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 (* transitive closcure (plus) *)
16 inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
17 |inj: ∀c. R a c → TC A R a c
18 |step : ∀b,c.TC A R a b → R b c → TC A R a c.
20 theorem trans_TC: ∀A,R,a,b,c.
21 TC A R a b → TC A R b c → TC A R a c.
22 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
25 theorem TC_idem: ∀A,R. exteqR … (TC A R) (TC A (TC A R)).
26 #A #R #a #b % /2/ #H (elim H) /2/
29 lemma monotonic_TC: ∀A,R,S. subR A R S → subR A (TC A R) (TC A S).
30 #A #R #S #subRS #a #b #H (elim H) /3/
33 lemma sub_TC: ∀A,R,S. subR A R (TC A S) → subR A (TC A R) (TC A S).
34 #A #R #S #Hsub #a #b #H (elim H) /3/
37 theorem sub_TC_to_eq: ∀A,R,S. subR A R S → subR A S (TC A R) →
38 exteqR … (TC A R) (TC A S).
39 #A #R #S #sub1 #sub2 #a #b % /2/
42 theorem TC_inv: ∀A,R. exteqR ?? (TC A (inv A R)) (inv A (TC A R)).
44 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_TC … H3) /2/
48 inductive star (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
49 |sstep: ∀b,c.star A R a b → R b c → star A R a c
52 lemma R_to_star: ∀A,R,a,b. R a b → star A R a b.
56 theorem trans_star: ∀A,R,a,b,c.
57 star A R a b → star A R b c → star A R a c.
58 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
61 theorem star_star: ∀A,R. exteqR … (star A R) (star A (star A R)).
62 #A #R #a #b % /2/ #H (elim H) /2/
65 lemma monotonic_star: ∀A,R,S. subR A R S → subR A (star A R) (star A S).
66 #A #R #S #subRS #a #b #H (elim H) /3/
69 lemma sub_star: ∀A,R,S. subR A R (star A S) →
70 subR A (star A R) (star A S).
71 #A #R #S #Hsub #a #b #H (elim H) /3/
74 theorem sub_star_to_eq: ∀A,R,S. subR A R S → subR A S (star A R) →
75 exteqR … (star A R) (star A S).
76 #A #R #S #sub1 #sub2 #a #b % /2/
79 theorem star_inv: ∀A,R.
80 exteqR ?? (star A (inv A R)) (inv A (star A R)).
82 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_star … H3) /2/
86 ∀A,R,x,y.star A R x y → x = y ∨ ∃z.R x z ∧ star A R z y.
87 #A #R #x #y #Hstar elim Hstar
88 [ #b #c #Hleft #Hright *
89 [ #H1 %2 @(ex_intro ?? c) % //
90 | * #x0 * #H1 #H2 %2 @(ex_intro ?? x0) % /2/ ]
94 (* right associative version of star *)
95 inductive starl (A:Type[0]) (R:relation A) : A → A → Prop ≝
96 |sstepl: ∀a,b,c.R a b → starl A R b c → starl A R a c
97 |refll: ∀a.starl A R a a.
99 lemma starl_comp : ∀A,R,a,b,c.
100 starl A R a b → R b c → starl A R a c.
101 #A #R #a #b #c #Hstar elim Hstar
102 [#a1 #b1 #c1 #Rab #sbc #Hind #a1 @(sstepl … Rab) @Hind //
103 |#a1 #Rac @(sstepl … Rac) //
107 lemma star_compl : ∀A,R,a,b,c.
108 R a b → star A R b c → star A R a c.
109 #A #R #a #b #c #Rab #Hstar elim Hstar
110 [#b1 #c1 #sbb1 #Rb1c1 #Hind @(sstep … Rb1c1) @Hind
115 lemma star_to_starl: ∀A,R,a,b.star A R a b → starl A R a b.
116 #A #R #a #b #Hs elim Hs //
117 #d #c #sad #Rdc #sad @(starl_comp … Rdc) //
120 lemma starl_to_star: ∀A,R,a,b.starl A R a b → star A R a b.
121 #A #R #a #b #Hs elim Hs // -Hs -b -a
122 #a #b #c #Rab #sbc #sbc @(star_compl … Rab) //
125 fact star_ind_l_aux: ∀A,R,a2. ∀P:predicate A.
127 (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
128 ∀a1,a. star … R a1 a → a = a2 → P a1.
129 #A #R #a2 #P #H1 #H2 #a1 #a #Ha1
130 elim (star_to_starl ???? Ha1) -a1 -a
131 [ #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
132 | #a #H destruct /2 width=1/
136 (* imporeved version of star_ind_l with "left_parameter" *)
137 lemma star_ind_l: ∀A,R,a2. ∀P:predicate A.
139 (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
140 ∀a1. star … R a1 a2 → P a1.
141 #A #R #a2 #P #H1 #H2 #a1 #Ha12
142 @(star_ind_l_aux … H1 H2 … Ha12) //
147 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
148 #R #A #a #b #TCH (elim TCH) /2/
151 lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
152 #A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
155 (* equiv -- smallest equivalence relation containing R *)
157 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
158 |inje: ∀a,b,c.equiv A R a b → R b c → equiv A R a c
159 |refle: ∀a,b.equiv A R a b
160 |syme: ∀a,b.equiv A R a b → equiv A R b a.
162 theorem trans_equiv: ∀A,R,a,b,c.
163 equiv A R a b → equiv A R b c → equiv A R a c.
164 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
167 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
171 lemma monotonic_equiv: ∀A,R,S. subR A R S → subR A (equiv A R) (equiv A S).
172 #A #R #S #subRS #a #b #H (elim H) /3/
175 lemma sub_equiv: ∀A,R,S. subR A R (equiv A S) →
176 subR A (equiv A R) (equiv A S).
177 #A #R #S #Hsub #a #b #H (elim H) /2/
180 theorem sub_equiv_to_eq: ∀A,R,S. subR A R S → subR A S (equiv A R) →
181 exteqR … (equiv A R) (equiv A S).
182 #A #R #S #sub1 #sub2 #a #b % /2/
185 (* well founded part of a relation *)
187 inductive WF (A:Type[0]) (R:relation A) : A → Prop ≝
188 | wf : ∀b.(∀a. R a b → WF A R a) → WF A R b.
190 lemma WF_antimonotonic: ∀A,R,S. subR A R S →
191 ∀a. WF A S a → WF A R a.
192 #A #R #S #subRS #a #HWF (elim HWF) #b
193 #H #Hind % #c #Rcb @Hind @subRS //