quox/tests/Tests/Typechecker.idr

module Tests.Typechecker

import Quox.Syntax
import Quox.Typechecker as Lib
import public TypingImpls
import TAP
import Quox.EffExtra


data Error'
  = TCError Typing.Error
  | WrongInfer (Term d n) (Term d n)
  | WrongQOut (QOutput n) (QOutput n)

export
ToInfo Error' where
  toInfo (TCError e) = toInfo e
  toInfo (WrongInfer good bad) =
    [("type", "WrongInfer"),
     ("wanted", prettyStr True good),
     ("got", prettyStr True bad)]
  toInfo (WrongQOut good bad) =
    [("type", "WrongQOut"),
     ("wanted", prettyStr True good),
     ("wanted", prettyStr True bad)]

0 M : Type -> Type
M = Eff [Except Error', DefsReader]

inj : TC a -> M a
inj = rethrow . mapFst TCError <=< lift . runExcept


reflTy : Term d n
reflTy =
  Pi_ Zero "A" (TYPE 0) $
  Pi_ One  "x" (BVT 0)  $
  Eq0 (BVT 1) (BVT 0) (BVT 0)

reflDef : Term d n
reflDef = [< "A","x"] :\\ [< "i"] :\\% BVT 0


fstTy : Term d n
fstTy =
  (Pi_ Zero "A" (TYPE 1) $
   Pi_ Zero "B" (Arr Any (BVT 0) (TYPE 1)) $
   Arr Any (Sig_ "x" (BVT 1) $ E $ BV 1 :@ BVT 0) (BVT 1))

fstDef : Term d n
fstDef =
  ([< "A","B","p"] :\\
    E (CasePair Any (BV 0) (SN $ BVT 2) (SY [< "x","y"] $ BVT 1)))

sndTy : Term d n
sndTy =
  (Pi_ Zero "A" (TYPE 1) $
   Pi_ Zero "B" (Arr Any (BVT 0) (TYPE 1)) $
   Pi_ Any  "p" (Sig_ "x" (BVT 1) $ E $ BV 1 :@ BVT 0) $
   E (BV 1 :@ E (F "fst" :@@ [BVT 2, BVT 1, BVT 0])))

sndDef : Term d n
sndDef =
  ([< "A","B","p"] :\\
    E (CasePair Any (BV 0)
        (SY [< "p"] $ E $ BV 2 :@ E (F "fst" :@@ [BVT 3, BVT 2, BVT 0]))
        (SY [< "x","y"] $ BVT 0)))


defGlobals : Definitions
defGlobals = fromList
  [("A",  mkPostulate gzero $ TYPE 0),
   ("B",  mkPostulate gzero $ TYPE 0),
   ("C",  mkPostulate gzero $ TYPE 1),
   ("D",  mkPostulate gzero $ TYPE 1),
   ("P",  mkPostulate gzero $ Arr Any (FT "A") (TYPE 0)),
   ("a",  mkPostulate gany  $ FT "A"),
   ("a'", mkPostulate gany  $ FT "A"),
   ("b",  mkPostulate gany  $ FT "B"),
   ("f",  mkPostulate gany  $ Arr One (FT "A") (FT "A")),
   ("g",  mkPostulate gany  $ Arr One (FT "A") (FT "B")),
   ("f2", mkPostulate gany  $ Arr One (FT "A") $ Arr One (FT "A") (FT "B")),
   ("p",  mkPostulate gany  $ Pi_ One "x" (FT "A") $ E $ F "P" :@ BVT 0),
   ("q",  mkPostulate gany  $ Pi_ One "x" (FT "A") $ E $ F "P" :@ BVT 0),
   ("refl", mkDef gany reflTy reflDef),
   ("fst",  mkDef gany fstTy fstDef),
   ("snd",  mkDef gany sndTy sndDef)]

parameters (label : String) (act : Lazy (M ()))
           {default defGlobals globals : Definitions}
  testTC : Test
  testTC = test label {e = Error', a = ()} $
    extract $ runExcept $ runReader globals act

  testTCFail : Test
  testTCFail = testThrows label (const True) $
    (extract $ runExcept $ runReader globals act) $> "()"


