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 //
198 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
199 R a1 a → TC … R a a2 → TC … R a1 a2.
202 lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
205 lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
206 P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
207 ∀a2. TC … R a1 a2 → P a2.
208 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
211 inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
212 |inj_dx: ∀a,c. R a c → TC_dx A R a c
213 |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
215 lemma TC_dx_strap: ∀A. ∀R: relation A.
216 ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
217 #A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
220 lemma TC_to_TC_dx: ∀A. ∀R: relation A.
221 ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
222 #A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
225 lemma TC_dx_to_TC: ∀A. ∀R: relation A.
226 ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
227 #A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
230 fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A.
231 (∀a1. R a1 a2 → P a1) →
232 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
233 ∀a1,a. TC … R a1 a → a = a2 → P a1.
234 #A #R #a2 #P #H1 #H2 #a1 #a #Ha1
235 elim (TC_to_TC_dx ???? Ha1) -a1 -a
236 [ #a #c #Hac #H destruct /2 width=1/
237 | #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
241 lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A.
242 (∀a1. R a1 a2 → P a1) →
243 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
244 ∀a1. TC … R a1 a2 → P a1.
245 #A #R #a2 #P #H1 #H2 #a1 #Ha12
246 @(TC_ind_dx_aux … H1 H2 … Ha12) //
249 lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R).
250 #A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/
253 lemma TC_star_ind_dx: ∀A,R. reflexive A R →
254 ∀a2. ∀P:predicate A. P a2 →
255 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
256 ∀a1. TC … R a1 a2 → P a1.
257 #A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
258 @(TC_ind_dx … P ? H … Ha12) /3 width=4/
261 lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
262 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/
265 (* ************ confluence of star *****************)
267 lemma star_strip: ∀A,R. confluent A R →
268 ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 →
269 ∃∃a. R a1 a & star … R a2 a.
270 #A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
271 #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
272 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
273 elim (HR … Ha1 … Ha0) -a /3 width=5/
276 lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R).
277 #A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
278 #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
279 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
280 elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/
283 (* relations on unboxed pairs ***********************************************)
285 inductive bi_TC (A,B) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝
286 |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d
287 |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.
289 lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2.
290 R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2.
291 #A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/
294 lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R →
295 bi_reflexive … (bi_TC … R).
298 inductive bi_TC_dx (A,B) (R:bi_relation A B): bi_relation A B ≝
299 |bi_inj_dx : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2
300 |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 →
301 bi_TC_dx A B R a1 b1 a2 b2.
303 lemma bi_TC_dx_strap: ∀A,B. ∀R: bi_relation A B.
304 ∀a1,a,a2,b1,b,b2. bi_TC_dx A B R a1 b1 a b →
305 R a b a2 b2 → bi_TC_dx A B R a1 b1 a2 b2.
306 #A #B #R #a1 #a #a2 #b1 #b #b2 #H1 elim H1 -a -b /3 width=4/
309 lemma bi_TC_to_bi_TC_dx: ∀A,B. ∀R: bi_relation A B.
310 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
311 bi_TC_dx … R a1 b1 a2 b2.
312 #A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a2 -b2 /2 width=4/
315 lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B.
316 ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 →
317 bi_TC … R a1 b1 a2 b2.
318 #A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/
321 fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B.
322 (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
323 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
324 ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1.
325 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1
326 elim (bi_TC_to_bi_TC_dx … a1 a b1 b H1) -a1 -a -b1 -b
327 [ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/
328 | #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/
332 lemma bi_TC_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B.
333 (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
334 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
335 ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
336 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #b1 #H12
337 @(bi_TC_ind_dx_aux ?????? H1 H2 … H12) //
340 lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R →
341 bi_symmetric A B (bi_TC … R).
342 #A #B #R #HR #a1 #a2 #b1 #b2 #H21
343 @(bi_TC_ind_dx … a2 b2 H21) -a2 -b2 /3 width=1/ /3 width=4/
346 lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R).
347 #A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/
350 definition bi_Conf3: ∀A,B,C. relation3 A B C → predicate (bi_relation A B) ≝
352 ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c.
