comments etc

lib/Quox/Equal.idr

@@ -37,6 +37,11 @@ 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
@@ -67,6 +72,14 @@ 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 =
@@ -97,6 +110,10 @@ parameters {auto _ : HasErr q m}
 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
@@ -106,6 +123,8 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
         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
@@ -119,29 +138,46 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
       compare0' ctx (TYPE _) s t = compareType ctx s t
 
       compare0' ctx ty@(Pi {arg, res, _}) s t {n} = local {mode := Equal} $
-        let ctx' = ctx :< arg
+        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
-        in
-        case (s, t) of
-          (Lam _ b1, Lam _ b2) => compare0 ctx' res.term b1.term b2.term
-          (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
 
       compare0' ctx ty@(Sig {fst, snd, _}) s t = local {mode := Equal} $
-        -- no η (no fst/snd for π ≱ 0), so…
-        -- [todo] η for π ≥ 0 maybe
         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 ✨
-      compare0' _ (Eq {}) _ _ = pure ()
+        --
+        -- 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
@@ -150,6 +186,8 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
         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
     compareType : TContext q 0 n -> (s, t : Term q 0 n) -> m ()
     compareType ctx s t = do
@@ -167,22 +205,37 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
                    (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
@@ -194,8 +247,9 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
       -- has been inlined by whnfD
       Elim.compare0 ctx e f
 
-    ||| assumes the elim is already typechecked! only does the work necessary
+    ||| performs the minimum work required to recompute the type of an elim.
-    ||| to calculate the overall type
+    |||
+    ||| ⚠ **assumes the elim is already typechecked.** ⚠
     private covering
     computeElimType : TContext q 0 n -> (e : Elim q 0 n) ->
                       (0 ne : NotRedex defs e) ->
@@ -227,6 +281,11 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
     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 =
@@ -244,6 +303,8 @@ parameters (defs : Definitions' q _) {auto _ : (CanEqual q m, Eq q)}
                   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 =

lib/Quox/Syntax/Qty.idr

@@ -45,8 +45,7 @@ interface Eq q => IsQty q where
   compat : Dec2 Compat
 
   ||| true if it is ok for this quantity to appear for the
-  ||| subject of a typing judgement. this is about the
+  ||| subject of a typing judgement [@qtt, §2.3].
-  ||| subject reduction stuff in atkey
   IsSubj : Pred q
   isSubj : Dec1 IsSubj
   zeroIsSubj : IsSubj zero

lib/Quox/Typechecker.idr

@@ -69,6 +69,8 @@ parameters {auto _ : IsQty q} {auto _ : CanTC q m}
   -- substitutions. both of them seem like the same amount of work for the
   -- computer but pushing is less work for the me
 
+  ||| "Ψ | Γ ⊢ σ · s ⇐ A ⊳ Σ"
+  |||
   ||| `check ctx sg subj ty` checks that in the context `ctx`, the term
   ||| `subj` has the type `ty`, with quantity `sg`. if so, returns the
   ||| quantities of all bound variables that it used.
@@ -79,12 +81,16 @@ parameters {auto _ : IsQty q} {auto _ : CanTC q m}
     let Element subj nc = pushSubsts subj in
     check' ctx sg subj nc ty
 
-  ||| `check0 ctx subj ty` checks a term in a zero context.
+  ||| "Ψ | Γ ⊢₀ s ⇐ A"
+  |||
+  ||| `check0 ctx subj ty` checks a term (as `check`) in a zero context.
   export covering %inline
   check0 : TyContext q d n -> Term q d n -> Term q d n -> m ()
   check0 ctx tm ty = ignore $ check ctx szero tm ty
     -- the output will always be 𝟎 because the subject quantity is 0
 
+  ||| "Ψ | Γ ⊢ σ · e ⇒ A ⊳ Σ"
+  |||
   ||| `infer ctx sg subj` infers the type of `subj` in the context `ctx`,
   ||| and returns its type and the bound variables it used.
   export covering %inline
@@ -205,7 +211,6 @@ parameters {auto _ : IsQty q} {auto _ : CanTC q m}
     funres <- infer ctx sg fun
     (qty, argty, res) <- expectPi !ask funres.type
     -- if Ψ | Γ ⊢ σ ⨴ π · s ⇐ A ⊳ Σ₂
-    --   (σ ⨴ 0 = 0; σ ⨴ π = σ otherwise)
     argout <- check ctx (subjMult sg qty) arg argty
     -- then Ψ | Γ ⊢ σ · f s ⇒ B[s] ⊳ Σ₁ + Σ₂
     pure $ InfRes {