matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn_alt2.etc

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  14
  15 include "basic_2/substitution/cofrees_lift.ma".
  16 include "basic_2/substitution/llpx_sn_alt1.ma".
  17
  18 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
  19
  20 (* alternative definition of llpx_sn (not recursive) *)
  21 definition llpx_sn_alt2: relation4 bind2 lenv term term → relation4 ynat term lenv lenv ≝
  22                          λR,d,T,L1,L2. |L1| = |L2| ∧
  23                          (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) →
  24                             ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
  25                             I1 = I2 ∧ R I1 K1 V1 V2
  26                          ).
  27
  28 (* Main properties **********************************************************)
  29
  30 lemma cpy_inv_nlift2_be: ∀G,L,U1,U2,d. ⦃G, L⦄ ⊢ U1 ▶[d, ∞] U2 → ∀i. d ≤ yinj i →
  31                          ∀K,s. ⇩[s, i, 1] L ≡ K →
  32                          (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) → (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥).
  33 #G #L #U1 #U2 #d #HU12 #i #Hdi #K #s #HLK #HnU2 #T1 #HTU1
  34 elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) /2 width=2 by/
  35 qed-.
  36
  37 theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2.
  38 #R #L1 #L2 #U1 #d #H elim (llpx_sn_inv_alt1 … H) -H
  39 #HL12 #IH @conj // -HL12
  40 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU1 #HLK1 #HLK2 elim (frees_inv_gen … HnU1) -HnU1
  41 #U2 #H generalize in match IH; -IH @(cpys_ind_dx … H) -U1
  42 [ #IH #HnU2 elim (IH … HnU2 … HLK1 HLK2) -L1 -L2 -U2 /2 width=1 by conj/
  43 | #U1 #U0 #HU10 #_ #IHU02 #IH #HnU2 @IHU02 /2 width=2 by/ -I1 -I2 -K1 -K2 -V1 -V2 -U2 -i
  44   #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH // -R -I2 -L2 -K2 -V2
  45   @(cpy_inv_nlift2_be … HU10) /2 width=3 by/