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added Σ types

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This commit is contained in:
Rachel Lambda Samuelsson 2022-07-21 04:18:50 +02:00
parent ab7d70d562
commit d1b27c826b
8 changed files with 186 additions and 71 deletions
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1
README.md
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@ -19,7 +19,6 @@ A dependently typed system
# TODO
* Fun types
* Σ
* Id
* Parser

66
src/Check.idr
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@ -22,20 +22,32 @@ mutual
check trs tys xpt' tr = do
xpt <- whnf xpt'
case tr of
TLam sc => case xpt of
VClos env (TPi a b) => do
v <- VGen <$> fresh
check (v :: trs) (VClos env a :: tys) (VClos (v :: env) b) sc
_ => oops "expected pi"
TLam sc => case xpt of
VClos env (TPi a b) => do
v <- VGen <$> fresh
check (v :: trs) (VClos env a :: tys) (VClos (v :: env) b) sc
_ => oops "expected pi"
TPi a b => case xpt of
VType => do
v <- VGen <$> fresh
(&&) <$> check trs tys VType a
<*> delay <$> check (v :: trs) (VClos trs a :: tys) VType b
-- pi and sigma could be inferred /shrug
TPi a b => case xpt of
VType => do
v <- VGen <$> fresh
guardS "Pi a" =<< check trs tys VType a
check (v :: trs) (VClos trs a :: tys) VType b
_ => oops "expected type"
TSigma a b => case xpt of
VType => do
v <- VGen <$> fresh
guardS "Σ a" =<< check trs tys VType a
check trs tys (VClos trs (TPi a TType)) b
_ => oops "expected type"
_ => oops "expected type"
TPair x y => case xpt of
(VClos env (TSigma a b)) => do
guardS "Pair a" =<< check trs tys (VClos env a) x
check trs tys (VClos env b `VApp` VClos trs x) y
_ => oops "expected sigma"
_ => convert xpt =<< infer trs tys tr
@ -46,7 +58,7 @@ mutual
infer trs tys (TApp f x) = infer trs tys f >>= whnf >>=
\case
VClos env (TPi a b) => do
guard =<< check trs tys (VClos env a) x
guardS "app x" =<< check trs tys (VClos env a) x
pure (VClos (VClos trs x :: env) b)
_ => oops "expected infer pi"
@ -57,27 +69,37 @@ mutual
infer trs tys TStar = pure VTop
infer trs tys TZero = pure VNat
infer trs tys (TSuc n) = do
guard =<< check trs tys VNat n
guardS "suc n" =<< check trs tys VNat n
pure VNat
infer trs tys (TTopInd c st) = do
guard =<< check trs tys (VClos trs (TPi TTop TType)) c
guard =<< check trs tys (VApp (VClos trs c) VStar) st
guardS "⊤ C" =<< check trs tys (VClos trs (TPi TTop TType)) c
guardS "⊤ ★" =<< check trs tys (VApp (VClos trs c) VStar) st
pure (VClos trs (TPi TTop (TApp (weakTr c) (TVar 0))))
infer trs tys (TBotInd c) = do
guard =<< check trs tys (VClos trs (TPi TBot TType)) c
guardS "⊥ C" =<< check trs tys (VClos trs (TPi TBot TType)) c
pure (VClos trs (TPi TBot (TApp (weakTr c) (TVar 0))))
infer trs tys (TNatInd c z s) = do
guard =<< check trs tys (VClos trs (TPi TNat TType)) c
