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/
258 inductive bi_TC (A,B:Type[0]) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝
259 |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d
260 |bi_step: ∀c,d,e,f. bi_TC A B R a b c d → R c d e f → bi_TC A B R a b e f.
262 lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2.
263 R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2.
264 #A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/
267 lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R →
268 bi_reflexive A B (bi_TC … R).
271 inductive bi_TC_dx (A,B:Type[0]) (R:bi_relation A B): bi_relation A B ≝
272 |bi_inj_dx : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2
273 |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 →
274 bi_TC_dx A B R a1 b1 a2 b2.
276 lemma bi_TC_dx_strap: ∀A,B. ∀R: bi_relation A B.
277 ∀a1,a,a2,b1,b,b2. bi_TC_dx A B R a1 b1 a b →
278 R a b a2 b2 → bi_TC_dx A B R a1 b1 a2 b2.
279 #A #B #R #a1 #a #a2 #b1 #b #b2 #H1 elim H1 -a -b /3 width=4/
282 lemma bi_TC_to_bi_TC_dx: ∀A,B. ∀R: bi_relation A B.
283 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
284 bi_TC_dx … R a1 b1 a2 b2.
285 #A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a2 -b2 /2 width=4/
288 lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B.
289 ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 →
290 bi_TC … R a1 b1 a2 b2.
291 #A #b #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/
294 fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B.
295 (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
296 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
297 ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1.
298 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1
299 elim (bi_TC_to_bi_TC_dx ??????? H1) -a1 -a -b1 -b
300 [ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/
301 | #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/
305 lemma bi_TC_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B.
306 (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
307 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
308 ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
309 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #b1 #H12
310 @(bi_TC_ind_dx_aux ?????? H1 H2 … H12) //
313 lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R →
314 bi_symmetric A B (bi_TC … R).
315 #A #B #R #HR #a1 #a2 #b1 #b2 #H21
316 @(bi_TC_ind_dx ?????????? H21) -a2 -b2 /3 width=1/ /3 width=4/
319 lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R).
320 #A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/
323 definition bi_Conf3: ∀A,B,C. relation3 A B C → bi_relation A B → Prop ≝ λA,B,C,S,R.
324 ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c.
326 lemma bi_TC_Conf3: ∀A,B,C,S,R. bi_Conf3 A B C S R → bi_Conf3 A B C S (bi_TC … R).
327 #A #B #C #S #R #HSR #c #a1 #b1 #Hab1 #a2 #b2 #H elim H -a2 -b2 /2 width=4/
330 lemma bi_TC_star_ind: ∀A,B,R. bi_reflexive A B R → ∀a1,b1. ∀P:relation2 A B.
331 P a1 b1 → (∀a,a2,b,b2. bi_TC … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
332 ∀a2,b2. bi_TC … R a1 b1 a2 b2 → P a2 b2.
333 #A #B #R #HR #a1 #b1 #P #H1 #IH #a2 #b2 #H12 elim H12 -a2 -b2 /3 width=5/
336 lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R →
337 ∀a2,b2. ∀P:relation2 A B. P a2 b2 →
338 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
339 ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
340 #A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12
341 @(bi_TC_ind_dx … P ? IH … H12) /3 width=5/
344 definition bi_star: ∀A,B,R. bi_relation A B ≝ λA,B,R,a1,b1,a2,b2.
345 (a1 = a2 ∧ b1 = b2) ∨ bi_TC A B R a1 b1 a2 b2.
347 lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R).
350 lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
351 bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
354 lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
355 R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
358 lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
359 R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
360 #A #B #R #a1 #a #a2 #b1 #b #b2 *
361 [ * #H1 #H2 destruct /2 width=1/
366 lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b →
367 bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
368 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
369 [ * #H1 #H2 destruct /2 width=1/
374 lemma bi_star_to_bi_TC_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
375 bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
376 #A #B #R #a1 #a #a2 #b1 #b #b2 *
377 [ * #H1 #H2 destruct /2 width=1/
382 lemma bi_TC_to_bi_star_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_TC A B R a1 b1 a b →
383 bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
384 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
385 [ * #H1 #H2 destruct /2 width=1/
390 lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R).
391 #A #B #R #a1 #a #b1 #b #H #a2 #b2 *
392 [ * #H1 #H2 destruct /2 width=1/
397 lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 →
398 (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
399 ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2.
400 #A #B #R #a1 #b1 #P #H #IH #a2 #b2 *
401 [ * #H1 #H2 destruct //
402 | #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
406 lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 →
407 (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) →
408 ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1.
409 #A #B #R #a2 #b2 #P #H #IH #a1 #b1 *
410 [ * #H1 #H2 destruct //
411 | #H12 @(bi_TC_ind_dx ?????????? H12) -a1 -b1 /2 width=5/ -H /3 width=5/