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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Fsub/part1a_inversion/".
+include "Fsub/defn.ma".
+
+(*** Lemma A.1 (Reflexivity) ***)
+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 : ∀G,T,U.G ⊢ T ⊴ U → ∀H.WFEnv H → incl ? G H → H ⊢ T ⊴ U.
+intros 4;elim H
+  [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_list_head
+        |destruct H13;apply in_list_cons;apply (H7 ? H10)]]]
+qed.
+
+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 JSubtype_Arrow_inv:
+ ∀G:list bound.∀T1,T2,T3:Typ.
+  ∀P:list bound → Typ → Prop.
+   (∀n,t1.
+    (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Arrow T2 T3) → P G t1 → P G (TFree n)) →
+   (∀t,t1. G ⊢ T2 ⊴ t → G ⊢ t1 ⊴ T3 → P G (Arrow t t1)) →
+    G ⊢ T1 ⊴ (Arrow T2 T3) → P G T1.
+ intros;
+ generalize in match (refl_eq ? (Arrow T2 T3));
+ generalize in match (refl_eq ? G);
+ elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
+  [1,2: destruct H6
+  |5: destruct H8
+  | lapply (H5 H6 H7); destruct; clear H5;
+    apply H;
+    assumption
+  | destruct;
+    clear H4 H6;
+    apply H1;
+    assumption
+  ]
+qed.
+
+lemma JSubtype_Forall_inv:
+ ∀G:list bound.∀T1,T2,T3:Typ.
+  ∀P:list bound → Typ → Prop.
+   (∀n,t1.
+    (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Forall T2 T3) → P G t1 → P G (TFree n)) →
+   (∀t,t1. G ⊢ T2 ⊴ t → (∀X. ¬(X ∈ fv_env G) → (mk_bound true X T2)::G ⊢ subst_type_nat t1 (TFree X) O ⊴ subst_type_nat T3 (TFree X) O)
+     → P G (Forall t t1)) →
+      G ⊢ T1 ⊴ (Forall T2 T3) → P G T1.
+ intros;
+ generalize in match (refl_eq ? (Forall T2 T3));
+ generalize in match (refl_eq ? G);
+ elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
+  [1,2: destruct H6
+  |4: destruct H8
+  | lapply (H5 H6 H7); destruct; clear H5;
+    apply H;
+    assumption
+  | destruct;
+    clear H4 H6;
+    apply H1;
+    assumption
+  ]
+qed.
+
+
+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
+  |apply (JSubtype_Arrow_inv ? ? ? ? ? ? ? H6); intros;
+    [ autobatch
+    | inversion H7;intros; destruct; autobatch depth=4 width=4 size=9]
+  |apply (JSubtype_Forall_inv ? ? ? ? ? ? ? H6); intros;
+    [ autobatch
+    | inversion H7;intros; destruct;
+       [ apply SA_Top
+          [ assumption
+          | apply WFT_Forall;
+             [ autobatch
+             | intros;lapply (H8 ? H11);
+               autobatch]]
+       | apply SA_All
+          [ autobatch
+          | intros;apply (H4 X);
+             [intro;apply H13;apply H5;assumption
+                 |intro;apply H13;apply H5;apply (WFT_to_incl ? ? ? H3);
+                  assumption
+                 |simplify;autobatch
+                 |apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
+                    [intros;apply H2
+                       [unfold;intros;lapply (H5 ? H15);rewrite > fv_append;
+                        autobatch
+                       |apply (JS_weakening ? ? ? H9)
+                          [autobatch
+                          |unfold;intros;autobatch]
+                       |assumption]
+                    |*:autobatch]
+                 |autobatch]]]]]
+qed.
+
+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 : ∀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.