X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FFsub%2Fpart1a.ma;h=48543ce18dd07ae6f6c1c65816de48e3f960791d;hb=b58315ef220a130a44acbf528cd6885ddadad642;hp=e12a64be4809093d647ba9f04a8906d42d2a9f83;hpb=247ec86cf6e8ba95843b408453afd57f7cf2d075;p=helm.git

diff --git a/helm/software/matita/library/Fsub/part1a.ma b/helm/software/matita/library/Fsub/part1a.ma
index e12a64be4..48543ce18 100644
--- a/helm/software/matita/library/Fsub/part1a.ma
+++ b/helm/software/matita/library/Fsub/part1a.ma
@@ -13,264 +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]]
-     |(*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 [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
-        |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 (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;rewrite > Hcut;apply in_Base
+        |destruct H13;rewrite < Hcut1 in H10;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 n.
-  (\forall G,T,Q,U.
-     (t_len Q) \leq n \to (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 
-     (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 
-              (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)]
+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; subst;
+    [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? ? H8 H9);autobatch
+    |inversion H11;intros; subst; 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 ⊢ (? ? ? %→%);subst
+     [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? H7 H8 H9 H10);reflexivity
+     |inversion H11;intros;subst
+        [apply SA_Top
+           [autobatch
+              |apply WFT_Forall
+                 [autobatch
+                 |intros;lapply (H4 ? H13);autobatch]]
            |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 s 0;simplify;intros;apply incl_nat_cons;assumption]]]
-  |intros 4;(*generalize in match H1;*)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;elim (H (pred m))
-           [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)
-              |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]
-     |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;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)]]]]
+              [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::l2) 
+                         ? ? ? ? ? H7 ? ? [])
+                    [intros;apply H9
+                       [unfold;intros;lapply (H8 ? H17);rewrite > fv_append;
+                        autobatch
+                       |apply (JS_weakening ? ? ? H7)
+                          [autobatch
+                          |unfold;intros;autobatch]
+                       |assumption]
+                    |*: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 (t_len U));apply (H2 ? ? ? ? ? H H1);constructor 1;
+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 (t_len Q));apply (H3 ? ? ? ? ? ? ? ? H H1);
-constructor 1;
+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.