1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basics/star1.ma".
16 include "ground_2/xoa/xoa_props.ma".
18 (* PROPERTIES OF RELATIONS **************************************************)
20 definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
21 ≝ λA,B,C,D,E.A→B→C→D→E→Prop.
23 definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
24 ≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
26 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
28 definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
29 ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
31 definition left_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
32 ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
34 definition right_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
35 ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
37 definition pw_confluent2: ∀A. relation A → relation A → predicate A ≝ λA,R1,R2,a0.
38 ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
39 ∃∃a. R2 a1 a & R1 a2 a.
41 definition confluent2: ∀A. relation (relation A) ≝ λA,R1,R2.
42 ∀a0. pw_confluent2 A R1 R2 a0.
44 definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
45 ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
46 ∃∃a. R2 a1 a & R1 a a2.
48 definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
49 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
50 ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
52 definition LTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝
55 definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
56 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
58 definition s_r_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
59 ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2.
61 definition s_rs_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
62 ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → LTC … R1 L1 T1 T2.
64 definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
65 ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
67 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
68 ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
69 ∃∃a. R2 a1 a & TC … R1 a2 a.
70 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
72 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
73 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
74 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
75 elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5 by step, ex2_intro/
79 lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
80 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
81 ∃∃a. TC … R2 a1 a & R1 a2 a.
82 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
84 elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
85 | #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
86 elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
87 elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3 by step, ex2_intro/
91 lemma TC_confluent2: ∀A,R1,R2.
92 confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
93 #A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
95 elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3 by inj, ex2_intro/
96 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
97 elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
98 elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5 by step, ex2_intro/
102 lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
103 ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
104 ∃∃a. R2 a1 a & TC … R1 a a2.
105 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
106 [ #a0 #Ha10 #a2 #Ha02
107 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
108 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
109 elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
110 elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/
114 lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
115 ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
116 ∃∃a. TC … R2 a1 a & R1 a a2.
117 #A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
118 [ #a2 #Ha02 #a1 #Ha10
119 elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
120 | #a #a2 #_ #Ha02 #IHa #a1 #Ha10
121 elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
122 elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3 by step, ex2_intro/
126 lemma TC_transitive2: ∀A,R1,R2.
127 transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
128 #A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
129 [ #a0 #Ha10 #a2 #Ha02
130 elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3 by inj, ex2_intro/
131 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
132 elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
133 elim (IHa … Ha0) -a /4 width=5 by step, ex2_intro/
137 definition NF: ∀A. relation A → relation A → predicate A ≝
138 λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
140 definition NF_dec: ∀A. relation A → relation A → Prop ≝
141 λA,R,S. ∀a1. NF A R S a1 ∨
142 ∃∃a2. R … a1 a2 & (S a2 a1 → ⊥).
144 inductive SN (A) (R,S:relation A): predicate A ≝
145 | SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
148 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
150 @SN_intro #a2 #HRa12 #HSa12
151 elim HSa12 -HSa12 /2 width=1 by/
154 lemma SN_to_NF: ∀A,R,S. NF_dec A R S →
156 ∃∃a2. star … R a1 a2 & NF A R S a2.
157 #A #R #S #HRS #a1 #H elim H -a1
158 #a1 #_ #IHa1 elim (HRS a1) -HRS /2 width=3 by srefl, ex2_intro/
159 * #a0 #Ha10 #Ha01 elim (IHa1 … Ha10 Ha01) -IHa1 -Ha01 /3 width=3 by star_compl, ex2_intro/
162 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
163 λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
165 inductive SN_sn (A) (R,S:relation A): predicate A ≝
166 | SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
169 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
171 @SN_sn_intro #a1 #HRa12 #HSa12
172 elim HSa12 -HSa12 /2 width=1 by/
175 lemma LTC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S.
176 #A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 /3 width=3 by inj/
177 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
178 lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3 by step/
181 lemma s_r_conf1_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (LTC … S) R.
182 #A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3 by/
185 lemma s_r_trans_LTC1: ∀A,B,S,R. s_r_confluent1 A B S R →
186 s_r_transitive A B S R → s_rs_transitive A B S R.
187 #A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /2 width=3 by/
188 #T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR … HT1 … HL12) -H2SR -HT1
189 /4 width=5 by s_r_conf1_LTC1, trans_TC/
192 lemma s_r_trans_LTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (LTC … R).
193 #A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /3 width=3 by inj/
196 lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (LTC … S) R →
197 s_rs_transitive A B S R.
