helm/software/matita/nlibrary/logic/equality.ma

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  14
  15 include "logic/connectives.ma".
  16 include "properties/relations.ma".
  17
  18 ninductive eq (A: Type[0]) (a: A) : A → CProp[0] ≝
  19  refl: eq A a a.
  20
  21 nlemma eq_rect_CProp0_r':
  22  ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
  23  #A; #a; #x; #p; ncases p; #P; #H; nassumption.
  24 nqed.
  25
  26 nlemma eq_rect_CProp0_r:
  27  ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
  28  #A; #a; #P; #p; #x0; #p0; napply (eq_rect_CProp0_r' ??? p0); nassumption.
  29 nqed.
  30
  31 interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
  32
  33 interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
  34
  35 ndefinition EQ: ∀A:Type[0]. equivalence_relation A.
  36  #A; napply mk_equivalence_relation
  37   [ napply eq
  38   | napply refl
  39   | #x; #y; #H; nrewrite < H; napply refl
  40   | #x; #y; #z; #Hyx; #Hxz; nrewrite < Hxz; nassumption]
  41 nqed.