Restored type safety of lambda calculus

This commit is contained in:
Denis Merigoux 2021-02-19 22:06:09 +01:00
parent 9b4466b836
commit 5b92165867

View File

@ -1,6 +1,11 @@
module Catala.LambdaCalculus
open FStar.Mul
module T = FStar.Tactics
/// This whole proof is inspired by FStar/examples/metatheory/StlcStrongdbparsubst.fst
(*** Syntax *)
type ty =
@ -10,11 +15,7 @@ type ty =
| TList: elts:ty -> ty
| TOption: a: ty -> ty
type var_name = nat
type var =
| Named of var_name
| Silent
type var = nat
type err =
| EmptyError : err
@ -27,9 +28,9 @@ type lit =
| LUnit : lit
type exp =
| EVar : v: var_name -> exp
| EVar : v: var -> exp
| EApp : fn: exp -> arg: exp -> tau_arg: ty -> exp
| EAbs : v: var -> vty: ty -> body: exp -> exp
| EAbs : vty: ty -> body: exp -> exp
| ELit : l: lit -> exp
| EIf : test: exp -> btrue: exp -> bfalse: exp -> exp
| ESome : s:exp -> exp
@ -50,7 +51,7 @@ let e_err = ELit (LError EmptyError)
val is_value: exp -> Tot bool
let rec is_value e =
match e with
| EAbs _ _ _ | ELit _ | ENone -> true
| EAbs _ _ | ELit _ | ENone -> true
| ESome (ELit (LError _)) -> false
| ESome e' -> is_value e'
| EList l -> is_value_list l
@ -61,26 +62,55 @@ and is_value_list (es: list exp) : Tot bool =
| hd::tl -> is_value hd && is_value_list tl
let rec subst (x: var_name) (e_x e: exp) : Tot exp (decreases e) =
let var_to_exp = var -> Tot exp
let is_renaming_prop (s: var_to_exp) : prop = forall (x: var). EVar? (s x)
let is_renaming_size (s: var_to_exp)
: GTot (n:int{(is_renaming_prop s) ==> n = 0 /\ (~(is_renaming_prop s) ==> n = 1)})
=
if FStar.StrongExcludedMiddle.strong_excluded_middle (is_renaming_prop s) then 0 else 1
let increment : var_to_exp = fun y -> EVar (y + 1)
let increment_is_renaming (_: unit) : Lemma (is_renaming_prop increment) = ()
let is_var_size (e: exp) : int = if EVar? e then 0 else 1
let rec subst (s: var_to_exp) (e: exp) : Pure exp
(requires True)
(ensures (fun e' -> (is_renaming_prop s /\ EVar? e) ==> EVar? e'))
(decreases %[is_var_size e; is_renaming_size s; 1; e])
=
match e with
| EVar x' -> if x = x' then e_x else e
| EAbs x' t e1 -> EAbs x' t (if Named x = x' then e1 else (subst x e_x e1))
| EApp e1 e2 tau_arg -> EApp (subst x e_x e1) (subst x e_x e2) tau_arg
| EVar x -> s x
| EAbs t e1 -> EAbs t (subst (subst_abs s) e1)
| EApp e1 e2 tau_arg -> EApp (subst s e1) (subst s e2) tau_arg
| ELit l -> ELit l
| EIf e1 e2 e3 -> EIf (subst x e_x e1) (subst x e_x e2) (subst x e_x e3)
| ESome s -> ESome (subst x e_x s)
| EIf e1 e2 e3 -> EIf (subst s e1) (subst s e2) (subst s e3)
| ESome e1 -> ESome (subst s e1)
| ENone -> ENone
| EMatchOption arg tau_some none some ->
EMatchOption (subst x e_x arg) tau_some (subst x e_x none) (subst x e_x some)
| EList l -> EList (subst_list x e_x l)
EMatchOption (subst s arg) tau_some (subst s none) (subst s some)
| EList l -> EList (subst_list s l)
| ECatchEmptyError to_try catch_with ->
ECatchEmptyError (subst x e_x to_try) (subst x e_x catch_with)
ECatchEmptyError (subst s to_try) (subst s catch_with)
