393 lines
14 KiB
Idris
393 lines
14 KiB
Idris
module Quox.Equal
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import public Quox.Syntax
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import public Quox.Definition
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import public Quox.Typing
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import public Control.Monad.Either
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import public Control.Monad.Reader
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import Data.Maybe
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public export
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record CmpContext where
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constructor MkCmpContext
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mode : EqMode
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public export
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0 HasCmpContext : (Type -> Type) -> Type
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HasCmpContext = MonadReader CmpContext
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public export
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0 CanEqual : (q : Type) -> (Type -> Type) -> Type
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CanEqual q m = (HasErr q m, HasCmpContext m)
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private %inline
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mode : HasCmpContext m => m EqMode
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mode = asks mode
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private %inline
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clashT : CanEqual q m => Term q d n -> Term q d n -> Term q d n -> m a
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clashT ty s t = throwError $ ClashT !mode ty s t
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private %inline
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clashE : CanEqual q m => Elim q d n -> Elim q d n -> m a
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clashE e f = throwError $ ClashE !mode e f
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||| true if a term is syntactically a type, or is neutral.
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|||
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||| this function *doesn't* push substitutions, because its main use is as a
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||| `So` argument to skip cases that are already known to be nonsense. and
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||| the substitutions have already been pushed.
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public export %inline
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isTyCon : (t : Term {}) -> Bool
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isTyCon (TYPE {}) = True
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isTyCon (Pi {}) = True
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isTyCon (Lam {}) = False
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isTyCon (Sig {}) = True
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isTyCon (Pair {}) = False
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isTyCon (Eq {}) = True
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isTyCon (DLam {}) = False
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isTyCon (E {}) = True
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isTyCon (CloT {}) = False
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isTyCon (DCloT {}) = False
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public export %inline
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sameTyCon : (s, t : Term q d n) ->
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(0 ts : So (isTyCon s)) => (0 tt : So (isTyCon t)) =>
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Bool
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sameTyCon (TYPE {}) (TYPE {}) = True
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sameTyCon (TYPE {}) _ = False
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sameTyCon (Pi {}) (Pi {}) = True
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sameTyCon (Pi {}) _ = False
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sameTyCon (Sig {}) (Sig {}) = True
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sameTyCon (Sig {}) _ = False
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sameTyCon (Eq {}) (Eq {}) = True
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sameTyCon (Eq {}) _ = False
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sameTyCon (E {}) (E {}) = True
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sameTyCon (E {}) _ = False
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parameters (defs : Definitions' q g)
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||| true if a type is known to be a subsingleton purely by its form.
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||| a subsingleton is a type with only zero or one possible values.
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||| equality/subtyping accepts immediately on values of subsingleton types.
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|||
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||| * a function type is a subsingleton if its codomain is.
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||| * a pair type is a subsingleton if both its elements are.
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||| * all equality types are subsingletons because uip is admissible by
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||| boundary separation.
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public export
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isSubSing : Term q 0 n -> Bool
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isSubSing ty =
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let Element ty nc = whnfD defs ty in
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case ty of
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TYPE _ => False
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Pi {res, _} => isSubSing res.term
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Lam {} => False
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Sig {fst, snd, _} => isSubSing fst && isSubSing snd.term
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Pair {} => False
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Eq {} => True
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DLam {} => False
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E (s :# _) => isSubSing s
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E _ => False
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parameters {auto _ : HasErr q m}
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export %inline
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ensure : (a -> Error q) -> (p : a -> Bool) -> (t : a) -> m (So (p t))
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ensure e p t = case nchoose $ p t of
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Left y => pure y
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Right _ => throwError $ e t
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export %inline
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ensureTyCon : HasErr q m => (t : Term q d n) -> m (So (isTyCon t))
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ensureTyCon = ensure NotType isTyCon
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parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
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mutual
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namespace Term
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||| `compare0 ctx ty s t` compares `s` and `t` at type `ty`, according to
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||| the current variance `mode`.
