(* ========================================================================= *) (* FLYSPECK - BOOK FORMALIZATION *) (* *) (* Authour : VU KHAC KY *) (* Book lemma: *) (* Chaper : Packing (Marchal Cells 3) *) (* *) (* ========================================================================= *) open Sphere;; open Euler_main_theorem;; open Pack_defs;; open Pack_concl;; open Pack1;; open Pack2;; open Packing3;; open Rogers;; open Vukhacky_tactics;; open Marchal_cells;; open Emnwuus;; open Marchal_cells_2;; open Marchal_cells_2_new;; open Urrphbz1;; open Lepjbdj;; open Hdtfnfz;; open Rvfxzbu;; open Sltstlo;; open Urrphbz2;; open Urrphbz3;; open Ynhyjit;; open Njiutiu;; open Tezffsk;; open Qzksykg;; open Ddzuphj;; open Ajripqn;; open Qzyzmjc;; open Upfzbzm_support_lemmas;; let TAKE_TAC = UP_ASM_TAC THEN REPEAT STRIP_TAC;; (* ==================================================================== *) (* Lemma 1 *) let HD_IN_ROGERS = prove ( `!V ul. saturated V /\ packing V /\ barV V 3 ul ==> HD ul IN rogers V ul`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[ROGERS_EXPLICIT] THEN MATCH_MP_TAC IN_SET_IMP_IN_CONVEX_HULL_SET THEN SET_TAC[]);; (* ==================================================================== *) (* Lemma 2 *) let ROGERS_SUBSET_VORONOI_CLOSED = prove_by_refinement ( `!V ul. saturated V /\ packing V /\ barV V 3 ul ==> rogers V ul SUBSET voronoi_closed V (HD ul)`, [(REPEAT STRIP_TAC); (ASM_SIMP_TAC[ROGERS_EXPLICIT]); (NEW_GOAL `voronoi_closed V (HD ul) = convex hull (voronoi_closed V ((HD ul):real^3))`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[CONVEX_HULL_EQ]); (REWRITE_TAC[Packing3.CONVEX_VORONOI_CLOSED]); (ONCE_ASM_REWRITE_TAC[]); (MATCH_MP_TAC Marchal_cells.CONVEX_HULL_SUBSET); (NEW_GOAL `?u0 u1 u2 u3. ul = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (REWRITE_TAC[GSYM OMEGA_LIST_N]); (REWRITE_WITH `voronoi_closed V (omega_list_n V ul 0) = voronoi_list V (truncate_simplex 0 ul)`); (ASM_REWRITE_TAC[OMEGA_LIST_N; TRUNCATE_SIMPLEX_EXPLICIT_0; VORONOI_LIST; VORONOI_SET; set_of_list; HD] THEN SET_TAC[]); (NEW_GOAL `omega_list_n V ul 0 IN voronoi_list V (truncate_simplex 0 ul)`); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (NEW_GOAL `omega_list_n V ul 1 IN voronoi_list V (truncate_simplex 0 ul)`); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (NEW_GOAL `omega_list_n V ul 2 IN voronoi_list V (truncate_simplex 0 ul)`); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (NEW_GOAL `omega_list_n V ul 3 IN voronoi_list V (truncate_simplex 0 ul)`); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (ASM_SET_TAC[])]);; (* ==================================================================== *) (* Lemma 3 *) let HD_IN_MCELL = prove_by_refinement ( `!V ul i r X . packing V /\ saturated V /\ barV V 3 ul /\ X = mcell i V ul /\ ~(X = {}) /\ ~(i = 0) ==> (HD ul IN X)`, [(REPEAT STRIP_TAC); (NEW_GOAL `?u0 u1 u2 u3. ul = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_REWRITE_TAC[HD]); (ASM_CASES_TAC `i = 1`); (ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell1]); (COND_CASES_TAC); (ASM_SIMP_TAC[IN_DIFF;IN_INTER; HD; TL; IN_CBALL; DIST_REFL; TRUNCATE_SIMPLEX_EXPLICIT_1; rcone_gt; rconesgn]); (REPEAT STRIP_TAC); (REWRITE_WITH `u0 = HD ul /\ [u0; u1; u2; u3:real^3] = ul`); (ASM_REWRITE_TAC[HD]); (MATCH_MP_TAC HD_IN_ROGERS); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC SQRT_POS_LE); (REAL_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[IN; IN_ELIM_THM; VECTOR_ARITH `(u0 - u0) dot (u1 - u0) = &0`; DIST_REFL]); (REAL_ARITH_TAC); (NEW_GOAL `F`); (NEW_GOAL `X:real^3->bool = {}`); (REWRITE_TAC[ASSUME `X = mcell i V ul`; ASSUME `i = 1`; MCELL_EXPLICIT; mcell1]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REFL_TAC); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `i = 2`); (ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell2]); (COND_CASES_TAC); (LET_TAC); (ASM_SIMP_TAC[IN_INTER; HD; TL; DIST_REFL; TRUNCATE_SIMPLEX_EXPLICIT_1; rcone_ge; rconesgn; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (REWRITE_TAC[VECTOR_ARITH `(u0 - u0) dot (u1 - u0) = &0`]); (REAL_ARITH_TAC); (REWRITE_TAC[GSYM NORM_POW_2; GSYM dist; REAL_POW_2]); (REWRITE_TAC[REAL_ARITH `a * a >= a * a * b <=> &0 <= a * a * (&1 - b)`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (ABBREV_TAC `s = hl (truncate_simplex 1 [u0; u1; u2; u3:real^3])`); (EXPAND_TAC "a"); (REWRITE_TAC[REAL_ARITH `&0 <= a - b <=> b <= a`]); (REWRITE_WITH `s / sqrt (&2) <= &1 <=> s <= &1 * sqrt (&2)`); (MATCH_MP_TAC REAL_LE_LDIV_EQ); (MATCH_MP_TAC SQRT_POS_LT THEN REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (REWRITE_WITH `[u0; u1; u2; u3:real^3] = ul`); (ASM_REWRITE_TAC[]); (ABBREV_TAC `m = mxi V ul`); (ABBREV_TAC `s3 = omega_list_n V ul 3`); (REWRITE_WITH `aff_ge {u0, u1:real^3} {m, s3} u0 <=> u0 IN aff_ge {u0, u1} {m, s3}`); (REWRITE_TAC[IN]); (NEW_GOAL `u0 IN convex hull {u0,u1,m,s3:real^3}`); (MATCH_MP_TAC IN_SET_IMP_IN_CONVEX_HULL_SET); (SET_TAC[]); (NEW_GOAL `convex hull {u0,u1,m,s3:real^3} SUBSET aff_ge {u0, u1} {m, s3}`); (REWRITE_TAC[Marchal_cells_2_new.CONVEX_HULL_4_SUBSET_AFF_GE_2_2]); (UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `X:real^3->bool = {}`); (REWRITE_TAC[ASSUME `X = mcell i V ul`; ASSUME `i = 2`; MCELL_EXPLICIT; mcell2]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REFL_TAC); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `i = 3`); (ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell3]); (COND_CASES_TAC); (REWRITE_TAC[set_of_list;TRUNCATE_SIMPLEX_EXPLICIT_2]); (MATCH_MP_TAC IN_SET_IMP_IN_CONVEX_HULL_SET); (SET_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `X:real^3->bool = {}`); (REWRITE_TAC[ASSUME `X = mcell i V ul`; ASSUME `i = 3`; MCELL_EXPLICIT; mcell3]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REFL_TAC); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `i >= 4`); (ASM_SIMP_TAC[MCELL_EXPLICIT; mcell4]); (COND_CASES_TAC); (REWRITE_TAC[set_of_list]); (MATCH_MP_TAC IN_SET_IMP_IN_CONVEX_HULL_SET); (SET_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `X:real^3->bool = {}`); (SIMP_TAC[ASSUME `X = mcell i V ul`; ASSUME `i >= 4`; MCELL_EXPLICIT; mcell4]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REFL_TAC); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (ASM_ARITH_TAC); (ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 4 *) let FINITE_MCELL_SET_lemma1 = prove_by_refinement ( `!V ul i r X . packing V /\ saturated V /\ barV V 3 ul /\ X = mcell i V ul /\ X SUBSET ball (vec 0, r) /\ ~(X = {}) ==> (!u. u IN set_of_list ul ==> u IN ball (vec 0, r + &6))`, [(REPEAT STRIP_TAC); (* Case i = 0 *) (ASM_CASES_TAC `i = 0`); (UNDISCH_TAC `X = mcell i V ul`THEN ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell0]); (STRIP_TAC); (NEW_GOAL `!s. s IN rogers V ul ==> dist (HD ul, s) < &2`); (REPEAT STRIP_TAC); (NEW_GOAL `rogers V ul SUBSET voronoi_closed V (HD ul)`); (ASM_SIMP_TAC[ROGERS_SUBSET_VORONOI_CLOSED]); (NEW_GOAL `s:real^3 IN voronoi_closed V (HD ul)`); (ASM_SET_TAC[]); (UNDISCH_TAC `saturated (V:real^3->bool)`); (REWRITE_TAC[saturated]); (REPEAT STRIP_TAC); (NEW_GOAL `?y. y IN V /\ dist (s, y:real^3) < &2`); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UNDISCH_TAC `s:real^3 IN voronoi_closed V (HD ul)` THEN REWRITE_TAC[voronoi_closed; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `dist (s,HD ul) <= dist (s,y:real^3)`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC); (NEW_GOAL `?s. s:real^3 IN X`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `dist (HD ul, s:real^3) < &2`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (NEW_GOAL `set_of_list (ul:(real^3)list) SUBSET ball (HD ul,&4)`); (MATCH_MP_TAC BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:real^3->bool`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Packing3.HD_IN_SET_OF_LIST); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `u:real^3 IN ball (HD ul, &4)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN_BALL]); (STRIP_TAC); (NEW_GOAL `dist (vec 0,u) <= dist (vec 0, s) + dist (s, u:real^3)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `dist (s, u) <= dist (s, HD ul) + dist (HD ul, u:real^3)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `dist (vec 0,u) <= dist (vec 0,s) + dist (s,HD ul) + dist (HD ul,u:real^3)`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `dist (s:real^3, HD ul) < &2`); (ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[]); (NEW_GOAL `dist (vec 0,s:real^3) < r`); (REWRITE_TAC[GSYM IN_BALL]); (ASM_SET_TAC[]); (ASM_REAL_ARITH_TAC); (* Case i != 0 *) (NEW_GOAL `?u0 u1 u2 u3. ul = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `(HD ul):real^3 IN X`); (MATCH_MP_TAC HD_IN_MCELL); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (NEW_GOAL `(HD ul):real^3 IN ball (vec 0, r)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN_BALL] THEN STRIP_TAC); (NEW_GOAL `dist (vec 0,u) <= dist (vec 0,HD ul) + dist (HD ul, u:real^3)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `set_of_list (ul:(real^3)list) SUBSET ball (HD ul,&4)`); (MATCH_MP_TAC BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:real^3->bool`); (ASM_REWRITE_TAC[HD; set_of_list]); (SET_TAC[]); (NEW_GOAL `u:real^3 IN ball (HD ul, &4)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN_BALL]); (STRIP_TAC); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 5 *) let FINITE_MCELL_SET_LEMMA_concl = `!V r. packing V /\ saturated V ==> FINITE {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}`;; let FINITE_MCELL_SET_LEMMA = prove_by_refinement ( FINITE_MCELL_SET_LEMMA_concl, [(REPEAT STRIP_TAC); (REWRITE_WITH `FINITE {X | X SUBSET ball (vec 0,r) /\ mcell_set V X} <=> FINITE {X | X SUBSET ball (vec 0,r) /\ mcell_set V X /\ ~(X = {})}`); (ASM_CASES_TAC `{} IN {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}`); (REWRITE_WITH `{X | X SUBSET ball (vec 0,r) /\ mcell_set V X} = {} INSERT {X | X SUBSET ball (vec 0,r) /\ mcell_set V X /\ ~(X = {})}`); (ONCE_REWRITE_TAC[SET_RULE `A = B <=> B SUBSET A /\ A SUBSET B`]); (REPEAT STRIP_TAC); (REWRITE_TAC[INSERT_SUBSET]); (ASM_REWRITE_TAC[]); (SET_TAC[]); (REWRITE_TAC[SUBSET_INSERT_DELETE]); (ONCE_REWRITE_TAC[SUBSET] THEN ONCE_REWRITE_TAC[IN_DELETE] THEN REWRITE_TAC[IN_ELIM_THM]); (SET_TAC[]); (REWRITE_TAC[FINITE_INSERT]); (REWRITE_WITH `{X | X SUBSET ball (vec 0,r) /\ mcell_set V X} = {X | X SUBSET ball (vec 0,r) /\ mcell_set V X /\ ~(X = {})}`); (ONCE_REWRITE_TAC[SET_RULE `A = B <=> B SUBSET A /\ A SUBSET B`]); (REPEAT STRIP_TAC); (SET_TAC[]); (ONCE_REWRITE_TAC[SUBSET] THEN ONCE_REWRITE_TAC[IN] THEN REWRITE_TAC[IN_ELIM_THM]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[]); (NEW_GOAL `{} IN {X | X SUBSET ball (vec 0,r) /\ mcell_set V X}`); (ONCE_REWRITE_TAC[IN] THEN REWRITE_TAC[IN_ELIM_THM]); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (ABBREV_TAC `S = {X | X SUBSET ball (vec 0,r) /\ mcell_set V X /\ ~(X = {})}`); (ABBREV_TAC `s1:(real^3->bool) = V INTER ball (vec 0, r + &6)`); (ABBREV_TAC `s2 = {ul:(real^3)list | ?u0 u1 u2 u3. u0 IN s1 /\ u1 IN s1 /\ u2 IN s1 /\ u3 IN s1 /\ ul = [u0; u1; u2; u3]}`); (ABBREV_TAC `s3 = {(i:num, ul:(real^3)list) | i IN 0..4 /\ ul IN s2}`); (NEW_GOAL`S SUBSET {X| ?t. t IN s3 /\ X = (\t. mcell (FST t) V (SND t)) t}`); (EXPAND_TAC "S" THEN EXPAND_TAC "s3"); (ONCE_REWRITE_TAC[SUBSET] THEN ONCE_REWRITE_TAC[IN] THEN REWRITE_TAC[IN_ELIM_THM; FST; SND; mcell_set]); (REPEAT STRIP_TAC); (EXISTS_TAC `((if i <= 4 then i else 4), ul:(real^3)list)`); (ASM_REWRITE_TAC[FST;SND]); (REPEAT STRIP_TAC); (EXISTS_TAC `(if i <= 4 then i else 4)` THEN EXISTS_TAC `ul:(real^3)list`); (REWRITE_TAC[]); (REPEAT STRIP_TAC); (COND_CASES_TAC); (ASM_REWRITE_TAC[IN_NUMSEG_0]); (ASM_REWRITE_TAC[IN_NUMSEG_0] THEN ARITH_TAC); (EXPAND_TAC "s2"); (NEW_GOAL `?u0 u1 u2 u3. ul = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC THEN ONCE_REWRITE_TAC[IN]); (REWRITE_TAC[IN_ELIM_THM]); (EXISTS_TAC `u0:real^3` THEN EXISTS_TAC `u1:real^3`); (EXISTS_TAC `u2:real^3` THEN EXISTS_TAC `u3:real^3`); (ASM_REWRITE_TAC[]); (NEW_GOAL `!u:real^3. u IN set_of_list ul ==> u IN s1`); (EXPAND_TAC "s1" THEN REPEAT STRIP_TAC); (REWRITE_TAC[IN_INTER]); (STRIP_TAC); (NEW_GOAL `set_of_list ul SUBSET (V:real^3->bool)`); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_MESON_TAC[IN]); (ASM_SET_TAC[]); (NEW_GOAL `!u:real^3. u IN set_of_list ul ==> u IN ball (vec 0,r + &6)`); (MATCH_MP_TAC FINITE_MCELL_SET_lemma1); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `i:num` THEN EXISTS_TAC `x:real^3->bool`); (ASM_MESON_TAC[IN]); (ASM_SIMP_TAC[]); (REPEAT STRIP_TAC); (FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (COND_CASES_TAC); (REFL_TAC); (NEW_GOAL `i >= 4`); (ASM_ARITH_TAC); (ASM_SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`]); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{X | ?t. t IN s3 /\ X = (\t. mcell (FST t) V (SND t)) t}`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (EXPAND_TAC "s3"); (MATCH_MP_TAC FINITE_PRODUCT); (REWRITE_TAC[FINITE_NUMSEG]); (EXPAND_TAC "s2"); (MATCH_MP_TAC FINITE_SET_LIST_LEMMA); (EXPAND_TAC "s1"); (MATCH_MP_TAC Packing3.KIUMVTC); (ASM_REWRITE_TAC[])]);; (* ==================================================================== *) (* Lemma 6 *) let CARD_BOUNDARY_INT_BALL_BOUND_1 = prove_by_refinement ( `!x k1 k2. &0 < k1 /\ &0 < k2 ==> (?C. !r. &1 <= r ==> &(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= C * r pow 2)`, [(REPEAT STRIP_TAC); (NEW_GOAL `?C. !r. k2 + sqrt (&3) <= r ==> &(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= C * r pow 2`); (MATCH_MP_TAC Vol1.bdt5_finiteness); (ASM_REWRITE_TAC[]); (TAKE_TAC); (ABBREV_TAC `D = &(CARD (int_ball x ((k2 + sqrt (&3)) + k1)))`); (EXISTS_TAC `max C D`); (REPEAT STRIP_TAC); (ASM_CASES_TAC `k2 + sqrt (&3) <= r`); (NEW_GOAL `&(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= C * r pow 2`); (ASM_SIMP_TAC[]); (NEW_GOAL `C * r pow 2 <= max C D * r pow 2`); (REWRITE_TAC[REAL_ARITH `a * b <= c * b <=> &0 <= (c - a) * b`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_LE_POW_2]); (REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `&(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= &(CARD (int_ball x (r + k1)))`); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (STRIP_TAC); (SET_TAC[]); (REWRITE_TAC[Vol1.finite_int_ball]); (NEW_GOAL `&(CARD (int_ball x (r + k1))) <= D * r pow 2`); (NEW_GOAL `&(CARD (int_ball x (r + k1))) <= D`); (EXPAND_TAC "D"); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (REWRITE_TAC[Vol1.finite_int_ball; int_ball; Pack1.hinhcau_ball; SUBSET; IN_INTER; IN_BALL]); (REWRITE_TAC[IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (ASM_SIMP_TAC[]); (NEW_GOAL `r + k1 <= (k2 + sqrt (&3)) + k1`); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `D <= D * r pow 2`); (REWRITE_TAC[REAL_ARITH `a <= a * b <=> &0 <= a * (b - &1)`]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (EXPAND_TAC "D" THEN REWRITE_TAC[REAL_OF_NUM_LE; LE_0]); (REWRITE_TAC[REAL_ARITH `&0 <= a - b <=> b <= a`]); (ONCE_REWRITE_TAC[REAL_ARITH `&1 = (&1) pow 2`]); (REWRITE_WITH `(&1) pow 2 <= r pow 2 <=> &1 <= r`); (ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN MATCH_MP_TAC Collect_geom.POW2_COND_LT); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `D * r pow 2 <= max C D * r pow 2`); (REWRITE_TAC[REAL_ARITH `a * b <= c * b <=> &0 <= (c - a) * b`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_LE_POW_2]); (REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 7 *) (* The following is a simplified version of Vol1.bdt5_finiteness, Vol1.bdt7_finiteness *) let CARD_BOUNDARY_INT_BALL_BOUND = prove_by_refinement ( `!x k1 k2. (?C. !r. &1 <= r ==> &(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= C * r pow 2)`, [(REPEAT STRIP_TAC); (ASM_CASES_TAC `&0 < k1 /\ &0 < k2`); (ASM_MESON_TAC[CARD_BOUNDARY_INT_BALL_BOUND_1]); (ASM_CASES_TAC `&0 < k2`); (NEW_GOAL `(?C. !r. &1 <= r ==> &(CARD (int_ball x (r + &1) DIFF int_ball x (r - k2))) <= C * r pow 2)`); (MATCH_MP_TAC CARD_BOUNDARY_INT_BALL_BOUND_1); (ASM_REAL_ARITH_TAC); (TAKE_TAC); (EXISTS_TAC `C:real`); (REPEAT STRIP_TAC); (NEW_GOAL `&(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= &(CARD (int_ball x (r + &1) DIFF int_ball x (r - k2)))`); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (SIMP_TAC[FINITE_DIFF; Vol1.finite_int_ball]); (MATCH_MP_TAC (SET_RULE `A SUBSET B ==> A DIFF C SUBSET B DIFF C`)); (REWRITE_TAC[int_ball; Pack1.hinhcau_ball; SUBSET; IN_INTER; IN_BALL]); (REWRITE_TAC[IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (ASM_SIMP_TAC[]); (NEW_GOAL `r + k1 <= r + &1`); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `&(CARD (int_ball x (r + &1) DIFF int_ball x (r - k2))) <= C * r pow 2`); (ASM_SIMP_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_CASES_TAC `&0 < k1`); (NEW_GOAL `(?C. !r. &1 <= r ==> &(CARD (int_ball x (r + k1) DIFF int_ball x (r - &1))) <= C * r pow 2)`); (MATCH_MP_TAC CARD_BOUNDARY_INT_BALL_BOUND_1); (ASM_REAL_ARITH_TAC); (TAKE_TAC); (EXISTS_TAC `C:real`); (REPEAT STRIP_TAC); (NEW_GOAL `&(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= &(CARD (int_ball x (r + k1) DIFF int_ball x (r - &1)))`); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (SIMP_TAC[FINITE_DIFF; Vol1.finite_int_ball]); (MATCH_MP_TAC (SET_RULE `B SUBSET A ==> C DIFF A SUBSET C DIFF B`)); (REWRITE_TAC[int_ball; Pack1.hinhcau_ball; SUBSET; IN_INTER; IN_BALL]); (REWRITE_TAC[IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (ASM_SIMP_TAC[]); (NEW_GOAL `r - &1 <= r - k2`); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `&(CARD (int_ball x (r + k1) DIFF int_ball x (r - &1))) <= C * r pow 2`); (ASM_SIMP_TAC[]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `(?C. !r. &1 <= r ==> &(CARD (int_ball x (r + &1) DIFF int_ball x (r - &1))) <= C * r pow 2)`); (MATCH_MP_TAC CARD_BOUNDARY_INT_BALL_BOUND_1); (ASM_REAL_ARITH_TAC); (TAKE_TAC); (EXISTS_TAC `C:real`); (REPEAT STRIP_TAC); (NEW_GOAL `&(CARD (int_ball x (r + k1) DIFF int_ball x (r - k2))) <= &(CARD (int_ball x (r + &1) DIFF int_ball x (r - &1)))`); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (SIMP_TAC[FINITE_DIFF; Vol1.finite_int_ball]); (REWRITE_TAC[IN_DIFF; int_ball; Pack1.hinhcau_ball; SUBSET; IN_INTER; IN_BALL]); (REWRITE_TAC[IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (ASM_SIMP_TAC[]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `(!i. 1 <= i /\i <= 3 ==>integer (x'$i)) /\ dist (x:real^3,x') < r - k2`); (REPEAT STRIP_TAC); (ASM_SIMP_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[]); (NEW_GOAL `&(CARD (int_ball x (r + &1) DIFF int_ball x (r - &1))) <= C * r pow 2`); (ASM_SIMP_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 8 *) let CARD_INJ_LE = prove_by_refinement ( `!s t f. FINITE s /\ FINITE t /\ INJ (f:A->B) s t ==> CARD s <= CARD t`, [(REWRITE_TAC[INJ] THEN REPEAT STRIP_TAC); (REWRITE_WITH `CARD s = CARD (IMAGE (f:A->B) s)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN MATCH_MP_TAC CARD_IMAGE_INJ); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC CARD_SUBSET); (ASM_REWRITE_TAC[IMAGE; SUBSET]); (UNDISCH_TAC `!x. x IN s ==> (f:A->B) x IN t`); (REWRITE_TAC[IN; IN_ELIM_THM]); (SET_TAC[])]);; (* ==================================================================== *) (* Lemma 9 *) let BOUNDARY_VOLUME = prove_by_refinement ( `!p:real^3 k1 k2. ?C. (!r. &1 <= r ==> vol (ball (p:real^3,r + k1) DIFF ball (p,r - k2)) <= C * r pow 2)`, [(REPEAT STRIP_TAC); (ABBREV_TAC `C = &4 / &3 * pi * (&3 * abs (k1 + k2) + &3 * k1 pow 2 + abs (k1 pow 3 + k2 pow 3))`); (ABBREV_TAC `D = vol (ball (p:real^3, abs (k1 + k2)))`); (ABBREV_TAC `E = max C (D:real)`); (EXISTS_TAC `max (E:real) (&0)`); (REPEAT STRIP_TAC); (ASM_CASES_TAC `r + k1 <= r - k2`); (REWRITE_WITH `ball (p,r + k1) DIFF ball (p:real^3,r - k2) = {}`); (MATCH_MP_TAC (SET_RULE `A SUBSET B ==> A DIFF B = {}`)); (REWRITE_TAC[SUBSET; IN_BALL] THEN REPEAT STRIP_TAC); (ASM_REAL_ARITH_TAC); (REWRITE_TAC[MEASURE_EMPTY]); (NEW_GOAL `&0 * r pow 2 <= max E (&0) * r pow 2`); (REWRITE_TAC[REAL_ARITH `a * x <= b * x <=> &0 <= (b - a) * x`]); (MATCH_MP_TAC REAL_LE_MUL); (ASM_REWRITE_TAC[REAL_LE_POW_2] THEN REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (REWRITE_WITH `vol (ball (p,r + k1) DIFF ball (p,r - k2)) = vol (ball (p,r + k1)) - vol (ball (p,r - k2))`); (MATCH_MP_TAC MEASURE_DIFF_SUBSET); (REWRITE_TAC[MEASURABLE_BALL; SUBSET; IN_BALL] THEN ASM_REAL_ARITH_TAC); (REWRITE_TAC[VOLUME_BALL]); (ASM_CASES_TAC `&0 <= r - k2`); (NEW_GOAL `&0 <= r + k1`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `vol (ball (p,r + k1)) - vol (ball (p,r - k2)) <= C * r pow 2`); (ASM_SIMP_TAC [VOLUME_BALL]); (EXPAND_TAC "C"); (REWRITE_TAC[REAL_ARITH `x * pi * a - x * pi * b <= (x * pi * c) * d <=> &0 <= (x * pi) * (c * d - (a - b))`]); (MATCH_MP_TAC REAL_LE_MUL); (ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_DIV; PI_POS_LE; REAL_ARITH `&0 <= &3 /\ &0 <= &4`]); (REWRITE_TAC[REAL_ARITH `(r + k1) pow 3 - (r - k2) pow 3 = &3 * (k1 + k2) * r pow 2 + &3 * (k1 pow 2 - k2 pow 2) * r + (k1 pow 3 + k2 pow 3)`]); (REWRITE_TAC[REAL_ADD_RDISTRIB]); (NEW_GOAL `(&3 * abs (k1 + k2)) * r pow 2 >= &3 * (k1 * r pow 2 + k2 * r pow 2)`); (REWRITE_TAC[REAL_ARITH `(&3 * abs (k1 + k2)) * r pow 2 >= &3 * (k1 * r pow 2 + k2 * r pow 2) <=> &0 <= &3 * r pow 2 * (abs (k1 + k2) - (k1 + k2))`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_ARITH `&0 <= &3`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_LE_POW_2]); (REAL_ARITH_TAC); (NEW_GOAL `(&3 * k1 pow 2) * r pow 2 >= &3 * (k1 pow 2 - k2 pow 2) * r`); (NEW_GOAL `(&3 * k1 pow 2) * r pow 2 >= (&3 * k1 pow 2) * r`); (REWRITE_TAC[REAL_ARITH `x * y pow 2 >= x * y <=> &0 <= x * y * (y - &1)`]); (ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_POW_2; REAL_ARITH `&1 <= a ==> &0 <= a`; REAL_ARITH `&1 <= a ==> &0 <= (a - &1)`; REAL_ARITH `&0 <= &3`]); (NEW_GOAL `(&3 * k1 pow 2) * r >= &3 * (k1 pow 2 - k2 pow 2) * r`); (REWRITE_TAC[REAL_ARITH `(&3 * x) * r >= &3 * (x - y) * r <=> &0 <= y * r`]); (ASM_SIMP_TAC[REAL_LE_MUL; REAL_LE_POW_2; REAL_ARITH `&1 <= a ==> &0 <= a`]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `abs (k1 pow 3 + k2 pow 3) * r pow 2 >= k1 pow 3 + k2 pow 3`); (NEW_GOAL `abs (k1 pow 3 + k2 pow 3) * r pow 2 >= abs (k1 pow 3 + k2 pow 3)`); (REWRITE_TAC[REAL_ARITH `x * y pow 2 >= x <=> &0 <= x * (y pow 2 - &1)`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_ABS_POS]); (REWRITE_TAC[REAL_ARITH `&0 <= a - &1 <=> &1 pow 2 <= a`]); (REWRITE_WITH `&1 pow 2 <= r pow 2 <=> &1 <= r`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Collect_geom.POW2_COND_LT); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `C * r pow 2 <= max E (&0) * r pow 2`); (REWRITE_TAC[REAL_ARITH `a * x <= b * x <=> &0 <= (b - a) * x`]); (MATCH_MP_TAC REAL_LE_MUL); (ASM_REWRITE_TAC[REAL_LE_POW_2] THEN ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `r < k2`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `vol (ball (p,r + k1)) - vol (ball (p,r - k2)) <= D * r pow 2`); (REWRITE_WITH `ball (p:real^3,r - k2) = {}`); (REWRITE_TAC[SET_RULE `s = {} <=> (!x. x IN s ==> F)`]); (REWRITE_TAC[IN_BALL] THEN GEN_TAC); (MP_TAC (MESON[DIST_POS_LE] `&0 <= dist (p, x:real^3)`)); (ASM_REAL_ARITH_TAC); (REWRITE_TAC[MEASURE_EMPTY; REAL_ARITH `a - &0 = a`]); (NEW_GOAL `vol (ball (p,r + k1)) <= D`); (EXPAND_TAC "D" THEN MATCH_MP_TAC MEASURE_SUBSET); (REWRITE_TAC[MEASURABLE_BALL; SUBSET; IN_BALL]); (REPEAT STRIP_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `D <= D * r pow 2`); (REWRITE_TAC[REAL_ARITH `x <= x * y pow 2 <=> &0 <= x * (y pow 2 - &1)`]); (MATCH_MP_TAC REAL_LE_MUL); (EXPAND_TAC "D"); (STRIP_TAC); (MATCH_MP_TAC MEASURE_POS_LE); (REWRITE_TAC[MEASURABLE_BALL]); (REWRITE_TAC[REAL_ARITH `&0 <= a - &1 <=> &1 pow 2 <= a`]); (REWRITE_WITH `&1 pow 2 <= r pow 2 <=> &1 <= r`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Collect_geom.POW2_COND_LT); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `D * r pow 2 <= max E (&0) * r pow 2`); (REWRITE_TAC[REAL_ARITH `a * x <= b * x <=> &0 <= (b - a) * x`]); (MATCH_MP_TAC REAL_LE_MUL); (ASM_REWRITE_TAC[REAL_LE_POW_2] THEN ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 10 *) let PACKING_BALL_BOUNDARY = prove_by_refinement ( `!V p:real^3 k1 k2. packing V ==> ?C. !r. &1 <= r ==> &(CARD (V INTER ball (p,r + k1) DIFF V INTER ball (p,r - k2))) <= C * r pow 2`, [(REPEAT STRIP_TAC); (NEW_GOAL `?C. (!r. &1 <= r ==> vol (ball (p:real^3,r + k1 + &1) DIFF ball (p,r - (k2 + &1))) <= C * r pow 2)`); (REWRITE_TAC[BOUNDARY_VOLUME]); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `C:real`); (REPEAT STRIP_TAC); (ABBREV_TAC `B = V INTER ball (p,r + k1) DIFF V INTER ball (p:real^3,r - k2)`); (ABBREV_TAC `f = (\u:real^3. ball (u, &1))`); (NEW_GOAL `FINITE (B:real^3->bool)`); (EXPAND_TAC "B"); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `V INTER ball (p:real^3,r + k1)`); (ASM_SIMP_TAC[FINITE_PACK_LEMMA] THEN ASM_SET_TAC[]); (NEW_GOAL `sum B (\u:real^3. vol (f u)) = &(CARD B) * &4 / &3 * pi`); (EXPAND_TAC "f"); (SIMP_TAC[VOLUME_BALL; REAL_ARITH `&0 <= &1`]); (ASM_SIMP_TAC [SUM_CONST]); (REAL_ARITH_TAC); (NEW_GOAL `sum B (\u:real^3. vol (f u)) = vol (UNIONS (IMAGE f B))`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC MEASURE_DISJOINT_UNIONS_IMAGE); (EXPAND_TAC "f" THEN REWRITE_TAC[MEASURABLE_BALL] THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. ~(x IN s /\ x IN t)`]); (REWRITE_TAC[IN_BALL]); (REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN STRIP_TAC); (NEW_GOAL `dist (u, y) <= dist (u, x) + dist (x, y:real^3)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `&2 <= dist (u, y:real^3)`); (UNDISCH_TAC `packing V` THEN REWRITE_TAC[packing] THEN STRIP_TAC); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `vol (UNIONS (IMAGE f (B:real^3->bool))) <= vol (ball (p,r + k1 + &1) DIFF ball (p,r - (k2 + &1)))`); (EXPAND_TAC "f" THEN REWRITE_TAC[IMAGE]); (MATCH_MP_TAC MEASURE_SUBSET); (REPEAT STRIP_TAC); (MATCH_MP_TAC MEASURABLE_UNIONS); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (ASM_REWRITE_TAC[]); (REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (ASM_REWRITE_TAC[MEASURABLE_BALL]); (ASM_SIMP_TAC[MEASURABLE_DIFF; MEASURABLE_BALL]); (EXPAND_TAC "B"); (REWRITE_TAC[SUBSET; IN_UNIONS; IN_BALL; IN_DIFF; IN_INTER]); (REWRITE_TAC[IN_ELIM_THM; IN]); (REPEAT STRIP_TAC); (NEW_GOAL `dist (p, x) <= dist (p, x') + dist (x', x:real^3)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `dist (x',x:real^3) < &1`); (UNDISCH_TAC `(t:real^3->bool) x`); (ASM_REWRITE_TAC[ball; IN_ELIM_THM]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `dist (p, x':real^3) < r - k2`); (NEW_GOAL `dist (p, x') <= dist (p, x) + dist (x, x':real^3)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `dist (x,x':real^3) < &1`); (ONCE_REWRITE_TAC[DIST_SYM]); (UNDISCH_TAC `(t:real^3->bool) x`); (ASM_REWRITE_TAC[ball; IN_ELIM_THM]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[]); (NEW_GOAL `&(CARD (B:real^3->bool)) <= &(CARD B) * &4 / &3 * pi`); (REWRITE_TAC [REAL_ARITH `a <= a * b <=> &0 <= a * (b - &1)`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_OF_NUM_LE; LE_0]); (NEW_GOAL `#3.14159 < pi`); (REWRITE_TAC[Flyspeck_constants.bounds]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `vol (ball (p:real^3,r + k1 + &1) DIFF ball (p,r - (k2 + &1))) <= C * r pow 2`); (ASM_SIMP_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 11 *) let MCELL_SUBSET_BALL_4 = prove_by_refinement ( `!V X . packing V /\ saturated V /\ mcell_set V X ==> (?p. X SUBSET ball (p, &4))`, [(REWRITE_TAC[mcell_set_2; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `X:real^3->bool = {}`); (EXISTS_TAC `(vec 0):real^3`); (ASM_MESON_TAC[EMPTY_SUBSET]); (* Case i = 0 *) (ASM_CASES_TAC `i = 0`); (UNDISCH_TAC `X = mcell i V ul`THEN ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell0]); (STRIP_TAC); (NEW_GOAL `!s. s IN rogers V ul ==> dist (HD ul, s) < &2`); (REPEAT STRIP_TAC); (NEW_GOAL `rogers V ul SUBSET voronoi_closed V (HD ul)`); (ASM_SIMP_TAC[ROGERS_SUBSET_VORONOI_CLOSED]); (NEW_GOAL `s:real^3 IN voronoi_closed V (HD ul)`); (ASM_SET_TAC[]); (UNDISCH_TAC `saturated (V:real^3->bool)`); (REWRITE_TAC[saturated]); (REPEAT STRIP_TAC); (NEW_GOAL `?y. y IN V /\ dist (s, y:real^3) < &2`); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UNDISCH_TAC `s:real^3 IN voronoi_closed V (HD ul)` THEN REWRITE_TAC[voronoi_closed; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `dist (s,HD ul) <= dist (s,y:real^3)`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC); (EXISTS_TAC `HD (ul:(real^3)list)`); (REWRITE_TAC[SUBSET; IN_BALL]); (REPEAT STRIP_TAC); (NEW_GOAL `dist (HD ul, x:real^3) < &2`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (ASM_REAL_ARITH_TAC); (* Case i = 4 *) (ASM_CASES_TAC `i = 4`); (UNDISCH_TAC `X = mcell i V ul` THEN ASM_SIMP_TAC[MCELL_EXPLICIT; mcell4; ARITH_RULE `4 >= 4`]); (COND_CASES_TAC); (STRIP_TAC); (EXISTS_TAC `omega_list V ul`); (NEW_GOAL `ball (omega_list V ul,&4) = convex hull (ball (omega_list V ul,&4))`); (ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_BALL]); (ONCE_ASM_REWRITE_TAC[]); (MATCH_MP_TAC CONVEX_HULL_SUBSET); (REWRITE_TAC[SUBSET; IN_BALL] THEN REPEAT STRIP_TAC); (REWRITE_WITH `omega_list V ul = circumcenter (set_of_list ul)`); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[IN]); (REWRITE_WITH `dist (circumcenter (set_of_list ul),x:real^3) = hl ul`); (NEW_GOAL `(!x:real^3. x IN set_of_list ul ==> dist (circumcenter (set_of_list ul),x) = hl ul)`); (MATCH_MP_TAC Rogers.HL_PROPERTIES); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (FIRST_ASSUM MATCH_MP_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `sqrt (&2) <= &2`); (REWRITE_WITH `sqrt (&2) <= &2 <=> sqrt (&2) pow 2 <= &2 pow 2`); (MATCH_MP_TAC Collect_geom.POW2_COND_LT); (ASM_SIMP_TAC[SQRT_POS_LT; REAL_ARITH `&0 < &2`]); (REWRITE_WITH `sqrt (&2) pow 2 = (&2)`); (MATCH_MP_TAC SQRT_POW_2); (REAL_ARITH_TAC); (REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (STRIP_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (* Case i = 1 and i = 2 and i = 3 *) (ABBREV_TAC `s = omega_list_n V ul (i - 1)`); (EXISTS_TAC `s:real^3`); (MATCH_MP_TAC (SET_RULE `(?x. A SUBSET x /\ x SUBSET B) ==> A SUBSET B`)); (EXISTS_TAC `UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1}`); (STRIP_TAC); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzksykg.QZKSYKG2); (ASM_REWRITE_TAC[SET_RULE `i IN {0,1,2,3,4} <=> i=0\/i=1\/i=2\/i=3\/i=4`]); (ASM_ARITH_TAC); (REWRITE_TAC[SUBSET; IN_UNIONS; IN; IN_ELIM_THM; IN_BALL]); (REPEAT STRIP_TAC); (ABBREV_TAC `vl:(real^3)list = left_action_list p ul`); (NEW_GOAL `rogers V vl SUBSET voronoi_closed V (HD vl)`); (MATCH_MP_TAC ROGERS_SUBSET_VORONOI_CLOSED); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_REWRITE_TAC[SET_RULE `i IN {0,1,2,3,4} <=> i=0\/i=1\/i=2\/i=3\/i=4`]); (ASM_ARITH_TAC); (ASM_SET_TAC[]); (* -------------------------------------------------------------------------- *) (NEW_GOAL `barV V 3 vl /\ (!j. i - 1 <= j /\ j <= 3 ==> omega_list_n V vl j = omega_list_n V ul j)`); (ASM_CASES_TAC `i <= 1`); (UNDISCH_TAC `p permutes 0..i - 1`); (REWRITE_WITH `i - 1 = 0`); (ASM_ARITH_TAC); (REWRITE_TAC[Packing3.PERMUTES_TRIVIAL]); (STRIP_TAC THEN EXPAND_TAC "vl" THEN REWRITE_TAC[ASSUME `p = I:num->num`; Packing3.LEFT_ACTION_LIST_I]); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Ynhyjit.YNHYJIT); (EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (REWRITE_TAC[SET_RULE `i IN {2,3,4} <=> i = 2 \/ i = 3 \/ i = 4`]); (ASM_ARITH_TAC); (UNDISCH_TAC `~(X = {}:real^3->bool)` THEN ASM_REWRITE_TAC[]); (ASM_CASES_TAC `i = 2`); (ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell2; ARITH_RULE `2 - 1 = 1`]); (COND_CASES_TAC); (MESON_TAC[]); (MESON_TAC[]); (ASM_CASES_TAC `i = 3`); (ASM_REWRITE_TAC[MCELL_EXPLICIT; mcell3; ARITH_RULE `3 - 1 = 2`]); (COND_CASES_TAC); (MESON_TAC[]); (MESON_TAC[]); (NEW_GOAL `F`); (ASM_ARITH_TAC); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `s IN voronoi_list V (truncate_simplex (i - 1) vl)`); (REWRITE_WITH `s = omega_list_n V vl (i - 1)`); (EXPAND_TAC "s"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (FIRST_ASSUM MATCH_MP_TAC); (ASM_ARITH_TAC); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_IN_VORONOI_LIST); (EXISTS_TAC `3`); (ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC); (NEW_GOAL `s:real^3 IN voronoi_closed V (HD vl)`); (NEW_GOAL `voronoi_list V (truncate_simplex (i - 1) vl) SUBSET voronoi_closed V (HD vl)`); (REWRITE_TAC[VORONOI_LIST; VORONOI_SET]); (NEW_GOAL `HD (vl:(real^3)list) IN set_of_list (truncate_simplex (i - 1) vl)`); (NEW_GOAL `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (REWRITE_WITH `HD (vl:(real^3)list) = HD (truncate_simplex (i - 1) vl)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (ASM_ARITH_TAC); (MATCH_MP_TAC Packing3.HD_IN_SET_OF_LIST); (NEW_GOAL `?u0 u1 u2 u3. vl = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_CASES_TAC `i = 1`); (ASM_REWRITE_TAC[ARITH_RULE `1 - 1 = 0`; TRUNCATE_SIMPLEX_EXPLICIT_0; LENGTH] THEN ARITH_TAC); (ASM_CASES_TAC `i = 2`); (ASM_REWRITE_TAC[ARITH_RULE `2 - 1 = 1`; TRUNCATE_SIMPLEX_EXPLICIT_1; LENGTH] THEN ARITH_TAC); (ASM_CASES_TAC `i = 3`); (ASM_REWRITE_TAC[ARITH_RULE `3 - 1 = 2`; TRUNCATE_SIMPLEX_EXPLICIT_2; LENGTH] THEN ARITH_TAC); (NEW_GOAL `F`); (ASM_ARITH_TAC); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_SET_TAC[]); (NEW_GOAL `dist (s, x:real^3) <= dist (s, HD vl) + dist (HD vl, x)`); (REWRITE_TAC[DIST_TRIANGLE]); (NEW_GOAL `dist (s:real^3, HD vl) < &2`); (ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[GSYM IN_BALL]); (NEW_GOAL `voronoi_closed V ((HD vl):real^3) SUBSET ball (HD vl,&2)`); (ASM_SIMP_TAC[Packing3.VORONOI_BALL2]); (ASM_SET_TAC[]); (NEW_GOAL `dist (HD vl, x:real^3) < &2`); (REWRITE_TAC[GSYM IN_BALL]); (NEW_GOAL `voronoi_closed V ((HD vl):real^3) SUBSET ball (HD vl,&2)`); (ASM_SIMP_TAC[Packing3.VORONOI_BALL2]); (ASM_SET_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 12 *) let HL_2 = prove_by_refinement ( `!u v:real^3. hl [u; v] = inv (&2) * dist (u, v)`, [(REWRITE_TAC[HL; set_of_list; radV] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC SELECT_UNIQUE THEN REPEAT STRIP_TAC THEN REWRITE_TAC[BETA_THM; CIRCUMCENTER_2; midpoint] THEN EQ_TAC); (REPEAT STRIP_TAC); (REWRITE_WITH `y = dist (inv (&2) % (u + v),v:real^3)`); (FIRST_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[MESON [IN] `{a, b} s <=> s IN {a, b}`] THEN SET_TAC[]); (NORM_ARITH_TAC); (REPEAT STRIP_TAC); (NEW_GOAL `w = u \/ w = v:real^3`); (UP_ASM_TAC THEN REWRITE_TAC[MESON [IN] `{a, b} s <=> s IN {a, b}`] THEN SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC); (ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 13 *) let HL_LE_SQRT2_IMP_BARV_1 = prove_by_refinement ( `!V u0 u1. saturated V /\ packing V /\ u0 IN V /\ u1 IN V /\ ~(u0 = u1) /\ hl [u0;u1] < sqrt (&2) ==> barV V 1 [u0;u1:real^3]`, [(REPEAT STRIP_TAC); (REWRITE_TAC[BARV; LENGTH; ARITH_RULE `SUC (SUC 0) = 1 + 1`]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `LENGTH (vl:(real^3)list) = 1`); (NEW_GOAL `initial_sublist vl [u0; u1:real^3] /\ LENGTH vl = 0 + 1`); (ASM_REWRITE_TAC[] THEN ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `initial_sublist vl [u0; u1:real^3] /\ LENGTH vl = 0 + 1 <=> truncate_simplex 0 [u0;u1] = vl /\ 0 + 1 <= LENGTH [u0;u1:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_0]); (REWRITE_TAC[VORONOI_NONDG; VORONOI_LIST; VORONOI_SET; set_of_list; LENGTH; SET_RULE `INTERS {voronoi_closed V v | v IN {u0}} = voronoi_closed V u0`]); (ASM_SIMP_TAC[Packing3.AFF_DIM_VORONOI_CLOSED]); (REPEAT STRIP_TAC); (ARITH_TAC); (ASM_SET_TAC[]); (ARITH_TAC); (ASM_CASES_TAC `LENGTH (vl:(real^3)list) = 2`); (NEW_GOAL `initial_sublist vl [u0; u1:real^3] /\ LENGTH vl = 1 + 1`); (ASM_REWRITE_TAC[] THEN ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `initial_sublist vl [u0; u1:real^3] /\ LENGTH vl = 1 + 1 <=> truncate_simplex 1 [u0;u1] = vl /\ 1 + 1 <= LENGTH [u0;u1:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1]); (REWRITE_TAC[VORONOI_NONDG]); (REWRITE_TAC[set_of_list; LENGTH]); (REPEAT STRIP_TAC); (ARITH_TAC); (ASM_SET_TAC[]); (REWRITE_WITH `aff_dim (voronoi_list V [u0; u1:real^3]) = aff_dim (INTERS {bis (HD [u0; u1]) v | v | v IN set_of_list [u0; u1]})`); (MATCH_MP_TAC Rogers.YIFVQDV_lemma_aff_dim); (ASM_REWRITE_TAC[set_of_list; AFFINE_INDEPENDENT_2] THEN ASM_SET_TAC[]); (REWRITE_TAC[set_of_list; SET_RULE `a IN {b,c} <=> a =b \/ a = c`; HD]); (REWRITE_TAC[SET_RULE `INTERS {bis u0 v | v | v = u0 \/ v = u1} = bis u0 u0 INTER bis u0 u1`]); (REWRITE_WITH `!a:real^3. bis a a = (:real^3)`); (REWRITE_TAC[bis] THEN SET_TAC[]); (REWRITE_TAC[SET_RULE `(:real^3) INTER s = s`]); (MATCH_MP_TAC (ARITH_RULE `a = &2 ==> a + &(SUC(SUC 0)) = (&4):int`)); (REWRITE_WITH `&2 = &(dimindex (:3)) - &1:int`); (REWRITE_TAC[DIMINDEX_3]); (ARITH_TAC); (REWRITE_TAC[Packing3.BIS_EQ_HYPERPLANE]); (MATCH_MP_TAC AFF_DIM_HYPERPLANE); (ASM_NORM_ARITH_TAC); (NEW_GOAL `F`); (NEW_GOAL `initial_sublist vl [u0; u1:real^3] /\ LENGTH vl = (LENGTH vl - 1) + 1`); (ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `initial_sublist vl [u0; u1:real^3] /\ LENGTH vl = (LENGTH vl - 1) + 1 <=> truncate_simplex (LENGTH vl - 1) [u0;u1] = vl /\ (LENGTH vl - 1) + 1 <= LENGTH [u0;u1:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (REWRITE_TAC[LENGTH]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 14 *) let RCONE_GE_SUBSET = prove_by_refinement ( `!a b u0:real^N u1. a <= b ==> rcone_ge u0 u1 b SUBSET rcone_ge u0 u1 a`, [(REPEAT STRIP_TAC); (REWRITE_TAC[rcone_ge; rconesgn; SUBSET; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `&0 <= dist (x,u0:real^N) * dist (u1,u0) * b - dist (x,u0) * dist (u1,u0) * a`); (REWRITE_TAC[REAL_ARITH `x * y * b - x * y * a = x * y * (b - a)`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 15 *) let RCONE_GT_SUBSET = prove_by_refinement ( `!a b u0:real^N u1. a <= b ==> rcone_gt u0 u1 b SUBSET rcone_gt u0 u1 a`, [(REPEAT STRIP_TAC); (REWRITE_TAC[rcone_gt; rconesgn; SUBSET; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `&0 <= dist (x,u0:real^N) * dist (u1,u0) * b - dist (x,u0) * dist (u1,u0) * a`); (REWRITE_TAC[REAL_ARITH `x * y * b - x * y * a = x * y * (b - a)`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC)]);; (* ===================================================================== *) (* The following lemmas are in the svn 126 of HOL *) (* To be done by John Harrison *) (* ===================================================================== *) (* ==================================================================== *) (* Lemma 16 *) let BOUNDED_SING = prove (`!a. bounded {a}`, REWRITE_TAC[BOUNDED_INSERT; BOUNDED_EMPTY]);; let SUBSPACE_BOUNDED_EQ_TRIVIAL = prove (`!s:real^N->bool. subspace s ==> (bounded s <=> s = {vec 0})`, REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[BOUNDED_SING] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `~(s = {a}) ==> a IN s ==> ?b. b IN s /\ ~(b = a)`)) THEN ASM_SIMP_TAC[SUBSPACE_0] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N` STRIP_ASSUME_TAC) THEN REWRITE_TAC[bounded; NOT_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN DISCH_THEN(MP_TAC o SPEC `(B + &1) / norm v % v:real^N`) THEN ASM_SIMP_TAC[SUBSPACE_MUL; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN REAL_ARITH_TAC);; let AFFINE_BOUNDED_EQ_TRIVIAL = prove (`!s:real^N->bool. affine s ==> (bounded s <=> s = {} \/ ?a. s = {a})`, GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN DISCH_THEN(X_CHOOSE_THEN `b:real^N` MP_TAC) THEN GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SUBSPACE_BOUNDED_EQ_TRIVIAL] THEN FIRST_ASSUM(MP_TAC o MATCH_MP SUBSPACE_0) THEN SET_TAC[]);; let AFFINE_BOUNDED_EQ_LOWDIM = prove (`!s:real^N->bool. affine s ==> (bounded s <=> aff_dim s <= &0)`, SIMP_TAC[AFF_DIM_GE; INT_ARITH `--(&1):int <= x ==> (x <= &0 <=> x = --(&1) \/ x = &0)`] THEN SIMP_TAC[AFF_DIM_EQ_0; AFF_DIM_EQ_MINUS1; AFFINE_BOUNDED_EQ_TRIVIAL]);; let BOUNDED_HYPERPLANE_EQ_TRIVIAL = prove (`!a b. bounded {x:real^N | a dot x = b} <=> if a = vec 0 then ~(b = &0) else dimindex(:N) = 1`, REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_REWRITE_TAC[DOT_LZERO] THENL [ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[EMPTY_GSPEC; BOUNDED_EMPTY] THEN REWRITE_TAC[NOT_BOUNDED_UNIV; SET_RULE `{x | T} = UNIV`]; ASM_SIMP_TAC[AFFINE_BOUNDED_EQ_LOWDIM; AFF_DIM_HYPERPLANE; AFFINE_HYPERPLANE] THEN REWRITE_TAC[INT_ARITH `a - &1:int <= &0 <=> a <= &1`; INT_OF_NUM_LE] THEN MATCH_MP_TAC(ARITH_RULE `1 <= n ==> (n <= 1 <=> n = 1)`) THEN REWRITE_TAC[DIMINDEX_GE_1]]);; let UNBOUNDED_HYPERPLANE = prove (`!a b. ~(a = vec 0) ==> ~(bounded {x:real^3 | a dot x = b})`, SIMP_TAC[BOUNDED_HYPERPLANE_EQ_TRIVIAL; DIMINDEX_3; ARITH]);; (* ==================================================================== *) (* Lemma 17 *) let DIHV_SYM_2 = prove (`!x y z t:real^3. dihV x y z t = dihV x y t z`, REWRITE_TAC[dihV] THEN REPEAT STRIP_TAC THEN REPEAT LET_TAC THEN ASM_REWRITE_TAC[arcV; DOT_SYM; REAL_ARITH `a * b = b * a`]);; (* ==================================================================== *) (* Lemma 18 *) let REAL_DIV_LE_1_TACTICS = prove_by_refinement ( `!m n. &0 < n /\ m <= n ==> m / n <= &1`, [(REPEAT STRIP_TAC); (REWRITE_WITH `m / n <= &1 <=> m <= &1 * n`); (MATCH_MP_TAC REAL_LE_LDIV_EQ); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 19 *) let REAL_DIV_GE_1_TACTICS = prove_by_refinement ( `!m n. &0 < n /\ n <= m ==> &1 <= m / n`, [(REPEAT STRIP_TAC); (REWRITE_WITH `&1 <= m / n <=> &1 * n <= m`); (MATCH_MP_TAC REAL_LE_RDIV_EQ); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 20 *) let REAL_DIV_LT_1_TACTICS = prove_by_refinement ( `!m n. &0 < n /\ m < n ==> m / n < &1`, [(REPEAT STRIP_TAC); (REWRITE_WITH `m / n < &1 <=> m < &1 * n`); (MATCH_MP_TAC REAL_LT_LDIV_EQ); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 21 *) let REAL_DIV_GT_1_TACTICS = prove_by_refinement ( `!m n. &0 < n /\ n < m ==> &1 < m / n`, [(REPEAT STRIP_TAC); (REWRITE_WITH `&1 < m / n <=> &1 * n < m`); (MATCH_MP_TAC REAL_LT_RDIV_EQ); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* The following lemmas help to prove that the *) (* definition of dihX is well-defined *) (* ==================================================================== *) (* ==================================================================== *) (* Lemma 22 *) let MCELL_ID_OMEGA_LIST_N = prove_by_refinement ( `!V i j ul vl. packing V /\ saturated V /\ barV V 3 ul /\ barV V 3 vl /\ (mcell i V ul = mcell j V vl) /\ ~(negligible (mcell i V ul)) /\ i IN {2, 3, 4} /\ j IN {2, 3, 4} ==> (i = j) /\ (!k. i - 1 <= k /\ k <= 3 ==> omega_list_n V ul k = omega_list_n V vl k)`, [(REPEAT GEN_TAC THEN STRIP_TAC); (NEW_GOAL `i = j /\ mcell i V ul = mcell j V vl`); (MATCH_MP_TAC Ajripqn.AJRIPQN); (ASM_REWRITE_TAC[SET_RULE `a INTER a = a` ]); (STRIP_TAC); (ASM_SET_TAC[]); (STRIP_TAC); (ASM_SET_TAC[]); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (ASM_CASES_TAC `i = 4`); (REPEAT STRIP_TAC); (NEW_GOAL `k = 3`); (ASM_ARITH_TAC); (ASM_REWRITE_TAC[]); (REWRITE_WITH `omega_list_n V ul 3 = omega_list V ul`); (REWRITE_TAC[OMEGA_LIST]); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[ARITH_RULE `(3 + 1) - 1 = 3`]); (REWRITE_WITH `omega_list_n V vl 3 = omega_list V vl`); (REWRITE_TAC[OMEGA_LIST]); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[ARITH_RULE `(3 + 1) - 1 = 3`]); (REWRITE_WITH `omega_list V ul = circumcenter (set_of_list ul)`); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[IN]); (MATCH_MP_TAC (MESON[] `(~A ==> F) ==> A`)); (STRIP_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (SIMP_TAC[ASSUME `i = 4`; MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (REWRITE_WITH `omega_list V vl = circumcenter (set_of_list vl)`); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[IN]); (MATCH_MP_TAC (MESON[] `(~A ==> F) ==> A`)); (STRIP_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[]); (REWRITE_WITH `j = 4`); (ASM_ARITH_TAC); (SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (AP_TERM_TAC); (REWRITE_WITH `set_of_list ul = set_of_list (vl:(real^3)list) <=> convex hull (set_of_list ul) = convex hull (set_of_list vl)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.CONVEX_HULL_EQ_EQ_SET_EQ); (STRIP_TAC); (NEW_GOAL `?u0 u1 u2 u3. ul = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_REWRITE_TAC[set_of_list] THEN STRIP_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (REWRITE_TAC[ASSUME `i = 4`]); (SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (ASM_REWRITE_TAC[set_of_list]); (MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE); (MATCH_MP_TAC COPLANAR_SUBSET); (EXISTS_TAC `affine hull {u0, u1, u2, u3:real^3}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (REWRITE_TAC[COPLANAR_AFFINE_HULL_COPLANAR]); (MATCH_MP_TAC Rogers.AFF_DIM_LE_2_IMP_COPLANAR); (MATCH_MP_TAC Njiutiu.AFF_DEPENDENT_AFF_DIM_4); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `?u0 u1 u2 u3. vl = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_REWRITE_TAC[set_of_list] THEN STRIP_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (REWRITE_TAC[ASSUME `mcell i V ul = mcell j V vl`]); (REWRITE_WITH `j = 4`); (ASM_ARITH_TAC); (SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (ASM_REWRITE_TAC[set_of_list]); (MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE); (MATCH_MP_TAC COPLANAR_SUBSET); (EXISTS_TAC `affine hull {u0, u1, u2, u3:real^3}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (REWRITE_TAC[COPLANAR_AFFINE_HULL_COPLANAR]); (MATCH_MP_TAC Rogers.AFF_DIM_LE_2_IMP_COPLANAR); (MATCH_MP_TAC Njiutiu.AFF_DEPENDENT_AFF_DIM_4); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (MP_TAC (ASSUME `mcell i V ul = mcell j V vl`)); (REWRITE_WITH `j = 4 /\ i = 4`); (ASM_ARITH_TAC); (SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (COND_CASES_TAC); (REWRITE_TAC[]); (NEW_GOAL `F`); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[]); (REWRITE_WITH `j = 4 /\ i = 4`); (ASM_ARITH_TAC); (SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (REWRITE_WITH `j = 4 /\ i = 4`); (ASM_ARITH_TAC); (SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`;mcell4]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (* Finish the case i = 4 *) (* ========================================================================= *) (NEW_GOAL `mcell i V ul SUBSET UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1}`); (MATCH_MP_TAC Qzksykg.QZKSYKG2); (ASM_REWRITE_TAC[] THEN ASM_SET_TAC[]); (ABBREV_TAC `X = mcell i V ul`); (NEW_GOAL `X INTER UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1} = X`); (ASM_SET_TAC[]); (NEW_GOAL `~negligible (X INTER UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1})`); (ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[INTER_UNIONS]); (NEW_GOAL `!s:(real^3->bool)->bool. ~negligible (UNIONS s) /\ FINITE s ==> (?t. t IN s /\ ~negligible t)`); (MESON_TAC[NEGLIGIBLE_UNIONS]); (REWRITE_TAC[SET_RULE `{X INTER x | x IN {rogers V (left_action_list p ul) | p permutes 0..i - 1}} = {X INTER rogers V (left_action_list p ul) | p permutes 0..i - 1}`]); (ABBREV_TAC `S = {X INTER rogers V (left_action_list p ul) | p permutes 0..i - 1}`); (STRIP_TAC); (NEW_GOAL `(?t. t IN (S:(real^3->bool)->bool) /\ ~negligible t)`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_REWRITE_TAC[]); (EXPAND_TAC "S"); (ABBREV_TAC `f = (\p. X INTER rogers V (left_action_list p ul))`); (REWRITE_WITH `{X INTER rogers V (left_action_list p ul) | p permutes 0..i - 1} = {f p | p permutes 0..i-1}`); (EXPAND_TAC "f" THEN REFL_TAC); (ONCE_REWRITE_TAC[SET_RULE `{f x | P x} = {f x | x IN {x | P x}}`]); (REWRITE_TAC[SET_RULE `{f p | p IN {x | x permutes 0..i - 1} } = {y | ?p. p IN {x | x permutes 0..i - 1} /\ y = f p}`]); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (MATCH_MP_TAC FINITE_PERMUTATIONS); (REWRITE_TAC[FINITE_NUMSEG]); (UP_ASM_TAC THEN EXPAND_TAC "S" THEN REWRITE_TAC[IN; IN_ELIM_THM] THEN STRIP_TAC); (NEW_GOAL `mcell j V vl SUBSET UNIONS {rogers V (left_action_list q vl) | q permutes 0..j - 1}`); (MATCH_MP_TAC Qzksykg.QZKSYKG2); (ASM_REWRITE_TAC[] THEN ASM_SET_TAC[]); (NEW_GOAL `t SUBSET UNIONS {rogers V (left_action_list q vl) | q permutes 0..j - 1}`); (ASM_SET_TAC[]); (NEW_GOAL `t INTER UNIONS {rogers V (left_action_list q vl) | q permutes 0..j - 1} = t`); (ASM_SET_TAC[]); (NEW_GOAL `~negligible (t INTER UNIONS {rogers V (left_action_list q vl) | q permutes 0..j - 1})`); (ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[INTER_UNIONS]); (REWRITE_TAC[SET_RULE `{t INTER x | x IN {rogers V (left_action_list q vl) | q permutes 0..j - 1}} = {t INTER rogers V (left_action_list q vl) | q permutes 0..j - 1}`]); (NEW_GOAL `!s:(real^3->bool)->bool. ~negligible (UNIONS s) /\ FINITE s ==> (?t. t IN s /\ ~negligible t)`); (MESON_TAC[NEGLIGIBLE_UNIONS]); (ABBREV_TAC `R = {t INTER rogers V (left_action_list q vl) | q permutes 0..j - 1}`); (STRIP_TAC); (NEW_GOAL `(?r. r IN (R:(real^3->bool)->bool) /\ ~negligible r)`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_REWRITE_TAC[]); (EXPAND_TAC "R"); (ABBREV_TAC `f = (\q. t INTER rogers V (left_action_list q vl))`); (REWRITE_WITH `{t INTER rogers V (left_action_list q vl) | q permutes 0..j - 1} = {f q | q permutes 0..j - 1}`); (EXPAND_TAC "f" THEN REFL_TAC); (ONCE_REWRITE_TAC[SET_RULE `{f x | P x} = {f x | x IN {x | P x}}`]); (REWRITE_TAC[SET_RULE `{f q | q IN {x | x permutes 0..j - 1} } = {y | ?q. q IN {x | x permutes 0..j - 1} /\ y = f q}`]); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (MATCH_MP_TAC FINITE_PERMUTATIONS); (REWRITE_TAC[FINITE_NUMSEG]); (UP_ASM_TAC THEN EXPAND_TAC "R" THEN REWRITE_TAC[IN; IN_ELIM_THM] THEN STRIP_TAC); (NEW_GOAL `rogers V (left_action_list p ul) = rogers V (left_action_list q vl)`); (MATCH_MP_TAC (MESON[] `(~A ==> F) ==> A`)); (STRIP_TAC); (NEW_GOAL `coplanar (rogers V (left_action_list p ul) INTER rogers V (left_action_list q vl))`); (MATCH_MP_TAC Rogers.DUUNHOR); (ASM_REWRITE_TAC[IN]); (STRIP_TAC); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_SET_TAC[]); (REWRITE_TAC[GSYM (ASSUME `X = mcell j V vl`)]); (STRIP_TAC); (UNDISCH_TAC `~NULLSET X`); (REWRITE_TAC[ASSUME `X:real^3->bool = {}`; NEGLIGIBLE_EMPTY]); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `vl:(real^3)list` THEN EXISTS_TAC `j:num`); (EXISTS_TAC `q:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_SET_TAC[]); (REWRITE_TAC[GSYM (ASSUME `X = mcell j V vl`)]); (STRIP_TAC); (UNDISCH_TAC `~NULLSET X`); (REWRITE_TAC[ASSUME `X:real^3->bool = {}`; NEGLIGIBLE_EMPTY]); (UNDISCH_TAC `~NULLSET r`); (REWRITE_TAC[]); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `rogers V (left_action_list p ul) INTER rogers V (left_action_list q vl)`); (STRIP_TAC); (ASM_SIMP_TAC[COPLANAR_IMP_NEGLIGIBLE]); (ASM_SET_TAC[]); (ABBREV_TAC `xl:(real^3)list = (left_action_list p ul)`); (ABBREV_TAC `yl:(real^3)list = (left_action_list q vl)`); (NEW_GOAL `barV V 3 xl /\ (!k. i - 1 <= k /\ k <= 3 ==> omega_list_n V xl k = omega_list_n V ul k)`); (MATCH_MP_TAC Ynhyjit.YNHYJIT); (EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (ASM_CASES_TAC `j = 2`); (UNDISCH_TAC `~NULLSET X`); (REWRITE_TAC[GSYM (ASSUME `mcell i V ul = X`); ASSUME `i = j:num`; MCELL_EXPLICIT; mcell2; ASSUME `j = 2`]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[ARITH_RULE `2 - 1 = 1`]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `j = 3`); (ASM_SET_TAC[]); (UNDISCH_TAC `~NULLSET X`); (REWRITE_TAC[GSYM (ASSUME `mcell i V ul = X`); ASSUME `i = j:num`; MCELL_EXPLICIT; mcell3; ASSUME `j = 3`]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[ARITH_RULE `3 - 1 = 2`]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `barV V 3 yl /\ (!k. j - 1 <= k /\ k <= 3 ==> omega_list_n V yl k = omega_list_n V vl k)`); (MATCH_MP_TAC Ynhyjit.YNHYJIT); (EXISTS_TAC `q:num->num`); (ASM_REWRITE_TAC[]); (ASM_CASES_TAC `j = 2`); (UNDISCH_TAC `~NULLSET X`); (REWRITE_TAC[ASSUME `X = mcell j V vl`; MCELL_EXPLICIT; mcell2; ASSUME `j = 2`]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[ARITH_RULE `2 - 1 = 1`]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `j = 3`); (ASM_SET_TAC[]); (UNDISCH_TAC `~NULLSET X`); (REWRITE_TAC[ASSUME `X = mcell j V vl`; MCELL_EXPLICIT; mcell3; ASSUME `j = 3`]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[ARITH_RULE `3 - 1 = 2`]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `!k. 0 <= k /\ k <= 3 ==> omega_list_n V xl k = omega_list_n V yl k`); (MATCH_MP_TAC Njiutiu.NJIUTIU); (ASM_REWRITE_TAC[]); (NEW_GOAL `aff_dim (rogers V yl) <= &(dimindex (:3))`); (REWRITE_TAC[AFF_DIM_LE_UNIV]); (UP_ASM_TAC THEN REWRITE_TAC[DIMINDEX_3]); (STRIP_TAC); (NEW_GOAL `~(aff_dim (rogers V yl) <= &2)`); (STRIP_TAC); (UNDISCH_TAC `~NULLSET r` THEN REWRITE_TAC[]); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `rogers V yl`); (STRIP_TAC); (MATCH_MP_TAC COPLANAR_IMP_NEGLIGIBLE); (ASM_SIMP_TAC[Rogers.AFF_DIM_LE_2_IMP_COPLANAR]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ARITH_TAC); (NEW_GOAL `(!k. j - 1 <= k /\ k <= 3 ==> omega_list_n V xl k = omega_list_n V yl k)`); (REPEAT STRIP_TAC); (FIRST_ASSUM MATCH_MP_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `j = 2 \/ j = 3 \/ j = 4`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN ARITH_TAC); (ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 23 *) let MCELL_ID_MXI = prove_by_refinement ( `!V i j ul vl. packing V /\ saturated V /\ barV V 3 ul /\ barV V 3 vl /\ HD ul = HD vl /\ mcell i V ul = mcell j V vl /\ ~NULLSET (mcell i V ul) /\ i IN {2, 3} /\ j IN {2, 3} ==> mxi V ul = mxi V vl`, [(REPEAT STRIP_TAC); (NEW_GOAL `i = j /\ (!k. i - 1 <= k /\ k <= 3 ==> omega_list_n V ul k = omega_list_n V vl k)`); (MATCH_MP_TAC MCELL_ID_OMEGA_LIST_N); (ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_CASES_TAC `hl (truncate_simplex 2 (ul:(real^3)list)) >= sqrt (&2)`); (ASM_CASES_TAC `hl (truncate_simplex 2 (vl:(real^3)list)) >= sqrt (&2)`); (REWRITE_WITH `mxi V ul = omega_list_n V ul 2`); (REWRITE_TAC[mxi] THEN COND_CASES_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REWRITE_WITH `mxi V vl = omega_list_n V vl 2`); (REWRITE_TAC[mxi] THEN COND_CASES_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[REAL_ARITH `~(a >= b) <=> a < b`]); (STRIP_TAC); (ABBREV_TAC `xl = truncate_simplex 2 (ul:(real^3)list)`); (ABBREV_TAC `yl = truncate_simplex 2 (vl:(real^3)list)`); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)) = set_of_list (truncate_simplex (i - 1) vl)`); (NEW_GOAL `V INTER mcell i V ul = set_of_list (truncate_simplex (i - 1) ul)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `V INTER mcell j V vl = set_of_list (truncate_simplex (j - 1) vl)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `hl (xl:(real^3)list) <= dist (omega_list V xl, HD xl)`); (MATCH_MP_TAC Rogers.WAUFCHE1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN] THEN EXPAND_TAC "xl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `hl yl = dist (omega_list V yl, HD yl)`); (MATCH_MP_TAC Rogers.WAUFCHE2); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "yl" THEN MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `omega_list V xl = omega_list V yl`); (REWRITE_TAC[OMEGA_LIST]); (REWRITE_WITH `LENGTH (xl:(real^3)list) = 3 /\ LENGTH (yl:(real^3)list) = 3`); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_TAC[ARITH_RULE `3 = 2 + 1`]); (STRIP_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_TAC[ARITH_RULE `3 - 1 = 2`]); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_WITH `truncate_simplex 2 ul = truncate_simplex (2 + 0) (ul:(real^3)list) /\ truncate_simplex 2 vl = truncate_simplex (2 + 0) (vl:(real^3)list)`); (REWRITE_TAC[ARITH_RULE `2 + 0 = 2`]); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) ul) 2 = omega_list_n V ul 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) vl) 2 = omega_list_n V vl 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (NEW_GOAL `HD (xl:(real^3)list) = HD yl`); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_WITH `HD (truncate_simplex 2 ul) = (HD ul):real^3`); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_WITH `HD (truncate_simplex 2 vl) = (HD vl):real^3`); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `hl (yl:(real^3)list) = dist (omega_list V xl, HD xl)`); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[]); (* ======================================================================= *) (UP_ASM_TAC THEN REWRITE_TAC[REAL_ARITH `~(a >= b) <=> a < b`]); (STRIP_TAC); (ASM_CASES_TAC `hl (truncate_simplex 2 (vl:(real^3)list)) >= sqrt (&2)`); (ABBREV_TAC `xl = truncate_simplex 2 (ul:(real^3)list)`); (ABBREV_TAC `yl = truncate_simplex 2 (vl:(real^3)list)`); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)) = set_of_list (truncate_simplex (i - 1) vl)`); (NEW_GOAL `V INTER mcell i V ul = set_of_list (truncate_simplex (i - 1) ul)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `V INTER mcell j V vl = set_of_list (truncate_simplex (j - 1) vl)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `hl (yl:(real^3)list) <= dist (omega_list V yl, HD yl)`); (MATCH_MP_TAC Rogers.WAUFCHE1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN] THEN