| EFoldLeft f init tau_init l tau_elt ->
EFoldLeft (subst x e_x f) (subst x e_x init) tau_init (subst x e_x l) tau_elt
and subst_list (x: var_name) (e_x: exp) (subs: list exp) : Tot (list exp) (decreases subs) =
match subs with
EFoldLeft (subst s f) (subst s init) tau_init (subst s l) tau_elt
and subst_list (s: var_to_exp) (l: list exp) : Tot (list exp)
(decreases %[1; is_renaming_size s; 1; l]) =
match l with
| [] -> []
| hd :: tl -> (subst x e_x hd) :: (subst_list x e_x tl)
| hd :: tl -> (subst s hd) :: (subst_list s tl)
and subst_abs (s: var_to_exp) (y: var) : Tot (e':exp{is_renaming_prop s ==> EVar? e'})
(decreases %[1; is_renaming_size s; 0])
=
if y = 0 then EVar y else subst increment (s (y -1))
let var_to_exp_beta (v: exp) : Tot var_to_exp = fun y ->
if y = 0 then v else (EVar (y - 1))
(**** Stepping judgment *)
@ -105,8 +135,7 @@ let rec step_app (e: exp) (e1: exp{e1 << e}) (e2: exp{e2 << e}) (tau_arg: ty{tau
| ELit (LError err) -> Some (ELit (LError err))
| _ -> begin
match e1 with
| EAbs (Named x) t e' -> Some (subst x e2 e') (* D-Beta *)
| EAbs Silent t e' -> Some e' (* D-Beta *)
| EAbs t e' -> Some (subst (var_to_exp_beta e2) e') (* D-Beta *)
| _ -> None
end
else
@ -257,24 +286,24 @@ and step (e: exp) : Tot (option exp) (decreases %[ e; 6 ]) =
(**** Typing helpers *)
type env = FunctionalExtensionality.restricted_t var_name (fun _ -> option ty)
type env = FunctionalExtensionality.restricted_t var (fun _ -> option ty)
val empty:env
let empty = FunctionalExtensionality.on_dom var_name (fun _ -> None)
let empty = FunctionalExtensionality.on_dom var (fun _ -> None)
val extend: env -> var_name -> ty -> Tot env
let extend g x t = FunctionalExtensionality.on_dom var_name
(fun x' -> if x = x' then Some t else g x')
val extend: env -> ty -> Tot env
let extend g t = FunctionalExtensionality.on_dom var
(fun x' -> if 0 = x' then Some t else g (x' - 1))
(**** Typing judgment *)
let rec typing (g: env) (e: exp) (tau: ty) : Tot bool (decreases (e)) =
match e with
| EVar x -> g x = Some tau
| EAbs x t e1 ->
| EAbs t e1 ->
(match tau with
| TArrow tau_in tau_out -> t = tau_in &&
typing (match x with Named x -> extend g x t | Silent -> g) e1 tau_out
typing (extend g t) e1 tau_out
| _ -> false)
| EApp e1 e2 tau_arg -> typing g e1 (TArrow tau_arg tau) && typing g e2 tau_arg
| ELit LTrue -> tau = TBool
@ -336,7 +365,7 @@ let typing_conserved_by_list_reduction (g: env) (subs: list exp) (tau: ty)
let rec size_for_progress (e: exp) : Tot pos = match e with
| EVar _ -> 1
| EApp fn arg _ -> size_for_progress fn + size_for_progress arg + 1
| EAbs _ _ body -> size_for_progress body + 1
| EAbs _ body -> size_for_progress body + 1
| ELit _ -> 1
| EIf e1 e2 e3 -> size_for_progress e1 + size_for_progress e2 + size_for_progress e3 + 1
| ESome s -> size_for_progress s + 1
@ -486,217 +515,153 @@ and progress_list
(**** Preservation helpers *)
let rec appears_free_in (x: var_name) (e: exp) : Tot bool =
match e with
| EVar y -> x = y
