From: Claudio Sacerdoti Coen Date: Sun, 20 Apr 2008 21:32:11 +0000 (+0000) Subject: Alternative prove using just one induction/inversion principle. X-Git-Tag: make_still_working~5305 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=f11b3533be1310039dd48d0d0e92f96a020bd5e1;p=helm.git Alternative prove using just one induction/inversion principle. --- diff --git a/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion2.ma b/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion2.ma new file mode 100644 index 000000000..a8dd6370f --- /dev/null +++ b/helm/software/matita/contribs/POPLmark/Fsub/part1a_inversion2.ma @@ -0,0 +1,153 @@ +(**************************************************************************) +(* ___ *) +(* ||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/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_inv: + ∀G:list bound.∀T1,T:Typ. + ∀P:list bound → Typ → Prop. + (∀t. WFEnv G → WFType G t → T=Top → P G t) → + (∀n. T=TFree n → P G (TFree n)) → + (∀n,t1. + (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ T → P G t1 → P G (TFree n)) → + (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 → G ⊢ s2 ⊴ t2 → T=Arrow t1 t2 → P G (Arrow s1 s2)) → + (∀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)) → + G ⊢ T1 ⊴ T → P G T1. + intros; + generalize in match (refl_eq ? T); + generalize in match (refl_eq ? G); + elim H5 in ⊢ (? ? ? % → ? ? ? % → %); + [1,2: destruct; autobatch + | rewrite < H9 in H6 H7 H8 ⊢ %; + rewrite < H10 in H7 H8; + autobatch + | rewrite < H10 in H6 H8 ⊢ %; + autobatch + | rewrite < H10 in H6 H8 ⊢ %; + apply (H4 t t1 t2 t3); 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_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;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.