586 lines
22 KiB
Idris
586 lines
22 KiB
Idris
module Tests.Equal
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import Quox.Equal
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import Quox.Typechecker
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import public TypingImpls
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import TAP
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import Quox.EffExtra
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import AstExtra
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defGlobals : Definitions
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defGlobals = fromList
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[("A", ^mkPostulate gzero (^TYPE 0)),
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("B", ^mkPostulate gzero (^TYPE 0)),
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("a", ^mkPostulate gany (^FT "A" 0)),
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("a'", ^mkPostulate gany (^FT "A" 0)),
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("b", ^mkPostulate gany (^FT "B" 0)),
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("f", ^mkPostulate gany (^Arr One (^FT "A" 0) (^FT "A" 0))),
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("id", ^mkDef gany (^Arr One (^FT "A" 0) (^FT "A" 0)) (^LamY "x" (^BVT 0))),
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("eq-AB", ^mkPostulate gzero (^Eq0 (^TYPE 0) (^FT "A" 0) (^FT "B" 0))),
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("two", ^mkDef gany (^Nat) (^Succ (^Succ (^Zero))))]
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parameters (label : String) (act : Equal ())
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{default defGlobals globals : Definitions}
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testEq : Test
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testEq = test label $ runEqual globals act
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testNeq : Test
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testNeq = testThrows label (const True) $ runTC globals act $> "()"
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parameters (ctx : TyContext d n)
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subT, equalT : Term d n -> Term d n -> Term d n -> TC ()
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subT ty s t = lift $ Term.sub noLoc ctx ty s t
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equalT ty s t = lift $ Term.equal noLoc ctx ty s t
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equalTy : Term d n -> Term d n -> TC ()
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equalTy s t = lift $ Term.equalType noLoc ctx s t
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subE, equalE : Elim d n -> Elim d n -> TC ()
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subE e f = lift $ Elim.sub noLoc ctx e f
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equalE e f = lift $ Elim.equal noLoc ctx e f
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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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"function types" :- [
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note "cumulativity",
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testEq "0.★₀ → ★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.★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 "0.★₁ → ★₀ ≠ 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.★₁ → ★₀ <: 1.★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 "1.★₀ → ★₀ <: 1.★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 "1.★₀ → ★₀ <: 1.★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 "1.A → B = 1.A → B" $
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let tm = ^Arr One (^FT "A" 0) (^FT "B" 0) in
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equalT empty (^TYPE 0) tm tm,
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testEq "1.A → B <: 1.A → B" $
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let tm = ^Arr One (^FT "A" 0) (^FT "B" 0) in
