2021-02-07 18:10:01 +03:00
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module Catala.Translation
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module L = Catala.LambdaCalculus
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module D = Catala.DefaultCalculus
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2021-02-08 00:38:04 +03:00
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(*** Translation definitions *)
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2021-02-08 12:17:38 +03:00
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(**** Helpers *)
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2021-02-07 18:10:01 +03:00
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let rec translate_ty (ty: D.ty) : Tot L.ty = match ty with
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| D.TBool -> L.TBool
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| D.TUnit -> L.TUnit
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| D.TArrow t1 t2 -> L.TArrow (translate_ty t1) (translate_ty t2)
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let translate_lit (l: D.lit) : Tot L.lit = match l with
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| D.LTrue -> L.LTrue
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| D.LFalse -> L.LFalse
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| D.LUnit -> L.LUnit
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| D.LEmptyError -> L.LError L.EmptyError
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| D.LConflictError -> L.LError L.ConflictError
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let process_exceptions_f (tau: L.ty) : Tot L.exp =
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let a = 0 in
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let e = 1 in
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let e' = 2 in
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let a' = 3 in
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let e'' = 4 in
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L.EAbs (L.Named a) (L.TOption tau) (L.EAbs (L.Named e) (L.TArrow L.TUnit tau) (
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L.EApp (L.EAbs (L.Named e') (L.TOption tau) (
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L.EMatchOption (L.EVar a) tau
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(L.EVar e')
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(L.EAbs (L.Named a') tau (
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L.EMatchOption (L.EVar e') tau
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(L.EVar a)
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(L.EAbs (L.Named e'') tau (L.ELit (L.LError L.ConflictError)))
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))
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))
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(L.ECatchEmptyError (L.ESome (L.EApp (L.EVar e) (L.ELit L.LUnit) L.TUnit)) L.ENone)
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(L.TOption tau)
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))
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let typ_process_exceptions_f (tau: L.ty)
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: Lemma (L.typing L.empty (process_exceptions_f tau)
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(L.TArrow (L.TOption tau) (L.TArrow (L.TArrow L.TUnit tau) (L.TOption tau))))
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=
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2021-02-07 21:35:00 +03:00
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assert_norm(L.typing L.empty (process_exceptions_f tau)
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(L.TArrow (L.TOption tau) (L.TArrow (L.TArrow L.TUnit tau) (L.TOption tau))))
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2021-02-07 18:10:01 +03:00
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2021-02-08 12:17:38 +03:00
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(**** Main translation *)
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2021-02-07 18:10:01 +03:00
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let rec translate_exp (e: D.exp) : Tot L.exp = match e with
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| D.EVar x -> L.EVar x
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| D.EApp e1 e2 tau_arg ->
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L.EApp (translate_exp e1) (translate_exp e2) (translate_ty tau_arg)
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| D.EAbs x ty body -> L.EAbs (L.Named x) (translate_ty ty) (translate_exp body)
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| D.ELit l -> L.ELit (translate_lit l)
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| D.EIf e1 e2 e3 -> L.EIf
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(translate_exp e1)
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(translate_exp e2)
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(translate_exp e3)
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| D.EDefault exceptions just cons tau ->
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let tau' = translate_ty tau in
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L.EMatchOption
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(L.EFoldLeft
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(process_exceptions_f tau')
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L.ENone (L.TOption tau')
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(L.EList (translate_exp_list exceptions)) (L.TArrow L.TUnit tau'))
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tau'
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(L.EIf
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(translate_exp just)
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(translate_exp cons)
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(L.ELit (L.LError L.EmptyError)))
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(L.EAbs (L.Named 0) tau' (L.EVar 0))
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and translate_exp_list (l: list D.exp) : Tot (list L.exp) =
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match l with
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| [] -> []
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| hd::tl -> (L.EAbs L.Silent L.TUnit (translate_exp hd))::(translate_exp_list tl)
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let translate_env (g: D.env) : Tot L.env =
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FunctionalExtensionality.on_dom L.var_name
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(fun v -> match g v with None -> None | Some t -> Some (translate_ty t))
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(*** Typing preservation *)
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(**** Helpers and lemmas *)
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let extend_translate_commute (g: D.env) (x: D.var) (tau: D.ty)
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: Lemma (L.extend (translate_env g) x (translate_ty tau) == translate_env (D.extend g x tau))
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=
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FunctionalExtensionality.extensionality L.var_name (fun _ -> option L.ty)
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(L.extend (translate_env g) x (translate_ty tau))
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(translate_env (D.extend g x tau))
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2021-02-08 14:24:51 +03:00
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let translate_empty_is_empty () : Lemma (translate_env D.empty == L.empty) =
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FunctionalExtensionality.extensionality L.var_name (fun _ -> option L.ty)
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(translate_env D.empty)
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L.empty