anys : {n : Nat} -> QContext n
anys {n = 0}   = [<]
anys {n = S n} = anys :< Any

ctx, ctx01 : {n : Nat} -> Context (\n => (BaseName, Term 0 n)) n ->
             TyContext 0 n
ctx tel = let (ns, ts) = unzip tel in
          MkTyContext new [<] ts ns anys
ctx01 tel = let (ns, ts) = unzip tel in
            MkTyContext ZeroIsOne [<] ts ns anys

empty01 : TyContext 0 0
empty01 = eqDim (K Zero) (K One) empty

inferredTypeEq : TyContext d n -> (exp, got : Term d n) -> M ()
inferredTypeEq ctx exp got =
  wrapErr (const $ WrongInfer exp got) $ inj $ equalType ctx exp got

qoutEq : (exp, got : QOutput n) -> M ()
qoutEq qout res = unless (qout == res) $ throw $ WrongQOut qout res

inferAs : TyContext d n -> (sg : SQty) -> Elim d n -> Term d n -> M ()
inferAs ctx@(MkTyContext {dctx, _}) sg e ty = do
  case !(inj $ infer ctx sg e) of
    Just res => inferredTypeEq ctx ty res.type
    Nothing  => pure ()

inferAsQ : TyContext d n -> (sg : SQty) ->
           Elim d n -> Term d n -> QOutput n -> M ()
inferAsQ ctx@(MkTyContext {dctx, _}) sg e ty qout = do
  case !(inj $ infer ctx sg e) of
    Just res => do
      inferredTypeEq ctx ty res.type
      qoutEq qout res.qout
    Nothing => pure ()

infer_ : TyContext d n -> (sg : SQty) -> Elim d n -> M ()
infer_ ctx sg e = ignore $ inj $ infer ctx sg e

checkQ : TyContext d n -> SQty ->
         Term d n -> Term d n -> QOutput n -> M ()
checkQ ctx@(MkTyContext {dctx, _}) sg s ty qout = do
  case !(inj $ check ctx sg s ty) of
    Just res => qoutEq qout res
    Nothing  => pure ()

check_ : TyContext d n -> SQty -> Term d n -> Term d n -> M ()
check_ ctx sg s ty = ignore $ inj $ check ctx sg s ty

checkType_ : TyContext d n -> Term d n -> Maybe Universe -> M ()
checkType_ ctx s u = inj $ checkType ctx s u


export
tests : Test
tests = "typechecker" :- [
  "universes" :- [
    testTC "0 · ★₀ ⇐ ★₁  # by checkType" $
      checkType_ empty (TYPE 0) (Just 1),
    testTC "0 · ★₀ ⇐ ★₁  # by check" $
      check_ empty szero (TYPE 0) (TYPE 1),
    testTC "0 · ★₀ ⇐ ★₂" $
      checkType_ empty (TYPE 0) (Just 2),
    testTC "0 · ★₀ ⇐ ★_" $
      checkType_ empty (TYPE 0) Nothing,
    testTCFail "0 · ★₁ ⇍ ★₀" $
      checkType_ empty (TYPE 1) (Just 0),
    testTCFail "0 · ★₀ ⇍ ★₀" $
      checkType_ empty (TYPE 0) (Just 0),
    testTC "0=1 ⊢ 0 · ★₁ ⇐ ★₀" $
      checkType_ empty01 (TYPE 1) (Just 0),
    testTCFail "1 · ★₀ ⇍ ★₁  # by check" $
      check_ empty sone (TYPE 0) (TYPE 1)
  ],