EXPAND_TAC "yl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `hl xl = dist (omega_list V xl, HD xl)`); (MATCH_MP_TAC Rogers.WAUFCHE2); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "xl" THEN MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `omega_list V xl = omega_list V yl`); (REWRITE_TAC[OMEGA_LIST]); (REWRITE_WITH `LENGTH (xl:(real^3)list) = 3 /\ LENGTH (yl:(real^3)list) = 3`); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_TAC[ARITH_RULE `3 = 2 + 1`]); (STRIP_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_TAC[ARITH_RULE `3 - 1 = 2`]); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_WITH `truncate_simplex 2 ul = truncate_simplex (2 + 0) (ul:(real^3)list) /\ truncate_simplex 2 vl = truncate_simplex (2 + 0) (vl:(real^3)list)`); (REWRITE_TAC[ARITH_RULE `2 + 0 = 2`]); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) ul) 2 = omega_list_n V ul 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) vl) 2 = omega_list_n V vl 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (NEW_GOAL `HD (xl:(real^3)list) = HD yl`); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_WITH `HD (truncate_simplex 2 ul) = (HD ul):real^3`); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_WITH `HD (truncate_simplex 2 vl) = (HD vl):real^3`); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `hl (xl:(real^3)list) = dist (omega_list V yl, HD yl)`); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[REAL_ARITH `~(a >= b) <=> a < b`]); (STRIP_TAC); (* ======================================================================= *) (NEW_GOAL `sqrt (&2) <= hl (ul:(real^3)list)`); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_CASES_TAC `i = 2`); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `i = 2`; mcell2]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `i = 3`); (ASM_SET_TAC[]); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `i = 3`; mcell3]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `sqrt (&2) <= hl (vl:(real^3)list)`); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[]); (ASM_CASES_TAC `j = 2`); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `j = 2`; mcell2]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `j = 3`); (ASM_SET_TAC[]); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `j = 3`; mcell3]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ABBREV_TAC `s2 = omega_list_n V ul 2`); (ABBREV_TAC `s3 = omega_list_n V ul 3`); (NEW_GOAL `s2 = omega_list_n V vl 2`); (EXPAND_TAC "s2" THEN FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (NEW_GOAL `s3 = omega_list_n V vl 3`); (EXPAND_TAC "s3" THEN FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (ABBREV_TAC `u0:real^3 = HD ul`); (NEW_GOAL `?s. between s (s2,s3) /\ dist (u0,s) = sqrt (&2) /\ mxi V ul = s`); (EXPAND_TAC "s2" THEN EXPAND_TAC "s3"); (MATCH_MP_TAC MXI_EXPLICIT); (NEW_GOAL `?v0 u1 u2 u3. ul = [v0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `u1:real^3` THEN EXISTS_TAC `u2:real^3` THEN EXISTS_TAC `u3:real^3`); (ASM_REWRITE_TAC[]); (REWRITE_WITH `v0:real^3 = HD ul`); (REWRITE_TAC[ASSUME `ul = [v0;u1;u2;u3:real^3]`; HD]); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?t. between t (s2,s3) /\ dist (u0,t) = sqrt (&2) /\ mxi V vl = t`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC MXI_EXPLICIT); (NEW_GOAL `?v0 u1 u2 u3. vl = [v0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `u1:real^3` THEN EXISTS_TAC `u2:real^3` THEN EXISTS_TAC `u3:real^3`); (ASM_REWRITE_TAC[HD]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `!n. between n (s2, s3:real^3) ==> dist (u0, n) pow 2 = dist (s2, n) pow 2 + dist (u0, s2) pow 2`); (REPEAT STRIP_TAC); (REWRITE_TAC[dist]); (MATCH_MP_TAC PYTHAGORAS); (REWRITE_TAC[orthogonal]); (ABBREV_TAC `zl = truncate_simplex 2 (ul:(real^3)list)`); (REWRITE_WITH `s2:real^3 = circumcenter (set_of_list zl)`); (REWRITE_WITH `s2 = omega_list V zl`); (EXPAND_TAC "s2" THEN EXPAND_TAC "zl"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (MATCH_MP_TAC Rogers.MHFTTZN4); (EXISTS_TAC `V:real^3->bool`); (EXISTS_TAC `2`); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `s2 IN voronoi_list V zl`); (REWRITE_WITH `s2 = omega_list V zl`); (EXPAND_TAC "s2" THEN EXPAND_TAC "zl"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.OMEGA_LIST_IN_VORONOI_LIST); (EXISTS_TAC `2`); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `s3 IN voronoi_list V zl`); (EXPAND_TAC "s3" THEN EXPAND_TAC "zl"); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3 /\ 3 <= 3`]); (NEW_GOAL `affine hull {s2, s3} SUBSET affine hull voronoi_list V zl`); (MATCH_MP_TAC Marchal_cells_2_new.AFFINE_SUBSET_KY_LEMMA); (ASM_SET_TAC[]); (NEW_GOAL `n IN affine hull {s2, s3:real^3}`); (NEW_GOAL `convex hull {s2,s3} SUBSET affine hull {s2,s3:real^3}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (NEW_GOAL `n IN convex hull {s2, s3:real^3}`); (REWRITE_TAC[GSYM BETWEEN_IN_CONVEX_HULL]); (ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1); (REWRITE_WITH `u0:real^3 = HD zl`); (EXPAND_TAC "zl" THEN EXPAND_TAC "u0"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.HD_IN_SET_OF_LIST); (REWRITE_WITH `LENGTH (zl:(real^3)list) = 2 + 1 /\ CARD (set_of_list zl) = 2 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (ARITH_TAC); (NEW_GOAL `dist (u0,s) pow 2 = dist (s2,s:real^3) pow 2 + dist (u0,s2) pow 2`); (ASM_SIMP_TAC[]); (NEW_GOAL `dist (u0,t) pow 2 = dist (s2,t:real^3) pow 2 + dist (u0,s2) pow 2`); (ASM_SIMP_TAC[]); (NEW_GOAL `dist (s2, s:real^3) = dist (s2, t)`); (REWRITE_TAC[DIST_EQ]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); (UNDISCH_TAC `between s (s2, s3:real^3)` THEN UNDISCH_TAC `between t (s2, s3:real^3)`); (REWRITE_TAC[BETWEEN_IN_CONVEX_HULL; CONVEX_HULL_2; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (UNDISCH_TAC `dist (s2,s) = dist (s2,t:real^3)`); (REWRITE_TAC[dist;ASSUME `s = u' % s2 + v' % s3:real^3`; ASSUME `t = u % s2 + v % s3:real^3` ]); (REWRITE_WITH `s2 - (u' % s2 + v' % s3) = (u' + v') % s2 - (u' % s2 + v' % s3:real^3)`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_WITH `s2 - (u % s2 + v % s3) = (u + v) % s2 - (u % s2 + v % s3:real^3)`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_TAC[VECTOR_ARITH `(u + v) % s - (u % s + v % t) = v % (s - t)`]); (REWRITE_TAC[NORM_MUL]); (REWRITE_WITH `abs v = v /\ abs v' = v'`); (ASM_SIMP_TAC[REAL_ABS_REFL]); (REWRITE_TAC[REAL_ARITH `a * b = c * b <=> (a - c) * b = &0`;REAL_ENTIRE]); (STRIP_TAC); (REWRITE_WITH `v = v':real /\ u = u':real`); (ASM_REAL_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[NORM_EQ_0]); (REWRITE_TAC[VECTOR_ARITH `a - b = vec 0 <=> a = b`] THEN STRIP_TAC); (NEW_GOAL `F`); (NEW_GOAL `hl (ul:(real^3)list) <= dist (s3,u0:real^3)`); (REWRITE_WITH `s3 = omega_list V ul /\ u0 = HD ul`); (STRIP_TAC); (REWRITE_TAC[GSYM (ASSUME `omega_list_n V ul 3 = s3`); OMEGA_LIST]); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[ARITH_RULE `(3 + 1) - 1 = 3`]); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Rogers.WAUFCHE1); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[IN]); (ABBREV_TAC `zl:(real^3)list = truncate_simplex 2 ul`); (NEW_GOAL `hl (zl:(real^3)list) = dist (s3, u0:real^3)`); (REWRITE_WITH `s3 = omega_list V zl /\ u0 = HD zl`); (STRIP_TAC); (REWRITE_TAC[GSYM (ASSUME `s2:real^3 = s3`); OMEGA_LIST]); (EXPAND_TAC "s2"); (REWRITE_WITH `LENGTH (zl:(real^3)list) = 2 + 1 /\ CARD (set_of_list zl) = 2 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (REWRITE_TAC[ARITH_RULE `(2 + 1) - 1 = 2`]); (EXPAND_TAC "zl"); (REWRITE_WITH `truncate_simplex 2 (ul:(real^3)list) = truncate_simplex (2 + 0) (ul:(real^3)list)`); (REWRITE_TAC[ARITH_RULE `2 + 0 = 2`]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (EXPAND_TAC "zl" THEN EXPAND_TAC "u0"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Rogers.WAUFCHE2); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 24 *) let AFFINE_HULL_3_INSERT = prove_by_refinement ( `!a:real^3 S. a IN affine hull S ==> affine hull (a INSERT S) = affine hull S`, [(REPEAT STRIP_TAC); (MATCH_MP_TAC AFFINE_HULLS_EQ); (ASM_REWRITE_TAC[SET_SUBSET_AFFINE_HULL; INSERT_SUBSET]); (MATCH_MP_TAC (SET_RULE `a:real^3 INSERT s SUBSET t ==> s SUBSET t`)); (REWRITE_TAC[SET_SUBSET_AFFINE_HULL])]);; (* ==================================================================== *) (* Lemma 25 *) let FINITE_EDGE_X2 = prove_by_refinement ( `!(V:real^3->bool) e:real^3->bool u0 u1. packing V /\ saturated V /\ e = {u0, u1} ==> FINITE {X | mcell_set V X /\ edgeX V X e}`, [(REPEAT STRIP_TAC); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{X | mcell_set V X /\ edgeX V X e /\ ~(X = {})} UNION {{}}`); (ONCE_REWRITE_TAC[SET_RULE `A /\ B <=> B /\ A`]); (STRIP_TAC); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM; IN_UNION]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[MESON[IN] `{{}} x <=> x IN {{}}`; SET_RULE `x IN {a} <=> x = a`]); (MESON_TAC[]); (REWRITE_TAC [FINITE_UNION; FINITE_SING]); (MATCH_MP_TAC FINITE_SUBSET); (ABBREV_TAC `f = (\(i, ul). mcell i V ul)`); (ABBREV_TAC `s = V INTER ball (u0:real^3, &4)`); (ABBREV_TAC `S = {y | ?v0 v1 v2 v3:real^3. v0 IN s /\ v1 IN s /\ v2 IN s /\ v3 IN s /\ y = [v0; v1; v2; v3]}`); (ABBREV_TAC `S1 = {i, (ul:(real^3)list) | i IN 0..4 /\ ul IN S}`); (EXISTS_TAC `{(f:num#(real^3)list->real^3->bool) x | x IN S1}`); (STRIP_TAC); (REWRITE_TAC[SET_RULE `{f x |x IN s } = {y | ?x. x IN s /\ y = f x}`]); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (EXPAND_TAC "S1"); (MATCH_MP_TAC FINITE_PRODUCT); (REWRITE_TAC[FINITE_NUMSEG]); (EXPAND_TAC "S"); (MATCH_MP_TAC Ajripqn.FINITE_SET_LIST_LEMMA); (EXPAND_TAC "s"); (MATCH_MP_TAC Pack2.KIUMVTC THEN ASM_SIMP_TAC[]); (EXPAND_TAC "f" THEN REWRITE_TAC[SUBSET; IN; IN_ELIM_THM; mcell_set_2]); (REPEAT STRIP_TAC); (EXISTS_TAC `(i:num, ul:(real^3)list)`); (ASM_REWRITE_TAC[]); (EXPAND_TAC "S1" THEN REWRITE_TAC[IN_ELIM_THM]); (EXISTS_TAC `i:num` THEN EXISTS_TAC `ul:(real^3)list`); (ASM_REWRITE_TAC[IN_NUMSEG_0]); (EXPAND_TAC "S" THEN REWRITE_TAC[IN; IN_ELIM_THM]); (NEW_GOAL `?v0 v1 v2 v3. ul = [v0; v1; v2; v3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN REPEAT STRIP_TAC); (EXISTS_TAC `v0:real^3` THEN EXISTS_TAC `v1:real^3` THEN EXISTS_TAC `v2:real^3` THEN EXISTS_TAC `v3:real^3`); (ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list ul SUBSET (s:real^3->bool)`); (EXPAND_TAC "s"); (REWRITE_TAC[SUBSET_INTER]); (STRIP_TAC); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzyzmjc.BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `~NULLSET x`); (UNDISCH_TAC `edgeX V x e`); (REWRITE_TAC[edgeX; IN_ELIM_THM; VX]); (COND_CASES_TAC); (REPEAT STRIP_TAC); (UNDISCH_TAC `{ } (u:real^3)`); (REWRITE_TAC[MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (ASM_MESON_TAC[]); (NEW_GOAL `VX V x = V INTER x`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (NEW_GOAL `V INTER (x:real^3->bool) = set_of_list (truncate_simplex (i - 1) ul)`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (REWRITE_TAC[GSYM (ASSUME `ul = [v0;v1;v2;v3:real^3]`); GSYM (ASSUME `x = mcell i V ul`)]); (ASM_REWRITE_TAC[]); (ASM_CASES_TAC `i = 0`); (NEW_GOAL `F`); (UNDISCH_TAC `edgeX V x e`); (REWRITE_TAC[edgeX; IN_ELIM_THM]); (REWRITE_WITH `VX V x = {}`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ_0); (ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (REPEAT STRIP_TAC); (UNDISCH_TAC `{ } (u:real^3)`); (REWRITE_TAC[MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN ARITH_TAC); (NEW_GOAL `u0 IN VX V x`); (UNDISCH_TAC `edgeX V x e`); (REWRITE_TAC[edgeX; IN_ELIM_THM; ASSUME `e = {u0, u1:real^3}`; IN]); (REPEAT STRIP_TAC); (ASM_SET_TAC[]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `ul = [v0; v1; v2; v3:real^3]`)]); (STRIP_TAC); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) ul) SUBSET set_of_list (ul:(real^3)list)`); (MATCH_MP_TAC Rogers.SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET); (ASM_REWRITE_TAC[LENGTH] THEN ASM_ARITH_TAC); (ASM_SET_TAC[]); (REWRITE_TAC[MESON[IN] `(s:real^3->bool) v <=> v IN s`]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[])]);; (* ==================================================================== *) (* Lemma 26 *) let CONTINUOUS_ON_LIFT_DOT2 = prove (`!f:real^M->real^N g s. f continuous_on s /\ g continuous_on s ==> (\x. lift(f x dot g x)) continuous_on s`, REPEAT GEN_TAC THEN SUBGOAL_THEN `bilinear (\x y:real^N. lift(x dot y))` MP_TAC THENL [REWRITE_TAC[bilinear; linear; DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL] THEN REWRITE_TAC[LIFT_ADD; LIFT_CMUL]; REWRITE_TAC[GSYM IMP_CONJ_ALT; GSYM CONJ_ASSOC] THEN DISCH_THEN(MP_TAC o MATCH_MP BILINEAR_CONTINUOUS_ON_COMPOSE) THEN REWRITE_TAC[]]);; let LIFT_MUL_CONTINUOUS_ON = prove (`!f:real^N->real g s. lift o f continuous_on s /\ lift o g continuous_on s ==> lift o (\x. f x * g x) continuous_on s`, REWRITE_TAC[o_DEF; LIFT_CMUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN ASM_REWRITE_TAC[o_DEF]);; let LIFT_DIV_CONTINUOUS_ON = prove (`!f:real^N->real g s. lift o f continuous_on s /\ lift o g continuous_on s /\ (!x. x IN s ==> ~(g x = &0))==> lift o (\x. f x / g x) continuous_on s`, REWRITE_TAC[o_DEF; LIFT_CMUL; real_div] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_MUL THEN ASM_REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN ASM_REWRITE_TAC[]);; let LIFT_DOT_CONTINUOUS_ON = prove (`!a:real^N b s. (lift o ((\x. (x - a) dot b))) continuous_on s`, REPEAT GEN_TAC THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_DOT2 THEN ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);; let LIFT_NORM_CONTINUOUS_ON = prove (`!a:real^N s. (lift o ((\x. norm (x - a)))) continuous_on s`, REPEAT GEN_TAC THEN REWRITE_TAC[o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);; (* ==================================================================== *) (* Lemma 27 *) let aff_ge_alt = new_definition `aff_ge_alt s t (v:real^3) <=> (?f q. FINITE q /\ q SUBSET t /\ v = lin_combo (s UNION q) f /\ (!w. q w ==> &0 <= f w) /\ sum (s UNION q) f = &1)`;; let smallest_angle_set = new_definition `!u0:real^3 u1 s. smallest_angle_set s u0 u1 = (@x. x IN s /\ (!y. y IN s ==> ((y - u0) dot (u1 - u0)) / (norm (y - u0) * norm (u1 - u0)) <= ((x - u0) dot (u1 - u0)) / (norm (x - u0) * norm (u1 - u0))))`;; let smallest_angle_line = new_definition `!a b c d:real^3. smallest_angle_line a b c d = smallest_angle_set (convex hull {a, b}) c d`;; let SMALLEST_ANGLE_LINE_EXISTS = prove_by_refinement ( `!a b u0 u1. ~(u0 = u1) /\ ~(u0 IN convex hull {a, b}) ==> (?x. (x IN convex hull {a, b:real^3} /\ (!y. y IN convex hull {a, b} ==> ((y - u0) dot (u1 - u0)) / (norm (y - u0) * norm (u1 - u0)) <= ((x - u0) dot (u1 - u0)) / (norm (x - u0) * norm (u1 - u0)))))`, [(REPEAT STRIP_TAC); (ABBREV_TAC `f = (\x:real^3. ((x - u0) dot (u1 - u0)) / (norm (x - u0) * norm (u1 - u0)))`); (REWRITE_WITH `!y:real^3. ((y - u0) dot (u1 - u0)) / (norm (y - u0) * norm (u1 - u0)) = f y`); (EXPAND_TAC "f" THEN REWRITE_TAC[]); (MATCH_MP_TAC CONTINUOUS_ATTAINS_SUP); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMP_COMPACT_CONVEX_HULL); (REWRITE_TAC[Geomdetail.FINITE6]); (STRIP_TAC); (REWRITE_TAC[CONVEX_HULL_EQ_EMPTY] THEN SET_TAC[]); (ABBREV_TAC `f1 = (\x. (x - u0) dot (u1 - u0:real^3))`); (ABBREV_TAC `f2 = (\x. norm (x - u0) * norm (u1 - u0:real^3))`); (REWRITE_WITH `f = (\x:real^3. f1 x / f2 x)`); (EXPAND_TAC "f" THEN EXPAND_TAC "f1" THEN EXPAND_TAC "f2"); (REWRITE_TAC[]); (MATCH_MP_TAC LIFT_DIV_CONTINUOUS_ON); (REPEAT STRIP_TAC); (EXPAND_TAC "f1"); (REWRITE_TAC[LIFT_DOT_CONTINUOUS_ON]); (ABBREV_TAC `f3 = (\x:real^3. norm (u1 - u0:real^3))`); (ABBREV_TAC `f4 = (\x:real^3. norm (x - u0:real^3))`); (REWRITE_WITH `f2 = (\x:real^3. f4 x * f3 x)`); (EXPAND_TAC "f2" THEN EXPAND_TAC "f3" THEN EXPAND_TAC "f4"); (REWRITE_TAC[]); (MATCH_MP_TAC LIFT_MUL_CONTINUOUS_ON); (EXPAND_TAC "f3" THEN EXPAND_TAC "f4"); (REWRITE_TAC[LIFT_NORM_CONTINUOUS_ON]); (REWRITE_TAC[o_DEF; CONTINUOUS_ON_CONST]); (UP_ASM_TAC THEN EXPAND_TAC "f2" THEN REWRITE_TAC[REAL_ENTIRE; NORM_EQ_0]); (ASM_REWRITE_TAC[VECTOR_ARITH `a - b = vec 0 <=> b = a`]); (STRIP_TAC); (ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 28 *) let SMALLEST_ANGLE_IN_CONVEX_HULL = prove_by_refinement ( `!m n p q x. ~(p = q) /\ ~(p IN convex hull {m,n}) /\ x = smallest_angle_line m n p q ==> x IN convex hull {m, n}`, [(REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[smallest_angle_line; smallest_angle_set]); (STRIP_TAC); (ABBREV_TAC `X = (\x:real^3. x IN convex hull {m, n} /\ (!y. y IN convex hull {m, n} ==> ((y - p) dot (q - p)) / (norm (y - p) * norm (q - p)) <= ((x - p) dot (q - p)) / (norm (x - p) * norm (q - p))))`); (NEW_GOAL `(X:real^3->bool) x`); (ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SELECT_AX); (EXPAND_TAC "X" THEN ASM_SIMP_TAC[SMALLEST_ANGLE_LINE_EXISTS]); (UP_ASM_TAC THEN EXPAND_TAC "X" THEN MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 29 *) let SMALLEST_ANGLE_LINE_PROPERTY = prove_by_refinement ( `!m n u0 u1 x y. ~(u0 = u1) /\ ~(u0 IN convex hull {m,n}) /\ x = smallest_angle_line m n u0 u1 /\ y IN convex hull {m, n} ==> ((y - u0) dot (u1 - u0)) / (norm (y - u0) * norm (u1 - u0)) <= ((x - u0) dot (u1 - u0)) / (norm (x - u0) * norm (u1 - u0))`, [(REPEAT STRIP_TAC); (UNDISCH_TAC `x = smallest_angle_line m n u0 u1`); (REWRITE_TAC[smallest_angle_line; smallest_angle_set]); (STRIP_TAC); (ABBREV_TAC `X = (\x:real^3. x IN convex hull {m, n} /\ (!y. y IN convex hull {m, n} ==> ((y - u0) dot (u1 - u0)) / (norm (y - u0) * norm (u1 - u0)) <= ((x - u0) dot (u1 - u0)) / (norm (x - u0) * norm (u1 - u0))))`); (NEW_GOAL `(X:real^3->bool) x`); (ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SELECT_AX); (EXPAND_TAC "X" THEN ASM_SIMP_TAC[SMALLEST_ANGLE_LINE_EXISTS]); (UP_ASM_TAC THEN EXPAND_TAC "X" THEN ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 30 *) let MCELL_SUBSET_BALL8_1 = prove_by_refinement ( `!v:real^3 ul i V. i<= 4 /\ packing V /\ saturated V /\ barV V 3 ul /\ v IN mcell i V ul ==> mcell i V ul SUBSET ball (v, &8)`, [(REPEAT STRIP_TAC); (NEW_GOAL `mcell i V ul SUBSET UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1}`); (MATCH_MP_TAC Qzksykg.QZKSYKG2); (ASM_REWRITE_TAC[SET_RULE `i IN {0,1,2,3,4}<=> i =0\/i=1\/i=2\/i=3\/i=4`]); (ASM_ARITH_TAC); (NEW_GOAL `UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1} SUBSET UNIONS {voronoi_closed V s | s IN set_of_list ul}`); (REWRITE_TAC[SUBSET; IN_UNIONS; IN;IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `voronoi_closed V ((HD (left_action_list p ul)):real^3)`); (STRIP_TAC); (EXISTS_TAC `(HD (left_action_list p ul)):real^3`); (REWRITE_TAC[MESON[IN] `(a:real^3->bool) s <=> s IN a`]); (REWRITE_WITH `set_of_list (ul:(real^3)list) = set_of_list (left_action_list p ul)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (ASM_CASES_TAC `i = 4`); (MATCH_MP_TAC Packing3.SET_OF_LIST_LEFT_ACTION_LIST); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[ARITH_RULE `3 + 1 = 4`]); (ASM_CASES_TAC `i = 3`); (MATCH_MP_TAC Marchal_cells_2_new.SET_OF_LIST_LEFT_ACTION_LIST_2); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[ARITH_RULE `(3 + 1) - 2 = 3 - 1 /\ 2 <= 3 + 1`]); (ASM_CASES_TAC `i = 2`); (MATCH_MP_TAC Marchal_cells_2_new.SET_OF_LIST_LEFT_ACTION_LIST_3); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[ARITH_RULE `(3 + 1) - 3 = 2 - 1 /\ 3 <= 3 + 1`]); (NEW_GOAL `i <= 1`); (ASM_ARITH_TAC); (NEW_GOAL `p permutes 0..0`); (ASM_MESON_TAC[ARITH_RULE `i <= 1 ==> i - 1 = 0`]); (UP_ASM_TAC THEN REWRITE_TAC[Packing3.PERMUTES_TRIVIAL] THEN STRIP_TAC); (ASM_REWRITE_TAC[Packing3.LEFT_ACTION_LIST_I]); (MATCH_MP_TAC Packing3.HD_IN_SET_OF_LIST); (REWRITE_TAC[Packing3.LENGTH_LEFT_ACTION_LIST]); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[MESON[IN] `(a:real^3->bool) s <=> s IN a`]); (ASM_REWRITE_TAC[]); (NEW_GOAL `rogers V (left_action_list p ul) SUBSET voronoi_closed V (HD (left_action_list p ul))`); (MATCH_MP_TAC ROGERS_SUBSET_VORONOI_CLOSED); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (REWRITE_TAC[SET_RULE `i IN {0,1,2,3,4} <=> i=0\/i=1\/i=2\/i=3\/i=4`]); (ASM_ARITH_TAC); (UNDISCH_TAC `v IN mcell i V ul` THEN SET_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `UNIONS {voronoi_closed V s | s IN set_of_list ul} SUBSET UNIONS {ball (s:real^3, &2) | s IN set_of_list ul}`); (NEW_GOAL `!s:real^3. voronoi_closed V s SUBSET ball (s, &2)`); (ASM_SIMP_TAC[Packing3.VORONOI_BALL2]); (UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `mcell i V ul SUBSET UNIONS {ball (s,&2) | s IN set_of_list ul}`); (ABBREV_TAC `s1 = UNIONS {rogers V (left_action_list p ul) | p permutes 0..i - 1}`); (ABBREV_TAC `s2 = UNIONS {voronoi_closed V (s:real^3) | s IN set_of_list ul}`); (ABBREV_TAC `s3 = UNIONS {ball (s:real^3,&2) | s IN set_of_list ul}`); (DEL_TAC THEN DEL_TAC THEN DEL_TAC); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[SUBSET; IN_UNIONS; IN; IN_ELIM_THM; IN_BALL]); (REPEAT STRIP_TAC); (NEW_GOAL `?t. (?s. set_of_list ul s /\ t = ball (s:real^3,&2)) /\ t x`); (ASM_SIMP_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `dist (s, x:real^3) < &2`); (REWRITE_TAC[GSYM IN_BALL]); (UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `?t. (?s. set_of_list ul s /\ t = ball (s:real^3,&2)) /\ t v`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `dist (s', v:real^3) < &2`); (REWRITE_TAC[GSYM IN_BALL]); (UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `dist (v, x:real^3) <= dist (s', v) + dist (s', s) + dist (s, x)`); (NORM_ARITH_TAC); (NEW_GOAL `dist (s', s:real^3) < &4`); (REWRITE_TAC[GSYM IN_BALL]); (NEW_GOAL `set_of_list ul SUBSET ball (s':real^3,&4)`); (MATCH_MP_TAC Qzyzmjc.BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:real^3->bool`); (ASM_REWRITE_TAC[IN]); (ASM_SET_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 31 *) let MCELL_SUBSET_BALL8 = prove_by_refinement ( `!v:real^3 ul i V. packing V /\ saturated V /\ barV V 3 ul /\ v IN mcell i V ul ==> mcell i V ul SUBSET ball (v, &8)`, [(REPEAT STRIP_TAC); (ASM_CASES_TAC `i <= 4`); (ASM_SIMP_TAC[MCELL_SUBSET_BALL8_1]); (UNDISCH_TAC `v IN mcell i V ul`); (REWRITE_WITH `mcell i V ul = mcell 4 V ul`); (ASM_SIMP_TAC[MCELL_EXPLICIT; ARITH_RULE `4 >= 4`; ARITH_RULE `~(i <= 4)==>i >= 4`]); (STRIP_TAC); (ASM_SIMP_TAC[MCELL_SUBSET_BALL8_1; ARITH_RULE `4 <= 4`])]);; (* ==================================================================== *) (* Lemma 32 *) let FINITE_MCELL_SET_LEMMA_2 = prove_by_refinement ( `!V r s:real^3. packing V /\ saturated V ==> FINITE {X | X SUBSET ball (s,r) /\ mcell_set V X}`, [(REPEAT STRIP_TAC); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{X | X SUBSET ball (vec 0,norm (s:real^3) + r) /\ mcell_set V X}`); (STRIP_TAC); (ASM_SIMP_TAC[FINITE_MCELL_SET_LEMMA]); (REWRITE_TAC[IN; SUBSET; IN_BALL; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (NEW_GOAL `dist (vec 0, x') <= norm s + dist (s, x':real^3)`); (NORM_ARITH_TAC); (NEW_GOAL `dist (s, x':real^3) < r`); (ASM_SIMP_TAC[]); (UP_ASM_TAC THEN UP_ASM_TAC THEN REAL_ARITH_TAC); (ASM_REWRITE_TAC[])]);; (* ==================================================================== *) (* Lemma 33 *) let CONIC_CAP_WEDGE_EQ_0 =prove_by_refinement ( `!v0 v1 a r w1 w2. a < &1 /\ &0 < r /\ vol (conic_cap v0 v1 r a INTER wedge v0 v1 w1 w2) = &0 ==> coplanar {v0,v1,w1, w2:real^3}`, [(REPEAT STRIP_TAC); (MATCH_MP_TAC (MESON[] `(~A==> F) ==> A`) THEN STRIP_TAC); (NEW_GOAL `~collinear {v0, v1, w1:real^3}`); (MATCH_MP_TAC NOT_COPLANAR_NOT_COLLINEAR); (EXISTS_TAC `w2:real^3` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `~collinear {v0, v1, w2:real^3}`); (MATCH_MP_TAC NOT_COPLANAR_NOT_COLLINEAR); (ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,b,d,c}`] THEN EXISTS_TAC `w1:real^3` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `vol (conic_cap v0 v1 r a INTER wedge v0 v1 w1 (w2:real^3)) = (if &1 < a \/ r < &0 then &0 else azim v0 v1 w1 w2 / &3 * (&1 - max a (-- &1)) * r pow 3)`); (ASM_MESON_TAC[VOLUME_CONIC_CAP_WEDGE]); (UP_ASM_TAC THEN COND_CASES_TAC); (STRIP_TAC THEN ASM_REAL_ARITH_TAC); (ASM_REWRITE_TAC[REAL_ARITH `&0 = a <=> a = &0`; REAL_ENTIRE; REAL_ARITH `a / &3 = &0 <=> a = &0`]); (REWRITE_TAC[REAL_ARITH `&1 - max a (-- &1) = &0 <=> a = &1`]); (REPEAT STRIP_TAC); (UNDISCH_TAC `~coplanar {v0, v1, w1, w2:real^3}`); (ASM_SIMP_TAC[AZIM_EQ_0_PI_IMP_COPLANAR]); (ASM_REAL_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[] THEN MATCH_MP_TAC REAL_POW_NZ); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 34 *) let CONIC_CAP_AFF_GT_EQ_0 =prove_by_refinement ( `!v0 v1 a r w1 w2. a < &1 /\ &0 < r /\ vol (conic_cap v0 v1 r a INTER aff_gt {v0, v1} {w1, w2}) = &0 ==> coplanar {v0,v1,w1, w2:real^3}`, [(REPEAT STRIP_TAC); (MATCH_MP_TAC (MESON[] `(~A==> F) ==> A`) THEN STRIP_TAC); (NEW_GOAL `~collinear {v0, v1, w1:real^3}`); (MATCH_MP_TAC NOT_COPLANAR_NOT_COLLINEAR); (EXISTS_TAC `w2:real^3` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `~collinear {v0, v1, w2:real^3}`); (MATCH_MP_TAC NOT_COPLANAR_NOT_COLLINEAR); (ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,b,d,c}`] THEN EXISTS_TAC `w1:real^3` THEN ASM_REWRITE_TAC[]); (ASM_CASES_TAC `azim v0 v1 w1 (w2:real^3) < pi`); (NEW_GOAL `wedge v0 v1 w1 w2 = aff_gt {v0, v1} {w1, w2:real^3}`); (ASM_MESON_TAC[WEDGE_LUNE]); (ASM_MESON_TAC[ CONIC_CAP_WEDGE_EQ_0]); (ASM_CASES_TAC `azim v0 v1 w1 (w2:real^3) > pi`); (NEW_GOAL `azim v0 v1 w2 (w1:real^3) = (if azim v0 v1 w1 w2 = &0 then &0 else &2 * pi - azim v0 v1 w1 w2)`); (MATCH_MP_TAC AZIM_COMPL); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN COND_CASES_TAC); (UNDISCH_TAC `azim v0 v1 w1 (w2:real^3) > pi` THEN ASM_REWRITE_TAC[]); (MESON_TAC[PI_POS_LE; REAL_ARITH `&0 > a <=> ~(&0 <= a)`]); (STRIP_TAC); (NEW_GOAL `azim v0 v1 w2 (w1:real^3) < pi`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `wedge v0 v1 w2 w1 = aff_gt {v0, v1} {w1, w2:real^3}`); (REWRITE_WITH `{w1, w2:real^3} = {w2, w1}`); (SET_TAC[]); (UNDISCH_TAC `~coplanar {v0, v1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,b,d,c}`]); (ASM_MESON_TAC[WEDGE_LUNE]); (UNDISCH_TAC `~coplanar {v0, v1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,b,d,c}`]); (ASM_MESON_TAC[ CONIC_CAP_WEDGE_EQ_0]); (NEW_GOAL `azim v0 v1 w1 (w2:real^3) = pi`); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[AZIM_EQ_0_PI_IMP_COPLANAR])]);; (* ==================================================================== *) (* Lemma 35 *) let CONIC_CAP_INTER_CONVEX_HULL_4_GT_0 = prove_by_refinement ( `!u0:real^3 u1 w1 w2 r a. &0 < r /\ a < &1 /\ &0 <= a /\ ~coplanar {u0, u1, w1, w2} ==> vol (conic_cap u0 u1 r a INTER convex hull {u0, u1, w1, w2}) > &0`, [(REPEAT STRIP_TAC); (NEW_GOAL `measurable (conic_cap u0 u1 r a INTER convex hull {u0, u1, w1, w2:real^3})`); (MATCH_MP_TAC MEASURABLE_INTER); (REWRITE_TAC[MEASURABLE_CONIC_CAP; MEASURABLE_CONVEX_HULL]); (SIMP_TAC[MEASURABLE_CONVEX_HULL;FINITE_IMP_BOUNDED;Geomdetail.FINITE6]); (REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 <= a /\ ~(a = &0)`]); (ASM_SIMP_TAC[MEASURE_POS_LE]); (REWRITE_WITH `vol (conic_cap u0 u1 r a INTER convex hull {u0, u1, w1, w2}) = &0 <=> NULLSET (conic_cap u0 u1 r a INTER convex hull {u0, u1, w1, w2})`); (ASM_SIMP_TAC[MEASURABLE_MEASURE_EQ_0]); (REPEAT STRIP_TAC); (NEW_GOAL `~(u0 = u1:real^3)`); (STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}` THEN REWRITE_TAC[]); (REWRITE_WITH `{u0, u1, w1, w2} = {u1:real^3, w1, w2}`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[COPLANAR_3]); (NEW_GOAL `?r1 a1. &0 < r1 /\ a1 < &1 /\ conic_cap u0 u1 r1 a1 INTER aff_ge {u0, u1} {w1, w2:real^3} = conic_cap u0 u1 r1 a1 INTER convex hull {u0, u1,w1, w2} /\ conic_cap u0 u1 r1 a1 SUBSET conic_cap u0 u1 r a`); (ABBREV_TAC `s = closest_point (affine hull {u1, w1, w2:real^3}) u0`); (ABBREV_TAC `r2 = dist (u0:real^3, s)`); (ABBREV_TAC `r1 = min r2 r`); (ABBREV_TAC `s2 = smallest_angle_line w1 (w2:real^3) u0 u1`); (ABBREV_TAC `a2 = ((s2 - u0:real^3) dot (u1 - u0)) /(norm (s2 - u0) * norm (u1 - u0))`); (ABBREV_TAC `a1 = max a2 a`); (EXISTS_TAC `r1:real` THEN EXISTS_TAC `a1:real`); (NEW_GOAL `&0 < r1 /\ a1 < &1`); (EXPAND_TAC "r1" THEN EXPAND_TAC "a1"); (REWRITE_TAC[REAL_ARITH `&0 < min a b <=> &0 < a /\ &0 < b`; REAL_ARITH `max a b < &1 <=> a < &1 /\ b < &1`;]); (ASM_REWRITE_TAC[]); (STRIP_TAC); (EXPAND_TAC "r2"); (MATCH_MP_TAC DIST_POS_LT); (STRIP_TAC); (NEW_GOAL `u0 IN affine hull {u1, w1, w2:real^3}`); (ASM_REWRITE_TAC[]); (DEL_TAC THEN EXPAND_TAC "s"); (MATCH_MP_TAC CLOSEST_POINT_IN_SET); (REWRITE_TAC[CLOSED_AFFINE_HULL; AFFINE_HULL_EQ_EMPTY] THEN SET_TAC[]); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (REWRITE_WITH `affine hull {u0, u1, w1, w2} = affine