X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fassembly%2Fcompiler%2Fenv_to_flatenv.ma;h=e9298cc908d9a86ae31ce4242869585f23a6a0f7;hb=759451f66c0009b12e5bcc9fe0c61f7ab5277057;hp=9f489b7924d7836d0e8d5b51916f27cd34adadc5;hpb=aa9c56aecab7c7f52de13fb1af9696446bccb047;p=helm.git

diff --git a/helm/software/matita/contribs/assembly/compiler/env_to_flatenv.ma b/helm/software/matita/contribs/assembly/compiler/env_to_flatenv.ma
index 9f489b792..e9298cc90 100755
--- a/helm/software/matita/contribs/assembly/compiler/env_to_flatenv.ma
+++ b/helm/software/matita/contribs/assembly/compiler/env_to_flatenv.ma
@@ -125,7 +125,9 @@ definition check_id_map ≝ λd.λmap:aux_map_type d.λname:aux_str_type.
 definition checkb_id_map ≝ λd.λmap:aux_map_type d.λname:aux_str_type.
  match get_id_map_aux d map name with [ None ⇒ false | Some _ ⇒ true ].
 
-lemma checkbidmap_to_checkidmap : ∀d.∀map:aux_map_type d.∀name.checkb_id_map d map name = true → check_id_map d map name.
+lemma checkbidmap_to_checkidmap :
+ ∀d.∀map:aux_map_type d.∀name.
+  checkb_id_map d map name = true → check_id_map d map name.
  unfold checkb_id_map;
  unfold check_id_map;
  intros 3;
@@ -141,7 +143,9 @@ definition checknot_id_map ≝ λd.λmap:aux_map_type d.λname:aux_str_type.
 definition checknotb_id_map ≝ λd.λmap:aux_map_type d.λname:aux_str_type.
  match get_id_map_aux d map name with [ None ⇒ true | Some _ ⇒ false ].
 
-lemma checknotbidmap_to_checknotidmap : ∀d.∀map:aux_map_type d.∀name.checknotb_id_map d map name = true → checknot_id_map d map name.
+lemma checknotbidmap_to_checknotidmap :
+ ∀d.∀map:aux_map_type d.∀name.
+  checknotb_id_map d map name = true → checknot_id_map d map name.
  unfold checknotb_id_map;
  unfold checknot_id_map;
  intros 3;
@@ -171,7 +175,8 @@ definition check_desc_flatEnv ≝ λfe:aux_flatEnv_type.λstr:aux_strId_type.
 definition checkb_desc_flatEnv ≝ λfe:aux_flatEnv_type.λstr:aux_strId_type.
  match get_desc_flatEnv_aux fe str with [ None ⇒ false | Some _ ⇒ true ].
 
-lemma checkbdescflatenv_to_checkdescflatenv : ∀fe,str.checkb_desc_flatEnv fe str = true → check_desc_flatEnv fe str.
+lemma checkbdescflatenv_to_checkdescflatenv :
+ ∀fe,str.checkb_desc_flatEnv fe str = true → check_desc_flatEnv fe str.
  unfold checkb_desc_flatEnv;
  unfold check_desc_flatEnv;
  intros 2;
@@ -187,7 +192,8 @@ definition checknot_desc_flatEnv ≝ λfe:aux_flatEnv_type.λstr:aux_strId_type.
 definition checknotb_desc_flatEnv ≝ λfe:aux_flatEnv_type.λstr:aux_strId_type.
  match get_desc_flatEnv_aux fe str with [ None ⇒ true | Some _ ⇒ false ].
 
-lemma checknotbdescflatenv_to_checknotdescflatenv : ∀fe,str.checknotb_desc_flatEnv fe str = true → checknot_desc_flatEnv fe str.
+lemma checknotbdescflatenv_to_checknotdescflatenv :
+ ∀fe,str.checknotb_desc_flatEnv fe str = true → checknot_desc_flatEnv fe str.
  unfold checknotb_desc_flatEnv;
  unfold checknot_desc_flatEnv;
  intros 2;
@@ -197,6 +203,79 @@ lemma checknotbdescflatenv_to_checknotdescflatenv : ∀fe,str.checknotb_desc_fla
  ].
 qed.
 
+(* leave: elimina la testa (il "cur" corrente) *)
+let rec leave_map d (map:aux_map_type (S d)) on map ≝
+ match map with
+  [ nil ⇒ []
+  | cons h t ⇒
+   (MAP_ELEM d
+             (get_name_mapElem (S d) h)
+             (get_max_mapElem (S d) h)
+             (cut_first_mapList ? (get_cur_mapElem (S d) h) (defined_mapList_get ??))
+   )::(leave_map d t)
+  ].
+
+(* enter: propaga in testa la vecchia testa (il "cur" precedente) *)
+let rec enter_map d (map:aux_map_type d) on map ≝
+ match map with
+  [ nil ⇒ []
+  | cons h t ⇒
+   (MAP_ELEM (S d)
+             (get_name_mapElem d h)
+             (get_max_mapElem d h)
+             (map_cons (S d)
+                       (get_first_mapList ? (get_cur_mapElem d h) (defined_mapList_get ??))
+                       (get_cur_mapElem d h))
+   )::(enter_map d t)
+  ].
