) <- expectPi defs !(computeElimType ctx f (noOr1 ne)) pure $ sub1 res (s :# arg) computeElimType ctx (CasePair {pair, ret, _}) _ = do pure $ sub1 ret pair computeElimType ctx (f :% p) ne = do (ty, _, _) <- expectEq defs !(computeElimType ctx f (noOr1 ne)) pure $ dsub1 ty p computeElimType ctx (_ :# ty) _ = do pure ty private covering replaceEnd : TContext q 0 n -> (e : Elim q 0 n) -> DimConst -> (0 ne : NotRedex defs e) -> m (Elim q 0 n) replaceEnd ctx e p ne = do (ty, l, r) <- expectEq defs !(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 : TContext q 0 n -> (e, f : Elim q 0 n) -> m () compare0 ctx e f = let Element e ne = whnfD defs e Element f nf = whnfD defs f in -- [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' : TContext q 0 n -> (e, f : Elim q 0 n) -> (0 ne : NotRedex defs e) -> (0 nf : NotRedex defs f) -> m () -- replace applied equalities with the appropriate end first -- e.g. e : Eq [i ⇒ A] s t ⊢ e 𝟎 = s : A‹𝟎/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' _ e@(F x) f@(F y) _ _ = unless (x == y) $ clashE e f compare0' _ e@(F _) f _ _ = clashE e f compare0' ctx e@(B i) f@(B j) _ _ = unless (i == j) $ clashE e f compare0' _ e@(B _) f _ _ = clashE e f compare0' ctx (e :@ s) (f :@ t) ne nf = local {mode := Equal} $ do compare0 ctx e f (_, arg, _) <- expectPi defs !(computeElimType ctx e (noOr1 ne)) Term.compare0 ctx arg s t compare0' _ e@(_ :@ _) f _ _ = clashE e f compare0' ctx (CasePair epi e _ eret _ _ ebody) (CasePair fpi f _ fret _ _ fbody) ne nf = local {mode := Equal} $ do compare0 ctx e f ety <- computeElimType ctx e (noOr1 ne) compareType (ctx :< ety) eret.term fret.term (fst, snd) <- expectSig defs ety Term.compare0 (ctx :< fst :< snd.term) (substCasePairRet ety eret) ebody.term fbody.term unless (epi == fpi) $ throwError $ ClashQ epi fpi compare0' _ e@(CasePair {}) f _ _ = clashE e f compare0' ctx (s :# a) (t :# _) _ _ = Term.compare0 ctx a s t 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' _ e@(_ :# _) f _ _ = clashE e f parameters {auto _ : (HasDefs' q _ m, HasErr q m, Eq q)} (ctx : TyContext q d n) parameters (mode : EqMode) namespace Term export covering compare : (ty, s, t : Term q d n) -> m () compare ty s t = do defs <- ask runReaderT {m} (MkCmpContext {mode}) $ for_ (splits ctx.dctx) $ \th => compare0 defs (map (/// th) ctx.tctx) (ty /// th) (s /// th) (t /// th) export covering compareType : (s, t : Term q d n) -> m () compareType s t = do defs <- ask runReaderT {m} (MkCmpContext {mode}) $ for_ (splits ctx.dctx) $ \th => compareType defs (map (/// th) ctx.tctx) (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 q d n) -> m () compare e f = do defs <- ask runReaderT {m} (MkCmpContext {mode}) $ for_ (splits ctx.dctx) $ \th => compare0 defs (map (/// th) ctx.tctx) (e /// th) (f /// th) namespace Term export covering %inline equal, sub, super : (ty, s, t : Term q d n) -> m () equal = compare Equal sub = compare Sub super = compare Super export covering %inline equalType, subtype, supertype : (s, t : Term q d n) -> m () equalType = compareType Equal subtype = compareType Sub supertype = compareType Super namespace Elim export covering %inline equal, sub, super : (e, f : Elim q d n) -> m () equal = compare Equal sub = compare Sub super = compare Super