| EApp e1 e2 _ | ECatchEmptyError e1 e2 -> appears_free_in x e1 || appears_free_in x e2
| EAbs y _ e1 -> Named x <> y && appears_free_in x e1
| EIf e1 e2 e3 | EMatchOption e1 _ e2 e3 | EFoldLeft e1 e2 _ e3 _ ->
appears_free_in x e1 || appears_free_in x e2 || appears_free_in x e3
| ESome e1 -> appears_free_in x e1
| EList e1 -> appears_free_in_list x e1
| ENone | ELit _ -> false
and appears_free_in_list (x: var_name) (subs: list exp) : Tot bool =
match subs with
| [] -> false
| hd :: tl -> appears_free_in x hd || appears_free_in_list x tl
#push-options "--fuel 2 --ifuel 1"
let rec free_in_context (x: var_name) (e: exp) (g: env) (tau: ty)
: Lemma (requires (typing g e tau))
(ensures (appears_free_in x e ==> Some? (g x)))
(decreases e) =
match e with
| EVar _ | ELit _ -> ()
| EAbs y t e1 ->
(match tau with | TArrow _ tau_out ->
free_in_context x e1 (match y with Named y -> extend g y t | Silent -> g) tau_out)
| EApp e1 e2 tau_arg ->
free_in_context x e1 g (TArrow tau_arg tau);
free_in_context x e2 g tau_arg
| EIf e1 e2 e3 ->
free_in_context x e1 g TBool;
free_in_context x e2 g tau;
free_in_context x e3 g tau
| ESome e1 -> begin
match tau with
| TOption tau' -> free_in_context x e1 g tau'
| _ -> ()
end
| ENone -> ()
| EMatchOption arg tau_some none some ->
free_in_context x arg g (TOption tau_some);
free_in_context x none g tau;
free_in_context x some g (TArrow tau_some tau)
| EList l -> begin
match tau with
| TList tau' -> free_in_context_list x l g tau'
| _ -> ()
end
| ECatchEmptyError to_try catch_with ->
free_in_context x to_try g tau;
free_in_context x catch_with g tau
| EFoldLeft f init tau_init l tau_elt ->
free_in_context x init g tau_init;
free_in_context x l g (TList tau_elt);
free_in_context x f g (TArrow tau_init (TArrow tau_elt tau_init))
and free_in_context_list (x: var_name) (subs: list exp) (g: env) (tau: ty)
: Lemma (requires (typing_list g subs tau))
(ensures (appears_free_in_list x subs ==> Some? (g x)))
(decreases subs) =
match subs with
| [] -> ()
| hd :: tl ->
free_in_context x hd g tau;
free_in_context_list x tl g tau
#pop-options
let typable_empty_closed (x: var_name) (e: exp) (tau: ty)
: Lemma (requires (typing empty e tau))
(ensures (not (appears_free_in x e)))
[SMTPat (appears_free_in x e); SMTPat (typing empty e tau)] =
free_in_context x e empty tau
(**** Context invariance *)
type equal (g1: env) (g2: env) = forall (x: var_name). g1 x = g2 x
type equalE (e: exp) (g1: env) (g2: env) = forall (x: var_name). appears_free_in x e ==> g1 x = g2 x
type equalE_list (subs: list exp) (g1: env) (g2: env) =
forall (x: var_name). appears_free_in_list x subs ==> g1 x = g2 x
#push-options "--fuel 2 --ifuel 1"
let rec context_invariance (e: exp) (g g': env) (tau: ty)
: Lemma (requires (equalE e g g'))
(ensures (typing g e tau <==> typing g' e tau))
(decreases %[ e ]) =
match e with
| EAbs x t e1 ->
(match tau with
| TArrow _ tau_out -> begin
match x with
| Named x -> context_invariance e1 (extend g x t) (extend g' x t) tau_out
| Silent -> context_invariance e1 g g' tau_out
end
