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Proof went until the end, now havee to remove admits

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This commit is contained in:
Denis Merigoux 2021-02-21 23:55:01 +01:00
parent b2d9407c2b
commit 1071f3232d
2 changed files with 176 additions and 55 deletions
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68
doc/formalization/Catala.DefaultCalculus.fst
View File

@ -89,6 +89,11 @@ type empty_count_result =
| OneNonEmpty of exp
| Conflict
type exceptions_count_result =
| NoStep
| IllFormed
| SomeStep of exp
let rec empty_count (acc: empty_count_result) (l: list exp) : Tot empty_count_result (decreases l) =
match l with
| [] -> acc
@ -178,14 +183,17 @@ and step_exceptions
(just: exp{just << e})
(cons: exp{cons << e})
(tau: ty)
: Tot (option exp) (decreases %[ e; 3 ]) =
if List.Tot.for_all (fun except -> is_value except) exceptions
: Tot exceptions_count_result (decreases %[ e; 3 ]) =
if List.Tot.for_all is_value exceptions
then
match empty_count AllEmpty exceptions with
| AllEmpty -> None
| OneNonEmpty e' -> Some e' (* D-DefaultOneException *)
| Conflict -> Some (ELit LConflictError) (* D-DefaultExceptionConflict *)
else step_exceptions_left_to_right e exceptions just cons tau
| AllEmpty -> NoStep
| OneNonEmpty e' -> SomeStep e' (* D-DefaultOneException *)
| Conflict -> SomeStep (ELit LConflictError) (* D-DefaultExceptionConflict *)
else
match step_exceptions_left_to_right e exceptions just cons tau with
| None -> IllFormed
| Some e' -> SomeStep e'
and step_default
(e: exp)
@ -195,22 +203,16 @@ and step_default
(tau: ty)
: Tot (option exp) (decreases %[ e; 4 ]) =
match step_exceptions e exceptions just cons tau with
| Some e' -> Some e'
| None ->
| IllFormed -> None
| SomeStep e' -> Some e'
| NoStep ->
if is_value just then
match just with
| ELit LConflictError -> Some c_err (* D-ContextConflictError *)
| ELit LEmptyError -> Some e_err (* D-ContextEmptyError *)
| _ ->
match just with
| ELit LTrue ->
if is_value cons
then Some cons
else
(match (step cons) with
| Some (ELit LConflictError) -> Some c_err (* D-ContextConflictError *)
| Some cons' -> Some (EDefault exceptions just cons' tau) (* D-DefaultTrueNoExceptions*)
| None -> None)
| ELit LTrue -> Some cons (* D-DefaultTrueNoExceptions*)
| ELit LFalse -> Some e_err (* D-DefaultFalseNoExceptions *)
| _ -> None
else
@ -282,7 +284,7 @@ let typing_conserved_by_list_reduction (g: env) (subs: list exp) (tau: ty)
(**** Progress theorem *)
#push-options "--fuel 2 --ifuel 1"
#push-options "--fuel 2 --ifuel 1 --z3rlimit 20"
let rec progress (e: exp) (tau: ty)
: Lemma (requires (typing empty e tau))
(ensures (is_value e \/ (Some? (step e))))
@ -305,14 +307,17 @@ and progress_default
(cons: exp{cons << e})
(tau: ty)
: Lemma (requires (
~(is_value e) /\
e == EDefault exceptions just cons tau /\ (typing empty e tau)
))
(ensures (Some? (step_default e exceptions just cons tau)))
(decreases %[ e; 2 ]) =
match step_exceptions e exceptions just cons tau with
| Some _ -> ()
| None ->
| IllFormed ->
assert(step_exceptions_left_to_right e exceptions just cons tau == None);
assert(~(List.Tot.for_all is_value exceptions));
progress_default_exceptions e exceptions just cons tau
| SomeStep _ -> ()