guard =<< check trs tys (VApp (VClos trs c) (VNatTr 0)) z
guard =<< check trs tys (VClos trs (TPi TNat
(TPi (TApp (weakTr c) (TVar 0))
(TApp (weakTr2 c) (TSuc (TVar (FS FZ))))))) s
guardS "ℕ C" =<< check trs tys (VClos trs (TPi TNat TType)) c
guardS "ℕ z" =<< check trs tys (VApp (VClos trs c) (VNatTr 0)) z
guardS "ℕ s" =<< check trs tys (VClos trs (TPi TNat
(TPi (TApp (weakTr c) (TVar 0))
(TApp (weakTr2 c) (TSuc (TVar (FS FZ))))))) s
pure (VClos trs (TPi TNat (TApp (weakTr c) (TVar 0))))
infer trs tys _ = oops "cannot infer type"
infer trs tys (TSigInd a b c f) = do
guardS "Σ A" =<< check trs tys VType a
guardS "Σ B" =<< check trs tys (VClos trs (TPi a TType)) b
guardS "Σ C" =<< check trs tys (VClos trs (TPi (TSigma a b) TType)) c
guardS "Σ f" =<< check trs tys (VClos trs (TPi a {-a:A-}
(TPi (weakTr b `TApp` TVar 0) {-b:Ba-}
(weakTr2 c `TApp` TPair (TVar (FS FZ)) (TVar 0))))) f
pure (VClos trs (TPi (TSigma a b) (weakTr c `TApp` TVar 0)))
infer trs tys x = oops ("cannot infer type" ++ show x)
public export
typecheck : Term 0 -> Term 0 -> Either String (Bool, List String)

46
src/Convert.idr
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@ -14,6 +14,8 @@ import Data.Vect
%default total
{- TODO: I should throw eval, eval, convert into a helper sometime -}
public export
convert : Value -> Value -> PI Bool
convert u1 u2 = do
@ -31,28 +33,28 @@ convert u1 u2 = do
(VNatTr n, VNatTr m) => pure (n == m)
(VApp f1 x1, VApp f2 x2) => (&&) <$> convert f1 f2
<*> delay <$> convert x1 x2
(VApp f1 x1, VApp f2 x2) => (&&) <$> convert f1 f2 <*> delay <$> convert x1 x2
(VGen k1, VGen k2) => pure (k1 == k2)
(VPair a1 b1, VPair a2 b2) => (&&) <$> convert a1 a2 <*> delay <$> convert b1 b2
(VClos env1 (TLam sc1), VClos env2 (TLam sc2)) => do
v <- VGen <$> fresh
convert (VClos (v :: env1) sc1)
(VClos (v :: env2) sc2)
convert (VClos (v :: env1) sc1) (VClos (v :: env2) sc2)
(VClos env1 (TPi a1 b1), VClos env2 (TPi a2 b2)) => do
v <- VGen <$> fresh
(&&) <$> convert (VClos env1 a1)
(VClos env2 a2)
<*> delay <$> convert (VClos (v :: env1) b1)
(VClos (v :: env2) b2)
guard =<< convert (VClos env1 a1) (VClos env2 a2)
convert (VClos (v :: env1) b1) (VClos (v :: env2) b2)
(VClos env1 (TTopInd c1 st1), VClos env2 (TTopInd c2 st2)) => do
c1' <- eval env1 c1
c2' <- eval env2 c2
guard =<< convert c1' c2'
st1' <- eval env1 st1
st2' <- eval env2 st2
(&&) <$> convert c1' c2' <*> delay <$> convert st1' st2'
convert st1' st2'
(VClos env1 (TBotInd c1), VClos env2 (TBotInd c2)) => do
c1' <- eval env1 c1
@ -63,12 +65,34 @@ convert u1 u2 = do
(VClos env1 (TNatInd c1 z1 s1), VClos env2 (TNatInd c2 z2 s2)) => do
c1' <- eval env1 c1
c2' <- eval env2 c2
guard =<< convert c1' c2'
z1' <- eval env1 z1
z2' <- eval env2 z2
guard =<< convert z1' z2'
s1' <- eval env1 s1
s2' <- eval env2 s2
