X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FPOPLmark%2FFsub%2Fdefn.ma;h=337fe26868c6f5663c82aa943f779aa16060d6a2;hb=3532e6714b7d0ee10b692e41c356866cfab5c646;hp=cd7bbfdfef92913da1205ea914c251e31cf09723;hpb=e2af8984bc1f706eb69b54c99870d8b64a6d75a7;p=helm.git

diff --git a/helm/software/matita/contribs/POPLmark/Fsub/defn.ma b/helm/software/matita/contribs/POPLmark/Fsub/defn.ma
index cd7bbfdfe..337fe2686 100644
--- a/helm/software/matita/contribs/POPLmark/Fsub/defn.ma
+++ b/helm/software/matita/contribs/POPLmark/Fsub/defn.ma
@@ -45,8 +45,11 @@ let rec subst_type_nat T U i ≝
 
 (*** definitions about lists ***)
 
+definition filter_types : list bound → list bound ≝
+  λG.(filter ? G (λB.match B with [mk_bound B X T ⇒ B])).
+
 definition fv_env : list bound → list nat ≝
-  λG.(map ? ? (λb.match b with [mk_bound B X T ⇒ X]) G).
+  λG.(map ? ? (λb.match b with [mk_bound B X T ⇒ X]) (filter_types G)).
 
 let rec fv_type T ≝
   match T with
@@ -92,13 +95,21 @@ inductive JSubtype : list bound → Typ → Typ → Prop ≝
                 (subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O)) →
              JSubtype G (Forall S1 S2) (Forall T1 T2).
 
-notation "hvbox(e ⊢ break ta ⊴  break tb)" 
+notation "mstyle color #007f00 (hvbox(e ⊢ break ta ⊴ break tb))" 
   non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.  
 interpretation "Fsub subtype judgement" 'subjudg e ta tb = (JSubtype e ta tb).
 
-notation > "hvbox(\Forall S.T)" 
+notation "mstyle color #007f00 (hvbox(e ⊢ ♦))" 
+  non associative with precedence 30 for @{ 'wfejudg $e }.  
+interpretation "Fsub WF env judgement" 'wfejudg e = (WFEnv e).
+
+notation "mstyle color #007f00 (hvbox(e ⊢ break t))" 
+  non associative with precedence 30 for @{ 'wftjudg $e $t }.  
+interpretation "Fsub WF type judgement" 'wftjudg e t = (WFType e t).
+
+notation > "\Forall S.T" 
   non associative with precedence 60 for @{ 'forall $S $T}.
-notation < "hvbox('All' \sub S. break T)" 
+notation < "hvbox(⊓ \sub S. break T)" 
   non associative with precedence 60 for @{ 'forall $S $T}.
 interpretation "universal type" 'forall S T = (Forall S T).
   
@@ -115,7 +126,7 @@ notation "hvbox(s break ⇛ t)"
   right associative with precedence 55 for @{ 'arrow $s $t }.
 interpretation "arrow type" 'arrow S T = (Arrow S T).
   
-notation "hvbox(S [# n ↦ T])"
+notation "hvbox(S [#n ↦ T])"
   non associative with precedence 80 for @{ 'substvar $S $T $n }.
 interpretation "subst bound var" 'substvar S T n = (subst_type_nat S T n).  
 
@@ -127,37 +138,32 @@ interpretation "subtyping bound" 'subtypebound X T = (mk_bound true X T).
 
 (*** theorems about lists ***)
 
-lemma boundinenv_natinfv : ∀x,G.(∃B,T.in_list ? (mk_bound B x T) G) →
-                              in_list ? x (fv_env G).
+lemma boundinenv_natinfv : ∀x,G.(∃T.!x ⊴ T ∈ G) → x ∈ (fv_env G).
 intros 2;elim G;decompose
-  [elim (not_in_list_nil ? ? H)
-  |elim (in_list_cons_case ? ? ? ? H1)
-     [rewrite < H2;simplify;apply in_list_head
-     |simplify;apply in_list_cons;apply H;autobatch]]
+  [elim (not_in_list_nil ? ? H1)
+  |elim (in_list_cons_case ? ? ? ? H2)
+     [rewrite < H1;simplify;apply in_list_head
+     |elim a;apply (bool_elim ? b);intro;simplify;try apply in_list_cons;
+      apply H;autobatch]]
 qed.
 
-lemma natinfv_boundinenv : ∀x,G.in_list ? x (fv_env G) →
-                              ∃B,T.in_list ? (mk_bound B x T) G.
+lemma natinfv_boundinenv : ∀x,G.x ∈ (fv_env G) → ∃T.!x ⊴ T ∈ G.
 intros 2;elim G 0
   [simplify;intro;lapply (not_in_list_nil ? ? H);elim Hletin
   |intros 3;
-   elim a;simplify in H1;elim (in_list_cons_case ? ? ? ? H1)
+   elim a;simplify in H1;elim b in H1;simplify in H1
+   [elim (in_list_cons_case ? ? ? ? H1)
      [rewrite < H2;autobatch
-     |elim (H H2);elim H3;apply ex_intro[apply a1];autobatch]]
+     |elim (H H2);autobatch]
+   |elim (H H1);autobatch]]
 qed.
 