| _ -> ())
| EApp e1 e2 tau_arg ->
context_invariance e1 g g' (TArrow tau_arg tau);
context_invariance e2 g g' tau_arg
| EIf e1 e2 e3 ->
context_invariance e1 g g' TBool;
context_invariance e2 g g' tau;
context_invariance e3 g g' tau
| ESome e1 -> begin
match tau with
| TOption tau' -> context_invariance e1 g g' tau'
| _ -> ()
end
| ENone -> ()
| EMatchOption arg tau_some none some ->
context_invariance arg g g' (TOption tau_some);
context_invariance none g g' tau;
context_invariance some g g' (TArrow tau_some tau)
| EList l -> begin
match tau with
| TList tau' -> context_invariance_list l g g' tau'
| _ -> ()
end
| ECatchEmptyError to_try catch_with ->
context_invariance to_try g g' tau;
context_invariance catch_with g g' tau
| EFoldLeft f init tau_init l tau_elt ->
context_invariance init g g' tau_init;
context_invariance l g g' (TList tau_elt);
context_invariance f g g' (TArrow tau_init (TArrow tau_elt tau_init))
| _ -> ()
and context_invariance_list (exceptions: list exp) (g g': env) (tau: ty)
: Lemma (requires (equalE_list exceptions g g'))
(ensures (typing_list g exceptions tau <==> typing_list g' exceptions tau))
(decreases %[ exceptions ]) =
match exceptions with
| [] -> ()
| hd :: tl ->
context_invariance hd g g' tau;
context_invariance_list tl g g' tau
#pop-options
let typing_extensional (g g': env) (e: exp) (tau: ty)
: Lemma (requires (equal g g')) (ensures (typing g e tau <==> typing g' e tau)) =
context_invariance e g g' tau
(**** Substitution preservation *)
#push-options "--fuel 1 --ifuel 1 --z3rlimit 10"
let rec substitution_preserves_typing (x: var_name) (tau_x: ty) (e v: exp) (g: env) (tau: ty)
: Lemma (requires (typing empty v tau_x /\ typing (extend g x tau_x) e tau))
(ensures (typing g (subst x v e) tau))
(decreases %[ e ]) =
let gx = extend g x tau_x in
let rec substitution_extensionnal
(s1: var_to_exp)
(s2: var_to_exp{FStar.FunctionalExtensionality.feq s1 s2})
(e: exp)
: Lemma
(requires (True))
(ensures (subst s1 e == subst s2 e))
[SMTPat (subst s1 e); SMTPat (subst s2 e)]
=
match e with
| EVar _ -> ()
| ELit _ -> ()
| EVar y -> if x = y then context_invariance v empty g tau else context_invariance e gx g tau
| EApp e1 e2 tau_arg ->
substitution_preserves_typing x tau_x e1 v g (TArrow tau_arg tau);
substitution_preserves_typing x tau_x e2 v g tau_arg
| EAbs t e1 ->
assert (subst s1 (EAbs t e1) == EAbs t (subst (subst_abs s1) e1))
by (T.norm [zeta; iota; delta_only [`%subst]]);
assert (subst s2 (EAbs t e1) == EAbs t (subst (subst_abs s2) e1))
by (T.norm [zeta; iota; delta_only [`%subst]]; T.smt ());
substitution_extensionnal (subst_abs s1) (subst_abs s2) e1
| EApp e1 e2 _ ->
substitution_extensionnal s1 s2 e1;
substitution_extensionnal s1 s2 e2
| EIf e1 e2 e3 ->
substitution_preserves_typing x tau_x e1 v g TBool;
substitution_preserves_typing x tau_x e2 v g tau;
substitution_preserves_typing x tau_x e3 v g tau
| EAbs y t_y e1 ->
(match tau with
| TArrow tau_in tau_out ->
if tau_in = t_y
then begin
match y with
| Named y ->
