X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FFsub%2Fpart1a.ma;h=7fb8fed9c14c5433fc8cbd503a8f71e02aedb655;hb=92165c07e82baa982ec06ff9ab325f97659a3900;hp=795546299e842f8c2f2824a7c102db04e548d5d1;hpb=c5608d2704215a7004d23f4801e19df3939f0285;p=helm.git

diff --git a/helm/software/matita/library/Fsub/part1a.ma b/helm/software/matita/library/Fsub/part1a.ma
index 795546299..7fb8fed9c 100644
--- a/helm/software/matita/library/Fsub/part1a.ma
+++ b/helm/software/matita/library/Fsub/part1a.ma
@@ -13,65 +13,20 @@
 (**************************************************************************)
 
 set "baseuri" "cic:/matita/Fsub/part1a/".
-include "logic/equality.ma".
-include "nat/nat.ma".
-include "datatypes/bool.ma".
-include "nat/compare.ma".
-include "Fsub/util.ma".
 include "Fsub/defn.ma".
 
 (*** Lemma A.1 (Reflexivity) ***)
 
-theorem JS_Refl : \forall T,G.(WFType G T) \to (WFEnv G) \to (JSubtype G T T).
-apply Typ_len_ind;intro;elim U
-  [(* FIXME *) generalize in match H1;intro;inversion H1
-     [intros;destruct H6
-     |intros;destruct H5
-     |*:intros;destruct H9]
-  |apply (SA_Refl_TVar ? ? H2);(*FIXME*)generalize in match H1;intro;
-   inversion H1
-     [intros;destruct H6;rewrite > Hcut;assumption
-     |intros;destruct H5
-     |*:intros;destruct H9]
-  |apply (SA_Top ? ? H2 H1)
-  |cut ((WFType G t) \land (WFType G t1))
-     [elim Hcut;apply SA_Arrow
-        [apply H2
-           [apply t_len_arrow1
-           |*:assumption]
-        |apply H2
-           [apply t_len_arrow2
-           |*:assumption]]
-     |(*FIXME*)generalize in match H3;intro;inversion H3
-        [intros;destruct H8
-        |intros;destruct H7
-        |intros;destruct H11;rewrite > Hcut;rewrite > Hcut1;split;assumption
-        |intros;destruct H11]]
-  |elim (fresh_name ((fv_type t1) @ (fv_env G)));
-   cut ((\lnot (in_list ? a (fv_type t1))) \land
-        (\lnot (in_list ? a (fv_env G))))
-     [elim Hcut;cut (WFType G t)
-        [apply (SA_All2 ? ? ? ? ? a ? H7 H6 H6)
-           [apply H2
-              [apply t_len_forall1
-              |*:assumption]
-           |apply H2
-              [rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
-               apply t_len_forall2
-              |(*FIXME*)generalize in match H3;intro;inversion H3
-                 [intros;destruct H11
-                 |intros;destruct H10
-                 |intros;destruct H14
-                 |intros;destruct H14;rewrite < Hcut2 in H11;
-                  rewrite < Hcut3 in H11;rewrite < H13;rewrite < H13 in H11;
-                  apply (H11 ? H7 H6)]
-              |apply WFE_cons;assumption]]
-        |(*FIXME*)generalize in match H3;intro;inversion H3
-           [intros;destruct H11
-           |intros;destruct H10
-           |intros;destruct H14
-           |intros;destruct H14;rewrite > Hcut1;assumption]]
-     |split;unfold;intro;apply H5;apply natinG_or_inH_to_natinGH;auto]]
+theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴  T.
+intros 3;elim H
+  [apply (SA_Refl_TVar l n H2 H1);
+  |apply (SA_Top l Top H1 (WFT_Top l));
+  |apply (SA_Arrow l t t1 t t1 (H2 H5) (H4 H5))
+  |apply (SA_All ? ? ? ? ? (H2 H5));intros;apply (H4 X H6)
+     [intro;apply H6;apply (fv_WFT (Forall t t1) X l)
+        [apply (WFT_Forall ? ? ? H1 H3)
+        |simplify;autobatch]
+     |autobatch]]
 qed.