+
+(* aggiungi descrittore *)
+let rec new_map_elem_from_depth_aux (d:nat) on d ≝
+ match d
+  return λd.map_list d
+ with
+  [ O ⇒ map_nil
+  | S n ⇒ map_cons n (None ?) (new_map_elem_from_depth_aux n)
+  ].
+
+let rec new_max_map_aux d (map:aux_map_type d) (name:aux_str_type) (max:nat) on map ≝
+ match map with
+  [ nil ⇒ []
+  | cons h t ⇒ match eqStr_str name (get_name_mapElem d h) with
+                [ true ⇒ (MAP_ELEM d name max (map_cons d (Some ? max) (cut_first_mapList (S d) (get_cur_mapElem d h) (defined_mapList_get ??))))::t
+                | false ⇒ h::(new_max_map_aux d t name max)
+                ] 
+  ].
+
+definition add_desc_env_flatEnv_map ≝
+λd.λmap:aux_map_type d.λstr.
+ match get_max_map_aux d map str with
+  [ None ⇒ map@[ MAP_ELEM d str O (map_cons d (Some ? O) (new_map_elem_from_depth_aux d)) ]
+  | Some x ⇒ new_max_map_aux d map str (S x)
+  ].
+
+definition add_desc_env_flatEnv_fe ≝
+λd.λmap:aux_map_type d.λfe.λstr.λc.λdesc.
+ fe@[ VAR_FLAT_ELEM (STR_ID str match get_max_map_aux d map str with [ None ⇒ O | Some x ⇒ (S x)])
+                    (DESC_ELEM c desc) ].
+
+let rec build_trasfEnv_env (d:nat) (trsf:list (Prod3T aux_str_type bool ast_type)) on trsf : aux_env_type d → aux_env_type d ≝
+ match trsf with
+  [ nil ⇒ (λx.x)
+  | cons h t ⇒ (λx.(build_trasfEnv_env d t) (add_desc_env d x (fst3T ??? h) (snd3T ??? h) (thd3T ??? h)))
+  ].
+
+let rec build_trasfEnv_mapFe (d:nat) (trsf:list (Prod3T aux_str_type bool ast_type)) on trsf :
+ Prod (aux_map_type d) aux_flatEnv_type → Prod (aux_map_type d) aux_flatEnv_type ≝
+ match trsf with
+  [ nil ⇒ (λx.x)
+  | cons h t ⇒ (λx.(build_trasfEnv_mapFe d t)
+                   (pair ?? (add_desc_env_flatEnv_map d (fst ?? x) (fst3T ??? h))
+                            (add_desc_env_flatEnv_fe d (fst ?? x) (snd ?? x) (fst3T ??? h) (snd3T ??? h) (thd3T ??? h))))
+  ].
+
+(* ********************************************* *)
+
 (* fe <= fe' *)
 definition eq_flatEnv_elem ≝
 λe1,e2.match e1 with
@@ -272,10 +351,11 @@ lemma le_to_leflatenv : ∀fe,x.le_flatEnv fe (fe@x) = true.
  ].
 qed.
 
-lemma leflatenv_to_check : ∀fe,fe',str.
- (le_flatEnv fe fe' = true) →
- (check_desc_flatEnv fe str) →
- (check_desc_flatEnv fe' str).
+lemma leflatenv_to_check :
+ ∀fe,fe',str.
+  (le_flatEnv fe fe' = true) →
+  (check_desc_flatEnv fe str) →
+  (check_desc_flatEnv fe' str).
  intros 4;
  cases (leflatenv_to_le fe fe' H);
  rewrite < H1;
@@ -290,10 +370,11 @@ lemma leflatenv_to_check : ∀fe,fe',str.
  ].
 qed.
 
-lemma leflatenv_to_get : ∀fe,fe',str.
- (le_flatEnv fe fe' = true) →
- (check_desc_flatEnv fe str) →
- (get_desc_flatEnv fe str = get_desc_flatEnv fe' str).
+lemma leflatenv_to_get :
+ ∀fe,fe',str.
+  (le_flatEnv fe fe' = true) →
+  (check_desc_flatEnv fe str) →
+  (get_desc_flatEnv fe str = get_desc_flatEnv fe' str).
  intros 4;
  cases (leflatenv_to_le fe fe' H);
  rewrite < H1;
@@ -318,118 +399,6 @@ lemma leflatenv_to_get : ∀fe,fe',str.
  ].
 qed.
 
-(* controllo di coerenza env <-> map *)
-let rec check_map_env_align_auxEnv (d:nat) (e:aux_env_type d) (str:aux_str_type) (res:nat) on e ≝
- match e with
-  [ env_nil h ⇒ λf.f h
-  | env_cons d' h t ⇒ λf.check_map_env_align_auxEnv d' t str (f h)
-  ] (λx.match get_desc_from_lev_env x str with [ None ⇒ S res | Some _ ⇒ O ]).
-
-let rec check_map_env_align_auxMap (d:nat) (map:map_list d) (res:nat) on map ≝
- match map with
-  [ map_nil ⇒ eqb res O
-  | map_cons d' h t ⇒ match eqb res O with
-   [ true ⇒ match h with
-    [ None ⇒ check_map_env_align_auxMap d' t O | Some _ ⇒ false ]
-   | false ⇒ match h with
-    [ None ⇒ false | Some _ ⇒ check_map_env_align_auxMap d' t (pred res) ]
-   ]
-  ].
