matita/matita/contribs/lambdadelta/basic_2/static/fle_fle.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/syntax/lveq_lveq.ma".
  16 include "basic_2/static/frees_frees.ma".
  17 include "basic_2/static/fle_fqup.ma".
  18
  19 (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma fle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
  24                        ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
  25                        ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 &
  26                                    L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
  27 #L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
  28 lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
  29 lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
  30 lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
  31 /2 width=6 by ex3_3_intro/
  32 qed-.
  33
  34 (* Main properties **********************************************************)
  35 (*
  36 theorem fle_trans: bi_transitive … fle.
  37 #L1 #L #T1 #T * #f1 #f #HT1 #HT #Hf1 #L2 #T2 * #g #f2 #Hg #HT2 #Hf2
  38 /5 width=8 by frees_mono, sle_trans, sle_eq_repl_back2, ex3_2_intro/
  39 qed-.
  40 *)
  41 theorem fle_bind_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
  42                      ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
  43 #L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
  44 elim (fle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
  45 elim (lveq_inj_void_sn … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
  46 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
  47 <tls_xn in H2n2; #H2n2
  48 /4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
  49 qed.
  50
  51 theorem fle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
  52                      ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
  53 #L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
  54 elim (fle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
  55 elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
  56 elim (sor_isfin_ex f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
  57 /4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
  58 qed.
  59 (*
  60 theorem fle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
  61                   ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
  62                   ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
  63 #L1 #L2 #V1 #V2 #HV #I1 #I2 #T1 #T2 #HT #p
  64 @fle_bind_sn
  65 [ @fle_bind_dx_sn //
  66 | @fle_bind_dx_dx
  67
  68
  69 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
  70 elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
  71 /4 width=12 by frees_bind, monotonic_sle_sor, sle_tl, ex3_2_intro/
  72 qed.
  73 *)
  74 theorem fle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
  75                   ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
  76                   ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
  77 /3 width=1 by fle_flat_sn, fle_flat_dx_dx, fle_flat_dx_sn/ qed-.