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2021-02-08 00:38:04 +03:00
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(**** Typing preservation theorem *)
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#push-options "--fuel 1 --ifuel 1 --z3rlimit 30"
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let rec translation_preserves_typ (g: D.env) (e: D.exp) (tau: D.ty) : Lemma
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(requires (D.typing g e tau))
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(ensures (L.typing (translate_env g) (translate_exp e) (translate_ty tau)))
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(decreases %[e; 1])
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=
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match e with
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| D.EVar _ -> ()
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| D.EApp e1 e2 tau_arg ->
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translation_preserves_typ g e1 (D.TArrow tau_arg tau);
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translation_preserves_typ g e2 tau_arg
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| D.EAbs x tau_arg body -> begin
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match tau with
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| D.TArrow tau_in tau_out ->
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if tau_in = tau_arg then begin
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translation_preserves_typ (D.extend g x tau_in) body tau_out;
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extend_translate_commute g x tau_in
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end else ()
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| _ -> ()
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end
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| D.ELit _ -> ()
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| D.EIf e1 e2 e3 ->
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translation_preserves_typ g e1 D.TBool;
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translation_preserves_typ g e2 tau;
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translation_preserves_typ g e3 tau
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| D.EDefault exceptions just cons tau_out ->
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if tau = tau_out then begin
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let tau' = translate_ty tau in
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translation_preserves_typ_exceptions g e exceptions tau;
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typ_process_exceptions_f tau';
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translation_preserves_typ g just D.TBool;
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translation_preserves_typ g cons tau;
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let result_exp = L.EMatchOption
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(L.EFoldLeft
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(process_exceptions_f tau')
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L.ENone (L.TOption tau')
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(L.EList (translate_exp_list exceptions)) (L.TArrow L.TUnit tau'))
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tau'
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(L.EIf
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(translate_exp just)
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(translate_exp cons)
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(L.ELit (L.LError L.EmptyError)))
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(L.EAbs (L.Named 0) tau' (L.EVar 0))
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in
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let open FStar.Tactics in
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assert(L.typing (translate_env g) result_exp tau') by begin
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compute ();
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smt ()
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end
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end else ()
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and translation_preserves_typ_exceptions
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(g: D.env)
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(e: D.exp)
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(exceptions: list D.exp{exceptions << e})
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(tau: D.ty)
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: Lemma
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(requires (D.typing_list g exceptions tau))
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(ensures (L.typing_list
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(translate_env g)
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(translate_exp_list exceptions)
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(L.TArrow L.TUnit (translate_ty tau))))
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(decreases %[e; 0; exceptions])
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=
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match exceptions with
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| [] -> ()
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| hd::tl ->
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translation_preserves_typ g hd tau;
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translation_preserves_typ_exceptions g e tl tau;
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let g' = translate_env g in
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let hd' = translate_exp hd in
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let tl' = translate_exp_list tl in
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let tau' = translate_ty tau in
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let thunked_tau' = L.TArrow L.TUnit tau' in
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assert(L.typing_list g' tl' thunked_tau');
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assert(L.typing g' hd' tau');
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assert(L.typing g' (L.EAbs L.Silent L.TUnit hd') thunked_tau')
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2021-02-08 00:38:04 +03:00
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#pop-options
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2021-02-08 14:24:51 +03:00
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(*** Translation correctness *)
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(**** Helpers *)
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let rec l_step_rec (e: L.exp) (fuel: nat) : Tot (option L.exp) (decreases fuel) =
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match L.step e with
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| None -> None
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| Some e' -> if fuel = 0 then Some e' else l_step_rec e' (fuel - 1)
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let multiple_l_steps (e1: L.exp) (e2: L.exp) (n: nat) = l_step_rec e1 n = Some e2
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let not_l_value = e:L.exp{not (L.is_value e)}
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let step_lift_commute_non_value (f:(L.exp -> not_l_value)) (e: L.exp) =
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if L.is_value e then true else
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match L.step e with
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| None -> L.step (f e) = None
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| Some e' -> L.step (f e) = Some (f e')
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let is_stepping_agnostic_lift (f:(L.exp -> not_l_value)) = forall (e: L.exp).