hull {u1:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[COPLANAR_AFFINE_HULL_COPLANAR; COPLANAR_3]); (EXPAND_TAC "a2"); (MATCH_MP_TAC REAL_DIV_LT_1_TACTICS); (NEW_GOAL `s2 IN convex hull {w1, w2:real^3}`); (MATCH_MP_TAC SMALLEST_ANGLE_IN_CONVEX_HULL); (EXISTS_TAC `u0:real^3` THEN EXISTS_TAC `u1:real^3`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (REWRITE_WITH `affine hull {u0, u1, w1, w2} = affine hull {u1:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (NEW_GOAL `convex hull {w1, w2} SUBSET affine hull {u1, w1, w2:real^3}`); (NEW_GOAL `convex hull {w1, w2} SUBSET affine hull {w1, w2:real^3}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (NEW_GOAL `affine hull {w1, w2} SUBSET affine hull {u1, w1, w2:real^3}`); (MATCH_MP_TAC Marchal_cells_2_new.AFFINE_SUBSET_KY_LEMMA); (SET_TAC[]); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (REWRITE_TAC[COPLANAR_3; COPLANAR_AFFINE_HULL_COPLANAR]); (NEW_GOAL `~(s2 = u0:real^3)`); (STRIP_TAC); (UNDISCH_TAC `s2 IN convex hull {w1, w2:real^3}` THEN ASM_REWRITE_TAC[]); (STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (REWRITE_WITH `affine hull {u0, u1, w1, w2} = affine hull {u1:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (NEW_GOAL `convex hull {w1, w2} SUBSET affine hull {u1, w1, w2:real^3}`); (NEW_GOAL `convex hull {w1, w2} SUBSET affine hull {w1, w2:real^3}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (NEW_GOAL `affine hull {w1, w2} SUBSET affine hull {u1, w1, w2:real^3}`); (MATCH_MP_TAC Marchal_cells_2_new.AFFINE_SUBSET_KY_LEMMA); (SET_TAC[]); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (REWRITE_TAC[COPLANAR_3; COPLANAR_AFFINE_HULL_COPLANAR]); (STRIP_TAC); (MATCH_MP_TAC REAL_LT_MUL); (ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_ARITH `a - b = vec 0 <=> a = b`]); (REWRITE_TAC[REAL_ARITH `a < b <=> (a <= b) /\ ~(a = b)`]); (STRIP_TAC); (REWRITE_TAC[NORM_CAUCHY_SCHWARZ]); (REWRITE_TAC[NORM_CAUCHY_SCHWARZ_EQ]); (STRIP_TAC); (ABBREV_TAC `t1 = norm (s2 - u0:real^3)`); (ABBREV_TAC `t2 = norm (u1 - u0:real^3)`); (NEW_GOAL `~(t2 = &0)`); (EXPAND_TAC "t2" THEN REWRITE_TAC[NORM_EQ_0; VECTOR_ARITH `a - b = vec 0 <=> a = b`]); (ASM_REWRITE_TAC[]); (NEW_GOAL `s2 = t1 / t2 % u1 + (t2 - t1) / t2 % u0:real^3`); (REWRITE_TAC[VECTOR_ARITH `t1 / t2 % u1 + (t2 - t1) / t2 % u0:real^3 = &1 / t2 % (t1 % u1 + (t2 - t1) % u0)`]); (REWRITE_WITH `t1 % u1 + (t2 - t1) % u0 = t2 % (s2:real^3)`); (UNDISCH_TAC `t1 % (u1 - u0) = t2 % (s2 - u0:real^3)` THEN VECTOR_ARITH_TAC); (REWRITE_TAC[VECTOR_ARITH `&1 / t2 % t2 % s2 = (t2 / t2) % s2`]); (REWRITE_WITH `t2 / t2 = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (VECTOR_ARITH_TAC); (UNDISCH_TAC `s2 IN convex hull {w1, w2:real^3}`); (ASM_REWRITE_TAC[IN; CONVEX_HULL_2; IN_ELIM_THM]); (STRIP_TAC); (ABBREV_TAC `t = t1 / t2`); (ABBREV_TAC `t' = (t1 - t2) / t2`); (NEW_GOAL `u1 IN affine hull {u0, w1, w2:real^3}`); (REWRITE_TAC[AFFINE_HULL_3; IN; IN_ELIM_THM]); (EXISTS_TAC `t'/ t` THEN EXISTS_TAC `u / t` THEN EXISTS_TAC `v / t`); (STRIP_TAC); (ASM_REWRITE_TAC[REAL_ARITH `a / t + b/ t + c / t = (a + b + c) / t `]); (REWRITE_WITH `t' + &1 = t`); (EXPAND_TAC "t'" THEN EXPAND_TAC "t"); (NEW_GOAL `t2 / t2 = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (UP_ASM_TAC THEN REAL_ARITH_TAC); (MATCH_MP_TAC REAL_DIV_REFL); (EXPAND_TAC "t"); (ONCE_REWRITE_TAC[REAL_ARITH `a / b = a * (&1 / b)`]); (REWRITE_TAC[REAL_ENTIRE]); (REWRITE_WITH `~(t1 = &0)`); (EXPAND_TAC "t1" THEN REWRITE_TAC[NORM_EQ_0; VECTOR_ARITH `a - b = vec 0 <=> a = b`] THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC Trigonometry2.NOT_EQ0_IMP_NEITHER_INVERT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[VECTOR_ARITH `t' / t % u0 + u / t % w1 + v / t % w2 = &1 / t % (t' % u0 + u % w1 + v % w2)`]); (REWRITE_WITH `t' % u0 + u % w1 + v % w2 = t % u1:real^3`); (EXPAND_TAC "t'"); (UNDISCH_TAC `t % u1 + (t2 - t1) / t2 % u0 = u % w1 + v % w2:real^3` THEN VECTOR_ARITH_TAC); (REWRITE_TAC[VECTOR_ARITH `&1 / t % t % u = t / t % u`]); (REWRITE_WITH `t / t = &1`); (MATCH_MP_TAC REAL_DIV_REFL); (EXPAND_TAC "t"); (ONCE_REWRITE_TAC[REAL_ARITH `a / b = a * (&1 / b)`]); (REWRITE_TAC[REAL_ENTIRE]); (REWRITE_WITH `~(t1 = &0)`); (EXPAND_TAC "t1" THEN REWRITE_TAC[NORM_EQ_0; VECTOR_ARITH `a - b = vec 0 <=> a = b`] THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC Trigonometry2.NOT_EQ0_IMP_NEITHER_INVERT); (ASM_REWRITE_TAC[]); (VECTOR_ARITH_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {b,a,c,d}`]); (REWRITE_WITH `affine hull {u1, u0, w1, w2} = affine hull {u0:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[COPLANAR_3; COPLANAR_AFFINE_HULL_COPLANAR]); (NEW_GOAL `conic_cap (u0:real^3) u1 r1 a1 SUBSET conic_cap u0 u1 r a`); (REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN MATCH_MP_TAC (SET_RULE `a SUBSET b /\ c SUBSET d ==> a INTER c SUBSET b INTER d`)); (STRIP_TAC); (MATCH_MP_TAC SUBSET_BALL); (EXPAND_TAC "r1" THEN REAL_ARITH_TAC); (MATCH_MP_TAC RCONE_GT_SUBSET); (EXPAND_TAC "a1" THEN REAL_ARITH_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[SET_RULE `b = a <=> a SUBSET b /\ b SUBSET a`]); (STRIP_TAC); (MATCH_MP_TAC (SET_RULE `a SUBSET b ==> x INTER a SUBSET x INTER b`)); (REWRITE_TAC[Marchal_cells_2_new.CONVEX_HULL_4_SUBSET_AFF_GE_2_2]); (REWRITE_TAC[SUBSET; IN_INTER]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (UP_ASM_TAC); (NEW_GOAL `DISJOINT {u0, u1:real^3} {w1, w2}`); (REWRITE_TAC[DISJOINT]); (ASM_CASES_TAC `w1 IN {u0, u1:real^3}`); (NEW_GOAL `F`); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_WITH `{u0, u1, w1, w2:real^3} = {u0, u1, w2:real^3}`); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[COPLANAR_3]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_CASES_TAC `w2 IN {u0, u1:real^3}`); (NEW_GOAL `F`); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_WITH `{u0, u1, w1, w2:real^3} = {u0, u1, w1:real^3}`); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[COPLANAR_3]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_SET_TAC[]); (ASM_SIMP_TAC[Marchal_cells_2.AFF_GE_2_2; CONVEX_HULL_4; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (EXISTS_TAC `t1:real` THEN EXISTS_TAC `t2:real` THEN EXISTS_TAC `t3:real` THEN EXISTS_TAC `t4:real`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `&0 <= t <=> ~(t < &0)`] THEN STRIP_TAC); (NEW_GOAL `!x. x IN affine hull {u1, w1, w2} ==> dist (u0,s:real^3) <= dist (u0,x:real^3)`); (REPEAT STRIP_TAC THEN EXPAND_TAC "s"); (MATCH_MP_TAC CLOSEST_POINT_LE); (ASM_REWRITE_TAC[CLOSED_AFFINE_HULL]); (NEW_GOAL `&1 < t2 + t3+ t4`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `~(t2 + t3+ t4 = &0)`); (ASM_REAL_ARITH_TAC); (ABBREV_TAC `y = t2 / (t2 + t3 + t4) % u1 + t3 / (t2 + t3 + t4) % w1 + t4 / (t2 + t3 + t4) % w2:real^3`); (NEW_GOAL `x = t1 % u0 + (t2 + t3 + t4) % (y:real^3)`); (ASM_REWRITE_TAC[] THEN EXPAND_TAC "y"); (REWRITE_TAC[VECTOR_ARITH `t1 % u0 + t2 % u1 + t3 % w1 + t4 % w2 = t1 % u0 + t % (t2 / t % u1 + t3 / t % w1 + t4 / t % w2) <=> t2 % u1 + t3 % w1 + t4 % w2 = t / t % (t2 % u1 + t3 % w1 + t4 % w2)`]); (REWRITE_WITH `(t2 + t3 + t4) / (t2 + t3 + t4) = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (VECTOR_ARITH_TAC); (NEW_GOAL `y IN affine hull {u1, w1, w2:real^3}`); (REWRITE_TAC[AFFINE_HULL_3; IN; IN_ELIM_THM]); (EXISTS_TAC `t2 / (t2 + t3 + t4)` THEN EXISTS_TAC `t3 / (t2 + t3 + t4)` THEN EXISTS_TAC `t4 / (t2 + t3 + t4)`); (ASM_REWRITE_TAC[REAL_ARITH `a/x + b/ x + c/ x = (a+b+c) / x`]); (ASM_SIMP_TAC[REAL_DIV_REFL]); (NEW_GOAL `dist (u0, s:real^3) <= dist (u0, y:real^3)`); (ASM_SIMP_TAC[]); (NEW_GOAL `dist (u0, s) <= dist (u0, x:real^3)`); (REWRITE_TAC[ASSUME `x = t1 % u0 + (t2 + t3 + t4) % y:real^3`; dist]); (REWRITE_WITH `u0 - (t1 % u0 + (t2 + t3 + t4) % y) = (t1 + t2 + t3 + t4) % u0 - (t1 % u0 + (t2 + t3 + t4) % y:real^3)`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_TAC[VECTOR_ARITH `(t1 + t) % u0 - (t1 % u0 + t % y) = t % (u0 - y)`; NORM_MUL]); (REWRITE_WITH `abs (t2 + t3 + t4) = t2 + t3 + t4`); (REWRITE_TAC[REAL_ABS_REFL] THEN ASM_REAL_ARITH_TAC); (REWRITE_TAC[GSYM dist]); (NEW_GOAL `dist (u0, y:real^3) <= (t2 + t3 + t4) * dist (u0, y)`); (REWRITE_TAC[REAL_ARITH `a <= b * a <=> &0 <= (b - &1) * a`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (NEW_GOAL `x IN ball (u0:real^3, r1)`); (UNDISCH_TAC `x IN conic_cap u0 (u1:real^3) r1 a1`); (REWRITE_TAC[conic_cap; NORMBALL_BALL] THEN SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN_BALL]); (EXPAND_TAC "r1" THEN EXPAND_TAC "r2"); (UP_ASM_TAC THEN REAL_ARITH_TAC); (* ========================================================================= *) (* OK here *) (REWRITE_TAC[REAL_ARITH `&0 <= t <=> ~(t < &0)`] THEN STRIP_TAC); (NEW_GOAL `!y. y IN convex hull {w1, w2:real^3} ==> ((y - u0) dot (u1 - u0)) / (norm (y - u0) * norm (u1 - u0)) <= ((s2 - u0) dot (u1 - u0)) / (norm (s2 - u0) * norm (u1 - u0))`); (REPEAT STRIP_TAC); (MATCH_MP_TAC SMALLEST_ANGLE_LINE_PROPERTY); (EXISTS_TAC `w1:real^3` THEN EXISTS_TAC `w2:real^3`); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (REWRITE_WITH `affine hull {u0, u1, w1, w2} = affine hull {u1:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (NEW_GOAL `convex hull {w1, w2:real^3} SUBSET affine hull {u1,w1, w2}`); (NEW_GOAL `convex hull {w1, w2:real^3} SUBSET affine hull {w1, w2}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (NEW_GOAL `affine hull {w1, w2:real^3} SUBSET affine hull {u1,w1, w2}`); (MATCH_MP_TAC Marchal_cells_2_new.AFFINE_SUBSET_KY_LEMMA THEN SET_TAC[]); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (REWRITE_TAC[COPLANAR_AFFINE_HULL_COPLANAR; COPLANAR_3]); (NEW_GOAL `~(t3+ t4 = &0)`); (STRIP_TAC); (NEW_GOAL `t3 = &0 /\ t4 = &0`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `x = t1 % u0 + t2 % u1:real^3`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (NEW_GOAL `x:real^3 IN rcone_gt u0 u1 a1`); (UNDISCH_TAC `x:real^3 IN conic_cap u0 u1 r1 a1`); (REWRITE_TAC[conic_cap] THEN SET_TAC[conic_cap]); (UP_ASM_TAC THEN REWRITE_TAC[rcone_gt; rconesgn; IN; IN_ELIM_THM] THEN STRIP_TAC); (NEW_GOAL `(x - u0) dot (u1 - u0:real^3) <= &0`); (REWRITE_TAC[ASSUME `x = t1 % u0 + t2 % u1:real^3`]); (REWRITE_WITH `(t1 % u0 + t2 % u1) - u0 = (t1 % u0 + t2 % u1) - (t1 + t2 + t3 + t4) % u0:real^3`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_TAC[ASSUME `t3 = &0 /\ t4 = &0`; VECTOR_ARITH `(t1 % u0 + t2 % u1) - (t1 + t2 + &0 + &0) % u0 = t2 % (u1 - u0)`; DOT_LMUL; REAL_ARITH `a * b <= &0 <=> &0 <= (--a) * b`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DOT_POS_LE] THEN ASM_REAL_ARITH_TAC); (NEW_GOAL `&0 <= dist (x,u0) * dist (u1,u0:real^3) * a1`); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIST_POS_LE]); (ASM_REAL_ARITH_TAC); (ASM_REAL_ARITH_TAC); (ABBREV_TAC `y = t3 / (t3 + t4) % w1 + t4 / (t3 + t4) % w2:real^3`); (NEW_GOAL `x = t1 % u0 + t2 % u1 + (t3 + t4) % (y:real^3)`); (ASM_REWRITE_TAC[] THEN EXPAND_TAC "y"); (REWRITE_TAC[VECTOR_ARITH `t1 % u0 + t2 % u1 + t3 % w1 + t4 % w2 = t1 % u0 + t2 % u1 + t % (t3 / t % w1 + t4 / t % w2) <=> t3 % w1 + t4 % w2 = t / t % (t3 % w1 + t4 % w2)`]); (REWRITE_WITH `(t3 + t4) / (t3 + t4) = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (VECTOR_ARITH_TAC); (NEW_GOAL `y IN convex hull {w1, w2:real^3}`); (REWRITE_TAC[CONVEX_HULL_2; IN; IN_ELIM_THM]); (EXISTS_TAC `t3 / (t3 + t4)` THEN EXISTS_TAC `t4 / (t3 + t4)`); (ASM_REWRITE_TAC[REAL_ARITH `a / x + b / x = (a + b) / x`]); (REPEAT STRIP_TAC); (MATCH_MP_TAC REAL_LE_DIV); (ASM_REAL_ARITH_TAC); (MATCH_MP_TAC REAL_LE_DIV); (ASM_REAL_ARITH_TAC); (ASM_SIMP_TAC[REAL_DIV_REFL]); (ABBREV_TAC `f = (\y:real^3. ((y - u0) dot (u1 - u0)) / (norm (y - u0:real^3) * norm (u1 - u0)))`); (NEW_GOAL `a1 < (f:real^3->real) x`); (NEW_GOAL `x:real^3 IN rcone_gt u0 u1 a1`); (UNDISCH_TAC `x:real^3 IN conic_cap u0 u1 r1 a1` THEN REWRITE_TAC[conic_cap] THEN SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[rcone_gt; rconesgn; IN; IN_ELIM_THM; dist]); (STRIP_TAC THEN EXPAND_TAC "f"); (REWRITE_WITH `a1 < ((x - u0) dot (u1 - u0)) / (norm (x - u0) * norm (u1 - u0)) <=> a1 * (norm (x - u0) * norm (u1 - u0)) < ((x - u0:real^3) dot (u1 - u0))`); (MATCH_MP_TAC REAL_LT_RDIV_EQ); (MATCH_MP_TAC REAL_LT_MUL); (REWRITE_TAC[NORM_POS_LE; REAL_ARITH `a < b <=> (a <= b) /\ ~(b = a)`]); (SIMP_TAC[NORM_EQ_0; VECTOR_ARITH `a - b = vec 0 <=> a = b`]); (ASM_REWRITE_TAC[]); (STRIP_TAC); (NEW_GOAL `u1 IN affine hull {u0, w1, w2:real^3}`); (REWRITE_TAC[IN; IN_ELIM_THM; AFFINE_HULL_3]); (EXISTS_TAC `(t2 + t3 + t4) / t2` THEN EXISTS_TAC `(--t3) / t2` THEN EXISTS_TAC `(--t4) / t2`); (REPEAT STRIP_TAC); (REWRITE_TAC[REAL_ARITH `(t2 + t3 + t4) / t2 + --t3 / t2 + --t4 / t2 = t2 / t2`]); (MATCH_MP_TAC REAL_DIV_REFL); (UNDISCH_TAC `t2 < &0` THEN REAL_ARITH_TAC); (REWRITE_WITH `(t2 + t3 + t4) = &1 - t1`); (UNDISCH_TAC `t1 + t2 + t3 + t4 = &1` THEN REAL_ARITH_TAC); (NEW_GOAL `u0 - t1 % u0 - t3 % w1 - t4 % w2:real^3 = t2 % u1`); (UP_ASM_TAC THEN VECTOR_ARITH_TAC); (ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - t1) / t2 % u0 + --t3 / t2 % u2 + --t4 / t2 % u3 = (&1 / t2) % (u0 - t1 % u0 - t3 % u2 - t4 % u3)`]); (REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_ARITH `&1 / a * a = a / a`]); (REWRITE_WITH `t2 / t2 = &1`); (MATCH_MP_TAC REAL_DIV_REFL); (UNDISCH_TAC `t2 < &0` THEN REAL_ARITH_TAC); (VECTOR_ARITH_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {b,a,c,d}`]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (REWRITE_WITH `affine hull {u1, u0, w1, w2} = affine hull {u0:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[COPLANAR_3; COPLANAR_AFFINE_HULL_COPLANAR]); (UP_ASM_TAC THEN REAL_ARITH_TAC); (NEW_GOAL `(f:real^3->real) y <= f s2`); (EXPAND_TAC "f"); (ASM_SIMP_TAC[]); (NEW_GOAL `&1 < t1 + t3+ t4`); (ASM_REAL_ARITH_TAC); (NEW_GOAL `~(t1 + t3+ t4 = &0)`); (ASM_REAL_ARITH_TAC); (ABBREV_TAC `w = t1 / (t1 + t3 + t4) % u0 + t3 / (t1 + t3 + t4) % w1 + t4 / (t1 + t3 + t4) % w2:real^3`); (NEW_GOAL `(f:real^3->real) y = f w`); (EXPAND_TAC "f"); (REWRITE_WITH `y:real^3 - u0 = &1 / (t3 + t4) % (t3 % w1 + t4 % w2 - (t3 + t4) % u0)`); (EXPAND_TAC "y"); (REWRITE_TAC[VECTOR_ARITH `(t3 / (t3 + t4) % u2 + t4 / (t3 + t4) % u3) - u0 = &1 / (t3 + t4) % (t3 % u2 + t4 % u3 - (t3 + t4) % u0) <=> (t3 + t4) / (t3 + t4) % u0 = u0`]); (REWRITE_WITH `(t3 + t4) / (t3 + t4) = &1`); (MATCH_MP_TAC REAL_DIV_REFL); (ASM_REWRITE_TAC[]); (VECTOR_ARITH_TAC); (REWRITE_TAC[NORM_MUL; DOT_LMUL]); (REWRITE_WITH `w:real^3 - u0 = &1 / (t1 + t3 + t4) % (t3 % w1 + t4 % w2 - (t3 + t4) % u0)`); (EXPAND_TAC "w"); (REWRITE_TAC[VECTOR_ARITH `(t1 / (t1 + t3 + t4) % u0 + t3 / (t1 + t3 + t4) % u2 + t4 / (t1 + t3 + t4) % u3) - u0 = &1 / (t1 + t3 + t4) % (t3 % u2 + t4 % u3 - (t3 + t4) % u0) <=> (t1 + t3 + t4) / (t1 + t3 + t4) % u0 = u0`]); (REWRITE_WITH `(t1 + t3 + t4) / (t1 + t3 + t4) = &1`); (MATCH_MP_TAC REAL_DIV_REFL); (ASM_REWRITE_TAC[]); (VECTOR_ARITH_TAC); (REWRITE_TAC[NORM_MUL; DOT_LMUL]); (REWRITE_WITH `abs (&1 / (t3 + t4)) = &1 / (t3 + t4)`); (REWRITE_TAC[REAL_ABS_REFL]); (ASM_SIMP_TAC[REAL_LE_DIV;REAL_LE_ADD; REAL_ARITH `&0 <= &1`]); (REWRITE_WITH `abs (&1 / (t1 + t3 + t4)) = &1 / (t1 + t3 + t4)`); (REWRITE_TAC[REAL_ABS_REFL]); (MATCH_MP_TAC REAL_LE_DIV THEN REWRITE_TAC[REAL_ARITH `&0 <= &1`]); (ASM_REAL_ARITH_TAC); (REWRITE_TAC[REAL_ARITH `(a * x) / ((a * y) * z) = (a * x) / (a * (y * z))`]); (ABBREV_TAC `b1 = norm (t3 % w1 + t4 % w2 - (t3 + t4) % u0) * norm (u1 - u0:real^3)`); (NEW_GOAL `~(b1 = &0)`); (EXPAND_TAC "b1" THEN ASM_REWRITE_TAC[REAL_ENTIRE; NORM_EQ_0; VECTOR_ARITH `(a - b = vec 0 <=> a = b)/\(a + b-c = vec 0 <=> a + b = c)`]); (STRIP_TAC); (NEW_GOAL `u0 IN convex hull {w1, w2:real^3}`); (REWRITE_TAC[CONVEX_HULL_2; IN; IN_ELIM_THM]); (EXISTS_TAC `t3 / (t3 + t4)` THEN EXISTS_TAC `t4 / (t3 + t4)`); (REPEAT STRIP_TAC); (MATCH_MP_TAC REAL_LE_DIV THEN ASM_REAL_ARITH_TAC); (MATCH_MP_TAC REAL_LE_DIV THEN ASM_REAL_ARITH_TAC); (ASM_SIMP_TAC[REAL_DIV_REFL; REAL_ARITH `a / x + b / x = (a + b) / x`]); (SIMP_TAC[VECTOR_ARITH `a/ x % u + b/ x % v = &1/ x % (a % u + b % v)`]); (REWRITE_TAC[ASSUME `t3 % w1 + t4 % w2 = (t3 + t4) % u0:real^3`; VECTOR_ARITH `&1 / a % (a % b) = (a / a) % b`]); (REWRITE_WITH `(t3 + t4) / (t3 + t4) = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (VECTOR_ARITH_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[]); (ONCE_REWRITE_TAC[GSYM COPLANAR_AFFINE_HULL_COPLANAR]); (REWRITE_WITH `affine hull {u0, u1, w1, w2} = affine hull {u1:real^3, w1, w2}`); (MATCH_MP_TAC AFFINE_HULL_3_INSERT); (NEW_GOAL `convex hull {w1, w2:real^3} SUBSET affine hull {u1,w1, w2}`); (NEW_GOAL `convex hull {w1, w2:real^3} SUBSET affine hull {w1, w2}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (NEW_GOAL `affine hull {w1, w2:real^3} SUBSET affine hull {u1,w1, w2}`); (MATCH_MP_TAC Marchal_cells_2_new.AFFINE_SUBSET_KY_LEMMA THEN SET_TAC[]); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (REWRITE_TAC[COPLANAR_AFFINE_HULL_COPLANAR; COPLANAR_3]); (NEW_GOAL `~(&1 / (t3 + t4) = &0)`); (MATCH_MP_TAC Trigonometry2.NOT_EQ0_IMP_NEITHER_INVERT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `(&1 / (t3 + t4) * ((t3 % w1 + t4 % w2 - (t3 + t4) % u0) dot (u1 - u0))) / (&1 / (t3 + t4) * b1) = ((t3 % w1 + t4 % w2 - (t3 + t4) % u0) dot (u1 - u0:real^3)) / b1`); (UP_ASM_TAC THEN UP_ASM_TAC THEN MESON_TAC[Trigonometry1.REAL_DIV_MUL2]); (NEW_GOAL `~(&1 / (t1 + t3 + t4) = &0)`); (MATCH_MP_TAC Trigonometry2.NOT_EQ0_IMP_NEITHER_INVERT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `(&1 / (t1 + t3 + t4) * ((t3 % w1 + t4 % w2 - (t3 + t4) % u0) dot (u1 - u0))) / (&1 / (t1 + t3 + t4) * b1) = ((t3 % w1 + t4 % w2 - (t3 + t4) % u0) dot (u1 - u0:real^3)) / b1`); (UP_ASM_TAC THEN UNDISCH_TAC `~(b1 = &0)` THEN MESON_TAC[Trigonometry1.REAL_DIV_MUL2]); (NEW_GOAL `(f:real^3->real) x <= f w`); (EXPAND_TAC "f"); (REWRITE_WITH `((x - u0) dot (u1 - u0:real^3)) / (norm (x - u0) * norm (u1 - u0)) <= ((w - u0) dot (u1 - u0)) / (norm (w - u0) * norm (u1 - u0)) <=> ((x - u0) dot (u1 - u0)) * (norm (w - u0) * norm (u1 - u0)) <= ((w - u0) dot (u1 - u0)) * (norm (x - u0) * norm (u1 - u0))`); (MATCH_MP_TAC RAT_LEMMA4); (STRIP_TAC); (MATCH_MP_TAC REAL_LT_MUL); (ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_ARITH `x - b = vec 0 <=> x = b`]); (REPEAT STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}` THEN REWRITE_TAC[]); (REWRITE_TAC[coplanar]); (EXISTS_TAC `u0:real^3` THEN EXISTS_TAC `w1:real^3` THEN EXISTS_TAC `w2:real^3`); (ONCE_REWRITE_TAC[SET_RULE `{u0, u1, u2, u3} = {u1, u0, u2, u3}`]); (MATCH_MP_TAC (SET_RULE `u0 IN S /\ b SUBSET S ==> (u0 INSERT b) SUBSET S`)); (REWRITE_TAC[SET_SUBSET_AFFINE_HULL]); (REWRITE_TAC[AFFINE_HULL_3; IN; IN_ELIM_THM]); (EXISTS_TAC `(t2 + t3 + t4) / t2` THEN EXISTS_TAC `(--t3) / t2` THEN EXISTS_TAC `(--t4) / t2`); (REPEAT STRIP_TAC); (REWRITE_TAC[REAL_ARITH `(t2 + t3 + t4) / t2 + --t3 / t2 + --t4 / t2 = t2 / t2`]); (MATCH_MP_TAC REAL_DIV_REFL); (UNDISCH_TAC `t2 < &0` THEN REAL_ARITH_TAC); (REWRITE_WITH `(t2 + t3 + t4) = &1 - t1`); (UNDISCH_TAC `t1 + t2 + t3 + t4 = &1` THEN REAL_ARITH_TAC); (NEW_GOAL `u0 - t1 % u0 - t3 % w1 - t4 % w2:real^3 = t2 % u1`); (UP_ASM_TAC THEN VECTOR_ARITH_TAC); (ASM_REWRITE_TAC[VECTOR_ARITH `(&1 - t1) / t2 % u0 + --t3 / t2 % u2 + --t4 / t2 % u3 = (&1 / t2) % (u0 - t1 % u0 - t3 % u2 - t4 % u3)`]); (REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_ARITH `&1 / a * a = a / a`]); (REWRITE_WITH `t2 / t2 = &1`); (MATCH_MP_TAC REAL_DIV_REFL); (UNDISCH_TAC `t2 < &0` THEN REAL_ARITH_TAC); (VECTOR_ARITH_TAC); (MATCH_MP_TAC REAL_LT_MUL); (REWRITE_TAC[REAL_ARITH `&0 < a <=> (&0 <= a) /\ ~(a = &0)`]); (REWRITE_TAC[NORM_POS_LE; NORM_EQ_0; VECTOR_ARITH `x - y= vec 0 <=>x = y`]); (STRIP_TAC); (EXPAND_TAC "w" THEN STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_TAC[coplanar]); (EXISTS_TAC `u1:real^3` THEN EXISTS_TAC `w1:real^3` THEN EXISTS_TAC `w2:real^3`); (MATCH_MP_TAC (SET_RULE `u0 IN S /\ b SUBSET S ==> (u0 INSERT b) SUBSET S`)); (REWRITE_TAC[SET_SUBSET_AFFINE_HULL]); (REWRITE_TAC[AFFINE_HULL_3; IN; IN_ELIM_THM]); (EXISTS_TAC `&0` THEN EXISTS_TAC `t3 / (t3 + t4)` THEN EXISTS_TAC `t4 / (t3 + t4)`); (REPEAT STRIP_TAC); (REWRITE_TAC[REAL_ARITH `&0 + t3 / (t3 + t4) + t4 / (t3 + t4) = (t3 + t4) / (t3 + t4)`]); (MATCH_MP_TAC REAL_DIV_REFL); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[VECTOR_ARITH `&0 % u1 + t3 / (t3 + t4) % u2 + t4 / (t3 + t4) % u3 = (&1 / (t3 + t4)) % (t3 % u2 + t4 % u3)`]); (UP_ASM_TAC THEN REWRITE_TAC[VECTOR_ARITH `t1 / x % u0 + t3 / x % u2 + t4 / x % u3 = (&1 / x) % (t1 % u0 + t3 % u2 + t4 % u3)`]); (REWRITE_WITH `&1 / (t1 + t3 + t4) % (t1 % u0 + t3 % w1 + t4 % w2) = u0 <=> t1 % u0 + t3 % w1 + t4 % w2 = (t1 + t3 + t4) % u0:real^3`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Collect_geom.CHANGE_SIDE); (ASM_REWRITE_TAC[]); (REWRITE_TAC[VECTOR_ARITH `t1 % u0 + t3 % u2 + t4 % u3 = (t1 + t3 + t4) % u0 <=> t3 % u2 + t4 % u3 = (t3 + t4) % u0`]); (STRIP_TAC THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_ARITH `&1 / a * a = a / a`]); (REWRITE_WITH `(t3 + t4) / (t3 + t4) = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (VECTOR_ARITH_TAC); (ASM_REWRITE_TAC[]); (REWRITE_WITH `x = t2 % u1 + (t1 + t3 + t4) % w:real^3`); (ASM_REWRITE_TAC[] THEN EXPAND_TAC "w"); (REWRITE_TAC[VECTOR_ARITH `x % (t1 /x % u0 + t3 / x % u2 + t4 /x % u3) = (x / x) % (t1 % u0 + t3 % u2 + t4 % u3)`]); (REWRITE_WITH `(t1 + t3 + t4) / (t1 + t3 + t4) = &1`); (ASM_SIMP_TAC[REAL_DIV_REFL]); (VECTOR_ARITH_TAC); (ABBREV_TAC `t = t1 + t3 + t4`); (REWRITE_WITH `(t2 % u1 + t % w) - u0:real^3 = (t2 % u1 + t % w) - (t1 + t2 + t3 + t4) % u0`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_WITH `t1 + t2 + t3 + t4 = t2 + t:real`); (EXPAND_TAC "t" THEN REAL_ARITH_TAC); (REWRITE_TAC[VECTOR_ARITH `(t2 % u1 + t % w) - (t2 + t) % u0 = t2 % (u1 - u0) + t % (w - u0)`]); (ABBREV_TAC `x1 = u1 - u0:real^3`); (ABBREV_TAC `x2 = w - u0:real^3`); (REWRITE_WITH `(t2 % x1 + t % x2) dot x1 = t2 * norm x1 pow 2 + t * x2 dot (x1:real^3)`); (REWRITE_TAC[NORM_POW_2]); (VECTOR_ARITH_TAC); (NEW_GOAL `t2 * norm x1 pow 2 * norm x2 * norm x1 <= t2 * (x2 dot x1) * norm x1 * norm (x1:real^3)`); (REWRITE_TAC[REAL_POW_2; REAL_ARITH `t2 * (x1 * x1) * x2 * x1 <= t2 * x3 * x1 * x1 <=> &0 <= (x1 pow 2) * (--t2) * (x2 * x1 - x3)`]); (MATCH_MP_TAC REAL_LE_MUL); (ASM_SIMP_TAC[REAL_LE_MUL; NORM_POS_LE]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[REAL_ARITH `&0 <= a - b <=> b <= a`]); (STRIP_TAC); (UNDISCH_TAC `t2 < &0` THEN REAL_ARITH_TAC); (REWRITE_TAC[NORM_CAUCHY_SCHWARZ]); (NEW_GOAL `t2 * (x2 dot x1) * norm x1 * norm x1 + t * (x2 dot x1) * norm x2 * norm x1 <= (x2 dot x1) * norm (t2 % x1 + t % x2) * norm (x1:real^3)`); (REWRITE_TAC[REAL_ARITH `t2 * x3 * x1 * x1 + t * x3 * x2 * x1 <= x3 * x4 * x1 <=> &0 <= (x1 * x3) * (x4 - t2 * x1 - t * x2)`]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[NORM_POS_LE]); (ASM_CASES_TAC `x2 dot (x1:real^3) < &0`); (NEW_GOAL `F`); (NEW_GOAL `(f:real^3->real) x <= &0`); (EXPAND_TAC "f"); (REWRITE_TAC[REAL_ARITH `a / b <= &0 <=> &0 <= (--a) / b`]); (MATCH_MP_TAC REAL_LE_DIV); (SIMP_TAC[NORM_POS_LE; REAL_LE_MUL]); (REWRITE_WITH `x = t2 % u1 + (t1 + t3 + t4) % w:real^3`); (ASM_REWRITE_TAC[] THEN EXPAND_TAC "w"); (REWRITE_TAC[VECTOR_ARITH `x % (t1 /x % u0 + t3 / x % u2 + t4 /x % u3) = (x / x) % (t1 % u0 + t3 % u2 + t4 % u3)`]); (EXPAND_TAC "t"); (REWRITE_WITH `(t1 + t3 + t4) / (t1 + t3 + t4) = &1`); (MATCH_MP_TAC REAL_DIV_REFL); (ASM_REWRITE_TAC[]); (VECTOR_ARITH_TAC); (REWRITE_TAC[REAL_ARITH `&0 <= --a <=> a <= &0`]); (REWRITE_WITH `(t2 % u1 + (t1 + t3 + t4) % w) - u0:real^3 = (t2 % u1 + (t1 + t3 + t4) % w) - (t1 + t2 + t3 + t4) % u0`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_WITH `(t2 % u1 + (t1 + t3 + t4) % w) - (t1 + t2 + t3 + t4) % u0 = t2 % x1 + t % x2:real^3`); (EXPAND_TAC "x1" THEN EXPAND_TAC "x2" THEN EXPAND_TAC "t" THEN VECTOR_ARITH_TAC); (NEW_GOAL `t % x2 dot (x1:real^3) <= &0`); (REWRITE_TAC[DOT_LMUL; REAL_ARITH `a * b <= &0 <=> &0 <= a * (--b)`]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (EXPAND_TAC "t" THEN UNDISCH_TAC `t2 < &0` THEN UNDISCH_TAC `t1 + t2 + t3 + t4 = &1` THEN REAL_ARITH_TAC); (UP_ASM_TAC THEN REAL_ARITH_TAC); (NEW_GOAL `t2 % x1 dot (x1:real^3) <= &0`); (REWRITE_TAC[DOT_LMUL; REAL_ARITH `a * b <= &0 <=> &0 <= (--a) * b`]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (EXPAND_TAC "t" THEN UNDISCH_TAC `t2 < &0` THEN UNDISCH_TAC `t1 + t2 + t3 + t4 = &1` THEN REAL_ARITH_TAC); (REWRITE_TAC[DOT_POS_LE]); (REWRITE_TAC[DOT_LADD]); (UP_ASM_TAC THEN UP_ASM_TAC THEN REAL_ARITH_TAC); (UP_ASM_TAC THEN UNDISCH_TAC `a1 < (f:real^3->real) x`); (EXPAND_TAC "a1" THEN UNDISCH_TAC `&0 <= a` THEN REAL_ARITH_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (UP_ASM_TAC THEN REAL_ARITH_TAC); (REWRITE_TAC [REAL_ARITH `&0 <= a - b * d - c <=> c <= a + (--b) * d`]); (REWRITE_WITH `t * norm (x2:real^3) = abs t * norm x2`); (AP_THM_TAC THEN AP_TERM_TAC); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[REAL_ABS_REFL]); (EXPAND_TAC "t" THEN UNDISCH_TAC `t2 < &0` THEN UNDISCH_TAC `t1 + t2 + t3 + t4 = &1` THEN REAL_ARITH_TAC); (REWRITE_WITH `(--t2) * norm (x1:real^3) = abs (--t2) * norm x1`); (AP_THM_TAC THEN AP_TERM_TAC); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[REAL_ABS_REFL]); (UNDISCH_TAC `t2 < &0` THEN REAL_ARITH_TAC); (REWRITE_TAC[GSYM NORM_MUL]); (REWRITE_WITH `norm (t % x2:real^3) = norm ((t2 % x1 + t % x2) + (--t2 % x1))`); (AP_TERM_TAC THEN VECTOR_ARITH_TAC); (REWRITE_TAC[NORM_TRIANGLE]); (UP_ASM_TAC THEN UP_ASM_TAC THEN REAL_ARITH_TAC); (NEW_GOAL `(f:real^3->real) x <= f s2`); (UNDISCH_TAC `(f:real^3->real) y <= f s2`); (UP_ASM_TAC THEN UP_ASM_TAC THEN REAL_ARITH_TAC); (NEW_GOAL `(f:real^3->real) s2 <= a1`); (EXPAND_TAC "f"); (ASM_REWRITE_TAC[]); (EXPAND_TAC "a1" THEN REAL_ARITH_TAC); (UP_ASM_TAC THEN UP_ASM_TAC THEN UNDISCH_TAC `a1 < (f:real^3->real) x`); (REAL_ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (* ========================================================================= *) (NEW_GOAL `NULLSET (conic_cap u0 u1 r1 a1 INTER aff_ge {u0:real^3, u1} {w1, w2})`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `conic_cap u0 u1 r a INTER convex hull {u0:real^3, u1, w1, w2}`); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `vol (conic_cap u0 u1 r1 a1 INTER aff_ge {u0, u1} {w1, w2}) = &0`); (ASM_SIMP_TAC[MEASURE_EQ_0]); (UP_ASM_TAC THEN REWRITE_WITH `vol (conic_cap u0 u1 r1 a1 INTER aff_ge {u0, u1} {w1, w2}) = vol (conic_cap u0 u1 r1 a1 INTER aff_gt {u0, u1} {w1, w2})`); (MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF); (REWRITE_WITH `conic_cap u0 u1 r1 a1 INTER aff_gt {u0, u1} {w1, w2} DIFF conic_cap u0 u1 r1 a1 INTER aff_ge {u0, u1:real^3} {w1, w2} = {}`); (MATCH_MP_TAC (SET_RULE `A SUBSET B ==> (C INTER A) DIFF (C INTER B) = {}`)); (REWRITE_TAC[AFF_GT_SUBSET_AFF_GE]); (REWRITE_TAC[SET_RULE `a UNION {} = a`]); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `aff_ge {u0, u1:real^3} {w1, w2} DIFF aff_gt {u0, u1} {w1, w2}`); (REWRITE_TAC[SET_RULE `A INTER B DIFF A INTER C SUBSET B DIFF C`]); (REWRITE_WITH `aff_ge {u0, u1:real^3} {w1, w2} = aff_gt {u0, u1} {w1, w2} UNION UNIONS {aff_ge {u0, u1} ({w1, w2} DELETE a) | a | a IN {w1, w2}}`); (MATCH_MP_TAC AFF_GE_AFF_GT_DECOMP); (REWRITE_TAC[Geomdetail.FINITE6]); (REWRITE_TAC[DISJOINT]); (ASM_CASES_TAC `w1 IN {u0, u1:real^3}`); (NEW_GOAL `F`); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_WITH `{u0, u1, w1, w2} = {u0, u1, w2:real^3}`); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[COPLANAR_3]); (UP_ASM_TAC THEN MESON_TAC[]); (ASM_CASES_TAC `w2 IN {u0, u1:real^3}`); (NEW_GOAL `F`); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_WITH `{u0, u1, w1, w2} = {u0, u1, w1:real^3}`); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[COPLANAR_3]); (UP_ASM_TAC THEN MESON_TAC[]); (UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `UNIONS {aff_ge {u0, u1:real^3} ({w1, w2} DELETE a) | a | a IN {w1, w2}}`); (STRIP_TAC); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `aff_ge {u0, u1:real^3} {w1} UNION aff_ge {u0, u1:real^3} {w2}`); (STRIP_TAC); (MATCH_MP_TAC NEGLIGIBLE_UNION); (STRIP_TAC); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `affine hull {u0, u1:real^3, w1}`); (STRIP_TAC); (REWRITE_TAC[NEGLIGIBLE_AFFINE_HULL_3]); (REWRITE_WITH `{u0,u1,w1:real^3} = {u0,u1} UNION {w1}`); (SET_TAC[]); (REWRITE_TAC[AFF_GE_SUBSET_AFFINE_HULL]); (MATCH_MP_TAC NEGLIGIBLE_SUBSET); (EXISTS_TAC `affine hull {u0, u1:real^3, w2}`); (STRIP_TAC); (REWRITE_TAC[NEGLIGIBLE_AFFINE_HULL_3]); (REWRITE_WITH `{u0,u1,w2:real^3} = {u0,u1} UNION {w2}`); (SET_TAC[]); (REWRITE_TAC[AFF_GE_SUBSET_AFFINE_HULL]); (REWRITE_TAC[SET_RULE `UNIONS {aff_ge {u0, u1} ({m, s3} DELETE a) | a | a IN {m, s3}} = aff_ge {u0, u1} ({m, s3} DELETE s3) UNION aff_ge {u0, u1} ({m, s3} DELETE m)`]); (MATCH_MP_TAC (SET_RULE `A SUBSET B /\ C SUBSET D ==> A UNION C SUBSET B UNION D`)); (STRIP_TAC); (MATCH_MP_TAC AFF_GE_MONO_RIGHT); (STRIP_TAC); (SET_TAC[]); (REWRITE_TAC[DISJOINT]); (ASM_CASES_TAC `w1 IN {u0, u1:real^3}`); (NEW_GOAL `F`); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_WITH `{u0, u1, w1, w2} = {u0, u1, w2:real^3}`); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[COPLANAR_3]); (UP_ASM_TAC THEN MESON_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (MATCH_MP_TAC AFF_GE_MONO_RIGHT); (STRIP_TAC); (SET_TAC[]); (REWRITE_TAC[DISJOINT]); (ASM_CASES_TAC `w2 IN {u0, u1:real^3}`); (NEW_GOAL `F`); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}`); (REWRITE_WITH `{u0, u1, w1, w2} = {u0, u1, w1:real^3}`); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[COPLANAR_3]); (UP_ASM_TAC THEN MESON_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (SET_TAC[]); (STRIP_TAC); (UNDISCH_TAC `~coplanar {u0, u1, w1, w2:real^3}` THEN REWRITE_TAC[]); (MATCH_MP_TAC CONIC_CAP_AFF_GT_EQ_0); (EXISTS_TAC `a1:real` THEN EXISTS_TAC `r1:real`); (ASM_REWRITE_TAC[])]);; (* ==================================================================== *) (* Lemma 36 *) let MEASURABLE_BALL_AFF_GE = prove (`!z r s t. measurable(ball(z,r) INTER aff_ge s t)`, MESON_TAC[MEASURABLE_CONVEX; CONVEX_INTER; CONVEX_AFF_GE; CONVEX_BALL; BOUNDED_INTER; BOUNDED_BALL]);; (* ==================================================================== *) (* Currently added lemmas *) (* Lemma 37 *) let FINITE_LIST_KY_LEMMA_2 = prove (`!s:A->bool. FINITE s ==> FINITE {y | ?u0 u1. u0 IN s /\ u1 IN s /\ y = [u0; u1]}`, REWRITE_TAC[SET_RULE `{y | ?u0 u1. u0 IN s /\ u1 IN s /\ y = [u0; u1]} = {[u0;u1] | u0 IN s /\ u1 IN s}`] THEN REPEAT (GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_REWRITE_TAC[]));; let FINITE_SET_PRODUCT_KY_LEMMA = prove ( `!s:A->bool. FINITE s ==> FINITE {{u0, u1:A} | u0 IN s /\ u1 IN s}`, REPEAT (GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_REWRITE_TAC[]));; (* ==================================================================== *) (* Lemma 38: BETA_PAIR_THM - formal proof by TCHales *) let pre_beta = prove_by_refinement (`!g u' v'. (?f. (!u v. f { u, v } = (g:A->A->B) u v)) ==> ((\ { u, v}. g u v) {u',v'} = g u' v')`, [REWRITE_TAC[GABS_DEF;GEQ_DEF]; SELECT_ELIM_TAC; MESON_TAC[]]);; let WELLDEFINED_FUNCTION_2 = prove_by_refinement( `(?f:C->D. (!x:A y:B. f(s x y) = t x y)) <=> (!x x' y y'. (s x y = s x' y') ==> t x y = t x' y')`, [ MATCH_MP_TAC EQ_TRANS ; EXISTS_TAC `?f:C->D. !z. !x:A y:B. (s x y = z) ==> f z = t x y`; CONJ_TAC; MESON_TAC[]; REWRITE_TAC[GSYM SKOLEM_THM]; MESON_TAC[]]);; let well_defined_unordered_pair = prove_by_refinement (`(?f. (!u v. f { u, v} = (g:A->A->B) u v)) <=> (! u v. g u v = g v u)`, [REWRITE_TAC[WELLDEFINED_FUNCTION_2]; NEW_GOAL `!u v x' y'. {u,v} = {x',y'} <=> (u:A = x' /\ v = y')\/ (u = y' /\ v = x')`; SET_TAC[]; ASM_REWRITE_TAC[]; MESON_TAC[]]);; let BETA_PAIR_THM = prove_by_refinement( `!g u' v'. (!u v. (g:A->A->B) u v = g v u) ==> ((\ { u, v}. g u v) {u',v'} = g u' v')`, [REPEAT STRIP_TAC; MATCH_MP_TAC pre_beta; REWRITE_TAC[well_defined_unordered_pair]; ASM_REWRITE_TAC[]]);; (* ==================================================================== *) (* Lemma 39 *) let DIHX_RANGE = prove_by_refinement ( `!V X u v. &0 <= dihX V X (u,v) /\ dihX V X (u,v) <= pi`, [(REWRITE_TAC[dihX] THEN REPEAT GEN_TAC); (COND_CASES_TAC); (REWRITE_TAC[PI_POS_LE; REAL_ARITH `a <= a`]); (REPEAT LET_TAC THEN COND_CASES_TAC); (REWRITE_TAC[dihu2; DIHV_RANGE]); (COND_CASES_TAC); (REWRITE_TAC[dihu3; DIHV_RANGE]); (COND_CASES_TAC); (REWRITE_TAC[dihu4; DIHV_RANGE]); (REWRITE_TAC[PI_POS_LE; REAL_ARITH `a <= a`])]);; let DIHX_LE_PI = prove (`!V X u v. dihX V X (u,v) <= pi`, MESON_TAC[DIHX_RANGE]);; (* ==================================================================== *) (* Lemma 40 *) let BOUNDED_VOLUME_MCELL = prove_by_refinement ( `!V. ?c. !X. saturated V /\ packing V /\ mcell_set V X ==> vol X <= c`, [(GEN_TAC); (ASM_CASES_TAC `saturated V /\ packing V`); (EXISTS_TAC `&4 / &3 * pi * &8 pow 3`); (REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[mcell_set_2; IN_ELIM_THM; IN]); (STRIP_TAC); (ASM_CASES_TAC `X:real^3->bool = {}`); (REWRITE_TAC[MEASURE_EMPTY; ASSUME `X:real^3->bool = {}`]); (REWRITE_TAC[REAL_ARITH `&0 <= &4 / &3 * pi * &8 pow 3 <=> &0 <= pi`]); (REWRITE_TAC[PI_POS_LE]); (UP_ASM_TAC THEN REWRITE_TAC[SET_RULE `~(x = {}) <=> (?v. v IN x)`]); (STRIP_TAC); (NEW_GOAL `vol X <= vol (ball (v, &8))`); (MATCH_MP_TAC MEASURE_SUBSET); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MEASURABLE_MCELL); (ASM_REWRITE_TAC[]); (REWRITE_TAC[MEASURABLE_BALL]); (ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MCELL_SUBSET_BALL8); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN SIMP_TAC[VOLUME_BALL; REAL_ARITH `&0 <= &8`]); (EXISTS_TAC `&0`); (REPEAT STRIP_TAC THEN ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 41 *) let LEFT_ACTION_LIST_2_EXISTS = prove_by_refinement ( `!u0 u1 u2 x y z d:A. CARD {u0, u1, u2, d} = 4 /\ {x, y, z} = {u0, u1, u2} ==> (?p. p permutes 0..2 /\ [x; y; z; d] = left_action_list p [u0; u1; u2; d])`, [(REPEAT STRIP_TAC); (REWRITE_TAC[left_action_list; LENGTH; TABLE_4; ARITH_RULE `SUC (SUC (SUC (SUC 0))) = 4`]); (NEW_GOAL `{u0, u1, u2:A} = {EL 0 [u0;u1;u2;d] , EL 1 [u0;u1;u2;d], EL 2 [u0;u1;u2;d]}`); (REWRITE_TAC[EL; HD; ARITH_RULE `2 = SUC 1 /\ 1 = SUC 0`; TL]); (NEW_GOAL `{x, y, z:A} = {EL 0 [u0;u1;u2;d] , EL 1 [u0;u1;u2;d], EL 2 [u0;u1;u2;d]}`); (ASM_REWRITE_TAC[]); (NEW_GOAL `?i1. i1 IN {0,1,2} /\ EL i1 [u0;u1;u2;d] = x:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `?i2. i2 IN {0,1,2} /\ EL i2 [u0;u1;u2;d] = y:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `?i3. i3 IN {0,1,2} /\ EL i3 [u0;u1;u2;d] = z:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `~(i1 = i2) /\ ~(i1 = i3) /\ ~(i2 = i3:num)`); (REPEAT STRIP_TAC); (NEW_GOAL `{u0,u1,u2,d:A} = {x, z,d}`); (ASM_SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,d:A} = 4` THEN ASM_REWRITE_TAC[]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `{u0,u1,u2,d:A} = {x, y,d}`); (ASM_SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,d:A} = 4` THEN ASM_REWRITE_TAC[]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `{u0,u1,u2,d:A} = {x, y,d}`); (ASM_SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,d:A} = 4` THEN ASM_REWRITE_TAC[]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (SWITCH_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN REPEAT STRIP_TAC); (NEW_GOAL `{i1, i2, i3} = {0,1,2}`); (ASM_SET_TAC[]); (ABBREV_TAC `p = (\i:num. if i = i1 then 0 else if i = i2 then 1 else if i = i3 then 2 else i)`); (NEW_GOAL `p i1 = 0 /\ p i2 = 1 /\ p (i3:num) = 2`); (ASM_MESON_TAC[]); (EXISTS_TAC `p:num->num`); (NEW_GOAL `p permutes 0..2`); (REWRITE_TAC[permutes; EXISTS_UNIQUE; IN_NUMSEG_0; ARITH_RULE `a <= 2 <=> a = 0 \/ a = 1 \/ a = 2`; SET_RULE `x = 0 \/ x = 1 \/ x = 2 <=> x IN {0,1,2}`]); (NEW_GOAL `(!x. ~(x IN {0, 1, 2}) ==> p x = x)`); (REPEAT STRIP_TAC); (EXPAND_TAC "p"); (COND_CASES_TAC); (ASM_SET_TAC[]); (COND_CASES_TAC); (ASM_SET_TAC[]); (COND_CASES_TAC); (ASM_SET_TAC[]); (REFL_TAC); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `y' = 0`); (EXISTS_TAC `i1:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (ASM_SET_TAC[]); (ASM_CASES_TAC `y' = 1`); (EXISTS_TAC `i2:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (ASM_SET_TAC[]); (ASM_CASES_TAC `y' = 2`); (EXISTS_TAC `i3:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (ASM_SET_TAC[]); (NEW_GOAL `~(y' IN {0,1,2})`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (EXISTS_TAC `y':num`); (ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (NEW_GOAL `(p:num->num) y'' = y''`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (ASM_MESON_TAC[]); (REWRITE_WITH `inverse (p:num->num) 0 = i1 /\ inverse p 1 = i2 /\ inverse p 2 = i3`); (UP_ASM_TAC THEN MESON_TAC[PERMUTES_INVERSE_EQ; ASSUME `(p:num->num) i1 = 0 /\ p i2 = 1 /\ p i3 = 2`]); (ASM_REWRITE_TAC[]); (REWRITE_WITH `inverse (p:num->num) 3 = 3`); (NEW_GOAL `p 3 = 3`); (MP_TAC (ASSUME `p permutes 0..2`) THEN REWRITE_TAC[permutes]); (STRIP_TAC); (FIRST_ASSUM MATCH_MP_TAC); (REWRITE_TAC[IN_NUMSEG_0] THEN ARITH_TAC); (UP_ASM_TAC THEN UP_ASM_TAC THEN MESON_TAC[PERMUTES_INVERSE_EQ]); (REWRITE_TAC[EL;HD; TL; ARITH_RULE `3 = SUC 2 /\ 2 = SUC 1 /\ 1 = SUC 0`])]);; (* ==================================================================== *) (* Lemma 42 *) let LEFT_ACTION_LIST_3_EXISTS = prove_by_refinement ( `!u0 u1 u2 u3 x y z t:A. CARD {u0, u1, u2, u3} = 4 /\ {x, y, z, t} = {u0, u1, u2, u3} ==> (?p. p permutes 0..3 /\ [x; y; z; t] = left_action_list p [u0; u1; u2; u3])`, [(REPEAT STRIP_TAC); (REWRITE_TAC[left_action_list; LENGTH; TABLE_4; ARITH_RULE `SUC (SUC (SUC (SUC 0))) = 4`]); (NEW_GOAL `{u0, u1, u2, u3:A} = {EL 0 [u0;u1;u2;u3], EL 1 [u0;u1;u2;u3], EL 2 [u0;u1;u2;u3], EL 3 [u0;u1;u2;u3]}`); (REWRITE_TAC[EL; HD; ARITH_RULE `3 = SUC 2 /\ 2 = SUC 1 /\ 1 = SUC 0`; TL]); (NEW_GOAL `{x, y, z, t:A} = {EL 0 [u0;u1;u2;u3], EL 1 [u0;u1;u2;u3], EL 2 [u0;u1;u2;u3], EL 3 [u0;u1;u2;u3]}`); (ASM_REWRITE_TAC[]); (NEW_GOAL `?i1. i1 IN {0,1,2,3} /\ EL i1 [u0;u1;u2;u3] = x:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `?i2. i2 IN {0,1,2,3} /\ EL i2 [u0;u1;u2;u3] = y:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `?i3. i3 IN {0,1,2,3} /\ EL i3 [u0;u1;u2;u3] = z:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `?i4. i4 IN {0,1,2,3} /\ EL i4 [u0;u1;u2;u3] = t:A`); (UP_ASM_TAC THEN SET_TAC[]); (FIRST_X_ASSUM CHOOSE_TAC THEN SWITCH_TAC); (NEW_GOAL `~(i1 = i2) /\ ~(i1 = i3) /\ ~(i1 = i4:num) /\ ~(i2 = i3) /\ ~(i2 = i4) /\ ~(i3 = i4)`); (REPEAT STRIP_TAC); (NEW_GOAL `x = y:A`); (ASM_MESON_TAC[]); (NEW_GOAL `{u0,u1,u2,u3:A} = {x, z,t}`); (UNDISCH_TAC `{x, y, z, t:A} = {u0, u1, u2, u3}` THEN UP_ASM_TAC THEN SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,u3:A} = 4` THEN REWRITE_TAC[ASSUME `{u0, u1, u2, u3} = {x, z, t:A}`]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `x = z:A`); (ASM_MESON_TAC[]); (NEW_GOAL `{u0,u1,u2,u3:A} = {x, y,t}`); (UNDISCH_TAC `{x, y, z, t:A} = {u0, u1, u2, u3}` THEN UP_ASM_TAC THEN SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,u3:A} = 4` THEN REWRITE_TAC[ASSUME `{u0, u1, u2, u3} = {x, y, t:A}`]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `x = t:A`); (ASM_MESON_TAC[]); (NEW_GOAL `{u0,u1,u2,u3:A} = {x, y, z}`); (UNDISCH_TAC `{x, y, z, t:A} = {u0, u1, u2, u3}` THEN UP_ASM_TAC THEN SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,u3:A} = 4` THEN REWRITE_TAC[ASSUME `{u0, u1, u2, u3} = {x, y, z:A}`]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `y = z:A`); (ASM_MESON_TAC[]); (NEW_GOAL `{u0,u1,u2,u3:A} = {x, z,t}`); (UNDISCH_TAC `{x, y, z, t:A} = {u0, u1, u2, u3}` THEN UP_ASM_TAC THEN SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,u3:A} = 4` THEN REWRITE_TAC[ASSUME `{u0, u1, u2, u3} = {x, z, t:A}`]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `y = t:A`); (ASM_MESON_TAC[]); (NEW_GOAL `{u0,u1,u2,u3:A} = {x, z,t}`); (UNDISCH_TAC `{x, y, z, t:A} = {u0, u1, u2, u3}` THEN UP_ASM_TAC THEN SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,u3:A} = 4` THEN REWRITE_TAC[ASSUME `{u0, u1, u2, u3} = {x, z, t:A}`]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (NEW_GOAL `z = t:A`); (ASM_MESON_TAC[]); (NEW_GOAL `{u0,u1,u2,u3:A} = {x, y,t}`); (UNDISCH_TAC `{x, y, z, t:A} = {u0, u1, u2, u3}` THEN UP_ASM_TAC THEN SET_TAC[]); (UNDISCH_TAC `CARD {u0,u1,u2,u3:A} = 4` THEN REWRITE_TAC[ASSUME `{u0, u1, u2, u3} = {x, y, t:A}`]); (MESON_TAC[Geomdetail.CARD3; ARITH_RULE `a <= 3 ==> ~(a = 4)`]); (SWITCH_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN REPEAT STRIP_TAC); (NEW_GOAL `{i1, i2, i3,i4} = {0,1,2,3}`); (REWRITE_TAC[SET_EQ_LEMMA]); (REPEAT STRIP_TAC); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (ABBREV_TAC `p = (\i:num. if i = i1 then 0 else if i = i2 then 1 else if i = i3 then 2 else if i = i4 then 3 else i)`); (NEW_GOAL `p i1 = 0 /\ p i2 = 1 /\ p (i3:num) = 2 /\ p i4 = 3`); (REPEAT STRIP_TAC); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (EXPAND_TAC "p" THEN COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (COND_CASES_TAC); (REFL_TAC); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (EXISTS_TAC `p:num->num`); (NEW_GOAL `p permutes 0..3`); (REWRITE_TAC[permutes; EXISTS_UNIQUE; IN_NUMSEG_0; ARITH_RULE `a <= 3 <=> a = 0 \/ a = 1 \/ a = 2 \/ a = 3`; SET_RULE `x = 0 \/ x = 1 \/ x = 2 \/ x = 3 <=> x IN {0,1,2,3}`]); (NEW_GOAL `(!x. ~(x IN {0, 1, 2, 3}) ==> p x = x)`); (REPEAT STRIP_TAC); (EXPAND_TAC "p"); (COND_CASES_TAC); (NEW_GOAL `F`); (UNDISCH_TAC `~(x' IN {0, 1, 2, 3})` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (COND_CASES_TAC); (NEW_GOAL `F`); (UNDISCH_TAC `~(x' IN {0, 1, 2, 3})` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (COND_CASES_TAC); (NEW_GOAL `F`); (UNDISCH_TAC `~(x' IN {0, 1, 2, 3})` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (COND_CASES_TAC); (NEW_GOAL `F`); (UNDISCH_TAC `~(x' IN {0, 1, 2, 3})` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (REFL_TAC); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `y' = 0`); (EXISTS_TAC `i1:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i4:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2,3})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 0` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (REPLICATE_TAC 4 UP_ASM_TAC THEN REWRITE_TAC[] THEN UNDISCH_TAC `{i1,i2,i3,i4} = {0,1,2,3}` THEN SET_TAC[]); (ASM_CASES_TAC `y' = 1`); (EXISTS_TAC `i2:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i4:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2,3})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 1` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (REPLICATE_TAC 4 UP_ASM_TAC THEN REWRITE_TAC[] THEN UNDISCH_TAC `{i1,i2,i3,i4} = {0,1,2,3}` THEN SET_TAC[]); (ASM_CASES_TAC `y' = 2`); (EXISTS_TAC `i3:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i4:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2,3})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 2` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (REPLICATE_TAC 4 UP_ASM_TAC THEN REWRITE_TAC[] THEN UNDISCH_TAC `{i1,i2,i3,i4} = {0,1,2,3}` THEN SET_TAC[]); (ASM_CASES_TAC `y' = 3`); (EXISTS_TAC `i4:num`); (ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 3` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 3` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 3` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ASM_MESON_TAC[]); (ASM_CASES_TAC `~(y'' IN {0,1,2,3})`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = 3` THEN ASM_SIMP_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (ASM_MESON_TAC[]); (REPLICATE_TAC 4 UP_ASM_TAC THEN REWRITE_TAC[] THEN UNDISCH_TAC `{i1,i2,i3,i4} = {0,1,2,3}` THEN SET_TAC[]); (NEW_GOAL `~(y' IN {0,1,2,3})`); (REPLICATE_TAC 4 UP_ASM_TAC THEN SET_TAC[]); (EXISTS_TAC `y':num`); (ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `y'' = i1:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i2:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i3:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_CASES_TAC `y'' = i4:num`); (NEW_GOAL `F`); (UNDISCH_TAC `(p:num->num) y'' = y'` THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (NEW_GOAL `(p:num->num) y'' = y''`); (FIRST_ASSUM MATCH_MP_TAC); (REPLICATE_TAC 4 UP_ASM_TAC THEN REWRITE_TAC[] THEN UNDISCH_TAC `{i1,i2,i3,i4} = {0,1,2,3}` THEN SET_TAC[]); (ASM_MESON_TAC[]); (REWRITE_WITH `inverse (p:num->num) 0 = i1 /\ inverse p 3 = i4 /\ inverse p 1 = i2 /\ inverse p 2 = i3`); (UP_ASM_TAC THEN MESON_TAC[PERMUTES_INVERSE_EQ; ASSUME `(p:num->num) i1 = 0 /\ p i2 = 1 /\ p i3 = 2 /\ p i4 = 3`]); (ASM_REWRITE_TAC[])]);; (* ==================================================================== *) (* Lemma 43 *) let lmfun_bounded = prove_by_refinement ( `!h. &0 <= h ==> lmfun h <= h0 / (h0 - &1)`, [(REPEAT STRIP_TAC THEN REWRITE_TAC[lmfun] THEN COND_CASES_TAC); (REWRITE_TAC[REAL_ARITH `(h0 - h) / (h0 - &1) <= h0 / (h0 - &1) <=> &0 <= h / (h0 - &1)`]); (MATCH_MP_TAC REAL_LE_DIV); (REWRITE_TAC[h0] THEN ASM_REAL_ARITH_TAC); (MATCH_MP_TAC REAL_LE_DIV); (REWRITE_TAC[h0] THEN REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 44 *) let lmfun_pos_le = prove_by_refinement ( `!h. &0 <= lmfun h`, [(REPEAT STRIP_TAC THEN REWRITE_TAC[lmfun] THEN COND_CASES_TAC); (MATCH_MP_TAC REAL_LE_DIV); (STRIP_TAC); (ASM_REAL_ARITH_TAC); (ASM_REWRITE_TAC[h0] THEN REAL_ARITH_TAC); (REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 45 *) let BARV_CARD_LEMMA = prove ( `!V ul k. packing V /\ barV V k ul ==> CARD (set_of_list ul) = k + 1`, MESON_TAC[Rogers.BARV_IMP_LENGTH_EQ_CARD]);; let BARV_LENGTH_LEMMA = prove ( `!V ul k. packing V /\ barV V k ul ==> LENGTH ul = k + 1`, MESON_TAC[Rogers.BARV_IMP_LENGTH_EQ_CARD]);; (* ==================================================================== *) (* Lemma 46 *) let MCELL_ID_MXI_2 = prove_by_refinement ( `!V i j ul vl. packing V /\ saturated V /\ barV V 3 ul /\ barV V 3 vl /\ mcell i V ul = mcell j V vl /\ ~NULLSET (mcell i V ul) /\ i IN {2, 3} /\ j IN {2, 3} ==> mxi V ul = mxi V vl`, [(REPEAT STRIP_TAC); (NEW_GOAL `i = j /\ (!k. i - 1 <= k /\ k <= 3 ==> omega_list_n V ul k = omega_list_n V vl k)`); (MATCH_MP_TAC MCELL_ID_OMEGA_LIST_N); (ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_CASES_TAC `hl (truncate_simplex 2 (ul:(real^3)list)) >= sqrt (&2)`); (ASM_CASES_TAC `hl (truncate_simplex 2 (vl:(real^3)list)) >= sqrt (&2)`); (REWRITE_WITH `mxi V ul = omega_list_n V ul 2`); (REWRITE_TAC[mxi] THEN COND_CASES_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (REWRITE_WITH `mxi V vl = omega_list_n V vl 2`); (REWRITE_TAC[mxi] THEN COND_CASES_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `F`); (ASM_MESON_TAC[]); (ASM_MESON_TAC[]); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[REAL_ARITH `~(a >= b) <=> a < b`]); (STRIP_TAC); (ABBREV_TAC `xl = truncate_simplex 2 (ul:(real^3)list)`); (ABBREV_TAC `yl = truncate_simplex 2 (vl:(real^3)list)`); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)) = set_of_list (truncate_simplex (i - 1) vl)`); (NEW_GOAL `V INTER mcell i V ul = set_of_list (truncate_simplex (i - 1) ul)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `V INTER mcell j V vl = set_of_list (truncate_simplex (j - 1) vl)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `hl (xl:(real^3)list) <= dist (omega_list V xl, HD xl)`); (MATCH_MP_TAC Rogers.WAUFCHE1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN] THEN EXPAND_TAC "xl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `!w:real^3. w IN set_of_list yl ==> dist (circumcenter (set_of_list yl),w) = hl yl`); (MATCH_MP_TAC Rogers.HL_PROPERTIES); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `2` THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "yl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `circumcenter (set_of_list yl) = omega_list V (yl:(real^3)list)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "yl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (STRIP_TAC); (NEW_GOAL `hl yl = dist (omega_list V yl, HD xl)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `HD (xl:(real^3)list) = HD (truncate_simplex (i - 1) ul)`); (REWRITE_WITH `HD (xl:(real^3)list) = HD ul`); (EXPAND_TAC "xl"); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `i IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `HD (xl:(real^3)list) IN set_of_list (truncate_simplex (i - 1) ul)`); (REWRITE_TAC[ASSUME `HD (xl:(real^3)list) = HD (truncate_simplex (i - 1) ul)`]); (MATCH_MP_TAC Packing3.BARV_IMP_HD_IN_SET_OF_LIST); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `i - 1`); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `j IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`] THEN ARITH_TAC); (UP_ASM_TAC); (NEW_GOAL `set_of_list (truncate_simplex (j - 1) vl) SUBSET set_of_list (yl:(real^3)list)`); (EXPAND_TAC "yl"); (MATCH_MP_TAC Rogers.TRUNCATE_SIMPLEX_SUBSET); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `j IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (UP_ASM_TAC THEN ASM_REWRITE_TAC[]); (SET_TAC[]); (NEW_GOAL `omega_list V xl = omega_list V yl`); (REWRITE_TAC[OMEGA_LIST]); (REWRITE_WITH `LENGTH (xl:(real^3)list) = 3 /\ LENGTH (yl:(real^3)list) = 3`); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_TAC[ARITH_RULE `3 = 2 + 1`]); (STRIP_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_TAC[ARITH_RULE `3 - 1 = 2`]); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_WITH `truncate_simplex 2 ul = truncate_simplex (2 + 0) (ul:(real^3)list) /\ truncate_simplex 2 vl = truncate_simplex (2 + 0) (vl:(real^3)list)`); (REWRITE_TAC[ARITH_RULE `2 + 0 = 2`]); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) ul) 2 = omega_list_n V ul 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) vl) 2 = omega_list_n V vl 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `hl (yl:(real^3)list) = dist (omega_list V xl, HD xl)`); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[]); (* ======================================================================= *) (UP_ASM_TAC THEN REWRITE_TAC[REAL_ARITH `~(a >= b) <=> a < b`]); (STRIP_TAC); (ASM_CASES_TAC `hl (truncate_simplex 2 (vl:(real^3)list)) >= sqrt (&2)`); (ABBREV_TAC `xl = truncate_simplex 2 (ul:(real^3)list)`); (ABBREV_TAC `yl = truncate_simplex 2 (vl:(real^3)list)`); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)) = set_of_list (truncate_simplex (i - 1) vl)`); (NEW_GOAL `V INTER mcell i V ul = set_of_list (truncate_simplex (i - 1) ul)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `V INTER mcell j V vl = set_of_list (truncate_simplex (j - 1) vl)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (NEW_GOAL `hl (yl:(real^3)list) <= dist (omega_list V yl, HD yl)`); (MATCH_MP_TAC Rogers.WAUFCHE1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN] THEN EXPAND_TAC "yl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `!w:real^3. w IN set_of_list xl ==> dist (circumcenter (set_of_list xl),w) = hl xl`); (MATCH_MP_TAC Rogers.HL_PROPERTIES); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `2` THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "xl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `circumcenter (set_of_list xl) = omega_list V (xl:(real^3)list)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "xl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (STRIP_TAC); (NEW_GOAL `hl xl = dist (omega_list V xl, HD yl)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `HD (yl:(real^3)list) = HD (truncate_simplex (i - 1) vl)`); (REWRITE_WITH `HD (yl:(real^3)list) = HD vl`); (EXPAND_TAC "yl"); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `i IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (NEW_GOAL `HD (yl:(real^3)list) IN set_of_list (truncate_simplex (i - 1) vl)`); (REWRITE_TAC[ASSUME `HD (yl:(real^3)list) = HD (truncate_simplex (i - 1) vl)`]); (MATCH_MP_TAC Packing3.BARV_IMP_HD_IN_SET_OF_LIST); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `i - 1`); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `j IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`] THEN ARITH_TAC); (UP_ASM_TAC); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) vl) SUBSET set_of_list (xl:(real^3)list)`); (EXPAND_TAC "xl"); (REWRITE_TAC[GSYM (ASSUME `set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)) = set_of_list (truncate_simplex (i - 1) vl)`)]); (MATCH_MP_TAC Rogers.TRUNCATE_SIMPLEX_SUBSET); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `i IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (UP_ASM_TAC THEN ASM_REWRITE_TAC[]); (SET_TAC[]); (NEW_GOAL `omega_list V xl = omega_list V yl`); (REWRITE_TAC[OMEGA_LIST]); (REWRITE_WITH `LENGTH (xl:(real^3)list) = 3 /\ LENGTH (yl:(real^3)list) = 3`); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_TAC[ARITH_RULE `3 = 2 + 1`]); (STRIP_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_TAC[ARITH_RULE `3 - 1 = 2`]); (EXPAND_TAC "xl" THEN EXPAND_TAC "yl"); (REWRITE_WITH `truncate_simplex 2 ul = truncate_simplex (2 + 0) (ul:(real^3)list) /\ truncate_simplex 2 vl = truncate_simplex (2 + 0) (vl:(real^3)list)`); (REWRITE_TAC[ARITH_RULE `2 + 0 = 2`]); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) ul) 2 = omega_list_n V ul 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (REWRITE_WITH `omega_list_n V (truncate_simplex (2 + 0) vl) 2 = omega_list_n V vl 2`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1 /\ CARD (set_of_list vl) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (NEW_GOAL `hl (xl:(real^3)list) = dist (omega_list V yl, HD yl)`); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[REAL_ARITH `~(a >= b) <=> a < b`]); (STRIP_TAC); (* ======================================================================= *) (NEW_GOAL `sqrt (&2) <= hl (ul:(real^3)list)`); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_CASES_TAC `i = 2`); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `i = 2`; mcell2]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `i = 3`); (ASM_SET_TAC[]); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `i = 3`; mcell3]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `sqrt (&2) <= hl (vl:(real^3)list)`); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[]); (ASM_CASES_TAC `j = 2`); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `j = 2`; mcell2]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `j = 3`); (ASM_SET_TAC[]); (REWRITE_TAC[MCELL_EXPLICIT; ASSUME `j = 3`; mcell3]); (COND_CASES_TAC); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ABBREV_TAC `s2 = omega_list_n V ul 2`); (ABBREV_TAC `s3 = omega_list_n V ul 3`); (NEW_GOAL `s2 = omega_list_n V vl 2`); (EXPAND_TAC "s2" THEN FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (NEW_GOAL `s3 = omega_list_n V vl 3`); (EXPAND_TAC "s3" THEN FIRST_ASSUM MATCH_MP_TAC); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_ARITH_TAC); (ABBREV_TAC `u0:real^3 = HD ul`); (NEW_GOAL `?s. between s (s2,s3) /\ dist (u0,s) = sqrt (&2) /\ mxi V ul = s`); (EXPAND_TAC "s2" THEN EXPAND_TAC "s3"); (MATCH_MP_TAC MXI_EXPLICIT); (NEW_GOAL `?v0 u1 u2 u3. ul = [v0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `u1:real^3` THEN EXISTS_TAC `u2:real^3` THEN EXISTS_TAC `u3:real^3`); (ASM_REWRITE_TAC[]); (REWRITE_WITH `v0:real^3 = HD ul`); (REWRITE_TAC[ASSUME `ul = [v0;u1;u2;u3:real^3]`; HD]); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ABBREV_TAC `v0:real^3 = HD vl`); (NEW_GOAL `?t. between t (s2,s3) /\ dist (v0,t) = sqrt (&2) /\ mxi V vl = t`); (ASM_REWRITE_TAC[]); (EXPAND_TAC "v0"); (MATCH_MP_TAC MXI_EXPLICIT); (NEW_GOAL `?w0 u1 u2 u3. vl = [w0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `u1:real^3` THEN EXISTS_TAC `u2:real^3` THEN EXISTS_TAC `u3:real^3`); (ASM_REWRITE_TAC[HD]); (EXPAND_TAC "v0"); (REWRITE_TAC[ASSUME `vl = [w0;u1;u2;u3:real^3]`; HD]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `dist (u0:real^3, t) = sqrt (&2)`); (NEW_GOAL `(t:real^3) IN voronoi_list V (truncate_simplex 2 vl)`); (NEW_GOAL `s2 IN voronoi_list V (truncate_simplex 2 vl)`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (NEW_GOAL `s3 IN voronoi_list V (truncate_simplex 2 vl)`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[] THEN ARITH_TAC); (NEW_GOAL `convex hull {s2, s3:real^3} SUBSET convex hull (voronoi_list V (truncate_simplex 2 vl))`); (MATCH_MP_TAC Marchal_cells.CONVEX_HULL_SUBSET THEN ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_WITH `convex hull voronoi_list V (truncate_simplex 2 vl) = voronoi_list V (truncate_simplex 2 vl)`); (REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_VORONOI_LIST]); (STRIP_TAC); (NEW_GOAL `t IN convex hull {s2, s3:real^3}`); (REWRITE_TAC[GSYM BETWEEN_IN_CONVEX_HULL]); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (NEW_GOAL `u0:real^3 IN set_of_list (truncate_simplex 2 vl)`); (EXPAND_TAC "u0"); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)) = set_of_list (truncate_simplex (i - 1) vl)`); (NEW_GOAL `V INTER mcell i V ul = set_of_list (truncate_simplex (i - 1) ul)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `V INTER mcell j V vl = set_of_list (truncate_simplex (j - 1) vl)`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (NEW_GOAL `i = 2 \/ i = 3`); (ASM_SET_TAC[]); (ASM_REWRITE_TAC[]); (REPEAT STRIP_TAC); (ASM_ARITH_TAC); (ASM_ARITH_TAC); (UNDISCH_TAC `~NULLSET (mcell