(* ========================================================================== *)
(* FLYSPECK - BOOK FORMALIZATION *)
(* *)
(* Chapter: Packing/Rogers simplex *)
(* Author: Alexey Solovyev *)
(* Date: 2010-08-12 *)
(* ========================================================================== *)
flyspeck_needs "packing/pack3.hl";;
flyspeck_needs "fan/HypermapAndFan.hl";;
module Rogers = struct
open Pack_defs;;
open Sphere;;
open Packing3;;
(*****************************************************************)
(*****************************************)
(* Faces *)
(*****************************************)
(*********************************************)
(* KHEJKCI *)
(*********************************************)
(* A(u, v) is a face of A+(u, v) *)
(* Generalized version of the target theorem *)
let KHEJKCI_GEN = prove(`!V k r ul vl. saturated V /\ packing V /\ barV V k ul /\ barV V r vl /\ initial_sublist ul vl ==>
(voronoi_list V vl)
face_of (voronoi_list V ul) `,
REWRITE_TAC[
INITIAL_SUBLIST] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?h tl. ul = CONS (h:real^3) tl` MP_TAC THENL
[
ASM_MESON_TAC[
BARV_CONS];
ALL_TAC
] THEN
STRIP_TAC THEN
SUBGOAL_THEN `voronoi_list (V:real^3->bool) ul =
voronoi_closed V h
INTER INTERS {bis h u | u
IN set_of_list tl}` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC
VORONOI_LIST_BIS THEN
ASM_MESON_TAC[
BARV_SUBSET];
ALL_TAC
] THEN
SUBGOAL_THEN `voronoi_list (V:real^3->bool) vl =
voronoi_closed V h
INTER INTERS {bis h u | u
IN set_of_list (
APPEND tl yl)}` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC
VORONOI_LIST_BIS THEN
ASM_MESON_TAC[
BARV_SUBSET;
APPEND];
ALL_TAC
] THEN
REWRITE_TAC[
face_of;
IN_SET_OF_LIST;
MEM_APPEND] THEN
REPEAT CONJ_TAC THENL
[
SET_TAC[];
MATCH_MP_TAC
CONVEX_INTER THEN CONJ_TAC THENL
[
ASM_MESON_TAC[
DRUQUFE];
ALL_TAC
] THEN
MATCH_MP_TAC
CONVEX_INTERS THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
MESON_TAC[
CONVEX_BIS];
ALL_TAC
] THEN
REPEAT GEN_TAC THEN
SIMP_TAC[
IN_INTER;
IN_INTERS;
IN_ELIM_THM] THEN
REPLICATE_TAC 2 (DISCH_THEN (CONJUNCTS_THEN2 (ASSUME_TAC o CONJUNCT1) MP_TAC)) THEN
STRIP_TAC THEN
SUBGOAL_THEN `!u:real^3. u
IN V ==> a:real^3
IN bis_le h u /\ b:real^3
IN bis_le h u` ASSUME_TAC THENL
[
REPLICATE_TAC 3 (POP_ASSUM (fun
th -> ALL_TAC)) THEN GEN_TAC THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[
voronoi_closed;
bis_le;
IN_ELIM_THM;
IN] THEN
MESON_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!u:real^3. (
MEM u tl ==> u
IN V) /\ (
MEM u yl ==> u
IN V)` ASSUME_TAC THENL
[
GEN_TAC THEN REPLICATE_TAC 6 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN
POP_ASSUM (MP_TAC o (fun
th -> MATCH_MP
BARV_SUBSET th)) THEN
POP_ASSUM (MP_TAC o (fun
th -> MATCH_MP
BARV_SUBSET th)) THEN
REWRITE_TAC[IMP_IMP; GSYM
CONJ_ASSOC] THEN
REWRITE_TAC[
IMP_CONJ_ALT] THEN
REPLICATE_TAC 2 (DISCH_THEN (fun
th -> REWRITE_TAC[
th])) THEN
REWRITE_TAC[
SUBSET;
IN_SET_OF_LIST;
MEM_APPEND;
MEM] THEN
SIMP_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!u:real^3. u
IN V /\ x
IN bis h u ==> a
IN bis h u /\ b
IN bis h u` ASSUME_TAC THENL
[
POP_ASSUM (fun
th -> ALL_TAC) THEN GEN_TAC THEN
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o (SPEC `u:real^3`)) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
MP_TAC (ISPECL [`h:real^3`; `u:real^3`]
BIS_FACE_OF_BIS_LE) THEN
REWRITE_TAC[
face_of] THEN
DISCH_THEN (CONJUNCTS_THEN2 (fun
th -> ALL_TAC) (MP_TAC o CONJUNCT2)) THEN
DISCH_THEN (MP_TAC o ISPECL [`a:real^3`; `b:real^3`; `x:real^3`]) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_MESON_TAC[]);;
let KHEJKCI = prove(`!V k ul. saturated V /\ packing V /\ barV V k ul ==>
((voronoi_list V ul)
face_of (
voronoi_closed V (
HD ul)) )`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM (MP_TAC o (fun
th -> MATCH_MP
BARV_CONS th)) THEN
STRIP_TAC THEN POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
REWRITE_TAC[GSYM
VORONOI_LIST_SING] THEN
MATCH_MP_TAC
KHEJKCI_GEN THEN
EXISTS_TAC `0` THEN EXISTS_TAC `k:num` THEN
SUBGOAL_THEN `initial_sublist [h:real^3] ul` ASSUME_TAC THENL
[
ASM_REWRITE_TAC[
INITIAL_SUBLIST;
APPEND] THEN MESON_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`; `[h:real^3]`]
BARV_INITIAL_SUBLIST) THEN
ASM_REWRITE_TAC[
LENGTH; ARITH_RULE `0 < SUC 0`; ARITH_RULE `SUC 0 - 1 = 0`]);;
let VORONOI_BARV_CANONICAL = prove(`!V k ul. packing V /\ saturated V /\ barV V k ul
==> ?K. FINITE K /\ voronoi_list V ul =
affine hull (voronoi_list V ul)
INTER INTERS K /\
(!a. a
IN K ==> (?v. v
IN V /\ ~(v =
HD ul) /\ (a =
bis_le v (
HD ul) \/ a =
bis_le (
HD ul) v))) /\
(!K'. K'
PSUBSET K ==> (voronoi_list V ul)
PSUBSET (
affine hull (voronoi_list V ul)
INTER INTERS K'))`,
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`]
BARV_CONS) THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`]
BARV_SUBSET) THEN
ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `h:real^3`; `t:(real^3)list`]
VORONOI_LIST_CANONICAL) THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
DISCH_THEN (LABEL_TAC "A") THEN
DISCH_TAC THEN REMOVE_THEN "A" MP_TAC THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]));;
let REAL_LINE_BOUNDED = prove(`!a b. (!t. t * a <= b) ==> a = &0`,
REPEAT STRIP_TAC THEN
DISJ_CASES_TAC (TAUT `a = &0 \/ ~(a = &0)`) THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `(b + &1) * inv(a)`) THEN
ASM_REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
REAL_ARITH_TAC);;
let HALFSPACE_EQ = prove(`!(a:real^N) b c d. {x | a
dot x <= b} = {x | c
dot x <= d} <=> (?t. c = t % a /\ d = t * b /\ &0 < t) \/ (a =
vec 0 /\ c =
vec 0 /\ ((&0 <= b /\ &0 <= d) \/ (b < &0 /\ d < &0)))`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN
EQ_TAC THENL
[
DISCH_TAC THEN
DISJ_CASES_TAC (TAUT `(a:real^N) =
vec 0 \/ ~(a =
vec 0)`) THENL
[
DISJ2_TAC THEN
ASM_REWRITE_TAC[] THEN
DISJ_CASES_TAC (TAUT `(c:real^N) =
vec 0 \/ ~(c =
vec 0)`) THEN ASM_REWRITE_TAC[] THENL
[
FIRST_X_ASSUM (MP_TAC o check (is_forall o concl)) THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_TAC "A" `~((c:real^N)
dot c = &0)` [ ASM_REWRITE_TAC[
DOT_EQ_0] ] THEN
DISJ_CASES_TAC (REAL_ARITH `&0 <= b \/ b < &0`) THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `((d + &1) * inv((c:real^N)
dot c)) % c`) THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
REAL_ARITH_TAC;
FIRST_X_ASSUM (MP_TAC o SPEC `(d * inv((c:real^N)
dot c)) % c`) THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
ASM_SIMP_TAC[REAL_ARITH `b < &0 ==> ((&0 <= b) <=> F)`] THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
REWRITE_TAC[
REAL_MUL_RID;
REAL_LE_REFL]
];
ALL_TAC
] THEN
DISJ1_TAC THEN
SUBGOAL_TAC "A" `~((a:real^N
dot a) = &0)` [ ASM_REWRITE_TAC[
DOT_EQ_0] ] THEN
SUBGOAL_THEN `!u. u*((a:real^N
dot a)*(c
dot c) - (a
dot c)*(a
dot c)) <= d*(a
dot a) - b*(a
dot c)` MP_TAC THENL
[
GEN_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `((b - u * (a:real^N
dot c)) * inv(a
dot a)) % a + u % c`) THEN
REWRITE_TAC[
DOT_RADD;
DOT_RMUL] THEN
REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV;
REAL_MUL_RID; REAL_ARITH `(b:real - a) + a = b`;
REAL_LE_REFL] THEN
DISCH_TAC THEN
ASSUME_TAC (ISPEC `a:real^N`
DOT_POS_LE) THEN
MP_TAC (SPECL [`(b - u * (a:real^N
dot c)) * inv(a
dot a) * (c
dot a) + u * (c
dot c)`; `d:real`; `a:real^N
dot a`]
REAL_LE_RMUL) THEN
ASM_REWRITE_TAC[
REAL_ADD_RDISTRIB] THEN
REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_FIELD `~(a:real^N
dot a = &0) ==> inv(a
dot a) * (c
dot a) * (a
dot a) = (c
dot a)`] THEN
REWRITE_TAC[
REAL_SUB_RDISTRIB;
REAL_SUB_LDISTRIB;
DOT_SYM] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o (fun
th -> MATCH_MP
REAL_LINE_BOUNDED th)) THEN
REWRITE_TAC[GSYM REAL_POW_2; REAL_ARITH `a - b = &0 <=> b = a`] THEN
DISCH_TAC THEN
SUBGOAL_THEN `?t. c = t % (a:real^N)` MP_TAC THENL
[
MP_TAC (SPECL [`a:real^N`; `c:real^N`]
DOT_CAUCHY_SCHWARZ_EQUAL) THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA] THEN
STRIP_TAC THENL
[
EXISTS_TAC `&0` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO];
ALL_TAC
] THEN
EXISTS_TAC `c':real` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
EXISTS_TAC `t:real` THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o check (is_forall o concl)) THEN
ASM_REWRITE_TAC[
DOT_LMUL] THEN
DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o SPEC `(b * inv(a:real^N
dot a)) % a`) THEN
FIRST_ASSUM (MP_TAC o SPEC `(d * inv(t) * inv(a:real^N
dot a)) % a`) THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
REWRITE_TAC[
REAL_MUL_RID;
REAL_LE_REFL] THEN
SUBGOAL_THEN `~(t = &0)` ASSUME_TAC THENL
[
DISJ_CASES_TAC (TAUT `~(t = &0) \/ t = &0`) THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o check (is_forall o concl)) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 * a = &0`] THEN
DISCH_TAC THEN
DISJ_CASES_TAC (REAL_ARITH `&0 <= d \/ d < &0`) THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `((b + &1) * inv((a:real^N)
dot a)) % a`) THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
REAL_ARITH_TAC;
FIRST_X_ASSUM (MP_TAC o SPEC `(b * inv((a:real^N)
dot a)) % a`) THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
ASM_SIMP_TAC[REAL_ARITH `b < &0 ==> ((&0 <= b) <=> F)`] THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
REWRITE_TAC[
REAL_MUL_RID;
REAL_LE_REFL]
];
ALL_TAC
] THEN
ASM_SIMP_TAC[REAL_FIELD `~(t = &0) ==> t * d * inv t = d`] THEN
ASM_REWRITE_TAC[
REAL_LE_REFL] THEN
SUBGOAL_THEN `&0 < t` ASSUME_TAC THENL
[
DISJ_CASES_TAC (REAL_ARITH `&0 < t \/ t <= &0`) THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[
REAL_LE_LT] THEN
DISCH_TAC THEN
DISJ_CASES_TAC (REAL_ARITH `t * b - d - &1 <= &0 \/ &0 < t * b - d - &1`) THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `((d + &1) * inv(t) * inv(a:real^N
dot a)) % a`) THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV;
REAL_MUL_RID; REAL_FIELD `~(t = &0) ==> (t * d * inv t = d)`] THEN
REWRITE_TAC[REAL_ARITH `d + &1 <= d <=> F`] THEN
SUBGOAL_THEN `(d + &1) * inv t <= b` MP_TAC THENL
[
MP_TAC (SPECL [`(d + &1) * inv t`; `b:real`; `t:real`]
REAL_NEG_LE_RMUL) THEN
ASM_REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ASM_SIMP_TAC[REAL_MUL_LINV;
REAL_MUL_RID] THEN
POP_ASSUM MP_TAC THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
MESON_TAC[];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `(b * inv(a:real^N
dot a)) % a`) THEN
REWRITE_TAC[
DOT_RMUL; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV;
REAL_MUL_RID;
REAL_LE_REFL] THEN
POP_ASSUM MP_TAC THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
DISCH_TAC THEN
SUBGOAL_THEN `d:real <= t * b` MP_TAC THENL
[
MP_TAC (SPECL [`t:real`; `(d:real) * inv t`; `b:real`]
REAL_LE_LMUL) THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < t ==> &0 <= t`; REAL_FIELD `~(t = &0) ==> t * d * inv t = d`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THENL
[
GEN_TAC THEN
ASM_REWRITE_TAC[
DOT_LMUL] THEN
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ];
GEN_TAC THEN
ASM_REWRITE_TAC[
DOT_LZERO];
GEN_TAC THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REAL_ARITH_TAC
]);;
let FACET_OF_POLYHEDRON_EXPLICIT_BIS = prove(`!(V:real^3->bool) K s u.
