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/star.ma".
16 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
17 R a1 a → TC … R a a2 → TC … R a1 a2.
20 lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
23 lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
24 P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
25 ∀a2. TC … R a1 a2 → P a2.
26 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
29 inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
30 |inj_dx: ∀a,c. R a c → TC_dx A R a c
31 |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
33 lemma TC_dx_strap: ∀A. ∀R: relation A.
34 ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
35 #A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
38 lemma TC_to_TC_dx: ∀A. ∀R: relation A.
39 ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
40 #A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
43 lemma TC_dx_to_TC: ∀A. ∀R: relation A.
44 ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
45 #A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
48 fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A.
49 (∀a1. R a1 a2 → P a1) →
50 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
51 ∀a1,a. TC … R a1 a → a = a2 → P a1.
52 #A #R #a2 #P #H1 #H2 #a1 #a #Ha1
53 elim (TC_to_TC_dx ???? Ha1) -a1 -a
54 [ #a #c #Hac #H destruct /2 width=1/
55 | #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
59 lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A.
60 (∀a1. R a1 a2 → P a1) →
61 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
62 ∀a1. TC … R a1 a2 → P a1.
63 #A #R #a2 #P #H1 #H2 #a1 #Ha12
64 @(TC_ind_dx_aux … H1 H2 … Ha12) //
67 lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R).
68 #A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/
71 lemma TC_star_ind_dx: ∀A,R. reflexive A R →
72 ∀a2. ∀P:predicate A. P a2 →
73 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
74 ∀a1. TC … R a1 a2 → P a1.
75 #A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
76 @(TC_ind_dx … P ? H … Ha12) /3 width=4/
79 lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
80 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/
83 (* ************ confluence of star *****************)
85 lemma star_strip: ∀A,R. confluent A R →
86 ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 →
87 ∃∃a. R a1 a & star … R a2 a.
88 #A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
89 #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
90 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
91 elim (HR … Ha1 … Ha0) -a /3 width=5/
94 lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R).
95 #A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
96 #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
97 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
98 elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/
101 (* relations on unboxed pairs ***********************************************)
103 inductive bi_TC (A,B) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝
104 |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d
105 |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.
107 lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2.
108 R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2.
109 #A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/
112 lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R →
113 bi_reflexive … (bi_TC … R).
116 inductive bi_TC_dx (A,B) (R:bi_relation A B): bi_relation A B ≝
117 |bi_inj_dx : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2
118 |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 →
119 bi_TC_dx A B R a1 b1 a2 b2.
121 lemma bi_TC_dx_strap: ∀A,B. ∀R: bi_relation A B.
122 ∀a1,a,a2,b1,b,b2. bi_TC_dx A B R a1 b1 a b →
123 R a b a2 b2 → bi_TC_dx A B R a1 b1 a2 b2.
124 #A #B #R #a1 #a #a2 #b1 #b #b2 #H1 elim H1 -a -b /3 width=4/
127 lemma bi_TC_to_bi_TC_dx: ∀A,B. ∀R: bi_relation A B.
128 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
129 bi_TC_dx … R a1 b1 a2 b2.
130 #A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a2 -b2 /2 width=4/
133 lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B.
134 ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 →
135 bi_TC … R a1 b1 a2 b2.
136 #A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/
139 fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B.
140 (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
141 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
142 ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1.
143 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1
144 elim (bi_TC_to_bi_TC_dx … a1 a b1 b H1) -a1 -a -b1 -b
145 [ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/
146 | #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/
150 lemma bi_TC_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B.
151 (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) →
152 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
153 ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
154 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #b1 #H12
155 @(bi_TC_ind_dx_aux ?????? H1 H2 … H12) //
158 lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R →
159 bi_symmetric A B (bi_TC … R).
160 #A #B #R #HR #a1 #a2 #b1 #b2 #H21
161 @(bi_TC_ind_dx … a2 b2 H21) -a2 -b2 /3 width=1/ /3 width=4/
164 lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R).
