X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FFsub%2Fpart1a.ma;h=8558725cc883569783da832862df8b0d1ed8e969;hb=725f187fc8da130059794317091e03f509727fd8;hp=0437f1e29f4f354b028296a150de833801392277;hpb=03c219fd478b4160ef1bc0b9e66520afeb6394ac;p=helm.git

diff --git a/helm/software/matita/library/Fsub/part1a.ma b/helm/software/matita/library/Fsub/part1a.ma
index 0437f1e29..8558725cc 100644
--- a/helm/software/matita/library/Fsub/part1a.ma
+++ b/helm/software/matita/library/Fsub/part1a.ma
@@ -13,239 +13,122 @@
 (**************************************************************************)
 
 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]]
-     |(* no shortcut? *)
-      (*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]]
-  |cut (WFType G t)
-     [lapply (H2 t ? ? Hcut H4)
-        [apply t_len_forall1
-        |apply (SA_All ? ? ? ? ? Hletin);intros;apply H2
-           [rewrite < eq_t_len_TFree_subst;
-            apply t_len_forall2
-           |generalize in match H3;intro;inversion H3
-              [intros;destruct H9
-              |intros;destruct H8
-              |intros;destruct H12
-              |intros;destruct H12;subst;apply H9
-                 [assumption
-                 |unfold;intro;apply H5;
-                  elim (fresh_name ((fv_env l)@(fv_type t3)));
-                  cut ((\lnot (in_list ? a (fv_env l))) \land
-                       (\lnot (in_list ? a (fv_type t3))))
-                    [elim Hcut1;lapply (H9 ? H13 H14);
-                     lapply (fv_WFT ? X ? Hletin1)
-                       [simplify in Hletin2;inversion Hletin2
-                          [intros;lapply (inj_head_nat ? ? ? ? H16);subst;
-                           elim (H14 H11)
-                          |intros;lapply (inj_tail ? ? ? ? ? H18);
-                           rewrite < Hletin3 in H15;assumption]
-                       |apply varinT_varinT_subst;
-                        assumption]
-                    |split;unfold;intro;apply H12;apply natinG_or_inH_to_natinGH
-                       [right;assumption
-                       |left;assumption]]]]
-           |apply WFE_cons;assumption]]
-     |(*FIXME*)generalize in match H3;intro;inversion H3
-        [intros;destruct H8
-        |intros;destruct H7
-        |intros;destruct H11
-        |intros;destruct H11;subst;assumption]]]
+theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴  T.
+intros 3.elim H
+  [apply SA_Refl_TVar [apply H2|assumption]
+  |apply SA_Top [assumption|apply WFT_Top]
+  |apply (SA_Arrow ? ? ? ? ? (H2 H5) (H4 H5))
+  |apply (SA_All ? ? ? ? ? (H2 H5));intros;apply (H4 ? H6)
+     [intro;apply H6;apply (fv_WFT ? ? ? (WFT_Forall ? ? ? H1 H3));
+      simplify;autobatch
+     |autobatch]]
 qed.
 
-(* 
+(*
  * A slightly more general variant to lemma A.2.2, where weakening isn't
  * defined as concatenation of any two disjoint environments, but as
  * set inclusion.
  *)
- 
-lemma JS_weakening : \forall G,T,U.(JSubtype G T U) \to
-                      \forall H.(WFEnv H) \to (incl ? G H) \to (JSubtype H T U).
+
+lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.WFEnv H → incl ? G H → H ⊢ T ⊴ U.
 intros 4;elim H
-  [apply (SA_Top ? ? H4);lapply (incl_bound_fv ? ? H5);
-   apply (WFT_env_incl ? ? H2 ? Hletin)
-  |apply (SA_Refl_TVar ? ? H4);lapply (incl_bound_fv ? ? H5);
-   apply (Hletin ? H2)
-  |lapply (H3 ? H5 H6);lapply (H6 ? H1);
-   apply (SA_Trans_TVar ? ? ? ? Hletin1 Hletin)
-  |lapply (H2 ? H6 H7);lapply (H4 ? H6 H7);
-   apply (SA_Arrow ? ? ? ? ? Hletin Hletin1)
-  |lapply (H2 ? H6 H7);apply (SA_All ? ? ? ? ? Hletin);intros;apply H4
-     [unfold;intro;apply H8;lapply (incl_bound_fv ? ? H7);apply (Hletin1 ? H9)
-     |apply WFE_cons
-        [1,2:assumption
-        |apply (JS_to_WFT1 ? ? ? Hletin)]
-     |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 (SA_Top ? ? H4);apply (WFT_env_incl ? ? H2 ? (incl_bound_fv ? ? H5))
+  |apply (SA_Refl_TVar ? ? H4);apply (incl_bound_fv ? ? H5 ? H2)
+  |apply (SA_Trans_TVar ? ? ? ? ? (H3 ? H5 H6));apply H6;assumption
+  |apply (SA_Arrow ? ? ? ? ? (H2 ? H6 H7) (H4 ? H6 H7))
+  |apply (SA_All ? ? ? ? ? (H2 ? H6 H7));intros;apply H4
+     [unfold;intro;apply H8;apply (incl_bound_fv ? ? H7 ? H9)
+     |apply (WFE_cons ? ? ? ? H6 H8);autobatch
+     |unfold;intros;inversion H9;intros
+        [destruct H11;apply in_Base
+        |destruct H13;apply in_Skip;apply (H7 ? H10)]]]
 qed.
 
