matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cofrees/cofrees_alt.etc

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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 nlift_frees: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥).
  23 #L #U #d #i #HnTU #H elim (cofrees_fwd_lift … H) -H /2 width=2 by/
  24 qed-.
  25
  26 lemma frees_inv_ge: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
  27                     (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
  28                     ∃∃I,K,W,j. d ≤ yinj j & j < i & ⇩[j]L ≡ K.ⓑ{I}W &
  29                                (K ⊢ i-j-1 ~ϵ 𝐅*[yinj 0]⦃W⦄ → ⊥) & (∀T. ⇧[j, 1] T ≡ U → ⊥).
  30 #L #U #d #i #Hdi #H @(frees_ind … H) -U /3 width=2 by or_introl/
  31 #U1 #U2 #HU12 #HU2 *
  32 [ #HnU2 elim (cpy_fwd_nlift2_ge … HU12 … HnU2) -HU12 -HnU2 /3 width=2 by or_introl/
  33   * /5 width=9 by nlift_frees, ex5_4_intro, or_intror/
  34 | * #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/
  35   * #I1 #K1 #W1 #j1 #Hdj1 #Hj12 #HLK1 #HnW1 #HnU1
  36   lapply (ldrop_conf_ge … HLK1 … HLK2 ?) -HLK2 /2 width=1 by lt_to_le/
  37   #HK12 lapply (ldrop_inv_drop1_lt … HK12 ?) /2 width=1 by lt_plus_to_minus_r/ -HK12
  38   #HK12
  39   @or_intror @(ex5_4_intro … HLK1 … HnU1) -HLK1 -HnU1 /2 width=3 by transitive_lt/
  40   @(frees_be … HK12 … HnW1) /2 width=1 by arith_k_sn/ -HK12 -HnW1
  41   >minus_plus in ⊢ (??(?(?%?)?)??→?); >minus_plus in ⊢ (??(?(??%)?)??→?); >arith_b1 /2 width=1 by/
  42 ]
  43 qed-.
  44
  45 lemma frees_ind_ge: ∀R:relation4 ynat nat lenv term.
  46                     (∀d,i,L,U. d ≤ yinj i → (∀T. ⇧[i, 1] T ≡ U → ⊥) → R d i L U) →
  47                     (∀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) →
  48                     ∀d,i,L,U. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R d i L U.
  49 #R #IH1 #IH2 #d #i #L #U
  50 generalize in match d; -d generalize in match i; -i
  51 @(f2_ind … rfw … L U) -L -U
  52 #n #IHn #L #U #Hn #i #d #Hdi #H elim (frees_inv_ge … H) -H /3 width=2 by/
  53 -IH1 * #I #K #W #j #Hdj #Hji #HLK #HnW #HnU destruct /4 width=12 by ldrop_fwd_rfw/
  54 qed-.