403 lines
15 KiB
Idris
403 lines
15 KiB
Idris
module Tests.Equal
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import Quox.Equal
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import Quox.Syntax.Qty.Three
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import public TypingImpls
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import TAP
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0 M : Type -> Type
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M = ReaderT (Definitions Three) (Either (Error Three))
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defGlobals : Definitions Three
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defGlobals = fromList
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[("A", mkAbstract Zero $ TYPE 0),
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("B", mkAbstract Zero $ TYPE 0),
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("a", mkAbstract Any $ FT "A"),
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("a'", mkAbstract Any $ FT "A"),
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("b", mkAbstract Any $ FT "B"),
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("f", mkAbstract Any $ Arr One (FT "A") (FT "A")),
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("id", mkDef Any (Arr One (FT "A") (FT "A")) (["x"] :\\ BVT 0)),
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("eq-ab", mkAbstract Zero $ Eq0 (TYPE 0) (FT "A") (FT "B"))]
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parameters (label : String) (act : Lazy (M ()))
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{default defGlobals globals : Definitions Three}
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testEq : Test
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testEq = test label $ runReaderT globals act
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testNeq : Test
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testNeq = testThrows label (const True) $ runReaderT globals act
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parameters (0 d : Nat) (ctx : TyContext Three d n)
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subTD, equalTD : Term Three d n -> Term Three d n -> Term Three d n -> M ()
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subTD ty s t = Term.sub ctx ty s t
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equalTD ty s t = Term.equal ctx ty s t
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subED, equalED : Elim Three d n -> Elim Three d n -> M ()
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subED e f = Elim.sub ctx e f
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equalED e f = Elim.equal ctx e f
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parameters (ctx : TyContext Three 0 n)
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subT, equalT : Term Three 0 n -> Term Three 0 n -> Term Three 0 n -> M ()
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subT = subTD 0 ctx
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equalT = equalTD 0 ctx
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subE, equalE : Elim Three 0 n -> Elim Three 0 n -> M ()
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subE = subED 0 ctx
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equalE = equalED 0 ctx
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export
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tests : Test
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tests = "equality & subtyping" :- [
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note #""s{t,…}" for term substs; "s‹p,…›" for dim substs"#,
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note #""0=1 ⊢ 𝒥" means that 𝒥 holds in an inconsistent dim context"#,
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"universes" :- [
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testEq "★₀ = ★₀" $
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equalT empty (TYPE 1) (TYPE 0) (TYPE 0),
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testNeq "★₀ ≠ ★₁" $
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equalT empty (TYPE 2) (TYPE 0) (TYPE 1),
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testNeq "★₁ ≠ ★₀" $
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equalT empty (TYPE 2) (TYPE 1) (TYPE 0),
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testEq "★₀ <: ★₀" $
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subT empty (TYPE 1) (TYPE 0) (TYPE 0),
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testEq "★₀ <: ★₁" $
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subT empty (TYPE 2) (TYPE 0) (TYPE 1),
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testNeq "★₁ ≮: ★₀" $
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subT empty (TYPE 2) (TYPE 1) (TYPE 0)
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],
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"pi" :- [