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step_lift_commute_non_value f e
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let stepping_agnostic_lift = f:(L.exp -> not_l_value){is_stepping_agnostic_lift f}
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let rec l_values_dont_step (e: L.exp) : Lemma
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(requires (L.is_value e))
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(ensures (L.step e = None))
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(decreases %[e; 1])
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=
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match e with
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| L.EAbs _ _ _ -> ()
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| L.ELit _ -> ()
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| L.ENone -> ()
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| L.EList [] -> ()
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| L.EList l -> l_values_dont_step_list e l
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| _ -> ()
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and l_values_dont_step_list (e: L.exp) (l: list L.exp{l << e /\ Cons? l}) : Lemma
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(requires (L.is_value_list l))
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(ensures (L.step_list e l = (if l = [] then Some [] else None)))
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(decreases %[e; 0; l])
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=
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match l with
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| [hd] -> l_values_dont_step hd
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| hd::tl ->
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l_values_dont_step hd;
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l_values_dont_step_list e tl
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#push-options "--z3rlimit 50 --fuel 1 --ifuel 1"
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let rec lift_multiple_l_steps
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(e1: L.exp)
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(e2: L.exp)
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(n: nat)
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(f : stepping_agnostic_lift)
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: Lemma
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(requires (multiple_l_steps e1 e2 n))
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(ensures (multiple_l_steps (f e1) (f e2) n))
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(decreases n)
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=
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match L.step e1 with
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| None -> ()
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| Some e1' ->
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if L.is_value e1 then begin
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l_values_dont_step e1
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end else if n = 0 then
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assert(L.step (f e1) = Some (f e2))
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else lift_multiple_l_steps e1' e2 (n-1) f
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#pop-options
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2021-02-08 15:51:56 +03:00
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2021-02-08 14:24:51 +03:00
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(**** Main theorems *)
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let translation_correctness_value (e: D.exp) : Lemma
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((D.is_value e) <==> (L.is_value (translate_exp e)))
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= ()
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#push-options "--fuel 2 --ifuel 1 --z3rlimit 50"
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let rec translation_correctness_step (e: D.exp) : Pure nat
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(requires (Some? (D.step e)))
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(ensures (fun n -> multiple_l_steps (translate_exp e) (translate_exp (Some?.v (D.step e))) n))
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=
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let e' = translate_exp e in
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let stepped_e = Some?.v (D.step e) in
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let stepped_e' = translate_exp stepped_e in
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match e with
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| D.EVar _ -> 0
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| D.ELit _ -> 0
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| D.EAbs _ _ _ -> 0
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| D.EIf e1 e2 e3 ->
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if not (D.is_value e1) then begin
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let e1' = translate_exp e1 in
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let e2' = translate_exp e2 in
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let e3' = translate_exp e3 in
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let stepped_e1 = Some?.v (D.step e1) in
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let stepped_e1' = translate_exp stepped_e1 in
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let n_e1 = translation_correctness_step e1 in
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lift_multiple_l_steps e1' stepped_e1' n_e1 (fun e1' -> L.EIf e1' e2' e3');
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n_e1
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end else admit()
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| _ -> admit()
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(*** Wrap-up theorem *)
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let translation_correctness (e: D.exp) (tau: D.ty)
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: Lemma
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(requires (D.typing D.empty e tau))
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(ensures (
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let e' = translate_exp e in
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let tau' = translate_ty tau in
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L.typing L.empty e' tau' /\ begin
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if D.is_value e then L.is_value e' else begin
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D.progress e tau;
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let stepped_e = Some?.v (D.step e) in
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let stepped_e' = translate_exp stepped_e in
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exists (n:nat). multiple_l_steps e' stepped_e' n
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end
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end
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))
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=
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let e' = translate_exp e in
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let tau' = translate_ty tau in
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translation_preserves_typ D.empty e tau;
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translate_empty_is_empty ();
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if D.is_value e then translation_correctness_value e else begin
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D.progress e tau;
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let n = translation_correctness_step e in
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()
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end
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