FINITE K /\ s =
affine hull s
INTER INTERS K /\
(!a. a
IN K ==> (?v. v
IN V /\ ~(v = u) /\ (a =
bis_le v u \/ a =
bis_le u v))) /\
(!K'. K'
PSUBSET K ==> s
PSUBSET (
affine hull s
INTER INTERS K'))
==> (!c. c
facet_of s <=> (?v. v
IN V /\ (
bis_le v u
IN K \/
bis_le u v
IN K) /\ c = s
INTER bis u v))`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?a b. !h:real^3->bool. h
IN K ==> ~(a h =
vec 0) /\ h = {x | a h
dot x <= b h}` MP_TAC THENL
[
POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM MP_TAC THEN
REPLICATE_TAC 2 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
DISCH_TAC THEN
REWRITE_TAC[GSYM
SKOLEM_THM] THEN
GEN_TAC THEN
REWRITE_TAC[
RIGHT_EXISTS_IMP_THM] THEN
DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `h:(real^3->bool)`) THEN
ASM_REWRITE_TAC[
BIS_LE_EQ_HALFSPACE] THEN STRIP_TAC THENL
[
EXISTS_TAC `&2 % (u - (v:real^3))` THEN
EXISTS_TAC `(u:real^3)
dot u - (v:real^3)
dot v` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_EQ_0; REAL_ARITH `&2 = &0 <=> F`;
VECTOR_SUB_EQ];
EXISTS_TAC `&2 % ((v:real^3) - u)` THEN
EXISTS_TAC `(v:real^3)
dot v - (u:real^3)
dot u` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_EQ_0; REAL_ARITH `&2 = &0 <=> F`;
VECTOR_SUB_EQ]
];
ALL_TAC
] THEN
STRIP_TAC THEN
MP_TAC (ISPECL [`s:(real^3->bool)`; `K:(real^3->bool)->bool`; `a:(real^3->bool)->real^3`; `b:(real^3->bool)->
real`]
FACET_OF_POLYHEDRON_EXPLICIT) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
EQ_TAC THENL
[
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `h:real^3->bool`) THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (is_forall o concl)) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `h:real^3->bool`) THEN
ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THENL
[
EXISTS_TAC `v:real^3` THEN
SUBGOAL_THEN `{x:real^3 | x | (a:(real^3->bool)->real^3) h
dot x = b h} = bis u v` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`(a:(real^3->bool)->real^3) h`; `(b:(real^3->bool)->
real) h`; `v:real^3`; `u:real^3`]
HALFSPACE_EQ_BIS_LE_IMP_HYPERPLANE_EQ_BIS) THEN
ASM_MESON_TAC[
BIS_SYM];
ALL_TAC
] THEN
ASM_MESON_TAC[];
EXISTS_TAC `v:real^3` THEN
SUBGOAL_THEN `{x:real^3 | x | (a:(real^3->bool)->real^3) h
dot x = b h} = bis u v` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`(a:(real^3->bool)->real^3) h`; `(b:(real^3->bool)->
real) h`; `u:real^3`; `v:real^3`]
HALFSPACE_EQ_BIS_LE_IMP_HYPERPLANE_EQ_BIS) THEN
ASM_MESON_TAC[];
ALL_TAC
] THEN
ASM_MESON_TAC[]
];
ALL_TAC
] THEN
STRIP_TAC THENL
[
ABBREV_TAC `h =
bis_le (v:real^3) u` THEN
FIRST_X_ASSUM (MP_TAC o SPEC `h:real^3->bool`) THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (is_forall o concl)) THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o SPEC `h:real^3->bool`) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
EXISTS_TAC `h:real^3->bool` THEN
SUBGOAL_THEN `{x:real^3 | x | (a:(real^3->bool)->real^3) h
dot x = b h} = bis u v` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`(a:(real^3->bool)->real^3) h`; `(b:(real^3->bool)->
real) h`; `v:real^3`; `u:real^3`]
HALFSPACE_EQ_BIS_LE_IMP_HYPERPLANE_EQ_BIS) THEN
ASM_MESON_TAC[
BIS_SYM];
ALL_TAC
] THEN
ASM_MESON_TAC[];
ABBREV_TAC `h =
bis_le (u:real^3) v` THEN
FIRST_X_ASSUM (MP_TAC o SPEC `h:real^3->bool`) THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (is_forall o concl)) THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o SPEC `h:real^3->bool`) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
EXISTS_TAC `h:real^3->bool` THEN
SUBGOAL_THEN `{x:real^3 | x | (a:(real^3->bool)->real^3) h
dot x = b h} = bis u v` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`(a:(real^3->bool)->real^3) h`; `(b:(real^3->bool)->
real) h`; `u:real^3`; `v:real^3`]
HALFSPACE_EQ_BIS_LE_IMP_HYPERPLANE_EQ_BIS) THEN
ASM_MESON_TAC[
BIS_SYM];
ALL_TAC
] THEN
ASM_MESON_TAC[]
]);;
let IDBEZAL = prove(`!V ul k F. saturated V /\ packing V /\ barV V k ul /\ (k < 3) ==>
(F
facet_of voronoi_list V ul <=>
(?vl. (F = voronoi_list V vl) /\ barV V (k+1) vl /\ (truncate_simplex k vl = ul)))`,
REPEAT STRIP_TAC THEN
EQ_TAC THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`]
VORONOI_BARV_CANONICAL) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
ABBREV_TAC `s = voronoi_list (V:real^3->bool) ul` THEN
MP_TAC (SPECL [`V:real^3->bool`; `K:(real^3->bool)->bool`; `s:real^3->bool`; `(
HD ul):real^3`]
FACET_OF_POLYHEDRON_EXPLICIT_BIS) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [ ASM_MESON_TAC[]; ALL_TAC ] THEN
DISCH_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `
aff_dim (F':real^3->bool) =
aff_dim (s:real^3->bool) - &1` ASSUME_TAC THENL
[
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
facet_of] THEN
SIMP_TAC[];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_THEN (CHOOSE_THEN MP_TAC) THEN
DISCH_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN (ASSUME_TAC o CONJUNCT2) THEN
DISCH_TAC THEN
EXISTS_TAC `
APPEND ul [v:real^3]` THEN
SUBGOAL_THEN `truncate_simplex k (
APPEND ul [v:real^3]) = ul` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
TRUNCATE_SIMPLEX] THEN
MATCH_MP_TAC CHOICE_LEMMA THEN
CONJ_TAC THENL
[
EXISTS_TAC `ul:(real^3)list` THEN
CONJ_TAC THENL [ ASM_MESON_TAC[
BARV]; ALL_TAC] THEN
REWRITE_TAC[
INITIAL_SUBLIST_APPEND];
ALL_TAC
] THEN
REPLICATE_TAC 9 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[
BARV] THEN
REPEAT STRIP_TAC THEN
ASSUME_TAC (ISPECL [`ul:(real^3)list`; `[v:real^3]`]
INITIAL_SUBLIST_APPEND) THEN
ASM_MESON_TAC[
INITIAL_SUBLIST_UNIQUE];
ALL_TAC
] THEN
SUBGOAL_THEN `F' = voronoi_list V (
APPEND ul [v:real^3])` ASSUME_TAC THENL
[
POP_ASSUM (fun
th -> ALL_TAC) THEN
REPLICATE_TAC 3 (POP_ASSUM MP_TAC) THEN
REPLICATE_TAC 5 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
FIRST_ASSUM (MP_TAC o (fun
th -> MATCH_MP
BARV_SUBSET th)) THEN
FIRST_ASSUM (MP_TAC o (fun
th -> MATCH_MP
BARV_CONS th)) THEN
STRIP_TAC THEN
FIRST_X_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
DISCH_TAC THEN DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
DISCH_TAC THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
MATCH_MP_TAC
VORONOI_LIST_INTER_BIS THEN
ASM_MESON_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `barV (V:real^3->bool) (k + 1) (
APPEND ul [v])` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
BARV] THEN CONJ_TAC THENL
[
REPLICATE_TAC 10 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
BARV;
LENGTH_APPEND] THEN
REWRITE_TAC[
LENGTH; SYM
ONE] THEN
SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
VORONOI_NONDG;
INITIAL_SUBLIST_APPEND_SING] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
EXPAND_TAC "s" THEN
POP_ASSUM (fun
th -> ALL_TAC) THEN
POP_ASSUM MP_TAC THEN
REPLICATE_TAC 5 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
POP_ASSUM MP_TAC THEN
FIRST_ASSUM (MP_TAC o (fun
th -> MATCH_MP
BARV_SUBSET th)) THEN
POP_ASSUM MP_TAC THEN
REPLICATE_TAC 2 (POP_ASSUM (fun
th -> ALL_TAC)) THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THENL
[
ASM_REWRITE_TAC[
LENGTH_APPEND;
LENGTH] THEN
ASM_SIMP_TAC [ARITH_RULE `k < 3 ==> (k + 1) + SUC 0 < 5`];
REWRITE_TAC[
SUBSET;
IN_SET_OF_LIST;
MEM_APPEND;
MEM] THEN
ASM_MESON_TAC[
SUBSET;
IN_SET_OF_LIST];
POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM (fun
th -> ALL_TAC) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
LENGTH_APPEND;
LENGTH] THEN
POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM MP_TAC THEN
POP_ASSUM MP_TAC THEN
POP_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 1`] THEN
ARITH_TAC
];
ALL_TAC
] THEN
POP_ASSUM ACCEPT_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
facet_of] THEN
REPEAT STRIP_TAC THENL
[
SUBGOAL_THEN `initial_sublist ul (vl:(real^3)list)` ASSUME_TAC THENL
[
SUBGOAL_TAC "A" `k + 1 <=
LENGTH (vl:(real^3)list)` [ ASM_MESON_TAC[
BARV; ARITH_RULE `k + 1 <= (k + 1) + 1`] ] THEN
ASM_MESON_TAC[
TRUNCATE_SIMPLEX_INITIAL_SUBLIST];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
KHEJKCI_GEN THEN
ASM_MESON_TAC[];
POP_ASSUM MP_TAC THEN POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM MP_TAC THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
STRIP_TAC THEN
POP_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
ASM_SIMP_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `k < 3 ==> (k + 1) + 1 < 5`; ARITH_RULE `0 < a + 1`] THEN
DISCH_THEN ((LABEL_TAC "A") o CONJUNCT2) THEN
DISCH_TAC THEN REMOVE_THEN "A" MP_TAC THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th;
AFF_DIM_EMPTY]) THEN
POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD] THEN
REWRITE_TAC[INT_ARITH `-- &1 + (((&k):int) + &1) + &1 = &4 <=> (&k):int = &3`] THEN
ONCE_REWRITE_TAC[
INT_OF_NUM_EQ] THEN
ACCEPT_TAC (ARITH_RULE `k < 3 ==> ~(k = 3)`);
POP_ASSUM (fun
th -> ALL_TAC) THEN
POP_ASSUM MP_TAC THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
ASM_SIMP_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < a + 1`] THEN
DISCH_THEN (CONJUNCTS_THEN2 (fun
th -> ALL_TAC) (ASSUME_TAC o CONJUNCT2)) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_SIMP_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < a + 1`] THEN
DISCH_THEN (CONJUNCTS_THEN2 (fun
th -> ALL_TAC) (ASSUME_TAC o CONJUNCT2)) THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD] THEN
INT_ARITH_TAC
]);;
let VORONOI_LIST_EQ_UNION_CONVEX_HULL_FACETS = prove(`!V ul k p. packing V /\ saturated V /\ barV V k ul /\ k < 3 /\ p
IN voronoi_list V ul
==> voronoi_list V ul =
UNIONS {
convex hull (p
INSERT voronoi_list V vl) | vl |
barV V (k + 1) vl /\ truncate_simplex k vl = ul}`,
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`{p:real^3}`; `voronoi_list V ul`]
POLYTOPE_UNION_CONVEX_HULL_FACETS) THEN
ANTS_TAC THENL
[
ASM_SIMP_TAC[
SUBSET;
IN_SING] THEN
REPEAT CONJ_TAC THENL
[
MATCH_MP_TAC
POLYTOPE_VORONOI_LIST THEN
ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[GSYM
LENGTH_EQ_NIL] THEN
UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV] THEN
ARITH_TAC;
UNDISCH_TAC `barV V k ul` THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 1`] THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD] THEN
UNDISCH_TAC `k < 3` THEN
REWRITE_TAC[GSYM
INT_OF_NUM_LT] THEN
INT_ARITH_TAC;
REWRITE_TAC[
EXTENSION;
IN_SING;
NOT_IN_EMPTY;
NOT_FORALL_THM] THEN
EXISTS_TAC `p:real^3` THEN REWRITE_TAC[]
];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> ONCE_REWRITE_TAC[
th]) THEN
AP_TERM_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `k:num`]
IDBEZAL) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> ONCE_REWRITE_TAC[
th]) THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN GEN_TAC THEN
REWRITE_TAC[GSYM
EXTENSION; SET_RULE `!f:real^3->bool. {p}
UNION f = p
INSERT f`] THEN
EQ_TAC THEN STRIP_TAC THENL
[
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[];
EXISTS_TAC `voronoi_list V vl` THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[]
]);;
let NUMSEG_SUBSET_INDUCT = prove(`!s a b. (a
IN s) /\ (!k. a <= k /\ SUC k <= b /\ k
IN s ==> SUC k
IN s) ==> a..b
SUBSET s`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[
SUBSET;
IN_NUMSEG] THEN
INDUCT_TAC THENL
[
REWRITE_TAC[ARITH_RULE `a <= 0 <=> a = 0`] THEN DISCH_TAC THEN
UNDISCH_TAC `a:num
IN s` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_TAC THEN
ASM_CASES_TAC `SUC x = a:num` THENL
[
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (ARITH_RULE `a <= SUC x /\ ~(SUC x = a) /\ SUC x <= b ==> a <= x /\ x <= b`) THEN
ASM_SIMP_TAC[]);;
let BARV_EXISTS = prove(`!V wl k. packing V /\ saturated V /\ k < 3 /\ barV V k wl ==> ?vl. barV V (SUC k) vl /\ truncate_simplex k vl = wl`,
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`voronoi_list V wl`]
POLYTOPE_FACET_EXISTS) THEN
ANTS_TAC THENL
[
CONJ_TAC THENL
[
MATCH_MP_TAC
POLYTOPE_VORONOI_LIST_BARV THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[
AFF_DIM_VORONOI_LIST] THEN
UNDISCH_TAC `k < 3` THEN REWRITE_TAC[GSYM
INT_OF_NUM_LT] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `wl:(real^3)list`; `k:num`; `f:real^3->bool`]
IDBEZAL) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[
ADD1]);;
let BARV_EXISTS_ALT = prove(`!V k. packing V /\ saturated V /\ k <= 3 ==> ?ul. barV V k ul`,
GEN_TAC THEN INDUCT_TAC THENL
[
STRIP_TAC THEN
MP_TAC (SPEC_ALL
TIWWFYQ) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
EXISTS_TAC `[v:real^3]` THEN
ASM_SIMP_TAC[
BARV_0];
ALL_TAC
] THEN
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o check (is_imp o concl)) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC k <= 3 ==> k <= 3`] THEN
STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `k:num`]
BARV_EXISTS) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC k <= 3 ==> k < 3`] THEN
STRIP_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[]);;
let GLTVHUM_lemma1 = prove(`!V ul j. packing V /\ saturated V /\ j < 3 /\ barV V j ul
==> {k | k
IN j..3 /\
voronoi_list V ul =
UNIONS {
convex hull ({omega_list_n V vl i | i
IN j..k-1}
UNION voronoi_list V vl) | vl |
barV V k vl /\ truncate_simplex j vl = ul}} = j..3`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
SUBSET_ANTISYM THEN
CONJ_TAC THENL
[
SIMP_TAC[
SUBSET;
IN_ELIM_THM];
ALL_TAC
] THEN
MATCH_MP_TAC
NUMSEG_SUBSET_INDUCT THEN
REWRITE_TAC[
SUBSET;
IN_NUMSEG;
IN_ELIM_THM;
LE_REFL] THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = j + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V j ul` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
SUBGOAL_THEN `!vl. barV V j vl /\ truncate_simplex j vl = ul <=> vl = ul` (LABEL_TAC "j") THENL
[
GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL
[
UNDISCH_TAC `barV V j vl` THEN UNDISCH_TAC `barV V j ul` THEN
REWRITE_TAC[
BARV] THEN REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`j:num`; `vl:(real^3)list`; `vl:(real^3)list`]
TRUNCATE_SIMPLEX_INITIAL_SUBLIST) THEN
ASM_REWRITE_TAC[
LE_REFL;
INITIAL_SUBLIST_REFL;
EQ_SYM_EQ];
ALL_TAC
] THEN
ASM_SIMP_TAC[
TRUNCATE_SIMPLEX_REFL];
ALL_TAC
] THEN
CONJ_TAC THENL
[
ASM_REWRITE_TAC[
SING_GSPEC_APP;
UNIONS_1] THEN
ASM_SIMP_TAC[
LT_IMP_LE] THEN
ASM_CASES_TAC `j = 0` THENL
[
SUBGOAL_THEN `?x. ul = [x:real^3]` CHOOSE_TAC THENL
[
EXISTS_TAC `
HD ul:real^3` THEN
MATCH_MP_TAC
LENGTH_1_LEMMA THEN
UNDISCH_TAC `
LENGTH (ul:(real^3)list) = j + 1` THEN ASM_REWRITE_TAC[ARITH];
ALL_TAC
] THEN
ASM_REWRITE_TAC[ARITH_RULE `0 - 1 = 0`; ARITH_RULE `j <= 0 <=> j = 0`; ARITH_RULE `0 <= i /\ i = 0 <=> i = 0`] THEN
REWRITE_TAC[
SING_GSPEC_APP;
VORONOI_LIST;
set_of_list;
VORONOI_SET;
INTERS_1;
IN_SING;
OMEGA_LIST_N;
HD;
SING_UNION_EQ_INSERT] THEN
SUBGOAL_THEN `x:real^3
INSERT voronoi_closed V x =
voronoi_closed V x` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[GSYM
ABSORPTION;
CENTER_IN_VORONOI_CELL];
ALL_TAC
] THEN
REWRITE_TAC[
EQ_SYM_EQ;
CONVEX_HULL_EQ;
CONVEX_VORONOI_CLOSED];
ALL_TAC
] THEN
ASM_SIMP_TAC[ARITH_RULE `~(j = 0) ==> (j <= i /\ i <= j - 1 <=> F)`] THEN
SUBGOAL_THEN `{omega_list_n V ul i | i | F} = {}` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
NOT_IN_EMPTY];
ALL_TAC
] THEN
REWRITE_TAC[
UNION_EMPTY;
EQ_SYM_EQ;
CONVEX_HULL_EQ;
CONVEX_VORONOI_LIST];
ALL_TAC
] THEN
REPEAT STRIP_TAC THENL
[
ASM_SIMP_TAC[ARITH_RULE `j <= k ==> j <= SUC k`];
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[ARITH_RULE `SUC k - 1 = k`] THEN
ABBREV_TAC `f = \vl:(real^3)list. {omega_list_n V vl i | j <= i /\ i <= k}
UNION voronoi_list V vl` THEN
SUBGOAL_THEN `!vl:(real^3)list. {omega_list_n V vl i | j <= i /\ i <= k}
UNION voronoi_list V vl = f vl` (fun
th -> REWRITE_TAC[
th]) THENL
[
EXPAND_TAC "f" THEN REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!P:(real^3)list->(real^3->bool).