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note #""𝐴 ⊸ 𝐵" for (1·𝐴) → 𝐵"#,
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note #""𝐴 ⇾ 𝐵" for (0·𝐴) → 𝐵"#,
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testEq "★₀ ⇾ ★₀ = ★₀ ⇾ ★₀" $
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let tm = Arr Zero (TYPE 0) (TYPE 0) in
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equalT empty (TYPE 1) tm tm,
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testEq "★₀ ⇾ ★₀ <: ★₀ ⇾ ★₀" $
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let tm = Arr Zero (TYPE 0) (TYPE 0) in
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subT empty (TYPE 1) tm tm,
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testNeq "★₁ ⊸ ★₀ ≠ ★₀ ⇾ ★₀" $
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let tm1 = Arr Zero (TYPE 1) (TYPE 0)
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tm2 = Arr Zero (TYPE 0) (TYPE 0) in
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equalT empty (TYPE 2) tm1 tm2,
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testEq "★₁ ⊸ ★₀ <: ★₀ ⊸ ★₀" $
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let tm1 = Arr One (TYPE 1) (TYPE 0)
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tm2 = Arr One (TYPE 0) (TYPE 0) in
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subT empty (TYPE 2) tm1 tm2,
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testEq "★₀ ⊸ ★₀ <: ★₀ ⊸ ★₁" $
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let tm1 = Arr One (TYPE 0) (TYPE 0)
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tm2 = Arr One (TYPE 0) (TYPE 1) in
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subT empty (TYPE 2) tm1 tm2,
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testEq "★₀ ⊸ ★₀ <: ★₀ ⊸ ★₁" $
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let tm1 = Arr One (TYPE 0) (TYPE 0)
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tm2 = Arr One (TYPE 0) (TYPE 1) in
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subT empty (TYPE 2) tm1 tm2,
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testEq "A ⊸ B = A ⊸ B" $
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let tm = Arr One (FT "A") (FT "B") in
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equalT empty (TYPE 0) tm tm,
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testEq "A ⊸ B <: A ⊸ B" $
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let tm = Arr One (FT "A") (FT "B") in
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subT empty (TYPE 0) tm tm,
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note "incompatible quantities",
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testNeq "★₀ ⊸ ★₀ ≠ ★₀ ⇾ ★₁" $
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let tm1 = Arr Zero (TYPE 0) (TYPE 0)
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tm2 = Arr Zero (TYPE 0) (TYPE 1) in
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equalT empty (TYPE 2) tm1 tm2,
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testNeq "A ⇾ B ≠ A ⊸ B" $
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let tm1 = Arr Zero (FT "A") (FT "B")
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tm2 = Arr One (FT "A") (FT "B") in
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equalT empty (TYPE 0) tm1 tm2,
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testNeq "A ⇾ B ≮: A ⊸ B" $
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let tm1 = Arr Zero (FT "A") (FT "B")
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tm2 = Arr One (FT "A") (FT "B") in
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subT empty (TYPE 0) tm1 tm2,
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testEq "0=1 ⊢ A ⇾ B = A ⊸ B" $
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let tm1 = Arr Zero (FT "A") (FT "B")
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tm2 = Arr One (FT "A") (FT "B") in
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equalT (MkTyContext ZeroIsOne [<]) (TYPE 0) tm1 tm2,
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todo "dependent function types",
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note "[todo] should π ≤ ρ ⊢ (ρ·A) → B <: (π·A) → B?"
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],
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"lambda" :- [
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testEq "λ x. [x] = λ x. [x]" $
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equalT empty (Arr One (FT "A") (FT "A"))
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(["x"] :\\ BVT 0)
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(["x"] :\\ BVT 0),
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testEq "λ x. [x] <: λ x. [x]" $
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subT empty (Arr One (FT "A") (FT "A"))
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(["x"] :\\ BVT 0)
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(["x"] :\\ BVT 0),
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testEq "λ x. [x] = λ y. [y]" $
