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[helm.git] / matita / contribs / POPLmark / Fsub / part1a_inversion3.ma

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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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "Fsub/defn2.ma".
+
+(*** Lemma A.1 (Reflexivity) ***)
+theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴  T.
+intros 3; elim H;
+ [1,2,3: autobatch
+ | apply SA_All;
+    [ autobatch
+    | 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;
+ [1,2,3,4: autobatch depth=4 width=4 size=7
+ | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));intros;
+   apply H4
+     [ intro; autobatch
+     | apply WFE_cons; autobatch
+     | unfold;intros; elim (in_list_cons_case ? ? ? ? H9); destruct; autobatch]]
+qed.
+
+lemma JSubtype_inv:
+ ∀G:list bound.∀T1,T:Typ.
+  ∀P:list bound → Typ → Typ → Prop.
+   (∀t. WFEnv G → WFType G t → T=Top → P G t Top) →
+   (∀n. WFEnv G → n ∈ fv_env G → T=TFree n → P G (TFree n) (TFree n)) →
+   (∀n,t1,t.
+    (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ t → P G t1 t → T=t → P G (TFree n) T) →
+   (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 → G ⊢ s2 ⊴ t2 → T=Arrow t1 t2 → P G (Arrow s1 s2) (Arrow t1 t2)) →
+   (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 →
+    (∀X. ¬(X ∈ fv_env G) → (mk_bound true X t1)::G ⊢ subst_type_nat s2 (TFree X) O ⊴ subst_type_nat t2 (TFree X) O)
+      → T=Forall t1 t2 → P G (Forall s1 s2) (Forall t1 t2)) →
+    G ⊢ T1 ⊴ T → P G T1 T.
+ intros;
+ generalize in match (refl_eq ? T);
+ generalize in match (refl_eq ? G);
+ elim H5 in ⊢ (? ? ? % → ? ? ? % → %); destruct; 
+  [1,2,3,4: autobatch depth=10 width=10 size=8
+  | apply H4; first [assumption | autobatch]]
+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; destruct;
+ [1,2,4: autobatch width=10 depth=10 size=8
+ | elim (decidable_eq_nat X n)
+    [apply (SA_Trans_TVar ? ? ? P); destruct;
+      [ autobatch
+      | rewrite > append_cons; apply H1;
+        lapply (WFE_bound_bound true X t1 U ? ? H3); destruct;
+          [1,3: autobatch
+          | rewrite < append_cons; autobatch
+          ]]
+    | apply (SA_Trans_TVar ? ? ? t1)
+      [ apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
+        intro; autobatch
+      | autobatch]]
+ | apply SA_All;
+    [ autobatch
+    | intros;
+      apply (H6 ? ? (mk_bound true X1 t2::l1))
+      [ rewrite > fv_env_extends; autobatch
+      | autobatch]]]
+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;
+  [ apply (JSubtype_inv ? ? ? ? ? ? ? ? ? H3); intros; destruct; autobatch
+  | inversion H3; intros; destruct; assumption
+  |*: apply (JSubtype_inv ? ? ? ? ? ? ? ? ? H6); intros; destruct;
+    [1,3: autobatch
+    |*: inversion H7; intros; destruct;
+      [1,2: autobatch depth=4 width=4 size=9
+      | apply SA_Top
+         [ assumption
+         | apply WFT_Forall;
+            [ autobatch
+            | intros;lapply (H8 ? H11);
+              autobatch]]
+      | apply SA_All
+         [ autobatch
+         | intros;apply (H4 X);
+            [intro; autobatch;
+            |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; 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.