 
 (* 
@@ -95,114 +50,168 @@ intros 4;elim H
      [unfold;intro;apply H8;lapply (incl_bound_fv ? ? H7);apply (Hletin1 ? H9)
      |apply WFE_cons
         [1,2:assumption
-        |lapply (incl_bound_fv ? ? H7);apply (WFT_env_incl ? ? ? ? Hletin1);
-         apply (JS_to_WFT1 ? ? ? H1)]
-     |unfold;intros;inversion H9
-        [intros;lapply (inj_head ? ? ? ? H11);rewrite > Hletin1;apply in_Base
-        |intros;lapply (inj_tail ? ? ? ? ? H13);rewrite < Hletin1 in H10;
-         apply in_Skip;apply (H7 ? H10)]]]
+        |apply (JS_to_WFT1 ? ? ? Hletin)]
+     |unfold;intros;elim (in_cons_case ? ? ? ? H9)
+        [rewrite > H10;apply in_Base
+        |elim H10;apply (in_Skip ? ? ? ? ? H11);apply (H7 ? H12)]]]
 qed.
 
+lemma decidable_eq_Typ : \forall S,T:Typ.(S = T) ∨ (S ≠ T).
+intro;elim S
+  [elim T
+     [elim (decidable_eq_nat n n1)
+        [rewrite > H;left;reflexivity
+        |right;intro;destruct H1;apply (H Hcut)]
+     |2,3:right;intro;destruct H
+     |*:right;intro;destruct H2]
+  |elim T
+     [2:elim (decidable_eq_nat n n1)
+        [rewrite > H;left;reflexivity
+        |right;intro;destruct H1;apply (H Hcut)]
+     |1,3:right;intro;destruct H
+     |*:right;intro;destruct H2]
+  |elim T
+     [3:left;reflexivity
+     |1,2:right;intro;destruct H
+     |*:right;intro;destruct H2]
+  |elim T
+     [1,2,3:right;intro;destruct H2
+     |elim (H t2)
+        [rewrite > H4;elim (H1 t3)
+           [rewrite > H5;left;reflexivity
+           |right;intro;apply H5;destruct H6;assumption]
+        |right;intro;apply H4;destruct H5;assumption]
+     |right;intro;destruct H4]
+  |elim T
+     [1,2,3:right;intro;destruct H2
+     |right;intro;destruct H4
+     |elim (H t2)
+        [rewrite > H4;elim (H1 t3)
+           [rewrite > H5;left;reflexivity
+           |right;intro;apply H5;destruct H6;assumption]
+        |right;intro;apply H4;destruct H5;assumption]]]
+qed.
+
+lemma decidable_eq_bound: ∀b1,b2:bound.(b1 = b2) ∨ (b1 ≠ b2).
+intros;elim b1;elim b2;elim (decidable_eq_nat n n1)
+  [rewrite < H;elim (decidable_eq_Typ t t1)
+     [rewrite < H1;elim (bool_to_decidable_eq b b3)
+        [rewrite > H2;left;reflexivity
+        |right;intro;destruct H3;apply (H2 Hcut)]
+     |right;intro;destruct H2;apply (H1 Hcut1)]
+  |right;intro;destruct H1;apply (H Hcut1)]
+qed.     
+
 (* Lemma A.3 (Transitivity and Narrowing) *)
 
-lemma JS_trans_narrow : \forall n.
-  (\forall G,T,Q,U.
-     (t_len Q) \leq n \to (JSubtype G T Q) \to (JSubtype G Q U) \to 
+lemma JS_trans_narrow : \forall Q.
+  (\forall G,T,U.
+     (JSubtype G T Q) \to (JSubtype G Q U) \to 
      (JSubtype G T U)) \land
-  (\forall G,H,X,P,Q,M,N.