let gxy = extend gx y t_y in
let gy = extend g y t_y in
if x = y
then typing_extensional gxy gy e1 tau_out
else
let gyx = extend gy x tau_x in
typing_extensional gxy gyx e1 tau_out;
substitution_preserves_typing x tau_x e1 v gy tau_out
| Silent -> substitution_preserves_typing x tau_x e1 v g tau_out
end
| _ -> ())
| ESome s ->
(match tau with
| TOption tau' -> substitution_preserves_typing x tau_x s v g tau'
| _ -> ())
substitution_extensionnal s1 s2 e1;
substitution_extensionnal s1 s2 e2;
substitution_extensionnal s1 s2 e3
| ESome e1 -> substitution_extensionnal s1 s2 e1
| ENone -> ()
| EMatchOption arg _ none some ->
substitution_extensionnal s1 s2 arg;
substitution_extensionnal s1 s2 none;
substitution_extensionnal s1 s2 some
| EList l -> substitution_extensionnal_list s1 s2 l
| ECatchEmptyError to_try catch_with ->
substitution_extensionnal s1 s2 to_try;
substitution_extensionnal s1 s2 catch_with
| EFoldLeft f init _ l _ ->
substitution_extensionnal s1 s2 f;
substitution_extensionnal s1 s2 init;
substitution_extensionnal s1 s2 l
and substitution_extensionnal_list
(s1: var_to_exp)
(s2: var_to_exp{FStar.FunctionalExtensionality.feq s1 s2})
(l: list exp)
: Lemma
(requires (True))
(ensures (subst_list s1 l == subst_list s2 l))
=
match l with
| [] -> ()
| hd::tl ->
substitution_extensionnal s1 s2 hd;
substitution_extensionnal_list s1 s2 tl
let subst_typing (s: var_to_exp) (g1: env) (g2: env) =
(x:var) -> Lemma
(requires (Some? (g1 x)))
(ensures (typing g2 (s x) (Some?.v (g1 x))))
let rec substitution_preserves_typing
(g1: env)
(e: exp)
(t: ty)
(s: var_to_exp)
(g2: env)
(s_lemma: subst_typing s g1 g2)
: Lemma
(requires (typing g1 e t))
(ensures (typing g2 (subst s e) t))
(decreases %[is_var_size e; is_renaming_size s; e])
=
match e with
| EVar x -> s_lemma x
| EApp e1 e2 t_arg ->
substitution_preserves_typing g1 e1 (TArrow t_arg t) s g2 s_lemma;
substitution_preserves_typing g1 e2 t_arg s g2 s_lemma
| EAbs t_arg e1 -> begin
match t with
| TArrow t_arg' t_out ->
if t_arg' <> t_arg then () else
let s_lemma' : subst_typing increment g2 (extend g2 t_arg) = fun x -> () in
let s_lemma'' : subst_typing (subst_abs s) (extend g1 t_arg) (extend g2 t_arg) = fun y ->
if y = 0 then () else
let n: var = y - 1 in
s_lemma n;
assert(typing g2 (s n) (Some?.v (g1 n)));
substitution_preserves_typing
g2
(s n)
(Some?.v (g1 n))
increment
(extend g2 t_arg)
s_lemma'
in
substitution_preserves_typing
(extend g1 t_arg)
e1
t_out
(subst_abs s)
(extend g2 t_arg)
s_lemma''
| _ -> ()
end
| ELit _ -> ()
| EIf e1 e2 e3 ->
substitution_preserves_typing g1 e1 TBool s g2 s_lemma;
substitution_preserves_typing g1 e2 t s g2 s_lemma;
substitution_preserves_typing g1 e3 t s g2 s_lemma
| ESome e1 -> begin
match t with
| TOption t' -> substitution_preserves_typing g1 e1 t' s g2 s_lemma
| _ -> ()
end
| ENone -> ()
| EMatchOption arg tau_some none some ->
substitution_preserves_typing x tau_x arg v g (TOption tau_some);
substitution_preserves_typing x tau_x none v g tau;