165 #A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/
168 definition bi_Conf3: ∀A,B,C. relation3 A B C → predicate (bi_relation A B) ≝
170 ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c.
172 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).
173 #A #B #C #S #R #HSR #c #a1 #b1 #Hab1 #a2 #b2 #H elim H -a2 -b2 /2 width=4/
176 lemma bi_TC_star_ind: ∀A,B,R. bi_reflexive A B R → ∀a1,b1. ∀P:relation2 A B.
177 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) →
178 ∀a2,b2. bi_TC … R a1 b1 a2 b2 → P a2 b2.
179 #A #B #R #HR #a1 #b1 #P #H1 #IH #a2 #b2 #H12 elim H12 -a2 -b2 /3 width=5/
182 lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R →
183 ∀a2,b2. ∀P:relation2 A B. P a2 b2 →
184 (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
185 ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
186 #A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12
187 @(bi_TC_ind_dx … IH … a1 b1 H12) /3 width=5/
190 definition bi_star: ∀A,B,R. bi_relation A B ≝
191 λA,B,R. bi_RC A B (bi_TC … R).
193 lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R).
196 lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
197 bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
200 lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
201 R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
204 lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
205 R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
206 #A #B #R #a1 #a #a2 #b1 #b #b2 *
208 | * #H1 #H2 destruct /2 width=1/
212 lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b →
213 bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
214 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
216 | * #H1 #H2 destruct /2 width=1/
220 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 →
221 bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
222 #A #B #R #a1 #a #a2 #b1 #b #b2 *
224 | * #H1 #H2 destruct /2 width=1/
228 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 →
229 bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
230 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
232 | * #H1 #H2 destruct /2 width=1/
236 lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R).
237 #A #B #R #a1 #a #b1 #b #H #a2 #b2 *
239 | * #H1 #H2 destruct /2 width=1/
243 lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 →
244 (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
245 ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2.
246 #A #B #R #a1 #b1 #P #H #IH #a2 #b2 *
247 [ #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
248 | * #H1 #H2 destruct //
252 lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 →
253 (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) →
254 ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1.
255 #A #B #R #a2 #b2 #P #H #IH #a1 #b1 *
256 [ #H12 @(bi_TC_ind_dx … a1 b1 H12) -a1 -b1 /2 width=5/ -H /3 width=5/
257 | * #H1 #H2 destruct //
261 (* relations on unboxed triples *********************************************)
263 inductive tri_TC (A,B,C) (R:tri_relation A B C) (a1:A) (b1:B) (c1:C): relation3 A B C ≝
264 |tri_inj : ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2
265 |tri_step: ∀a,a2,b,b2,c,c2.
266 tri_TC A B C R a1 b1 c1 a b c → R a b c a2 b2 c2 →
267 tri_TC A B C R a1 b1 c1 a2 b2 c2.
269 lemma tri_TC_strap: ∀A,B,C. ∀R:tri_relation A B C. ∀a1,a,a2,b1,b,b2,c1,c,c2.
270 R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 →
271 tri_TC … R a1 b1 c1 a2 b2 c2.
272 #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/
275 lemma tri_TC_reflexive: ∀A,B,C,R. tri_reflexive A B C R →
276 tri_reflexive … (tri_TC … R).
279 inductive tri_TC_dx (A,B,C) (R:tri_relation A B C): tri_relation A B C ≝
280 |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
281 |tri_step_dx : ∀a1,a,a2,b1,b,b2,c1,c,c2.
282 R a1 b1 c1 a b c → tri_TC_dx A B C R a b c a2 b2 c2 →
283 tri_TC_dx A B C R a1 b1 c1 a2 b2 c2.
285 lemma tri_TC_dx_strap: ∀A,B,C. ∀R: tri_relation A B C.
286 ∀a1,a,a2,b1,b,b2,c1,c,c2.