  "function types" :- [
    note "A, B : ★₀;  C, D : ★₁;  P : A ⇾ ★₀",
    testTC "0 · A ⊸ B ⇐ ★₀" $
      check_ empty szero (Arr One (FT "A") (FT "B")) (TYPE 0),
    note "subtyping",
    testTC "0 · A ⊸ B ⇐ ★₁" $
      check_ empty szero (Arr One (FT "A") (FT "B")) (TYPE 1),
    testTC "0 · C ⊸ D ⇐ ★₁" $
      check_ empty szero (Arr One (FT "C") (FT "D")) (TYPE 1),
    testTCFail "0 · C ⊸ D ⇍ ★₀" $
      check_ empty szero (Arr One (FT "C") (FT "D")) (TYPE 0),
    testTC "0 · (1·x : A) → P x ⇐ ★₀" $
      check_ empty szero
        (Pi_ One "x" (FT "A") $ E $ F "P" :@ BVT 0)
        (TYPE 0),
    testTCFail "0 · A ⊸ P ⇍ ★₀" $
      check_ empty szero (Arr One (FT "A") $ FT "P") (TYPE 0),
    testTC "0=1 ⊢ 0 · A ⊸ P ⇐ ★₀" $
      check_ empty01 szero (Arr One (FT "A") $ FT "P") (TYPE 0)
  ],

  "pair types" :- [
    note #""A × B" for "(_ : A) × B""#,
    testTC "0 · A × A ⇐ ★₀" $
      check_ empty szero (FT "A" `And` FT "A") (TYPE 0),
    testTCFail "0 · A × P ⇍ ★₀" $
      check_ empty szero (FT "A" `And` FT "P") (TYPE 0),
    testTC "0 · (x : A) × P x ⇐ ★₀" $
      check_ empty szero
        (Sig_ "x" (FT "A") $ E $ F "P" :@ BVT 0) (TYPE 0),
    testTC "0 · (x : A) × P x ⇐ ★₁" $
      check_ empty szero
        (Sig_ "x" (FT "A") $ E $ F "P" :@ BVT 0) (TYPE 1),
    testTC "0 · (A : ★₀) × A ⇐ ★₁" $
      check_ empty szero (Sig_ "A" (TYPE 0) $ BVT 0) (TYPE 1),
    testTCFail "0 · (A : ★₀) × A ⇍ ★₀" $
      check_ empty szero (Sig_ "A" (TYPE 0) $ BVT 0) (TYPE 0),
    testTCFail "1 · A × A ⇍ ★₀" $
      check_ empty sone (FT "A" `And` FT "A") (TYPE 0)
  ],

  "enum types" :- [
    testTC "0 · {} ⇐ ★₀" $ check_ empty szero (enum []) (TYPE 0),
    testTC "0 · {} ⇐ ★₃" $ check_ empty szero (enum []) (TYPE 3),
    testTC "0 · {a,b,c} ⇐ ★₀" $
      check_ empty szero (enum ["a", "b", "c"]) (TYPE 0),
    testTC "0 · {a,b,c} ⇐ ★₃" $
      check_ empty szero (enum ["a", "b", "c"]) (TYPE 3),
    testTCFail "1 · {} ⇍ ★₀" $ check_ empty sone (enum []) (TYPE 0),
    testTC "0=1 ⊢ 1 · {} ⇐ ★₀" $ check_ empty01 sone (enum []) (TYPE 0)
  ],

  "free vars" :- [
    note "A : ★₀",
    testTC "0 · A ⇒ ★₀" $
      inferAs empty szero (F "A") (TYPE 0),
    testTC "0 · [A] ⇐ ★₀" $
      check_ empty szero (FT "A") (TYPE 0),
    note "subtyping",
    testTC "0 · [A] ⇐ ★₁" $
      check_ empty szero (FT "A") (TYPE 1),
    note "(fail) runtime-relevant type",
    testTCFail "1 · A ⇏ ★₀" $
      infer_ empty sone (F "A"),
    note "refl : (0·A : ★₀) → (1·x : A) → (x ≡ x : A) ≔ (λ A x ⇒ δ _ ⇒ x)",
    testTC "1 · refl ⇒ ⋯" $ inferAs empty sone (F "refl") reflTy,
    testTC "1 · [refl] ⇐ ⋯" $ check_ empty sone (FT "refl") reflTy
  ],

  "bound vars" :- [
    testTC "x : A ⊢ 1 · x ⇒ A ⊳ 1·x" $
      inferAsQ {n = 1} (ctx [< ("x", FT "A")]) sone
        (BV 0) (FT "A") [< One],
    testTC "x : A ⊢ 1 · [x] ⇐ A ⊳ 1·x" $
      checkQ {n = 1} (ctx [< ("x", FT "A")]) sone (BVT 0) (FT "A") [< One],
    note "f2 : A ⊸ A ⊸ B",
    testTC "x : A ⊢ 1 · f2 [x] [x] ⇒ B ⊳ ω·x" $
      inferAsQ {n = 1} (ctx [< ("x", FT "A")]) sone
        (F "f2" :@@ [BVT 0, BVT 0]) (FT "B") [< Any]
  ],