| NoStep ->
if is_value just then
(is_bool_value_cannot_be_abs empty just;
match just, cons with
@ -322,6 +327,27 @@ and progress_default
| ELit LEmptyError, _ -> ()
| ELit LConflictError, _ -> ())
else progress just TBool
and progress_default_exceptions
(e: exp)
(exceptions: list exp {exceptions << e})
(just: exp{just << e})
(cons: exp{cons << e})
(tau: ty)
: Lemma (requires (
typing_list empty exceptions tau /\
step_exceptions_left_to_right e exceptions just cons tau == None /\
~(List.Tot.for_all is_value exceptions)
))
(ensures (False))
(decreases %[ e; 1; exceptions ])
=
match exceptions with
| [] -> ()
| hd::tl ->
progress hd tau;
if is_value hd then begin
progress_default_exceptions e tl just cons tau
end else ()
#pop-options
(*** Preservation *)
@ -504,7 +530,7 @@ let rec preservation (e: exp) (tau: ty)
match empty_count AllEmpty exceptions with
| AllEmpty ->
begin if not (is_value just) then preservation just TBool else match just with
| ELit LTrue -> if not (is_value cons) then preservation cons tau
| ELit LTrue -> ()
| _ -> ()
end
| OneNonEmpty e' -> empty_count_preserves_type AllEmpty exceptions empty tau

163
doc/formalization/Catala.Translation.fst
View File

@ -810,14 +810,16 @@ let translation_correctness_exceptions_step
Some? (D.step de) /\
de == D.EDefault dexceptions djust dcons dtau /\
D.typing D.empty de dtau /\
D.step de == D.step_exceptions de dexceptions djust dcons dtau
(match D.step de, D.step_exceptions de dexceptions djust dcons dtau with
| Some de', D.SomeStep de'' -> de' == de''
| _ -> False)
))
(ensures (fun (n1, target_e, n2) ->
translation_preserves_empty_typ de dtau;
let lexceptions = translate_exp_list dexceptions in
let ljust = translate_exp djust in
let lcons = translate_exp dcons in
let Some de' = D.step_exceptions de dexceptions djust dcons dtau in
let D.SomeStep de' = D.step_exceptions de dexceptions djust dcons dtau in
let le' = translate_exp de' in
D.preservation de dtau;
let ltau = translate_ty dtau in
@ -835,7 +837,49 @@ let translation_correctness_exceptions_step
end else
translation_correctness_exceptions_left_to_right_step
de dexceptions djust dcons dtau L.ENone rec_lemma
#pop-options
let final_default_subexp
(tau: L.ty)
(just: typed_l_exp L.TBool) (cons: typed_l_exp tau)
: Tot (typed_l_exp tau)
=
L.EIf just cons (L.ELit (L.LError L.EmptyError))
#push-options "--fuel 2 --ifuel 1 --z3rlimit 150"
let translation_correctness_exceptions_no_exceptions_triggered
(de: D.exp)
(dexceptions: list D.exp {dexceptions << de})
(djust: D.exp{djust << de})
(dcons: D.exp{dcons << de})
(dtau: D.ty)
: Pure nat
(requires (
D.typing D.empty de dtau /\
D.typing_list D.empty dexceptions dtau /\
D.typing D.empty djust D.TBool /\
D.typing D.empty dcons dtau /\
List.Tot.for_all D.is_value dexceptions /\
(D.step de == D.step_default de dexceptions djust dcons dtau) /\
(D.step_exceptions de dexceptions djust dcons dtau == D.NoStep)
))
(ensures (fun n ->
translate_empty_is_empty ();
translation_preserves_typ_exceptions D.empty de dexceptions dtau;
translation_preserves_empty_typ djust D.TBool;
translation_preserves_empty_typ dcons dtau;
let lexceptions = translate_exp_list dexceptions in
let ljust = translate_exp djust in
let lcons = translate_exp dcons in
