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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  14
  15 include "basic_2/static/lsubf.ma".
  16
  17 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
  18
  19 (* Properties with context-sensitive free variables *************************)
  20
  21 axiom lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 → ∀f,L1. ⦃L1, f⦄ ⫃𝐅* ⦃L2, f2⦄ →
  22                          ∃∃f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 & f1 ⊆ f.
  23 (*
  24 #f2 #L2 #T #H elim H -f2 -L2 -T
  25 [ #f2 #I #Hf2 #f #L1 #H elim (lsubf_inv_atom2 … H) -H
  26   #H #_ destruct /3 width=3 by frees_atom, sle_isid_sn, ex2_intro/
  27 | #f2 #I #K2 #W #s #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  28   [ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
  29   destruct elim (IH … H12) -K2
  30   /3 width=3 by frees_sort, sle_inv_tl_dx, ex2_intro/
  31 | #f2 #I #K2 #W #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  32   [ #K1 #H elim (sle_inv_nx … H ??) -H [ <tl_next_rew |*: // ]
  33     #g2 #_ #H1 #H12 #H2 destruct elim (IH … H12) -K2
  34     /3 width=7 by frees_zero, sle_next, ex2_intro/
  35   | #g #K1 #V #Hg <tl_next_rew #Hf lapply (sor_sym … Hf) -Hf
  36     #Hf #H elim (sle_inv_nx … H ??) -H [|*: // ]
  37     #g2 #_ #H1 #H12 #H2 #H3 destruct elim (IH … H12) -K2
  38     #f1 #Hf1 elim (sor_isfin_ex … f1 g ??)
  39     /5 width=10 by frees_fwd_isfin, frees_flat, frees_zero, monotonic_sle_sor, sor_inv_sle_dx, sor_sym, sor_sle_sn, sle_next, ex2_intro/
  40   ]
  41 | #f2 #I #K2 #W #i #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  42   [ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
  43   destruct elim (IH … H12) -K2
  44   /3 width=3 by frees_lref, sle_inv_tl_dx, ex2_intro/
  45 | #f2 #I #K2 #W #l #_ #IH #f #L1 #H elim (lsubf_inv_pair2 … H) -H *
  46   [ #K1 #_ #H12 #H | #g #K1 #V #Hg #Hf #_ #H12 #H1 #H2 ]
  47   destruct elim (IH … H12) -K2
  48   /3 width=3 by frees_gref, sle_inv_tl_dx, ex2_intro/
  49 | #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f #L1 #H12
  50 | #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f #L1 #H12
  51 *)