  "lambda" :- [
    note "linear & unrestricted identity",
    testTC "1 · (λ x ⇒ x) ⇐ A ⊸ A" $
      check_ empty sone ([< "x"] :\\ BVT 0) (Arr One (FT "A") (FT "A")),
    testTC "1 · (λ x ⇒ x) ⇐ A → A" $
      check_ empty sone ([< "x"] :\\ BVT 0) (Arr Any (FT "A") (FT "A")),
    note "(fail) zero binding used relevantly",
    testTCFail "1 · (λ x ⇒ x) ⇍ A ⇾ A" $
      check_ empty sone ([< "x"] :\\ BVT 0) (Arr Zero (FT "A") (FT "A")),
    note "(but ok in overall erased context)",
    testTC "0 · (λ x ⇒ x) ⇐ A ⇾ A" $
      check_ empty szero ([< "x"] :\\ BVT 0) (Arr Zero (FT "A") (FT "A")),
    testTC "1 · (λ A x ⇒ refl A x) ⇐ ⋯  # (type of refl)" $
      check_ empty sone
        ([< "A", "x"] :\\ E (F "refl" :@@ [BVT 1, BVT 0]))
        reflTy,
    testTC "1 · (λ A x ⇒ δ i ⇒ x) ⇐ ⋯  # (def. and type of refl)" $
      check_ empty sone reflDef reflTy
  ],

  "pairs" :- [
    testTC "1 · (a, a) ⇐ A × A" $
      check_ empty sone (Pair (FT "a") (FT "a")) (FT "A" `And` FT "A"),
    testTC "x : A ⊢ 1 · (x, x) ⇐ A × A ⊳ ω·x" $
      checkQ (ctx [< ("x", FT "A")]) sone
        (Pair (BVT 0) (BVT 0)) (FT "A" `And` FT "A") [< Any],
    testTC "1 · (a, δ i ⇒ a) ⇐ (x : A) × (x ≡ a)" $
      check_ empty sone
        (Pair (FT "a") ([< "i"] :\\% FT "a"))
        (Sig_ "x" (FT "A") $ Eq0 (FT "A") (BVT 0) (FT "a"))
  ],