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subT empty (^TYPE 0) tm tm,
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note "incompatible quantities",
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testNeq "1.★₀ → ★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 "0.A → B ≠ 1.A → B" $
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let tm1 = ^Arr Zero (^FT "A" 0) (^FT "B" 0)
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tm2 = ^Arr One (^FT "A" 0) (^FT "B" 0) in
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equalT empty (^TYPE 0) tm1 tm2,
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testNeq "0.A → B ≮: 1.A → B" $
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let tm1 = ^Arr Zero (^FT "A" 0) (^FT "B" 0)
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tm2 = ^Arr One (^FT "A" 0) (^FT "B" 0) in
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subT empty (^TYPE 0) tm1 tm2,
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testEq "0=1 ⊢ 0.A → B = 1.A → B" $
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let tm1 = ^Arr Zero (^FT "A" 0) (^FT "B" 0)
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tm2 = ^Arr One (^FT "A" 0) (^FT "B" 0) in
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equalT empty01 (^TYPE 0) tm1 tm2,
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todo "dependent function types"
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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" 0) (^FT "A" 0))
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(^LamY "x" (^BVT 0))
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(^LamY "x" (^BVT 0)),
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testEq "λ x ⇒ x <: λ x ⇒ x" $
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subT empty (^Arr One (^FT "A" 0) (^FT "A" 0))
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(^LamY "x" (^BVT 0))
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(^LamY "x" (^BVT 0)),
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testEq "λ x ⇒ x = λ y ⇒ y" $
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equalT empty (^Arr One (^FT "A" 0) (^FT "A" 0))
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(^LamY "x" (^BVT 0))
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(^LamY "y" (^BVT 0)),
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testEq "λ x ⇒ x <: λ y ⇒ y" $
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subT empty (^Arr One (^FT "A" 0) (^FT "A" 0))
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(^LamY "x" (^BVT 0))
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(^LamY "y" (^BVT 0)),
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testNeq "λ x y ⇒ x ≠ λ x y ⇒ y" $
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equalT empty
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(^Arr One (^FT "A" 0) (^Arr One (^FT "A" 0) (^FT "A" 0)))
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(^LamY "x" (^LamY "y" (^BVT 1)))
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(^LamY "x" (^LamY "y" (^BVT 0))),
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testEq "λ x ⇒ a = λ x ⇒ a (Y vs N)" $
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equalT empty
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(^Arr Zero (^FT "B" 0) (^FT "A" 0))
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(^LamY "x" (^FT "a" 0))
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(^LamN (^FT "a" 0)),
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testEq "λ x ⇒ f x = f (η)" $
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equalT empty
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(^Arr One (^FT "A" 0) (^FT "A" 0))
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(^LamY "x" (E $ ^App (^F "f" 0) (^BVT 0)))
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(^FT "f" 0)
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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 gzero (^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" 0) (^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 d n -> Term d n -> Elim d n