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equalT empty (Arr One (FT "A") (FT "A"))
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(["x"] :\\ BVT 0)
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(["y"] :\\ BVT 0),
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testEq "λ x. [x] <: λ y. [y]" $
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equalT empty (Arr One (FT "A") (FT "A"))
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(["x"] :\\ BVT 0)
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(["y"] :\\ BVT 0),
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testNeq "λ x y. [x] ≠ λ x y. [y]" $
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equalT empty (Arr One (FT "A") $ Arr One (FT "A") (FT "A"))
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(["x", "y"] :\\ BVT 1)
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(["x", "y"] :\\ BVT 0),
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testEq "λ x. [a] = λ x. [a] (Y vs N)" $
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equalT empty (Arr Zero (FT "B") (FT "A"))
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(Lam $ SY ["x"] $ FT "a")
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(Lam $ SN $ FT "a"),
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testEq "λ x. [f [x]] = [f] (η)" $
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equalT empty (Arr One (FT "A") (FT "A"))
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(["x"] :\\ E (F "f" :@ BVT 0))
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(FT "f")
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],
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"eq type" :- [
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testEq "(★₀ ≡ ★₀ : ★₁) = (★₀ ≡ ★₀ : ★₁)" $
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let tm = Eq0 (TYPE 1) (TYPE 0) (TYPE 0) in
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equalT empty (TYPE 2) tm tm,
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testEq "A ≔ ★₁ ⊢ (★₀ ≡ ★₀ : ★₁) = (★₀ ≡ ★₀ : A)"
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{globals = fromList [("A", mkDef zero (TYPE 2) (TYPE 1))]} $
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equalT empty (TYPE 2)
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(Eq0 (TYPE 1) (TYPE 0) (TYPE 0))
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(Eq0 (FT "A") (TYPE 0) (TYPE 0)),
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todo "dependent equality types"
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],
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"equalities and uip" :-
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let refl : Term q d n -> Term q d n -> Elim q d n
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refl a x = (DLam $ S ["_"] $ N x) :# (Eq0 a x x)
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in
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[
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note #""refl [A] x" is an abbreviation for "(δ i. x) ∷ (x ≡ x : A)""#,
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note "binds before ∥ are globals, after it are BVs",
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testEq "refl [A] a = refl [A] a" $
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equalE empty (refl (FT "A") (FT "a")) (refl (FT "A") (FT "a")),
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testEq "p : (a ≡ a' : A), q : (a ≡ a' : A) ∥ ⊢ p = q (free)"
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{globals =
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let def = mkAbstract Zero $ Eq0 (FT "A") (FT "a") (FT "a'") in
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defGlobals `mergeLeft` fromList [("p", def), ("q", def)]} $
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equalE empty (F "p") (F "q"),
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testEq "∥ x : (a ≡ a' : A), y : (a ≡ a' : A) ⊢ x = y (bound)" $
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let ty : forall n. Term Three 0 n := Eq0 (FT "A") (FT "a") (FT "a'") in
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equalE (MkTyContext new [< ty, ty]) (BV 0) (BV 1),
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testEq "∥ x : [(a ≡ a' : A) ∷ Type 0], y : [ditto] ⊢ x = y" $
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let ty : forall n. Term Three 0 n
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:= E (Eq0 (FT "A") (FT "a") (FT "a'") :# TYPE 0) in
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equalE (MkTyContext new [< ty, ty]) (BV 0) (BV 1),
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testEq "E ≔ a ≡ a' : A, EE ≔ E ∥ x : EE, y : EE ⊢ x = y"
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{globals = defGlobals `mergeLeft` fromList
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[("E", mkDef zero (TYPE 0) (Eq0 (FT "A") (FT "a") (FT "a'"))),
