From: Claudio Sacerdoti Coen Date: Tue, 16 Oct 2007 21:39:28 +0000 (+0000) Subject: Interesting. Do we need many more inversion lemmas? (Or a different and X-Git-Tag: 0.4.95@7852~110 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=1b6e2d62ab142ce452d93e6d9acd63bc19baf148;p=helm.git Interesting. Do we need many more inversion lemmas? (Or a different and dumbier inversion tactic?). --- diff --git a/matita/library/Fsub/part1a_inversion.ma b/matita/library/Fsub/part1a_inversion.ma new file mode 100644 index 000000000..463ceca95 --- /dev/null +++ b/matita/library/Fsub/part1a_inversion.ma @@ -0,0 +1,156 @@ +(**************************************************************************) +(* ___ *) +(* ||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.