-lemma incl_bound_fv : ∀l1,l2.incl ? l1 l2 → incl ? (fv_env l1) (fv_env l2).
+lemma incl_bound_fv : ∀l1,l2.l1 ⊆ l2 → (fv_env l1) ⊆ (fv_env l2).
 intros;unfold in H;unfold;intros;apply boundinenv_natinfv;
 lapply (natinfv_boundinenv ? ? H1);decompose;autobatch depth=4;
 qed.
 
-lemma incl_cons : ∀x,l1,l2.incl ? l1 l2 → incl nat (x :: l1) (x :: l2).
-intros.unfold in H.unfold.intros.elim (in_list_cons_case ? ? ? ? H1)
-  [applyS in_list_head|autobatch]
-qed.
-
-lemma WFT_env_incl : ∀G,T.WFType G T → 
-                       ∀H.incl ? (fv_env G) (fv_env H) → WFType H T.
+lemma WFT_env_incl : ∀G,T.(G ⊢ T) → ∀H.fv_env G ⊆ fv_env H → (H ⊢ T).
 intros 3.elim H
   [apply WFT_TFree;unfold in H3;apply (H3 ? H1)
   |apply WFT_Top
@@ -166,19 +172,21 @@ intros 3.elim H
      [apply (H2 ? H6)
      |intros;apply (H4 ? ? H8)
         [unfold;intro;autobatch
-        |simplify;apply (incl_cons ? ? ? H6)]]]
+        |simplify;apply (incl_cons ???? H6)]]]
 qed.
 
-lemma fv_env_extends : ∀H,x,B,C,T,U,G.
-                          fv_env (H @ ((mk_bound B x T) :: G)) = 
-                          fv_env (H @ ((mk_bound C x U) :: G)).
-intros;elim H
-  [reflexivity|elim a;simplify;rewrite > H1;reflexivity]
+lemma fv_env_extends : ∀H,x,T,U,G,B.
+                         fv_env (H @ mk_bound B x T :: G) = 
+                         fv_env (H @ mk_bound B x U :: G).
+intros 5;elim H;elim B
+  [1,2:reflexivity
+  |*:elim a;elim b;simplify;lapply (H1 true);lapply (H1 false);
+   try rewrite > Hletin;try rewrite > Hletin1;reflexivity]
 qed.
 
 lemma lookup_env_extends : ∀G,H,B,C,D,T,U,V,x,y.
-            in_list ? (mk_bound D y V) (H @ ((mk_bound C x U) :: G)) → y ≠ x → 
-            in_list ? (mk_bound D y V) (H @ ((mk_bound B x T) :: G)).
+            mk_bound D y V ∈ H @ mk_bound C x U :: G → y ≠ x → 
+            mk_bound D y V ∈ H @ mk_bound B x T :: G.
 intros 10;elim H
   [simplify in H1;elim (in_list_cons_case ? ? ? ? H1)
      [destruct H3;elim H2;reflexivity
@@ -188,8 +196,7 @@ intros 10;elim H
      |apply (in_list_cons ? ? ? ? (H1 H4 H3))]]
 qed.
 
-lemma in_FV_subst : ∀x,T,U,n.in_list ? x (fv_type T) →
-                                in_list ? x (fv_type (subst_type_nat T U n)).
+lemma in_FV_subst : ∀x,T,U,n.x ∈ fv_type T → x ∈ fv_type (subst_type_nat T U n).
 intros 3;elim T
   [simplify in H;elim (not_in_list_nil ? ? H)
   |2,3:simplify;simplify in H;assumption
@@ -199,7 +206,7 @@ qed.
 
 (*** lemma on fresh names ***)
 
-lemma fresh_name : ∀l:list nat.∃n.¬in_list ? n l.
+lemma fresh_name : ∀l:list nat.∃n.n ∉ l.
 cut (∀l:list nat.∃n.∀m.n ≤ m → ¬ in_list ? m l);intros
   [lapply (Hcut l);elim Hletin;apply ex_intro;autobatch
   |elim l
@@ -215,8 +222,7 @@ qed.
 
 (*** lemmata on well-formedness ***)
 
-lemma fv_WFT : ∀T,x,G.WFType G T → in_list ? x (fv_type T) →
-                    in_list ? x (fv_env G).
+lemma fv_WFT : ∀T,x,G.(G ⊢ T) → x ∈ fv_type T → x ∈ fv_env G.
 intros 4.elim H
   [simplify in H2;elim (in_list_cons_case ? ? ? ? H2)
      [applyS H1|elim (not_in_list_nil ? ? H3)]
@@ -238,11 +244,11 @@ qed.
 
 (*** lemmata relating subtyping and well-formedness ***)
 
-lemma JS_to_WFE : ∀G,T,U.G ⊢ T ⊴ U → WFEnv G.
+lemma JS_to_WFE : ∀G,T,U.G ⊢ T ⊴ U → G ⊢ ♦.
 intros;elim H;assumption.
 qed.
 