let ltau = translate_ty dtau in
let l_err : typed_l_exp ltau = L.ELit (L.LError L.ConflictError) in
build_default_translation_typing lexceptions L.ENone ljust lcons ltau L.empty;
take_l_steps ltau (build_default_translation lexceptions L.ENone ljust lcons ltau) n
== Some (final_default_subexp ltau ljust lcons)
))
(decreases dexceptions)
=
admit()
#pop-options
#push-options "--fuel 2 --ifuel 1 --z3rlimit 50"
@ -864,41 +908,41 @@ let rec translation_correctness_step (de: D.exp) (dtau: D.ty)
| D.ELit _ -> 0, le, 0
| D.EAbs _ _ -> 0, le, 0
| D.EIf de1 de2 de3 ->
let le1 = translate_exp de1 in
let le2 = translate_exp de2 in
let le3 = translate_exp de3 in
if not (D.is_value de1) then begin
let de1' = Some?.v (D.step de1) in
D.preservation de1 D.TBool;
translation_preserves_empty_typ de1 D.TBool;
translation_preserves_empty_typ de2 dtau;
translation_preserves_empty_typ de3 dtau;
translation_preserves_empty_typ de1' D.TBool;
let le1' : typed_l_exp L.TBool = translate_exp de1' in
let n1_e1, target_e1, n2_e1 = translation_correctness_step de1 D.TBool in
assert(take_l_steps L.TBool le1 n1_e1 == Some target_e1);
assert(take_l_steps L.TBool le1' n2_e1 == Some target_e1);
lift_multiple_l_steps L.TBool ltau le1 target_e1 n1_e1
(if_cond_lift ltau le2 le3);
lift_multiple_l_steps L.TBool ltau le1' target_e1 n2_e1
(if_cond_lift ltau le2 le3);
n1_e1, if_cond_lift ltau le2 le3 target_e1, n2_e1
end else (1, le', 0)
let le1 = translate_exp de1 in
let le2 = translate_exp de2 in
let le3 = translate_exp de3 in
if not (D.is_value de1) then begin
let de1' = Some?.v (D.step de1) in
D.preservation de1 D.TBool;
translation_preserves_empty_typ de1 D.TBool;
translation_preserves_empty_typ de2 dtau;
translation_preserves_empty_typ de3 dtau;
translation_preserves_empty_typ de1' D.TBool;
let le1' : typed_l_exp L.TBool = translate_exp de1' in
let n1_e1, target_e1, n2_e1 = translation_correctness_step de1 D.TBool in
assert(take_l_steps L.TBool le1 n1_e1 == Some target_e1);
assert(take_l_steps L.TBool le1' n2_e1 == Some target_e1);
lift_multiple_l_steps L.TBool ltau le1 target_e1 n1_e1
(if_cond_lift ltau le2 le3);
lift_multiple_l_steps L.TBool ltau le1' target_e1 n2_e1
(if_cond_lift ltau le2 le3);
n1_e1, if_cond_lift ltau le2 le3 target_e1, n2_e1
end else (1, le', 0)
| D.EApp de1 de2 dtau_arg ->
let le1 = translate_exp de1 in
let le2 = translate_exp de2 in
let ltau_arg = translate_ty dtau_arg in
if not (D.is_value de1) then begin
let de1' = Some?.v (D.step de1) in
let le1' = translate_exp de1' in
let n1_e1, target_e1, n2_e1 = translation_correctness_step de1 (D.TArrow dtau_arg dtau) in
assert(take_l_steps (L.TArrow ltau_arg ltau) le1 n1_e1 == Some target_e1);
assert(take_l_steps (L.TArrow ltau_arg ltau) le1' n2_e1 == Some target_e1);
lift_multiple_l_steps (L.TArrow ltau_arg ltau) ltau le1 target_e1 n1_e1
(app_f_lift ltau_arg ltau le2);
lift_multiple_l_steps (L.TArrow ltau_arg ltau) ltau le1' target_e1 n2_e1
(app_f_lift ltau_arg ltau le2);
n1_e1, app_f_lift ltau_arg ltau le2 target_e1, n2_e1
let de1' = Some?.v (D.step de1) in
let le1' = translate_exp de1' in
let n1_e1, target_e1, n2_e1 = translation_correctness_step de1 (D.TArrow dtau_arg dtau) in
assert(take_l_steps (L.TArrow ltau_arg ltau) le1 n1_e1 == Some target_e1);