-
-(* esprimendolo in lingua umana il vincolo che controlla e':
-   ∀name ϵ map.
-    la map_list "cur" deve avere la seguente struttura
-     - Some _ ; ... ; Some _ ; None ; ... None
-     - il numero di Some _ deve essere pari a (d + 1 - x) con x
-       ricavato scandendo in avanti tutti gli ambienti e
-       numerando quanti ambienti CONSECUTIVI non contengono la definizione di name
-
-   ex: scandendo e in avanti si trova la seguente sequenza di check per il nome "pippo"
-       no si no si NO NO
-       quindi sapendo che d=5 (6 partendo da 0) e che check_env_align_aux ha restituito 2
-       sappiamo che la mappa alla voce "pippo" deve avere la seguente struttura scandita in avanti
-       Some _ ; Some _ ; Some _ ; Some _ ; None ; None
-       cioe' 5+1-2 Some seguiti da solo None
-
-   NB: e' solo meta' perche' cosi' si asserisce map ≤ env
-   
-*)
-let rec check_map_env_align (d:nat) (e:aux_env_type d) (map:aux_map_type d) on map ≝
- match map with
-  [ nil ⇒ true
-  | cons h t ⇒ (check_map_env_align_auxMap (S d) (get_cur_mapElem d h) (d + 1 - (check_map_env_align_auxEnv d e (get_name_mapElem d h) O)))⊗
-               (check_map_env_align d e t)
-  ].
-
-let rec check_env_map_align_aux (d:nat) (map:aux_map_type d) (le:list var_elem) on le ≝
- match le with
-  [ nil ⇒ true
-  | cons h t ⇒ match get_id_map_aux d map (get_name_var h) with
-   [ None ⇒ false | Some _ ⇒ check_env_map_align_aux d map t ]
-  ].
-
-(* esprimendolo in lingua umana il vincolo che controlla e':
-   ∀name ϵ e.name ϵ map
-   
-   NB: e' l'altra meta' perche' cosi' asserisce env ≤ map
-*)
-let rec check_env_map_align (de:nat) (e:aux_env_type de) (dm:nat) (map:aux_map_type dm) on e ≝
- match e with
-  [ env_nil h ⇒ check_env_map_align_aux dm map h
-  | env_cons d' h t ⇒ (check_env_map_align_aux dm map h)⊗(check_env_map_align d' t dm map)
-  ].
-
-(* invariante *)
-definition env_to_flatEnv_inv ≝
- λd.λe:aux_env_type d.λmap:aux_map_type d.λfe:aux_flatEnv_type.
-  ∀str.
-   check_desc_env d e str →
-   (((check_env_map_align d e d map)⊗(check_map_env_align d e map)) = true ∧
-    check_id_map d map str ∧
-    check_desc_flatEnv fe (STR_ID str (get_id_map d map str)) ∧
-    get_desc_env d e str = get_desc_flatEnv fe (STR_ID str (get_id_map d map str))).
-
-lemma inv_to_checkdescflatenv :
- ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.
- env_to_flatEnv_inv d e map fe →
- (∀str.check_desc_env d e str → check_desc_flatEnv fe (STR_ID str (get_id_map d map str))).
- intros 7;
- unfold env_to_flatEnv_inv in H:(%); 
- lapply (H str H1);
- apply (proj2 ?? (proj1 ?? Hletin));
-qed.
-
-lemma env_map_flatEnv_empty_aux : env_to_flatEnv_inv O empty_env empty_map empty_flatEnv.
- unfold env_to_flatEnv_inv;
- unfold empty_env;
- unfold empty_map;
- unfold empty_flatEnv;
- simplify;
- intros;
- elim H.
-qed.
-
-lemma leflatenv_to_inv :
- ∀d.∀e.∀map:aux_map_type d.∀fe,fe'.
-  le_flatEnv fe fe' = true →
-  env_to_flatEnv_inv d e map fe →
-  env_to_flatEnv_inv d e map fe'.
- intros 6;
- unfold env_to_flatEnv_inv;
- intros;
- split;
- [ split;
-   [ lapply (H1 str H2);
-     apply (proj1 ?? (proj1 ?? Hletin))
-   | lapply (H1 str H2);
-     apply (leflatenv_to_check fe fe' ? H (proj2 ?? (proj1 ?? Hletin)))
-   ]
- | lapply (H1 str H2);
-   rewrite < (leflatenv_to_get fe fe' ? H (proj2 ?? (proj1 ?? Hletin)));
-   apply (proj2 ?? Hletin)
- ].
-qed.
-
 lemma leflatenv_trans :
  ∀fe,fe',fe''.
   le_flatEnv fe fe' = true →
@@ -445,138 +414,6 @@ lemma leflatenv_trans :
  apply (le_to_leflatenv fe ?).
 qed.