i V ul)`); (ASM_REWRITE_TAC[NEGLIGIBLE_EMPTY]); (ASM_MESON_TAC[]); (NEW_GOAL `HD (ul:(real^3)list) = HD (truncate_simplex (i - 1) ul)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `i IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (NEW_GOAL `HD (ul:(real^3)list) IN set_of_list (truncate_simplex (i - 1) ul)`); (REWRITE_TAC[ASSUME `HD (ul:(real^3)list) = HD (truncate_simplex (i - 1) ul)`]); (MATCH_MP_TAC Packing3.BARV_IMP_HD_IN_SET_OF_LIST); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `i - 1`); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `j IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`] THEN ARITH_TAC); (UP_ASM_TAC THEN ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list (truncate_simplex (j - 1) vl) SUBSET set_of_list (truncate_simplex 2 (vl:(real^3)list))`); (MATCH_MP_TAC Rogers.TRUNCATE_SIMPLEX_SUBSET); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `j IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (UP_ASM_TAC THEN ASM_REWRITE_TAC[]); (SET_TAC[]); (NEW_GOAL `v0:real^3 IN set_of_list (truncate_simplex 2 vl)`); (EXPAND_TAC "v0"); (REWRITE_WITH `(HD vl):real^3 = HD (truncate_simplex 2 vl)`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (vl:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UNDISCH_TAC `j IN {2,3}` THEN REWRITE_TAC[SET_RULE `x IN {2,3} <=> x = 2 \/ x = 3`]); (ARITH_TAC); (MATCH_MP_TAC Packing3.BARV_IMP_HD_IN_SET_OF_LIST); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `2`); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `t:real^3 IN voronoi_closed V u0`); (UNDISCH_TAC `t:real^3 IN voronoi_list V (truncate_simplex 2 vl)`); (REWRITE_TAC[VORONOI_LIST; VORONOI_SET;]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN; IN_ELIM_THM; voronoi_closed]); (STRIP_TAC); (NEW_GOAL `dist (u0:real^3, t) <= dist (v0:real^3, t)`); (ONCE_REWRITE_TAC[DIST_SYM]); (FIRST_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "v0"); (REWRITE_TAC[MESON[IN] `(V:real^3->bool) s <=> s IN V`]); (NEW_GOAL `HD (vl:(real^3)list) IN set_of_list vl`); (MATCH_MP_TAC Packing3.BARV_IMP_HD_IN_SET_OF_LIST); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list (vl:(real^3)list) SUBSET V`); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (NEW_GOAL `t:real^3 IN voronoi_closed V v0`); (UNDISCH_TAC `t:real^3 IN voronoi_list V (truncate_simplex 2 vl)`); (REWRITE_TAC[VORONOI_LIST; VORONOI_SET;]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN; IN_ELIM_THM; voronoi_closed]); (STRIP_TAC); (NEW_GOAL `dist (v0:real^3, t) <= dist (u0:real^3, t)`); (ONCE_REWRITE_TAC[DIST_SYM]); (FIRST_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "u0"); (REWRITE_TAC[MESON[IN] `(V:real^3->bool) s <=> s IN V`]); (NEW_GOAL `HD (ul:(real^3)list) IN set_of_list ul`); (MATCH_MP_TAC Packing3.BARV_IMP_HD_IN_SET_OF_LIST); (EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list (ul:(real^3)list) SUBSET V`); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (ASM_REAL_ARITH_TAC); (ASM_REWRITE_TAC[]); (NEW_GOAL `!n. between n (s2, s3:real^3) ==> dist (u0, n) pow 2 = dist (s2, n) pow 2 + dist (u0, s2) pow 2`); (REPEAT STRIP_TAC); (REWRITE_TAC[dist]); (MATCH_MP_TAC PYTHAGORAS); (REWRITE_TAC[orthogonal]); (ABBREV_TAC `zl = truncate_simplex 2 (ul:(real^3)list)`); (REWRITE_WITH `s2:real^3 = circumcenter (set_of_list zl)`); (REWRITE_WITH `s2 = omega_list V zl`); (EXPAND_TAC "s2" THEN EXPAND_TAC "zl"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Rogers.XNHPWAB1); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (MATCH_MP_TAC Rogers.MHFTTZN4); (EXISTS_TAC `V:real^3->bool`); (EXISTS_TAC `2`); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `s2 IN voronoi_list V zl`); (REWRITE_WITH `s2 = omega_list V zl`); (EXPAND_TAC "s2" THEN EXPAND_TAC "zl"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.OMEGA_LIST_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.OMEGA_LIST_IN_VORONOI_LIST); (EXISTS_TAC `2`); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (NEW_GOAL `s3 IN voronoi_list V zl`); (EXPAND_TAC "s3" THEN EXPAND_TAC "zl"); (MATCH_MP_TAC Rogers.OMEGA_LIST_N_IN_VORONOI_LIST_GEN); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3 /\ 3 <= 3`]); (NEW_GOAL `affine hull {s2, s3} SUBSET affine hull voronoi_list V zl`); (MATCH_MP_TAC Marchal_cells_2_new.AFFINE_SUBSET_KY_LEMMA); (ASM_SET_TAC[]); (NEW_GOAL `n IN affine hull {s2, s3:real^3}`); (NEW_GOAL `convex hull {s2,s3} SUBSET affine hull {s2,s3:real^3}`); (REWRITE_TAC[CONVEX_HULL_SUBSET_AFFINE_HULL]); (NEW_GOAL `n IN convex hull {s2, s3:real^3}`); (REWRITE_TAC[GSYM BETWEEN_IN_CONVEX_HULL]); (ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1); (REWRITE_WITH `u0:real^3 = HD zl`); (EXPAND_TAC "zl" THEN EXPAND_TAC "u0"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Packing3.HD_IN_SET_OF_LIST); (REWRITE_WITH `LENGTH (zl:(real^3)list) = 2 + 1 /\ CARD (set_of_list zl) = 2 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (ARITH_TAC); (NEW_GOAL `dist (u0,s) pow 2 = dist (s2,s:real^3) pow 2 + dist (u0,s2) pow 2`); (ASM_SIMP_TAC[]); (NEW_GOAL `dist (u0,t) pow 2 = dist (s2,t:real^3) pow 2 + dist (u0,s2) pow 2`); (ASM_SIMP_TAC[]); (NEW_GOAL `dist (s2, s:real^3) = dist (s2, t)`); (REWRITE_TAC[DIST_EQ]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); (UNDISCH_TAC `between s (s2, s3:real^3)` THEN UNDISCH_TAC `between t (s2, s3:real^3)`); (REWRITE_TAC[BETWEEN_IN_CONVEX_HULL; CONVEX_HULL_2; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (UNDISCH_TAC `dist (s2,s) = dist (s2,t:real^3)`); (REWRITE_TAC[dist;ASSUME `s = u' % s2 + v' % s3:real^3`; ASSUME `t = u % s2 + v % s3:real^3` ]); (REWRITE_WITH `s2 - (u' % s2 + v' % s3) = (u' + v') % s2 - (u' % s2 + v' % s3:real^3)`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_WITH `s2 - (u % s2 + v % s3) = (u + v) % s2 - (u % s2 + v % s3:real^3)`); (ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC); (REWRITE_TAC[VECTOR_ARITH `(u + v) % s - (u % s + v % t) = v % (s - t)`]); (REWRITE_TAC[NORM_MUL]); (REWRITE_WITH `abs v = v /\ abs v' = v'`); (ASM_SIMP_TAC[REAL_ABS_REFL]); (REWRITE_TAC[REAL_ARITH `a * b = c * b <=> (a - c) * b = &0`;REAL_ENTIRE]); (STRIP_TAC); (REWRITE_WITH `v = v':real /\ u = u':real`); (ASM_REAL_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_TAC[NORM_EQ_0]); (REWRITE_TAC[VECTOR_ARITH `a - b = vec 0 <=> a = b`] THEN STRIP_TAC); (NEW_GOAL `F`); (NEW_GOAL `hl (ul:(real^3)list) <= dist (s3,u0:real^3)`); (REWRITE_WITH `s3 = omega_list V ul /\ u0 = HD ul`); (STRIP_TAC); (REWRITE_TAC[GSYM (ASSUME `omega_list_n V ul 3 = s3`); OMEGA_LIST]); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[ARITH_RULE `(3 + 1) - 1 = 3`]); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Rogers.WAUFCHE1); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[IN]); (ABBREV_TAC `zl:(real^3)list = truncate_simplex 2 ul`); (NEW_GOAL `hl (zl:(real^3)list) = dist (s3, u0:real^3)`); (REWRITE_WITH `s3 = omega_list V zl /\ u0 = HD zl`); (STRIP_TAC); (REWRITE_TAC[GSYM (ASSUME `s2:real^3 = s3`); OMEGA_LIST]); (EXPAND_TAC "s2"); (REWRITE_WITH `LENGTH (zl:(real^3)list) = 2 + 1 /\ CARD (set_of_list zl) = 2 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (REWRITE_TAC[ARITH_RULE `(2 + 1) - 1 = 2`]); (EXPAND_TAC "zl"); (REWRITE_WITH `truncate_simplex 2 (ul:(real^3)list) = truncate_simplex (2 + 0) (ul:(real^3)list)`); (REWRITE_TAC[ARITH_RULE `2 + 0 = 2`]); (MATCH_MP_TAC Packing3.OMEGA_LIST_N_LEMMA); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (EXPAND_TAC "zl" THEN EXPAND_TAC "u0"); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (MATCH_MP_TAC Rogers.WAUFCHE2); (EXISTS_TAC `2` THEN ASM_REWRITE_TAC[IN]); (EXPAND_TAC "zl"); (MATCH_MP_TAC Packing3.TRUNCATE_SIMPLEX_BARV); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[ARITH_RULE `2 <= 3`]); (ASM_REAL_ARITH_TAC); (ASM_MESON_TAC[])]);; (* ==================================================================== *) (* Lemma 47 *) let DIHV_SYM_3 = prove_by_refinement ( `!a b c d x y z t. {a,b:real^3} = {c,d} /\ {x,y} = {z,t} ==> dihV a b x y = dihV c d z t`, [(REWRITE_TAC[SET_RULE `{a,b} = {c,d} <=> (a = c /\ b = d) \/ (a = d /\ b = c)`]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[DIHV_SYM_2]); (ASM_REWRITE_TAC[DIHV_SYM_2; DIHV_SYM]); (ASM_REWRITE_TAC[DIHV_SYM_2; DIHV_SYM])]);; (* ==================================================================== *) (* Lemma 48 *) let DIHX_SYM = prove_by_refinement ( `!V X u v. packing V /\ saturated V /\ mcell_set V X /\ {u, v} IN edgeX V X ==> dihX V X (u,v) = dihX V X (v,u)`, [(REPEAT STRIP_TAC); (REWRITE_TAC[dihX]); (COND_CASES_TAC); (COND_CASES_TAC); (REFL_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (DEL_TAC THEN REWRITE_TAC[cell_params_d] THEN REPEAT LET_TAC); (UNDISCH_TAC `{u, v} IN edgeX V X` THEN REWRITE_TAC[edgeX; IN;IN_ELIM_THM]); (REWRITE_WITH `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (ASM_REWRITE_TAC[]); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set;IN_ELIM_THM;IN]); (MESON_TAC[]); (STRIP_TAC); (MP_TAC (ASSUME `(V INTER X) (u':real^3)`) THEN MP_TAC (ASSUME `(V INTER X) (v':real^3)`)); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set_2;IN_ELIM_THM;IN]); (STRIP_TAC); (REWRITE_TAC[ASSUME `X = mcell i V ul''`]); (REWRITE_WITH `(V INTER mcell i V ul'') = set_of_list (truncate_simplex (i-1) ul'')`); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `X = mcell i V ul''`)]); (STRIP_TAC); (REWRITE_TAC[ARITH_RULE `1 <= i <=> ~(i = 0)`]); (STRIP_TAC); (MP_TAC (ASSUME `(V INTER X) (u':real^3)`)); (REWRITE_WITH `V INTER X:real^3->bool = {}`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ_0); (ASM_REWRITE_TAC[]); (REWRITE_TAC[MESON[IN] `{} x <=> x IN {}`]); (SET_TAC[]); (UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `i <= 1`); (NEW_GOAL `?w0 w1 w2 w3. ul'' = [w0;w1;w2;w3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (NEW_GOAL `i - 1 = 0`); (UNDISCH_TAC `i <= 1` THEN ARITH_TAC); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_0; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REWRITE_TAC[SET_RULE `a IN {b} <=> a = b`]); (UNDISCH_TAC `~(u' = v':real^3)`); (SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[ARITH_RULE `~(i <= 1) <=> 2 <= i`] THEN STRIP_TAC); (* ========================================================================== *) (NEW_GOAL `?p. p permutes 0..i - 1 /\ initial_sublist [u:real^3; v] (left_action_list p ul'')`); (NEW_GOAL `?w0 w1 w2 w3. ul'' = [w0;w1;w2;w3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_CASES_TAC `i = 2`); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (NEW_GOAL `i - 1 = 1`); (UNDISCH_TAC `i = 2` THEN ARITH_TAC); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `u = w0:real^3`); (NEW_GOAL `v = w1:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UNDISCH_TAC `~(u' = v':real^3)` THEN UNDISCH_TAC `{u,v} = {u',v':real^3}` THEN SET_TAC[]); (EXISTS_TAC `I:num->num`); (ASM_REWRITE_TAC[PERMUTES_I; Packing3.LEFT_ACTION_LIST_I]); (REWRITE_WITH `initial_sublist [w0; w1] [w0; w1; w2; w3] /\ LENGTH [w0;w1:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1; LENGTH] THEN ARITH_TAC); (NEW_GOAL `v = w0:real^3 /\ u = w1:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UNDISCH_TAC `~(u' = v':real^3)` THEN UNDISCH_TAC `{u,v} = {u',v':real^3}` THEN SET_TAC[]); (ASM_REWRITE_TAC[]); (NEW_GOAL `?p. p permutes 0..1 /\ [w1; w0; w2; w3:real^3] = left_action_list p ul''`); (MATCH_MP_TAC Qzksykg.TWO_REARRANGEMENT_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC); (EXISTS_TAC `p:num->num` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM (ASSUME `[w1; w0; w2; w3] = left_action_list p [w0; w1; w2; w3:real^3]`)]); (REWRITE_WITH `initial_sublist [w1; w0] [w1; w0; w2; w3] /\ LENGTH [w1;w0:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1; LENGTH] THEN ARITH_TAC); (ASM_CASES_TAC `i = 3`); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (ASM_REWRITE_TAC[ARITH_RULE `3 - 1 = 2`; TRUNCATE_SIMPLEX_EXPLICIT_2; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REPEAT STRIP_TAC); (NEW_GOAL `u IN {w0, w1, w2} /\ v IN {w0, w1, w2:real^3} /\ ~(u = v)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?a. {u,v, a} = {w0, w1, w2:real^3}`); (NEW_GOAL `?w:real^3. w IN {w0, w1, w2} DIFF {u, v}`); (REWRITE_TAC[SET_RULE `(?x. x IN s) <=> ~(s = {})`]); (REWRITE_WITH `{w0, w1, w2} DIFF {u, v:real^3} = {} <=> CARD ({w0, w1, w2} DIFF {u, v}) = 0`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC CARD_EQ_0); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{w0, w1, w2:real^3}`); (REWRITE_TAC[Geomdetail.FINITE6] THEN SET_TAC[]); (NEW_GOAL `CARD {w0, w1, w2} = CARD ({w0, w1, w2} DIFF {u, v:real^3}) + CARD {u, v}`); (MATCH_MP_TAC Hypermap.CARD_MINUS_DIFF_TWO_SET); (ASM_REWRITE_TAC[Geomdetail.FINITE6]); (UP_ASM_TAC THEN REWRITE_WITH `CARD ({w0, w1, w2:real^3}) = 3`); (ASM_CASES_TAC `CARD {w0,w1,w2:real^3} <= 2`); (NEW_GOAL `F`); (NEW_GOAL `CARD {w0, w1, w2, w3:real^3} = 4`); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `CARD {w0, w1, w2, w3} = CARD {w3, w0,w1,w2:real^3}`); (AP_TERM_TAC THEN SET_TAC[]); (NEW_GOAL `CARD ({w3, w0, w1, w2}) <= SUC (CARD {w0, w1, w2:real^3})`); (SIMP_TAC[Geomdetail.CARD_CLAUSES_IMP; Geomdetail.FINITE6]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ARITH_TAC); (ASM_MESON_TAC[]); (NEW_GOAL `CARD {w0, w1, w2:real^3} <= 3`); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ARITH_TAC); (SUBGOAL_THEN `CARD {u,v:real^3} <= 2` MP_TAC); (REWRITE_TAC[Geomdetail.CARD2]); (ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `w:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?p. p permutes 0..2 /\ [u; v; a; w3] = left_action_list p [w0; w1; w2; w3:real^3]`); (MATCH_MP_TAC LEFT_ACTION_LIST_2_EXISTS); (ASM_REWRITE_TAC[]); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `[u; v; a; w3:real^3] = left_action_list p [w0; w1; w2; w3]`)]); (REWRITE_WITH `initial_sublist [u; v] [u; v; a; w3] /\ LENGTH [u; v:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST; LENGTH; TRUNCATE_SIMPLEX_EXPLICIT_1] THEN ARITH_TAC); (NEW_GOAL `i = 4`); (ASM_ARITH_TAC); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (ASM_REWRITE_TAC[ARITH_RULE `4 - 1 = 3`; TRUNCATE_SIMPLEX_EXPLICIT_3; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REPEAT STRIP_TAC); (NEW_GOAL `u IN {w0, w1, w2, w3} /\ v IN {w0, w1, w2, w3:real^3} /\ ~(u = v)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?a b. {u,v,a,b} = {w0, w1, w2, w3:real^3}`); (NEW_GOAL `?w:real^3. w IN {w0, w1, w2, w3} DIFF {u, v}`); (REWRITE_TAC[SET_RULE `(?x. x IN s) <=> ~(s = {})`]); (REWRITE_WITH `{w0, w1, w2, w3} DIFF {u, v:real^3} = {} <=> CARD ({w0, w1, w2, w3} DIFF {u, v}) = 0`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC CARD_EQ_0); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{w0, w1, w2, w3:real^3}`); (REWRITE_TAC[Geomdetail.FINITE6] THEN SET_TAC[]); (NEW_GOAL `CARD {w0, w1, w2, w3} = CARD ({w0, w1, w2, w3} DIFF {u, v:real^3}) + CARD {u, v}`); (MATCH_MP_TAC Hypermap.CARD_MINUS_DIFF_TWO_SET); (ASM_REWRITE_TAC[Geomdetail.FINITE6]); (UP_ASM_TAC THEN REWRITE_WITH `CARD ({w0, w1, w2, w3:real^3}) = 3 + 1`); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `CARD {u, v:real^3} <= 2`); (REWRITE_TAC[Geomdetail.CARD2]); (UP_ASM_TAC THEN ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?m:real^3. m IN {w0, w1, w2, w3} DIFF {u, v, w}`); (REWRITE_TAC[SET_RULE `(?x. x IN s) <=> ~(s = {})`]); (REWRITE_WITH `{w0, w1, w2, w3} DIFF {u, v, w:real^3} = {} <=> CARD ({w0, w1, w2, w3} DIFF {u, v, w}) = 0`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC CARD_EQ_0); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{w0, w1, w2, w3:real^3}`); (REWRITE_TAC[Geomdetail.FINITE6] THEN SET_TAC[]); (NEW_GOAL `CARD ({w0, w1, w2, w3} DIFF {u,v,w:real^3}) = CARD {w0, w1, w2, w3} - CARD {u,v,w}`); (MATCH_MP_TAC CARD_DIFF); (ASM_REWRITE_TAC[Geomdetail.FINITE6]); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN REWRITE_WITH `CARD ({w0, w1, w2, w3:real^3}) = 3 + 1`); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `CARD {u, v, w:real^3} <= 3`); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `w:real^3` THEN EXISTS_TAC `m:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?p. p permutes 0..3 /\ [u; v; a; b] = left_action_list p [w0; w1; w2; w3:real^3]`); (MATCH_MP_TAC LEFT_ACTION_LIST_3_EXISTS); (ASM_REWRITE_TAC[]); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `[u; v; a; b:real^3] = left_action_list p [w0; w1; w2; w3]`)]); (REWRITE_WITH `initial_sublist [u; v] [u; v; a; b] /\ LENGTH [u; v:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST; LENGTH; TRUNCATE_SIMPLEX_EXPLICIT_1] THEN ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (* ========================================================================== *) (NEW_GOAL `?q. q permutes 0..i - 1 /\ initial_sublist [v:real^3; u] (left_action_list q ul'')`); (NEW_GOAL `?w0 w1 w2 w3. ul'' = [w0;w1;w2;w3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_CASES_TAC `i = 2`); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (NEW_GOAL `i - 1 = 1`); (UNDISCH_TAC `i = 2` THEN ARITH_TAC); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `v = w0:real^3`); (NEW_GOAL `u = w1:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UNDISCH_TAC `~(u' = v':real^3)` THEN UNDISCH_TAC `{u,v} = {u',v':real^3}` THEN SET_TAC[]); (EXISTS_TAC `I:num->num`); (ASM_REWRITE_TAC[PERMUTES_I; Packing3.LEFT_ACTION_LIST_I]); (REWRITE_WITH `initial_sublist [w0; w1] [w0; w1; w2; w3] /\ LENGTH [w0;w1:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1; LENGTH] THEN ARITH_TAC); (NEW_GOAL `u = w0:real^3 /\ v = w1:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UNDISCH_TAC `~(u' = v':real^3)` THEN UNDISCH_TAC `{u,v} = {u',v':real^3}` THEN SET_TAC[]); (ASM_REWRITE_TAC[]); (NEW_GOAL `?q. q permutes 0..1 /\ [w1; w0; w2; w3:real^3] = left_action_list q ul''`); (MATCH_MP_TAC Qzksykg.TWO_REARRANGEMENT_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC); (EXISTS_TAC `q:num->num` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM (ASSUME `[w1; w0; w2; w3] = left_action_list q [w0; w1; w2; w3:real^3]`)]); (REWRITE_WITH `initial_sublist [w1; w0] [w1; w0; w2; w3] /\ LENGTH [w1;w0:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_1; LENGTH] THEN ARITH_TAC); (ASM_CASES_TAC `i = 3`); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (ASM_REWRITE_TAC[ARITH_RULE `3 - 1 = 2`; TRUNCATE_SIMPLEX_EXPLICIT_2; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REPEAT STRIP_TAC); (NEW_GOAL `u IN {w0, w1, w2} /\ v IN {w0, w1, w2:real^3} /\ ~(u = v)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?a. {u,v, a} = {w0, w1, w2:real^3}`); (NEW_GOAL `?w:real^3. w IN {w0, w1, w2} DIFF {u, v}`); (REWRITE_TAC[SET_RULE `(?x. x IN s) <=> ~(s = {})`]); (REWRITE_WITH `{w0, w1, w2} DIFF {u, v:real^3} = {} <=> CARD ({w0, w1, w2} DIFF {u, v}) = 0`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC CARD_EQ_0); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{w0, w1, w2:real^3}`); (REWRITE_TAC[Geomdetail.FINITE6] THEN SET_TAC[]); (NEW_GOAL `CARD {w0, w1, w2} = CARD ({w0, w1, w2} DIFF {u, v:real^3}) + CARD {u, v}`); (MATCH_MP_TAC Hypermap.CARD_MINUS_DIFF_TWO_SET); (ASM_REWRITE_TAC[Geomdetail.FINITE6]); (UP_ASM_TAC THEN REWRITE_WITH `CARD ({w0, w1, w2:real^3}) = 3`); (ASM_CASES_TAC `CARD {w0,w1,w2:real^3} <= 2`); (NEW_GOAL `F`); (NEW_GOAL `CARD {w0, w1, w2, w3:real^3} = 4`); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `CARD {w0, w1, w2, w3} = CARD {w3, w0,w1,w2:real^3}`); (AP_TERM_TAC THEN SET_TAC[]); (NEW_GOAL `CARD ({w3, w0, w1, w2}) <= SUC (CARD {w0, w1, w2:real^3})`); (SIMP_TAC[Geomdetail.CARD_CLAUSES_IMP; Geomdetail.FINITE6]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ARITH_TAC); (ASM_MESON_TAC[]); (NEW_GOAL `CARD {w0, w1, w2:real^3} <= 3`); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN UP_ASM_TAC THEN ARITH_TAC); (SUBGOAL_THEN `CARD {u,v:real^3} <= 2` MP_TAC); (REWRITE_TAC[Geomdetail.CARD2]); (ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `w:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UP_ASM_TAC THEN REWRITE_WITH `{u,v,a} = {v, u, a:real^3}`); (SET_TAC[]); (STRIP_TAC); (NEW_GOAL `?q. q permutes 0..2 /\ [v; u; a; w3] = left_action_list q [w0; w1; w2; w3:real^3]`); (MATCH_MP_TAC LEFT_ACTION_LIST_2_EXISTS); (ASM_REWRITE_TAC[]); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `q:num->num`); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `[v; u; a; w3:real^3] = left_action_list q [w0; w1; w2; w3]`)]); (REWRITE_WITH `initial_sublist [v; u] [v; u; a; w3] /\ LENGTH [v; u:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST; LENGTH; TRUNCATE_SIMPLEX_EXPLICIT_1] THEN ARITH_TAC); (NEW_GOAL `i = 4`); (ASM_ARITH_TAC); (UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (v':real^3)` THEN UNDISCH_TAC `set_of_list (truncate_simplex (i - 1) ul'') (u':real^3)`); (ASM_REWRITE_TAC[ARITH_RULE `4 - 1 = 3`; TRUNCATE_SIMPLEX_EXPLICIT_3; set_of_list; MESON[IN] `(A:real^3->bool) b <=> b IN A`]); (REPEAT STRIP_TAC); (NEW_GOAL `u IN {w0, w1, w2, w3} /\ v IN {w0, w1, w2, w3:real^3} /\ ~(u = v)`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?a b. {u,v,a,b} = {w0, w1, w2, w3:real^3}`); (NEW_GOAL `?w:real^3. w IN {w0, w1, w2, w3} DIFF {u, v}`); (REWRITE_TAC[SET_RULE `(?x. x IN s) <=> ~(s = {})`]); (REWRITE_WITH `{w0, w1, w2, w3} DIFF {u, v:real^3} = {} <=> CARD ({w0, w1, w2, w3} DIFF {u, v}) = 0`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC CARD_EQ_0); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{w0, w1, w2, w3:real^3}`); (REWRITE_TAC[Geomdetail.FINITE6] THEN SET_TAC[]); (NEW_GOAL `CARD {w0, w1, w2, w3} = CARD ({w0, w1, w2, w3} DIFF {u, v:real^3}) + CARD {u, v}`); (MATCH_MP_TAC Hypermap.CARD_MINUS_DIFF_TWO_SET); (ASM_REWRITE_TAC[Geomdetail.FINITE6]); (UP_ASM_TAC THEN REWRITE_WITH `CARD ({w0, w1, w2, w3:real^3}) = 3 + 1`); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `CARD {u, v:real^3} <= 2`); (REWRITE_TAC[Geomdetail.CARD2]); (UP_ASM_TAC THEN ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?m:real^3. m IN {w0, w1, w2, w3} DIFF {u, v, w}`); (REWRITE_TAC[SET_RULE `(?x. x IN s) <=> ~(s = {})`]); (REWRITE_WITH `{w0, w1, w2, w3} DIFF {u, v, w:real^3} = {} <=> CARD ({w0, w1, w2, w3} DIFF {u, v, w}) = 0`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC CARD_EQ_0); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{w0, w1, w2, w3:real^3}`); (REWRITE_TAC[Geomdetail.FINITE6] THEN SET_TAC[]); (NEW_GOAL `CARD ({w0, w1, w2, w3} DIFF {u,v,w:real^3}) = CARD {w0, w1, w2, w3} - CARD {u,v,w}`); (MATCH_MP_TAC CARD_DIFF); (ASM_REWRITE_TAC[Geomdetail.FINITE6]); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN REWRITE_WITH `CARD ({w0, w1, w2, w3:real^3}) = 3 + 1`); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (NEW_GOAL `CARD {u, v, w:real^3} <= 3`); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `w:real^3` THEN EXISTS_TAC `m:real^3`); (UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UP_ASM_TAC THEN REWRITE_WITH `{u,v,a,b} = {v, u, a, b:real^3}`); (SET_TAC[]); (STRIP_TAC); (NEW_GOAL `?q. q permutes 0..3 /\ [v; u; a; b] = left_action_list q [w0; w1; w2; w3:real^3]`); (MATCH_MP_TAC LEFT_ACTION_LIST_3_EXISTS); (ASM_REWRITE_TAC[]); (REWRITE_WITH `{w0, w1, w2, w3:real^3} = set_of_list ul''`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `LENGTH (ul'':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul'') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (EXISTS_TAC `q:num->num`); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `[v; u; a; b:real^3] = left_action_list q [w0; w1; w2; w3]`)]); (REWRITE_WITH `initial_sublist [v; u] [v; u; a; b] /\ LENGTH [v; u:real^3] = 1 + 1`); (REWRITE_TAC[GSYM Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST; LENGTH; TRUNCATE_SIMPLEX_EXPLICIT_1] THEN ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (* ========================================================================= *) (ABBREV_TAC `P = (\(k, ul). k <= 4 /\ ul IN barV V 3 /\ X = mcell k V ul /\ initial_sublist [v; u] ul)`); (NEW_GOAL `(P:num#(real^3)list->bool) (k, ul)`); (REWRITE_TAC[GSYM (ASSUME `(@) (P:num#(real^3)list->bool) = k,ul`)]); (MATCH_MP_TAC SELECT_AX); (EXISTS_TAC `(i:num, left_action_list q (ul'':(real^3)list))`); (EXPAND_TAC "P" THEN REWRITE_TAC[BETA_THM]); (ASM_REWRITE_TAC[IN]); (REPEAT STRIP_TAC); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `ul'':(real^3)list`); (EXISTS_TAC `i:num` THEN EXISTS_TAC `q:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_REWRITE_TAC[GSYM Ajripqn.UP_TO_4_KY_LEMMA]); (REWRITE_TAC[GSYM (ASSUME `X = mcell i V ul''`)]); (UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Rvfxzbu.RVFXZBU); (ASM_REWRITE_TAC[GSYM Ajripqn.UP_TO_4_KY_LEMMA]); (ABBREV_TAC `Q = (\(k, ul). k <= 4 /\ ul IN barV V 3 /\ X = mcell k V ul /\ initial_sublist [u; v] ul)`); (NEW_GOAL `(Q:num#(real^3)list->bool) (k', ul')`); (REWRITE_TAC[GSYM (ASSUME `(@) (Q:num#(real^3)list->bool) = k',ul'`)]); (MATCH_MP_TAC SELECT_AX); (EXISTS_TAC `(i:num, left_action_list p (ul'':(real^3)list))`); (EXPAND_TAC "Q" THEN REWRITE_TAC[BETA_THM]); (ASM_REWRITE_TAC[IN]); (REPEAT STRIP_TAC); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `ul'':(real^3)list`); (EXISTS_TAC `i:num` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_REWRITE_TAC[GSYM Ajripqn.UP_TO_4_KY_LEMMA]); (REWRITE_TAC[GSYM (ASSUME `X = mcell i V ul''`)]); (UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Rvfxzbu.RVFXZBU); (ASM_REWRITE_TAC[GSYM Ajripqn.UP_TO_4_KY_LEMMA]); (UP_ASM_TAC THEN EXPAND_TAC "Q" THEN REWRITE_TAC[] THEN DEL_TAC); (UP_ASM_TAC THEN EXPAND_TAC "P" THEN REWRITE_TAC[IN] THEN DEL_TAC); (REPEAT STRIP_TAC); (NEW_GOAL `i = k:num`); (REWRITE_WITH `i = k /\ mcell i V ul'' = mcell k V ul`); (MATCH_MP_TAC Ajripqn.AJRIPQN); (ASM_REWRITE_TAC[GSYM Ajripqn.UP_TO_4_KY_LEMMA]); (REWRITE_TAC[GSYM (ASSUME `X = mcell i V ul''`); GSYM (ASSUME `X = mcell k V ul`); SET_RULE `a INTER a = a`]); (ASM_REWRITE_TAC[]); (NEW_GOAL `i = k':num`); (REWRITE_WITH `i = k' /\ mcell i V ul'' = mcell k' V ul'`); (MATCH_MP_TAC Ajripqn.AJRIPQN); (ASM_REWRITE_TAC[GSYM Ajripqn.UP_TO_4_KY_LEMMA]); (REWRITE_TAC[GSYM (ASSUME `X = mcell i V ul''`); GSYM (ASSUME `i = k:num`); GSYM (ASSUME `X = mcell k' V ul'`); SET_RULE `a INTER a = a`]); (ASM_REWRITE_TAC[]); (NEW_GOAL `i = k /\ (!k. i - 1 <= k /\ k <= 3 ==> omega_list_n V ul k = omega_list_n V ul' k)`); (MATCH_MP_TAC MCELL_ID_OMEGA_LIST_N); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_MESON_TAC[]); (STRIP_TAC); (ASM_MESON_TAC[]); (REWRITE_TAC[SET_RULE `a IN {m,n,p} <=> a = m \/ a = n \/ a = p`]); (ASM_ARITH_TAC); (UP_ASM_TAC THEN STRIP_TAC); (REWRITE_WITH `k' = k:num`); (ASM_MESON_TAC[]); (ASM_CASES_TAC `i < 4`); (NEW_GOAL `mxi V ul' = mxi V ul`); (MATCH_MP_TAC MCELL_ID_MXI_2); (EXISTS_TAC `k':num` THEN EXISTS_TAC `k:num`); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `X = mcell k' V ul'`)]); (REWRITE_TAC[ASSUME `X = mcell k V ul`; ASSUME `~NULLSET X`]); (REWRITE_TAC[SET_RULE `a IN {b,c} <=> a = b \/ a = c`]); (ASM_ARITH_TAC); (NEW_GOAL `HD ul = HD [v; u:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_INITIAL_SUBLIST); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (NEW_GOAL `HD ul' = HD [u; v:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_INITIAL_SUBLIST); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (NEW_GOAL `EL 1 ul = EL 1 ([v; u:real^3])`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_WITH `[v; u:real^3] = truncate_simplex 1 ul`); (REWRITE_WITH `truncate_simplex 1 ul = [v;u:real^3] /\ 1 + 1 <= LENGTH ul`); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (MATCH_MP_TAC Packing3.EL_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `EL 1 ul' = EL 1 ([u; v:real^3])`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_WITH `[u; v:real^3] = truncate_simplex 1 ul'`); (REWRITE_WITH `truncate_simplex 1 ul' =[u;v:real^3] /\ 1 + 1 <= LENGTH ul'`); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (MATCH_MP_TAC Packing3.EL_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (COND_CASES_TAC); (COND_CASES_TAC); (REWRITE_TAC[dihu2]); (REWRITE_WITH `omega_list_n V ul 3 = omega_list_n V ul' 3`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_ARITH_TAC); (ASM_REWRITE_TAC[EL; HD; TL; ARITH_RULE `1 = SUC 0`; DIHV_SYM]); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (DEL_TAC THEN DEL_TAC); (REWRITE_TAC[dihu3]); (ASM_REWRITE_TAC[EL;HD;TL; ARITH_RULE `1 = SUC 0`]); (REWRITE_WITH `EL 2 ul' = EL 2 (ul:(real^3)list)`); (NEW_GOAL `set_of_list (truncate_simplex 2 (ul:(real^3)list)) = set_of_list (truncate_simplex 2 ul')`); (REWRITE_TAC[ARITH_RULE `2 = 3 - 1`]); (REWRITE_WITH `set_of_list (truncate_simplex (3 - 1) ul) = (V:real^3->bool) INTER X`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[ASSUME `X = mcell k V ul`; ASSUME `k = 3`]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[ARITH_RULE `1 <= 3 /\ 3 <= 4`]); (REWRITE_TAC[GSYM (ASSUME `X = mcell k V ul`); GSYM (ASSUME `k = 3`)]); (UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (REWRITE_TAC[ASSUME `X = mcell k' V ul'`]); (REWRITE_WITH `k' = 3`); (ASM_ARITH_TAC); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[ARITH_RULE `1 <= 3 /\ 3 <= 4`]); (REWRITE_WITH `3 = k'`); (ASM_ARITH_TAC); (REWRITE_TAC[GSYM (ASSUME `X = mcell k' V ul'`)]); (UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (NEW_GOAL `?y0 y1 y2 y3. ul = [y0;y1;y2;y3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?z0 z1 z2 z3. ul' = [z0;z1;z2;z3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UNDISCH_TAC `EL 1 ul = EL 1 [v; u:real^3]` THEN UNDISCH_TAC `HD ul = HD [v; u:real^3]` THEN ASM_REWRITE_TAC[EL;HD;TL; ARITH_RULE `2 = SUC 1 /\ 1 = SUC 0`]); (REPEAT STRIP_TAC); (UNDISCH_TAC `EL 1 ul' = EL 1 [u; v:real^3]` THEN UNDISCH_TAC `HD ul' = HD [u; v:real^3]` THEN ASM_REWRITE_TAC[EL;HD;TL; ARITH_RULE `2 = SUC 1 /\ 1 = SUC 0`]); (REPEAT STRIP_TAC); (UNDISCH_TAC `set_of_list (truncate_simplex 2 (ul:(real^3)list)) = set_of_list (truncate_simplex 2 ul')`); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_2; set_of_list]); (NEW_GOAL `~(y2 IN {u, v:real^3})`); (STRIP_TAC); (NEW_GOAL `CARD (set_of_list (ul:(real^3)list)) <= 3`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH ` {v, u, y2, y3} = {v,u, y3:real^3}`); (ASM_SET_TAC[]); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN REWRITE_WITH`CARD (set_of_list (ul:(real^3)list)) = 3 + 1`); (MATCH_MP_TAC BARV_CARD_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[DIHV_SYM]); (UP_ASM_TAC THEN MESON_TAC[]); (NEW_GOAL `F`); (ASM_ARITH_TAC); (ASM_MESON_TAC[]); (NEW_GOAL `k = 4`); (ASM_ARITH_TAC); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_ARITH_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (NEW_GOAL `F`); (ASM_ARITH_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (COND_CASES_TAC); (ASM_MESON_TAC[]); (COND_CASES_TAC); (ASM_MESON_TAC[]); (COND_CASES_TAC); (REWRITE_TAC[dihu4]); (* ========================================================================= *) (MATCH_MP_TAC DIHV_SYM_3); (REWRITE_TAC[EL]); (NEW_GOAL `HD ul = HD [v; u:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_INITIAL_SUBLIST); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (NEW_GOAL `HD ul' = HD [u; v:real^3]`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Packing3.HD_INITIAL_SUBLIST); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (NEW_GOAL `EL 1 ul = EL 1 ([v; u:real^3])`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_WITH `[v; u:real^3] = truncate_simplex 1 ul`); (REWRITE_WITH `truncate_simplex 1 ul = [v;u:real^3] /\ 1 + 1 <= LENGTH ul`); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (MATCH_MP_TAC Packing3.EL_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1 /\ CARD (set_of_list ul) = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `EL 1 ul' = EL 1 ([u; v:real^3])`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_WITH `[u; v:real^3] = truncate_simplex 1 ul'`); (REWRITE_WITH `truncate_simplex 1 ul' =[u;v:real^3] /\ 1 + 1 <= LENGTH ul'`); (REWRITE_TAC[Packing3.TRUNCATE_SIMPLEX_INITIAL_SUBLIST]); (ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC); (MATCH_MP_TAC Packing3.EL_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul':(real^3)list) = 3 + 1 /\ CARD (set_of_list ul') = 3 + 1`); (MATCH_MP_TAC Rogers.BARV_IMP_LENGTH_EQ_CARD); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `?y0 y1 y2 y3. ul = [y0;y1;y2;y3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (NEW_GOAL `?z0 z1 z2 z3. ul' = [z0;z1;z2;z3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (UNDISCH_TAC `EL 1 ul = EL 1 [v; u:real^3]` THEN UNDISCH_TAC `HD ul = HD [v; u:real^3]` THEN UNDISCH_TAC `EL 1 ul' = EL 1 [u; v:real^3]` THEN UNDISCH_TAC `HD ul' = HD [u; v:real^3]` THEN ASM_REWRITE_TAC[EL;HD;TL; ARITH_RULE `3 = SUC 2 /\2 = SUC 1 /\ 1 = SUC 0`]); (REPEAT STRIP_TAC); (ASM_SET_TAC[]); (NEW_GOAL `~(z2 IN {u,v:real^3})`); (STRIP_TAC); (NEW_GOAL `CARD (set_of_list (ul':(real^3)list)) <= 3`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `{u, v, z2, z3} = {v,u, z3:real^3}`); (ASM_SET_TAC[]); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN REWRITE_WITH`CARD (set_of_list (ul':(real^3)list)) = 3 + 1`); (MATCH_MP_TAC BARV_CARD_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `~(z3 IN {u,v:real^3})`); (STRIP_TAC); (NEW_GOAL `CARD (set_of_list (ul':(real^3)list)) <= 3`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `{u, v, z2, z3} = {v,u, z2:real^3}`); (ASM_SET_TAC[]); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN REWRITE_WITH`CARD (set_of_list (ul':(real^3)list)) = 3 + 1`); (MATCH_MP_TAC BARV_CARD_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `~(y2 IN {u,v:real^3})`); (STRIP_TAC); (NEW_GOAL `CARD (set_of_list (ul:(real^3)list)) <= 3`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `{v, u, y2, y3} = {v,u, y3:real^3}`); (ASM_SET_TAC[]); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN REWRITE_WITH`CARD (set_of_list (ul:(real^3)list)) = 3 + 1`); (MATCH_MP_TAC BARV_CARD_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `~(y3 IN {u,v:real^3})`); (STRIP_TAC); (NEW_GOAL `CARD (set_of_list (ul:(real^3)list)) <= 3`); (ASM_REWRITE_TAC[set_of_list]); (REWRITE_WITH `{v, u, y2, y3} = {v,u, y2:real^3}`); (ASM_SET_TAC[]); (REWRITE_TAC[Geomdetail.CARD3]); (UP_ASM_TAC THEN REWRITE_WITH`CARD (set_of_list (ul:(real^3)list)) = 3 + 1`); (MATCH_MP_TAC BARV_CARD_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (NEW_GOAL `set_of_list (truncate_simplex 3 (ul:(real^3)list)) = set_of_list (truncate_simplex 3 ul')`); (REWRITE_WITH `set_of_list (truncate_simplex 3 (ul:(real^3)list)) = V INTER X`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (REWRITE_TAC[ASSUME `X = mcell k V ul`; ASSUME `k = 4`; ARITH_RULE `3 = 4 - 1`]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (REWRITE_TAC[ARITH_RULE `1 <= 4 /\ 4 <= 4`]); (REWRITE_TAC[GSYM (ASSUME `X = mcell k V ul`); GSYM (ASSUME `k = 4`)]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (REWRITE_TAC[ASSUME `X = mcell k' V ul'`]); (REWRITE_WITH `k' = 4`); (ASM_ARITH_TAC); (REWRITE_TAC[ARITH_RULE `3 = 4 - 1`]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (REWRITE_TAC[ARITH_RULE `1 <= 4 /\ 4 <= 4`]); (REWRITE_WITH `4 = k'`); (ASM_ARITH_TAC); (REWRITE_TAC[GSYM (ASSUME `X = mcell k' V ul'`)]); (REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[]); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN UNDISCH_TAC `~NULLSET X` THEN MESON_TAC[NEGLIGIBLE_EMPTY]); (UP_ASM_TAC); (ASM_REWRITE_TAC[TRUNCATE_SIMPLEX_EXPLICIT_3; set_of_list]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (REFL_TAC)]);; (* ==================================================================== *) (* Lemma 49 *) let gammaY = new_definition `!V X f. gammaY V X f = (\({u, v}). if {u, v} IN edgeX V X then dihX V X (u,v) * f (hl [u; v]) else &0)`;; let gamma_y_lmfun_bound = prove_by_refinement ( `!V. packing V /\ saturated V ==> (?d. !X e. mcell_set V X /\ edgeX V X e ==> gammaY V X lmfun e <= d)`, [(REPEAT STRIP_TAC); (EXISTS_TAC `pi * h0 / (h0 - &1)`); (REPEAT STRIP_TAC); (ABBREV_TAC `f = (\u v. if {u, v} IN edgeX V X then dihX V X (u,v) * lmfun (hl [u; v]) else &0)`); (REWRITE_WITH `gammaY V X lmfun = (\({u, v}). f u v)`); (EXPAND_TAC "f" THEN REWRITE_TAC[gammaY]); (NEW_GOAL `!u v. (f:real^3->real^3->real) u v = f v u`); (EXPAND_TAC "f" THEN REWRITE_TAC[BETA_THM]); (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]); (REPEAT GEN_TAC THEN COND_CASES_TAC); (COND_CASES_TAC); (REWRITE_WITH `dihX V X (u,v) = dihX V X (v,u)`); (MATCH_MP_TAC DIHX_SYM); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (REFL_TAC); (UNDISCH_TAC `edgeX V X e` THEN REWRITE_TAC[IN;IN_ELIM_THM; edgeX]); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_WITH `(\({u, v}). f u v) {u, v} = (f:real^3->real^3->real) u v`); (MATCH_MP_TAC BETA_PAIR_THM); (ASM_REWRITE_TAC[]); (EXPAND_TAC "f"); (COND_CASES_TAC); (NEW_GOAL `dihX V X (u,v) * lmfun (hl [u;v]) <= pi * lmfun (hl [u;v])`); (REWRITE_TAC[REAL_ARITH `a * x <= b * x <=> &0 <= (b - a) * x`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[lmfun_pos_le; REAL_ARITH `&0 <= a - b <=> b <= a`; DIHX_LE_PI]); (NEW_GOAL `pi * lmfun (hl [u:real^3;v]) <= pi * h0 / (h0 - &1)`); (REWRITE_TAC[REAL_ARITH `a * x <= a * y <=> &0 <= (y - x) * a`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[PI_POS_LE; REAL_ARITH `&0 <= a - b <=> b <= a`]); (MATCH_MP_TAC lmfun_bounded); (REWRITE_TAC[HL_2; REAL_ARITH `&0 <= inv (&2) * x <=> &0 <= x`; DIST_POS_LE]); (UP_ASM_TAC THEN UP_ASM_TAC THEN REAL_ARITH_TAC); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[PI_POS_LE]); (MATCH_MP_TAC REAL_LE_DIV); (REWRITE_TAC[h0] THEN REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 50 *) let BETA_SET_2_THM = prove (`!g:(A->bool)->B u0 v0. (\({u, v}). g {u, v}) {u0, v0} = g {u0, v0}`, GEN_TAC THEN REWRITE_TAC[GABS_DEF;GEQ_DEF] THEN CONV_TAC SELECT_CONV THEN EXISTS_TAC `g:(A->bool)->B` THEN REWRITE_TAC[]);; (* ==================================================================== *) (* Lemma 51 *) (* Formal proof by John Harrison *) let SUM_PAIR_2_SET = prove (`!f s:real^N->bool d. FINITE s ==> sum {(m,n) | m IN s /\ n IN s /\ ~(m = n) /\ dist(m,n) <= d} (\(m,n). f {m, n}) = &2 * sum {{m, n} | m IN s /\ n IN s /\ ~(m = n) /\ dist(m,n) <= d} f`, REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`\(m:real^N,n). {m,n}`;`(\(m,n). f {m,n}):real^N#real^N->real`; `{(m,n) | m IN s /\ n IN s /\ ~(m:real^N = n) /\ dist(m,n) <= d}`; `{{m,n} | m IN s /\ n IN s /\ ~(m:real^N = n) /\ dist(m,n) <= d}`] SUM_GROUP) THEN ANTS_TAC THENL [CONJ_TAC THENL [ONCE_REWRITE_TAC[SET_RULE `m IN s /\ n IN s /\ P m n <=> m IN s /\ n IN {n | n IN s /\ P m n}`] THEN MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_SIMP_TAC[FINITE_RESTRICT]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN SET_TAC[]]; DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN MAP_EVERY X_GEN_TAC [`m:real^N`; `n:real^N`] THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum {(m:real^N,n), (n,m)} (\(m,n). f {m,n})` THEN CONJ_TAC THENL [AP_THM_TAC THEN AP_TERM_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[FORALL_PAIR_THM] THEN REWRITE_TAC[IN_ELIM_THM; PAIR_EQ; IN_INSERT; NOT_IN_EMPTY; SET_RULE `{a,b} = {c,d} <=> a = c /\ b = d \/ a = d /\ b = c`] THEN ASM_MESON_TAC[DIST_SYM]; SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ASM_REWRITE_TAC[IN_SING; NOT_IN_EMPTY; PAIR_EQ; REAL_ADD_RID] THEN REWRITE_TAC[INSERT_AC] THEN REAL_ARITH_TAC]]);; (* ==================================================================== *) (* Lemma 52 *) open Flyspeck_constants;; let H0_LT_SQRT2 = prove_by_refinement (`h0 < sqrt (&2)`, [REWRITE_TAC[h0; GSYM sqrt2]; NEW_GOAL `#1.414213 < sqrt2`; REWRITE_TAC[bounds]; ASM_REAL_ARITH_TAC]);; (* ==================================================================== *) (* Lemma 53 *) let gamma_y_pos_le = prove_by_refinement ( `!V X e. packing V /\ saturated V /\ mcell_set V X /\ edgeX V X e ==> &0 <= gammaY V X lmfun e`, [(REPEAT STRIP_TAC); (REWRITE_TAC[gammaY]); (NEW_GOAL `?u v. (VX V X u /\ VX V X v /\ ~(u = v)) /\ e = {u, v}`); (UP_ASM_TAC THEN REWRITE_TAC[edgeX; IN_ELIM_THM]); (UP_ASM_TAC THEN STRIP_TAC); (ASM_REWRITE_TAC[]); (ABBREV_TAC `f = (\u v. if {u, v} IN edgeX V X then dihX V X (u,v) * lmfun (hl [u; v]) else &0)`); (REWRITE_WITH `(\({u, v}). if {u, v} IN edgeX V X then dihX V X (u,v) * lmfun (hl [u; v]) else &0) = (\({u, v}). f u v)`); (EXPAND_TAC "f" THEN REWRITE_TAC[]); (REWRITE_WITH `(\({u:real^3, v}). f u v) {u, v} = (f:real^3->real^3->real) u v`); (MATCH_MP_TAC BETA_PAIR_THM); (EXPAND_TAC "f" THEN REPEAT STRIP_TAC); (REPEAT COND_CASES_TAC); (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]); (REWRITE_WITH `dihX V X (u',v') = dihX V X (v',u')`); (MATCH_MP_TAC DIHX_SYM); (ASM_REWRITE_TAC[]); (NEW_GOAL `F`); (UP_ASM_TAC THEN REWRITE_TAC[]); (ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (NEW_GOAL `F`); (UP_ASM_TAC THEN REWRITE_TAC[]); (ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (REFL_TAC); (EXPAND_TAC "f"); (COND_CASES_TAC); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[DIHX_POS;lmfun_pos_le]); (REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 54 *) let CARD_LIST_klemma = prove_by_refinement ( `!s t. FINITE (s:A->bool) /\ FINITE t ==> CARD {CONS u0 y1 | u0 IN s /\ y1 IN t} = CARD s * CARD t`, [(REPEAT STRIP_TAC); (ABBREV_TAC `S = {u0:A, y1:(A)list| u0 IN s /\ y1 IN t}`); (ABBREV_TAC `f = (\(u0:A, y1:(A)list). CONS u0 y1)`); (REWRITE_WITH `{CONS u0 y1 | u0 IN s /\ y1 IN t} = (IMAGE (f:A#(A)list->(A)list) S)`); (EXPAND_TAC "f" THEN EXPAND_TAC "S" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[SET_RULE `A = B <=> A SUBSET B /\ B SUBSET A`]); (STRIP_TAC); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `(u0:A,y1:(A)list)`); (ASM_REWRITE_TAC[]); (EXISTS_TAC `u0:A` THEN EXISTS_TAC `y1:(A)list` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[ASSUME `x' = u0:A,y1:(A)list`] THEN STRIP_TAC); (EXISTS_TAC `u0:A` THEN EXISTS_TAC `y1:(A)list` THEN ASM_REWRITE_TAC[]); (REWRITE_WITH `CARD (IMAGE (f:A#(A)list->(A)list) S) = CARD S`); (MATCH_MP_TAC CARD_IMAGE_INJ); (STRIP_TAC); (EXPAND_TAC "S" THEN EXPAND_TAC "f" THEN REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[ASSUME `x = u0:A,y1:(A)list`; ASSUME `y = u0':A,y1':(A)list`; CONS_11]); (STRIP_TAC THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "S" THEN MATCH_MP_TAC FINITE_PRODUCT THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "S" THEN MATCH_MP_TAC CARD_PRODUCT THEN ASM_REWRITE_TAC[])]);; (* ==================================================================== *) (* Lemma 55 *) let CARD_LIST_klemma_2 = prove_by_refinement ( `!s t. FINITE (s:A->bool) /\ FINITE t ==> CARD {[u0;y1] | u0 IN s /\ y1 IN t} = CARD s * CARD t`, [(REPEAT STRIP_TAC); (ABBREV_TAC `S = {u0:A, y1:A| u0 IN s /\ y1 IN t}`); (ABBREV_TAC `f = (\(u0:A, y1:A). [u0; y1])`); (REWRITE_WITH `{[u0; y1] | u0 IN s /\ y1 IN t} = (IMAGE (f:A#A->(A)list) S)`); (EXPAND_TAC "f" THEN EXPAND_TAC "S" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[SET_RULE `A = B <=> A SUBSET B /\ B SUBSET A`]); (STRIP_TAC); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `(u0:A,y1:A)`); (ASM_REWRITE_TAC[]); (EXISTS_TAC `u0:A` THEN EXISTS_TAC `y1:A` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[ASSUME `x' = u0:A,y1:A`] THEN STRIP_TAC); (EXISTS_TAC `u0:A` THEN EXISTS_TAC `y1:A` THEN ASM_REWRITE_TAC[]); (REWRITE_WITH `CARD (IMAGE (f:A#A->(A)list) S) = CARD S`); (MATCH_MP_TAC CARD_IMAGE_INJ); (STRIP_TAC); (EXPAND_TAC "S" THEN EXPAND_TAC "f" THEN REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (UP_ASM_TAC THEN REWRITE_TAC[ASSUME `x = u0:A,y1:A`; ASSUME `y = u0':A,y1':A`]); (REWRITE_TAC[PAIR_EQ]); (STRIP_TAC); (REWRITE_WITH `u0 = HD [u0':A; y1']`); (UP_ASM_TAC THEN MESON_TAC[HD]); (REWRITE_TAC[HD]); (REWRITE_WITH `y1 = HD (TL [u0':A; y1'])`); (UP_ASM_TAC THEN MESON_TAC[HD; TL]); (REWRITE_TAC[HD; TL]); (EXPAND_TAC "S" THEN MATCH_MP_TAC FINITE_PRODUCT THEN ASM_REWRITE_TAC[]); (EXPAND_TAC "S" THEN MATCH_MP_TAC CARD_PRODUCT THEN ASM_REWRITE_TAC[])]);; (* ==================================================================== *) (* Lemma 56 *) let FINITE_LIST_klemma = prove ( `!s t. FINITE (s:A->bool) /\ FINITE t ==> FINITE {CONS u0 y1 | u0 IN s /\ y1 IN t}`, REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_REWRITE_TAC[]);; (* ==================================================================== *) (* Lemma 57 *) let CARD_LIST_4_klemma = prove_by_refinement ( `!s:A->bool. FINITE s ==> CARD {y | ?u0 u1 u2 u3. u0 IN s /\ u1 IN s /\ u2 IN s /\ u3 IN s /\ y = [u0; u1; u2; u3]} = CARD s * CARD s * CARD s * CARD s`, [(REPEAT STRIP_TAC); (REWRITE_TAC[SET_RULE `{y | ?u0 u1 u2 u3. u0 IN s /\ u1 IN s /\ u2 IN s /\ u3 IN s /\ y = [u0; u1; u2; u3]} = {CONS u0 y1 | u0 IN s /\ y1 IN {CONS u1 y2 | u1 IN s /\ y2 IN {[u2;u3] | u2 IN s /\ u3 IN s}}}`]); (NEW_GOAL `FINITE {[u2:A; u3] | u2 IN s /\ u3 IN s}`); (MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_REWRITE_TAC[]); (NEW_GOAL `FINITE {CONS u1 y2 | (u1:A) IN s /\ y2 IN {[u2; u3] | u2 IN s /\ u3 IN s}}`); (MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_REWRITE_TAC[]); (NEW_GOAL `FINITE {CONS u0 y1 | u0 IN s /\ y1 IN {CONS u1 y2 | u1 IN s /\ y2 IN {[u2:A; u3] | u2 IN s /\ u3 IN s}}}`); (MATCH_MP_TAC FINITE_PRODUCT_DEPENDENT THEN ASM_REWRITE_TAC[]); (REWRITE_WITH `CARD {CONS u0 y1 | u0 IN s /\ y1 IN {CONS u1 y2 | u1 IN s /\ y2 IN {[u2:A; u3] | u2 IN s /\ u3 IN s}}} = CARD s * CARD {CONS u1 y2 | u1 IN s /\ y2 IN {[u2; u3] | u2 IN s /\ u3 IN s}}`); (ASM_SIMP_TAC[CARD_LIST_klemma]); (REWRITE_WITH `CARD {CONS u1 y2 | u1 IN s /\ y2 IN {[u2:A; u3] | u2 IN s /\ u3 IN s}} = CARD s * CARD {[u2; u3] | u2 IN s /\ u3 IN s}`); (ASM_SIMP_TAC[CARD_LIST_klemma]); (REWRITE_WITH `CARD {[u2:A; u3] | u2 IN s /\ u3 IN s} = CARD s * CARD s`); (ASM_SIMP_TAC[CARD_LIST_klemma_2])]);; (* ==================================================================== *) (* Lemma 58 *) let BOUNDS_VGEN_klemma = prove_by_refinement( `!u0 V r. &0 <= r /\ packing V ==> &(CARD (V INTER ball (u0,r))) <= (r + &1) pow 3`, [(REPEAT STRIP_TAC); (NEW_GOAL `sum (V INTER ball (u0,r)) (\x. vol (ball (x:real^3, &1))) = & (CARD (V INTER ball (u0,r))) * (&4 / &3 * pi)`); (NEW_GOAL `&0 <= &1`); (REAL_ARITH_TAC); (REWRITE_WITH `(\x. vol (ball (x,&1))) = (\x. &4 / &3 * pi)`); (REWRITE_TAC[FUN_EQ_THM]); (ASM_SIMP_TAC[VOLUME_BALL] THEN REAL_ARITH_TAC); (MATCH_MP_TAC SUM_CONST); (ASM_SIMP_TAC[FINITE_PACK_LEMMA]); (UP_ASM_TAC); (ABBREV_TAC `f = (\x:real^3. ball (x, &1))`); (REWRITE_WITH `(\x. vol (ball (x:real^3,&1))) = (\x. vol (f x))`); (REWRITE_TAC[FUN_EQ_THM] THEN STRIP_TAC); (EXPAND_TAC "f" THEN REWRITE_TAC[]); (REWRITE_WITH `sum (V INTER ball (u0:real^3,r)) (\x. vol (f x)) = vol (UNIONS (IMAGE f (V INTER ball (u0,r))))`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC MEASURE_DISJOINT_UNIONS_IMAGE); (EXPAND_TAC "f" THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_SIMP_TAC[FINITE_PACK_LEMMA]); (REWRITE_TAC[MEASURABLE_BALL]); (REWRITE_TAC[DISJOINT; SET_RULE `a = {} <=> ~(?s. s IN a)`]); (REWRITE_TAC[IN_INTER; IN_BALL]); (STRIP_TAC); (UNDISCH_TAC `packing V` THEN REWRITE_TAC[packing]); (REPEAT STRIP_TAC); (NEW_GOAL `&2 <= dist (x,y:real^3)`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (NEW_GOAL `dist (x,y) <= dist (x,s) + dist (y,s:real^3)`); (NORM_ARITH_TAC); (ASM_REAL_ARITH_TAC); (STRIP_TAC); (NEW_GOAL `vol (UNIONS (IMAGE f (V INTER ball (u0:real^3,r)))) <= vol (ball (u0, r + &1))`); (MATCH_MP_TAC MEASURE_SUBSET); (EXPAND_TAC "f" THEN REWRITE_TAC[IMAGE; MEASURABLE_BALL]); (STRIP_TAC); (MATCH_MP_TAC MEASURABLE_UNIONS); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (ASM_SIMP_TAC[FINITE_PACK_LEMMA]); (REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (ASM_REWRITE_TAC[MEASURABLE_BALL]); (REWRITE_TAC[UNIONS_SUBSET; IN_INTER; IN_BALL; IN_ELIM_THM; SUBSET]); (REPEAT STRIP_TAC); (NEW_GOAL `dist (u0, x') <= dist (u0,x:real^3) + dist (x, x')`); (NORM_ARITH_TAC); (NEW_GOAL `dist (x,x':real^3) < &1`); (REWRITE_TAC[GSYM IN_BALL] THEN ASM_MESON_TAC[]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `&(CARD (V INTER ball (u0,r))) * &4 / &3 * pi <= vol (ball (u0,r + &1))`); (ASM_REAL_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `vol (ball (u0, r + &1)) = &4 / &3 * pi * (r + &1) pow 3`); (NEW_GOAL `&0 <= r + &1`); (ASM_REAL_ARITH_TAC); (SIMP_TAC[VOLUME_BALL; ASSUME `&0 <= r + &1`]); (REWRITE_TAC[REAL_ARITH `a * &4 / &3 * pi <= &4 / &3 * pi * b <=> &0 <= pi * (b - a)`]); (STRIP_TAC); (NEW_GOAL `&0 <= (r + &1) pow 3 - &(CARD (V INTER ball (u0:real^3,r)))`); (MATCH_MP_TAC Real_ext.REAL_PROP_NN_LCANCEL); (EXISTS_TAC `pi`); (ASM_REWRITE_TAC[PI_POS]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 59 *) let BOUNDS_V4_klemma = prove_by_refinement ( `!u0 V. packing V ==> CARD (V INTER ball (u0, &4)) <= 125`, [(REPEAT STRIP_TAC); (NEW_GOAL `sum (V INTER ball (u0,&4)) (\x. vol (ball (x:real^3, &1))) = & (CARD (V INTER ball (u0,&4))) * (&4 / &3 * pi)`); (NEW_GOAL `&0 <= &1`); (REAL_ARITH_TAC); (REWRITE_WITH `(\x. vol (ball (x,&1))) = (\x. &4 / &3 * pi)`); (REWRITE_TAC[FUN_EQ_THM]); (ASM_SIMP_TAC[VOLUME_BALL] THEN REAL_ARITH_TAC); (MATCH_MP_TAC SUM_CONST); (ASM_SIMP_TAC[FINITE_PACK_LEMMA]); (UP_ASM_TAC); (ABBREV_TAC `f = (\x:real^3. ball (x, &1))`); (REWRITE_WITH `(\x. vol (ball (x:real^3,&1))) = (\x. vol (f x))`); (REWRITE_TAC[FUN_EQ_THM] THEN STRIP_TAC); (EXPAND_TAC "f" THEN REWRITE_TAC[]); (REWRITE_WITH `sum (V INTER ball (u0:real^3,&4)) (\x. vol (f x)) = vol (UNIONS (IMAGE f (V INTER ball (u0,&4))))`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC MEASURE_DISJOINT_UNIONS_IMAGE); (EXPAND_TAC "f" THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC); (ASM_SIMP_TAC[FINITE_PACK_LEMMA]); (REWRITE_TAC[MEASURABLE_BALL]); (REWRITE_TAC[DISJOINT; SET_RULE `a = {} <=> ~(?s. s IN a)`]); (REWRITE_TAC[IN_INTER; IN_BALL]); (STRIP_TAC); (UNDISCH_TAC `packing V` THEN REWRITE_TAC[packing]); (REPEAT STRIP_TAC); (NEW_GOAL `&2 <= dist (x,y:real^3)`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_SET_TAC[]); (NEW_GOAL `dist (x,y) <= dist (x,s) + dist (y,s:real^3)`); (NORM_ARITH_TAC); (ASM_REAL_ARITH_TAC); (STRIP_TAC); (NEW_GOAL `vol (UNIONS (IMAGE f (V INTER ball (u0:real^3,&4)))) <= vol (ball (u0, &5))`); (MATCH_MP_TAC MEASURE_SUBSET); (EXPAND_TAC "f" THEN REWRITE_TAC[IMAGE; MEASURABLE_BALL]); (STRIP_TAC); (MATCH_MP_TAC MEASURABLE_UNIONS); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMAGE_EXPAND); (ASM_SIMP_TAC[FINITE_PACK_LEMMA]); (REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (ASM_REWRITE_TAC[MEASURABLE_BALL]); (REWRITE_TAC[UNIONS_SUBSET; IN_INTER; IN_BALL; IN_ELIM_THM; SUBSET]); (REPEAT STRIP_TAC); (NEW_GOAL `dist (u0, x') <= dist (u0,x:real^3) + dist (x, x')`); (NORM_ARITH_TAC); (NEW_GOAL `dist (x,x':real^3) < &1`); (REWRITE_TAC[GSYM IN_BALL] THEN ASM_MESON_TAC[]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `&(CARD (V INTER ball (u0,&4))) * &4 / &3 * pi <= vol (ball (u0,&5))`); (ASM_REAL_ARITH_TAC); (UP_ASM_TAC THEN REWRITE_WITH `vol (ball (u0,&5)) = &4 / &3 * pi * (&5) pow 3`); (SIMP_TAC[VOLUME_BALL; REAL_ARITH `&0 <= &5`]); (REWRITE_TAC[REAL_ARITH `a * &4 / &3 * pi <= &4 / &3 * pi * b <=> &0 <= pi * (b - a)`]); (STRIP_TAC); (NEW_GOAL `&0 <= &5 pow 3 - &(CARD (V INTER ball (u0:real^3,&4)))`); (MATCH_MP_TAC Real_ext.REAL_PROP_NN_LCANCEL); (EXISTS_TAC `pi`); (ASM_REWRITE_TAC[PI_POS]); (REWRITE_TAC[GSYM REAL_OF_NUM_LE]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 60 *) let CARD_MCELL_CONTAINS_POINT_klemma = prove_by_refinement ( `?c. (!V u0. saturated V /\ packing V /\ u0 IN V ==> CARD {X | mcell_set V X /\ VX V X u0} <= c)`, [(EXISTS_TAC `1220703125`); (REPEAT STRIP_TAC); (ABBREV_TAC `S = {ul,i | barV V 3 ul /\ i IN 0..4 /\ u0 IN set_of_list ul}`); (NEW_GOAL `FINITE (S:(real^3)list#num->bool)`); (REWRITE_WITH `S = {ul,i | ul IN {ul | barV V 3 ul /\ u0 IN set_of_list ul} /\ i IN 0..4}`); (EXPAND_TAC "S"); (SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (REWRITE_TAC[FINITE_NUMSEG]); (MATCH_MP_TAC FINITE_SUBSET); (ABBREV_TAC `T1 = V INTER ball (u0:real^3, &4)`); (NEW_GOAL `FINITE (T1:real^3->bool)`); (EXPAND_TAC "T1" THEN MATCH_MP_TAC FINITE_PACK_LEMMA); (ASM_REWRITE_TAC[]); (EXISTS_TAC `{ul | ?v0 v1 v2 v3. v0 IN T1 /\ v1 IN T1 /\ v2 IN T1 /\ v3 IN T1 /\ ul = [v0;v1;v2;v3:real^3]}`); (STRIP_TAC); (MATCH_MP_TAC Ajripqn.FINITE_SET_LIST_LEMMA THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `?v0 v1 v2 v3. x = [v0:real^3; v1; v2; v3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC THEN EXISTS_TAC `v0:real^3` THEN EXISTS_TAC `v1:real^3` THEN EXISTS_TAC `v2:real^3` THEN EXISTS_TAC `v3:real^3`); (ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list (x:(real^3)list) SUBSET T1`); (EXPAND_TAC "T1" THEN REWRITE_TAC[SUBSET_INTER]); (STRIP_TAC); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzyzmjc.BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:(real^3->bool)`); (ASM_REWRITE_TAC[]); (UNDISCH_TAC `set_of_list (x:(real^3)list) u0` THEN ASM_REWRITE_TAC[IN]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (ABBREV_TAC `f = (\(ul,i). mcell i V ul)`); (NEW_GOAL `{X | mcell_set V X /\ VX V X u0} SUBSET (IMAGE (f:(real^3)list#num->real^3->bool) S) `); (EXPAND_TAC "S" THEN EXPAND_TAC "f" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[mcell_set_2; SUBSET; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `ul:(real^3)list,i:num`); (REWRITE_TAC[]); (ASM_REWRITE_TAC[]); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (REWRITE_TAC[MESON[IN] `(0..4) i <=> i IN 0..4`]); (ASM_REWRITE_TAC[IN_NUMSEG_0]); (REWRITE_TAC[MESON[IN] `(A:real^3->bool) i <=> i IN A`]); (ASM_REWRITE_TAC[IN_NUMSEG_0]); (NEW_GOAL `(u0:real^3) IN V INTER x`); (REWRITE_WITH `V INTER x = VX V x`); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (SWITCH_TAC THEN UP_ASM_TAC THEN ASM_REWRITE_TAC[VX]); (REWRITE_TAC[MESON[IN] `(A:real^3->bool) i <=> i IN A`]); (SET_TAC[]); (ASM_REWRITE_TAC[IN]); (UP_ASM_TAC THEN REWRITE_WITH `V INTER (x:real^3->bool) = set_of_list (truncate_simplex (i - 1) ul)`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[]); (NEW_GOAL `VX V x = V INTER x`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (SWITCH_TAC THEN UP_ASM_TAC THEN ASM_REWRITE_TAC[VX]); (REWRITE_TAC[MESON[IN] `(A:real^3->bool) i <=> i IN A`]); (SET_TAC[]); (REPEAT STRIP_TAC); (ASM_CASES_TAC `i = 0`); (NEW_GOAL `F`); (UNDISCH_TAC `VX V x u0` THEN ASM_REWRITE_TAC[]); (ASM_SIMP_TAC [Lepjbdj.LEPJBDJ_0]); (REWRITE_TAC[MESON[IN] `(A:real^3->bool) i <=> i IN A`]); (SET_TAC[]); (ASM_MESON_TAC[]); (ASM_ARITH_TAC); (UNDISCH_TAC `VX V x u0` THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[MESON[IN] `(A:real^3->bool) i <=> i IN A`]); (SET_TAC[]); (REWRITE_TAC[MESON[IN] `(A:real^3->bool) i <=> i IN A`]); (MATCH_MP_TAC (SET_RULE `A SUBSET B ==> (u0 IN A ==> u0 IN B)`)); (MATCH_MP_TAC Rogers.SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool`); (ASM_REWRITE_TAC[]); (ASM_ARITH_TAC); (MATCH_MP_TAC (ARITH_RULE `(?b. a:num <= b /\ b <= c) ==> a <= c`)); (EXISTS_TAC `CARD (IMAGE (f:(real^3)list#num->real^3->bool) S)`); (STRIP_TAC); (MATCH_MP_TAC CARD_SUBSET); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC FINITE_IMAGE); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC (ARITH_RULE `(?b. a:num <= b /\ b <= c) ==> a <= c`)); (EXISTS_TAC `CARD (S:(real^3)list#num->bool)`); (STRIP_TAC); (MATCH_MP_TAC CARD_IMAGE_LE); (ASM_REWRITE_TAC[]); (REWRITE_WITH `S = {ul,i |ul IN {ul | barV V 3 ul /\ u0 IN set_of_list ul}/\ i IN 0..4}`); (EXPAND_TAC "S" THEN SET_TAC[]); (REWRITE_WITH `CARD {ul,i |ul IN {ul | barV V 3 ul /\ u0 IN set_of_list ul}/\ i IN 0..4} = CARD {ul | barV V 3 ul /\ u0 IN set_of_list ul} * CARD (0..4)`); (MATCH_MP_TAC CARD_PRODUCT); (REWRITE_TAC[FINITE_NUMSEG]); (MATCH_MP_TAC FINITE_SUBSET); (ABBREV_TAC `T1 = V INTER ball (u0:real^3, &4)`); (NEW_GOAL `FINITE (T1:real^3->bool)`); (EXPAND_TAC "T1" THEN MATCH_MP_TAC FINITE_PACK_LEMMA); (ASM_REWRITE_TAC[]); (EXISTS_TAC `{ul | ?v0 v1 v2 v3. v0 IN T1 /\ v1 IN T1 /\ v2 IN T1 /\ v3 IN T1 /\ ul = [v0;v1;v2;v3:real^3]}`); (STRIP_TAC); (MATCH_MP_TAC Ajripqn.FINITE_SET_LIST_LEMMA THEN ASM_REWRITE_TAC[]); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `?v0 v1 v2 v3. x = [v0:real^3; v1; v2; v3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC THEN EXISTS_TAC `v0:real^3` THEN EXISTS_TAC `v1:real^3` THEN EXISTS_TAC `v2:real^3` THEN EXISTS_TAC `v3:real^3`); (ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list (x:(real^3)list) SUBSET T1`); (EXPAND_TAC "T1" THEN REWRITE_TAC[SUBSET_INTER]); (STRIP_TAC); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzyzmjc.BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:(real^3->bool)`); (ASM_REWRITE_TAC[]); (UNDISCH_TAC `set_of_list (x:(real^3)list) u0` THEN ASM_REWRITE_TAC[IN]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (REWRITE_TAC[CARD_NUMSEG; ARITH_RULE `(4 + 1) - 0 = 5`]); (REWRITE_TAC[ARITH_RULE `a * 5 <= 1220703125 <=> a <= 244140625`]); (ABBREV_TAC `T1 = V INTER ball (u0:real^3, &4)`); (NEW_GOAL `FINITE (T1:real^3->bool)`); (EXPAND_TAC "T1" THEN MATCH_MP_TAC FINITE_PACK_LEMMA); (ASM_REWRITE_TAC[]); (NEW_GOAL `{ul | barV V 3 ul /\ u0 IN set_of_list ul} SUBSET {ul | ?v0 v1 v2 