matita/matita/contribs/lambdadelta/basic_2/syntax/voids.ma

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  14
  15 include "basic_2/notation/relations/rvoidstar_4.ma".
  16 include "basic_2/syntax/lenv.ma".
  17
  18 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
  19
  20 inductive voids: bi_relation nat lenv ≝
  21 | voids_atom   : voids 0 (⋆) 0 (⋆)
  22 | voids_pair_sn: ∀I1,I2,K1,K2,V1,n. voids n K1 n K2 →
  23                  voids 0 (K1.ⓑ{I1}V1) 0 (K2.ⓘ{I2})
  24 | voids_pair_dx: ∀I1,I2,K1,K2,V2,n. voids n K1 n K2 →
  25                  voids 0 (K1.ⓘ{I1}) 0 (K2.ⓑ{I2}V2)
  26 | voids_void_sn: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 →
  27                  voids (⫯n1) (K1.ⓧ) n2 K2
  28 | voids_void_dx: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 →
  29                  voids n1 K1 (⫯n2) (K2.ⓧ)
  30 .
  31
  32 interpretation "equivalence up to exclusion binders (local environment)"
  33    'RVoidStar n1 L1 n2 L2 = (voids n1 L1 n2 L2).
  34
  35 (* Basic properties *********************************************************)
  36
  37 lemma voids_refl: ∀L. ∃n. ⓧ*[n]L ≋ ⓧ*[n]L.
  38 #L elim L -L /2 width=2 by ex_intro, voids_atom/
  39 #L #I * #n #IH cases I -I /3 width=2 by ex_intro, voids_pair_dx/
  40 * /4 width=2 by ex_intro, voids_void_sn, voids_void_dx/
  41 qed-.
  42
  43 lemma voids_sym: bi_symmetric … voids.
  44 #n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
  45 /2 width=2 by voids_atom, voids_pair_sn, voids_pair_dx, voids_void_sn, voids_void_dx/
  46 qed-.
  47
  48 (*
  49
  50
  51
  52
  53 lemma voids_zero: ∀L. L = ⓧ*[0]L.
  54 // qed.
  55
  56 lemma voids_succ: ∀L,n. (ⓧ*[n]L).ⓧ = ⓧ*[⫯n]L.
  57 // qed.
  58
  59 (* Advanced properties ******************************************************)
  60
  61 lemma voids_next: ∀n,L. ⓧ*[n](L.ⓧ) = ⓧ*[⫯n]L.
  62 #n elim n -n //
  63 qed.
  64
  65 (* Main inversion properties ************************************************)
  66
  67 theorem voids_inj: ∀n. injective … (λL. ⓧ*[n]L).
  68 #n elim n -n //
  69 #n #IH #L1 #L2
  70 <voids_succ <voids_succ #H
  71 elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
  72 /2 width=1 by/
  73 qed-.
  74 *)