b1 <- (&&) <$> convert c1' c2' <*> delay <$> convert z1' z2'
guard b1
convert s1' s2'
(VClos env1 (TSigma a1 b1), VClos env2 (TSigma a2 b2)) => do
a1' <- eval env1 a1
a2' <- eval env2 a2
guard =<< convert a1' a2'
b1' <- eval env1 b1
b2' <- eval env2 b2
convert b1' b2'
(VClos env1 (TSigInd a1 b1 c1 f1), VClos env2 (TSigInd a2 b2 c2 f2)) => do
a1' <- eval env1 a1
a2' <- eval env2 a2
guard =<< convert a1' a2'
b1' <- eval env1 b1
b2' <- eval env2 b2
guard =<< convert b1' b2'
c1' <- eval env1 c1
c2' <- eval env2 c2
guard =<< convert c1' c2'
f1' <- eval env1 f1
f2' <- eval env2 f2
convert f1' f2'
_ => pure False

6
src/Misc.idr
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@ -30,6 +30,12 @@ public export
oops : String -> PI a
oops = left
public export
guardS : String -> Bool -> PI ()
guardS str True = pure ()
guardS str False = oops str
public export
fresh : PI Nat
fresh = do

39
src/Normalize.idr
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@ -17,10 +17,10 @@ mutual
public export
-- no computational rule for ⊥
app : Value -> Value -> PI Value
app (VClos env (TLam sc)) x = eval (x :: env) sc
app (VClos env (TLam sc)) x = eval (x :: env) sc
app (VClos env (TTopInd c st)) VTop = eval env st
app f@(VClos env (TTopInd c st)) x = logS ("⊤-ind applied to " ++ show x)
app (VClos env (TTopInd c st)) VTop = eval env st
app f@(VClos env (TTopInd c st)) x = logS ("⊤-ind applied to " ++ show x)
>> pure (VApp f x)
app (VClos env (TNatInd _ z s)) (VNatTr n) = do
@ -35,8 +35,16 @@ mutual
sn <- app s (VNatTr n)
app sn rec
app f@(VClos env (TNatInd _ z s)) x = logS ("ℕ-ind applied to " ++ show x)
>> pure (VApp f x)
app f@(VClos env (TNatInd _ z s)) x = logS ("ℕ-ind applied to " ++ show x)
>> pure (VApp f x)
app (VClos env (TSigInd _ _ c f)) (VPair a b) = do
f' <- eval env f
fa <- app f' a
app fa b
app f@(VClos env (TSigInd _ _ c p)) x = logS ("Σ-ind applied to " ++ show x)
>> pure (VApp f x)
app f x = pure (VApp f x)
@ -47,21 +55,26 @@ mutual
f' <- eval env f
x' <- eval env x
assert_total (app f' x') -- :(
eval env TType = pure VType
eval env TTop = pure VTop
eval env TStar = pure VStar
eval env TBot = pure VBot
eval env TNat = pure VNat
eval env TZero = pure (VNatTr 0)
eval env TType = pure VType
eval env TTop = pure VTop
eval env TStar = pure VStar
eval env TBot = pure VBot
eval env TNat = pure VNat
eval env TZero = pure (VNatTr 0)
eval env (TSuc n) = do
eval env (TSuc n) = do
n' <- eval env n
case n' of
VNatTr n => pure (VNatTr (S n))
x => logS ("suc applied to " ++ show x)
>> pure (VClos env (TSuc n))
eval env tr = pure (VClos env tr)
eval env (TPair a b) = do
a' <- eval env a
b' <- eval env b
pure (VPair a' b')
eval env tr = pure (VClos env tr)
public export
whnf : Value -> PI Value

47
src/Term.idr
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@ -26,13 +26,17 @@ data Term : (_ : Index) -> Type where
