matita/matita/contribs/lambdadelta/basic_2/static/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 "ground_2/relocation/rtmap_id.ma".
  16 include "basic_2/notation/relations/subseteq_4.ma".
  17 include "basic_2/syntax/lveq.ma".
  18 include "basic_2/static/frees.ma".
  19
  20 (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
  21
  22 definition fle: bi_relation lenv term ≝ λL1,T1,L2,T2.
  23                 ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 &
  24                                L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
  25
  26 interpretation "free variables inclusion (restricted closure)"
  27    'SubSetEq L1 T1 L2 T2 = (fle L1 T1 L2 T2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma fle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄.
  32 #L elim (lveq_refl L)
  33 /3 width=8 by frees_sort, sle_refl, ex4_4_intro/
  34 qed.
  35
  36 lemma fle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄.
  37 #L elim (lveq_refl L)
  38 /3 width=8 by frees_gref, sle_refl, ex4_4_intro/
  39 qed.
  40
  41 (* Basic inversion lemmas ***************************************************)
  42 (*
  43 fact fle_inv_voids_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
  44                         ∀K1,K2,n1,n2. |K1| = |K2| → L1 = ⓧ*[n1]K1 → L2 = ⓧ*[n2]K2 →
  45                         ∃∃f1,f2. ⓧ*[n1]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & ⓧ*[n2]K2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
  46 #L1 #L2 #T1 #T2 * -L1 -L2
  47 #f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #HL12 #Hf12 #Y1 #Y2 #x1 #x2 #HY12 #H1 #H2 destruct
  48 >H1 in Hf1; >H2 in Hf2; #Hf2 #Hf1
  49 @(ex3_2_intro … Hf1 Hf2) -Hf1 -Hf2
  50
  51 elim (voids_inj_length … H1) // -H -HL12 -HY #H1 #H2 destruct
  52 /2 width=5 by ex3_2_intro/
  53 qed-.
  54
  55 lemma fle_inv_voids_sn: ∀L1,L2,T1,T2,n. ⦃ⓧ*[n]L1, T1⦄ ⊆ ⦃L2, T2⦄ → |L1| = |L2| →
  56                         ∃∃f1,f2. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n]f1 ⊆ f2.
  57 /2 width=3 by fle_inv_voids_sn_aux/ qed-.
  58 *)