 
-(* enter: propaga in testa la vecchia testa (il "cur" precedente) *)
-let rec enter_map d (map:aux_map_type d) on map ≝
- match map with
-  [ nil ⇒ []
-  | cons h t ⇒
-   (MAP_ELEM (S d)
-             (get_name_mapElem d h)
-             (get_max_mapElem d h)
-             (map_cons (S d)
-                       (get_first_mapList ? (get_cur_mapElem d h) (defined_mapList_get ??))
-                       (get_cur_mapElem d h))
-   )::(enter_map d t)
-  ].
-
-lemma getidmap_to_enter :
- ∀d.∀m:aux_map_type d.∀str.
-  get_id_map_aux (S d) (enter_map d m) str = get_id_map_aux d m str.
- intros;
- elim m;
- [ simplify; reflexivity
- | simplify; rewrite > H; reflexivity
- ]
-qed.
-
-(* leave: elimina la testa (il "cur" corrente) *)
-let rec leave_map d (map:aux_map_type (S d)) on map ≝
- match map with
-  [ nil ⇒ []
-  | cons h t ⇒
-   (MAP_ELEM d
-             (get_name_mapElem (S d) h)
-             (get_max_mapElem (S d) h)
-             (cut_first_mapList ? (get_cur_mapElem (S d) h) (defined_mapList_get ??))
-   )::(leave_map d t)
-  ].
-
-(* aggiungi descrittore *)
-let rec new_map_elem_from_depth_aux (d:nat) on d ≝
- match d
-  return λd.map_list d
- with
-  [ O ⇒ map_nil
-  | S n ⇒ map_cons n (None ?) (new_map_elem_from_depth_aux n)
-  ].
-
-let rec new_max_map_aux d (map:aux_map_type d) (name:aux_str_type) (max:nat) on map ≝
- match map with
-  [ nil ⇒ []
-  | cons h t ⇒ match eqStr_str name (get_name_mapElem d h) with
-                [ true ⇒ MAP_ELEM d name max (map_cons d (Some ? max) (cut_first_mapList (S d) (get_cur_mapElem d h) (defined_mapList_get ??)))
-                | false ⇒ h
-                ]::(new_max_map_aux d t name max) 
-  ].
-
-definition add_desc_env_flatEnv_map ≝
-λd.λmap:aux_map_type d.λstr.
- match get_max_map_aux d map str with
-  [ None ⇒ map@[ MAP_ELEM d str O (map_cons d (Some ? O) (new_map_elem_from_depth_aux d)) ]
-  | Some x ⇒ new_max_map_aux d map str (S x)
-  ].
-
-definition add_desc_env_flatEnv_fe ≝
-λd.λmap:aux_map_type d.λfe.λstr.λc.λdesc.
- fe@[ VAR_FLAT_ELEM (STR_ID str match get_max_map_aux d map str with [ None ⇒ O | Some x ⇒ (S x)])
-                    (DESC_ELEM c desc) ].
-
-let rec build_trasfEnv_env (d:nat) (trsf:list (Prod3T aux_str_type bool ast_type)) on trsf : aux_env_type d → aux_env_type d ≝
- match trsf with
-  [ nil ⇒ (λx.x)
-  | cons h t ⇒ (λx.(build_trasfEnv_env d t) (add_desc_env d x (fst3T ??? h) (snd3T ??? h) (thd3T ??? h)))
-  ].
-
-let rec build_trasfEnv_mapFe (d:nat) (trsf:list (Prod3T aux_str_type bool ast_type)) on trsf :
- Prod (aux_map_type d) aux_flatEnv_type → Prod (aux_map_type d) aux_flatEnv_type ≝
- match trsf with
-  [ nil ⇒ (λx.x)
-  | cons h t ⇒ (λx.(build_trasfEnv_mapFe d t)
-                   (pair ?? (add_desc_env_flatEnv_map d (fst ?? x) (fst3T ??? h))
-                            (add_desc_env_flatEnv_fe d (fst ?? x) (snd ?? x) (fst3T ??? h) (snd3T ??? h) (thd3T ??? h))))
-  ].
-
-(* avanzamento dell'invariante *)
-axiom env_map_flatEnv_enter_aux : 
- ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.
-  env_to_flatEnv_inv d e map fe →
-  env_to_flatEnv_inv (S d) (enter_env d e) (enter_map d map) fe.
-
-axiom env_map_flatEnv_add_aux : 
- ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.∀trsf.
-  env_to_flatEnv_inv d e map fe →
-  env_to_flatEnv_inv d
-                     (build_trasfEnv_env d trsf e)
-                     (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe)))
-                     (snd ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))).
-
-lemma env_map_flatEnv_addNil_aux :
- ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.
-  env_to_flatEnv_inv d e map fe →
-  env_to_flatEnv_inv d
-                     (build_trasfEnv_env d [] e)
-                     (fst ?? (build_trasfEnv_mapFe d [] (pair ?? map fe)))
-                     (snd ?? (build_trasfEnv_mapFe d [] (pair ?? map fe))).
- intros;
- simplify;
- apply H.
-qed.
-
-lemma env_map_flatEnv_addSingle_aux :
- ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.∀name,const,desc.
-  env_to_flatEnv_inv d e map fe →
-  env_to_flatEnv_inv d
-                     (add_desc_env d e name const desc)
-                     (fst ?? (build_trasfEnv_mapFe d [ tripleT ??? name const desc ] (pair ?? map fe)))
-                     (snd ?? (build_trasfEnv_mapFe d [ tripleT ??? name const desc ] (pair ?? map fe))).