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|||
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||| ⚠ **assumes that `s`, `t` have already been checked against `ty`**. ⚠
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export covering %inline
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compare0 : TContext q 0 n -> (ty, s, t : Term q 0 n) -> m ()
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compare0 ctx ty s t = do
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let Element ty nty = whnfD defs ty
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Element s ns = whnfD defs s
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Element t nt = whnfD defs t
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tty <- ensureTyCon ty
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compare0' ctx ty s t
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||| converts an elim "Γ ⊢ e" to "Γ, x ⊢ e x", for comparing with
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||| a lambda "Γ ⊢ λx ⇒ t" that has been converted to "Γ, x ⊢ t".
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private %inline
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toLamBody : Elim q d n -> Term q d (S n)
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toLamBody e = E $ weakE e :@ BVT 0
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private covering
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compare0' : TContext q 0 n ->
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(ty, s, t : Term q 0 n) ->
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(0 nty : NotRedex defs ty) => (0 tty : So (isTyCon ty)) =>
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(0 ns : NotRedex defs s) => (0 nt : NotRedex defs t) =>
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m ()
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compare0' ctx (TYPE _) s t = compareType ctx s t
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compare0' ctx ty@(Pi {arg, res, _}) s t {n} = local {mode := Equal} $
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case (s, t) of
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-- Γ, x : A ⊢ s = t : B
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-- -----------------------------------------
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-- Γ ⊢ (λx ⇒ s) = (λx ⇒ t) : (π·x : A) → B
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(Lam _ b1, Lam _ b2) => compare0 ctx' res.term b1.term b2.term
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-- Γ, x : A ⊢ s = e x : B
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-- ----------------------------------
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-- Γ ⊢ (λx ⇒ s) = e : (π·x : A) → B
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(E e, Lam _ b) => eta e b
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(Lam _ b, E e) => eta e b
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(E e, E f) => Elim.compare0 ctx e f
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_ => throwError $ WrongType ty s t
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where
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ctx' : TContext q 0 (S n)
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ctx' = ctx :< arg
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eta : Elim q 0 n -> ScopeTerm q 0 n -> m ()
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eta e (TUsed b) = compare0 ctx' res.term (toLamBody e) b
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eta e (TUnused _) = clashT ty s t
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compare0' ctx ty@(Sig {fst, snd, _}) s t = local {mode := Equal} $
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case (s, t) of
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-- Γ ⊢ s₁ = t₁ : A Γ ⊢ s₂ = t₂ : B{s₁/x}
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-- -------------------------------------------
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-- Γ ⊢ (s₁,t₁) = (s₂,t₂) : (x : A) × B
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--
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-- [todo] η for π ≥ 0 maybe
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(Pair sFst sSnd, Pair tFst tSnd) => do
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compare0 ctx fst sFst tFst
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compare0 ctx (sub1 snd (sFst :# fst)) sSnd tSnd
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_ => throwError $ WrongType ty s t
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compare0' _ (Eq {}) _ _ =
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-- ✨ uip ✨
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--
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-- Γ ⊢ e = f : Eq [i ⇒ A] s t
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pure ()
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compare0' ctx ty@(E _) s t = do
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-- a neutral type can only be inhabited by neutral values
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-- e.g. an abstract value in an abstract type, bound variables, …
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E e <- pure s | _ => throwError $ WrongType ty s t
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E f <- pure t | _ => throwError $ WrongType ty s t
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Elim.compare0 ctx e f
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||| compares two types, using the current variance `mode` for universes.
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||| fails if they are not types, even if they would happen to be equal.