UNIONS {P vl | barV V (SUC k) vl /\ truncate_simplex j vl = ul} =
UNIONS {
UNIONS {P vl | vl | barV V (SUC k) vl /\ truncate_simplex k vl = wl} | wl | barV V k wl /\ truncate_simplex j wl = ul}` (fun
th -> REWRITE_TAC[
th]) THENL
[
GEN_TAC THEN
REWRITE_TAC[
EXTENSION;
IN_UNIONS] THEN GEN_TAC THEN
REWRITE_TAC[GSYM
EXTENSION;
IN_ELIM_THM;
IN_UNIONS] THEN
EQ_TAC THEN STRIP_TAC THENL
[
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
EXISTS_TAC `
UNIONS {(P vl'):(real^3->bool) | vl' | barV V (SUC k) vl' /\ truncate_simplex k vl' = truncate_simplex k vl}` THEN
CONJ_TAC THENL
[
EXISTS_TAC `truncate_simplex k vl:(real^3)list` THEN
REWRITE_TAC[] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `SUC k` THEN ASM_REWRITE_TAC[ARITH_RULE `k <= SUC k`];
ALL_TAC
] THEN
MP_TAC (ISPECL [`vl:(real^3)list`; `j:num`; `k:num`]
TRUNCATE_TRUNCATE_SIMPLEX) THEN
UNDISCH_TAC `barV V (SUC k) vl` THEN SIMP_TAC[
BARV] THEN DISCH_TAC THEN
ASM_SIMP_TAC[ARITH_RULE `k + 1 <= SUC k + 1`];
ALL_TAC
] THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM] THEN
EXISTS_TAC `(P (vl:(real^3)list)):real^3->bool` THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM] THEN
STRIP_TAC THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
EXISTS_TAC `(P (vl:(real^3)list)):real^3->bool` THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `truncate_simplex j wl = ul:(real^3)list` THEN
MP_TAC (ISPECL [`vl:(real^3)list`; `j:num`; `k:num`]
TRUNCATE_TRUNCATE_SIMPLEX) THEN
UNDISCH_TAC `barV V (SUC k) vl` THEN SIMP_TAC[
BARV] THEN DISCH_TAC THEN
ASM_SIMP_TAC[ARITH_RULE `k + 1 <= SUC k + 1`];
ALL_TAC
] THEN
ABBREV_TAC `g = \vl:(real^3)list.
convex hull (omega_list_n V vl k
INSERT voronoi_list V vl)` THEN
SUBGOAL_THEN `!vl:(real^3)list.
convex hull (f vl):real^3->bool =
convex hull ({omega_list_n V vl i | j <= i /\ i <= k - 1}
UNION g vl)` (fun
th -> REWRITE_TAC[
th]) THENL
[
GEN_TAC THEN
EXPAND_TAC "g" THEN
ONCE_REWRITE_TAC[GSYM
CONV_UNION_lemma] THEN
ONCE_REWRITE_TAC[GSYM
SING_UNION_EQ_INSERT] THEN
REWRITE_TAC[GSYM
UNION_ASSOC] THEN
SUBGOAL_THEN `{omega_list_n V vl i | j <= i /\ i <= k - 1}
UNION {omega_list_n V vl k} = {omega_list_n V vl i | j <= i /\ i <= k}` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
EXTENSION;
IN_UNION;
IN_ELIM_THM;
IN_SING] THEN
GEN_TAC THEN EQ_TAC THENL
[
STRIP_TAC THENL
[
EXISTS_TAC `i:num` THEN
ASM_SIMP_TAC[ARITH_RULE `i <= k - 1 ==> i <= k`];
ALL_TAC
] THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[
LE_REFL];
ALL_TAC
] THEN
STRIP_TAC THEN
ASM_CASES_TAC `i = k:num` THENL
[
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISJ1_TAC THEN
EXISTS_TAC `i:num` THEN
ASM_SIMP_TAC[ARITH_RULE `i <= k /\ ~(i = k) ==> i <= k - 1`];
ALL_TAC
] THEN
EXPAND_TAC "f" THEN REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!wl. barV V k wl ==>
UNIONS {g vl | vl | barV V (SUC k) vl /\ truncate_simplex k vl = wl} = voronoi_list V wl` (LABEL_TAC "wl") THENL
[
REPEAT STRIP_TAC THEN
ABBREV_TAC `p = omega_list_n V wl k` THEN
SUBGOAL_THEN `
UNIONS {g vl | barV V (SUC k) vl /\ truncate_simplex k vl = wl} =
UNIONS {
convex hull ((p:real^3)
INSERT voronoi_list V vl) | vl | barV V (SUC k) vl /\ truncate_simplex k vl = wl}` (fun
th -> REWRITE_TAC[
th]) THENL
[
AP_TERM_TAC THEN
EXPAND_TAC "g" THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN GEN_TAC THEN
REWRITE_TAC[GSYM
EXTENSION] THEN
SUBGOAL_THEN `!vl. barV V (SUC k) vl /\ truncate_simplex k vl = wl ==> omega_list_n V vl k = p` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `k:num`; `0:num`]
OMEGA_LIST_N_LEMMA) THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = k + 2 /\
LENGTH (wl:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k wl` THEN UNDISCH_TAC `barV V (SUC k) vl` THEN
SIMP_TAC[
BARV; ARITH_RULE `k + 2 = SUC k + 1`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[ARITH_RULE `k + 0 + 1 <= k + 2`] THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[
th;
ADD_0]);
ALL_TAC
] THEN
EQ_TAC THENL
[
STRIP_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
ADD1] THEN
MATCH_MP_TAC (GSYM
VORONOI_LIST_EQ_UNION_CONVEX_HULL_FACETS) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC k <= 3 ==> k < 3`] THEN
EXPAND_TAC "p" THEN
SUBGOAL_THEN `voronoi_list V wl = voronoi_list V (truncate_simplex k wl):real^3->bool` (fun
th -> ONCE_REWRITE_TAC[
th]) THENL
[
SUBGOAL_THEN `truncate_simplex k wl = wl:(real^3)list` (fun
th -> REWRITE_TAC[
th]) THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_REFL THEN
UNDISCH_TAC `barV V k wl` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
MATCH_MP_TAC
OMEGA_LIST_N_IN_VORONOI_LIST THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[
LE_REFL];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
AP_TERM_TAC THEN
SUBGOAL_THEN `!wl. barV V k wl ==>
UNIONS {
convex hull ({omega_list_n V vl i | j <= i /\ i <= k - 1}
UNION g vl) | vl | barV V (SUC k) vl /\ truncate_simplex k vl = wl} =
convex hull ({omega_list_n V wl i | j <= i /\ i <= k - 1}
UNION voronoi_list V wl)` (LABEL_TAC "wl2") THENL
[
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `
LENGTH (wl:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k wl` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
ABBREV_TAC `S = {omega_list_n V wl i | j <= i /\ i <= k - 1}` THEN
SUBGOAL_THEN `!vl. barV V (SUC k) vl /\ truncate_simplex k vl = wl ==> {omega_list_n V vl i | j <= i /\ i <= k - 1} = S` (LABEL_TAC "vl") THENL
[
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `!i. j <= i /\ i <= k - 1 ==> omega_list_n V wl i = omega_list_n V vl i` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = k + 2` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V (SUC k) vl` THEN SIMP_TAC[
BARV; ARITH_RULE `k + 2 = SUC k + 1`];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `i:num`; `k - i:num`]
OMEGA_LIST_N_LEMMA) THEN
ASM_SIMP_TAC[ARITH_RULE `i <= k - 1 ==> i + k - i + 1 = k + 1 /\ i + k - i = k`; ARITH_RULE `k + 1 <= k + 2`];
ALL_TAC
] THEN
EXPAND_TAC "S" THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN
GEN_TAC THEN EQ_TAC THEN STRIP_TAC THENL
[
EXISTS_TAC `i:num` THEN ASM_SIMP_TAC[];
EXISTS_TAC `i:num` THEN ASM_SIMP_TAC[]
];
ALL_TAC
] THEN
MP_TAC (ISPECL [`{(g vl):real^3->bool | vl | barV V (SUC k) vl /\ truncate_simplex k vl = wl}`; `S:real^3->bool`]
CONVEX_HULL_UNION_UNIONS) THEN
ANTS_TAC THENL
[
ASM_SIMP_TAC[
CONVEX_VORONOI_LIST] THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
MP_TAC (SPEC_ALL
BARV_EXISTS) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC k <= 3 ==> k < 3`] THEN
STRIP_TAC THEN
EXISTS_TAC `(g (vl:(real^3)list)):real^3->bool` THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[] THEN
DISCH_THEN (fun
th -> ALL_TAC) THEN
AP_TERM_TAC THEN
REWRITE_TAC[
EXTENSION] THEN GEN_TAC THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
EQ_TAC THENL
[
STRIP_TAC THEN
EXISTS_TAC `(g (vl:(real^3)list)):real^3->bool` THEN
ASM_SIMP_TAC[] THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION] THEN GEN_TAC THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
EQ_TAC THENL
[
STRIP_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_SIMP_TAC[];
STRIP_TAC THEN
EXISTS_TAC `wl:(real^3)list` THEN
ASM_SIMP_TAC[]
]);;
let GLTVHUM = prove(`!V (u0:real^3) p. packing V /\ saturated V /\ (u0
IN V) ==>
(p
IN voronoi_closed V u0 <=>
(?vl. vl
IN barV V 3 /\ p
IN rogers V vl /\ (truncate_simplex 0 vl = [u0])))`,
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `[u0:real^3]`; `0`]
GLTVHUM_lemma1) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[ARITH_RULE `0 < 3`] THEN
ASM_SIMP_TAC[
BARV_0];
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION] THEN
DISCH_THEN (MP_TAC o SPEC `3`) THEN
REWRITE_TAC[
IN_NUMSEG;
LE_REFL;
LE_0;
IN_ELIM_THM] THEN
SUBGOAL_THEN `voronoi_list V [u0:real^3] =
voronoi_closed V u0` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
VORONOI_LIST;
set_of_list;
VORONOI_SET;
IN_SING;
SING_GSPEC_APP;
INTERS_1];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM] THEN
SUBGOAL_THEN `!vl. barV V 3 vl ==>
convex hull ({omega_list_n V vl i | i <= 2}
UNION voronoi_list V vl) = rogers V vl` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
ROGERS;
IMAGE_LEMMA] THEN
AP_TERM_TAC THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = 4` ASSUME_TAC THENL
[
POP_ASSUM MP_TAC THEN SIMP_TAC[
BARV; ARITH_RULE `3 + 1 = 4`];
ALL_TAC
] THEN
SUBGOAL_THEN `voronoi_list V vl = {omega_list_n V vl 3}` (fun
th -> REWRITE_TAC[
th]) THENL
[
SUBGOAL_THEN `
aff_dim (voronoi_list V vl:real^3->bool) = &0` MP_TAC THENL
[
ONCE_REWRITE_TAC[INT_ARITH `&0 =
int_of_num 3 - &3`] THEN
MATCH_MP_TAC
AFF_DIM_VORONOI_LIST THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
AFF_DIM_EQ_0] THEN
STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `3`; `3`]
OMEGA_LIST_N_IN_VORONOI_LIST) THEN
ASM_REWRITE_TAC[
LE_REFL] THEN
SUBGOAL_THEN `truncate_simplex 3 vl = vl:(real^3)list` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_REFL THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
ASM_SIMP_TAC[
IN_SING];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
EXTENSION;
IN_UNION] THEN GEN_TAC THEN
REWRITE_TAC[
IN_SING;
IN_ELIM_THM] THEN
EQ_TAC THENL
[
STRIP_TAC THENL
[
EXISTS_TAC `i:num` THEN
ASM_SIMP_TAC[ARITH_RULE `i <= 2 ==> i < 4`];
ALL_TAC
] THEN
EXISTS_TAC `3` THEN
ASM_REWRITE_TAC[ARITH_RULE `3 < 4`];
ALL_TAC
] THEN
STRIP_TAC THEN
ASM_CASES_TAC `x' = 3` THENL
[
UNDISCH_TAC `x = omega_list_n V vl x'` THEN ASM_SIMP_TAC[];
ALL_TAC
] THEN
DISJ1_TAC THEN
EXISTS_TAC `x':num` THEN
ASM_SIMP_TAC[ARITH_RULE `x' < 4 /\ ~(x' = 3) ==> x' <= 2`];
ALL_TAC
] THEN
REWRITE_TAC[ARITH_RULE `3 - 1 = 2`] THEN
EQ_TAC THEN STRIP_TAC THENL
[
POP_ASSUM MP_TAC THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
UNDISCH_TAC `vl
IN barV V 3` THEN DISCH_THEN (ASSUME_TAC o REWRITE_RULE[
IN]) THEN
EXISTS_TAC `rogers V vl` THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `vl:(real^3)list` THEN
ASM_SIMP_TAC[]);;
let ODIGPXU_lemma = prove(`!P f f' p0 p (q : real^N) t s.
polyhedron P /\ p0
IN P /\ ~(p0
IN f
UNION f') /\
f
facet_of P /\ f'
facet_of P /\
p
IN f /\ q
IN f' /\
&0 < t /\ &0 < s /\
(&1 - t) % p0 + t % p = (&1 - s) % p0 + s % q ==> s <= t`,
REPEAT STRIP_TAC THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
VECTOR_SUB_RDISTRIB] THEN
REWRITE_TAC[VECTOR_ARITH `a - t % (p0 : real^N) + t % p = a - s % p0 + s % q <=> t % (p - p0) = s % (q - p0)`] THEN
MP_TAC (REAL_ARITH `s <= t \/ t < s`) THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o AP_TERM `\x : real^N. inv t % x`) THEN REWRITE_TAC[
VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < t ==> ~(t = &0)`; REAL_MUL_LINV;
VECTOR_MUL_LID] THEN
ABBREV_TAC `r = inv t * s` THEN
SUBGOAL_THEN `&1 < r` ASSUME_TAC THENL
[
MP_TAC (SPECL [`&1`; `r :
real`; `t :
real`]
REAL_LT_LMUL_EQ) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[GSYM
th]) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
REWRITE_TAC[REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[
REAL_MUL_RID; REAL_ARITH `&0 < t ==> ~(t = &0)`;
REAL_MUL_RINV; REAL_MUL_LID];
ALL_TAC
] THEN
MP_TAC (SPECL[`P : real^N -> bool`; `f' : real^N -> bool`]
FACET_OF_POLYHEDRON) THEN
ASM_REWRITE_TAC[] THEN
REPEAT (DISCH_THEN (CHOOSE_THEN MP_TAC)) THEN REWRITE_TAC[
SUBSET] THEN
STRIP_TAC THEN
SUBGOAL_THEN `(a : real^N)
dot p0 < b` ASSUME_TAC THENL
[
REWRITE_TAC[REAL_ARITH `x < y <=> x <= y /\ ~(x = y)`] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `p0 : real^N`) THEN
ASM_REWRITE_TAC[
IN_ELIM_THM] THEN SIMP_TAC[] THEN DISCH_THEN (fun
th -> ALL_TAC) THEN
UNDISCH_TAC `~(p0 : real^N
IN f
UNION f')` THEN
ASM_REWRITE_TAC[CONTRAPOS_THM;
IN_UNION] THEN DISCH_TAC THEN
DISJ2_TAC THEN ASM_REWRITE_TAC[
IN_INTER;
IN_ELIM_THM];
ALL_TAC
] THEN
REWRITE_TAC[VECTOR_ARITH `p - p0 = x <=> p = p0 + x : real^N`] THEN DISCH_TAC THEN
MP_TAC (ISPECL[`f : real^N -> bool`; `P : real^N -> bool`]
FACET_OF_IMP_SUBSET) THEN
ASM_REWRITE_TAC[
SUBSET] THEN DISCH_THEN (MP_TAC o SPEC `p : real^N`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> FIRST_X_ASSUM (MP_TAC o (fun
th2 -> MATCH_MP
th2 th))) THEN
REWRITE_TAC[
IN_ELIM_THM;
DOT_RADD;
DOT_RMUL;
DOT_RSUB] THEN
REPLICATE_TAC 3 (POP_ASSUM MP_TAC) THEN REWRITE_TAC[
EXTENSION] THEN
DISCH_THEN (MP_TAC o SPEC `q : real^N`) THEN ASM_REWRITE_TAC[
IN_INTER;
IN_ELIM_THM] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN DISCH_TAC THEN DISCH_THEN (fun
th -> ALL_TAC) THEN
REWRITE_TAC[REAL_ARITH `x + r * (b - x) <= b <=> (r - &1) * b <= (r - &1) * x`] THEN
ASM_SIMP_TAC[REAL_ARITH `&1 < r ==> &0 < r - &1`;
REAL_LE_LMUL_EQ] THEN
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let ODIGPXU = prove(`!P f f' p0 p (q : real^N) t s.
