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Type safety of default calculus restoed

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Denis Merigoux 2021-02-19 19:01:23 +01:00
parent 6662a07bfd
commit 9b4466b836
1 changed files with 86 additions and 2 deletions
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88
doc/formalization/Catala.DefaultCalculus.fst
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@ -3,6 +3,8 @@ module Catala.DefaultCalculus
open FStar.StrongExcludedMiddle
module T = FStar.Tactics
/// This whole proof is inspired by FStar/examples/metatheory/StlcStrongdbparsubst.fst
(*** Syntax *)
type ty =
@ -235,7 +237,7 @@ val empty:env
let empty = FunctionalExtensionality.on_dom var (fun _ -> None)
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')
let extend g t = FunctionalExtensionality.on_dom var (fun x' -> if 0 = x' then Some t else g (x' - 1))
(**** Typing judgment *)
@ -391,6 +393,83 @@ and substitution_extensionnal_list
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
| EDefault exceptions just cons tau ->
if tau <> t then () else
substitution_preserves_typing_list g1 exceptions t s g2 s_lemma;
substitution_preserves_typing g1 just TBool s g2 s_lemma;
substitution_preserves_typing g1 cons t s g2 s_lemma
and substitution_preserves_typing_list
(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 g1 hd t s g2 s_lemma;
substitution_preserves_typing_list g1 tl t s g2 s_lemma
(**** Preservation theorem *)
#push-options "--fuel 3 --ifuel 1 --z3rlimit 20"
@ -411,7 +490,12 @@ let rec preservation (e: exp) (tau: ty)
if is_value e2
then
match e1 with
| EAbs x _ ebody -> substitution_preserves_typing x tau_arg ebody e2 empty tau
| 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)