2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/relations.ma".
14 (********** relations **********)
16 inductive star (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
17 |inj: ∀b,c.star A R a b → R b c → star A R a c
20 theorem trans_star: ∀A,R,a,b,c.
21 star A R a b → star A R b c → star A R a c.
22 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
25 theorem star_star: ∀A,R. exteqR … (star A R) (star A (star A R)).
26 #A #R #a #b % /2/ #H (elim H) /2/
29 definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
31 lemma monotonic_star: ∀A,R,S. subR A R S → subR A (star A R) (star A S).
32 #A #R #S #subRS #a #b #H (elim H) /3/
35 lemma sub_star: ∀A,R,S. subR A R (star A S) →
36 subR A (star A R) (star A S).
37 #A #R #S #Hsub #a #b #H (elim H) /3/
40 theorem sub_star_to_eq: ∀A,R,S. subR A R S → subR A S (star A R) →
41 exteqR … (star A R) (star A S).
42 #A #R #S #sub1 #sub2 #a #b % /2/
45 (* equiv -- smallest equivalence relation containing R *)
47 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
48 |inje: ∀a,b,c.equiv A R a b → R b c → equiv A R a c
49 |refle: ∀a,b.equiv A R a b
50 |syme: ∀a,b.equiv A R a b → equiv A R b a.
52 theorem trans_equiv: ∀A,R,a,b,c.
53 equiv A R a b → equiv A R b c → equiv A R a c.
54 #A #R #a #b #c #Hab #Hbc (inversion Hbc) /2/
57 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
61 lemma monotonic_equiv: ∀A,R,S. subR A R S → subR A (equiv A R) (equiv A S).
62 #A #R #S #subRS #a #b #H (elim H) /3/
65 lemma sub_equiv: ∀A,R,S. subR A R (equiv A S) →
66 subR A (equiv A R) (equiv A S).
67 #A #R #S #Hsub #a #b #H (elim H) /2/
70 theorem sub_equiv_to_eq: ∀A,R,S. subR A R S → subR A S (equiv A R) →
71 exteqR … (equiv A R) (equiv A S).
72 #A #R #S #sub1 #sub2 #a #b % /2/
75 (* well founded part of a relation *)
77 inductive WF (A:Type[0]) (R:relation A) : A → Prop ≝
78 | wf : ∀b.(∀a. R a b → WF A R a) → WF A R b.
80 lemma WF_antimonotonic: ∀A,R,S. subR A R S →
81 ∀a. WF A S a → WF A R a.
82 #A #R #S #subRS #a #HWF (elim HWF) #b
83 #H #Hind % #c #Rcb @Hind @subRS //