polyhedron P /\ p0
IN P /\ ~(p0
IN f
UNION f') /\
f
facet_of P /\ f'
facet_of P /\
p
IN f /\ q
IN f' /\
&0 < t /\ &0 < s /\
(&1 - t) % p0 + t % p = (&1 - s) % p0 + s % q
==> s = t`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[REAL_ARITH `s = t <=> s <= t /\ t <= s`] THEN
STRIP_TAC THEN MATCH_MP_TAC
ODIGPXU_lemma THENL
[
MAP_EVERY EXISTS_TAC [`P : real^N -> bool`; `f : real^N -> bool`; `f' : real^N -> bool`] THEN
MAP_EVERY EXISTS_TAC [`p0 : real^N`; `p : real^N`; `q : real^N`] THEN
ASM_REWRITE_TAC[];
MAP_EVERY EXISTS_TAC [`P : real^N -> bool`; `f' : real^N -> bool`; `f : real^N -> bool`] THEN
MAP_EVERY EXISTS_TAC [`p0 : real^N`; `q : real^N`; `p : real^N`] THEN
ASM_REWRITE_TAC[
UNION_COMM]
]);;
let OMEGA_LIST_N_EQ = prove(`!V ul i j. omega_list_n V ul i
IN voronoi_list V (truncate_simplex (SUC i) ul)
==> omega_list_n V ul (SUC i) = omega_list_n V ul i`,
REPEAT GEN_TAC THEN ABBREV_TAC `X = voronoi_list V (truncate_simplex (SUC i) ul)` THEN
ASM_REWRITE_TAC[
OMEGA_LIST_N] THEN DISCH_TAC THEN
MP_TAC (ISPECL[`X:real^3 -> bool`; `omega_list_n V ul i`]
CLOSEST_POINT_EXISTS) THEN
ANTS_TAC THENL
[
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXPAND_TAC "X" THEN REWRITE_TAC[
CLOSED_VORONOI_LIST] THEN
EXISTS_TAC `omega_list_n V ul i` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
POP_ASSUM (MP_TAC o SPEC `omega_list_n V ul i`) THEN
ASM_REWRITE_TAC[
DIST_REFL;
DIST_LE_0] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]));;
let OMEGA_LIST_N_IN_FACET = prove(`!V ul k i. packing V /\ saturated V /\ barV V k ul /\ i < k
==> ?F. F
facet_of voronoi_list V (truncate_simplex i ul) /\
voronoi_list V (truncate_simplex (i + 1) ul) = F /\
(!j. i < j /\ j <= k ==> omega_list_n V ul j
IN F)`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `
FF = voronoi_list V (truncate_simplex (i + 1) ul)` THEN
EXISTS_TAC `
FF : real^3 -> bool` THEN REWRITE_TAC[] THEN
CONJ_TAC THENL
[
MP_TAC (SPECL[`V:real^3->bool`; `truncate_simplex i ul : (real^3)list`; `i:num`; `FF:real^3->bool`]
IDBEZAL) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MP_TAC (SPEC_ALL
BARV_IMP_K_LE_3) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ARITH_RULE `i < k /\ k <= 3 ==> i < 3`) THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN ASM_SIMP_TAC[
LT_IMP_LE];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
EXISTS_TAC `truncate_simplex (i + 1) ul : (real^3)list` THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN ASM_SIMP_TAC[ARITH_RULE `i < k ==> i + 1 <= k`];
MP_TAC (ISPECL[`ul:(real^3)list`; `i:num`; `i + 1`]
TRUNCATE_TRUNCATE_SIMPLEX) THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[ARITH_RULE `i <= i + 1`] THEN
UNDISCH_TAC `barV V k ul` THEN REWRITE_TAC[
BARV] THEN DISCH_TAC THEN
ASM_SIMP_TAC[ARITH_RULE `i < k ==> (i + 1) + 1 <= k + 1`]
];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPECL[`V:real^3->bool`; `ul:(real^3)list`; `k:num`; `j:num`]
OMEGA_LIST_N_IN_VORONOI_LIST) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `voronoi_list V (truncate_simplex j ul)
SUBSET FF` ASSUME_TAC THENL
[
EXPAND_TAC "FF" THEN REWRITE_TAC[
VORONOI_LIST;
VORONOI_SET] THEN
REWRITE_TAC[
SUBSET;
IN_INTERS;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `v:real^3` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[
IN_SET_OF_LIST;
MEM_EXISTS_EL] THEN
UNDISCH_TAC `barV V k ul` THEN REWRITE_TAC[
BARV] THEN DISCH_TAC THEN
MP_TAC (ISPECL[`i + 1`; `ul:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
MP_TAC (ISPECL[`j:num`; `ul:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[ARITH_RULE `j + 1 <= k + 1 <=> j <= k`] THEN
ASM_SIMP_TAC[ARITH_RULE `i < k ==> i + 1 <= k`] THEN
REPLICATE_TAC 2 (DISCH_THEN (fun
th -> REWRITE_TAC[
th])) THEN
DISCH_THEN (X_CHOOSE_THEN `r:num` MP_TAC) THEN STRIP_TAC THEN POP_ASSUM MP_TAC THEN
MP_TAC (ISPECL[`ul:(real^3)list`; `i + 1`; `r:num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[ARITH_RULE `i < k ==> (i + 1) + 1 <= k + 1`] THEN
ASM_SIMP_TAC[ARITH_RULE `r < (i + 1) + 1 ==> r <= i + 1`] THEN
REPEAT DISCH_TAC THEN
EXISTS_TAC `r:num` THEN
MP_TAC (ARITH_RULE `j <= k /\ i < j /\ r < (i + 1) + 1 ==> r < j + 1`) THEN
ASM_SIMP_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL[`ul:(real^3)list`; `j:num`; `r:num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[ARITH_RULE `j + 1 <= k + 1 <=> j <= k`] THEN
ASM_SIMP_TAC[ARITH_RULE `r < j + 1 ==> r <= j`];
ALL_TAC
] THEN
DISCH_TAC THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `voronoi_list V (truncate_simplex j ul)` THEN
ASM_REWRITE_TAC[]);;
let OMEGA_LIST_N_IN_VORONOI_LIST_GEN = prove(`!V ul k i j. packing V /\ saturated V /\ barV V k ul /\
i <= j /\ j <= k ==>
omega_list_n V ul j
IN voronoi_list V (truncate_simplex i ul)`,
REPEAT STRIP_TAC THEN
UNDISCH_TAC `i <= j:num` THEN REWRITE_TAC[
LE_LT] THEN STRIP_TAC THENL
[
MP_TAC (SPEC_ALL
OMEGA_LIST_N_IN_FACET) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL
[
MATCH_MP_TAC (ARITH_RULE `i < j /\ j <= k ==> i < k : num`) THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `F':real^3->bool` THEN
CONJ_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
FACET_OF_IMP_SUBSET THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
OMEGA_LIST_N_IN_VORONOI_LIST THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[]);;
let TRUNCATE_SIMPLEX_SUBSET = prove(`!(ul:(A)list) i j. j <= i /\ i + 1 <=
LENGTH ul ==>
set_of_list (truncate_simplex j ul)
SUBSET set_of_list (truncate_simplex i ul)`,
REWRITE_TAC[
SUBSET;
IN_SET_OF_LIST;
MEM_EXISTS_EL] THEN REPEAT STRIP_TAC THEN
EXISTS_TAC `i':num` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
MP_TAC (SPECL[`i:num`; `ul:(A)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
MP_TAC (SPECL[`j:num`; `ul:(A)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
MP_TAC (ARITH_RULE `!x. j <= i /\ i + 1 <= x ==> j + 1 <= x`) THEN
MP_TAC (ARITH_RULE `i' < j + 1 /\ j <= i ==> i' < i + 1`) THEN
ASM_SIMP_TAC[] THEN REPEAT DISCH_TAC THEN
MP_TAC (SPECL[`ul:(A)list`; `i:num`; `i':num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[ARITH_RULE `i' < i + 1 ==> i' <= i`] THEN DISCH_TAC THEN
MP_TAC (SPECL[`ul:(A)list`; `j:num`; `i':num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[ARITH_RULE `i' < j + 1 ==> i' <= j`]);;
let OMEGA_LIST_N_EQ_GEN = prove(`!V ul k i j. packing V /\ saturated V /\ barV V k ul /\
i < j /\ j <= k /\ omega_list_n V ul i
IN voronoi_list V (truncate_simplex j ul)
==> omega_list_n V ul (SUC i) = omega_list_n V ul i`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
OMEGA_LIST_N_EQ THEN
MATCH_MP_TAC
IN_TRANS THEN EXISTS_TAC `voronoi_list V (truncate_simplex j ul)` THEN
ASM_REWRITE_TAC[
VORONOI_LIST] THEN
MATCH_MP_TAC
VORONOI_SET_SUBSET THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_SUBSET THEN
ASM_SIMP_TAC[ARITH_RULE `i < j ==> SUC i <= j`] THEN
UNDISCH_TAC `barV V k ul` THEN REWRITE_TAC[
BARV] THEN
ASM_SIMP_TAC[ARITH_RULE `j <= k ==> j + 1 <= k + 1`]);;
let CARD_LE_3 = prove(`!s. ~(s = {}) /\ FINITE s /\
CARD s <= 3 ==> ?x y z : A. s = {x, y, z}`,
REPEAT STRIP_TAC THEN
MP_TAC (ARITH_RULE `!n. n <= 3 ==> n = 0 \/ n = 1 \/ n = 2 \/ n = 3`) THEN
DISCH_THEN (MP_TAC o SPEC `
CARD (s:A->bool)`) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN POP_ASSUM MP_TAC THEN UNDISCH_TAC `FINITE (s:A->bool)` THEN REWRITE_TAC[IMP_IMP; GSYM
HAS_SIZE; num_CONV `3`; num_CONV `2`; num_CONV `1`;
HAS_SIZE_CLAUSES] THENL
[
DISCH_TAC THEN UNDISCH_TAC `~((s:A->bool) = {})` THEN ASM_REWRITE_TAC[];
STRIP_TAC THEN POP_ASSUM MP_TAC THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN
REPLICATE_TAC 3 (EXISTS_TAC `a:A`) THEN SET_TAC[];
REPEAT (DISCH_THEN (CHOOSE_THEN MP_TAC)) THEN
DISCH_THEN (CONJUNCTS_THEN2 MP_TAC (LABEL_TAC "A")) THEN
STRIP_TAC THEN REMOVE_THEN "A" MP_TAC THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN
EXISTS_TAC `a:A` THEN EXISTS_TAC `a':A` THEN EXISTS_TAC `a':A` THEN SET_TAC[];
REPEAT (DISCH_THEN (CHOOSE_THEN MP_TAC)) THEN
DISCH_THEN (CONJUNCTS_THEN2 MP_TAC (LABEL_TAC "A")) THEN
REPEAT (DISCH_THEN (CHOOSE_THEN MP_TAC)) THEN
DISCH_THEN (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
STRIP_TAC THEN REMOVE_THEN "A" MP_TAC THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN
EXISTS_TAC `a:A` THEN EXISTS_TAC `a':A` THEN EXISTS_TAC `a'':A` THEN REWRITE_TAC[]
]);;
let ROGERS_AFF_DIM_FULL = prove(`!V ul. barV V 3 ul /\
aff_dim (rogers V ul) = &3
==> !i j. i < 4 /\ j < 4 /\ ~(i = j) ==> ~(omega_list_n V ul i = omega_list_n V ul j)`,
REWRITE_TAC[
ROGERS;
AFF_DIM_CONVEX_HULL;
BARV; ARITH] THEN REPEAT GEN_TAC THEN
REWRITE_TAC[GSYM IMP_IMP] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_THEN (fun
th -> ALL_TAC) THEN REPEAT STRIP_TAC THEN
ABBREV_TAC `S =
IMAGE (omega_list_n V ul) {j | j < 4}` THEN
ABBREV_TAC `a = omega_list_n V ul i` THEN
ABBREV_TAC `b = omega_list_n V ul j` THEN
SUBGOAL_THEN `S
DELETE a
DELETE b
SUBSET IMAGE (omega_list_n V ul) {k | k < 4 /\ ~(k = i) /\ ~(k = j)}` MP_TAC THENL
[
REPLICATE_TAC 3 (POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th])) THEN
REWRITE_TAC[
SUBSET;
IN_DELETE;
IN_IMAGE;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
EXISTS_TAC `x':num` THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[
POP_ASSUM (fun
th -> ALL_TAC) THEN POP_ASSUM MP_TAC THEN ASM_SIMP_TAC[CONTRAPOS_THM];
POP_ASSUM MP_TAC THEN ASM_SIMP_TAC[CONTRAPOS_THM]
];
ALL_TAC
] THEN
SUBGOAL_THEN `FINITE (S:real^3->bool)` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN
MATCH_MP_TAC
FINITE_IMAGE THEN REWRITE_TAC[
FINITE_NUMSEG_LT];
ALL_TAC
] THEN
DISCH_TAC THEN
SUBGOAL_THEN `
CARD (S
DELETE a
DELETE (b:real^3)) <= 2` MP_TAC THENL
[
MATCH_MP_TAC
LE_TRANS THEN
EXISTS_TAC `
CARD (
IMAGE (omega_list_n V ul) {k | k < 4 /\ ~(k = i) /\ ~(k = j)})` THEN
SUBGOAL_THEN `{k | k < 4 /\ ~(k = i) /\ ~(k = j)}
HAS_SIZE 2` MP_TAC THENL
[
SUBGOAL_THEN `{k | k < 4 /\ ~(k = i) /\ ~(k = j)} = {k | k < 4}
DELETE i
DELETE j` ASSUME_TAC THENL
[
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
IN_DELETE;
CONJ_ACI];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
HAS_SIZE;
FINITE_DELETE;
FINITE_NUMSEG_LT] THEN
MP_TAC (ISPECL[`j:num`; `{k | k < 4}
DELETE i`]
CARD_DELETE) THEN
ASM_REWRITE_TAC[
FINITE_DELETE;
FINITE_NUMSEG_LT;
IN_DELETE;
IN_ELIM_THM] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
MP_TAC (ISPECL[`i:num`; `{k | k < 4}`]
CARD_DELETE) THEN
ASM_REWRITE_TAC[
FINITE_NUMSEG_LT;
IN_ELIM_THM] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
CARD_NUMSEG_LT] THEN ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
HAS_SIZE] THEN STRIP_TAC THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
CARD_SUBSET THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
FINITE_IMAGE THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC
CARD_IMAGE_LE THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (ISPECL[`S:real^3->bool`; `a:real^3`]