-(* Lemma A.3 (Transitivity and Narrowing) *)
+theorem narrowing:∀X,G,G1,U,P,M,N.
+  G1 ⊢ P ⊴ U → (∀G2,T.G2@G1 ⊢ U ⊴ T → G2@G1 ⊢ P ⊴ T) → G ⊢ M ⊴ N →
+  ∀l.G=l@(mk_bound true X U::G1) → l@(mk_bound true X P::G1) ⊢ M ⊴ N.
+intros 10.elim H2
+  [apply SA_Top
+    [rewrite > H5 in H3;
+     apply (WFE_Typ_subst ? ? ? ? ? ? ? H3 (JS_to_WFT1 ? ? ? H))
+    |rewrite > H5 in H4;apply (WFT_env_incl ? ? H4);apply incl_fv_env]
+  |apply SA_Refl_TVar
+    [rewrite > H5 in H3;apply (WFE_Typ_subst ? ? ? ? ? ? ? H3);
+     apply (JS_to_WFT1 ? ? ? H)
+    |rewrite > H5 in H4;rewrite < fv_env_extends;apply H4]
+  |elim (decidable_eq_nat X n)
+    [apply (SA_Trans_TVar ? ? ? P)
+      [rewrite < H7;elim l1;simplify
+        [constructor 1|constructor 2;assumption]
+      |rewrite > append_cons;apply H1;
+       lapply (WFE_bound_bound true n t1 U ? ? H3)
+        [apply (JS_to_WFE ? ? ? H4)
+        |rewrite < Hletin;rewrite < append_cons;apply (H5 ? H6)
+        |rewrite < H7;rewrite > H6;elim l1;simplify
+          [constructor 1|constructor 2;assumption]]]
+    |apply (SA_Trans_TVar ? ? ? t1)
+      [rewrite > H6 in H3;apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
+       unfold;intro;apply H7;symmetry;assumption
+      |apply (H5 ? H6)]]
+  |apply (SA_Arrow ? ? ? ? ? (H4 ? H7) (H6 ? H7))
+  |apply (SA_All ? ? ? ? ? (H4 ? H7));intros;
+   apply (H6 ? ? (mk_bound true X1 t2::l1))
+      [rewrite > H7;rewrite > fv_env_extends;apply H8
+      |simplify;rewrite < H7;reflexivity]]
+qed.
 
-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,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)).
-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))
-        [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;constructor 2;assumption]
-                   |apply H7
-                      [lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin);
-                       apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros;
-                       elim l1;simplify;constructor 2;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
-                            |constructor 2;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)
-     |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;apply SA_Arrow
-              [rewrite > H12 in H2;rewrite < Hcut in H8;
-               lapply (H6 t2)
-                 [elim Hletin;apply (H15 ? ? ? H8 H2)
-                 |apply (t_len_arrow1 t2 t3)]
-              |rewrite > H12 in H4;rewrite < Hcut1 in H10;
-               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)
-           |intros;lapply (H4 ? H13);lapply (JS_to_WFT1 ? ? ? Hletin);
-            apply (WFT_env_incl ? ? Hletin1);simplify;unfold;intros;
-            assumption]
-        |intros;destruct H11
-        |intros;destruct H12
-        |intros;destruct H13
-        |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
+lemma JS_trans_prova: ∀T,G1.WFType G1 T →
+∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
+intros 3;elim H;clear H; try autobatch;
+  [rewrite > (JSubtype_Top ? ? H3);autobatch
+  |generalize in match H7;generalize in match H4;generalize in match H2;
+   generalize in match H5;clear H7 H4 H2 H5;
+   generalize in match (refl_eq ? (Arrow t t1));
+   elim H6 in ⊢ (? ? ? %→%); clear H6; intros; destruct;
+    [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? ? H8 H9);autobatch
+    |inversion H11;intros; destruct; autobatch depth=4 width=4 size=9;
+    ]
+  |generalize in match H7;generalize in match H4;generalize in match H2;
+   generalize in match H5;clear H7 H4 H2 H5;
+   generalize in match (refl_eq ? (Forall t t1));elim H6 in ⊢ (? ? ? %→%);destruct;
+     [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? H7 H8 H9 H10);reflexivity
+     |inversion H11;intros;destruct;
+        [apply SA_Top
+           [autobatch
+              |apply WFT_Forall
+                 [autobatch
+                 |intros;lapply (H4 ? H13);autobatch]]
+           |apply SA_All
+              [autobatch paramodulation
+              |intros;apply (H10 X)
+                 [intro;apply H15;apply H8;assumption
+                 |intro;apply H15;apply H8;apply (WFT_to_incl ? ? ? H3);
+                  assumption
+                 |simplify;autobatch
+                 |apply (narrowing X (mk_bound true X t::l1)
+                         ? ? ? ? ? H7 ? ? [])
+                    [intros;apply H9
+                       [unfold;intros;lapply (H8 ? H17);rewrite > fv_append;
+                        autobatch
+                       |apply (JS_weakening ? ? ? H7)
+                          [autobatch
+                          |unfold;intros;autobatch]
                        |assumption]
-                    |apply t_len_forall1]
-                 |apply (H10 ? H12)]
-              |rewrite < eq_t_len_TFree_subst;
-               apply t_len_forall2]]]]]
+                    |*:autobatch]
+                 |autobatch]]]]]
 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;autobatch;
+theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
+intros 5;apply (JS_trans_prova ? G);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;autobatch;
+theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
+                       (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
+                       (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
+intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
+intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
+     [autobatch|unfold;intros;autobatch]
 qed.