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refl a x = ^Ann (^DLam (SN x)) (^Eq0 a x x)
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in
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[
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note "binds before ∥ are globals, after it are BVs",
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note #"refl A x is an abbreviation for "(δ i ⇒ x) ∷ (x ≡ x : A)""#,
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testEq "refl A a = refl A a" $
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equalE empty
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(refl (^FT "A" 0) (^FT "a" 0))
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(refl (^FT "A" 0) (^FT "a" 0)),
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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 = ^mkPostulate gzero
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0))
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in defGlobals `mergeLeft` fromList [("p", def), ("q", def)]} $
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equalE empty (^F "p" 0) (^F "q" 0),
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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 0 n :=
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^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0) in
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equalE (extendTyN [< (Any, "x", ty), (Any, "y", ty)] empty)
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(^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 0 n :=
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E $ ^Ann (^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0)) (^TYPE 0) in
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equalE (extendTyN [< (Any, "x", ty), (Any, "y", ty)] empty)
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(^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 gzero (^TYPE 0)
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0))),
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("EE", ^mkDef gzero (^TYPE 0) (^FT "E" 0))]} $
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equalE
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(extendTyN [< (Any, "x", ^FT "EE" 0), (Any, "y", ^FT "EE" 0)] empty)
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(^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 gzero (^TYPE 0)
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0))),
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("EE", ^mkDef gzero (^TYPE 0) (^FT "E" 0))]} $
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equalE
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(extendTyN [< (Any, "x", ^FT "EE" 0), (Any, "y", ^FT "E" 0)] empty)
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(^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 gzero (^TYPE 0)
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0)))]} $
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equalE (extendTyN [< (Any, "x", ^FT "E" 0), (Any, "y", ^FT "E" 0)] empty)
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(^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 gzero (^TYPE 0)
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0)))]} $
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let ty : forall n. Term 0 n := ^Sig (^FT "E" 0) (SN $ ^FT "E" 0) in
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equalE (extendTyN [< (Any, "x", ty), (Any, "y", ty)] empty)
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(^BV 0) (^BV 1),
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testEq "E ≔ a ≡ a' : A, W ≔ E × E ∥ x : W, y : E×E ⊢ x = y"
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{globals = defGlobals `mergeLeft` fromList
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[("E", ^mkDef gzero (^TYPE 0)
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a'" 0))),
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("W", ^mkDef gzero (^TYPE 0) (^And (^FT "E" 0) (^FT "E" 0)))]} $
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equalE