-     (t_len Q) \leq n \to 
+  (\forall G,H,X,P,M,N.
      (JSubtype (H @ ((mk_bound true X Q) :: G)) M N) \to
      (JSubtype G P Q) \to
      (JSubtype (H @ ((mk_bound true X P) :: G)) M N)).
-intro;apply (nat_elim1 n);intros 2;
-cut (\forall G,T,Q.(JSubtype G T Q) \to 
-        \forall U.(t_len Q \leq m) \to (JSubtype G Q U) \to (JSubtype G T U))
-  [cut (\forall G,M,N.(JSubtype G M N) \to
-           \forall G1,X,Q,G2,P.
-              (G = G2 @ ((mk_bound true X Q) :: G1)) \to (t_len Q) \leq m \to
-              (JSubtype G1 P Q) \to 
+apply Typ_len_ind;intros 2;
+cut (\forall G,T,P. 
+           (JSubtype G T U) \to 
+           (JSubtype G U P) \to 
+           (JSubtype G T P))
+  [split
+     [assumption
+     |cut (\forall G,M,N.(JSubtype G M N) \to
+           \forall G1,X,G2,P.
+              (G = G2 @ ((mk_bound true X U) :: G1)) \to 
+              (JSubtype G1 P U) \to 
               (JSubtype (G2 @ ((mk_bound true X P) :: G1)) M N))
-     [split
-        [intros;apply (Hcut ? ? ? H2 ? H1 H3)
-        |intros;apply (Hcut1 ? ? ? H3 ? ? ? ? ? ? H2 H4);reflexivity]
-     |intros 9;cut (incl ? (fv_env (G2 @ ((mk_bound true X Q)::G1)))
-                   (fv_env (G2 @ ((mk_bound true X P)::G1))))
-        [intros;
-(*                 [rewrite > H6 in H2;lapply (JS_to_WFT1 ? ? ? H8);
-                  apply (WFE_Typ_subst ? ? ? ? ? ? ? H2 Hletin) *)
-         generalize in match Hcut1;generalize in match H2;
-         generalize in match H1;generalize in match H4;
-         generalize in match G1;generalize in match G2;elim H1
-           [apply SA_Top
-              [rewrite > H9 in H5;lapply (JS_to_WFT1 ? ? ? H7);
-               apply (WFE_Typ_subst ? ? ? ? ? ? ? H5 Hletin)
-              |rewrite > H9 in H6;apply (WFT_env_incl ? ? H6);elim l
-                 [simplify;unfold;intros;assumption
-                 |simplify;apply (incl_nat_cons ? ? ? H11)]]
-           |apply SA_Refl_TVar
-              [rewrite > H9 in H5;lapply (JS_to_WFT1 ? ? ? H7);
-               apply (WFE_Typ_subst ? ? ? ? ? ? ? H5 Hletin)
-              |apply H10;rewrite < H9;assumption]           
-           |elim (decidable_eq_nat X n1)
-              [apply (SA_Trans_TVar ? ? ? P)
-                 [rewrite < H12;elim l
-                    [simplify;apply in_Base
-                    |simplify;apply in_Skip;assumption]
-                 |lapply (JS_to_WFE ? ? ? H9);rewrite > H10 in Hletin;
-                  rewrite > H10 in H5;
-                  lapply (WFE_bound_bound ? ? ? Q ? Hletin H5)
-                    [lapply (H7 ? ? H8 H6 H10 H11);rewrite > Hletin1 in Hletin2;
-                     apply (Hcut ? ? ? ? ? H3 Hletin2);
-                     lapply (JS_to_WFE ? ? ? Hletin2);
-                     apply (JS_weakening ? ? ? H8 ? Hletin3);unfold;intros;
-                     elim l;simplify;apply in_Skip;assumption
-                    |rewrite > H12;elim l
-                       [simplify;apply in_Base
-                       |simplify;apply in_Skip;assumption]]]
-              |rewrite > H10 in H5;apply (SA_Trans_TVar ? ? ? t1)
-                 [apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H5);unfold;
-                  intro;apply H12;symmetry;assumption
-                 |apply (H7 ? ? H8 H6 H10 H11)]]
-           |apply SA_Arrow
-              [apply (H6 ? ? H9 H5 H11 H12)
-              |apply (H8 ? ? H9 H7 H11 H12)]
-           |apply SA_All
