2 (* ========================================================================= *)
3 (* FLYSPECK - BOOK FORMALIZATION *)
5 (* Authour : VU KHAC KY *)
6 (* Book lemma: RDWKARC *)
7 (* Chaper : Packing (Clusters) *)
9 (* ========================================================================= *)
10 (* FILES NEED TO BE LOADED *)
12 (* ========================================================================= *)
14 module Rdwkarc = struct
19 open Vukhacky_tactics;;
22 (*-------------------------------------------------------------------------- *)
25 `~kepler_conjecture /\ (!V. cell_cluster_estimate V) /\ TSKAJXY_statement
26 ==> (?V. packing V /\ V SUBSET ball_annulus /\ ~lmfun_ineq_center V)`;;
29 (* ------------------------------------------------------------------------- *)
30 (* The following lemmas are necessary for the main theorem RDWKARC *)
31 (* ------------------------------------------------------------------------- *)
34 let JGXZYGW_KY = prove_by_refinement (
37 (?A. fcc_compatible A S /\ negligible_fun_0 A S)
40 (UNIONS {ball (v,&1) | v IN S} INTER ball (vec 0,r)) /
41 vol (ball (vec 0,r)) <=
42 pi / sqrt (&18) + c / r)`,
43 [(MP_TAC JGXZYGW THEN DISCH_THEN (LABEL_TAC "asm1"));
45 (REWRITE_TAC[negligible_fun_0]);
46 (USE_THEN "asm1" (MP_TAC o SPEC `S:real^3->bool`));
47 (DISCH_THEN (LABEL_TAC "asm2"));
48 (USE_THEN "asm2" (MP_TAC o SPEC `(vec 0):real^3`));
51 (* ------------------------------------------------------------------------- *)
53 let PACKING_SUBSET = prove_by_refinement (
54 `!V S. packing V /\ S SUBSET V ==> packing S`,
55 [(REPEAT GEN_TAC THEN REWRITE_TAC[packing;SUBSET;IN_ELIM_THM]);
58 `!u:real^3 v. V u /\ V v /\ ~(u = v) ==> &2 <= dist (u,v)`) );
60 (REWRITE_WITH `V (u:real^3) /\ V v <=> u IN V /\ v IN V`);
63 (* Break into smaller subgoals *)
65 (MATCH_MP_TAC(ASSUME `!(x:real^3). x IN S ==> x IN V`) );
66 (ASM_REWRITE_TAC[IN]);
67 (MATCH_MP_TAC(ASSUME `!(x:real^3). x IN S ==> x IN V`) );
68 (ASM_REWRITE_TAC[IN])]);;
71 (* ------------------------------------------------------------------------ *)
73 let PACKING_TRANS = prove_by_refinement (
74 `! V (x:real^3). packing V ==> packing {u | (u + x) IN V}`,
76 [(REPEAT GEN_TAC THEN REWRITE_TAC[packing;IN_ELIM_THM]);
78 (ABBREV_TAC `u' = (u:real^3) + x`);
79 (ABBREV_TAC `v' = (v:real^3) + x`);
80 (NEW_GOAL `V (u':real^3) /\ V v' /\ ~(u' = v')`);
83 (ASM_REWRITE_TAC[GSYM IN]);
84 (ASM_REWRITE_TAC[GSYM IN]);
85 (NEW_GOAL `u = v:real^3`);
86 (REPLICATE_TAC 3 UP_ASM_TAC );