FINITE_DELETE) THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN ASM_SIMP_TAC[
CARD_DELETE;
IN_DELETE] THEN
SUBGOAL_THEN `b:real^3
IN S` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN EXPAND_TAC "b" THEN
REWRITE_TAC[
IN_IMAGE;
IN_ELIM_THM] THEN EXISTS_TAC `j:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (ISPECL[`S:real^3->bool`; `b:real^3`] Hypermap.CARD_ATLEAST_1) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_SIMP_TAC[GSYM
INT_OF_NUM_LE; GSYM
INT_OF_NUM_SUB] THEN
MP_TAC (ISPEC `S:real^3->bool`
AFF_DIM_LE_CARD) THEN ASM_REWRITE_TAC[] THEN
INT_ARITH_TAC);;
let VORONOI_LIST_AFF_DIM = prove(`!V ul k i. barV V k ul /\ i <= k
==>
aff_dim (voronoi_list V (truncate_simplex i ul)) = &3 - &i`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `barV V i (truncate_simplex i ul)` MP_TAC THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `truncate_simplex i ul : (real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < i + 1`] THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD] THEN
INT_ARITH_TAC);;
let DUUNHOR = prove(`!V ul vl. packing V /\ saturated V /\
ul
IN barV V 3 /\ vl
IN barV V 3 /\
~(rogers V ul = rogers V vl) ==>
coplanar (rogers V ul
INTER rogers V vl)`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `
LENGTH (ul : (real^3)list) = 4 /\
LENGTH (vl : (real^3)list) = 4` ASSUME_TAC THENL
[
REPLICATE_TAC 3 (POP_ASSUM MP_TAC) THEN SIMP_TAC[
BARV; ARITH];
ALL_TAC
] THEN
ASM_CASES_TAC `~(
aff_dim (rogers V ul) = &3)` THENL
[
MATCH_MP_TAC
COPLANAR_SUBSET THEN
EXISTS_TAC `rogers V ul` THEN REWRITE_TAC[
INTER_SUBSET] THEN
MATCH_MP_TAC
AFF_DIM_LE_2_IMP_COPLANAR THEN
MP_TAC (ISPEC `rogers V ul`
AFF_DIM_LE_UNIV) THEN REWRITE_TAC[DIMINDEX_3] THEN
POP_ASSUM MP_TAC THEN INT_ARITH_TAC;
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[NOT_CLAUSES] THEN
ASM_CASES_TAC `~(
aff_dim (rogers V vl) = &3)` THENL
[
DISCH_THEN (fun
th -> ALL_TAC) THEN MATCH_MP_TAC
COPLANAR_SUBSET THEN
EXISTS_TAC `rogers V vl` THEN REWRITE_TAC[
INTER_SUBSET] THEN
MATCH_MP_TAC
AFF_DIM_LE_2_IMP_COPLANAR THEN
MP_TAC (ISPEC `rogers V vl`
AFF_DIM_LE_UNIV) THEN REWRITE_TAC[DIMINDEX_3] THEN
POP_ASSUM MP_TAC THEN INT_ARITH_TAC;
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[NOT_CLAUSES] THEN
SUBGOAL_THEN `?k. k <= 3 /\ (!i. i < k ==> omega_list_n V ul i = omega_list_n V vl i /\
voronoi_list V (truncate_simplex i ul) = voronoi_list V (truncate_simplex i vl)) /\
~(voronoi_list V (truncate_simplex k ul) = voronoi_list V (truncate_simplex k vl))` MP_TAC THENL
[
ONCE_REWRITE_TAC[GSYM NOT_CLAUSES] THEN UNDISCH_TAC `~(rogers V ul = rogers V vl)` THEN
REWRITE_TAC[CONTRAPOS_THM;
NOT_EXISTS_THM; DE_MORGAN_THM;
NOT_FORALL_THM; NOT_IMP] THEN
REWRITE_TAC[
ROGERS] THEN DISCH_TAC THEN
AP_TERM_TAC THEN
SUBGOAL_THEN `!i. i < 4 ==> omega_list_n V ul i = omega_list_n V vl i /\ voronoi_list V (truncate_simplex i ul) = voronoi_list V (truncate_simplex i vl)` ASSUME_TAC THENL
[
MATCH_MP_TAC
num_WF THEN INDUCT_TAC THEN ASM_REWRITE_TAC[
OMEGA_LIST_N] THENL
[
REPLICATE_TAC 2 (DISCH_THEN (fun
th -> ALL_TAC)) THEN
POP_ASSUM (MP_TAC o SPEC `0`) THEN
REWRITE_TAC[ARITH_RULE `~(0 <= 3) <=> F`; ARITH_RULE `i < 0 <=> F`; ARITH_RULE `0 < 4`] THEN
REPLICATE_TAC 3 (POP_ASSUM MP_TAC) THEN
DISCH_THEN (LABEL_TAC "A") THEN DISCH_THEN (LABEL_TAC "B") THEN DISCH_TAC THEN
USE_THEN "A" (MP_TAC o MATCH_MP
BARV_SUBSET) THEN
USE_THEN "B" (MP_TAC o MATCH_MP
BARV_SUBSET) THEN
REWRITE_TAC[
SUBSET] THEN
USE_THEN "A" (MP_TAC o MATCH_MP
BARV_IMP_HD_IN_SET_OF_LIST) THEN DISCH_TAC THEN
USE_THEN "B" (MP_TAC o MATCH_MP
BARV_IMP_HD_IN_SET_OF_LIST) THEN DISCH_TAC THEN
DISCH_THEN (MP_TAC o SPEC `
HD vl : real^3`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
DISCH_THEN (MP_TAC o SPEC `
HD ul : real^3`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPEC `ul:(real^3)list`
TRUNCATE_0_EQ_HEAD) THEN
MP_TAC (ISPEC `vl:(real^3)list`
TRUNCATE_0_EQ_HEAD) THEN
ASM_REWRITE_TAC[ARITH] THEN
REPLICATE_TAC 2 (DISCH_THEN (fun
th -> REWRITE_TAC[
th])) THEN
REWRITE_TAC[
VORONOI_LIST_SING] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
VORONOI_CLOSED_EQ_LEMMA THEN
EXISTS_TAC `V : real^3 -> bool` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
POP_ASSUM (fun
th -> ALL_TAC) THEN
DISCH_THEN (LABEL_TAC "A") THEN DISCH_TAC THEN
REMOVE_THEN "A" MP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `SUC i`) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC i < 4 ==> SUC i <= 3`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[
DISCH_THEN (MP_TAC o SPEC `i':num`) THEN
MP_TAC (ARITH_RULE `SUC i < 4 /\ i' < SUC i ==> i' < 4`) THEN
ASM_SIMP_TAC[];
DISCH_THEN (MP_TAC o SPEC `i':num`) THEN
MP_TAC (ARITH_RULE `SUC i < 4 /\ i' < SUC i ==> i' < 4`) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o SPEC `i:num`) THEN
ASM_SIMP_TAC[ARITH_RULE `i < SUC i`; ARITH_RULE `SUC i < 4 ==> i < 4`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
EXTENSION] THEN GEN_TAC THEN
REWRITE_TAC[
IN_IMAGE;
IN_ELIM_THM] THEN EQ_TAC THENL
[
DISCH_THEN (CHOOSE_THEN ASSUME_TAC) THEN
EXISTS_TAC `x' : num` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `x':num`) THEN ASM_SIMP_TAC[];
DISCH_THEN (CHOOSE_THEN ASSUME_TAC) THEN
EXISTS_TAC `x' : num` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `x':num`) THEN ASM_SIMP_TAC[];
];
ALL_TAC
] THEN
DISCH_THEN (CHOOSE_THEN STRIP_ASSUME_TAC) THEN
DISCH_TAC THEN DISCH_TAC THEN
ABBREV_TAC `X = voronoi_list V (truncate_simplex k ul)
INTER voronoi_list V (truncate_simplex k vl)` THEN
ABBREV_TAC `Y =
IMAGE (omega_list_n V ul) {j | j < k}` THEN
SUBGOAL_THEN `rogers V ul
INTER rogers V vl
SUBSET convex hull (Y
UNION X)` MP_TAC THENL
[
REWRITE_TAC[
SUBSET;
IN_INTER] THEN X_GEN_TAC `w:real^3` THEN
ABBREV_TAC `YX =
convex hull (Y
UNION X) : real^3->bool` THEN
ASM_REWRITE_TAC[
ROGERS;
CONVEX_HULL_FINITE] THEN
ABBREV_TAC `L1 =
IMAGE (omega_list_n V ul) {j | j < 4}` THEN
ABBREV_TAC `L2 =
IMAGE (omega_list_n V vl) {j | j < 4}` THEN
REWRITE_TAC[GSYM IMP_IMP;
IN_ELIM_THM] THEN
DISCH_THEN (X_CHOOSE_THEN `ss : real^3 ->
real` STRIP_ASSUME_TAC) THEN
DISCH_THEN (X_CHOOSE_THEN `tt : real^3 ->
real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `FINITE (L1:real^3->bool) /\ FINITE (L2:real^3->bool)` ASSUME_TAC THENL
[
CONJ_TAC THEN EXPAND_TAC "L1" THEN EXPAND_TAC "L2" THEN MATCH_MP_TAC
FINITE_IMAGE THEN REWRITE_TAC[
FINITE_NUMSEG_LT];
ALL_TAC
] THEN
SUBGOAL_THEN `Y
SUBSET (L1 : real^3->bool) /\ Y
SUBSET (L2 : real^3->bool)` ASSUME_TAC THENL
[
MAP_EVERY EXPAND_TAC ["L1";
let ORTHOGONAL_TO_SPAN_EXISTS = prove(`!(s:real^N->bool) t. s
SUBSET span t /\
dim s <
dim t ==>
?v. ~(v =
vec 0) /\ v
IN span t /\ (!x. x
IN s ==> x
dot v = &0)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `span s
SUBSET span (t:real^N->bool)` ASSUME_TAC THENL
[
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM
SPAN_SPAN] THEN
MATCH_MP_TAC
SPAN_MONO THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `?y:real^N. y
IN span t /\ ~(y
IN span s)` CHOOSE_TAC THENL
[
UNDISCH_TAC `
dim (s:real^N->bool) <
dim (t:real^N->bool)` THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[
NOT_EXISTS_THM; TAUT `~(A /\ ~B) <=> (A ==> B)`; GSYM
SUBSET] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP;
SUBSET_ANTISYM_EQ] THEN
DISCH_TAC THEN
MATCH_MP_TAC (ARITH_RULE `a:num = b ==> ~(a < b)`) THEN
MATCH_MP_TAC
SPAN_EQ_DIM THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (SPEC `span (s:real^N->bool)`
ORTHOGONAL_BASIS_SUBSPACE) THEN
REWRITE_TAC[
SUBSPACE_SPAN;
DIM_SPAN] THEN
STRIP_TAC THEN
ABBREV_TAC `v:real^N = (y -
vsum b (\v. (v
dot y) / (v
dot v) % v))` THEN
EXISTS_TAC `v:real^N` THEN
REPEAT CONJ_TAC THENL
[
POP_ASSUM (LABEL_TAC "A") THEN
DISCH_TAC THEN REMOVE_THEN "A" MP_TAC THEN
ASM_REWRITE_TAC[
VECTOR_SUB_EQ] THEN
DISCH_TAC THEN
SUBGOAL_THEN `y:real^N
IN span b` MP_TAC THENL
[
REWRITE_TAC[
SPAN_EXPLICIT;
IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`b:real^N->bool`; `\v:real^N. (v
dot y) / (v
dot v)`] THEN
ASM_REWRITE_TAC[
EQ_SYM_EQ;
SUBSET_REFL] THEN
CONJ_TAC THENL
[
UNDISCH_TAC `b:real^N->bool
HAS_SIZE dim (s:real^N->bool)` THEN
SIMP_TAC[
HAS_SIZE];
ALL_TAC
] THEN
POP_ASSUM ACCEPT_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[];
EXPAND_TAC "v" THEN
MATCH_MP_TAC
SPAN_SUB THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `span (s:real^N->bool)` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
SPAN_VSUM THEN
UNDISCH_TAC `b:real^N->bool
HAS_SIZE dim (s:real^N->bool)` THEN
SIMP_TAC[
HAS_SIZE] THEN DISCH_TAC THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
SPAN_MUL THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `b:real^N->bool` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`b:real^N->bool`; `y:real^N`; `x:real^N`]
GRAM_SCHMIDT_STEP) THEN
ASM_REWRITE_TAC[
orthogonal] THEN
ANTS_TAC THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `s:real^N->bool` THEN
ASM_REWRITE_TAC[
SPAN_INC]);;
let INDEPENDENT_EXPLICIT_NUMSEG = prove(`!(v:num->real^N) f n. (!i j. i
IN (1..n) /\ j
IN (1..n) /\ v i = v j ==> i = j) /\
independent (
IMAGE v (1..n)) /\
vsum (1..n) (\i. f i % v i) =
vec 0
==> (!i. i
IN 1..n ==> f i = &0)`,
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`v:num->real^N`; `1..n`]
INJECTIVE_ON_LEFT_INVERSE) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
SUBGOAL_THEN `
vsum (1..n) (\i. f i % v i) =
vsum (
IMAGE v (1..n)) (\x:real^N. f (g x) % x)` MP_TAC THENL
[
MP_TAC (ISPECL [`\i:num. f i % v i:real^N`; `1..n`]
VSUM_RESTRICT) THEN
REWRITE_TAC[
FINITE_NUMSEG] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
SUBGOAL_THEN `(\x:num. if x
IN 1..n then f x % v x else
vec 0) = (\i:num. if i
IN 1..n then ((\x:real^N. f (g x) % x) o v) i else
vec 0)` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
FUN_EQ_THM;
o_THM] THEN
GEN_TAC THEN COND_CASES_TAC THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `x:num`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]);
ALL_TAC
] THEN
REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (ISPECL [`(\x:real^N. f ((g:real^N->num) x) % x) o (v:num->real^N)`; `1..n`]
VSUM_RESTRICT) THEN
REWRITE_TAC[
FINITE_NUMSEG] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
MATCH_MP_TAC (GSYM
VSUM_IMAGE) THEN
ASM_REWRITE_TAC[
FINITE_NUMSEG] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_TAC THEN
UNDISCH_TAC `
vsum (1..n) (\i. f i % v i) =
vec 0:real^N` THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
MP_TAC (ISPEC `
IMAGE (v:num->real^N) (1..n)`
INDEPENDENT_EXPLICIT) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
DISCH_THEN (fun
th -> (FIRST_X_ASSUM (MP_TAC o (fun
th2 -> MATCH_MP
th2 th)))) THEN
DISCH_THEN (MP_TAC o SPEC `(v:num->real^N) i`) THEN
REWRITE_TAC[
IN_IMAGE] THEN
ANTS_TAC THENL
[
EXISTS_TAC `i:num` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[]);;
let UNIQUE_SOLUTION_lemma = prove(`!(S:real^N->bool) b.
