matita/matita/contribs/lambdadelta/basic_2/etc/voids/lveq_voids.etc

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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 (* requires <key name="ex">3 6</key> *)
  16 include "ground_2/xoa/xoa2.ma".
  17 include "basic_2/syntax/voids_length.ma".
  18 include "basic_2/syntax/lveq.ma".
  19
  20 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
  21
  22 (* Inversion lemmas with extension with exclusion binders *******************)
  23
  24 lemma lveq_inv_voids: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
  25                       ∨∨ ∧∧ ⓧ*[n1]⋆ = L1 & ⓧ*[n2]⋆ = L2
  26                        | ∃∃I1,I2,K1,K2,V1,n. K1 ≋ⓧ*[n, n] K2 & ⓧ*[n1](K1.ⓑ{I1}V1) = L1 & ⓧ*[n2](K2.ⓘ{I2}) = L2
  27                        | ∃∃I1,I2,K1,K2,V2,n. K1 ≋ⓧ*[n, n] K2 & ⓧ*[n1](K1.ⓘ{I1}) = L1 & ⓧ*[n2](K2.ⓑ{I2}V2) = L2.
  28 #L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2
  29 [ /3 width=1 by conj, or3_intro0/
  30 |2,3: /3 width=9 by or3_intro1, or3_intro2, ex3_6_intro/
  31 |4,5: #K1 #K2 #n1 #n2 #HK12 * *
  32    /3 width=9 by conj, or3_intro0, or3_intro1, or3_intro2, ex3_6_intro/
  33 ]
  34 qed-.
  35
  36 (* Eliminators with extension with exclusion binders ************************)
  37
  38 lemma lveq_ind_voids: ∀R:bi_relation lenv nat. (
  39                          ∀n1,n2. R (ⓧ*[n1]⋆) n1 (ⓧ*[n2]⋆) n2
  40                       ) → (
  41                          ∀I1,I2,K1,K2,V1,n1,n2,n. K1 ≋ⓧ*[n, n] K2 → R K1 n K2 n →
  42                          R (ⓧ*[n1]K1.ⓑ{I1}V1) n1 (ⓧ*[n2]K2.ⓘ{I2}) n2
  43                       ) → (
  44                          ∀I1,I2,K1,K2,V2,n1,n2,n. K1 ≋ⓧ*[n, n] K2 → R K1 n K2 n →
  45                          R (ⓧ*[n1]K1.ⓘ{I1}) n1 (ⓧ*[n2]K2.ⓑ{I2}V2) n2
  46                       ) →
  47                       ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → R L1 n1 L2 n2.
  48 #R #IH1 #IH2 #IH3 #L1 #L2 @(f2_ind ?? length2 ?? L1 L2) -L1 -L2
  49 #m #IH #L1 #L2 #Hm #n1 #n2 #H destruct
  50 elim (lveq_inv_voids … H) -H * //
  51 #I1 #I2 #K1 #K2 #V #n #HK #H1 #H2 destruct
  52 /4 width=3 by lt_plus/
  53 qed-.
  54
  55 (*
  56
  57 (* Properties with extension with exclusion binders *************************)
  58
  59 lemma lveq_voids_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
  60                      ∀m1. ⓧ*[m1]L1 ≋ⓧ*[m1+n1, n2] L2.
  61 #L1 #L2 #n1 #n2 #HL12 #m1 elim m1 -m1 /2 width=1 by lveq_void_sn/
  62 qed-.
  63
  64 lemma lveq_voids_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
  65                      ∀m2. L1 ≋ⓧ*[n1, m2+n2] ⓧ*[m2]L2.
  66 #L1 #L2 #n1 #n2 #HL12 #m2 elim m2 -m2 /2 width=1 by lveq_void_dx/
  67 qed-.
  68
  69 lemma lveq_voids: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
  70                   ∀m1,m2. ⓧ*[m1]L1 ≋ⓧ*[m1+n1, m2+n2] ⓧ*[m2]L2.
  71 /3 width=1 by lveq_voids_dx, lveq_voids_sn/ qed-.
  72
  73 lemma lveq_voids_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
  74                        ∀n1,n2. ⓧ*[n1]L1 ≋ⓧ*[n1, n2] ⓧ*[n2]L2.
  75 #L1 #L2 #HL12 #n1 #n2
  76 >(plus_n_O … n1) in ⊢ (?%???); >(plus_n_O … n2) in ⊢ (???%?);
  77 /2 width=1 by lveq_voids/ qed-.
  78
  79 (* Inversion lemmas with extension with exclusion binders *******************)
  80
  81 lemma lveq_inv_voids_sn: ∀L1,L2,n1,n2,m1. ⓧ*[m1]L1 ≋ⓧ*[m1+n1, n2] L2 →
  82                          L1 ≋ⓧ*[n1, n2] L2.
  83 #L1 #L2 #n1 #n2 #m1 elim m1 -m1 /3 width=1 by lveq_inv_void_sn/
  84 qed-.
  85
  86 lemma lveq_inv_voids_dx: ∀L1,L2,n1,n2,m2. L1 ≋ⓧ*[n1, m2+n2] ⓧ*[m2]L2 →
  87                          L1 ≋ⓧ*[n1, n2] L2.
  88 #L1 #L2 #n1 #n2 #m2 elim m2 -m2 /3 width=1 by lveq_inv_void_dx/
  89 qed-.
  90
  91 lemma lveq_inv_voids: ∀L1,L2,n1,n2,m1,m2. ⓧ*[m1]L1 ≋ⓧ*[m1+n1, m2+n2] ⓧ*[m2]L2 →
  92                       L1 ≋ⓧ*[n1, n2] L2.
  93 /3 width=3 by lveq_inv_voids_dx, lveq_inv_voids_sn/ qed-.
  94
  95 lemma lveq_inv_voids_zero: ∀L1,L2,n1,n2. ⓧ*[n1]L1 ≋ⓧ*[n1, n2] ⓧ*[n2]L2 →
  96                            L1 ≋ⓧ*[0, 0] L2.
  97 /2 width=3 by lveq_inv_voids/ qed-.
  98
  99 *)