- intros;
- apply (env_map_flatEnv_add_aux d e map fe [ tripleT ??? name const desc ]);
- apply H.
-qed.
-
-lemma env_map_flatEnv_addJoin_aux :
- ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe:aux_flatEnv_type.∀name,const,desc,trsf.
-  env_to_flatEnv_inv d
-                     (build_trasfEnv_env d trsf (add_desc_env d e name const desc))
-                     map fe →
-  env_to_flatEnv_inv d
-                     (build_trasfEnv_env d ([ tripleT ??? name const desc ]@trsf) e)
-                     map fe.
- intros;
- simplify;
- apply H.
-qed.
-
 lemma env_map_flatEnv_add_aux_fe_al :
  ∀trsf.∀d.∀m:aux_map_type d.∀a,l.
   snd ?? (build_trasfEnv_mapFe d trsf (pair ?? m (a::l))) =
@@ -637,10 +474,26 @@ lemma buildtrasfenvadd_to_le :
  reflexivity.
 qed.
 
-axiom env_map_flatEnv_leave_aux :
- ∀d.∀e:aux_env_type (S d).∀map:aux_map_type (S d).∀fe.∀trsf.
-  env_to_flatEnv_inv (S d) (build_trasfEnv_env (S d) trsf e) map fe →
-  env_to_flatEnv_inv d (leave_env d (build_trasfEnv_env (S d) trsf e)) (leave_map d map) fe.
+(* ********************************************* *)
+
+(* miscellanea *)
+lemma leave_env_enter_env : ∀d.∀e:aux_env_type d.leave_env d (enter_env d e) = e.
+ intros;
+ elim e;
+ reflexivity.
+qed.
+
+lemma leave_map_enter_map : ∀d.∀map. leave_map d (enter_map d map) = map.
+ intros;
+ elim map;
+  [ reflexivity
+  | simplify;
+    rewrite > H;
+    cases a;
+    simplify;
+    reflexivity
+  ]
+qed.
 
 lemma leave_add_enter_env_aux :
  ∀d.∀a:aux_env_type d.∀trsf.∀c.
@@ -664,6 +517,439 @@ lemma leave_add_enter_env :
  reflexivity.
 qed.
 
+(* ********************************************* *)
+
+(* invariante a un livello *)
+definition env_to_flatEnv_inv_one_level ≝
+ λd.λe:aux_env_type d.λmap:aux_map_type d.λfe:aux_flatEnv_type.
+  ∀str.
+   check_desc_env d e str →
+    check_id_map d map str ∧
+    check_desc_flatEnv fe (STR_ID str (get_id_map d map str)) ∧
+    get_desc_env d e str = get_desc_flatEnv fe (STR_ID str (get_id_map d map str)).
+
+(* invariante su tutti i livelli *)
+let rec env_to_flatEnv_inv (d:nat) : aux_env_type d → aux_map_type d → aux_flatEnv_type → Prop ≝
+ match d
+  return λd.aux_env_type d → aux_map_type d → aux_flatEnv_type → Prop
+ with
+  [ O ⇒ λe:aux_env_type O.λmap:aux_map_type O.λfe:aux_flatEnv_type.
+   env_to_flatEnv_inv_one_level O e map fe
+  | S n ⇒ λe:aux_env_type (S n).λmap:aux_map_type (S n).λfe:aux_flatEnv_type.
+   env_to_flatEnv_inv_one_level (S n) e map fe ∧
+   env_to_flatEnv_inv n (leave_env n e) (leave_map n map) fe
+  ].
+
+(* invariante full -> invariante a un livello *)
+lemma env_to_flatEnv_inv_last :
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.
+  env_to_flatEnv_inv d e map fe →
+  env_to_flatEnv_inv_one_level d e map fe.
+ intro;
+ cases d;
+ intros;
+  [ apply H
+  | simplify in H;
+    apply (proj1 ?? H)
+  ]
+qed.
+
+(* invariante caso base *)
+lemma env_map_flatEnv_empty_aux : env_to_flatEnv_inv O empty_env empty_map empty_flatEnv.
+ unfold env_to_flatEnv_inv;
+ unfold empty_env;
+ unfold empty_map;
+ unfold empty_flatEnv;
+ simplify;
+ intros;
+ elim H.
+qed.
+
+(* invariante & leflatenv *)
+lemma leflatenv_to_inv :
+ ∀d.∀e.∀map:aux_map_type d.∀fe,fe'.
+  le_flatEnv fe fe' = true →
+  env_to_flatEnv_inv d e map fe →
+  env_to_flatEnv_inv d e map fe'.