-              [apply (H6 ? ? H9 H5 H11 H12)
-              |intros;apply (H8 ? ? (mk_bound true X1 t2::l) l1)
-                 [unfold;intro;apply H13;rewrite > H11 in H14;apply (H12 ? H14)
-                 |assumption
-                 |apply H7;rewrite > H11;unfold;intro;apply H13;apply (H12 ? H14)
-                 |simplify;rewrite < H11;reflexivity
-                 |simplify;apply incl_nat_cons;assumption]]]
-        |elim G2 0
-           [simplify;unfold;intros;assumption
-           |intro;elim t 0;simplify;intros;apply incl_nat_cons;assumption]]]
-  |intros 4;(*generalize in match H1;*)elim H1
+        [intros;apply (Hcut1 ? ? ? H2 ? ? ? ? ? H3);reflexivity
+        |intros;cut (incl ? (fv_env (G2 @ ((mk_bound true X U)::G1)))
+                    (fv_env (G2 @ ((mk_bound true X P)::G1))))
+           [intros;generalize in match H2;generalize in match Hcut1;
+            generalize in match Hcut;generalize in match G2;elim H1
+              [apply SA_Top
+                 [rewrite > H8 in H4;lapply (JS_to_WFT1 ? ? ? H3);
+                  apply (WFE_Typ_subst ? ? ? ? ? ? ? H4 Hletin)
+                 |rewrite > H8 in H5;apply (WFT_env_incl ? ? H5 ? H7)]
+              |apply SA_Refl_TVar
+                 [rewrite > H8 in H4;apply (WFE_Typ_subst ? ? ? ? ? ? ? H4);
+                  apply (JS_to_WFT1 ? ? ? H3)
+                 |rewrite > H8 in H5;apply (H7 ? H5)]
+              |elim (decidable_eq_nat X n)
+                 [apply (SA_Trans_TVar ? ? ? P)
+                    [rewrite < H10;elim l1
+                      [simplify;constructor 1
+                      |simplify;elim (decidable_eq_bound (mk_bound true X P) t2)
+                         [rewrite < H12;apply in_Base
+                         |apply (in_Skip ? ? ? ? ? H12);assumption]]
+                   |apply H7
+                      [lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin);
+                       apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros;
+                       elim l1
+                         [simplify;
+                          elim (decidable_eq_bound x (mk_bound true X P))
+                            [rewrite < H12;apply in_Base
+                            |apply (in_Skip ? ? ? ? ? H12);assumption]
+                         |simplify;elim (decidable_eq_bound x t2)
+                            [rewrite < H13;apply in_Base
+                            |apply (in_Skip ? ? ? ? ? H13);assumption]]
+                      |lapply (WFE_bound_bound true n t1 U ? ? H4)
+                         [apply (JS_to_WFE ? ? ? H5)
+                         |rewrite < Hletin;apply (H6 ? H7 H8 H9)
+                         |rewrite > H9;rewrite > H10;elim l1;simplify
+                            [constructor 1
+                            |elim (decidable_eq_bound (mk_bound true n U) t2)
+                               [rewrite > H12;apply in_Base
+                               |apply (in_Skip ? ? ? ? ? H12);assumption]]]]]
+                |apply (SA_Trans_TVar ? ? ? t1)
+                   [rewrite > H9 in H4;
+                    apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H4);
+                    unfold;intro;apply H10;symmetry;assumption
+                   |apply (H6 ? H7 H8 H9)]]
+             |apply SA_Arrow
+                [apply (H5 ? H8 H9 H10)
+                |apply (H7 ? H8 H9 H10)]
+             |apply SA_All
