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/
92 ∀A,R,x,y.star A R x y → x = y ∨ ∃z.R x z ∧ star A R z y.
93 #A #R #x #y #Hstar elim Hstar
94 [ #b #c #Hleft #Hright *
95 [ #H1 %2 @(ex_intro ?? c) % //
96 | * #x0 * #H1 #H2 %2 @(ex_intro ?? x0) % /2/ ]
100 (* right associative version of star *)
101 inductive starl (A:Type[0]) (R:relation A) : A → A → Prop ≝
102 |injl: ∀a,b,c.R a b → starl A R b c → starl A R a c
103 |refll: ∀a.starl A R a a.
105 lemma starl_comp : ∀A,R,a,b,c.
106 starl A R a b → R b c → starl A R a c.
107 #A #R #a #b #c #Hstar elim Hstar
108 [#a1 #b1 #c1 #Rab #sbc #Hind #a1 @(injl … Rab) @Hind //
109 |#a1 #Rac @(injl … Rac) //
113 lemma star_compl : ∀A,R,a,b,c.
114 R a b → star A R b c → star A R a c.
115 #A #R #a #b #c #Rab #Hstar elim Hstar
116 [#b1 #c1 #sbb1 #Rb1c1 #Hind @(inj … Rb1c1) @Hind
121 lemma star_to_starl: ∀A,R,a,b.star A R a b → starl A R a b.
122 #A #R #a #b #Hs elim Hs //
123 #d #c #sad #Rdc #sad @(starl_comp … Rdc) //
126 lemma starl_to_star: ∀A,R,a,b.starl A R a b → star A R a b.
127 #A #R #a #b #Hs elim Hs // -Hs -b -a
128 #a #b #c #Rab #sbc #sbc @(star_compl … Rab) //
132 ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
134 (∀a,b,c.R a b → star A R b c → Q b c → Q a c) →
135 ∀a,b.star A R a b → Q a b.
136 #A #R #Q #H1 #H2 #a #b #H0
137 elim (star_to_starl ???? H0) // -H0 -b -a
138 #a #b #c #Rab #slbc @H2 // @starl_to_star //
143 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
144 #R #A #a #b #TCH (elim TCH) /2/
147 lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
148 #A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
151 (* equiv -- smallest equivalence relation containing R *)
153 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
154 |inje: ∀a,b,c.equiv A R a b → R b c → equiv A R a c
155 |refle: ∀a,b.equiv A R a b
156 |syme: ∀a,b.equiv A R a b → equiv A R b a.
158 theorem trans_equiv: ∀A,R,a,b,c.
159 equiv A R a b → equiv A R b c → equiv A R a c.
160 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
163 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
167 lemma monotonic_equiv: ∀A,R,S. subR A R S → subR A (equiv A R) (equiv A S).
168 #A #R #S #subRS #a #b #H (elim H) /3/
171 lemma sub_equiv: ∀A,R,S. subR A R (equiv A S) →
172 subR A (equiv A R) (equiv A S).
173 #A #R #S #Hsub #a #b #H (elim H) /2/
176 theorem sub_equiv_to_eq: ∀A,R,S. subR A R S → subR A S (equiv A R) →
177 exteqR … (equiv A R) (equiv A S).
178 #A #R #S #sub1 #sub2 #a #b % /2/
181 (* well founded part of a relation *)
183 inductive WF (A:Type[0]) (R:relation A) : A → Prop ≝
184 | wf : ∀b.(∀a. R a b → WF A R a) → WF A R b.
186 lemma WF_antimonotonic: ∀A,R,S. subR A R S →
187 ∀a. WF A S a → WF A R a.
188 #A #R #S #subRS #a #HWF (elim HWF) #b
189 #H #Hind % #c #Rcb @Hind @subRS //
192 (* added from lambda_delta *)
194 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
195 R a1 a → TC … R a a2 → TC … R a1 a2.
198 lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
201 lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
202 P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
203 ∀a2. TC … R a1 a2 → P a2.
204 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
207 inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
208 |inj_dx: ∀a,c. R a c → TC_dx A R a c
209 |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
211 lemma TC_dx_strap: ∀A. ∀R: relation A.
212 ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
213 #A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
216 lemma TC_to_TC_dx: ∀A. ∀R: relation A.
217 ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
218 #A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
221 lemma TC_dx_to_TC: ∀A. ∀R: relation A.
222 ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
223 #A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
226 fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A.
227 (∀a1. R a1 a2 → P a1) →
228 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
229 ∀a1,a. TC … R a1 a → a = a2 → P a1.
230 #A #R #a2 #P #H1 #H2 #a1 #a #Ha1
231 elim (TC_to_TC_dx ???? Ha1) -a1 -a
232 [ #a #c #Hac #H destruct /2 width=1/
233 | #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
237 lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A.
238 (∀a1. R a1 a2 → P a1) →
239 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
240 ∀a1. TC … R a1 a2 → P a1.
241 #A #R #a2 #P #H1 #H2 #a1 #Ha12
242 @(TC_ind_dx_aux … H1 H2 … Ha12) //
245 lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R).
246 #A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/
249 lemma TC_star_ind_dx: ∀A,R. reflexive A R →
250 ∀a2. ∀P:predicate A. P a2 →
251 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
252 ∀a1. TC … R a1 a2 → P a1.
253 #A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
254 @(TC_ind_dx … P ? H … Ha12) /3 width=4/
257 definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
258 ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
260 lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
261 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/