  "unpairing" :- [
    testTC "x : A × A ⊢ 1 · (case1 x return B of (l,r) ⇒ f2 l r) ⇒ B ⊳ 1·x" $
      inferAsQ (ctx [< ("x", FT "A" `And` FT "A")]) sone
        (CasePair One (BV 0) (SN $ FT "B")
          (SY [< "l", "r"] $ E $ F "f2" :@@ [BVT 1, BVT 0]))
        (FT "B") [< One],
    testTC "x : A × A ⊢ 1 · (caseω x return B of (l,r) ⇒ f2 l r) ⇒ B ⊳ ω·x" $
      inferAsQ (ctx [< ("x", FT "A" `And` FT "A")]) sone
        (CasePair Any (BV 0) (SN $ FT "B")
          (SY [< "l", "r"] $ E $ F "f2" :@@ [BVT 1, BVT 0]))
        (FT "B") [< Any],
    testTC "x : A × A ⊢ 0 · (caseω x return B of (l,r) ⇒ f2 l r) ⇒ B ⊳ 0·x" $
      inferAsQ (ctx [< ("x", FT "A" `And` FT "A")]) szero
        (CasePair Any (BV 0) (SN $ FT "B")
          (SY [< "l", "r"] $ E $ F "f2" :@@ [BVT 1, BVT 0]))
        (FT "B") [< Zero],
    testTCFail "x : A × A ⊢ 1 · (case0 x return B of (l,r) ⇒ f2 l r) ⇏" $
      infer_ (ctx [< ("x", FT "A" `And` FT "A")]) sone
        (CasePair Zero (BV 0) (SN $ FT "B")
          (SY [< "l", "r"] $ E $ F "f2" :@@ [BVT 1, BVT 0])),
    testTC "x : A × B ⊢ 1 · (caseω x return A of (l,r) ⇒ l) ⇒ A ⊳ ω·x" $
      inferAsQ (ctx [< ("x", FT "A" `And` FT "B")]) sone
        (CasePair Any (BV 0) (SN $ FT "A")
          (SY [< "l", "r"] $ BVT 1))
        (FT "A") [< Any],
    testTC "x : A × B ⊢ 0 · (case1 x return A of (l,r) ⇒ l) ⇒ A ⊳ 0·x" $
      inferAsQ (ctx [< ("x", FT "A" `And` FT "B")]) szero
        (CasePair One (BV 0) (SN $ FT "A")
          (SY [< "l", "r"] $ BVT 1))
        (FT "A") [< Zero],
    testTCFail "x : A × B ⊢ 1 · (case1 x return A of (l,r) ⇒ l) ⇏" $
      infer_ (ctx [< ("x", FT "A" `And` FT "B")]) sone
        (CasePair One (BV 0) (SN $ FT "A")
          (SY [< "l", "r"] $ BVT 1)),
    note "fst : (0·A : ★₁) → (0·B : A ↠ ★₁) → ((x : A) × B x) ↠ A",
    note "    ≔ (λ A B p ⇒ caseω p return A of (x, y) ⇒ x)",
    testTC "0 · ‹type of fst› ⇐ ★₂" $
      check_ empty szero fstTy (TYPE 2),
    testTC "1 · ‹def of fst› ⇐ ‹type of fst›" $
      check_ empty sone fstDef fstTy,
    note "snd : (0·A : ★₁) → (0·B : A ↠ ★₁) → (ω·p : (x : A) × B x) → B (fst A B p)",
    note "    ≔ (λ A B p ⇒ caseω p return p ⇒ B (fst A B p) of (x, y) ⇒ y)",
    testTC "0 · ‹type of snd› ⇐ ★₂" $
      check_ empty szero sndTy (TYPE 2),
    testTC "1 · ‹def of snd› ⇐ ‹type of snd›" $
      check_ empty sone sndDef sndTy,
    testTC "0 · snd ★₀ (λ x ⇒ x) ⇒ (ω·p : (A : ★₀) × A) → fst ★₀ (λ x ⇒ x) p" $
      inferAs empty szero
        (F "snd" :@@ [TYPE 0, [< "x"] :\\ BVT 0])
        (Pi_ Any "A" (Sig_ "A" (TYPE 0) $ BVT 0) $
                (E $ F "fst" :@@ [TYPE 0, [< "x"] :\\ BVT 0, BVT 0]))
  ],

  "enums" :- [
    testTC "1 · 'a ⇐ {a}" $
      check_ empty sone (Tag "a") (enum ["a"]),
    testTC "1 · 'a ⇐ {a, b, c}" $
      check_ empty sone (Tag "a") (enum ["a", "b", "c"]),
    testTCFail "1 · 'a ⇍ {b, c}" $
      check_ empty sone (Tag "a") (enum ["b", "c"]),
    testTC "0=1 ⊢ 1 · 'a ⇐ {b, c}" $
      check_ empty01 sone (Tag "a") (enum ["b", "c"])
  ],

  "enum matching" :- [
    testTC "ω.x : {tt} ⊢ 1 · case1 x return {tt} of { 'tt ⇒ 'tt } ⇒ {tt}" $
      inferAs (ctx [< ("x", enum ["tt"])]) sone
        (CaseEnum One (BV 0) (SN (enum ["tt"])) $
          singleton "tt" (Tag "tt"))
        (enum ["tt"]),
    testTCFail "ω.x : {tt} ⊢ 1 · case1 x return {tt} of { 'ff ⇒ 'tt } ⇏" $
      infer_ (ctx [< ("x", enum ["tt"])]) sone
        (CaseEnum One (BV 0) (SN (enum ["tt"])) $
          singleton "ff" (Tag "tt"))
  ],