TSuc : Term n -> Term n -- successor
TNatInd : Term n -> Term n -> Term n -> Term n -- : (x : ℕ) → C x
TSigma : Term n -> Term n -> Term n -- Sum type (Σ _ : A, B _)
TPair : Term n -> Term n -> Term n -- Sum constructor _,_
TSigInd : Term n -> Term n -> Term n -> Term n -> Term n -- A B C f : (x : Σ _ : A , B _) → C x
TLam : Term (S n) -> Term n -- Lambda abstraction (λ _ . Scope)
TPi : Term n -> Term (S n) -> Term n -- Pi type (∏ _ : A . B _ )
TApp : Term n -> Term n -> Term n -- Appliction
TVar : Fin n -> Term n -- Variable
TVar : Fin n -> Term n -- Variable
infixl 2 `TApp`
infixl 3 `TApp`
public export
Show (Term n) where
@ -50,6 +54,10 @@ Show (Term n) where
show (TSuc n) = "suc (" ++ show n ++ ")"
show (TNatInd c z s) = "ℕ-ind (" ++ show c ++ ") (" ++ show z ++ ") (" ++ show s ++ ")"
show (TSigma a b) = "TSigma (" ++ show a ++ ") (" ++ show b ++ ")"
show (TPair a b) = "TPair (" ++ show a ++ ") (" ++ show b ++ ")"
show (TSigInd a b c f) = "Σ-ind (" ++ show a ++ ") (" ++ show b ++ ") (" ++ show c ++ ") (" ++ show f ++ ")"
show (TLam sc) = "TLam (" ++ show sc ++ ")"
show (TPi ty sc) = "TPi (" ++ show ty ++ ") (" ++ show sc ++ ")"
@ -61,22 +69,25 @@ weakTr : Term n -> Term (S n)
weakTr = go 0
where
go : {0 n : Nat} -> Fin (S n) -> Term n -> Term (S n)
go n TType = TType
go n TTop = TTop
go n TStar = TTop
go n (TTopInd c st) = TTopInd (go n c) (go n st)
go n TBot = TBot
go n (TBotInd c) = TBotInd (go n c)
go n TNat = TNat
go n TZero = TZero
go n (TSuc m) = TSuc (go n m)
go n (TNatInd c z s) = TNatInd (go n c) (go n z) (go n s)
go n (TLam sc) = TLam (go (FS n) sc)
go n (TPi ty sc) = TPi (go n ty) (go (FS n) sc)
go n (TApp f x) = TApp (go n f) (go n x)
go n (TVar i) = if weaken i < n
then TVar (weaken i)
else TVar (FS i)
go n TType = TType
go n TTop = TTop
go n TStar = TTop
go n (TTopInd c st) = TTopInd (go n c) (go n st)
go n TBot = TBot
go n (TBotInd c) = TBotInd (go n c)
go n TNat = TNat
go n TZero = TZero
go n (TSuc m) = TSuc (go n m)
go n (TNatInd c z s) = TNatInd (go n c) (go n z) (go n s)
go n (TSigma a b) = TSigma (go n a) (go n b)
go n (TPair a b) = TPair (go n a) (go n b)
go n (TSigInd a b c f) = TSigInd (go n a) (go n b) (go n c) (go n f)
go n (TLam sc) = TLam (go (FS n) sc)
go n (TPi ty sc) = TPi (go n ty) (go (FS n) sc)
go n (TApp f x) = TApp (go n f) (go n x)
go n (TVar i) = if weaken i < n
then TVar (weaken i)
else TVar (FS i)
public export
weakTr2 : Term n -> Term (2+n)

47
src/Tests.idr
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@ -22,11 +22,11 @@ test_id = typecheck (TLam (TLam (TVar 0)))
{- λA. λB. λf. λx. f x : ∏ (A : Type) ∏ (B : A → Type) ∏ (f : ∏ (x : A) B x) ∏ (x : A) B x -}
test_app : Either String (Bool, List String)
test_app = typecheck (TLam (TLam (TLam (TLam (TApp (TVar 1) (TVar 0))))))