198 #A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
199 elim (TC_idem … (S L1) … T1 T2)
203 lemma s_rs_to_s_r_trans: ∀A,B,S,R. s_rs_transitive A B S R →
204 s_r_transitive A B (LTC … S) R.
205 #A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
206 elim (TC_idem … (S L1) … T1 T2)
210 lemma s_rs_trans_TC1: ∀A,B,S,R. s_rs_transitive A B S R →
211 s_rs_transitive A B (LTC … S) R.
212 #A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
213 elim (TC_idem … (S L1) … T1 T2)
214 elim (TC_idem … (S L2) … T1 T2)
215 #_ #H1 #H2 #_ @H2 @HSR /3 width=3 by/
218 (* relations on unboxed pairs ***********************************************)
220 lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
221 ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
222 ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
223 #A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
225 elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
226 | #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2
227 elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4 by ex2_2_intro, bi_step/
231 lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
232 bi_confluent A B (bi_TC … R).
233 #A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
234 [ #a1 #b1 #H01 #a2 #b2 #H02
235 elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
236 | #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02
237 elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20
238 elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7 by ex2_2_intro, bi_step/
242 lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
243 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
245 ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
246 #A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4 by ex2_2_intro, or_intror/
249 lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
250 ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
252 ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
253 #A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
254 [ /2 width=1 by or_introl/
255 | #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4 by ex2_2_intro, or_intror/ (**) (* auto fails without #_ *)
259 (* relations on unboxed triples *********************************************)
261 definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
262 λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
263 ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
265 lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
266 /3 width=1 by and3_intro, or_intror/ qed.
268 definition tri_star: ∀A,B,C,R. tri_relation A B C ≝
269 λA,B,C,R. tri_RC A B C (tri_TC … R).
271 lemma tri_star_tri_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_star … R).
274 lemma tri_TC_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
275 tri_TC A B C R a1 b1 c1 a2 b2 c2 →
276 tri_star A B C R a1 b1 c1 a2 b2 c2.
277 /2 width=1 by or_introl/ qed.
279 lemma tri_R_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
280 R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
281 /3 width=1 by tri_TC_to_tri_star, tri_inj/ qed.
283 lemma tri_star_strap1: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
284 tri_star A B C R a1 b1 c1 a b c →
285 R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
286 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
287 [ /3 width=5 by tri_TC_to_tri_star, tri_step/
288 | * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
292 lemma tri_star_strap2: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c →
293 tri_star A B C R a b c a2 b2 c2 →
294 tri_star A B C R a1 b1 c1 a2 b2 c2.
295 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
296 [ /3 width=5 by tri_TC_to_tri_star, tri_TC_strap/
297 | * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
301 lemma tri_star_to_tri_TC_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
302 tri_star A B C R a1 b1 c1 a b c →
303 tri_TC A B C R a b c a2 b2 c2 →
304 tri_TC A B C R a1 b1 c1 a2 b2 c2.
305 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
306 [ /2 width=5 by tri_TC_transitive/
307 | * #H1 #H2 #H3 destruct /2 width=1 by/
311 lemma tri_TC_to_tri_star_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
312 tri_TC A B C R a1 b1 c1 a b c →
313 tri_star A B C R a b c a2 b2 c2 →
314 tri_TC A B C R a1 b1 c1 a2 b2 c2.
315 #A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
316 [ /2 width=5 by tri_TC_transitive/
317 | * #H1 #H2 #H3 destruct /2 width=1 by/
321 lemma tri_tansitive_tri_star: ∀A,B,C,R. tri_transitive A B C (tri_star … R).
322 #A #B #C #R #a1 #a #b1 #b #c1 #c #H #a2 #b2 #c2 *
323 [ /3 width=5 by tri_star_to_tri_TC_to_tri_TC, tri_TC_to_tri_star/
324 | * #H1 #H2 #H3 destruct /2 width=1 by/
328 lemma tri_star_ind: ∀A,B,C,R,a1,b1,c1. ∀P:relation3 A B C. P a1 b1 c1 →
329 (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
330 ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
331 #A #B #C #R #a1 #b1 #c1 #P #H #IH #a2 #b2 #c2 *
332 [ #H12 elim H12 -a2 -b2 -c2 /3 width=6 by tri_TC_to_tri_star/
333 | * #H1 #H2 #H3 destruct //
337 lemma tri_star_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
338 (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
339 ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
340 #A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 *
341 [ #H12 @(tri_TC_ind_dx … a1 b1 c1 H12) -a1 -b1 -c1 /3 width=6 by tri_TC_to_tri_star/
342 | * #H1 #H2 #H3 destruct //