independent S ==> ?!p. p
IN span S /\ (!x. x
IN S ==> p
dot x = b x)`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `n =
dim (span (S:real^N->bool))` THEN
SUBGOAL_THEN `?w:num->real^N. S =
IMAGE w (1..n) /\ (!i j. i
IN (1..n) /\ j
IN (1..n) /\ w i = w j ==> i = j)` MP_TAC THENL
[
SUBGOAL_THEN `FINITE S /\
CARD (S:real^N->bool) =
dim (span S)` ASSUME_TAC THENL
[
REWRITE_TAC[GSYM
HAS_SIZE] THEN
MATCH_MP_TAC
BASIS_HAS_SIZE_DIM THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (ISPEC `S:real^N->bool`
FINITE_INDEX_NUMSEG) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
EXISTS_TAC `f:num->real^N` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
ABBREV_TAC `f:real^N->real^N = (\p:real^N. (
lambda i. if i <= n then p
dot (w i) else &0))` THEN
ABBREV_TAC `P:real^N->bool =
IMAGE f (span (S:real^N->bool))` THEN
ABBREV_TAC `t:real^N->bool =
IMAGE basis (1..n)` THEN
SUBGOAL_THEN `P:real^N->bool
SUBSET span t` ASSUME_TAC THENL
[
REPLICATE_TAC 2 (POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th])) THEN
REWRITE_TAC[
SUBSET;
IN_SPAN_IMAGE_BASIS] THEN GEN_TAC THEN
EXPAND_TAC "f" THEN
REWRITE_TAC[
IN_IMAGE;
IN_NUMSEG] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (is_forall o concl)) THEN
ASM_SIMP_TAC[
LAMBDA_BETA] THEN
POP_ASSUM MP_TAC THEN ASM_SIMP_TAC[DE_MORGAN_THM];
ALL_TAC
] THEN
SUBGOAL_THEN `
subspace (P:real^N->bool)` ASSUME_TAC THENL
[
REWRITE_TAC[
subspace] THEN
EXPAND_TAC "P" THEN REWRITE_TAC[
IN_IMAGE] THEN
REPEAT CONJ_TAC THENL
[
EXISTS_TAC `
vec 0:real^N` THEN
REWRITE_TAC[
SPAN_0] THEN
EXPAND_TAC "f" THEN
REWRITE_TAC[
DOT_LZERO] THEN
VECTOR_ARITH_TAC;
REPEAT STRIP_TAC THEN
EXISTS_TAC `x' + x'':real^N` THEN
CONJ_TAC THENL
[
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "f" THEN
REWRITE_TAC[
DOT_LADD] THEN
SIMP_TAC[
CART_EQ;
VECTOR_ADD_COMPONENT;
LAMBDA_BETA] THEN
ARITH_TAC;
ALL_TAC
] THEN
MATCH_MP_TAC
SPAN_ADD THEN
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC THEN
EXISTS_TAC `c % x':real^N` THEN
CONJ_TAC THENL
[
ASM_REWRITE_TAC[] THEN EXPAND_TAC "f" THEN
SIMP_TAC[
CART_EQ;
VECTOR_MUL_COMPONENT;
LAMBDA_BETA;
DOT_LMUL] THEN
ARITH_TAC;
ALL_TAC
] THEN
MATCH_MP_TAC
SPAN_MUL THEN
ASM_REWRITE_TAC[]
];
ALL_TAC
] THEN
SUBGOAL_THEN `P:real^N->bool = span t` ASSUME_TAC THENL
[
ASM_CASES_TAC `P:real^N->bool = span t` THEN ASM_REWRITE_TAC[] THEN
MP_TAC (SPECL [`P:real^N->bool`; `t:real^N->bool`]
ORTHOGONAL_TO_SPAN_EXISTS) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
DIM_PSUBSET THEN
REWRITE_TAC[
PSUBSET] THEN
SUBGOAL_THEN `span P = P:real^N->bool` (fun
th -> REWRITE_TAC[
th]) THENL
[
ASM_REWRITE_TAC[
SPAN_EQ_SELF];
ALL_TAC
] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
ABBREV_TAC `p:real^N =
vsum (1..n) (\i. (v:real^N)$i % w i)` THEN
POP_ASSUM (LABEL_TAC "p") THEN
SUBGOAL_THEN `(f:real^N->real^N) p
IN P` ASSUME_TAC THENL
[
EXPAND_TAC "P" THEN
REWRITE_TAC[
IN_IMAGE] THEN
EXISTS_TAC `p:real^N` THEN ASM_REWRITE_TAC[] THEN
EXPAND_TAC "p" THEN
MATCH_MP_TAC
SPAN_VSUM THEN
REWRITE_TAC[
FINITE_NUMSEG] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
SPAN_MUL THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
IMAGE (w:num->real^N) (1..n)` THEN
REWRITE_TAC[
SPAN_INC;
IN_IMAGE] THEN
EXISTS_TAC `x:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `p:real^N
dot p = &0` MP_TAC THENL
[
REMOVE_THEN "p" (fun
th -> GEN_REWRITE_TAC (PAT_CONV `\p:real^N. a
dot p = &0`) [SYM
th]) THEN
SIMP_TAC[
FINITE_NUMSEG;
DOT_RSUM] THEN
REWRITE_TAC[
DOT_RMUL] THEN
SUBGOAL_THEN `
sum (1..n) (\y. v$y * (p
dot w y)) = (v:real^N)
dot ((f:real^N->real^N) p)` (fun
th -> REWRITE_TAC[
th]) THENL
[
GEN_REWRITE_TAC RAND_CONV[
dot] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `
sum (1..dimindex (:N)) (\i. (v:real^N)$i * (p:real^N
dot w i))` THEN
CONJ_TAC THENL
[
SUBGOAL_THEN `1..dimindex (:N) = (1..n)
UNION (n + 1..dimindex (:N))` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC (GSYM
NUMSEG_COMBINE_R) THEN
EXPAND_TAC "n" THEN
REWRITE_TAC[
DIM_SUBSET_UNIV; ARITH_RULE `1 <= a + 1`];
ALL_TAC
] THEN
MATCH_MP_TAC (GSYM
SUM_UNION_RZERO) THEN
REWRITE_TAC[
FINITE_NUMSEG;
IN_NUMSEG] THEN
GEN_TAC THEN STRIP_TAC THEN
MATCH_MP_TAC
EQ_SYM THEN
REWRITE_TAC[
REAL_ENTIRE] THEN
DISJ1_TAC THEN
UNDISCH_TAC `v:real^N
IN span t` THEN
EXPAND_TAC "t" THEN
REWRITE_TAC[
IN_SPAN_IMAGE_BASIS] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_NUMSEG] THEN
MATCH_MP_TAC (ARITH_RULE `n + 1 <= x ==> 1 <= x`) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
SUM_EQ THEN
REWRITE_TAC[
IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
REPEAT (FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (is_forall o concl))) THEN
EXPAND_TAC "f" THEN ASM_SIMP_TAC[
LAMBDA_BETA] THEN
COND_CASES_TAC THEN REWRITE_TAC[] THEN
REWRITE_TAC[
REAL_MUL_RZERO;
REAL_ENTIRE] THEN
DISJ1_TAC THEN
UNDISCH_TAC `v:real^N
IN span t` THEN
EXPAND_TAC "t" THEN
REWRITE_TAC[
IN_SPAN_IMAGE_BASIS] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_NUMSEG];
ALL_TAC
] THEN
REWRITE_TAC[
DOT_SYM] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
DOT_EQ_0] THEN
DISCH_TAC THEN
SUBGOAL_THEN `!i. i
IN 1..n ==> (v:real^N)$i = &0` ASSUME_TAC THENL
[
MATCH_MP_TAC
INDEPENDENT_EXPLICIT_NUMSEG THEN
EXISTS_TAC `w:num->real^N` THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `S:real^N->bool =
IMAGE w (1..n)` THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
SUBGOAL_THEN `v:real^N =
vec 0` MP_TAC THENL
[
REWRITE_TAC[
CART_EQ;
VEC_COMPONENT] THEN
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `i <= n:num` THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_NUMSEG];
ALL_TAC
] THEN
UNDISCH_TAC `v:real^N
IN span t` THEN
EXPAND_TAC "t" THEN
REWRITE_TAC[
IN_SPAN_IMAGE_BASIS] THEN
DISCH_THEN (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[
IN_NUMSEG; DE_MORGAN_THM];
ALL_TAC
] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ABBREV_TAC `r:real^N =
lambda i. if i <= n then b ((w:num->real^N) i) else &0` THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (is_forall o concl)) THEN
SUBGOAL_THEN `?p. p
IN span S /\ f (p:real^N) = r:real^N` MP_TAC THENL
[
SUBGOAL_THEN `r:real^N
IN P` MP_TAC THENL
[
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "t" THEN
REWRITE_TAC[
IN_SPAN_IMAGE_BASIS] THEN
REWRITE_TAC[
IN_NUMSEG; DE_MORGAN_THM] THEN
REPEAT STRIP_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN
EXPAND_TAC "r" THEN
ASM_SIMP_TAC[
LAMBDA_BETA];
ALL_TAC
] THEN
EXPAND_TAC "P" THEN
REWRITE_TAC[
IN_IMAGE] THEN
STRIP_TAC THEN
EXISTS_TAC `x:real^N` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
REWRITE_TAC[
EXISTS_UNIQUE] THEN
EXISTS_TAC `p:real^N` THEN
SUBGOAL_THEN `p:real^N
IN span S /\ (!x. x
IN S ==> p
dot x = b x)` ASSUME_TAC THENL
[
ASM_REWRITE_TAC[] THEN
GEN_TAC THEN REWRITE_TAC[
IN_IMAGE;
IN_NUMSEG] THEN
DISCH_THEN (X_CHOOSE_THEN `i:num` ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `f (p:real^N) = r:real^N` THEN
DISCH_THEN (MP_TAC o AP_TERM `(\v:real^N. v$i)`) THEN
BETA_TAC THEN
EXPAND_TAC "f" THEN EXPAND_TAC "r" THEN
SUBGOAL_THEN `1 <= i /\ i <=
dimindex (:N)` MP_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
LE_TRANS THEN
EXISTS_TAC `n:num` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "n" THEN
REWRITE_TAC[
DIM_SUBSET_UNIV];
ALL_TAC
] THEN
SIMP_TAC[
LAMBDA_BETA] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
FIRST_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[GSYM
VECTOR_SUB_EQ] THEN
MATCH_MP_TAC
ORTHOGONAL_TO_ALL_IMP_ZERO THEN
EXISTS_TAC `S:real^N->bool` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
SPAN_SUB THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[
DOT_LSUB;
REAL_SUB_REFL]);;
let UNIQUE_SOLUTION_AFFINE_INDEPENDENT = prove(`!(S:real^N->bool) b. ~(S = {}) /\ ~affine_dependent S
==> (?!p. p
IN affine hull S /\
(!x y. x
IN S /\ y
IN S ==> p
dot (x - y) = b x - b y))`,
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
REPEAT GEN_TAC THEN
DISCH_THEN (CONJUNCTS_THEN2 (X_CHOOSE_THEN `v0:real^N` ASSUME_TAC) ASSUME_TAC) THEN
ABBREV_TAC `R:real^N->bool = {y - v0 | y | y
IN S
DELETE v0}` THEN
MP_TAC (SPECL [`R:real^N->bool`; `\v:real^N. (b:real^N->
real) (v + v0) - b v0 - v0
dot v`]
UNIQUE_SOLUTION_lemma) THEN
ANTS_TAC THENL
[
MP_TAC (SPEC `S:real^N->bool`
AFFINE_INDEPENDENT_IMP_INDEPENDENT) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `v0:real^N`) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
EXISTS_UNIQUE] THEN
STRIP_TAC THEN
EXISTS_TAC `p + v0:real^N` THEN
SUBGOAL_THEN `span (R:real^N->bool) =
affine hull IMAGE (\x. --v0 + x) S` ASSUME_TAC THENL
[
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `span ((v0 - v0:real^N)
INSERT R)` THEN
CONJ_TAC THENL [ REWRITE_TAC[
VECTOR_SUB_REFL;
SPAN_INSERT_0]; ALL_TAC ] THEN
SUBGOAL_THEN `(v0 - v0)
INSERT R =
IMAGE (\x:real^N. --v0 + x) S` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
EXTENSION;
IN_INSERT;
IN_IMAGE; VECTOR_ARITH `--v0 + x = x - v0:real^N`] THEN
EXPAND_TAC "R" THEN
REWRITE_TAC[
IN_ELIM_THM;
IN_DELETE] THEN
ASM SET_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC (GSYM
AFFINE_HULL_EQ_SPAN) THEN
MATCH_MP_TAC
HULL_INC THEN
REWRITE_TAC[
IN_IMAGE] THEN
EXISTS_TAC `v0:real^N` THEN
ASM_REWRITE_TAC[VECTOR_ARITH `--v0 + v0 =
vec 0:real^N`];
ALL_TAC
] THEN
CONJ_TAC THENL
[
CONJ_TAC THENL
[
UNDISCH_TAC `p:real^N
IN span R` THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
REWRITE_TAC[VECTOR_ARITH `p = --v0 + x <=> p + v0:real^N = x`] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!x:real^N. x
IN S ==> (p + v0)
dot (x - v0) = b x - b v0` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `x = v0:real^N` THENL
[
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
DOT_RZERO;
REAL_SUB_REFL];
ALL_TAC
] THEN
REWRITE_TAC[
DOT_LADD] THEN
SUBGOAL_THEN `p:real^N
dot (x - v0) = b ((x - v0) + v0) - b v0 - v0
dot (x - v0)` (fun
th -> REWRITE_TAC[
th]) THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXPAND_TAC "R" THEN
REWRITE_TAC[
IN_ELIM_THM;
IN_DELETE] THEN
EXISTS_TAC `x:real^N` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[VECTOR_ARITH `x - v0 + v0 = x:real^N`] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `a - b = (a - v0:real^N) - (b - v0)`] THEN
ONCE_REWRITE_TAC[
DOT_RSUB] THEN
FIRST_ASSUM (MP_TAC o SPEC `x:real^N`) THEN
FIRST_ASSUM (MP_TAC o SPEC `y:real^N`) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REPEAT (DISCH_THEN (fun
th -> REWRITE_TAC[
th])) THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
GEN_TAC THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `y - v0:real^N`) THEN
POP_ASSUM STRIP_ASSUME_TAC THEN
ANTS_TAC THENL
[
CONJ_TAC THENL
[
ASM_REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
EXISTS_TAC `y:real^N` THEN
ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
ALL_TAC
] THEN
EXPAND_TAC "R" THEN
REWRITE_TAC[
IN_IMAGE;
IN_DELETE;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
DOT_LSUB; VECTOR_ARITH `y - v0 + v0:real^N = y`] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`y':real^N`; `v0:real^N`]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]);
ALL_TAC
] THEN
VECTOR_ARITH_TAC);;
let QXSKIIT = prove(`!(vf:A->real^N) b . FINITE (
IMAGE vf (:A)) /\
~affine_dependent (
IMAGE vf (:A)) /\ (!i j. (vf i = vf j) ==> (b i = b j))
==> (?!p. p
IN affine hull (
IMAGE vf (:A)) /\ (!i j. (p
dot (vf i - vf j) = (b i - b j))))`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `S =
IMAGE (vf:A->real^N) (:A)` THEN
SUBGOAL_THEN `?h:real^N->real. !i:A. h (vf i) = b i` CHOOSE_TAC THENL
[
EXISTS_TAC `\x:real^N. if x
IN S then b (
inverse (vf:A->real^N) x) else &0` THEN
EXPAND_TAC "S" THEN
REWRITE_TAC[
IN_IMAGE;
IN_UNIV] THEN
GEN_TAC THEN
SUBGOAL_THEN `?x:A. vf i = (vf x):real^N` (fun
th -> REWRITE_TAC[
th]) THENL
[
EXISTS_TAC `i:A` THEN REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
inverse] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
MESON_TAC[];
ALL_TAC
] THEN
MP_TAC (SPECL [`S:real^N->bool`; `\x:real^N. (h:real^N->
real) x`]
UNIQUE_SOLUTION_AFFINE_INDEPENDENT) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXPAND_TAC "S" THEN
REWRITE_TAC[
IN_IMAGE] THEN
MP_TAC
UNIV_NOT_EMPTY THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`(vf:A->real^N) x`; `x:A`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[] THEN
SUBGOAL_THEN `!p. (!x y:real^N. x
IN S /\ y
IN S ==> p
dot (x - y) = h x - h y) <=> (!i j:A. p
dot (vf i - vf j) = b i - b j)` MP_TAC THENL
[
GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
[
POP_ASSUM (MP_TAC o SPECL [`(vf:A->real^N) i`; `(vf:A->real^N) j`]) THEN
EXPAND_TAC "S" THEN
ASM_REWRITE_TAC[
IN_IMAGE;
IN_UNIV] THEN
DISCH_THEN MATCH_MP_TAC THEN
CONJ_TAC THENL [ EXISTS_TAC `i:A`; EXISTS_TAC `j:A` ] THEN REWRITE_TAC[];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
EXPAND_TAC "S" THEN
REWRITE_TAC[
IN_IMAGE;
IN_UNIV] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th])
);;
let CIRCUMCENTER_LEMMA = prove(`!S:real^N->bool. ~(S = {}) /\ ~affine_dependent S ==>
?!p. p
IN affine hull S /\
(?c. !w. w
IN S ==> c =
dist (p,w))`,
GEN_TAC THEN DISCH_THEN (MP_TAC o MATCH_MP
CIRCUMCENTER_lemma) THEN
SUBGOAL_THEN `!p. (!x y:real^N. x
IN S /\ y
IN S ==>
dist (p,x) =
dist (p,y)) <=> (?c. !w. w
IN S ==> c =
dist (p,w))` ASSUME_TAC THENL
[
GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
[
ASM_CASES_TAC `S:real^N->bool = {}` THEN ASM_REWRITE_TAC[
NOT_IN_EMPTY] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
STRIP_TAC THEN
EXISTS_TAC `
dist (p,x:real^N)` THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `c:real` THEN
REWRITE_TAC[
EQ_SYM_EQ] THEN
CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[]);;
let OAPVION2 = prove(`!(S:real^N->bool). ~affine_dependent S ==>
(!w. w
IN S ==> (radV S =
dist(circumcenter S, w)))`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `~(S:real^N->bool = {})` ASSUME_TAC THENL
[
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `w:real^N` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (SPEC `S:real^N->bool`
CIRCUMCENTER_LEMMA) THEN
ASM_REWRITE_TAC[
IN; radV;
EXISTS_UNIQUE] THEN
STRIP_TAC THEN
SUBGOAL_THEN `circumcenter S = p:real^N` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[circumcenter] THEN
MATCH_MP_TAC
SELECT_UNIQUE THEN
ASM_MESON_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
SELECT_UNIQUE THEN
X_GEN_TAC `r:real` THEN
EQ_TAC THENL
[
REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `w:real^N`) THEN
UNDISCH_TAC `w:real^N
IN S` THEN
SIMP_TAC[
IN];
ALL_TAC
] THEN
BETA_TAC THEN
REMOVE_ASSUM THEN
FIRST_ASSUM (MP_TAC o SPEC `w:real^N`) THEN
UNDISCH_TAC `w:real^N
IN S` THEN
SIMP_TAC[
IN] THEN
DISCH_TAC THEN DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN DISCH_TAC THEN
ASM_REWRITE_TAC[]);;
let OAPVION3 = prove(`!(S:real^N->bool). ~affine_dependent S ==>
(!p. (p
IN affine hull S) /\ (?c. !w. (w
IN S) ==> (
dist(p,w) = c)) ==> (p = circumcenter S))`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `~(S:real^N->bool = {})` ASSUME_TAC THENL
[
DISCH_TAC THEN
UNDISCH_TAC `p:real^N
IN affine hull S` THEN
ASM_REWRITE_TAC[
AFFINE_HULL_EMPTY;
NOT_IN_EMPTY];
ALL_TAC
] THEN
MP_TAC (SPEC `S:real^N->bool`
CIRCUMCENTER_LEMMA) THEN
ASM_REWRITE_TAC[
EXISTS_UNIQUE; circumcenter] THEN
REPEAT STRIP_TAC THEN
REPEAT (POP_ASSUM MP_TAC) THEN REWRITE_TAC[
IN] THEN REPEAT DISCH_TAC THEN
MATCH_MP_TAC (GSYM
SELECT_UNIQUE) THEN
BETA_TAC THEN GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL
[
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `c:real` THEN
UNDISCH_TAC `!w:real^N. S w ==>
dist (p,w) = c` THEN
SIMP_TAC[
EQ_SYM_EQ];
ALL_TAC
] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `p':real^N` THEN
CONJ_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
EQ_SYM THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `c:real` THEN
UNDISCH_TAC `!w:real^N. S w ==>
dist (p,w) = c` THEN
SIMP_TAC[
EQ_SYM_EQ]);;
let CIRCUMCENTER_NOT_EQ = prove(`!S:real^N->bool. ~affine_dependent S /\ 1 <
CARD S ==>
!x. x
IN S ==> ~(circumcenter S = x)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?y. y
IN S
DELETE (x:real^N)` MP_TAC THENL
[
REWRITE_TAC[
MEMBER_NOT_EMPTY] THEN
DISCH_THEN (MP_TAC o AP_TERM `\s:real^N->bool.