354 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).
355 #A #B #C #S #R #HSR #c #a1 #b1 #Hab1 #a2 #b2 #H elim H -a2 -b2 /2 width=4/
358 lemma bi_TC_star_ind: ∀A,B,R. bi_reflexive A B R → ∀a1,b1. ∀P:relation2 A B.
359 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) →
360 ∀a2,b2. bi_TC … R a1 b1 a2 b2 → P a2 b2.
361 #A #B #R #HR #a1 #b1 #P #H1 #IH #a2 #b2 #H12 elim H12 -a2 -b2 /3 width=5/
364 lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R →
365 ∀a2,b2. ∀P:relation2 A B. P a2 b2 →
366 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
367 ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
368 #A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12
369 @(bi_TC_ind_dx … IH … a1 b1 H12) /3 width=5/
372 definition bi_star: ∀A,B,R. bi_relation A B ≝
373 λA,B,R. bi_RC A B (bi_TC … R).
375 lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R).
378 lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
379 bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
382 lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
383 R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
386 lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
387 R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
388 #A #B #R #a1 #a #a2 #b1 #b #b2 *
390 | * #H1 #H2 destruct /2 width=1/
394 lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b →
395 bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
396 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
398 | * #H1 #H2 destruct /2 width=1/
402 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 →
403 bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
404 #A #B #R #a1 #a #a2 #b1 #b #b2 *
406 | * #H1 #H2 destruct /2 width=1/
410 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 →
411 bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
412 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
414 | * #H1 #H2 destruct /2 width=1/
418 lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R).
419 #A #B #R #a1 #a #b1 #b #H #a2 #b2 *
421 | * #H1 #H2 destruct /2 width=1/
425 lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 →
426 (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
427 ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2.
428 #A #B #R #a1 #b1 #P #H #IH #a2 #b2 *
429 [ #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
430 | * #H1 #H2 destruct //
434 lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 →
435 (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) →
436 ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1.
437 #A #B #R #a2 #b2 #P #H #IH #a1 #b1 *
438 [ #H12 @(bi_TC_ind_dx … a1 b1 H12) -a1 -b1 /2 width=5/ -H /3 width=5/
439 | * #H1 #H2 destruct //
443 (* relations on unboxed triples *********************************************)
445 inductive tri_TC (A,B,C) (R:tri_relation A B C) (a1:A) (b1:B) (c1:C): relation3 A B C ≝
446 |tri_inj : ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2
447 |tri_step: ∀a,a2,b,b2,c,c2.
448 tri_TC A B C R a1 b1 c1 a b c → R a b c a2 b2 c2 →
449 tri_TC A B C R a1 b1 c1 a2 b2 c2.
451 lemma tri_TC_strap: ∀A,B,C. ∀R:tri_relation A B C. ∀a1,a,a2,b1,b,b2,c1,c,c2.
452 R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 →
453 tri_TC … R a1 b1 c1 a2 b2 c2.
454 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #HR #H elim H -a2 -b2 -c2 /2 width=5/ /3 width=5/
457 lemma tri_TC_reflexive: ∀A,B,C,R. tri_reflexive A B C R →
458 tri_reflexive … (tri_TC … R).
461 inductive tri_TC_dx (A,B,C) (R:tri_relation A B C): tri_relation A B C ≝
462 |tri_inj_dx : ∀a1,a2,b1,b2,c1,c2. R a1 b1 c1 a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2
463 |tri_step_dx : ∀a1,a,a2,b1,b,b2,c1,c,c2.
464 R a1 b1 c1 a b c → tri_TC_dx A B C R a b c a2 b2 c2 →
465 tri_TC_dx A B C R a1 b1 c1 a2 b2 c2.
467 lemma tri_TC_dx_strap: ∀A,B,C. ∀R: tri_relation A B C.
468 ∀a1,a,a2,b1,b,b2,c1,c,c2.