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("EE", mkDef zero (TYPE 0) (FT "E"))]} $
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equalE (MkTyContext new [< FT "EE", FT "EE"]) (BV 0) (BV 1),
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testEq "E ≔ a ≡ a' : A, EE ≔ E ∥ x : EE, y : E ⊢ x = y"
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{globals = defGlobals `mergeLeft` fromList
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[("E", mkDef zero (TYPE 0) (Eq0 (FT "A") (FT "a") (FT "a'"))),
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("EE", mkDef zero (TYPE 0) (FT "E"))]} $
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equalE (MkTyContext new [< FT "EE", FT "E"]) (BV 0) (BV 1),
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testEq "E ≔ a ≡ a' : A ∥ x : E, y : E ⊢ x = y"
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{globals = defGlobals `mergeLeft` fromList
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[("E", mkDef zero (TYPE 0) (Eq0 (FT "A") (FT "a") (FT "a'")))]} $
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equalE (MkTyContext new [< FT "E", FT "E"]) (BV 0) (BV 1),
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testEq "E ≔ a ≡ a' : A ∥ x : (E×E), y : (E×E) ⊢ x = y"
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{globals = defGlobals `mergeLeft` fromList
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[("E", mkDef zero (TYPE 0) (Eq0 (FT "A") (FT "a") (FT "a'")))]} $
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let ty : forall n. Term Three 0 n
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:= Sig (FT "E") $ S ["_"] $ N $ FT "E" in
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equalE (MkTyContext new [< ty, ty]) (BV 0) (BV 1),
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testEq "E ≔ a ≡ a' : A, F ≔ E × E ∥ x : F, y : F ⊢ x = y"
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{globals = defGlobals `mergeLeft` fromList
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[("E", mkDef zero (TYPE 0) (Eq0 (FT "A") (FT "a") (FT "a'"))),
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("W", mkDef zero (TYPE 0) (FT "E" `And` FT "E"))]} $
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equalE (MkTyContext new [< FT "W", FT "W"]) (BV 0) (BV 1)
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],
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"term closure" :- [
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testEq "[#0]{} = [#0] : A" $
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equalT (MkTyContext new [< FT "A"]) (FT "A")
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(CloT (BVT 0) id)
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(BVT 0),
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testEq "[#0]{a} = [a] : A" $
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equalT empty (FT "A")
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(CloT (BVT 0) (F "a" ::: id))
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(FT "a"),
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testEq "[#0]{a,b} = [a] : A" $
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equalT empty (FT "A")
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(CloT (BVT 0) (F "a" ::: F "b" ::: id))
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(FT "a"),
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testEq "[#1]{a,b} = [b] : A" $
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equalT empty (FT "A")
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(CloT (BVT 1) (F "a" ::: F "b" ::: id))
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(FT "b"),
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testEq "(λy. [#1]){a} = λy. [a] : B ⇾ A (N)" $
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equalT empty (Arr Zero (FT "B") (FT "A"))
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(CloT (Lam $ S ["y"] $ N $ BVT 0) (F "a" ::: id))
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(Lam $ S ["y"] $ N $ FT "a"),
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testEq "(λy. [#1]){a} = λy. [a] : B ⇾ A (Y)" $
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equalT empty (Arr Zero (FT "B") (FT "A"))
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(CloT (["y"] :\\ BVT 1) (F "a" ::: id))
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(["y"] :\\ FT "a")
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],
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"term d-closure" :- [
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testEq "★₀‹𝟎› = ★₀ : ★₁" $
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equalTD 1 empty (TYPE 1) (DCloT (TYPE 0) (K Zero ::: id)) (TYPE 0),
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testEq "(δ i. a)‹𝟎› = (δ i. a) : (a ≡ a : A)" $
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equalTD 1 empty
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(Eq0 (FT "A") (FT "a") (FT "a"))