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(extendTyN [< (Any, "x", ^FT "W" 0),
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(Any, "y", ^And (^FT "E" 0) (^FT "E" 0))] empty)
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(^BV 0) (^BV 1)
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],
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"term closure" :- [
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note "bold numbers for de bruijn indices",
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testEq "𝟎{} = 𝟎 : A" $
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equalT (extendTy Any "x" (^FT "A" 0) empty)
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(^FT "A" 0)
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(CloT (Sub (^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" 0)
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(CloT (Sub (^BVT 0) (^F "a" 0 ::: id)))
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(^FT "a" 0),
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testEq "𝟎{a,b} = a : A" $
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equalT empty (^FT "A" 0)
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(CloT (Sub (^BVT 0) (^F "a" 0 ::: ^F "b" 0 ::: id)))
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(^FT "a" 0),
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testEq "𝟏{a,b} = b : A" $
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equalT empty (^FT "A" 0)
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(CloT (Sub (^BVT 1) (^F "a" 0 ::: ^F "b" 0 ::: id)))
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(^FT "b" 0),
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testEq "(λy ⇒ 𝟏){a} = λy ⇒ a : 0.B → A (N)" $
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equalT empty (^Arr Zero (^FT "B" 0) (^FT "A" 0))
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(CloT (Sub (^LamN (^BVT 0)) (^F "a" 0 ::: id)))
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(^LamN (^FT "a" 0)),
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testEq "(λy ⇒ 𝟏){a} = λy ⇒ a : 0.B → A (Y)" $
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equalT empty (^Arr Zero (^FT "B" 0) (^FT "A" 0))
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(CloT (Sub (^LamY "y" (^BVT 1)) (^F "a" 0 ::: id)))
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(^LamY "y" (^FT "a" 0))
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],
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"term d-closure" :- [
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testEq "★₀‹0› = ★₀ : ★₁" $
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equalT (extendDim "𝑗" empty)
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(^TYPE 1) (DCloT (Sub (^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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equalT (extendDim "𝑗" empty)
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(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a" 0))
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(DCloT (Sub (^DLamN (^FT "a" 0)) (^K Zero ::: id)))
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(^DLamN (^FT "a" 0)),
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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 gany (^TYPE 1) (^TYPE 0)),
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("B", ^mkDef gany (^TYPE 1) (^TYPE 0))]
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au_ba = fromList
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[("A", ^mkDef gany (^TYPE 1) (^TYPE 0)),
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("B", ^mkDef gany (^TYPE 1) (^FT "A" 0))]
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in [
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testEq "A = A" $
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equalE empty (^F "A" 0) (^F "A" 0),
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testNeq "A ≠ B" $
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equalE empty (^F "A" 0) (^F "B" 0),
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testEq "0=1 ⊢ A = B" $