469 tri_TC_dx A B C R a1 b1 c1 a b c →
470 R a b c a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2.
471 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H1 elim H1 -a -b -c /3 width=5/
474 lemma tri_TC_to_tri_TC_dx: ∀A,B,C. ∀R: tri_relation A B C.
475 ∀a1,a2,b1,b2,c1,c2. tri_TC … R a1 b1 c1 a2 b2 c2 →
476 tri_TC_dx … R a1 b1 c1 a2 b2 c2.
477 #A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a2 -b2 -c2 /2 width=1/ /2 width=5/
480 lemma tri_TC_dx_to_tri_TC: ∀A,B,C. ∀R: tri_relation A B C.
481 ∀a1,a2,b1,b2,c1,c2. tri_TC_dx … R a1 b1 c1 a2 b2 c2 →
482 tri_TC … R a1 b1 c1 a2 b2 c2.
483 #A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a1 -a2 -b1 -b2 -c1 -c2
484 /2 width=1/ /2 width=5/
487 fact tri_TC_ind_dx_aux: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C.
488 (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2→ P a1 b1 c1) →
489 (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
490 ∀a1,a,b1,b,c1,c. tri_TC … R a1 b1 c1 a b c → a = a2 → b = b2 → c = c2 → P a1 b1 c1.
491 #A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #a #b1 #b #c1 #c #H1
492 elim (tri_TC_to_tri_TC_dx … a1 a b1 b c1 c H1) -a1 -a -b1 -b -c1 -c
493 [ #a1 #x #b1 #y #c1 #z #H1 #Hx #Hy #Hz destruct /2 width=1/
494 | #a1 #a #x #b1 #b #y #c1 #c #z #H1 #H #IH #Hx #Hy #Hz destruct /3 width=6/
498 lemma tri_TC_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C.
499 (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2 → P a1 b1 c1) →
500 (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
501 ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
502 #A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #b1 #c1 #H12
503 @(tri_TC_ind_dx_aux ???????? H1 H2 … H12) //
506 lemma tri_TC_symmetric: ∀A,B,C,R. tri_symmetric A B C R →
507 tri_symmetric … (tri_TC … R).
508 #A #B #C #R #HR #a1 #a2 #b1 #b2 #c1 #c2 #H21
509 @(tri_TC_ind_dx … a2 b2 c2 H21) -a2 -b2 -c2 /3 width=1/ /3 width=5/
512 lemma tri_TC_transitive: ∀A,B,C,R. tri_transitive A B C (tri_TC … R).
513 #A #B #C #R #a1 #a #b1 #b #c1 #c #H elim H -a -b -c /2 width=5/ /3 width=5/
516 definition tri_Conf4: ∀A,B,C,D. relation4 A B C D → predicate (tri_relation A B C) ≝
518 ∀d,a1,b1,c1. S a1 b1 c1 d → ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → S a2 b2 c2 d.
520 lemma tri_TC_Conf4: ∀A,B,C,D,S,R.
521 tri_Conf4 A B C D S R → tri_Conf4 A B C D S (tri_TC … R).
522 #A #B #C #D #S #R #HSR #d #a1 #b1 #c1 #Habc1 #a2 #b2 #c2 #H elim H -a2 -b2 -c2
526 lemma tri_TC_star_ind: ∀A,B,C,R. tri_reflexive A B C R →
527 ∀a1,b1,c1. ∀P:relation3 A B C.
528 P a1 b1 c1 → (∀a,a2,b,b2,c,c2. tri_TC … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
529 ∀a2,b2,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
530 #A #B #C #R #HR #a1 #b1 #c1 #P #H1 #IH #a2 #b2 #c2 #H12 elim H12 -a2 -b2 -c2
531 /2 width=6/ /3 width=6/
534 lemma tri_TC_star_ind_dx: ∀A,B,C,R. tri_reflexive A B C R →
535 ∀a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
536 (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
537 ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
538 #A #B #C #R #HR #a2 #b2 #c2 #P #H2 #IH #a1 #b1 #c1 #H12
539 @(tri_TC_ind_dx … IH … a1 b1 c1 H12) /3 width=6/