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(DCloT (["i"] :\\% FT "a") (K Zero ::: id))
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(["i"] :\\% FT "a"),
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note "it is hard to think of well-typed terms with big dctxs"
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],
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"free var" :-
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let au_bu = fromList
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[("A", mkDef Any (TYPE (U 1)) (TYPE (U 0))),
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("B", mkDef Any (TYPE (U 1)) (TYPE (U 0)))]
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au_ba = fromList
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[("A", mkDef Any (TYPE (U 1)) (TYPE (U 0))),
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("B", mkDef Any (TYPE (U 1)) (FT "A"))]
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in [
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testEq "A = A" $
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equalE empty (F "A") (F "A"),
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testNeq "A ≠ B" $
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equalE empty (F "A") (F "B"),
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testEq "0=1 ⊢ A = B" $
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equalE (MkTyContext ZeroIsOne [<]) (F "A") (F "B"),
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testEq "A : ★₁ ≔ ★₀ ⊢ A = (★₀ ∷ ★₁)" {globals = au_bu} $
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equalE empty (F "A") (TYPE 0 :# TYPE 1),
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testEq "A : ★₁ ≔ ★₀ ⊢ [A] = ★₀" {globals = au_bu} $
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equalT empty (TYPE 1) (FT "A") (TYPE 0),
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testEq "A ≔ ★₀, B ≔ ★₀ ⊢ A = B" {globals = au_bu} $
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equalE empty (F "A") (F "B"),
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testEq "A ≔ ★₀, B ≔ A ⊢ A = B" {globals = au_ba} $
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equalE empty (F "A") (F "B"),
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testEq "A <: A" $
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subE empty (F "A") (F "A"),
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testNeq "A ≮: B" $
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subE empty (F "A") (F "B"),
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testEq "A : ★₃ ≔ ★₀, B : ★₃ ≔ ★₂ ⊢ A <: B"
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{globals = fromList [("A", mkDef Any (TYPE 3) (TYPE 0)),
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("B", mkDef Any (TYPE 3) (TYPE 2))]} $
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subE empty (F "A") (F "B"),
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note "(A and B in different universes)",
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testEq "A : ★₁ ≔ ★₀, B : ★₃ ≔ ★₂ ⊢ A <: B"
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{globals = fromList [("A", mkDef Any (TYPE 1) (TYPE 0)),
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("B", mkDef Any (TYPE 3) (TYPE 2))]} $
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subE empty (F "A") (F "B"),
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testEq "0=1 ⊢ A <: B" $
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subE (MkTyContext ZeroIsOne [<]) (F "A") (F "B")
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],
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"bound var" :- [
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testEq "#0 = #0" $
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equalE (MkTyContext new [< TYPE 0]) (BV 0) (BV 0),
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testEq "#0 <: #0" $
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subE (MkTyContext new [< TYPE 0]) (BV 0) (BV 0),
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testNeq "#0 ≠ #1" $
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equalE (MkTyContext new [< TYPE 0, TYPE 0]) (BV 0) (BV 1),
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testNeq "#0 ≮: #1" $
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subE (MkTyContext new [< TYPE 0, TYPE 0]) (BV 0) (BV 1),
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testEq "0=1 ⊢ #0 = #1" $
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equalE (MkTyContext ZeroIsOne [< TYPE 0, TYPE 0]) (BV 0) (BV 1)
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],
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"application" :- [
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testEq "f [a] = f [a]" $
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equalE empty (F "f" :@ FT "a") (F "f" :@ FT "a"),