  "equality types" :- [
    testTC "0 · ℕ ≡ ℕ : ★₀ ⇐ Type" $
      checkType_ empty (Eq0 (TYPE 0) Nat Nat) Nothing,
    testTC "0 · ℕ ≡ ℕ : ★₀ ⇐ ★₁" $
      check_ empty szero (Eq0 (TYPE 0) Nat Nat) (TYPE 1),
    testTCFail "1 · ℕ ≡ ℕ : ★₀ ⇍ ★₁" $
      check_ empty sone (Eq0 (TYPE 0) Nat Nat) (TYPE 1),
    testTC "0 · ℕ ≡ ℕ : ★₀ ⇐ ★₂" $
      check_ empty szero (Eq0 (TYPE 0) Nat Nat) (TYPE 2),
    testTC "0 · ℕ ≡ ℕ : ★₁ ⇐ ★₂" $
      check_ empty szero (Eq0 (TYPE 1) Nat Nat) (TYPE 2),
    testTCFail "0 · ℕ ≡ ℕ : ★₁ ⇍ ★₁" $
      check_ empty szero (Eq0 (TYPE 1) Nat Nat) (TYPE 1),
    testTCFail "0 ≡ 'beep : {beep} ⇍ Type" $
      checkType_ empty (Eq0 (enum ["beep"]) Zero (Tag "beep")) Nothing,
    testTC "ab : A ≡ B : ★₀, x : A, y : B ⊢ 0 · Eq [i ⇒ ab i] x y ⇐ ★₀" $
      check_ (ctx [< ("ab", Eq0 (TYPE 0) (FT "A") (FT "B")),
                     ("x", FT "A"), ("y", FT "B")]) szero
        (Eq (SY [< "i"] $ E $ BV 2 :% BV 0) (BVT 1) (BVT 0))
        (TYPE 0),
    testTCFail "ab : A ≡ B : ★₀, x : A, y : B ⊢ Eq [i ⇒ ab i] y x ⇍ Type" $
      checkType_ (ctx [< ("ab", Eq0 (TYPE 0) (FT "A") (FT "B")),
                         ("x", FT "A"), ("y", FT "B")])
        (Eq (SY [< "i"] $ E $ BV 2 :% BV 0) (BVT 0) (BVT 1))
        Nothing
  ],

  "equalities" :- [
    testTC "1 · (δ i ⇒ a) ⇐ a ≡ a" $
      check_ empty sone (DLam $ SN $ FT "a")
        (Eq0 (FT "A") (FT "a") (FT "a")),
    testTC "0 · (λ p q ⇒ δ i ⇒ p) ⇐ (ω·p q : a ≡ a') → p ≡ q" $
      check_ empty szero
        ([< "p","q"] :\\ [< "i"] :\\% BVT 1)
        (Pi_ Any "p" (Eq0 (FT "A") (FT "a") (FT "a")) $
         Pi_ Any "q" (Eq0 (FT "A") (FT "a") (FT "a")) $
         Eq0 (Eq0 (FT "A") (FT "a") (FT "a")) (BVT 1) (BVT 0)),
    testTC "0 · (λ p q ⇒ δ i ⇒ q) ⇐ (ω·p q : a ≡ a') → p ≡ q" $
      check_ empty szero
        ([< "p","q"] :\\ [< "i"] :\\% BVT 0)
        (Pi_ Any "p" (Eq0 (FT "A") (FT "a") (FT "a")) $
         Pi_ Any "q" (Eq0 (FT "A") (FT "a") (FT "a")) $
         Eq0 (Eq0 (FT "A") (FT "a") (FT "a")) (BVT 1) (BVT 0))
  ],

  "natural numbers" :- [
    testTC "0 · ℕ ⇐ ★₀" $ check_ empty szero Nat (TYPE 0),
    testTC "0 · ℕ ⇐ ★₇" $ check_ empty szero Nat (TYPE 7),
    testTCFail "1 · ℕ ⇍ ★₀" $ check_ empty sone Nat (TYPE 0),
    testTC "1 · zero ⇐ ℕ" $ check_ empty sone Zero Nat,
    testTCFail "1 · zero ⇍ ℕ×ℕ" $ check_ empty sone Zero (Nat `And` Nat),
    testTC "ω·n : ℕ ⊢ 1 · succ n ⇐ ℕ" $
      check_ (ctx [< ("n", Nat)]) sone (Succ (BVT 0)) Nat,
    testTC "1 · λ n ⇒ succ n ⇐ 1.ℕ → ℕ" $
      check_ empty sone ([< "n"] :\\ Succ (BVT 0)) (Arr One Nat Nat),
    todo "nat elim"
  ],

