matita/matita/contribs/lambdadelta/basic_2/substitution/cofrees_alt.ma

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  14
  15 include "basic_2/relocation/cpy_nlift.ma".
  16 include "basic_2/substitution/cofrees_lift.ma".
  17
  18 (* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
  19
  20 (* Alternative definition of frees_ge ***************************************)
  21
  22 lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
  23                     (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
  24                     ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
  25                                (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
  26 #L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
  27 #U1 #U2 #HU12 #HU2 *
  28 [ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
  29   * /5 width=9 by cofrees_fwd_nlift, ex5_4_intro, or_intror/
  30 | * #I2 #K2 #W2 #j2 #Hdj2 #Hj2i #HLK2 #HnW2 #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /4 width=9 by ex5_4_intro, or_intror/
  31   * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
  32   lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
  33   #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
  34   #HK12
  35   @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
  36   @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
  37   >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
  38 ]
  39 qed-.
  40
  41 lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
  42                     (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
  43                     (∀d,i,j,I,L,K,W,U. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W → (K ⊢ i-j-1 ~ϵ 𝐅*[0]⦃W⦄ → ⊥) → (∀T. ⇧[j, 1] T ≡ U → ⊥) → R 0 (i-j-1) K W → R d i L U) →
  44                     ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
  45 #R #IH1 #IH2 #d #i #L #U
  46 generalize in match d; -d generalize in match i; -i
  47 @(f2_ind … rfw … L U) -L -U
  48 #n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
  49 -IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
  50 qed-.