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
  19 inductive eq (A:Type) (x:A) : A \to Prop \def
  20     refl_eq : eq A x x.
  21
  22 (*CSC: the URI must disappear: there is a bug now *)
  23 interpretation "leibnitz's equality"
  24    'eq x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
  25
  26 theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
  27 simplify.intros.apply refl_eq.
  28 qed.
  29
  30 theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
  31 simplify.intros.elim H. apply refl_eq.
  32 qed.
  33
  34 theorem sym_eq : \forall A:Type.\forall x,y:A. x=y  \to y=x
  35 \def symmetric_eq.
  36
  37 theorem transitive_eq : \forall A:Type. transitive A (eq A).
  38 simplify.intros.elim H1.assumption.
  39 qed.
  40
  41 theorem trans_eq : \forall A:Type.\forall x,y,z:A. x=y  \to y=z \to x=z
  42 \def transitive_eq.
  43
  44 theorem eq_elim_r:
  45  \forall A:Type.\forall x:A. \forall P: A \to Prop.
  46    P x \to \forall y:A. y=x \to P y.
  47 intros. elim sym_eq ? ? ? H1.assumption.
  48 qed.
  49
  50 default "equality"
  51  cic:/matita/logic/equality/eq.ind
  52  cic:/matita/logic/equality/sym_eq.con
  53  cic:/matita/logic/equality/trans_eq.con
  54  cic:/matita/logic/equality/eq_ind.con
  55  cic:/matita/logic/equality/eq_elim_r.con.
  56
  57 theorem eq_f: \forall  A,B:Type.\forall f:A\to B.
  58 \forall x,y:A. x=y \to f x = f y.
  59 intros.elim H.reflexivity.
  60 qed.
  61
  62 theorem eq_f2: \forall  A,B,C:Type.\forall f:A\to B \to C.
  63 \forall x1,x2:A. \forall y1,y2:B.
  64 x1=x2 \to y1=y2 \to f x1 y1 = f x2 y2.
  65 intros.elim H1.elim H.reflexivity.
  66 qed.