helm/matita/library/logic/equality.ma

   1 (**************************************************************************)
   2 (*       ___                                                                *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
   8 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU Lesser General Public License Version 2.1         *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/logic/equality/".
  16
  17 include "higher_order_defs/relations.ma".
  18 include "logic/connectives.ma".
  19
  20 inductive eq (A:Type) (x:A) : A \to Prop \def
  21     refl_eq : eq A x x.
  22
  23 (*CSC: the URI must disappear: there is a bug now *)
  24 interpretation "leibnitz's equality"
  25    'eq x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
  26 (*CSC: the URI must disappear: there is a bug now *)
  27 interpretation "leibnitz's non-equality"
  28   'neq x y = (cic:/matita/logic/connectives/Not.con
  29     (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y)).
  30
  31 theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
  32 simplify.intros.apply refl_eq.
  33 qed.
  34
  35 theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
  36 simplify.intros.elim H. apply refl_eq.
  37 qed.
  38
  39 theorem sym_eq : \forall A:Type.\forall x,y:A. x=y  \to y=x
  40 \def symmetric_eq.
  41
  42 theorem transitive_eq : \forall A:Type. transitive A (eq A).
  43 simplify.intros.elim H1.assumption.
  44 qed.
  45
  46 theorem trans_eq : \forall A:Type.\forall x,y,z:A. x=y  \to y=z \to x=z
  47 \def transitive_eq.
  48
  49 theorem eq_elim_r:
  50  \forall A:Type.\forall x:A. \forall P: A \to Prop.
  51    P x \to \forall y:A. y=x \to P y.
  52 intros. elim sym_eq ? ? ? H1.assumption.
  53 qed.
  54
  55 default "equality"
  56  cic:/matita/logic/equality/eq.ind
  57  cic:/matita/logic/equality/sym_eq.con
  58  cic:/matita/logic/equality/trans_eq.con
  59  cic:/matita/logic/equality/eq_ind.con
  60  cic:/matita/logic/equality/eq_elim_r.con.
  61
  62 theorem eq_f: \forall  A,B:Type.\forall f:A\to B.
  63 \forall x,y:A. x=y \to f x = f y.
  64 intros.elim H.reflexivity.
  65 qed.
  66
  67 theorem eq_f2: \forall  A,B,C:Type.\forall f:A\to B \to C.
  68 \forall x1,x2:A. \forall y1,y2:B.
  69 x1=x2 \to y1=y2 \to f x1 y1 = f x2 y2.
  70 intros.elim H1.elim H.reflexivity.
  71 qed.