+                [apply (H5 ? H8 H9 H10)
+                |intros;apply (H7 ? ? (mk_bound true X1 t2::l1) H8)
+                   [rewrite > H10;cut ((fv_env (l1@(mk_bound true X P::G1))) =
+                                       (fv_env (l1@(mk_bound true X U::G1))))
+                      [unfold;intro;apply H11;rewrite > Hcut2;assumption
+                      |elim l1
+                         [simplify;reflexivity
+                         |elim t4;simplify;rewrite > H12;reflexivity]]
+                   |simplify;apply (incl_nat_cons ? ? ? H9)
+                   |simplify;rewrite < H10;reflexivity]]]
+          |cut ((fv_env (G2@(mk_bound true X U::G1))) =
+                (fv_env (G2@(mk_bound true X P::G1))))
+             [rewrite > Hcut1;unfold;intros;assumption
+             |elim G2
+                [simplify;reflexivity
+                |elim t;simplify;rewrite > H4;reflexivity]]]]]
+  |intros 4;generalize in match H;elim H1
      [inversion H5
         [intros;rewrite < H8;apply (SA_Top ? ? H2 H3)
         |intros;destruct H9
         |intros;destruct H10
         |*:intros;destruct H11]
      |assumption
-     |apply (SA_Trans_TVar ? ? ? ? H2);apply (H4 ? H5 H6)
+     |apply (SA_Trans_TVar ? ? ? ? H2);apply (H4 H5 H6)
      |inversion H7
         [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Arrow
            [apply (JS_to_WFT2 ? ? ? H2)
            |apply (JS_to_WFT1 ? ? ? H4)]
         |intros;destruct H11
         |intros;destruct H12
-        |intros;destruct H13;elim (H (pred m))
-           [apply SA_Arrow
+        |intros;destruct H13;apply SA_Arrow
               [rewrite > H12 in H2;rewrite < Hcut in H8;
-               apply (H15 ? ? ? ? ? H8 H2);lapply (t_len_arrow1 t2 t3);
-               unfold in Hletin;lapply (trans_le ? ? ? Hletin H6);
-               apply (t_len_pred ? ? Hletin1)
+               lapply (H6 t2)
+                 [elim Hletin;apply (H15 ? ? ? H8 H2)
+                 |apply (t_len_arrow1 t2 t3)]
               |rewrite > H12 in H4;rewrite < Hcut1 in H10;
-               apply (H15 ? ? ? ? ? H4 H10);lapply (t_len_arrow2 t2 t3);
-               unfold in Hletin;lapply (trans_le ? ? ? Hletin H6);
-               apply (t_len_pred ? ? Hletin1)]
-           |apply (pred_m_lt_m ? ? H6)]
-        |intros;destruct H13]
+               lapply (H6 t3)
+                 [elim Hletin;apply (H15 ? ? ? H4 H10)
+                 |apply (t_len_arrow2 t2 t3)]]
+           |intros;destruct H13]
      |inversion H7
         [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Forall
            [apply (JS_to_WFT2 ? ? ? H2)
@@ -212,65 +221,32 @@ cut (\forall G,T,Q.(JSubtype G T Q) \to
         |intros;destruct H11
         |intros;destruct H12
         |intros;destruct H13
-        |intros;destruct H13;elim (H (pred m))
-           [elim (fresh_name ((fv_env e1) @ (fv_type t1) @ (fv_type t7)));
-            cut ((\lnot (in_list ? a (fv_env e1))) \land 
-                 (\lnot (in_list ? a (fv_type t1))) \land
-                 (\lnot (in_list ? a (fv_type t7))))
-              [elim Hcut2;elim H18;clear Hcut2 H18;apply (SA_All2 ? ? ? ? ? a)
-                 [rewrite < Hcut in H8;rewrite > H12 in H2;
-                  apply (H15 ? ? ? ? ? H8 H2);lapply (t_len_forall1 t2 t3);
-                  unfold in Hletin;lapply (trans_le ? ? ? Hletin H6);
-                  apply (t_len_pred ? ? Hletin1)
-                 |5:lapply (H10 ? H20);rewrite > H12 in H5;