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equalE empty01 (^F "A" 0) (^F "B" 0),
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testEq "A : ★₁ ≔ ★₀ ⊢ A = (★₀ ∷ ★₁)" {globals = au_bu} $
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equalE empty (^F "A" 0) (^Ann (^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" 0) (^TYPE 0),
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testEq "A ≔ ★₀, B ≔ ★₀ ⊢ A = B" {globals = au_bu} $
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equalE empty (^F "A" 0) (^F "B" 0),
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testEq "A ≔ ★₀, B ≔ A ⊢ A = B" {globals = au_ba} $
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equalE empty (^F "A" 0) (^F "B" 0),
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testEq "A <: A" $
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subE empty (^F "A" 0) (^F "A" 0),
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testNeq "A ≮: B" $
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subE empty (^F "A" 0) (^F "B" 0),
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testEq "A : ★₃ ≔ ★₀, B : ★₃ ≔ ★₂ ⊢ A <: B"
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{globals = fromList [("A", ^mkDef gany (^TYPE 3) (^TYPE 0)),
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("B", ^mkDef gany (^TYPE 3) (^TYPE 2))]} $
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subE empty (^F "A" 0) (^F "B" 0),
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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 gany (^TYPE 1) (^TYPE 0)),
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("B", ^mkDef gany (^TYPE 3) (^TYPE 2))]} $
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subE empty (^F "A" 0) (^F "B" 0),
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testEq "0=1 ⊢ A <: B" $
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subE empty01 (^F "A" 0) (^F "B" 0)
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],
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"bound var" :- [
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note "bold numbers for de bruijn indices",
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testEq "𝟎 = 𝟎" $
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equalE (extendTy Any "A" (^TYPE 0) empty) (^BV 0) (^BV 0),
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testEq "𝟎 <: 𝟎" $
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subE (extendTy Any "A" (^TYPE 0) empty) (^BV 0) (^BV 0),
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testNeq "𝟎 ≠ 𝟏" $
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equalE (extendTyN [< (Any, "A", ^TYPE 0), (Any, "B", ^TYPE 0)] empty)
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(^BV 0) (^BV 1),
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testNeq "𝟎 ≮: 𝟏" $
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subE (extendTyN [< (Any, "A", ^TYPE 0), (Any, "B", ^TYPE 0)] empty)
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(^BV 0) (^BV 1),
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testEq "0=1 ⊢ 𝟎 = 𝟏" $
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equalE (extendTyN [< (Any, "A", ^TYPE 0), (Any, "B", ^TYPE 0)] empty01)
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(^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 (^App (^F "f" 0) (^FT "a" 0)) (^App (^F "f" 0) (^FT "a" 0)),
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testEq "f a <: f a" $
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subE empty (^App (^F "f" 0) (^FT "a" 0)) (^App (^F "f" 0) (^FT "a" 0)),
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testEq "(λ x ⇒ x ∷ 1.A → A) a = ((a ∷ A) ∷ A) (β)" $
|
||
equalE empty
|
||
(^App (^Ann (^LamY "x" (^BVT 0)) (^Arr One (^FT "A" 0) (^FT "A" 0)))
|
||
(^FT "a" 0))
|
||
(^Ann (E $ ^Ann (^FT "a" 0) (^FT "A" 0)) (^FT "A" 0)),
|
||
testEq "(λ x ⇒ x ∷ A ⊸ A) a = a (βυ)" $
|
||
equalE empty
|
||
(^App (^Ann (^LamY "x" (^BVT 0)) (^Arr One (^FT "A" 0) (^FT "A" 0)))
|
||
(^FT "a" 0))
|
||
(^F "a" 0),
|
||
testEq "((λ g ⇒ g a) ∷ 1.(1.A → A) → A) f = ((λ y ⇒ f y) ∷ 1.A → A) a # β↘↙" $
|
||