CARD s`) THEN
REWRITE_TAC[
CARD_CLAUSES] THEN
MP_TAC (ISPECL [`x:real^N`; `S:real^N->bool`]
CARD_DELETE) THEN
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE] THEN
UNDISCH_TAC `1 <
CARD (S:real^N->bool)` THEN
ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
IN_DELETE] THEN
STRIP_TAC THEN
MP_TAC (SPEC `S:real^N->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o SPEC `x:real^N`) THEN
REWRITE_TAC[
DIST_REFL] THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `y:real^N`) THEN
ASM_REWRITE_TAC[
EQ_SYM_EQ;
DIST_EQ_0]);;
let CIRCUMCENTER_TRANSLATION = prove(`!s (a:real^N). ~(s = {}) /\ ~affine_dependent s ==>
circumcenter (
IMAGE (\x. a + x) s) = a + circumcenter s`,
REPEAT STRIP_TAC THEN
MP_TAC (SPEC `
IMAGE (\x:real^N. a + x) s`
OAPVION3) THEN
ASM_REWRITE_TAC[
AFFINE_DEPENDENT_TRANSLATION_EQ] THEN
DISCH_THEN (fun
th -> MATCH_MP_TAC (GSYM
th)) THEN
CONJ_TAC THENL
[
REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
EXISTS_TAC `circumcenter (s:real^N->bool)` THEN
MP_TAC (SPEC `s:real^N->bool`
OAPVION1) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
EXISTS_TAC `radV (s:real^N->bool)` THEN
REWRITE_TAC[
IN_IMAGE] THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPEC `s:real^N->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
dist] THEN
AP_TERM_TAC THEN
VECTOR_ARITH_TAC);;
let RADV_TRANSLATION = prove(`!s (a:real^N). ~affine_dependent s ==>
radV (
IMAGE (\x. a + x) s) = radV s`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `s:real^N->bool = {}` THENL
[
ASM_REWRITE_TAC[
IMAGE_CLAUSES];
ALL_TAC
] THEN
MP_TAC (SPEC `
IMAGE (\x:real^N. a + x) s`
OAPVION2) THEN
ASM_REWRITE_TAC[
AFFINE_DEPENDENT_TRANSLATION_EQ] THEN
FIRST_ASSUM MP_TAC THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
STRIP_TAC THEN
DISCH_THEN (MP_TAC o SPEC `a + x:real^N`) THEN
ANTS_TAC THENL
[
REWRITE_TAC[
IN_IMAGE] THEN
EXISTS_TAC `x:real^N` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ASM_SIMP_TAC[
CIRCUMCENTER_TRANSLATION;
dist; VECTOR_ARITH `(a + b) - (a + x) = b - x:real^N`] THEN
REWRITE_TAC[GSYM
dist] THEN
MATCH_MP_TAC
EQ_SYM THEN
MP_TAC (SPEC `s:real^N->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[]);;
let MHFTTZN_lemma = prove(`!V ul k. packing V /\ barV V k ul ==>
affine hull voronoi_list V ul =
INTERS {bis (
HD ul) u | u | u
IN set_of_list ul}`,
REPEAT GEN_TAC THEN
SPEC_TAC (`ul:(real^3)list`, `ul:(real^3)list`) THEN
SPEC_TAC (`k:num`, `k:num`) THEN
INDUCT_TAC THENL
[
REWRITE_TAC[
BARV;
VORONOI_NONDG; ARITH] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `UNIV:real^3->bool` THEN
REWRITE_TAC[
VORONOI_LIST;
VORONOI_SET] THEN
MP_TAC (ISPEC `ul:(real^3)list`
LENGTH_1_LEMMA) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> ONCE_REWRITE_TAC[
th]) THEN
REWRITE_TAC[
set_of_list;
IN_SING;
HD] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
CONTAINS_BALL_AFFINE_HULL THEN
SUBGOAL_THEN `
INTERS {
voronoi_closed V v | v:real^3 =
HD ul} =
voronoi_closed V (
HD ul)` (fun
th -> REWRITE_TAC[
th]) THENL
[
SET_TAC[];
ALL_TAC
] THEN
EXISTS_TAC `(
HD ul):real^3` THEN
MATCH_MP_TAC
VORONOI_CLOSED_CONTAINS_BALL THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
INTERS {bis (
HD ul) u | u | u:real^3 =
HD ul} = bis (
HD ul) (
HD ul)` (fun
th -> REWRITE_TAC[
th]) THENL
[
SET_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[bis] THEN
SET_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `SUC k <= 3` ASSUME_TAC THENL
[
MATCH_MP_TAC
BARV_IMP_K_LE_3 THEN
EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `ul:(real^3)list` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ABBREV_TAC `vl:(real^3)list =
BUTLAST ul` THEN
POP_ASSUM (LABEL_TAC "vl") THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = k + 2` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V (SUC k) ul` THEN
REWRITE_TAC[
BARV] THEN
ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `truncate_simplex k (ul:(real^3)list) = vl` (LABEL_TAC "vl2") THENL
[
MP_TAC (ISPEC `ul:(real^3)list`
TRUNCATE_SIMPLEX_EQ_BUTLAST) THEN
ASM_REWRITE_TAC[ARITH_RULE `2 <= k + 2`; ARITH_RULE `(k + 2) - 2 = k`];
ALL_TAC
] THEN
SUBGOAL_THEN `barV V k vl` ASSUME_TAC THENL
[
REMOVE_THEN "vl2" (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `SUC k` THEN
ASM_SIMP_TAC[ARITH_RULE `k <= SUC k`];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `
HD vl = (
HD ul):real^3` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HD_TRUNCATE_SIMPLEX THEN
UNDISCH_TAC `barV V (SUC k) ul` THEN
REWRITE_TAC[
BARV] THEN
ARITH_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
ABBREV_TAC `A0 =
INTERS {bis (
HD ul) u | u | u:real^3
IN set_of_list vl}` THEN
ABBREV_TAC `A1 =
INTERS {bis (
HD ul) u | u | u:real^3
IN set_of_list ul}` THEN
SUBGOAL_THEN `voronoi_list V ul = voronoi_list V vl
INTER bis (
HD ul) (
LAST ul)` (LABEL_TAC "B1") THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `
LAST ul:real^3`; `
HD vl:real^3`; `
TL vl:(real^3)list`]
VORONOI_LIST_INTER_BIS) THEN
SUBGOAL_THEN `
APPEND vl [
LAST ul:real^3] = ul` (fun
th -> REWRITE_TAC[
th]) THENL
[
REMOVE_THEN "vl" (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC
APPEND_BUTLAST_LAST THEN
ASM_REWRITE_TAC[GSYM
LENGTH_EQ_NIL] THEN
ARITH_TAC;
ALL_TAC
] THEN
UNDISCH_TAC `
HD vl =
HD ul:real^3` THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
DISCH_THEN (fun
th -> MATCH_MP_TAC (GSYM
th)) THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `vl:(real^3)list`]
BARV_SUBSET) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
set_of_list (ul:(real^3)list)` THEN
MP_TAC (SPECL [`V:real^3->bool`; `SUC k`; `ul:(real^3)list`]
BARV_SUBSET) THEN
ASM_SIMP_TAC[] THEN
DISCH_THEN (fun
th -> ALL_TAC) THEN
REWRITE_TAC[
IN_SET_OF_LIST] THEN
MP_TAC (ISPEC `ul:(real^3)list`
LAST_EL) THEN
ASM_REWRITE_TAC[GSYM
LENGTH_EQ_NIL; ARITH_RULE `~(k + 2 = 0)`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
MATCH_MP_TAC
MEM_EL THEN
ASM_REWRITE_TAC[] THEN
ARITH_TAC;
ALL_TAC
] THEN
MP_TAC (ISPEC `vl:(real^3)list`
LENGTH_IMP_CONS) THEN
ANTS_TAC THENL
[
UNDISCH_TAC `barV V k vl` THEN
REWRITE_TAC[
BARV] THEN
ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[
HD;
TL];
ALL_TAC
] THEN
SUBGOAL_THEN `A0
INTER bis ((
HD ul):real^3) (
LAST ul) = A1` (LABEL_TAC "A1") THENL
[
EXPAND_TAC "A0" THEN EXPAND_TAC "A1" THEN
SUBGOAL_THEN `
set_of_list ul =
set_of_list vl
UNION {
LAST ul:real^3}` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPEC `ul:(real^3)list`
APPEND_BUTLAST_LAST) THEN
ASM_REWRITE_TAC[GSYM
LENGTH_EQ_NIL; ARITH_RULE `~(k + 2 = 0)`] THEN
DISCH_THEN (fun
th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SYM
th]) THEN
REWRITE_TAC[
SET_OF_LIST_APPEND;
set_of_list];
ALL_TAC
] THEN
SET_TAC[];
ALL_TAC
] THEN
ASM_CASES_TAC `A1 = A0:real^3->bool` THENL
[
SUBGOAL_THEN `voronoi_list V ul = voronoi_list V vl` (fun
th -> ASM_REWRITE_TAC[
th]) THEN
SUBGOAL_THEN `voronoi_list V vl = voronoi_list V vl
INTER A0` MP_TAC THENL
[
MATCH_MP_TAC (SET_RULE `A
SUBSET B ==> A = A
INTER B:A->bool`) THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `
affine hull voronoi_list V vl` THEN
ASM_REWRITE_TAC[
HULL_SUBSET;
SUBSET_REFL];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
ONCE_ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[
INTER_ASSOC];
ALL_TAC
] THEN
SUBGOAL_THEN `
affine (A1:real^3->bool)` ASSUME_TAC THENL
[
REMOVE_THEN "A1" (fun
th -> ALL_TAC) THEN
EXPAND_TAC "A1" THEN
MATCH_MP_TAC
AFFINE_INTERS THEN
REWRITE_TAC[
IN_ELIM_THM;
BIS_EQ_HYPERPLANE] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
AFFINE_HYPERPLANE];
ALL_TAC
] THEN
SUBGOAL_THEN `A1:real^3->bool =
affine hull A1` (fun
th -> ONCE_REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC
EQ_SYM THEN
ASM_REWRITE_TAC[
AFFINE_HULL_EQ];
ALL_TAC
] THEN
MATCH_MP_TAC
AFF_DIM_EQ_AFFINE_HULL THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `
affine hull voronoi_list V ul` THEN
ASM_REWRITE_TAC[
HULL_SUBSET] THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `
affine hull (voronoi_list V vl)
INTER affine hull (bis (
HD ul) (
LAST ul))` THEN
ASM_REWRITE_TAC[
AFF_INTER_SUBSET_INTER_AFF] THEN
SUBGOAL_THEN `
affine hull bis ((
HD ul):real^3) (
LAST ul) = bis (
HD ul) (
LAST ul)` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
AFFINE_HULL_EQ] THEN
REWRITE_TAC[
BIS_EQ_HYPERPLANE;
AFFINE_HYPERPLANE];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
SUBSET_REFL];
ALL_TAC
] THEN
SUBGOAL_THEN `
aff_dim (voronoi_list V ul) = &(3 - SUC k)` (fun
th -> REWRITE_TAC[
th]) THENL
[
UNDISCH_TAC `barV V (SUC k) ul` THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
STRIP_TAC THEN
POP_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 2`] THEN
ASM_SIMP_TAC[GSYM
INT_OF_NUM_SUB] THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD;
ADD1] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
aff_dim (A1:real^3->bool) =
aff_dim (A0:real^3->bool) - &1 \/
aff_dim A1 = -- &1` MP_TAC THENL
[
USE_THEN "A1" (fun
th -> REWRITE_TAC[SYM
th]) THEN
REWRITE_TAC[
BIS_EQ_HYPERPLANE] THEN
ABBREV_TAC `a:real^3 = &2 % (
LAST ul -
HD ul)` THEN
ABBREV_TAC `b =
LAST ul
dot LAST ul -
HD ul
dot HD (ul:(real^3)list)` THEN
MP_TAC (ISPECL [`a:real^3`; `b:real`; `A0:real^3->bool`]
AFF_DIM_AFFINE_INTER_HYPERPLANE) THEN
ANTS_TAC THENL
[
EXPAND_TAC "A0" THEN
MATCH_MP_TAC
AFFINE_INTERS THEN
REWRITE_TAC[
IN_ELIM_THM;
BIS_EQ_HYPERPLANE] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
AFFINE_HYPERPLANE];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
COND_CASES_TAC THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN
EXPAND_TAC "a" THEN EXPAND_TAC "b" THEN
REWRITE_TAC[GSYM
BIS_EQ_HYPERPLANE] THEN
DISCH_TAC THEN
SUBGOAL_THEN `A0 = A1:real^3->bool` MP_TAC THENL
[
REMOVE_THEN "A1" (fun
th -> REWRITE_TAC[SYM
th]) THEN
POP_ASSUM MP_TAC THEN
SET_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[
UNDISCH_TAC `
affine hull voronoi_list V vl = A0:real^3->bool` THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
UNDISCH_TAC `barV V k vl` THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG;
AFF_DIM_AFFINE_HULL] THEN
STRIP_TAC THEN
POP_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 1`] THEN
ASM_SIMP_TAC[GSYM
INT_OF_NUM_SUB] THEN
ASM_REWRITE_TAC[GSYM
INT_OF_NUM_ADD;
ADD1] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
ASM_SIMP_TAC[GSYM
INT_OF_NUM_SUB] THEN
UNDISCH_TAC `SUC k <= 3` THEN
REWRITE_TAC[GSYM
INT_OF_NUM_LE] THEN
INT_ARITH_TAC);;
let XYOFCGX_lemma0 = prove(`!V S p w. packing V /\ S
SUBSET V /\ ~affine_dependent S /\
(p = circumcenter S) /\ (radV S <
sqrt(&2)) /\
w
IN V
DIFF S /\
dist (p,w) <= radV S /\ 1 <
CARD S
==> (!u. u
IN S ==>
pi / &2 <
arcV p w u) /\
(!u v. u
IN S /\ v
IN S /\ ~(u = v) ==>
pi / &2 <
arcV p u v)`,
REPEAT GEN_TAC THEN
REWRITE_TAC[packing;
ARCV_GT_PI2] THEN
DISCH_THEN STRIP_ASSUME_TAC THEN
SUBGOAL_THEN `!p u w:real^3. ~(p = u) /\ ~(p = w) /\
dist (p,u) <
sqrt(&2) /\
dist (p,w) <
sqrt(&2) /\ &2 <=
dist(u,w) ==>
cos(
arcV p u w) < &0` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
ARCV_ANGLE] THEN
MP_TAC (ISPECL [`p':real^3`; `u:real^3`; `w':real^3`]
LAW_OF_COSINES) THEN
ABBREV_TAC `d0 =
dist(u:real^3,w')` THEN
ABBREV_TAC `d1 =
dist(p':real^3,u)` THEN
ABBREV_TAC `d2 =
dist(p':real^3,w')` THEN
ABBREV_TAC `x =
cos(
angle (u:real^3,p',w'))` THEN
SUBGOAL_THEN `(d1 pow 2 + d2 pow 2) - d0 pow 2 < &0` ASSUME_TAC THENL
[
REWRITE_TAC[REAL_ARITH `a - b < &0 <=> a < b`] THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `&4` THEN
CONJ_TAC THENL
[
REWRITE_TAC[REAL_ARITH `&4 = &2 + &2`] THEN
MATCH_MP_TAC
REAL_LT_ADD2 THEN
MP_TAC (SPEC `&2`
SQRT_WORKS) THEN
REWRITE_TAC[REAL_ARITH `&0 <= &2`] THEN
DISCH_THEN (CONJUNCTS_THEN2 ASSUME_TAC (fun
th -> ONCE_REWRITE_TAC[SYM
th])) THEN
REWRITE_TAC[GSYM
REAL_LT_SQUARE_ABS] THEN
POP_ASSUM MP_TAC THEN
ONCE_REWRITE_TAC[GSYM
REAL_ABS_REFL] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
EXPAND_TAC "d1" THEN EXPAND_TAC "d2" THEN
REWRITE_TAC[
dist;
REAL_ABS_NORM] THEN
ASM_REWRITE_TAC[GSYM
dist];
ALL_TAC
] THEN
REWRITE_TAC[REAL_ARITH `&4 = &2 pow 2`] THEN
REWRITE_TAC[GSYM
REAL_LE_SQUARE_ABS; REAL_ARITH `abs(&2) = &2`] THEN
EXPAND_TAC "d0" THEN
REWRITE_TAC[
dist;
REAL_ABS_NORM] THEN
ASM_REWRITE_TAC[GSYM
dist];
ALL_TAC
] THEN
REWRITE_TAC[REAL_ARITH `d0 = d12 - x <=> x = d12 - d0:real`] THEN
DISCH_TAC THEN
MATCH_MP_TAC
REAL_LT_LCANCEL_IMP THEN
EXISTS_TAC `&2 * d1 * d2` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
REAL_LT_MUL THEN
REWRITE_TAC[REAL_ARITH `&0 < &2`] THEN
MATCH_MP_TAC
REAL_LT_MUL THEN
EXPAND_TAC "d1" THEN EXPAND_TAC "d2" THEN
ASM_REWRITE_TAC[GSYM
DIST_NZ];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
REAL_MUL_RZERO; GSYM REAL_MUL_ASSOC];
ALL_TAC
] THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
SUBGOAL_THEN `!x. x
IN S ==>
dist(p:real^3,x) <
sqrt(&2)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `x:real^3`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!u:real^3. u
IN S ==> V u /\ ~(u = w)` (LABEL_TAC "A") THENL
[
REPEAT STRIP_TAC THENL
[
ONCE_REWRITE_TAC[GSYM
IN] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
UNDISCH_TAC `u:real^3
IN S` THEN
UNDISCH_TAC `w:real^3
IN V
DIFF S` THEN
ASM_SIMP_TAC[
IN_DIFF];
ALL_TAC
] THEN
SUBGOAL_THEN `(V:real^3->bool) w` ASSUME_TAC THENL
[
UNDISCH_TAC `w:real^3
IN V
DIFF S` THEN
SIMP_TAC[
IN_DIFF;
IN];
ALL_TAC
] THEN
CONJ_TAC THEN REPEAT STRIP_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
SUBGOAL_THEN `~(circumcenter S = u:real^3)` ASSUME_TAC THENL
[
MP_TAC (ISPEC `S:real^3->bool`
CIRCUMCENTER_NOT_EQ) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
REPEAT CONJ_TAC THENL
[
DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `u:real^3`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `u:real^3`) THEN
ASM_REWRITE_TAC[
REAL_NOT_LT] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `&2` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
REAL_LE_LSQRT THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `radV (S:real^3->bool)` THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `p = circumcenter S:real^3` THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]);
REMOVE_THEN "A" (fun
th -> ALL_TAC) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `u:real^3`) THEN
ASM_SIMP_TAC[];
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_SIMP_TAC[]
];
ALL_TAC
] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_SIMP_TAC[
CIRCUMCENTER_NOT_EQ] THEN
UNDISCH_TAC `p:real^3 = circumcenter S` THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_SIMP_TAC[]);;
let XYOFCGX_2 = prove(`!V S p. packing V /\ S
SUBSET V /\ ~affine_dependent S /\
p = circumcenter S /\ radV S <
sqrt(&2) /\
CARD S = 2 ==>
(!u v. u
IN S /\ v
IN (V
DIFF S) ==>
dist (v,p) >
dist (u,p))`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `
dist (v,p:real^3) >
dist(u,p)` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN PURE_REWRITE_TAC[
real_gt;
REAL_NOT_LT;
DIST_SYM] THEN DISCH_TAC THEN
SUBGOAL_THEN `
dist (p:real^3,v) <= radV (S:real^3->bool)` ASSUME_TAC THENL
[
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `
dist (p,u:real^3)` THEN
POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
REAL_LE_LT] THEN DISJ2_TAC THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> MATCH_MP_TAC (GSYM
th)) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `S:real^3->bool`; `p:real^3`; `v:real^3`]
XYOFCGX_lemma0) THEN
ASM_REWRITE_TAC[ARITH_RULE `1 < 2`] THEN
MP_TAC (ISPEC `S:real^3->bool`
CARD_2_IMP_DOUBLE) THEN
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE] THEN
STRIP_TAC THEN
DISCH_THEN (CONJUNCTS_THEN2 ASSUME_TAC (fun
th -> ALL_TAC)) THEN
FIRST_ASSUM (MP_TAC o SPEC `a:real^3`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `b:real^3`) THEN
ASM_REWRITE_TAC[
IN_INSERT;
CIRCUMCENTER_2;
ARCV_ANGLE] THEN
REPEAT DISCH_TAC THEN
MP_TAC (ISPECL [`a:real^3`; `b:real^3`; `
midpoint (a:real^3,b)`; `v:real^3`]
ANGLES_ALONG_LINE) THEN
ASM_REWRITE_TAC[
BETWEEN_MIDPOINT;
MIDPOINT_EQ_ENDPOINT] THEN
ONCE_REWRITE_TAC[
ANGLE_SYM] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REAL_ARITH_TAC);;
let ANGLE_GT_PI2 = prove(`!a b c:real^N.