+ intro;
+ elim d;
+ [ simplify;
+   unfold env_to_flatEnv_inv_one_level;
+   intros;
+   split;
+   [ split;
+     [ lapply (H1 str H2);
+       apply (proj1 ?? (proj1 ?? Hletin))
+     | lapply (H1 str H2);
+       apply (leflatenv_to_check fe fe' ? H (proj2 ?? (proj1 ?? Hletin)))
+     ]
+   | lapply (H1 str H2);
+     rewrite < (leflatenv_to_get fe fe' ? H (proj2 ?? (proj1 ?? Hletin)));
+     apply (proj2 ?? Hletin)
+   ]
+ | simplify in H2 ⊢ %;
+   cases H2;
+   split;
+   [ unfold env_to_flatEnv_inv_one_level;
+     intros;
+     split;
+     [ split;
+       [ lapply (H3 str H5);
+         apply (proj1 ?? (proj1 ?? Hletin))
+       | lapply (H3 str H5);
+         apply (leflatenv_to_check fe fe' ? H1 (proj2 ?? (proj1 ?? Hletin)))
+       ]
+     | lapply (H3 str H5);
+       rewrite < (leflatenv_to_get fe fe' ? H1 (proj2 ?? (proj1 ?? Hletin)));
+       apply (proj2 ?? Hletin)
+     ]
+   | apply (H ???? H1 H4)
+   ]
+ ].
+qed.
+
+(* invariante & enter *)
+lemma getidmap_to_enter :
+ ∀d.∀m:aux_map_type d.∀str.
+  get_id_map_aux (S d) (enter_map d m) str = get_id_map_aux d m str.
+ intros;
+ elim m;
+ [ simplify; reflexivity
+ | simplify; rewrite > H; reflexivity
+ ]
+qed.
+
+lemma checkdescenter_to_checkdesc :
+ ∀d.∀e:aux_env_type d.∀str.
+  check_desc_env (S d) (enter_env d e) str →
+  check_desc_env d e str.
+ intros;
+ unfold enter_env in H:(%);
+ simplify in H:(%);
+ apply H.
+qed.
+
+
+lemma env_map_flatEnv_enter_aux : 
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.
+  env_to_flatEnv_inv d e map fe →
+  env_to_flatEnv_inv (S d) (enter_env d e) (enter_map d map) fe.
+ intro;
+ elim d;
+ [ simplify in H;
+   split;
+   [ unfold env_to_flatEnv_inv_one_level;
+     intros;
+     split;
+     [ split;
+       [ unfold check_id_map;
+         rewrite > (getidmap_to_enter ???);
+         apply (proj1 ?? (proj1 ?? (H str (checkdescenter_to_checkdesc ??? H1))))
+       | unfold get_id_map;
+         rewrite > (getidmap_to_enter ???);
+         apply (proj2 ?? (proj1 ?? (H str (checkdescenter_to_checkdesc ??? H1))))
+       ]
+     | unfold get_id_map;
+       rewrite > (getidmap_to_enter ???);
+       apply (proj2 ?? (H str (checkdescenter_to_checkdesc ??? H1)))
+     ]
+   | rewrite > leave_env_enter_env;
+     rewrite > leave_map_enter_map;
+     change with (env_to_flatEnv_inv_one_level O e map fe);
+     apply H
+   ]
+ | split;
+   [ cases H1;
+     simplify in H2;
+     unfold env_to_flatEnv_inv_one_level;
+     intros;
+     split;
+     [ split;
+       [ unfold check_id_map;
+         rewrite > (getidmap_to_enter ???);
+         apply (proj1 ?? (proj1 ?? (H2 str (checkdescenter_to_checkdesc ??? H4))))
+       | unfold get_id_map;
+         rewrite > (getidmap_to_enter ???);
+         apply (proj2 ?? (proj1 ?? (H2 str (checkdescenter_to_checkdesc ??? H4))))
+       ]
+     | unfold get_id_map;
+       rewrite > (getidmap_to_enter ???);
+       apply (proj2 ?? (H2 str (checkdescenter_to_checkdesc ??? H4)))
+     ]
+   | rewrite > leave_env_enter_env;
+     rewrite > leave_map_enter_map;
+     change with (env_to_flatEnv_inv (S n) e map fe);
+     apply H1
+   ]
+ ]
+qed.
+
+(* invariante & leave *)
+lemma env_map_flatEnv_leave_aux :
+ ∀d.∀e:aux_env_type (S d).∀map:aux_map_type (S d).∀fe.∀trsf.
+  env_to_flatEnv_inv (S d) (build_trasfEnv_env (S d) trsf e) map fe →
+  env_to_flatEnv_inv d (leave_env d (build_trasfEnv_env (S d) trsf e)) (leave_map d map) fe.
+ intros;
+ simplify in H;
+ cases H;
+ apply H2.
+qed.
+
+(* invariante & add *)
+definition occurs_in_trsf ≝
+λtrsf:list (Prod3T aux_str_type bool ast_type).λstr.
+ fold_right_list ?? (λhe,b.eqStr_str (fst3T ??? he) str ⊕ b) false trsf.
+
+lemma occurs_in_subTrsf_r :
+ ∀a,l,str.
+  occurs_in_trsf (a::l) str = false →
+  occurs_in_trsf l str = false.
+ intros;
+ simplify in H:(%);
+ rewrite > (orb_false_false_r ?? H);
+ reflexivity.
+qed.
+
+lemma occurs_in_subTrsf_l :
+ ∀a,l,str.
+  occurs_in_trsf (a::l) str = false →
+  eqStr_str (fst3T ??? a) str = false.
+ intros;
+ simplify in H:(%);
+ rewrite > (orb_false_false ?? H);
+ reflexivity.
+qed.