let a = ^FT "A" 0; a2a = ^Arr One a a; aa2a = ^Arr One a2a a in
|
||
equalE empty
|
||
(^App (^Ann (^LamY "g" (E $ ^App (^BV 0) (^FT "a" 0))) aa2a)
|
||
(^FT "f" 0))
|
||
(^App (^Ann (^LamY "y" (E $ ^App (^F "f" 0) (^BVT 0))) a2a)
|
||
(^FT "a" 0)),
|
||
testEq "((λ x ⇒ x) ∷ 1.A → A) a <: a" $
|
||
subE empty
|
||
(^App (^Ann (^LamY "x" (^BVT 0)) (^Arr One (^FT "A" 0) (^FT "A" 0)))
|
||
(^FT "a" 0))
|
||
(^F "a" 0),
|
||
note "id : A ⊸ A ≔ λ x ⇒ x",
|
||
testEq "id a = a" $ equalE empty (^App (^F "id" 0) (^FT "a" 0)) (^F "a" 0),
|
||
testEq "id a <: a" $ subE empty (^App (^F "id" 0) (^FT "a" 0)) (^F "a" 0)
|
||
],
|
||
|
||
"dim application" :- [
|
||
testEq "eq-AB @0 = eq-AB @0" $
|
||
equalE empty
|
||
(^DApp (^F "eq-AB" 0) (^K Zero))
|
||
(^DApp (^F "eq-AB" 0) (^K Zero)),
|
||
testNeq "eq-AB @0 ≠ eq-AB @1" $
|
||
equalE empty
|
||
(^DApp (^F "eq-AB" 0) (^K Zero))
|
||
(^DApp (^F "eq-AB" 0) (^K One)),
|
||
testEq "𝑖 | ⊢ eq-AB @𝑖 = eq-AB @𝑖" $
|
||
equalE
|
||
(extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 0))
|
||
(^DApp (^F "eq-AB" 0) (^BV 0)),
|
||
testNeq "𝑖 | ⊢ eq-AB @𝑖 ≠ eq-AB @0" $
|
||
equalE (extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 0))
|
||
(^DApp (^F "eq-AB" 0) (^K Zero)),
|
||
testEq "𝑖, 𝑖=0 | ⊢ eq-AB @𝑖 = eq-AB @0" $
|
||
equalE (eqDim (^BV 0) (^K Zero) $ extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 0))
|
||
(^DApp (^F "eq-AB" 0) (^K Zero)),
|
||
testNeq "𝑖, 𝑖=1 | ⊢ eq-AB @𝑖 ≠ eq-AB @0" $
|
||
equalE (eqDim (^BV 0) (^K One) $ extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 0))
|
||
(^DApp (^F "eq-AB" 0) (^K Zero)),
|
||
testNeq "𝑖, 𝑗 | ⊢ eq-AB @𝑖 ≠ eq-AB @𝑗" $
|
||
equalE (extendDim "𝑗" $ extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 1))
|
||
(^DApp (^F "eq-AB" 0) (^BV 0)),
|
||
testEq "𝑖, 𝑗, 𝑖=𝑗 | ⊢ eq-AB @𝑖 = eq-AB @𝑗" $
|
||
equalE (eqDim (^BV 0) (^BV 1) $ extendDim "𝑗" $ extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 1))
|
||
(^DApp (^F "eq-AB" 0) (^BV 0)),
|
||
testEq "𝑖, 𝑗, 𝑖=0, 𝑗=0 | ⊢ eq-AB @𝑖 = eq-AB @𝑗" $
|
||
equalE
|
||
(eqDim (^BV 0) (^K Zero) $ eqDim (^BV 1) (^K Zero) $
|
||
extendDim "𝑗" $ extendDim "𝑖" empty)
|
||
(^DApp (^F "eq-AB" 0) (^BV 1))
|
||
(^DApp (^F "eq-AB" 0) (^BV 0)),
|
||
testEq "0=1 | ⊢ eq-AB @𝑖 = eq-AB @𝑗" $
|
||
equalE (extendDim "𝑗" $ extendDim "𝑖" empty01)
|
||
(^DApp (^F "eq-AB" 0) (^BV 1))
|
||
(^DApp (^F "eq-AB" 0) (^BV 0)),
|
||
testEq "eq-AB @0 = A" $
|
||
equalE empty (^DApp (^F "eq-AB" 0) (^K Zero)) (^F "A" 0),
|
||
testEq "eq-AB @1 = B" $
|
||
equalE empty (^DApp (^F "eq-AB" 0) (^K One)) (^F "B" 0),
|
||
testEq "((δ i ⇒ a) ∷ a ≡ a : A) @0 = a" $
|
||
equalE empty
|
||
(^DApp (^Ann (^DLamN (^FT "a" 0))
|
||
(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a" 0)))
|
||
(^K Zero))
|
||
(^F "a" 0),
|
||
testEq "((δ i ⇒ a) ∷ a ≡ a : A) @0 = ((δ i ⇒ a) ∷ a ≡ a : A) @1" $
|
||
equalE empty
|
||
(^DApp (^Ann (^DLamN (^FT "a" 0))
|
||
(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a" 0)))
|
||
(^K Zero))
|
||
(^DApp (^Ann (^DLamN (^FT "a" 0))
|
||
(^Eq0 (^FT "A" 0) (^FT "a" 0) (^FT "a" 0)))
|
||
(^K One))
|
||
],
|
||
|
||
"annotation" :- [
|
||
testEq "(λ x ⇒ f x) ∷ 1.A → A = f ∷ 1.A → A" $
|
||
equalE empty
|
||
(^Ann (^LamY "x" (E $ ^App (^F "f" 0) (^BVT 0)))
|
||
(^Arr One (^FT "A" 0) (^FT "A" 0)))
|
||
(^Ann (^FT "f" 0) (^Arr One (^FT "A" 0) (^FT "A" 0))),
|
||
testEq "f ∷ 1.A → A = f" $
|
||
equalE empty
|
||
(^Ann (^FT "f" 0) (^Arr One (^FT "A" 0) (^FT "A" 0)))
|
||
(^F "f" 0),
|
||
testEq "(λ x ⇒ f x) ∷ 1.A → A = f" $
|
||
equalE empty
|
||
(^Ann (^LamY "x" (E $ ^App (^F "f" 0) (^BVT 0)))
|
||
(^Arr One (^FT "A" 0) (^FT "A" 0)))
|
||
(^F "f" 0)
|
||
],
|
||
|
||
"natural type" :- [
|
||
testEq "ℕ = ℕ" $ equalTy empty (^Nat) (^Nat),
|
||
testEq "ℕ = ℕ : ★₀" $ equalT empty (^TYPE 0) (^Nat) (^Nat),
|
||
testEq "ℕ = ℕ : ★₆₉" $ equalT empty (^TYPE 69) (^Nat) (^Nat),
|
||
testNeq "ℕ ≠ {}" $ equalTy empty (^Nat) (^enum []),
|
||
testEq "0=1 ⊢ ℕ = {}" $ equalTy empty01 (^Nat) (^enum [])
|
||
],
|
||
|
||
"natural numbers" :- [
|
||
testEq "0 = 0" $ equalT empty (^Nat) (^Zero) (^Zero),
|
||
testEq "succ two = succ two" $
|
||
