1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/logic/equality/".
17 include "higher_order_defs/relations.ma".
19 inductive eq (A:Type) (x:A) : A \to Prop \def
22 theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
23 simplify.intros.apply refl_eq.
26 theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
27 simplify.intros.elim H. apply refl_eq.
30 theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y \to eq A y x
33 theorem transitive_eq : \forall A:Type. transitive A (eq A).
34 simplify.intros.elim H1.assumption.
37 theorem trans_eq : \forall A:Type.\forall x,y,z:A. eq A x y \to eq A y z \to eq A x z
41 \forall A:Type.\forall x:A. \forall P: A \to Prop.
42 P x \to \forall y:A. eq A y x \to P y.
43 intros. elim sym_eq ? ? ? H1.assumption.
47 cic:/matita/logic/equality/eq.ind
48 cic:/matita/logic/equality/sym_eq.con
49 cic:/matita/logic/equality/trans_eq.con
50 cic:/matita/logic/equality/eq_ind.con
51 cic:/matita/logic/equality/eq_elim_r.con.
53 theorem eq_f: \forall A,B:Type.\forall f:A\to B.
54 \forall x,y:A. eq A x y \to eq B (f x) (f y).
55 intros.elim H.reflexivity.
58 theorem eq_f2: \forall A,B,C:Type.\forall f:A\to B \to C.
59 \forall x1,x2:A. \forall y1,y2:B.
60 eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
61 intros.elim H1.elim H.reflexivity.