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export covering %inline
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compareType : TContext q 0 n -> (s, t : Term q 0 n) -> m ()
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compareType ctx s t = do
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let Element s ns = whnfD defs s
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Element t nt = whnfD defs t
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ts <- ensureTyCon s
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tt <- ensureTyCon t
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st <- either pure (const $ clashT (TYPE UAny) s t) $
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nchoose $ sameTyCon s t
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compareType' ctx s t
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private covering
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compareType' : TContext q 0 n -> (s, t : Term q 0 n) ->
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(0 ns : NotRedex defs s) => (0 ts : So (isTyCon s)) =>
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(0 nt : NotRedex defs t) => (0 tt : So (isTyCon t)) =>
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(0 st : So (sameTyCon s t)) =>
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m ()
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-- equality is the same as subtyping, except with the
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-- "≤" in the TYPE rule being replaced with "="
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compareType' ctx (TYPE k) (TYPE l) =
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-- 𝓀 ≤ ℓ
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-- ----------------------
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-- Γ ⊢ Type 𝓀 <: Type ℓ
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expectModeU !mode k l
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compareType' ctx (Pi {qty = sQty, arg = sArg, res = sRes, _})
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(Pi {qty = tQty, arg = tArg, res = tRes, _}) = do
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-- Γ ⊢ A₁ :> A₂ Γ, x : A₁ ⊢ B₁ <: B₂
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-- ----------------------------------------
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-- Γ ⊢ (π·x : A₁) → B₁ <: (π·x : A₂) → B₂
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expectEqualQ sQty tQty
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local {mode $= flip} $ compareType ctx sArg tArg -- contra
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compareType (ctx :< sArg) sRes.term tRes.term
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compareType' ctx (Sig {fst = sFst, snd = sSnd, _})
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(Sig {fst = tFst, snd = tSnd, _}) = do
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-- Γ ⊢ A₁ <: A₂ Γ, x : A₁ ⊢ B₁ <: B₂
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-- --------------------------------------
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-- Γ ⊢ (x : A₁) × B₁ <: (x : A₂) × B₂
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compareType ctx sFst tFst
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compareType (ctx :< sFst) sSnd.term tSnd.term
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compareType' ctx (Eq {ty = sTy, l = sl, r = sr, _})
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(Eq {ty = tTy, l = tl, r = tr, _}) = do
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-- Γ ⊢ A₁‹ε/i› <: A₂‹ε/i›
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-- Γ ⊢ l₁ = l₂ : A₁‹𝟎/i› Γ ⊢ r₁ = r₂ : A₁‹𝟏/i›
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-- ------------------------------------------------
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-- Γ ⊢ Eq [i ⇒ A₁] l₁ r₂ <: Eq [i ⇒ A₂] l₂ r₂
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compareType ctx sTy.zero tTy.zero
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compareType ctx sTy.one tTy.one
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local {mode := Equal} $ do
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Term.compare0 ctx sTy.zero sl tl
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Term.compare0 ctx sTy.one sr tr
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compareType' ctx (E e) (E f) = do
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-- no fanciness needed here cos anything other than a neutral
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-- has been inlined by whnfD
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Elim.compare0 ctx e f
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||| performs the minimum work required to recompute the type of an elim.
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|||
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||| ⚠ **assumes the elim is already typechecked.** ⚠
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private covering
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computeElimType : TContext q 0 n -> (e : Elim q 0 n) ->
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(0 ne : NotRedex defs e) ->
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m (Term q 0 n)
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computeElimType ctx (F x) _ = do
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defs <- lookupFree' defs x
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pure $ defs.type.get
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computeElimType ctx (B i) _ = do
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pure $ ctx !! i
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computeElimType ctx (f :@ s) ne = do
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(_, arg, res) <- expectPi defs !(computeElimType ctx f (noOr1 ne))
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pure $ sub1 res (s :# arg)
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computeElimType ctx (CasePair {pair, ret, _}) _ = do
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pure $ sub1 ret pair
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computeElimType ctx (f :% p) ne = do
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(ty, _, _) <- expectEq defs !(computeElimType ctx f (noOr1 ne))
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pure $ dsub1 ty p
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computeElimType ctx (_ :# ty) _ = do
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pure ty
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private covering
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replaceEnd : TContext q 0 n ->
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(e : Elim q 0 n) -> DimConst -> (0 ne : NotRedex defs e) ->
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m (Elim q 0 n)
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replaceEnd ctx e p ne = do
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(ty, l, r) <- expectEq defs !(computeElimType ctx e ne)
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pure $ ends l r p :# dsub1 ty (K p)
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namespace Elim
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-- [fixme] the following code ends up repeating a lot of work in the
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-- computeElimType calls. the results should be shared better
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||| compare two eliminations according to the given variance `mode`.
