--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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;rewrite > Hcut;apply in_Base
+ |destruct H13;rewrite < Hcut1 in H10;apply in_Skip;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));
+ change in ⊢ (% → ?) with (Arrow T2 T3 = Arrow T2 T3);
+ generalize in match (refl_eq ? G);
+ change in ⊢ (% → ?) with (G = G);
+ elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
+ [1,2: destruct H6
+ |5: destruct H8
+ | lapply (H5 H6 H7); subst; clear H5;
+ apply H;
+ assumption
+ | subst;
+ 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; 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
+ [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 : ∀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.
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