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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "ground_2/relocation/rtmap_id.ma".
  16 include "basic_2/notation/relations/subseteq_4.ma".
  17 include "basic_2/syntax/voids_length.ma".
  18 include "basic_2/static/frees.ma".
  19
  20 (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
  21
  22 inductive fle (T1) (T2): relation lenv ≝
  23 | fle_intro: ∀f1,f2,L1,L2,n1,n2. ⓧ*[n1]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → ⓧ*[n2]L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
  24              |L1| = |L2| → ⫱*[n1]f1 ⊆ ⫱*[n2]f2 → fle T1 T2 (ⓧ*[n1]L1) (ⓧ*[n2]L2)
  25 .
  26
  27 interpretation "free variables inclusion (restricted closure)"
  28    'SubSetEq L1 T1 L2 T2 = (fle T1 T2 L1 L2).
  29
  30 (* Basic properties *********************************************************)
  31
  32 lemma fle_sort: ∀L1,L2. |L1| = |L2| → ∀s1,s2. ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
  33 /3 width=5 by frees_sort, sle_refl, fle_intro/ qed.
  34
  35 lemma fle_gref: ∀L1,L2. |L1| = |L2| → ∀l1,l2. ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
  36 /3 width=5 by frees_gref, sle_refl, fle_intro/ qed.
  37
  38 (* Basic inversion lemmas ***************************************************)
  39
  40 fact fle_inv_voids_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
  41                         ∀K1,K2,n1,n2. |K1| = |K2| → L1 = ⓧ*[n1]K1 → L2 = ⓧ*[n2]K2 →
  42                         ∃∃f1,f2. ⓧ*[n1]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & ⓧ*[n2]K2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
  43 #L1 #L2 #T1 #T2 * -L1 -L2
  44 #f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #HL12 #Hf12 #Y1 #Y2 #x1 #x2 #HY12 #H1 #H2 destruct
  45 >H1 in Hf1; >H2 in Hf2; #Hf2 #Hf1
  46 @(ex3_2_intro … Hf1 Hf2) -Hf1 -Hf2
  47
  48 elim (voids_inj_length … H1) // -H -HL12 -HY #H1 #H2 destruct
  49 /2 width=5 by ex3_2_intro/
  50 qed-.
  51
  52 lemma fle_inv_voids_sn: ∀L1,L2,T1,T2,n. ⦃ⓧ*[n]L1, T1⦄ ⊆ ⦃L2, T2⦄ → |L1| = |L2| →
  53                         ∃∃f1,f2. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n]f1 ⊆ f2.
  54 /2 width=3 by fle_inv_voids_sn_aux/ qed-.