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) //
126 ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
128 (∀a,b,c.R a b → star A R b c → Q b c → Q a c) →
129 ∀a,b.star A R a b → Q a b.
130 #A #R #Q #H1 #H2 #a #b #H0
131 elim (star_to_starl ???? H0) // -H0 -b -a
132 #a #b #c #Rab #slbc @H2 // @starl_to_star //
137 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
138 #R #A #a #b #TCH (elim TCH) /2/
141 lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
142 #A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
145 (* equiv -- smallest equivalence relation containing R *)
147 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
148 |inje: ∀a,b,c.equiv A R a b → R b c → equiv A R a c
149 |refle: ∀a,b.equiv A R a b
150 |syme: ∀a,b.equiv A R a b → equiv A R b a.
152 theorem trans_equiv: ∀A,R,a,b,c.
153 equiv A R a b → equiv A R b c → equiv A R a c.
154 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
157 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
161 lemma monotonic_equiv: ∀A,R,S. subR A R S → subR A (equiv A R) (equiv A S).
162 #A #R #S #subRS #a #b #H (elim H) /3/
165 lemma sub_equiv: ∀A,R,S. subR A R (equiv A S) →
166 subR A (equiv A R) (equiv A S).
167 #A #R #S #Hsub #a #b #H (elim H) /2/
170 theorem sub_equiv_to_eq: ∀A,R,S. subR A R S → subR A S (equiv A R) →
171 exteqR … (equiv A R) (equiv A S).
172 #A #R #S #sub1 #sub2 #a #b % /2/
175 (* well founded part of a relation *)
177 inductive WF (A:Type[0]) (R:relation A) : A → Prop ≝
178 | wf : ∀b.(∀a. R a b → WF A R a) → WF A R b.
180 lemma WF_antimonotonic: ∀A,R,S. subR A R S →
181 ∀a. WF A S a → WF A R a.
182 #A #R #S #subRS #a #HWF (elim HWF) #b
183 #H #Hind % #c #Rcb @Hind @subRS //
186 (* added from lambda_delta *)
188 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
189 R a1 a → TC … R a a2 → TC … R a1 a2.
192 lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
195 lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
196 P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
197 ∀a2. TC … R a1 a2 → P a2.
198 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
201 inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
202 |inj_dx: ∀a,c. R a c → TC_dx A R a c
203 |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
205 lemma TC_dx_strap: ∀A. ∀R: relation A.
206 ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
207 #A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
210 lemma TC_to_TC_dx: ∀A. ∀R: relation A.
211 ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
212 #A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
215 lemma TC_dx_to_TC: ∀A. ∀R: relation A.
216 ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
217 #A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
220 fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A.
221 (∀a1. R a1 a2 → P a1) →
222 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
223 ∀a1,a. TC … R a1 a → a = a2 → P a1.
224 #A #R #a2 #P #H1 #H2 #a1 #a #Ha1
225 elim (TC_to_TC_dx ???? Ha1) -a1 -a
226 [ #a #c #Hac #H destruct /2 width=1/
227 | #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
231 lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A.
232 (∀a1. R a1 a2 → P a1) →
233 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
234 ∀a1. TC … R a1 a2 → P a1.
235 #A #R #a2 #P #H1 #H2 #a1 #Ha12
236 @(TC_ind_dx_aux … H1 H2 … Ha12) //
239 lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R).
240 #A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/
243 lemma TC_star_ind_dx: ∀A,R. reflexive A R →
244 ∀a2. ∀P:predicate A. P a2 →
245 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
246 ∀a1. TC … R a1 a2 → P a1.
247 #A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
248 @(TC_ind_dx … P ? H … Ha12) /3 width=4/
251 definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
252 ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
254 lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
255 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/