matita/matita/contribs/lambdadelta/basic_2/multiple/frees_leq.ma

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  14
  15 include "basic_2/substitution/ldrop_leq.ma".
  16 include "basic_2/multiple/frees.ma".
  17
  18 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
  19
  20 (* Properties on equivalence for local environments *************************)
  21
  22 lemma leq_frees_trans: ∀L2,U,d,i. L2 ⊢ i ϵ 𝐅*[d]⦃U⦄ →
  23                        ∀L1. L1 ≃[d, ∞] L2 → L1 ⊢ i ϵ 𝐅*[d]⦃U⦄.
  24 #L2 #U #d #i #H elim H -L2 -U -d -i /3 width=2 by frees_eq/
  25 #I2 #L2 #K2 #U #W2 #d #i #j #Hdj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
  26 elim (leq_ldrop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
  27 lapply (leq_inv_O_Y … HK12) -HK12 #H destruct /3 width=9 by frees_be/
  28 qed-.
  29
  30 lemma frees_leq_conf: ∀L1,U,d,i. L1 ⊢ i ϵ 𝐅*[d]⦃U⦄ →
  31                       ∀L2. L1 ≃[d, ∞] L2 → L2 ⊢ i ϵ 𝐅*[d]⦃U⦄.
  32 /3 width=3 by leq_sym, leq_frees_trans/ qed-.