-                  lapply (H5 ? H20 (subst_type_O t5 (TFree a)))
-                    [apply (H15 ? ? ? ? ? ? Hletin)
-                       [rewrite < Hcut1;rewrite > subst_O_nat;
-                        rewrite < eq_t_len_TFree_subst;
-                        lapply (t_len_forall2 t2 t3);unfold in Hletin2;
-                        lapply (trans_le ? ? ? Hletin2 H6);
-                        apply (t_len_pred ? ? Hletin3)
-                       |rewrite < Hcut in H8;
-                        apply (H16 e1 (nil ?) a t6 t2 ? ? ? Hletin1 H8);
-                        lapply (t_len_forall1 t2 t3);unfold in Hletin2;
-                        lapply (trans_le ? ? ? Hletin2 H6);
-                        apply (t_len_pred ? ? Hletin3)]
-                    |rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
-                     lapply (t_len_forall2 t2 t3);unfold in Hletin1;
-                     lapply (trans_le ? ? ? Hletin1 H6);
-                     apply (trans_le ? ? ? ? Hletin2);constructor 2;
-                     constructor 1
-                    |rewrite > Hcut1;rewrite > H12 in H4;
-                     lapply (H4 ? H20);rewrite < Hcut1;apply JS_Refl
-                       [apply (JS_to_WFT2 ? ? ? Hletin1)
-                       |apply (JS_to_WFE ? ? ? Hletin1)]]
-                 |*:assumption]
-              |split
-                 [split
-                    [unfold;intro;apply H17;
-                     apply (natinG_or_inH_to_natinGH ? (fv_env e1));right;
-                     assumption
-                    |unfold;intro;apply H17;
-                     apply (natinG_or_inH_to_natinGH 
-                               ((fv_type t1) @ (fv_type t7)));left;
-                     apply natinG_or_inH_to_natinGH;right;assumption]
-                 |unfold;intro;apply H17;
-                  apply (natinG_or_inH_to_natinGH 
-                            ((fv_type t1) @ (fv_type t7)));left;
-                  apply natinG_or_inH_to_natinGH;left;assumption]]
-           |apply (pred_m_lt_m ? ? H6)]]]]
+        |intros;destruct H13;subst;apply SA_All
+           [lapply (H6 t4)
+              [elim Hletin;apply (H12 ? ? ? H8 H2)
+              |apply t_len_forall1]
+           |intros;(*destruct H12;*)subst;
+            lapply (H6 (subst_type_nat t5 (TFree X) O))
+              [elim Hletin;apply H13
+                 [lapply (H6 t4)
+                    [elim Hletin1;apply (H16 l1 [] X t6);
+                       [simplify;apply H4;assumption
+                       |assumption]
+                    |apply t_len_forall1]
+                 |apply (H10 ? H12)]
+              |rewrite < eq_t_len_TFree_subst;
+               apply t_len_forall2]]]]]
 qed.
 
 theorem JS_trans : \forall G,T,U,V.(JSubtype G T U) \to 
                                    (JSubtype G U V) \to
                                    (JSubtype G T V).
-intros;elim (JS_trans_narrow (t_len U));apply (H2 ? ? ? ? ? H H1);constructor 1;
+intros;elim JS_trans_narrow;autobatch;
 qed.
 
 theorem JS_narrow : \forall G1,G2,X,P,Q,T,U.
                        (JSubtype (G2 @ (mk_bound true X Q :: G1)) T U) \to
                        (JSubtype G1 P Q) \to
                        (JSubtype (G2 @ (mk_bound true X P :: G1)) T U).
-intros;elim (JS_trans_narrow (t_len Q));apply (H3 ? ? ? ? ? ? ? ? H H1);
-constructor 1;
+intros;elim JS_trans_narrow;autobatch;
 qed.