+
+axiom get_id_map_aux_add_not_occurs :
+ ∀d.∀map:aux_map_type d.∀fe,trsf,str.
+  occurs_in_trsf trsf str = false →
+  get_id_map_aux d map str =
+  get_id_map_aux d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str.
+
+lemma get_id_map_add_not_occurs :
+ ∀d.∀map:aux_map_type d.∀fe,trsf,str.
+  occurs_in_trsf trsf str = false →
+  get_id_map d map str =
+  get_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str.
+ intros;
+ unfold get_id_map;
+ rewrite < (get_id_map_aux_add_not_occurs ????? H);
+ reflexivity.
+qed.
+
+lemma check_id_map_not_occurs :
+ ∀d.∀map:aux_map_type d.∀fe.∀trsf.∀str.
+  occurs_in_trsf trsf str = false →
+  check_id_map d map str →
+  check_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str.
+ intros;
+ whd in H1 ⊢ %;
+ rewrite < (get_id_map_aux_add_not_occurs ????? H);
+ apply H1.
+qed.
+
+lemma check_desc_flatEnv_not_occurs :
+ ∀d.∀map:aux_map_type d.∀fe.∀trsf.∀str.
+  occurs_in_trsf trsf str = false →
+  check_desc_flatEnv fe (STR_ID str (get_id_map d map str)) →
+  check_desc_flatEnv (snd ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe)))
+                     (STR_ID str (get_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str)).
+ intros;
+ rewrite < (get_id_map_add_not_occurs ????? H);
+ apply (leflatenv_to_check ??? (buildtrasfenvadd_to_le ????) H1).
+qed.
+
+lemma get_desc_flatEnv_not_occurs :
+ ∀d.∀map:aux_map_type d.∀fe.∀trsf.∀str.
+  occurs_in_trsf trsf str = false →
+  check_desc_flatEnv fe (STR_ID str (get_id_map d map str)) →
+  get_desc_flatEnv (snd ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe)))
+                   (STR_ID str (get_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str)) =
+  get_desc_flatEnv fe (STR_ID str (get_id_map d map str)).
+ intros;
+ rewrite < (get_id_map_add_not_occurs ????? H);
+ rewrite < (leflatenv_to_get ??? (buildtrasfenvadd_to_le ????) H1);
+ reflexivity.
+qed.
+
+axiom get_desc_env_aux_not_occurs :
+ ∀d.∀e:aux_env_type d.∀trsf.∀str.
+  occurs_in_trsf trsf str = false →
+  get_desc_env_aux d (build_trasfEnv_env d trsf e) str =
+  get_desc_env_aux d e str.
+
+lemma get_desc_env_not_occurs :
+ ∀d.∀e:aux_env_type d.∀trsf.∀str.
+  occurs_in_trsf trsf str = false →
+  get_desc_env d (build_trasfEnv_env d trsf e) str =
+  get_desc_env d e str.
+ intros;
+ unfold get_desc_env;
+ rewrite > (get_desc_env_aux_not_occurs ???? H);
+ reflexivity.
+qed.
+
+lemma check_desc_env_not_occurs :
+ ∀d.∀e:aux_env_type d.∀trsf.∀str.
+  occurs_in_trsf trsf str = false →
+  check_desc_env d (build_trasfEnv_env d trsf e) str →
+  check_desc_env d e str.
+ intros 5;
+ unfold check_desc_env;
+ intro;
+ rewrite < (get_desc_env_aux_not_occurs ???? H);
+ apply H1.
+qed.
+
+axiom get_id_map_aux_add_occurs :
+ ∀d.∀map:aux_map_type d.∀fe,trsf,str.
+  occurs_in_trsf trsf str = true →
+  ∃x.get_id_map_aux d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str = Some ? x.
+
+lemma check_id_map_occurs :
+ ∀d.∀map:aux_map_type d.∀fe.∀trsf.∀str.
+  occurs_in_trsf trsf str = true →
+  check_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str.
+ intros;
+ cases (get_id_map_aux_add_occurs d map fe trsf str H);
+ unfold check_id_map;
+ rewrite > H1;
+ simplify;
+ apply I.
+qed.
+
+axiom check_desc_flatEnv_occurs :
+ ∀d.∀map:aux_map_type d.∀fe.∀trsf.∀str. 
+  occurs_in_trsf trsf str = true →
+  check_desc_flatEnv (snd ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe)))
+                     (STR_ID str (get_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str)).
+
+axiom get_desc_env_eq_get_desc_flatEnv_occurs :
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.∀trsf.∀str.
+ occurs_in_trsf trsf str = true →
+ get_desc_env d (build_trasfEnv_env d trsf e) str =
+ get_desc_flatEnv (snd ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe)))
+                  (STR_ID str (get_id_map d (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))) str)).
+
+axiom leave_env_build_trasfEnv :
+ ∀d.∀e:aux_env_type (S d).∀trsf.
+  leave_env d (build_trasfEnv_env (S d) trsf e) = leave_env d e.
+
+axiom leave_map_build_trasfEnv_mapFe:
+ ∀d.∀map:aux_map_type (S d).∀fe.∀trsf.
+  leave_map d (fst ?? (build_trasfEnv_mapFe (S d) trsf (pair ?? map fe))) =
+  leave_map d map.