test_app = typecheck (TLam (TLam (TLam (TLam (TVar 1 `TApp` TVar 0)))))
(TPi TType
(TPi (TPi (TVar 0) TType)
(TPi (TPi (TVar 1) (TApp (TVar 1) (TVar 0)))
(TPi (TVar 2) (TApp (TVar 2) (TVar 0))))))
(TPi (TPi (TVar 1) (TVar 1 `TApp` TVar 0))
(TPi (TVar 2) (TVar 2 `TApp` TVar 0)))))
{- λf. λx. f x ≃ λf. λx. (λy. f y) x -}
eta_test : Either String (Bool, List String)
@ -34,14 +34,14 @@ eta_test = resolve action
where
action : PI Bool
action = do
x <- eval ctx0 (TLam (TLam (TApp (TVar 1) (TVar 0))))
y <- eval ctx0 (TLam (TLam (TApp (TLam (TApp (TVar 2) (TVar 0))) (TVar 0))))
x <- eval ctx0 (TLam (TLam (TVar 1 `TApp` TVar 0)))
y <- eval ctx0 (TLam (TLam (TLam (TVar 2 `TApp` TVar 0) `TApp` TVar 0)))
convert x y
addition : Term 0
addition = TNatInd (TLam (TPi TNat TNat))
(TLam (TVar 0))
(TLam {-n-} (TLam {-n+-} (TLam {-m-} (TSuc (TApp (TVar 1) (TVar 0))))))
(TLam {-n-} (TLam {-n+-} (TLam {-m-} (TSuc (TVar 1 `TApp` TVar 0)))))
addition_type : Either String (Bool, List String)
addition_type = typecheck addition (TPi TNat (TPi TNat TNat))
@ -80,3 +80,38 @@ unit_test = typecheck TStar TTop
absurd_test : Either String (Bool, List String)
absurd_test = typecheck (TLam (TBotInd (TLam (TVar 1)))) (TPi TType (TPi TBot (TVar 1)))
pr1ty : Term 0
pr1ty = TPi TType {- A : Type -}
(TPi (TPi (TVar 0) TType) {- B : A → Type -}
(TPi (TSigma (TVar 1) (TVar 0)) {- Σ A B -}
(TVar 2)))
pr1 : Term 0
pr1 = TLam {- A : Type -}
(TLam {- B : A → Type -}
(TSigInd (TVar 1) (TVar 0) (TLam {-ΣAB-} (TVar 2)) (TLam (TLam (TVar 1)))))
pr1_test : Either String (Bool, List String)
pr1_test = typecheck pr1 pr1ty
pr2ty : Term 0
pr2ty = TPi TType {- A : Type -}
(TPi (TPi (TVar 0) TType) {- B : A → Type -}
(TPi (TSigma (TVar 1) (TVar 0)) {- Σ A B -}
(TVar 1 `TApp` (TSigInd (TVar 2) (TVar 1) (TLam (TVar 3)) (TLam (TLam (TVar 1))) `TApp` TVar 0))))
pr2 : Term 0
pr2 = TLam {- A : Type -}
(TLam {- B : A → Type -}
(TSigInd (TVar 1)
(TVar 0)
(TLam {-ΣAB-}
(TVar 1 `TApp` (TSigInd (TVar 2) (TVar 1) (TLam (TVar 3)) (TLam (TLam (TVar 1))) `TApp` TVar 0)))
(TLam (TLam (TVar 0)))))
pr2ty_test : Either String (Bool, List String)
pr2ty_test = typecheck pr2ty TType
pr2_test : Either String (Bool, List String)
pr2_test = typecheck pr2 pr2ty

5
src/Value.idr
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@ -20,10 +20,14 @@ mutual
VNat : Value
VNatTr : Nat -> Value
VPair : Value -> Value -> Value
VGen : Nat -> Value
VApp : Value -> Value -> Value
VClos : Ctx n -> Term n -> Value
infixl 2 `VApp`
public export
Ctx : Index -> Type
Ctx i = Vect i Value
@ -39,6 +43,7 @@ Show Value where
show VStar = "VStar"
show VBot = "VBot"
show VNat = "VNat"
show (VPair a b) = "VPair (" ++ show a ++ ") (" ++ show b ++ ")"
show (VNatTr n) = "V" ++ show n
show (VGen i) = "VGen " ++ show i
show (VApp f x) = "VApp (" ++ show f ++ ") (" ++ show x ++ ")"