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 definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
18 definition inv ≝ λA.λR:relation A.λa,b.R b a.
20 (* transitive closcure (plus) *)
22 inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
23 |inj: ∀c. R a c → TC A R a c
24 |step : ∀b,c.TC A R a b → R b c → TC A R a c.
26 theorem trans_TC: ∀A,R,a,b,c.
27 TC A R a b → TC A R b c → TC A R a c.
28 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
31 theorem TC_idem: ∀A,R. exteqR … (TC A R) (TC A (TC A R)).
32 #A #R #a #b % /2/ #H (elim H) /2/
35 lemma monotonic_TC: ∀A,R,S. subR A R S → subR A (TC A R) (TC A S).
36 #A #R #S #subRS #a #b #H (elim H) /3/
39 lemma sub_TC: ∀A,R,S. subR A R (TC A S) → subR A (TC A R) (TC A S).
40 #A #R #S #Hsub #a #b #H (elim H) /3/
43 theorem sub_TC_to_eq: ∀A,R,S. subR A R S → subR A S (TC A R) →
44 exteqR … (TC A R) (TC A S).
45 #A #R #S #sub1 #sub2 #a #b % /2/
48 theorem TC_inv: ∀A,R. exteqR ?? (TC A (inv A R)) (inv A (TC A R)).
50 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_TC … H3) /2/
54 inductive star (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
55 |inj: ∀b,c.star A R a b → R b c → star A R a c
58 lemma R_to_star: ∀A,R,a,b. R a b → star A R a b.
62 theorem trans_star: ∀A,R,a,b,c.
63 star A R a b → star A R b c → star A R a c.
64 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
67 theorem star_star: ∀A,R. exteqR … (star A R) (star A (star A R)).
68 #A #R #a #b % /2/ #H (elim H) /2/
71 lemma monotonic_star: ∀A,R,S. subR A R S → subR A (star A R) (star A S).
72 #A #R #S #subRS #a #b #H (elim H) /3/
75 lemma sub_star: ∀A,R,S. subR A R (star A S) →
76 subR A (star A R) (star A S).
77 #A #R #S #Hsub #a #b #H (elim H) /3/
80 theorem sub_star_to_eq: ∀A,R,S. subR A R S → subR A S (star A R) →
81 exteqR … (star A R) (star A S).
82 #A #R #S #sub1 #sub2 #a #b % /2/
85 theorem star_inv: ∀A,R.
86 exteqR ?? (star A (inv A R)) (inv A (star A R)).
88 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_star … H3) /2/
93 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
94 #R #A #a #b #TCH (elim TCH) /2/
97 lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
98 #A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
101 (* equiv -- smallest equivalence relation containing R *)
103 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
104 |inje: ∀a,b,c.equiv A R a b → R b c → equiv A R a c
105 |refle: ∀a,b.equiv A R a b
106 |syme: ∀a,b.equiv A R a b → equiv A R b a.
108 theorem trans_equiv: ∀A,R,a,b,c.
109 equiv A R a b → equiv A R b c → equiv A R a c.
110 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
113 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
117 lemma monotonic_equiv: ∀A,R,S. subR A R S → subR A (equiv A R) (equiv A S).
118 #A #R #S #subRS #a #b #H (elim H) /3/
121 lemma sub_equiv: ∀A,R,S. subR A R (equiv A S) →
122 subR A (equiv A R) (equiv A S).
123 #A #R #S #Hsub #a #b #H (elim H) /2/
126 theorem sub_equiv_to_eq: ∀A,R,S. subR A R S → subR A S (equiv A R) →
127 exteqR … (equiv A R) (equiv A S).
128 #A #R #S #sub1 #sub2 #a #b % /2/
131 (* well founded part of a relation *)
133 inductive WF (A:Type[0]) (R:relation A) : A → Prop ≝
134 | wf : ∀b.(∀a. R a b → WF A R a) → WF A R b.
136 lemma WF_antimonotonic: ∀A,R,S. subR A R S →
137 ∀a. WF A S a → WF A R a.
138 #A #R #S #subRS #a #HWF (elim HWF) #b
139 #H #Hind % #c #Rcb @Hind @subRS //
142 (* added from lambda_delta *)
144 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
145 R a1 a → TC … R a a2 → TC … R a1 a2.
148 lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
151 lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
152 P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
153 ∀a2. TC … R a1 a2 → P a2.
154 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/