+
+lemma env_map_flatEnv_add_aux : 
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.∀trsf.
+  env_to_flatEnv_inv d e map fe →
+  env_to_flatEnv_inv d
+                     (build_trasfEnv_env d trsf e)
+                     (fst ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe)))
+                     (snd ?? (build_trasfEnv_mapFe d trsf (pair ?? map fe))).
+ intro;
+ cases d;
+ [ intros;
+   simplify in H;
+   simplify;
+   intros 2;
+   apply (bool_elim ? (occurs_in_trsf trsf str));
+   [ intro;
+     split
+     [ split
+       [ apply (check_id_map_occurs ????? H2)
+       | apply (check_desc_flatEnv_occurs ????? H2)
+       ]
+     | apply (get_desc_env_eq_get_desc_flatEnv_occurs ?????? H2)
+     ]
+   | intro;
+     cases (H str (check_desc_env_not_occurs ???? H2 H1));
+     split
+     [ split
+       [ apply (check_id_map_not_occurs ????? H2 (proj1 ?? H3))
+       | apply (check_desc_flatEnv_not_occurs ????? H2 (proj2 ?? H3))
+       ]
+     | rewrite > (get_desc_flatEnv_not_occurs ????? H2 (proj2 ?? H3));
+       rewrite > (get_desc_env_not_occurs ???? H2);
+       apply H4
+     ]
+   ]
+ | intros;
+   simplify in H;
+   cases H;
+   unfold env_to_flatEnv_inv_one_level in H1;
+   split;
+   [ intros 2;
+     apply (bool_elim ? (occurs_in_trsf trsf str));
+     [ intro;
+       split
+       [ split
+         [ apply (check_id_map_occurs ????? H4)
+         | apply (check_desc_flatEnv_occurs ????? H4)
+         ]
+       | apply (get_desc_env_eq_get_desc_flatEnv_occurs ?????? H4)
+       ]
+     | intro;
+       split;
+       [ split
+         [ apply (check_id_map_not_occurs ????? H4 (proj1 ?? (proj1 ?? (H1 str (check_desc_env_not_occurs ???? H4 H3)))))
+         | apply (check_desc_flatEnv_not_occurs ????? H4 (proj2 ?? (proj1 ?? (H1 str (check_desc_env_not_occurs ???? H4 H3)))))
+         ]
+       | rewrite > (get_desc_flatEnv_not_occurs ????? H4 (proj2 ?? (proj1 ?? (H1 str (check_desc_env_not_occurs ???? H4 H3)))));
+         rewrite > (get_desc_env_not_occurs ???? H4);
+         apply (proj2 ?? (H1 str (check_desc_env_not_occurs ???? H4 H3)))
+       ]
+     ]
+   | change with (env_to_flatEnv_inv n (leave_env n (build_trasfEnv_env (S n) trsf e))
+                 (leave_map n (fst ?? (build_trasfEnv_mapFe (S n) trsf (pair ?? map fe))))
+                 (snd ?? (build_trasfEnv_mapFe (S n) trsf (pair ?? map fe))));
+     rewrite > leave_map_build_trasfEnv_mapFe;
+     rewrite > leave_env_build_trasfEnv;
+     apply (leflatenv_to_inv ?????? H2);
+     apply buildtrasfenvadd_to_le
+   ]
+ ].
+qed.
+
+lemma env_map_flatEnv_addNil_aux :
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.
+  env_to_flatEnv_inv d e map fe →
+  env_to_flatEnv_inv d
+                     (build_trasfEnv_env d [] e)
+                     (fst ?? (build_trasfEnv_mapFe d [] (pair ?? map fe)))
+                     (snd ?? (build_trasfEnv_mapFe d [] (pair ?? map fe))).
+ intros;
+ simplify;
+ apply H.
+qed.
+
+lemma env_map_flatEnv_addSingle_aux :
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe.∀name,const,desc.
+  env_to_flatEnv_inv d e map fe →
+  env_to_flatEnv_inv d
+                     (add_desc_env d e name const desc)
+                     (fst ?? (build_trasfEnv_mapFe d [ tripleT ??? name const desc ] (pair ?? map fe)))
+                     (snd ?? (build_trasfEnv_mapFe d [ tripleT ??? name const desc ] (pair ?? map fe))).
+ intros;
+ apply (env_map_flatEnv_add_aux d e map fe [ tripleT ??? name const desc ]);
+ apply H.
+qed.
+
+lemma env_map_flatEnv_addJoin_aux :
+ ∀d.∀e:aux_env_type d.∀map:aux_map_type d.∀fe:aux_flatEnv_type.∀name,const,desc,trsf.
+  env_to_flatEnv_inv d
+                     (build_trasfEnv_env d trsf (add_desc_env d e name const desc))
+                     map fe →
+  env_to_flatEnv_inv d
+                     (build_trasfEnv_env d ([ tripleT ??? name const desc ]@trsf) e)
+                     map fe.
+ intros;
+ simplify;
+ apply H.
+qed.
+
+(* ********************************************* *)
+
 lemma newid_from_init :
  ∀d.∀e:aux_env_type d.∀name,const,desc.
   ast_id d (add_desc_env d e name const desc) const desc.