equalT empty (^Nat) (^Succ (^FT "two" 0)) (^Succ (^FT "two" 0)),
|
||
testNeq "succ two ≠ two" $
|
||
equalT empty (^Nat) (^Succ (^FT "two" 0)) (^FT "two" 0),
|
||
testNeq "0 ≠ 1" $
|
||
equalT empty (^Nat) (^Zero) (^Succ (^Zero)),
|
||
testEq "0=1 ⊢ 0 = 1" $
|
||
equalT empty01 (^Nat) (^Zero) (^Succ (^Zero))
|
||
],
|
||
|
||
"natural elim" :- [
|
||
testEq "caseω 0 return {a,b} of {zero ⇒ 'a; succ _ ⇒ 'b} = 'a" $
|
||
equalT empty
|
||
(^enum ["a", "b"])
|
||
(E $ ^CaseNat Any Zero (^Ann (^Zero) (^Nat))
|
||
(SN $ ^enum ["a", "b"])
|
||
(^Tag "a")
|
||
(SN $ ^Tag "b"))
|
||
(^Tag "a"),
|
||
testEq "caseω 1 return {a,b} of {zero ⇒ 'a; succ _ ⇒ 'b} = 'b" $
|
||
equalT empty
|
||
(^enum ["a", "b"])
|
||
(E $ ^CaseNat Any Zero (^Ann (^Succ (^Zero)) (^Nat))
|
||
(SN $ ^enum ["a", "b"])
|
||
(^Tag "a")
|
||
(SN $ ^Tag "b"))
|
||
(^Tag "b"),
|
||
testEq "caseω 4 return ℕ of {0 ⇒ 0; succ n ⇒ n} = 3" $
|
||
equalT empty
|
||
(^Nat)
|
||
(E $ ^CaseNat Any Zero (^Ann (^makeNat 4) (^Nat))
|
||
(SN $ ^Nat)
|
||
(^Zero)
|
||
(SY [< "n", ^BN Unused] $ ^BVT 1))
|
||
(^makeNat 3)
|
||
],
|
||
|
||
todo "pair types",
|
||
|
||
"pairs" :- [
|
||
testEq "('a, 'b) = ('a, 'b) : {a} × {b}" $
|
||
equalT empty
|
||
(^And (^enum ["a"]) (^enum ["b"]))
|
||
(^Pair (^Tag "a") (^Tag "b"))
|
||
(^Pair (^Tag "a") (^Tag "b")),
|
||
testNeq "('a, 'b) ≠ ('b, 'a) : {a,b} × {a,b}" $
|
||
equalT empty
|
||
(^And (^enum ["a", "b"]) (^enum ["a", "b"]))
|
||
(^Pair (^Tag "a") (^Tag "b"))
|
||
(^Pair (^Tag "b") (^Tag "a")),
|
||
testEq "0=1 ⊢ ('a, 'b) = ('b, 'a) : {a,b} × {a,b}" $
|
||
equalT empty01
|
||
(^And (^enum ["a", "b"]) (^enum ["a", "b"]))
|
||
(^Pair (^Tag "a") (^Tag "b"))
|
||
(^Pair (^Tag "b") (^Tag "a")),
|
||
testEq "0=1 ⊢ ('a, 'b) = ('b, 'a) : ℕ" $
|
||
equalT empty01
|
||
(^Nat)
|
||
(^Pair (^Tag "a") (^Tag "b"))
|
||
(^Pair (^Tag "b") (^Tag "a"))
|
||
],
|
||
|
||
todo "pair elim",
|
||
|
||
todo "enum types",
|
||
todo "enum",
|
||
todo "enum elim",
|
||
|
||
todo "box types",
|
||
todo "boxes",
|
||
todo "box elim",
|
||
|
||
"elim closure" :- [
|
||
note "bold numbers for de bruijn indices",
|
||
testEq "𝟎{a} = a" $
|
||
equalE empty (CloE (Sub (^BV 0) (^F "a" 0 ::: id))) (^F "a" 0),
|
||
testEq "𝟏{a} = 𝟎" $
|
||
equalE (extendTy Any "x" (^FT "A" 0) empty)
|
||
(CloE (Sub (^BV 1) (^F "a" 0 ::: id))) (^BV 0)
|
||
],
|
||
|
||
"elim d-closure" :- [
|
||
note "bold numbers for de bruijn indices",
|
||
note "0·eq-AB : (A ≡ B : ★₀)",
|
||
testEq "(eq-AB @𝟎)‹0› = eq-AB @0" $
|
||
equalE empty
|
||
(DCloE (Sub (^DApp (^F "eq-AB" 0) (^BV 0)) (^K Zero ::: id)))
|
||
(^DApp (^F "eq-AB" 0) (^K Zero)),
|
||
testEq "(eq-AB @𝟎)‹0› = A" $
|
||
equalE empty
|
||
(DCloE (Sub (^DApp (^F "eq-AB" 0) (^BV 0)) (^K Zero ::: id)))
|
||
(^F "A" 0),
|
||
testEq "(eq-AB @𝟎)‹1› = B" $
|
||
equalE empty
|
||
(DCloE (Sub (^DApp (^F "eq-AB" 0) (^BV 0)) (^K One ::: id)))
|
||
(^F "B" 0),
|
||
testNeq "(eq-AB @𝟎)‹1› ≠ A" $
|
||
equalE empty
|
||
(DCloE (Sub (^DApp (^F "eq-AB" 0) (^BV 0)) (^K One ::: id)))
|
||
(^F "A" 0),
|
||
testEq "(eq-AB @𝟎)‹𝟎,0› = (eq-AB 𝟎)" $
|
||
equalE (extendDim "𝑖" empty)
|
||
(DCloE (Sub (^DApp (^F "eq-AB" 0) (^BV 0)) (^BV 0 ::: ^K Zero ::: id)))
|
||
(^DApp (^F "eq-AB" 0) (^BV 0)),
|
||
testNeq "(eq-AB 𝟎)‹0› ≠ (eq-AB 0)" $
|
||
equalE (extendDim "𝑖" empty)
|
||
(DCloE (Sub (^DApp (^F "eq-AB" 0) (^BV 0)) (^BV 0 ::: ^K Zero ::: id)))
|
||
(^DApp (^F "eq-AB" 0) (^K Zero)),
|
||
testEq "𝟎‹0› = 𝟎 # term and dim vars distinct" $
|
||
equalE
|
||
(extendTy Any "x" (^FT "A" 0) empty)
|
||
(DCloE (Sub (^BV 0) (^K Zero ::: id))) (^BV 0),
|
||
testEq "a‹0› = a" $
|
||
equalE empty
|
||
(DCloE (Sub (^F "a" 0) (^K Zero ::: id))) (^F "a" 0),
|
||
testEq "(f a)‹0› = f‹0› a‹0›" $
|
||
let th = ^K Zero ::: id in
|
||
equalE empty
|
||
(DCloE (Sub (^App (^F "f" 0) (^FT "a" 0)) th))
|
||
(^App (DCloE (Sub (^F "f" 0) th)) (DCloT (Sub (^FT "a" 0) th)))
|
||
],
|
||
|
||
"clashes" :- [
|
||
testNeq "★₀ ≠ 0.★₀ → ★₀" $
|
||
equalT empty (^TYPE 1) (^TYPE 0) (^Arr Zero (^TYPE 0) (^TYPE 0)),
|
||
testEq "0=1 ⊢ ★₀ = 0.★₀ → ★₀" $
|
||
equalT empty01 (^TYPE 1) (^TYPE 0) (^Arr Zero (^TYPE 0) (^TYPE 0)),
|
||
todo "others"
|
||
]
|
||
]
|