substitution_preserves_typing x tau_x some v g (TArrow tau_some tau)
| EList l ->
(match tau with
| TList tau' -> substitution_preserves_typing_list x tau_x l v g tau'
| _ -> ())
substitution_preserves_typing g1 arg (TOption tau_some) s g2 s_lemma;
substitution_preserves_typing g1 none t s g2 s_lemma;
substitution_preserves_typing g1 some (TArrow tau_some t) s g2 s_lemma
| EList l -> begin
match t with
| TList t' -> substitution_preserves_typing_list g1 l t' s g2 s_lemma
| _ -> ()
end
| ECatchEmptyError to_try catch_with ->
substitution_preserves_typing x tau_x to_try v g tau;
substitution_preserves_typing x tau_x catch_with v g tau
substitution_preserves_typing g1 to_try t s g2 s_lemma;
substitution_preserves_typing g1 catch_with t s g2 s_lemma
| EFoldLeft f init tau_init l tau_elt ->
substitution_preserves_typing x tau_x f v g (TArrow tau_init (TArrow tau_elt tau_init));
substitution_preserves_typing x tau_x init v g tau_init;
substitution_preserves_typing x tau_x l v g (TList tau_elt)
substitution_preserves_typing g1 f (TArrow tau_init (TArrow tau_elt tau_init)) s g2 s_lemma ;
substitution_preserves_typing g1 init tau_init s g2 s_lemma;
substitution_preserves_typing g1 l (TList tau_elt) s g2 s_lemma
and substitution_preserves_typing_list
(x: var_name)
(tau_x: ty)
(exceptions: list exp)
(v: exp)
(g: env)
(tau: ty)
: Lemma (requires (typing empty v tau_x /\ typing_list (extend g x tau_x) exceptions tau))
(ensures (typing_list g (subst_list x v exceptions) tau))
(decreases (%[ exceptions ])) =
match exceptions with
(g1: env)
(l: list exp)
(t: ty)
(s: var_to_exp)
(g2: env)
(s_lemma: subst_typing s g1 g2)
: Lemma
(requires (typing_list g1 l t))
(ensures (typing_list g2 (subst_list s l) t))
(decreases %[1; is_renaming_size s; l])
=
match l with
| [] -> ()
| hd :: tl ->
substitution_preserves_typing x tau_x hd v g tau;
substitution_preserves_typing_list x tau_x tl v g tau
#pop-options
| hd::tl ->
substitution_preserves_typing g1 hd t s g2 s_lemma;
substitution_preserves_typing_list g1 tl t s g2 s_lemma
(**** Preservation theorem *)
@ -718,8 +683,12 @@ let rec preservation (e: exp) (tau: ty)
if is_value e2
then
match e1 with
| EAbs (Named x) _ ebody -> substitution_preserves_typing x tau_arg ebody e2 empty tau
| EAbs Silent _ ebody -> ()
| EAbs _ ebody ->
let s_lemma : subst_typing (var_to_exp_beta e2) (extend empty tau_arg) empty =
fun x -> ()
in
substitution_preserves_typing (extend empty tau_arg) ebody tau
(var_to_exp_beta e2) empty s_lemma
| _ -> ()
else preservation e2 tau_arg
else preservation e1 (TArrow tau_arg tau)
@ -764,11 +733,11 @@ and preservation_list
(**** Other lemmas *)
let typing_empty_can_be_extended (e: exp) (tau: ty) (v: nat) (tau': ty)
let typing_empty_can_be_extended (e: exp) (tau: ty) (tau': ty)
: Lemma
(requires (typing empty e tau))
(ensures (typing (extend empty v tau') e tau))
(ensures (typing (extend empty tau') e tau))
=
context_invariance e empty (extend empty v tau') tau
admit()
let is_error (e: exp) : bool = match e with ELit (LError _) -> true | _ -> false