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testEq "f [a] <: f [a]" $
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subE empty (F "f" :@ FT "a") (F "f" :@ FT "a"),
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testEq "(λ x. [x] ∷ A ⊸ A) a = ([a ∷ A] ∷ A) (β)" $
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equalE empty
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(((["x"] :\\ BVT 0) :# Arr One (FT "A") (FT "A")) :@ FT "a")
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(E (FT "a" :# FT "A") :# FT "A"),
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testEq "(λ x. [x] ∷ A ⊸ A) a = a (βυ)" $
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equalE empty
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(((["x"] :\\ BVT 0) :# Arr One (FT "A") (FT "A")) :@ FT "a")
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(F "a"),
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testEq "(λ g. [g [a]] ∷ ⋯)) [f] = (λ y. [f [y]] ∷ ⋯) [a] (β↘↙)" $
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let a = FT "A"; a2a = (Arr One a a) in
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equalE empty
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(((["g"] :\\ E (BV 0 :@ FT "a")) :# Arr One a2a a) :@ FT "f")
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(((["y"] :\\ E (F "f" :@ BVT 0)) :# a2a) :@ FT "a"),
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testEq "(λ x. [x] ∷ A ⊸ A) a <: a" $
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subE empty
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(((["x"] :\\ BVT 0) :# (Arr One (FT "A") (FT "A"))) :@ FT "a")
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(F "a"),
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note "id : A ⊸ A ≔ λ x. [x]",
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testEq "id [a] = a" $ equalE empty (F "id" :@ FT "a") (F "a"),
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testEq "id [a] <: a" $ subE empty (F "id" :@ FT "a") (F "a")
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],
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todo "dim application",
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"annotation" :- [
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testEq "(λ x. f [x]) ∷ A ⊸ A = [f] ∷ A ⊸ A" $
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equalE empty
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((["x"] :\\ E (F "f" :@ BVT 0)) :# Arr One (FT "A") (FT "A"))
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(FT "f" :# Arr One (FT "A") (FT "A")),
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testEq "[f] ∷ A ⊸ A = f" $
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equalE empty (FT "f" :# Arr One (FT "A") (FT "A")) (F "f"),
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testEq "(λ x. f [x]) ∷ A ⊸ A = f" $
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equalE empty
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((["x"] :\\ E (F "f" :@ BVT 0)) :# Arr One (FT "A") (FT "A"))
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(F "f")
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],
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"elim closure" :- [
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||
testEq "#0{a} = a" $
|
||
equalE empty (CloE (BV 0) (F "a" ::: id)) (F "a"),
|
||
testEq "#1{a} = #0" $
|
||
equalE (MkTyContext new [< FT "A"])
|
||
(CloE (BV 1) (F "a" ::: id)) (BV 0)
|
||
],
|
||
|
||
"elim d-closure" :- [
|
||
note "0·eq-ab : (A ≡ B : ★₀)",
|
||
testEq "(eq-ab #0)‹𝟎› = eq-ab 𝟎" $
|
||
equalED 1 empty
|
||
(DCloE (F "eq-ab" :% BV 0) (K Zero ::: id))
|
||
(F "eq-ab" :% K Zero),
|
||
testEq "(eq-ab #0)‹𝟎› = A" $
|
||
equalED 1 empty (DCloE (F "eq-ab" :% BV 0) (K Zero ::: id)) (F "A"),
|
||
testEq "(eq-ab #0)‹𝟏› = B" $
|
||
equalED 1 empty (DCloE (F "eq-ab" :% BV 0) (K One ::: id)) (F "B"),
|
||
testNeq "(eq-ab #0)‹𝟏› ≠ A" $
|
||
equalED 1 empty (DCloE (F "eq-ab" :% BV 0) (K One ::: id)) (F "A"),
|
||
testEq "(eq-ab #0)‹#0,𝟎› = (eq-ab #0)" $
|
||
equalED 2 empty
|
||
(DCloE (F "eq-ab" :% BV 0) (BV 0 ::: K Zero ::: id))
|
||
(F "eq-ab" :% BV 0),
|
||
testNeq "(eq-ab #0)‹𝟎› ≠ (eq-ab 𝟎)" $
|
||
equalED 2 empty
|
||
(DCloE (F "eq-ab" :% BV 0) (BV 0 ::: K Zero ::: id))
|
||
(F "eq-ab" :% K Zero),
|
||
testEq "#0‹𝟎› = #0 # term and dim vars distinct" $
|
||
equalED 1 (MkTyContext new [< FT "A"])
|
||
(DCloE (BV 0) (K Zero ::: id)) (BV 0),
|
||
testEq "a‹𝟎› = a" $
|
||
equalED 1 empty (DCloE (F "a") (K Zero ::: id)) (F "a"),
|
||
testEq "(f [a])‹𝟎› = f‹𝟎› [a]‹𝟎›" $
|
||
let th = K Zero ::: id in
|
||
equalED 1 empty
|
||
(DCloE (F "f" :@ FT "a") th)
|
||
(DCloE (F "f") th :@ DCloT (FT "a") th)
|
||
],
|
||
|
||
"clashes" :- [
|
||
testNeq "★₀ ≠ ★₀ ⇾ ★₀" $
|
||
equalT empty (TYPE 1) (TYPE 0) (Arr Zero (TYPE 0) (TYPE 0)),
|
||
testEq "0=1 ⊢ ★₀ = ★₀ ⇾ ★₀" $
|
||
equalT (MkTyContext ZeroIsOne [<])
|
||
(TYPE 1) (TYPE 0) (Arr Zero (TYPE 0) (TYPE 0)),
|
||
todo "others"
|
||
]
|
||
]
|