-lemma JS_to_WFT : ∀G,T,U.G ⊢ T ⊴ U → WFType G T ∧ WFType G U.
+lemma JS_to_WFT : ∀G,T,U.G ⊢ T ⊴ U → (G ⊢ T) ∧ (G ⊢ U).
 intros;elim H
   [1,2:autobatch
   |split 
@@ -256,18 +262,18 @@ intros;elim H
       apply (WFT_env_incl ? ? H10);simplify;unfold;intros;assumption]]
 qed.
 
-lemma JS_to_WFT1 : ∀G,T,U.G ⊢ T ⊴ U → WFType G T.
+lemma JS_to_WFT1 : ∀G,T,U.G ⊢ T ⊴ U → G ⊢ T.
 intros;elim (JS_to_WFT ? ? ? H);assumption;
 qed.
 
-lemma JS_to_WFT2 : ∀G,T,U.G ⊢ T ⊴ U → WFType G U.
+lemma JS_to_WFT2 : ∀G,T,U.G ⊢ T ⊴ U → G ⊢ U.
 intros;elim (JS_to_WFT ? ? ? H);assumption;
 qed.
 
-lemma WFE_Typ_subst : ∀H,x,B,C,T,U,G.
-                      WFEnv (H @ ((mk_bound B x T) :: G)) → WFType G U →
-                      WFEnv (H @ ((mk_bound C x U) :: G)).
-intros 7;elim H 0
+lemma WFE_Typ_subst : ∀H,x,B,T,U,G.
+                      H @ mk_bound B x T :: G ⊢ ♦ → (G ⊢ U) →
+                      H @ mk_bound B x U :: G ⊢ ♦.
+intros 6;elim H 0
   [simplify;intros;inversion H1;intros
      [elim (nil_cons ? G (mk_bound B x T) H3)
      |destruct H7;autobatch]
@@ -276,15 +282,14 @@ intros 7;elim H 0
      [destruct H5
      |destruct H9;apply WFE_cons
         [apply (H1 H5 H3)
-        |rewrite < (fv_env_extends ? x B C T U); assumption
+        |rewrite < (fv_env_extends ? x T U); assumption
         |apply (WFT_env_incl ? ? H8);
-         rewrite < (fv_env_extends ? x B C T U);unfold;intros;
+         rewrite < (fv_env_extends ? x T U);unfold;intros;
          assumption]]]
 qed.
 
-lemma WFE_bound_bound : ∀B,x,T,U,G.WFEnv G → in_list ? (mk_bound B x T) G →
-                        in_list ? (mk_bound B x U) G → T = U.
-intros 6;elim H
+lemma WFE_bound_bound : ∀x,T,U,G.G ⊢ ♦ → !x ⊴  T ∈ G → !x ⊴ U ∈ G → T = U.
+intros 5;elim H
   [lapply (not_in_list_nil ? ? H1);elim Hletin
   |elim (in_list_cons_case ? ? ? ? H6)
      [destruct H7;destruct;elim (in_list_cons_case ? ? ? ? H5)
@@ -295,9 +300,9 @@ intros 6;elim H
         |apply (H2 H8 H7)]]]
 qed.
 
-lemma WFT_to_incl: ∀G,T,U.(∀X.¬in_list ? X (fv_env G) → ¬in_list ? X (fv_type U) →
-    WFType (mk_bound true X T::G) (subst_type_nat U (TFree X) O))
-  → incl ? (fv_type U) (fv_env G).
+lemma WFT_to_incl: ∀G,T,U.(∀X.X ∉ fv_env G → X ∉ fv_type U →
+    (mk_bound true X T::G ⊢ (subst_type_nat U (TFree X) O))) →
+    fv_type U ⊆ fv_env G.
 intros;elim (fresh_name (fv_type U@fv_env G));lapply(H a)
   [unfold;intros;lapply (fv_WFT ? x ? Hletin)
      [simplify in Hletin1;inversion Hletin1;intros
@@ -308,11 +313,12 @@ intros;elim (fresh_name (fv_type U@fv_env G));lapply(H a)
 qed.
 
 lemma incl_fv_env: ∀X,G,G1,U,P.
-      incl ? (fv_env (G1@(mk_bound true X U::G))) 
-             (fv_env (G1@(mk_bound true X P::G))).
+      fv_env (G1@ !X ⊴ U::G) ⊆ fv_env (G1@ !X ⊴ P::G).
 intros.rewrite < fv_env_extends.apply incl_A_A.
 qed.
 
 lemma fv_append : ∀G,H.fv_env (G @ H) = fv_env G @ fv_env H.
-intro;elim G;simplify;autobatch paramodulation;
+intro;elim G;simplify;
+[reflexivity
+|elim a;simplify;elim b;simplify;lapply (H H1);rewrite > Hletin;reflexivity]
 qed.
\ No newline at end of file