matita/matita/contribs/lambdadelta/ground/lib/subset_equivalence.ma

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  14
  15 include "ground/notation/relations/arroweq_2.ma".
  16 include "ground/lib/subset_inclusion.ma".
  17
  18 (* EQUIVALENCE FOR SUBSETS **************************************************)
  19
  20 definition subset_eq (A): relation2 𝒫❨A❩ 𝒫❨A❩ ≝
  21            λu1,u2. ∧∧ u1 ⊆ u2 & u2 ⊆ u1.
  22
  23 interpretation
  24   "equivalence (subset)"
  25   'ArrowEq u1 u2 = (subset_eq ? u1 u2).
  26
  27 (* Basic destructions *******************************************************)
  28
  29 lemma subset_in_eq_repl_back (A) (a:A):
  30       ∀u1. a ϵ u1 → ∀u2. u1 ⇔ u2 → a ϵ u2.
  31 #A #a #u1 #Hu1 #u2 *
  32 /2 width=1 by/
  33 qed-.
  34
  35 lemma subset_in_eq_repl_fwd (A) (a:A):
  36       ∀u1. a ϵ u1 → ∀u2. u2 ⇔ u1 → a ϵ u2.
  37 #A #a #u1 #Hu1 #u2 *
  38 /2 width=1 by/
  39 qed-.
  40
  41 (* Basic constructions ******************************************************)
  42
  43 lemma subset_eq_refl (A):
  44       reflexive … (subset_eq A).
  45 /2 width=1 by conj/ qed.
  46
  47 lemma subset_eq_sym (A):
  48       symmetric … (subset_eq A).
  49 #A #u1 #u2 * /2 width=1 by conj/
  50 qed-.
  51
  52 (* Main constructions *******************************************************)
  53
  54 theorem subset_eq_trans (A):
  55         Transitive … (subset_eq A).
  56 #A #u1 #u2 * #H12 #H21 #u3 * #H23 #H32
  57 /3 width=5 by subset_le_trans, conj/
  58 qed-.