assert(take_l_steps (L.TArrow ltau_arg ltau) le1' n2_e1 == Some target_e1);
lift_multiple_l_steps (L.TArrow ltau_arg ltau) ltau le1 target_e1 n1_e1
(app_f_lift ltau_arg ltau le2);
lift_multiple_l_steps (L.TArrow ltau_arg ltau) ltau le1' target_e1 n2_e1
(app_f_lift ltau_arg ltau le2);
n1_e1, app_f_lift ltau_arg ltau le2 target_e1, n2_e1
end else begin match de1 with
| D.ELit D.LConflictError -> 1, le', 0
| D.ELit D.LEmptyError -> 1, le', 0
@ -924,10 +968,61 @@ let rec translation_correctness_step (de: D.exp) (dtau: D.ty)
| D.EDefault dexceptions djust dcons dtau' ->
if dtau' <> dtau then 0, le', 0 else begin
match D.step_exceptions de dexceptions djust dcons dtau with
| Some _ ->
| D.SomeStep _ ->
translation_correctness_exceptions_step de dexceptions djust dcons dtau
(fun df tf -> translation_correctness_step df tf)
| None -> admit()
| D.NoStep ->
let n1_final =
translation_correctness_exceptions_no_exceptions_triggered
de dexceptions djust dcons dtau
in
let ltau = translate_ty dtau in
let ljust = translate_exp djust in
let lcons = translate_exp dcons in
let l_err = L.ELit (L.LError L.EmptyError) in
assert(take_l_steps ltau le n1_final = Some (final_default_subexp ltau ljust lcons));
if not (D.is_value djust) then begin
let djust' = Some?.v (D.step djust) in
D.preservation djust D.TBool;
translation_preserves_empty_typ djust D.TBool;
translation_preserves_empty_typ dcons dtau;
translation_preserves_empty_typ djust' D.TBool;
let ljust' : typed_l_exp L.TBool = translate_exp djust' in
let n1_just, target_just, n2_just = translation_correctness_step djust D.TBool in
assert(take_l_steps L.TBool ljust n1_just == Some target_just);
assert(take_l_steps L.TBool ljust' n2_just == Some target_just);
lift_multiple_l_steps L.TBool ltau ljust target_just n1_just
(if_cond_lift ltau lcons l_err);
lift_multiple_l_steps L.TBool ltau ljust' target_just n2_just
(if_cond_lift ltau lcons l_err);
take_l_steps_transitive ltau le (final_default_subexp ltau ljust lcons) n1_final n1_just;
assert(take_l_steps ltau le (n1_final + n1_just) == Some (L.EIf target_just lcons l_err));
assert(de' == D.EDefault dexceptions djust' dcons dtau);
assume(D.step_exceptions de dexceptions djust dcons dtau ==
D.step_exceptions de' dexceptions djust' dcons dtau);
// Should be easy enough, djust does not influence the result
let n2_final =
translation_correctness_exceptions_no_exceptions_triggered
de' dexceptions djust' dcons dtau
in
take_l_steps_transitive ltau le' (final_default_subexp ltau ljust' lcons) n2_final n2_just;
n1_final + n1_just, L.EIf target_just lcons l_err, n2_final + n2_just
end else begin
D.is_bool_value_cannot_be_abs D.empty djust;
match djust with
| D.ELit D.LTrue ->
take_l_steps_transitive ltau le (final_default_subexp ltau ljust lcons) n1_final 1;
n1_final + 1, lcons, 0
| D.ELit D.LFalse ->
take_l_steps_transitive ltau le (final_default_subexp ltau ljust lcons) n1_final 1;
n1_final + 1, l_err, 0
| D.ELit D.LConflictError ->
take_l_steps_transitive ltau le (final_default_subexp ltau ljust lcons) n1_final 1;
n1_final + 1, L.ELit (L.LError L.ConflictError), 0
| D.ELit D.LEmptyError ->
take_l_steps_transitive ltau le (final_default_subexp ltau ljust lcons) n1_final 1;
n1_final + 1, l_err, 0
end
end
(*** Wrap-up theorem *)