matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_append.ma

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  14
  15 include "basic_2A/substitution/drop_append.ma".
  16 include "basic_2A/multiple/frees.ma".
  17
  18 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
  19
  20 (* Properties on append for local environments ******************************)
  21
  22 lemma frees_append: ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → i ≤ |L2| →
  23                     ∀L1. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
  24 #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
  25 #I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
  26 lapply (drop_fwd_length_minus2 … HLK2) normalize #H0
  27 lapply (drop_O1_append_sn_le … HLK2 … L1) -HLK2
  28 [ -I -L1 -K2 -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
  29 | #HLK2 @(frees_be … HnU HLK2) // -HnU -HLK2 @IHW -IHW
  30   >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
  31 ]
  32 qed.
  33
  34 (* Inversion lemmas on append for local environments ************************)
  35
  36 fact frees_inv_append_aux: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ → ∀L1,L2. L = L1 @@ L2 →
  37                            i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
  38 #L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/
  39 #Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
  40 elim (drop_O1_lt (Ⓕ) L2 j) [2: -Z -Y -L1 -X -U -l /2 width=3 by lt_to_le_to_lt/ ]
  41 #I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 … HLK2) normalize #H0
  42 lapply (drop_O1_inv_append1_le … HLY … HLK2) -HLY
  43 [ -Z -I -Y -K2 -L1 -X -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
  44 | normalize #H destruct
  45   @(frees_be … HnU HLK2) -HnU -HLK2 // @IHW -IHW //
  46   >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
  47 ]
  48 qed-.
  49
  50 lemma frees_inv_append: ∀L1,L2,U,l,i. L1 @@ L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
  51                         i ≤ |L2| → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
  52 /2 width=4 by frees_inv_append_aux/ qed-.