v3. v0 IN T1 /\ v1 IN T1 /\ v2 IN T1 /\ v3 IN T1 /\ ul = [v0;v1;v2;v3]}`); (REWRITE_TAC[SUBSET; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `?v0 v1 v2 v3. x = [v0:real^3; v1; v2; v3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC THEN EXISTS_TAC `v0:real^3` THEN EXISTS_TAC `v1:real^3` THEN EXISTS_TAC `v2:real^3` THEN EXISTS_TAC `v3:real^3`); (ASM_REWRITE_TAC[]); (NEW_GOAL `set_of_list (x:(real^3)list) SUBSET T1`); (EXPAND_TAC "T1" THEN REWRITE_TAC[SUBSET_INTER]); (STRIP_TAC); (MATCH_MP_TAC Packing3.BARV_SUBSET); (EXISTS_TAC `3` THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC Qzyzmjc.BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:(real^3->bool)`); (ASM_REWRITE_TAC[]); (UNDISCH_TAC `set_of_list (x:(real^3)list) u0` THEN ASM_REWRITE_TAC[IN]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN SET_TAC[]); (MATCH_MP_TAC (ARITH_RULE `(?x. a <= x /\ x <= b:num) ==> a <= b`)); (EXISTS_TAC `CARD {ul | ?v0 v1 v2 v3. v0 IN T1 /\ v1 IN T1 /\ v2 IN T1 /\ v3 IN T1 /\ ul = [v0:real^3; v1; v2; v3]}`); (STRIP_TAC); (MATCH_MP_TAC CARD_SUBSET); (ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FINITE_SET_LIST_LEMMA); (ASM_REWRITE_TAC[]); (REWRITE_WITH `CARD {y | ?u0 u1 u2 u3. u0 IN T1 /\ u1 IN T1 /\ u2 IN T1 /\ u3 IN T1 /\ y = [u0:real^3; u1; u2; u3:real^3]} = CARD T1 * CARD T1 * CARD T1 * CARD T1`); (ASM_SIMP_TAC [CARD_LIST_4_klemma]); (NEW_GOAL `CARD (T1:real^3->bool) <= 125`); (EXPAND_TAC "T1"); (MATCH_MP_TAC BOUNDS_V4_klemma); (ASM_REWRITE_TAC[]); (NEW_GOAL `CARD (T1:real^3->bool) * CARD T1 <= 125 * 125`); (MATCH_MP_TAC LE_MULT2); (ASM_REWRITE_TAC[]); (NEW_GOAL `CARD (T1:real^3->bool) * CARD T1 * CARD T1 <= 125 * 125 * 125`); (MATCH_MP_TAC LE_MULT2); (ASM_REWRITE_TAC[]); (NEW_GOAL `CARD (T1:real^3->bool) * CARD T1 * CARD T1 * CARD T1 <= 125 * 125 * 125 * 125`); (MATCH_MP_TAC LE_MULT2); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 61 *) let BOUND_BETA_BUMP = prove_by_refinement ( `?c. !V X e. saturated V /\ packing V /\ mcell_set V X /\ critical_edgeX V X e ==> beta_bump V e X <= c`, [(EXISTS_TAC `&614400`); (REWRITE_TAC[mcell_set_2; IN_ELIM_THM; IN; beta_bump]); (REPEAT STRIP_TAC THEN LET_TAC); (UP_ASM_TAC THEN REWRITE_TAC[cell_params]); (ONCE_REWRITE_TAC[EQ_SYM_EQ]); (STRIP_TAC); (ABBREV_TAC `P = (\(k,ul'). k <= 4 /\ ul' IN barV V 3 /\ mcell i V ul = mcell k V ul')`); (NEW_GOAL `(P:(num#(real^3)list->bool)) (k,ul')`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC SELECT_AX); (EXISTS_TAC `(i:num,ul:(real^3)list)`); (EXPAND_TAC "P"); (REWRITE_TAC[BETA_THM]); (ASM_REWRITE_TAC[IN]); (UP_ASM_TAC THEN EXPAND_TAC "P"); (REWRITE_TAC[BETA_THM; IN] THEN STRIP_TAC); (ABBREV_TAC `S = {e',e'',p,vl | 4 = k /\ {e', e''} = critical_edgeX V X /\ e' = e /\ p permutes 0..3 /\ left_action_list p ul' = vl /\ {EL 0 vl, EL 1 vl} = e' /\ {EL 2 vl, EL 3 vl} = e''}`); (NEW_GOAL `?u0 u1 u2 u3. ul = [u0;u1;u2;u3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (ABBREV_TAC `K1 = {us:real^3->bool | us SUBSET set_of_list ul}`); (NEW_GOAL `FINITE (K1:(real^3->bool)->bool)`); (EXPAND_TAC "K1" THEN MATCH_MP_TAC FINITE_POWERSET); (ASM_REWRITE_TAC[set_of_list; Geomdetail.FINITE6]); (NEW_GOAL `critical_edgeX V X SUBSET K1`); (EXPAND_TAC "K1" THEN REWRITE_TAC[critical_edgeX; edgeX; SUBSET; IN; IN_ELIM_THM]); (REPEAT STRIP_TAC); (NEW_GOAL `(x:real^3->bool) SUBSET set_of_list ul`); (REWRITE_TAC[ASSUME `x = {u, v:real^3}`; ASSUME `{u, v} = {u', v':real^3}`]); (NEW_GOAL `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (STRIP_TAC); (NEW_GOAL `VX V X = {}`); (ASM_REWRITE_TAC[VX]); (UNDISCH_TAC `VX V X u'` THEN ASM_REWRITE_TAC[MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (NEW_GOAL `V INTER (X:real^3->bool) = set_of_list (truncate_simplex (i - 1) ul)`); (REWRITE_TAC[ASSUME `X = mcell i V ul`]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_CASES_TAC `i = 0`); (NEW_GOAL `F`); (NEW_GOAL `VX V X = {}`); (REWRITE_TAC[ASSUME `VX V X = V INTER X`; ASSUME `X = mcell i V ul`; ASSUME `i = 0`]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ_0); (ASM_REWRITE_TAC[]); (UNDISCH_TAC `VX V X u'` THEN REWRITE_TAC[ASSUME `VX V X = {}`; MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (ASM_MESON_TAC[]); (ASM_ARITH_TAC); (REWRITE_TAC[GSYM (ASSUME `mcell i V ul = mcell k V ul'`); GSYM (ASSUME `X = mcell i V ul`)]); (STRIP_TAC); (NEW_GOAL `NULLSET X`); (REWRITE_TAC[NEGLIGIBLE_EMPTY; ASSUME `X = {}:real^3->bool`]); (NEW_GOAL `VX V X = {}`); (REWRITE_TAC[VX; ASSUME `NULLSET X`]); (UNDISCH_TAC `VX V X u'` THEN REWRITE_TAC[MESON[IN] `{} x <=> x IN {}`; ASSUME `VX V X = {}`] THEN SET_TAC[]); (NEW_GOAL `set_of_list (truncate_simplex (i - 1) ul) SUBSET set_of_list (ul:(real^3)list)`); (MATCH_MP_TAC Rogers.SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET); (ASM_REWRITE_TAC[LENGTH] THEN ASM_ARITH_TAC); (ASM_SET_TAC[]); (ASM_SET_TAC[]); (NEW_GOAL `FINITE (S:(real^3->bool)#(real^3->bool)#(num->num)#(real^3)list->bool)`); (EXPAND_TAC "S"); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{e',e'',p,vl | {e', e''} = critical_edgeX V X /\ p permutes 0..3 /\ left_action_list p ul' = vl:(real^3)list}`); (STRIP_TAC); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{e',e'',p,vl | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3 /\ left_action_list p ul' = vl:(real^3)list}`); (STRIP_TAC); (ABBREV_TAC `S_temp = {e',e'',p | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3}`); (ABBREV_TAC `f_temp = (\(e':real^3->bool,e'':real^3->bool,p). e',e'',p, (left_action_list p (ul':(real^3)list)))`); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `IMAGE (f_temp:(real^3->bool)#(real^3->bool)#(num->num)->(real^3->bool)#(real^3->bool)#(num->num)#(real^3)list) S_temp`); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMAGE); (REWRITE_WITH `S_temp = {e':real^3->bool,u | e' IN K1 /\ u IN {e'', p | e'' IN K1 /\p permutes 0..3}}`); (EXPAND_TAC "S_temp"); (SET_TAC[]); (ABBREV_TAC `K2 = {e'':real^3->bool, p | e'' IN K1 /\p permutes 0..3}`); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (EXPAND_TAC "f_temp" THEN EXPAND_TAC "S_temp" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[IN; SUBSET; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `e':real^3->bool,e'':real^3->bool,p:num->num`); (ASM_REWRITE_TAC[]); (EXISTS_TAC `e':real^3->bool` THEN EXISTS_TAC `e'':real^3->bool` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN SET_TAC[]); (SET_TAC[]); (* ------------------------------------------------------------------------ *) (* Finish proving S is FINITE *) (* ------------------------------------------------------------------------ *) (MATCH_MP_TAC (REAL_ARITH `(?x. a <= x /\ x <= b) ==> a <= b`)); (EXISTS_TAC `sum (S:(real^3->bool)#(real^3->bool)#(num->num)#(real^3)list->bool) (\x. &100)`); (STRIP_TAC); (MATCH_MP_TAC SUM_LE); (ASM_REWRITE_TAC[]); (EXPAND_TAC "S" THEN REWRITE_TAC[IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM (ASSUME `4 = k`); REAL_ARITH `a / &4 <= &100 <=> a <= &400`]); (REWRITE_TAC[bump]); (REWRITE_TAC[REAL_ARITH `#0.005 * (&1 - a) - #0.005 * (&1 - b) <= &400 <=> b - a <= &80000`]); (MATCH_MP_TAC (REAL_ARITH `(&0 <= a /\ b <= c ==> b - a <= c)`)); (STRIP_TAC); (MATCH_MP_TAC REAL_LE_DIV); (REWRITE_TAC[Real_ext.REAL_LE_POW_2]); (REWRITE_TAC[hplus; h0; REAL_ARITH `a / (#1.3254 - #1.26) pow 2 <= &80000 <=> a <= #342.1728`]); (NEW_GOAL `barV V 3 vl`); (MATCH_MP_TAC Qzksykg.QZKSYKG1); (EXISTS_TAC `ul':(real^3)list`); (EXISTS_TAC `4` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[ARITH_RULE `4 - 1 = 3`]); (STRIP_TAC); (SET_TAC[]); (REWRITE_TAC[GSYM (ASSUME `mcell i V ul = mcell k V ul'`); GSYM (ASSUME `X = mcell i V ul`)]); (STRIP_TAC); (NEW_GOAL `critical_edgeX V X = {}`); (REWRITE_TAC[critical_edgeX ; edgeX; IN; IN_ELIM_THM; SET_RULE `X = {} <=> ~(?s. s IN X)`]); (STRIP_TAC); (NEW_GOAL `NULLSET X`); (ASM_MESON_TAC[NEGLIGIBLE_EMPTY]); (UNDISCH_TAC `VX V X v'` THEN REWRITE_TAC[VX; ASSUME `NULLSET X`]); (REWRITE_TAC[MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (UNDISCH_TAC `critical_edgeX V X e`); (REWRITE_TAC[ASSUME `critical_edgeX V X = {}`; MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (NEW_GOAL `?v0 v1 v2 v3. vl = [v0;v1;v2;v3:real^3]`); (MATCH_MP_TAC BARV_3_EXPLICIT); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN REPEAT STRIP_TAC); (ASM_REWRITE_TAC[EL; ARITH_RULE `3 = SUC 2 /\ 2 = SUC 1 /\ 1 = SUC 0`; HD; TL]); (REWRITE_TAC[ARITH_RULE `SUC (SUC 0) = 2`]); (NEW_GOAL `(hl [v2; v3:real^3] - #1.26) pow 2 <= (&10) pow 2`); (REWRITE_TAC[GSYM REAL_LE_SQUARE_ABS; REAL_ARITH `abs (&10) = &10`]); (ASM_CASES_TAC `&0 <= hl [v2; v3:real^3] - #1.26`); (REWRITE_WITH `abs (hl [v2; v3:real^3] - #1.26) = hl [v2; v3] - #1.26`); (ASM_REAL_ARITH_TAC); (REWRITE_TAC[HL_2; REAL_ARITH `inv (&2) * a - #1.26 <= &10 <=> a <= #22.52`]); (NEW_GOAL `set_of_list vl SUBSET ball (v2:real^3,&4)`); (MATCH_MP_TAC Qzyzmjc.BARV_3_IMP_FINITE_lemma2); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[set_of_list]); (SET_TAC[]); (UP_ASM_TAC THEN ASM_REWRITE_TAC[set_of_list] THEN STRIP_TAC); (SUBGOAL_THEN `v3:real^3 IN ball (v2, &k)` MP_TAC); (UP_ASM_TAC THEN SET_TAC[]); (REWRITE_TAC[IN_BALL; GSYM (ASSUME `4 = k`)]); (REAL_ARITH_TAC); (REWRITE_WITH `abs (hl [v2; v3] - #1.26) = #1.26 - hl [v2; v3:real^3]`); (ASM_REAL_ARITH_TAC); (MATCH_MP_TAC (REAL_ARITH `&0 <= a ==> #1.26 - a <= &10`)); (REWRITE_TAC[HL_2] THEN NORM_ARITH_TAC); (UP_ASM_TAC THEN REAL_ARITH_TAC); (ASM_SIMP_TAC[SUM_CONST]); (EXPAND_TAC "S"); (MATCH_MP_TAC (REAL_ARITH `(?s. x <= s /\ s <= b) ==> x <= b`)); (EXISTS_TAC `& (CARD {e',e'',p,vl | {e', e''} = critical_edgeX V X /\ p permutes 0..3 /\ left_action_list p ul' = vl:(real^3)list}) * &100 `); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `a * &100 <= b * &100 <=> a <= b`]); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (STRIP_TAC); (SET_TAC[]); (MATCH_MP_TAC FINITE_SUBSET); (*****) (EXISTS_TAC `{e',e'',p,vl | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3 /\ left_action_list p ul' = vl:(real^3)list}`); (STRIP_TAC); (ABBREV_TAC `S_temp = {e',e'',p | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3}`); (ABBREV_TAC `f_temp = (\(e':real^3->bool,e'':real^3->bool,p). e',e'',p, (left_action_list p (ul':(real^3)list)))`); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `IMAGE (f_temp:(real^3->bool)#(real^3->bool)#(num->num)->(real^3->bool)#(real^3->bool)#(num->num)#(real^3)list) S_temp`); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMAGE); (REWRITE_WITH `S_temp = {e':real^3->bool,u | e' IN K1 /\ u IN {e'', p | e'' IN K1 /\p permutes 0..3}}`); (EXPAND_TAC "S_temp"); (SET_TAC[]); (ABBREV_TAC `K2 = {e'':real^3->bool, p | e'' IN K1 /\p permutes 0..3}`); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (EXPAND_TAC "f_temp" THEN EXPAND_TAC "S_temp" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[IN; SUBSET; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `e':real^3->bool,e'':real^3->bool,p:num->num`); (ASM_REWRITE_TAC[]); (EXISTS_TAC `e':real^3->bool` THEN EXISTS_TAC `e'':real^3->bool` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (SWITCH_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (MATCH_MP_TAC (REAL_ARITH `(?s. x <= s /\ s <= b) ==> x <= b`)); (EXISTS_TAC `& (CARD {e',e'',p,vl | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3 /\ left_action_list p ul' = vl:(real^3)list}) * &100 `); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `a * &100 <= b * &100 <=> a <= b`]); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (STRIP_TAC); (SWITCH_TAC THEN UP_ASM_TAC THEN SET_TAC[]); (* **** *) (ABBREV_TAC `S_temp = {e',e'',p | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3}`); (ABBREV_TAC `f_temp = (\(e':real^3->bool,e'':real^3->bool,p). e',e'',p, (left_action_list p (ul':(real^3)list)))`); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `IMAGE (f_temp:(real^3->bool)#(real^3->bool)#(num->num)->(real^3->bool)#(real^3->bool)#(num->num)#(real^3)list) S_temp`); (STRIP_TAC); (MATCH_MP_TAC FINITE_IMAGE); (REWRITE_WITH `S_temp = {e':real^3->bool,u | e' IN K1 /\ u IN {e'', p | e'' IN K1 /\p permutes 0..3}}`); (EXPAND_TAC "S_temp"); (SET_TAC[]); (ABBREV_TAC `K2 = {e'':real^3->bool, p | e'' IN K1 /\p permutes 0..3}`); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (EXPAND_TAC "f_temp" THEN EXPAND_TAC "S_temp" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[IN; SUBSET; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `e':real^3->bool,e'':real^3->bool,p:num->num`); (ASM_REWRITE_TAC[]); (EXISTS_TAC `e':real^3->bool` THEN EXISTS_TAC `e'':real^3->bool` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC (REAL_ARITH `(?s. x <= s /\ s <= b) ==> x <= b`)); (ABBREV_TAC `S_temp = {e',e'',p | (e':real^3->bool) IN K1 /\ e'' IN K1 /\ p permutes 0..3}`); (ABBREV_TAC `f_temp = (\(e':real^3->bool,e'':real^3->bool,p). e',e'',p, (left_action_list p (ul':(real^3)list)))`); (EXISTS_TAC `& (CARD (IMAGE (f_temp:(real^3->bool)#(real^3->bool)#(num->num)->(real^3->bool)#(real^3->bool)#(num->num)#(real^3)list) S_temp)) * &100`); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `a * &100 <= b * &100 <=> a <= b`]); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_SUBSET); (STRIP_TAC); (EXPAND_TAC "f_temp" THEN EXPAND_TAC "S_temp" THEN REWRITE_TAC[IMAGE]); (REWRITE_TAC[IN; SUBSET; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `e':real^3->bool,e'':real^3->bool,p:num->num`); (ASM_REWRITE_TAC[]); (EXISTS_TAC `e':real^3->bool` THEN EXISTS_TAC `e'':real^3->bool` THEN EXISTS_TAC `p:num->num`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC FINITE_IMAGE); (REWRITE_WITH `S_temp = {e':real^3->bool,u | e' IN K1 /\ u IN {e'', p | e'' IN K1 /\p permutes 0..3}}`); (EXPAND_TAC "S_temp"); (SET_TAC[]); (ABBREV_TAC `K2 = {e'':real^3->bool, p | e'' IN K1 /\p permutes 0..3}`); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (MATCH_MP_TAC (REAL_ARITH `(?s. x <= s /\ s <= b) ==> x <= b`)); (EXISTS_TAC `& (CARD (S_temp:(real^3->bool)#(real^3->bool)#(num->num)->bool)) * &100`); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `a * &100 <= b * &100 <=> a <= b`]); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_IMAGE_LE); (REWRITE_WITH `S_temp = {e':real^3->bool,u | e' IN K1 /\ u IN {e'', p | e'' IN K1 /\p permutes 0..3}}`); (EXPAND_TAC "S_temp"); (SET_TAC[]); (ABBREV_TAC `K2 = {e'':real^3->bool, p | e'' IN K1 /\p permutes 0..3}`); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (REWRITE_WITH `S_temp = {e':real^3->bool,u | e' IN K1 /\ u IN {e'', p | e'' IN K1 /\p permutes 0..3}}`); (EXPAND_TAC "S_temp"); (SET_TAC[]); (ABBREV_TAC `K2 = {e'':real^3->bool, p | e'' IN K1 /\p permutes 0..3}`); (REWRITE_WITH `CARD {e':real^3->bool,u:(real^3->bool)#(num->num) | e' IN K1 /\ u IN K2} = CARD K1 * CARD K2`); (MATCH_MP_TAC CARD_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (REWRITE_WITH `CARD (K2:(real^3->bool)#(num->num)->bool) = CARD (K1:(real^3->bool)->bool) * CARD {p | p permutes 0..3}`); (REWRITE_WITH `K2 = {e'':real^3->bool,p | e'' IN K1 /\ p IN {p | p permutes 0..3}}`); (EXPAND_TAC "K2" THEN SET_TAC[]); (MATCH_MP_TAC CARD_PRODUCT); (ASM_REWRITE_TAC[]); (REWRITE_TAC[GSYM NUMSEG_LE; ARITH_RULE `a <= 3 <=> a < 4`; Marchal_cells_2_new.SET_OF_0_TO_3; Upfzbzm_support_lemmas.FINITE_PERMUTE_4]); (REWRITE_WITH `CARD (K1:(real^3->bool)->bool) = 16`); (REWRITE_WITH `16 = 2 EXP (CARD (set_of_list (ul:(real^3)list)))`); (REWRITE_WITH `CARD (set_of_list (ul:(real^3)list)) = 3 + 1`); (MATCH_MP_TAC BARV_CARD_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ARITH_TAC); (EXPAND_TAC "K1"); (MATCH_MP_TAC CARD_POWERSET); (ASM_REWRITE_TAC[set_of_list; Geomdetail.FINITE6]); (REWRITE_WITH `CARD {p | p permutes 0..3} = FACT (CARD (0..3))`); (MATCH_MP_TAC CARD_PERMUTATIONS); (REWRITE_TAC[FINITE_NUMSEG]); (REWRITE_TAC[CARD_NUMSEG; ARITH_RULE `16 * 16 * FACT ((3 + 1) - 0) = 6144`]); (REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 62 *) let MCELL_SUBSET_BALL8_2 = prove ( `!V X v. packing V /\ saturated V /\ mcell_set V X /\ v IN X ==> X SUBSET ball (v,&8)`, REWRITE_TAC[mcell_set; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN UP_ASM_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC MCELL_SUBSET_BALL8 THEN ASM_REWRITE_TAC[] THEN ASM_SET_TAC[]);; (* ==================================================================== *) (* Lemma 63 *) let critical_edge_subset_mcell = prove_by_refinement ( `!V X x. packing V /\ saturated V /\ mcell_set V X /\ critical_edgeX V X x ==> x SUBSET X`, [(REWRITE_TAC[mcell_set_2; critical_edgeX; edgeX; IN_ELIM_THM; IN]); (REPEAT STRIP_TAC); (NEW_GOAL `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (STRIP_TAC THEN UNDISCH_TAC `VX V X u'` THEN ASM_REWRITE_TAC[VX; MESON[IN] `{} u <=> u IN {}`] THEN SET_TAC[]); (ASM_SET_TAC[])]);; (* ==================================================================== *) (* Lemma 64 *) let EDGEX_SUBSET_MCELL = prove_by_refinement ( `!V X e. packing V /\ saturated V /\ mcell_set V X /\ edgeX V X e ==> e SUBSET X`, [(REWRITE_TAC[edgeX; IN; IN_ELIM_THM; mcell_set] THEN REPEAT STRIP_TAC); (NEW_GOAL `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC `VX V X u` THEN ASM_REWRITE_TAC[VX; MESON[IN] `{} x <=> x IN {}`] THEN SET_TAC[]); (ASM_SET_TAC[])]);; let CRITICAL_EDGEX_SUBSET_MCELL = prove ( `!V X e. packing V /\ saturated V /\ mcell_set V X /\ critical_edgeX V X e ==> e SUBSET X`, (REPEAT STRIP_TAC) THEN (MATCH_MP_TAC EDGEX_SUBSET_MCELL) THEN (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]) THEN (UP_ASM_TAC THEN REWRITE_TAC[critical_edgeX] THEN SET_TAC[]));; (* ==================================================================== *) (* Lemma 65 *) let FINITE_VX = prove_by_refinement ( `!V X. packing V /\ saturated V /\ mcell_set V X ==> FINITE (VX V X)`, [(REWRITE_TAC[mcell_set_2; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (ASM_CASES_TAC `VX V X = {}`); (ASM_REWRITE_TAC[FINITE_EMPTY]); (REWRITE_WITH `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC `~(VX V X = {})` THEN ASM_REWRITE_TAC[VX]); (ASM_CASES_TAC `V INTER (X:real^3->bool) = {}`); (ASM_REWRITE_TAC[FINITE_EMPTY]); (REWRITE_WITH `(V:real^3->bool) INTER X = set_of_list (truncate_simplex (i-1) ul)`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_CASES_TAC `i = 0`); (NEW_GOAL `F`); (UNDISCH_TAC `~(V INTER (X:real^3->bool) = {})`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ_0); (ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_ARITH_TAC); (STRIP_TAC); (UNDISCH_TAC `~(VX V X = {})` THEN ASM_REWRITE_TAC[VX; NEGLIGIBLE_EMPTY]); (REWRITE_TAC[FINITE_SET_OF_LIST])]);; (* ==================================================================== *) (* Lemma 66 *) let CARD_EDGEX_LE_16 = prove_by_refinement ( `!V X. packing V /\ saturated V /\ mcell_set V X ==> CARD (edgeX V X) <= 16`, [(REWRITE_TAC[edgeX] THEN REPEAT STRIP_TAC); (ABBREV_TAC `S = {u, v | VX V X u /\ VX V X v /\ ~(u = v)}`); (ABBREV_TAC `f = (\(u,v:real^3). {u,v})`); (MATCH_MP_TAC (ARITH_RULE `(?x:num. y <= x /\ x <= z) ==> y <= z`)); (EXISTS_TAC `CARD (IMAGE (f:(real^3#real^3)->(real^3->bool)) S)`); (STRIP_TAC); (MATCH_MP_TAC CARD_SUBSET); (STRIP_TAC); (EXPAND_TAC "f" THEN EXPAND_TAC "S" THEN REWRITE_TAC[IMAGE; SUBSET; IN; IN_ELIM_THM] THEN REPEAT STRIP_TAC); (EXISTS_TAC `u:real^3,v:real^3` THEN ASM_REWRITE_TAC[]); (EXISTS_TAC `u:real^3` THEN EXISTS_TAC `v:real^3` THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC FINITE_IMAGE); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{u,v | u IN VX V X /\ v IN VX V X}`); (STRIP_TAC); (MATCH_MP_TAC FINITE_PRODUCT); (REWRITE_TAC[] THEN MATCH_MP_TAC FINITE_VX); (ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (MATCH_MP_TAC (ARITH_RULE `(?x:num. y <= x /\ x <= z) ==> y <= z`)); (EXISTS_TAC `CARD (S:(real^3#real^3)->bool)`); (STRIP_TAC); (MATCH_MP_TAC CARD_IMAGE_LE); (MATCH_MP_TAC FINITE_SUBSET); (EXISTS_TAC `{u,v | u IN VX V X /\ v IN VX V X}`); (STRIP_TAC); (MATCH_MP_TAC FINITE_PRODUCT); (REWRITE_TAC[] THEN MATCH_MP_TAC FINITE_VX); (ASM_REWRITE_TAC[]); (ASM_SET_TAC[]); (MATCH_MP_TAC (ARITH_RULE `(?x:num. y <= x /\ x <= z) ==> y <= z`)); (EXISTS_TAC `CARD {u,v | u IN VX V X /\ v IN VX V X}`); (STRIP_TAC); (MATCH_MP_TAC CARD_SUBSET); (STRIP_TAC); (ASM_SET_TAC[]); (MATCH_MP_TAC FINITE_PRODUCT); (REWRITE_TAC[] THEN MATCH_MP_TAC FINITE_VX); (ASM_REWRITE_TAC[]); (REWRITE_WITH `CARD {u,v | u IN VX V X /\ v IN VX V X} = CARD (VX V X) * CARD (VX V X)`); (MATCH_MP_TAC CARD_PRODUCT THEN ASM_SIMP_TAC[FINITE_VX]); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set_2; IN; IN_ELIM_THM]); (STRIP_TAC); (ASM_CASES_TAC `VX V X = {}`); (ASM_REWRITE_TAC[CARD_CLAUSES] THEN ARITH_TAC); (REWRITE_WITH `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC `~(VX V X = {})` THEN ASM_REWRITE_TAC[VX]); (ASM_CASES_TAC `V INTER (X:real^3->bool) = {}`); (ASM_REWRITE_TAC[CARD_CLAUSES] THEN ARITH_TAC); (REWRITE_WITH `(V:real^3->bool) INTER X = set_of_list (truncate_simplex (i-1) ul)`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ); (ASM_REWRITE_TAC[]); (STRIP_TAC); (ASM_CASES_TAC `i = 0`); (NEW_GOAL `F`); (UNDISCH_TAC `~(V INTER (X:real^3->bool) = {})`); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Lepjbdj.LEPJBDJ_0); (ASM_REWRITE_TAC[]); (ASM_MESON_TAC[]); (ASM_ARITH_TAC); (STRIP_TAC); (UNDISCH_TAC `~(VX V X = {})` THEN ASM_REWRITE_TAC[VX; NEGLIGIBLE_EMPTY]); (NEW_GOAL `CARD (set_of_list (truncate_simplex (i - 1) ul)) <= LENGTH (truncate_simplex (i - 1) (ul:(real^3)list))`); (REWRITE_TAC[CARD_SET_OF_LIST_LE]); (UP_ASM_TAC THEN REWRITE_WITH `LENGTH (truncate_simplex (i - 1) (ul:(real^3)list)) = (i - 1 + 1)`); (MATCH_MP_TAC Packing3.LENGTH_TRUNCATE_SIMPLEX); (REWRITE_WITH `LENGTH (ul:(real^3)list) = 3 + 1`); (MATCH_MP_TAC BARV_LENGTH_LEMMA); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (ASM_ARITH_TAC); (STRIP_TAC); (ABBREV_TAC `s = CARD (set_of_list (truncate_simplex (i - 1) (ul:(real^3)list)))`); (NEW_GOAL `s <= 4`); (ASM_ARITH_TAC); (NEW_GOAL `s * s <= s * 4 /\ s * 4 <= 4 * 4`); (ASM_REWRITE_TAC[LE_MULT_LCANCEL; LE_MULT_RCANCEL]); (ASM_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 67 *) let gamma_y_lmfun_bound2 = prove_by_refinement ( `?d. (!V X e. packing V /\ saturated V /\ mcell_set V X /\ edgeX V X e ==> gammaY V X lmfun e <= d)`, [ (EXISTS_TAC `pi * h0 / (h0 - &1)`); (REPEAT STRIP_TAC); (ABBREV_TAC `f = (\u v. if {u, v} IN edgeX V X then dihX V X (u,v) * lmfun (hl [u; v]) else &0)`); (REWRITE_WITH `gammaY V X lmfun = (\({u, v}). f u v)`); (EXPAND_TAC "f" THEN REWRITE_TAC[gammaY]); (NEW_GOAL `!u v. (f:real^3->real^3->real) u v = f v u`); (EXPAND_TAC "f" THEN REWRITE_TAC[BETA_THM]); (REWRITE_TAC[HL; set_of_list; SET_RULE `{a,b} = {b,a}`]); (REPEAT GEN_TAC THEN COND_CASES_TAC); (COND_CASES_TAC); (REWRITE_WITH `dihX V X (u,v) = dihX V X (v,u)`); (MATCH_MP_TAC DIHX_SYM); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN MESON_TAC[]); (COND_CASES_TAC); (UP_ASM_TAC THEN MESON_TAC[]); (REFL_TAC); (UNDISCH_TAC `edgeX V X e` THEN REWRITE_TAC[IN;IN_ELIM_THM; edgeX]); (STRIP_TAC); (ASM_REWRITE_TAC[]); (REWRITE_WITH `(\({u, v}). f u v) {u, v} = (f:real^3->real^3->real) u v`); (MATCH_MP_TAC BETA_PAIR_THM); (ASM_REWRITE_TAC[]); (EXPAND_TAC "f"); (COND_CASES_TAC); (NEW_GOAL `dihX V X (u,v) * lmfun (hl [u;v]) <= pi * lmfun (hl [u;v])`); (REWRITE_TAC[REAL_ARITH `a * x <= b * x <=> &0 <= (b - a) * x`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[lmfun_pos_le; REAL_ARITH `&0 <= a - b <=> b <= a`; DIHX_LE_PI]); (NEW_GOAL `pi * lmfun (hl [u:real^3;v]) <= pi * h0 / (h0 - &1)`); (REWRITE_TAC[REAL_ARITH `a * x <= a * y <=> &0 <= (y - x) * a`]); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[PI_POS_LE; REAL_ARITH `&0 <= a - b <=> b <= a`]); (MATCH_MP_TAC lmfun_bounded); (REWRITE_TAC[HL_2; REAL_ARITH `&0 <= inv (&2) * x <=> &0 <= x`; DIST_POS_LE]); (UP_ASM_TAC THEN UP_ASM_TAC THEN REAL_ARITH_TAC); (MATCH_MP_TAC REAL_LE_MUL); (REWRITE_TAC[PI_POS_LE]); (MATCH_MP_TAC REAL_LE_DIV); (REWRITE_TAC[h0] THEN REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 68 *) let BOUND_GAMMA_X_lmfun = prove_by_refinement ( `?c. (!V X. packing V /\ saturated V /\ mcell_set V X ==> gammaX V X lmfun <= c)`, [(MP_TAC gamma_y_lmfun_bound2 THEN STRIP_TAC); (ABBREV_TAC `e = max d (&0)`); (EXISTS_TAC `&4 / &3 * pi * (&8) pow 3 + (&8 * mm2 / pi) * &16 * e`); (REPEAT STRIP_TAC); (REWRITE_TAC[gammaX]); (MATCH_MP_TAC (REAL_ARITH `a <= x /\ &0 <= b /\ c <= y ==> a - b + c <= x + y`)); (REPEAT STRIP_TAC); (ASM_CASES_TAC `X:real^3->bool = {}`); (ASM_REWRITE_TAC[MEASURE_EMPTY; REAL_ARITH `&0 <= &4 / &3 * pi * &8 pow 3 <=> &0 <= pi`]); (REWRITE_TAC[PI_POS_LE]); (NEW_GOAL `?p:real^3. p IN X`); (ASM_SET_TAC[]); (UP_ASM_TAC THEN STRIP_TAC); (REWRITE_WITH `&4 / &3 * pi * &8 pow 3 = vol (ball (p, &8))`); (ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN MATCH_MP_TAC VOLUME_BALL); (REAL_ARITH_TAC); (MATCH_MP_TAC MEASURE_SUBSET); (REWRITE_TAC[MEASURABLE_BALL]); (STRIP_TAC); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set; IN; IN_ELIM_THM]); (STRIP_TAC); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC MEASURABLE_MCELL); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC MCELL_SUBSET_BALL8_2); (EXISTS_TAC `V:real^3->bool` THEN ASM_REWRITE_TAC[]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `&0 <= &2 * a <=> &0 <= a`]); (MATCH_MP_TAC REAL_LE_DIV); (STRIP_TAC); (NEW_GOAL `#1.012080 < mm1`); (REWRITE_TAC[Flyspeck_constants.bounds]); (ASM_REAL_ARITH_TAC); (REWRITE_TAC[PI_POS_LE]); (REWRITE_TAC[total_solid]); (MATCH_MP_TAC SUM_POS_LE); (REPEAT STRIP_TAC); (MATCH_MP_TAC FINITE_VX); (ASM_REWRITE_TAC[]); (REWRITE_TAC[sol]); (NEW_GOAL `eventually_radial (x:real^3) X`); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set; IN; IN_ELIM_THM]); (STRIP_TAC); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Urrphbz2.URRPHBZ2); (ASM_REWRITE_TAC[]); (NEW_GOAL `VX V X = V INTER X`); (MATCH_MP_TAC Hdtfnfz.HDTFNFZ); (EXISTS_TAC `ul:(real^3)list` THEN EXISTS_TAC `i:num`); (ASM_REWRITE_TAC[]); (STRIP_TAC THEN UNDISCH_TAC `x IN VX V X`); (ASM_REWRITE_TAC[VX] THEN SET_TAC[]); (ASM_SET_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[eventually_radial]); (STRIP_TAC); (REWRITE_WITH `sol x X = &3 * vol (X INTER normball x r) / r pow 3`); (MATCH_MP_TAC sol); (REWRITE_TAC[GSYM Marchal_cells_2_new.RADIAL_VS_RADIAL_NORM; NORMBALL_BALL]); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC MEASURABLE_INTER); (REWRITE_TAC[MEASURABLE_BALL]); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set; IN; IN_ELIM_THM]); (STRIP_TAC); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC MEASURABLE_MCELL); (ASM_REWRITE_TAC[]); (REWRITE_TAC[REAL_ARITH `&0 <= &3 * a <=> &0 <= a`]); (MATCH_MP_TAC REAL_LE_DIV); (STRIP_TAC); (MATCH_MP_TAC MEASURE_POS_LE); (REWRITE_TAC[NORMBALL_BALL]); (MATCH_MP_TAC MEASURABLE_INTER); (REWRITE_TAC[MEASURABLE_BALL]); (UNDISCH_TAC `mcell_set V X` THEN REWRITE_TAC[mcell_set; IN; IN_ELIM_THM]); (STRIP_TAC); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC MEASURABLE_MCELL); (ASM_REWRITE_TAC[]); (MATCH_MP_TAC Real_ext.REAL_PROP_NN_POW); (ASM_REAL_ARITH_TAC); (ONCE_REWRITE_TAC[REAL_ARITH `a <= b <=> &0 <= b - a`]); (REWRITE_TAC[REAL_ARITH `a * x - a * y = a * (x - y)`]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (REWRITE_TAC[REAL_ARITH `&0 <= &8 * a <=> &0 <= a`]); (MATCH_MP_TAC REAL_LE_DIV); (STRIP_TAC); (NEW_GOAL `#0.02541 < mm2`); (REWRITE_TAC[Flyspeck_constants.bounds]); (ASM_REAL_ARITH_TAC); (REWRITE_TAC[PI_POS_LE]); (REWRITE_TAC[GSYM gammaY]); (ONCE_REWRITE_TAC[REAL_ARITH `&0 <= b - a <=> a <= b`]); (NEW_GOAL `sum (edgeX V X) (gammaY V X lmfun) <= sum (edgeX V X) (\x. e)`); (MATCH_MP_TAC SUM_LE); (REWRITE_TAC[Upfzbzm_support_lemmas.FINITE_edgeX]); (REPEAT STRIP_TAC); (NEW_GOAL `gammaY V X lmfun x <= d`); (FIRST_ASSUM MATCH_MP_TAC); (ASM_REWRITE_TAC[]); (UP_ASM_TAC THEN REWRITE_TAC[IN]); (ASM_REAL_ARITH_TAC); (NEW_GOAL `sum (edgeX V X) (\x. e) <= &16 * e`); (REWRITE_WITH `sum (edgeX V X) (\x. e) = &(CARD (edgeX V X)) * e`); (MATCH_MP_TAC SUM_CONST); (REWRITE_TAC[Upfzbzm_support_lemmas.FINITE_edgeX]); (ONCE_REWRITE_TAC[REAL_ARITH `a <= b <=> &0 <= b - a`]); (REWRITE_TAC[REAL_ARITH `a * x - b * x = x * (a - b)`]); (MATCH_MP_TAC REAL_LE_MUL); (STRIP_TAC); (ASM_REAL_ARITH_TAC); (ONCE_REWRITE_TAC[REAL_ARITH `&0 <= b - a <=> a <= b`]); (REWRITE_TAC[REAL_OF_NUM_LE]); (MATCH_MP_TAC CARD_EDGEX_LE_16); (ASM_REWRITE_TAC[]); (ASM_REAL_ARITH_TAC)]);; (* ==================================================================== *) (* Lemma 69 *) (* ==================================================================== *) (* Lemma 70 *) (* ==================================================================== *) (* Lemma 71 *)