  "natural elim" :- [
    note "1 · λ n ⇒ case1 n return ℕ of { zero ⇒ 0; succ n ⇒ n }",
    note "    ⇐ 1.ℕ → ℕ",
    testTC "pred" $
      check_ empty sone
        ([< "n"] :\\ E (CaseNat One Zero (BV 0) (SN Nat)
            Zero (SY [< "n", Unused] $ BVT 1)))
        (Arr One Nat Nat),
    note "1 · λ m n ⇒ case1 m return ℕ of { zero ⇒ n; succ _, 1.p ⇒ succ p }",
    note "    ⇐ 1.ℕ → 1.ℕ → 1.ℕ",
    testTC "plus" $
      check_ empty sone
        ([< "m", "n"] :\\ E (CaseNat One One (BV 1) (SN Nat)
          (BVT 0) (SY [< Unused, "p"] $ Succ $ BVT 0)))
        (Arr One Nat $ Arr One Nat Nat)
  ],

  "box types" :- [
    testTC "0 · [0.ℕ] ⇐ ★₀" $
      check_ empty szero (BOX Zero Nat) (TYPE 0),
    testTC "0 · [0.★₀] ⇐ ★₁" $
      check_ empty szero (BOX Zero (TYPE 0)) (TYPE 1),
    testTCFail "0 · [0.★₀] ⇍ ★₀" $
      check_ empty szero (BOX Zero (TYPE 0)) (TYPE 0)
  ],

  todo "box values",
  todo "box elim",

  "misc" :- [
    note "0·A : Type, 0·P : A → Type, ω·p : (1·x : A) → P x",
    note "⊢",
    note "1 · λ x y xy ⇒ δ i ⇒ p (xy i)",
    note "  ⇐ (0·x y : A) → (1·xy : x ≡ y) → Eq [i ⇒ P (xy i)] (p x) (p y)",
    testTC "cong" $
      check_ empty sone
        ([< "x", "y", "xy"] :\\ [< "i"] :\\% E (F "p" :@ E (BV 0 :% BV 0)))
        (Pi_ Zero "x"  (FT "A") $
         Pi_ Zero "y"  (FT "A") $
         Pi_ One  "xy" (Eq0 (FT "A") (BVT 1) (BVT 0)) $
         Eq_ "i" (E $ F "P" :@ E (BV 0 :% BV 0))
             (E $ F "p" :@ BVT 2) (E $ F "p" :@ BVT 1)),
    note "0·A : Type, 0·P : ω·A → Type,",
    note "ω·p q : (1·x : A) → P x",
    note "⊢",
    note "1 · λ eq ⇒ δ i ⇒ λ x ⇒ eq x i",
    note "  ⇐ (1·eq : (1·x : A) → p x ≡ q x) → p ≡ q",
    testTC "funext" $
      check_ empty sone
        ([< "eq"] :\\ [< "i"] :\\% [< "x"] :\\ E (BV 1 :@ BVT 0 :% BV 0))
        (Pi_ One "eq"
          (Pi_ One "x" (FT "A")
            (Eq0 (E $ F "P" :@ BVT 0)
                 (E $ F "p" :@ BVT 0) (E $ F "q" :@ BVT 0)))
          (Eq0 (Pi_ Any "x" (FT "A") $ E $ F "P" :@ BVT 0) (FT "p") (FT "q"))),
    todo "absurd (when coerce is in)"
    -- absurd : (`true ≡ `false : {true, false}) ⇾ (0·A : ★₀) → A ≔
    --   λ e ⇒
    --     case coerce [i ⇒ case e @i return ★₀ of {`true ⇒ {tt}; `false ⇒ {}}]
    --            @0 @1 `tt
    --     return A
    --     of { }
  ]
]