91 (REWRITE_WITH `dist (u:real^3, v) = dist (u', v':real^3)`);
93 (EXPAND_TAC "u'" THEN EXPAND_TAC "v'");
98 (UP_ASM_TAC THEN ASM_REWRITE_TAC[])]);;
101 (* ------------------------------------------------------------------------- *)
104 let SATURATED_TRANS = prove_by_refinement (
105 `!V (x:real^3). saturated V ==> saturated {u | u + x IN V}`,
107 [(REPEAT GEN_TAC THEN REWRITE_TAC[saturated]);
108 (DISCH_THEN (LABEL_TAC "asm1"));
110 (USE_THEN "asm1" (MP_TAC o SPEC `(x':real^3) + x`));
111 (DEL_TAC THEN DISCH_TAC);
112 (FIRST_X_ASSUM CHOOSE_TAC);
113 (EXISTS_TAC `y - (x:real^3)`);
114 (REWRITE_TAC[IN_ELIM_THM; VECTOR_ARITH `y - x + x = (y:real^3)`]);
117 (NEW_GOAL `dist (x',y - x) = dist (x'+ x,y:real^3)`);
123 (ASM_MESON_TAC[])]);;
126 (* ------------------------------------------------------------------------- *)
129 let RADV_TRANS_EQ = prove (
130 `!u v:real^3 x. ~(u = v) ==> radV {u, v} = radV {u + x, v + x}`,
131 REWRITE_TAC[GSYM set_of_list; GSYM HL; HL_2] THEN NORM_ARITH_TAC);;
133 (* ========================================================================= *)
134 (* MAIN THEOREM RDWKARC *)
135 (* ========================================================================= *)
137 let RDWKARC = prove_by_refinement (RDWKARC_concl,
138 [ (REWRITE_TAC[kepler_conjecture]);
140 `~(!V. packing V /\ saturated V
143 (UNIONS {ball (v,&1) | v IN V} INTER ball (vec 0,r)) /
144 vol (ball (vec 0,r)) <=
145 pi / sqrt (&18) + c / r)) <=>
146 (?V. packing V /\ saturated V
149 (UNIONS {ball (v,&1) | v IN V} INTER ball (vec 0,r)) /
150 vol (ball (vec 0,r)) <=
151 pi / sqrt (&18) + c / r))`);
156 (NEW_GOAL `~(lmfun_inequality (V:real^3->bool))`);
158 (NEW_GOAL `(?G. negligible_fun_0 G V /\ fcc_compatible G V)`);
159 (ASM_MESON_TAC[UPFZBZM]);
160 (NEW_GOAL `(?c. !r. &1 <= r
161 ==> vol (UNIONS {ball (v,&1) | v IN V} INTER ball (vec 0,r)) /
162 vol (ball (vec 0,r)) <=
163 pi / sqrt (&18) + c / r)`);
164 (MATCH_MP_TAC JGXZYGW_KY);
166 (CHOOSE_TAC (ASSUME `?G. negligible_fun_0 G V /\ fcc_compatible G V`));
167 (EXISTS_TAC `G:real^3->real`);
171 (* ---------------------------------------------------------------------- *)
173 (UP_ASM_TAC THEN REWRITE_TAC[lmfun_inequality]);
176 ==> sum {v | v IN V /\ ~(u = v) /\ dist (u,v) <= &2 * h0}
177 (\v. lmfun (hl [u; v])) <=
180 /\ ~(sum {v | v IN V /\ ~(u = v) /\ dist (u,v) <= &2 * h0}
181 (\v. lmfun (hl [u; v])) <=
184 (REWRITE_TAC[REAL_ARITH `~(a <= b) <=> b < a`]);
186 (FIRST_X_ASSUM CHOOSE_TAC);