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|||
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||| ⚠ **assumes that they have both been typechecked, and have
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||| equal types.** ⚠
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export covering %inline
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compare0 : TContext q 0 n -> (e, f : Elim q 0 n) -> m ()
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compare0 ctx e f =
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let Element e ne = whnfD defs e
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Element f nf = whnfD defs f
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in
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-- [fixme] there is a better way to do this "isSubSing" stuff for sure
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unless (isSubSing defs !(computeElimType ctx e ne)) $
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compare0' ctx e f ne nf
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private covering
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compare0' : TContext q 0 n ->
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(e, f : Elim q 0 n) ->
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(0 ne : NotRedex defs e) -> (0 nf : NotRedex defs f) ->
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m ()
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-- replace applied equalities with the appropriate end first
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-- e.g. e : Eq [i ⇒ A] s t ⊢ e 𝟎 = s : A‹𝟎/i›
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--
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-- [todo] maybe have typed whnf and do this (and η???) there instead
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compare0' ctx (e :% K p) f ne nf =
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compare0 ctx !(replaceEnd ctx e p $ noOr1 ne) f
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compare0' ctx e (f :% K q) ne nf =
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compare0 ctx e !(replaceEnd ctx f q $ noOr1 nf)
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compare0' _ e@(F x) f@(F y) _ _ = unless (x == y) $ clashE e f
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compare0' _ e@(F _) f _ _ = clashE e f
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compare0' ctx e@(B i) f@(B j) _ _ = unless (i == j) $ clashE e f
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compare0' _ e@(B _) f _ _ = clashE e f
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compare0' ctx (e :@ s) (f :@ t) ne nf =
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local {mode := Equal} $ do
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compare0 ctx e f
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(_, arg, _) <- expectPi defs !(computeElimType ctx e (noOr1 ne))
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Term.compare0 ctx arg s t
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compare0' _ e@(_ :@ _) f _ _ = clashE e f
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compare0' ctx (CasePair epi e _ eret _ _ ebody)
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(CasePair fpi f _ fret _ _ fbody) ne nf =
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local {mode := Equal} $ do
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compare0 ctx e f
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ety <- computeElimType ctx e (noOr1 ne)
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compareType (ctx :< ety) eret.term fret.term
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(fst, snd) <- expectSig defs ety
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Term.compare0 (ctx :< fst :< snd.term) (substCasePairRet ety eret)
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ebody.term fbody.term
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unless (epi == fpi) $ throwError $ ClashQ epi fpi
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compare0' _ e@(CasePair {}) f _ _ = clashE e f
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compare0' ctx (s :# a) (t :# _) _ _ = Term.compare0 ctx a s t
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compare0' ctx (s :# a) f _ _ = Term.compare0 ctx a s (E f)
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compare0' ctx e (t :# b) _ _ = Term.compare0 ctx b (E e) t
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compare0' _ e@(_ :# _) f _ _ = clashE e f
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parameters {auto _ : (HasDefs' q _ m, HasErr q m, Eq q)} (ctx : TyContext q d n)
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parameters (mode : EqMode)
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namespace Term
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export covering
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compare : (ty, s, t : Term q d n) -> m ()
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compare ty s t = do
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defs <- ask
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runReaderT {m} (MkCmpContext {mode}) $
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for_ (splits ctx.dctx) $ \th =>
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compare0 defs (map (/// th) ctx.tctx)
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(ty /// th) (s /// th) (t /// th)
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export covering
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compareType : (s, t : Term q d n) -> m ()
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compareType s t = do
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defs <- ask
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runReaderT {m} (MkCmpContext {mode}) $
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for_ (splits ctx.dctx) $ \th =>
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compareType defs (map (/// th) ctx.tctx) (s /// th) (t /// th)
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namespace Elim
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||| you don't have to pass the type in but the arguments must still be
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||| of the same type!!
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export covering %inline
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compare : (e, f : Elim q d n) -> m ()
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compare e f = do
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defs <- ask
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runReaderT {m} (MkCmpContext {mode}) $
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for_ (splits ctx.dctx) $ \th =>
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compare0 defs (map (/// th) ctx.tctx) (e /// th) (f /// th)
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namespace Term
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export covering %inline
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equal, sub, super : (ty, s, t : Term q d n) -> m ()
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equal = compare Equal
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sub = compare Sub
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super = compare Super
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export covering %inline
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equalType, subtype, supertype : (s, t : Term q d n) -> m ()
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equalType = compareType Equal
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subtype = compareType Sub
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supertype = compareType Super
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namespace Elim
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export covering %inline
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equal, sub, super : (e, f : Elim q d n) -> m ()
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equal = compare Equal
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sub = compare Sub
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super = compare Super
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