287 tri_TC_dx A B C R a1 b1 c1 a b c →
288 R a b c a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2.
289 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H1 elim H1 -a -b -c /3 width=5/
292 lemma tri_TC_to_tri_TC_dx: ∀A,B,C. ∀R: tri_relation A B C.
293 ∀a1,a2,b1,b2,c1,c2. tri_TC … R a1 b1 c1 a2 b2 c2 →
294 tri_TC_dx … R a1 b1 c1 a2 b2 c2.
295 #A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a2 -b2 -c2 /2 width=1/ /2 width=5/
298 lemma tri_TC_dx_to_tri_TC: ∀A,B,C. ∀R: tri_relation A B C.
299 ∀a1,a2,b1,b2,c1,c2. tri_TC_dx … R a1 b1 c1 a2 b2 c2 →
300 tri_TC … R a1 b1 c1 a2 b2 c2.
301 #A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a1 -a2 -b1 -b2 -c1 -c2
302 /2 width=1/ /2 width=5/
305 fact tri_TC_ind_dx_aux: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C.
306 (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2→ P a1 b1 c1) →
307 (∀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) →
308 ∀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.
309 #A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #a #b1 #b #c1 #c #H1
310 elim (tri_TC_to_tri_TC_dx … a1 a b1 b c1 c H1) -a1 -a -b1 -b -c1 -c
311 [ #a1 #x #b1 #y #c1 #z #H1 #Hx #Hy #Hz destruct /2 width=1/
312 | #a1 #a #x #b1 #b #y #c1 #c #z #H1 #H #IH #Hx #Hy #Hz destruct /3 width=6/
316 lemma tri_TC_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C.
317 (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2 → P a1 b1 c1) →
318 (∀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) →
319 ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
320 #A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #b1 #c1 #H12
321 @(tri_TC_ind_dx_aux ???????? H1 H2 … H12) //
324 lemma tri_TC_symmetric: ∀A,B,C,R. tri_symmetric A B C R →
325 tri_symmetric … (tri_TC … R).
326 #A #B #C #R #HR #a1 #a2 #b1 #b2 #c1 #c2 #H21
327 @(tri_TC_ind_dx … a2 b2 c2 H21) -a2 -b2 -c2 /3 width=1/ /3 width=5/
330 lemma tri_TC_transitive: ∀A,B,C,R. tri_transitive A B C (tri_TC … R).
331 #A #B #C #R #a1 #a #b1 #b #c1 #c #H elim H -a -b -c /2 width=5/ /3 width=5/
334 definition tri_Conf4: ∀A,B,C,D. relation4 A B C D → predicate (tri_relation A B C) ≝
336 ∀d,a1,b1,c1. S a1 b1 c1 d → ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → S a2 b2 c2 d.
338 lemma tri_TC_Conf4: ∀A,B,C,D,S,R.
339 tri_Conf4 A B C D S R → tri_Conf4 A B C D S (tri_TC … R).
340 #A #B #C #D #S #R #HSR #d #a1 #b1 #c1 #Habc1 #a2 #b2 #c2 #H elim H -a2 -b2 -c2
344 lemma tri_TC_star_ind: ∀A,B,C,R. tri_reflexive A B C R →
345 ∀a1,b1,c1. ∀P:relation3 A B C.
346 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) →
347 ∀a2,b2,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
348 #A #B #C #R #HR #a1 #b1 #c1 #P #H1 #IH #a2 #b2 #c2 #H12 elim H12 -a2 -b2 -c2
349 /2 width=6/ /3 width=6/
352 lemma tri_TC_star_ind_dx: ∀A,B,C,R. tri_reflexive A B C R →
353 ∀a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
354 (∀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) →
355 ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
356 #A #B #C #R #HR #a2 #b2 #c2 #P #H2 #IH #a1 #b1 #c1 #H12
357 @(tri_TC_ind_dx … IH … a1 b1 c1 H12) /3 width=6/