189 (ABBREV_TAC `V' = {v:real^3 | v + u IN V}`);
190 (EXISTS_TAC `V':real^3->bool INTER ball_annulus`);
191 (ASM_REWRITE_TAC[INTER_SUBSET]);
193 (NEW_GOAL `packing (V':real^3->bool)`);
194 (EXPAND_TAC "V'" THEN ASM_MESON_TAC[PACKING_TRANS]);
196 (ASM_MESON_TAC[PACKING_SUBSET;INTER_SUBSET]);
198 (* -------------------------------------------------------------------------- *)
200 (REWRITE_TAC[lmfun_ineq_center]);
201 (REWRITE_TAC[REAL_ARITH `~(a <= b) <=> b < a`]);
202 (EXPAND_TAC "V'" THEN REWRITE_TAC[ball_annulus]);
205 `sum ({v | v + u IN V} INTER (cball (vec 0,&2 * h0) DIFF ball (vec 0,&2)))
206 (\v. lmfun (hl [vec 0; v])) =
207 sum {v:real^3 | v IN V /\ ~(u = v) /\ dist (u,v) <= &2 * h0}
208 (\v. lmfun (hl [u; v]))`);
209 (MATCH_MP_TAC SUM_EQ_GENERAL_INVERSES);
210 (REWRITE_TAC[IN_ELIM_THM]);
211 (EXISTS_TAC `(\x:real^3. x + u)`);
212 (EXISTS_TAC `(\x:real^3. x - u)`);
214 (REWRITE_TAC[IN_ELIM_THM;cball;INTER]);
216 (ASM_REWRITE_TAC[VECTOR_ARITH `y:real^3 - u + u = y`]);
217 (REWRITE_TAC[DIFF;IN_ELIM_THM;ball]);
218 (UP_ASM_TAC THEN REWRITE_TAC[dist]);
219 (DISCH_TAC THEN CONJ_TAC);
220 (UP_ASM_TAC THEN NORM_ARITH_TAC);
221 (REWRITE_WITH `norm (vec 0 - (y:real^3 - u)) = dist (u,y)`);
224 (REWRITE_TAC[REAL_ARITH `~(a < b) <=> b <= a`]);
225 (NEW_GOAL `V u /\ V y /\ ~(u = (y:real^3))`);
228 (ONCE_REWRITE_TAC[GSYM IN]);
230 (ONCE_REWRITE_TAC[GSYM IN]);
233 (ASM_MESON_TAC[packing]);
234 (REWRITE_TAC[BETA_THM]);
236 (REWRITE_TAC[BETA_THM]);
237 (UP_ASM_TAC THEN REWRITE_TAC[INTER;IN_ELIM_THM]);
239 (UP_ASM_TAC THEN REWRITE_TAC[IN_ELIM_THM]);
240 (REWRITE_TAC[VECTOR_ARITH `(u = x + u:real^3) <=> (x = vec 0)`]);
241 (UP_ASM_TAC THEN REWRITE_TAC[INTER; DIFF;IN_ELIM_THM;ball]);
243 (NEW_GOAL `dist(vec 0, x:real^3) = &0`);
244 (ASM_REWRITE_TAC[dist]);
246 (ASM_REAL_ARITH_TAC);
247 (REWRITE_TAC[BETA_THM;dist]);
248 (UP_ASM_TAC THEN REWRITE_TAC[INTER; DIFF;IN_ELIM_THM;ball]);
249 (REWRITE_TAC[IN_ELIM_THM;cball]);
251 (REWRITE_TAC[BETA_THM]);
253 (REWRITE_TAC[BETA_THM]);
256 (REWRITE_WITH `!u v:real^3. set_of_list [u; v] = {u , v}`);
257 (REWRITE_TAC[set_of_list]);
258 (UP_ASM_TAC THEN REWRITE_TAC[INTER; DIFF;IN_ELIM_THM;ball]);
260 (NEW_GOAL `~(x:real^3 = vec 0)`);
262 (NEW_GOAL `dist(vec 0, x:real^3) = &0`);
263 (ASM_REWRITE_TAC[dist]);
265 (ASM_REAL_ARITH_TAC);
266 (REWRITE_WITH `radV {u:real^3, x + u} = radV {vec 0 + u, x + u}`);
267 (MESON_TAC[VECTOR_ARITH `!u. u = vec 0 + u`]);
268 (ASM_MESON_TAC[RADV_TRANS_EQ]);
269 (ASM_REWRITE_TAC[]) ]);;
272 (* Finish the Lemma *)