pi / &2 <
angle (a,b,c) <=> (a - b)
dot (c - b) < &0`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[
angle;
vector_angle] THEN
ASM_CASES_TAC `a - b =
vec 0:real^N \/ c - b =
vec 0` THEN ASM_REWRITE_TAC[] THENL
[
REWRITE_TAC[
REAL_LT_REFL;
REAL_NOT_LT;
REAL_LE_LT] THEN
DISJ2_TAC THEN
POP_ASSUM DISJ_CASES_TAC THEN ASM_REWRITE_TAC[
DOT_LZERO;
DOT_RZERO];
ALL_TAC
] THEN
ABBREV_TAC `x = (a - b:real^N)
dot (c - b)` THEN
ABBREV_TAC `y =
norm (a - b:real^N) *
norm (c - b)` THEN
SUBGOAL_THEN `-- &1 <= x / y /\ x / y <= &1` ASSUME_TAC THENL
[
EXPAND_TAC "x" THEN EXPAND_TAC "y" THEN
MATCH_MP_TAC Trigonometry1.NORM_CAUCHY_SCHWARZ_FRAC THEN
ASM_REWRITE_TAC[GSYM DE_MORGAN_THM];
ALL_TAC
] THEN
EQ_TAC THENL
[
REWRITE_TAC[GSYM
ACS_0] THEN
MP_TAC (SPECL [`&0`; `x / y`]
ACS_MONO_LT_EQ) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[REAL_ARITH `abs a <= &1 <=> -- &1 <= a /\ a <= &1`] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[
REAL_NOT_LT] THEN DISCH_TAC THEN
MATCH_MP_TAC
REAL_LE_DIV THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "y" THEN
MATCH_MP_TAC
REAL_LE_MUL THEN
REWRITE_TAC[
NORM_POS_LE];
ALL_TAC
] THEN
REWRITE_TAC[GSYM
ACS_0] THEN DISCH_TAC THEN
MATCH_MP_TAC
ACS_MONO_LT THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 <= &1`] THEN
ONCE_REWRITE_TAC[REAL_ARITH `x / y < &0 <=> &0 < --x / y`] THEN
MATCH_MP_TAC
REAL_LT_DIV THEN
ASM_REWRITE_TAC[
REAL_NEG_GT0] THEN
EXPAND_TAC "y" THEN
MATCH_MP_TAC
REAL_LT_MUL THEN
ASM_REWRITE_TAC[
NORM_POS_LT; GSYM DE_MORGAN_THM]);;
let STRICT_CYCLIC_IMP_FAN = prove(`!V p. FINITE V /\ 2 <=
CARD V /\
(!v w. v
IN V /\ w
IN V /\ ~(v = w) ==> &0 <
azim (
vec 0) p v w)
==>
FAN (
vec 0, V
UNION {p}, {{p, v} | v | v
IN V})`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?v w:real^3. v
IN V /\ w
IN V /\ ~(v = w)` MP_TAC THENL
[
MP_TAC (ISPEC `V:real^3->bool`
CHOOSE_SUBSET) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `2`) THEN
ASM_REWRITE_TAC[
LE_REFL;
HAS_SIZE_2_EXISTS] THEN
REPEAT STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`x:real^3`; `y:real^3`] THEN
ASM_REWRITE_TAC[] THEN
CONJ_TAC THEN MATCH_MP_TAC
IN_TRANS THEN EXISTS_TAC `t:real^3->bool` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==> ~collinear {
vec 0, p, v}` (LABEL_TAC "c") THENL
[
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `v:real^3 = v'` THENL
[
FIRST_X_ASSUM (MP_TAC o SPECL [`v:real^3`; `w:real^3`]) THEN
ASM_REWRITE_TAC[
azim_def;
REAL_LT_REFL];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`v':real^3`; `v:real^3`]) THEN
ASM_REWRITE_TAC[
azim_def;
REAL_LT_REFL];
ALL_TAC
] THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==>
DISJOINT {
vec 0, p} {v} /\
DISJOINT {
vec 0} {p, v}` (LABEL_TAC "d") THENL
[
REPEAT STRIP_TAC THENL
[
MATCH_MP_TAC Collect_geom.COLLINEAR_DISJOINT3 THEN
FIRST_X_ASSUM (MP_TAC o SPEC `v':real^3`) THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[Collect_geom.PER_SET3];
REWRITE_TAC[
DISJOINT;
INTER_ACI] THEN
REWRITE_TAC[GSYM
DISJOINT] THEN
MATCH_MP_TAC Collect_geom.COLLINEAR_DISJOINT3 THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[]
];
ALL_TAC
] THEN
SUBGOAL_THEN `~(p:real^3 =
vec 0) /\ ~(p
IN V) /\ ~(
vec 0
IN V)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THENL
[
REMOVE_THEN "c" (MP_TAC o SPEC `v:real^3`) THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA_ALT];
REMOVE_THEN "c" (MP_TAC o SPEC `p:real^3`) THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA_ALT] THEN
DISJ2_TAC THEN EXISTS_TAC `&1` THEN
REWRITE_TAC[
VECTOR_MUL_LID];
REMOVE_THEN "c" (MP_TAC o SPEC `
vec 0:real^3`) THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA_ALT] THEN
DISJ2_TAC THEN EXISTS_TAC `&0` THEN
REWRITE_TAC[
VECTOR_MUL_LZERO]
];
ALL_TAC
] THEN
POP_ASSUM STRIP_ASSUME_TAC THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==> ~(v = p)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN POP_ASSUM (ASSUME_TAC o SYM) THEN
UNDISCH_TAC `~(p:real^3
IN V)` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[Fan_defs.FAN] THEN
REPEAT STRIP_TAC THENL
[
(* UNIONS E SUBSET (V UNION {p}) *)
REWRITE_TAC[
SUBSET;
IN_UNIONS;
IN_UNION;
IN_ELIM_THM;
IN_SING] THEN
REPEAT STRIP_TAC THEN
POP_ASSUM MP_TAC THEN
ASM_REWRITE_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[];
(* graph E *)
REWRITE_TAC[Fan_defs.graph;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
HAS_SIZE_2_EXISTS;
IN_INSERT;
NOT_IN_EMPTY] THEN
MAP_EVERY EXISTS_TAC [`p:real^3`; `v':real^3`] THEN
REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[
EQ_SYM_EQ] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
(* FINITE V /\ ~(V = {}) *)
ASM_REWRITE_TAC[Fan_defs.fan1;
FINITE_UNION;
FINITE_SING;
SUBSET_EMPTY] THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `p:real^3` THEN
REWRITE_TAC[
IN_UNION;
IN_SING];
(* ~(vec 0 IN V) *)
ASM_REWRITE_TAC[Fan_defs.fan2;
IN_UNION;
IN_SING];
(* ~collinear {vec 0, e} *)
REWRITE_TAC[Fan_defs.fan6;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `{
vec 0:real^3}
UNION {p, v'} = {
vec 0, p, v'}` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
EXTENSION;
IN_UNION;
IN_INSERT;
NOT_IN_EMPTY];
ALL_TAC
] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
(* fan7 *)
REWRITE_TAC[Fan_defs.fan7;
IN_UNION;
IN_ELIM_THM;
IN_SING] THEN
SUBGOAL_THEN `!t1 t2 r1 r2 x y v w:real^3. v
IN V /\ w
IN V /\ x =
t1 % p + t2 % v /\ y = r1 % p + r2 % w /\ &0 < t2 /\ &0 < r2 ==>
azim (
vec 0) p x y =
azim (
vec 0) p v w` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o SPEC `v:real^3`) THEN
SUBGOAL_THEN `~collinear {
vec 0, p, x:real^3} /\ ~collinear {
vec 0, p, y}` ASSUME_TAC THENL
[
ASM_REWRITE_TAC[
COLLINEAR_LEMMA_ALT; DE_MORGAN_THM;
NOT_EXISTS_THM] THEN
REWRITE_TAC[VECTOR_ARITH `
t1 % p + t2 % v = c % p <=> t2 % v = (c -
t1) % p:real^3`] THEN
CONJ_TAC THEN GEN_TAC THENL
[
DISCH_THEN (MP_TAC o AP_TERM `\v:real^3. inv(t2) % v`) THEN
ASM_SIMP_TAC[
VECTOR_MUL_ASSOC; REAL_ARITH `&0 < a ==> ~(a = &0)`; REAL_MUL_LINV] THEN
REMOVE_THEN "c" (MP_TAC o SPEC `v':real^3`) THEN
ASM_SIMP_TAC[
COLLINEAR_LEMMA_ALT;
VECTOR_MUL_LID;
NOT_EXISTS_THM];
DISCH_THEN (MP_TAC o AP_TERM `\v:real^3. inv(r2) % v`) THEN
ASM_SIMP_TAC[
VECTOR_MUL_ASSOC; REAL_ARITH `&0 < a ==> ~(a = &0)`; REAL_MUL_LINV] THEN
REMOVE_THEN "c" (MP_TAC o SPEC `w':real^3`) THEN
ASM_SIMP_TAC[
COLLINEAR_LEMMA_ALT;
VECTOR_MUL_LID;
NOT_EXISTS_THM]
];
ALL_TAC
] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `
azim (
vec 0) p v' y` THEN
CONJ_TAC THENL
[
ONCE_REWRITE_TAC[
AZIM_EQ_SYM] THEN
MATCH_MP_TAC
EQ_SYM THEN
MP_TAC (SPECL [`
vec 0:real^3`; `p:real^3`; `y:real^3`; `v':real^3`; `x:real^3`]
AZIM_EQ) THEN
ASM_SIMP_TAC[] THEN DISCH_TAC THEN
ASM_SIMP_TAC[
AFF_GT_2_1;
IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`&1 -
t1 - t2`; `t1:real`; `t2:real`] THEN
ASM_REWRITE_TAC[
VECTOR_MUL_RZERO;
VECTOR_ADD_LID] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
MATCH_MP_TAC
EQ_SYM THEN
MP_TAC (SPECL [`
vec 0:real^3`; `p:real^3`; `v':real^3`; `w':real^3`; `y:real^3`]
AZIM_EQ) THEN
ASM_SIMP_TAC[] THEN DISCH_TAC THEN
ASM_SIMP_TAC[
AFF_GT_2_1;
IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`&1 - r1 - r2`; `r1:real`; `r2:real`] THEN
ASM_REWRITE_TAC[
VECTOR_MUL_RZERO;
VECTOR_ADD_LID] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `!v w:real^3. v
IN V /\ w
IN V ==> aff_ge {
vec 0} {p, v}
INTER aff_ge {
vec 0} {p, w} = if (v = w) then aff_ge {
vec 0} {p, v} else aff_ge {
vec 0} {p}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
INTER_ACI] THEN
ASM_SIMP_TAC[Fan.AFF_GE_1_2;
HALFLINE] THEN
REWRITE_TAC[
VECTOR_MUL_RZERO;
VECTOR_ADD_LID;
EXTENSION;
IN_ELIM_THM;
IN_INTER] THEN
GEN_TAC THEN EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `t3 = &0` THENL
[
EXISTS_TAC `t2:real` THEN
UNDISCH_TAC `x = t2 % p + t3 % v':real^3` THEN
ASM_REWRITE_TAC[] THEN
VECTOR_ARITH_TAC;
ALL_TAC
] THEN
ASM_CASES_TAC `t3' = &0` THENL
[
EXISTS_TAC `t2':real` THEN
UNDISCH_TAC `x = t2' % p + t3' % w':real^3` THEN
ASM_REWRITE_TAC[] THEN
VECTOR_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
azim (
vec 0) p x x =
azim (
vec 0) p v' w'` MP_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
MAP_EVERY EXISTS_TAC [`t2:real`; `t3:real`; `t2':real`; `t3':real`] THEN
ASM_REWRITE_TAC[
REAL_LT_LE] THEN
UNDISCH_TAC `x = t2 % p + t3 % v':real^3` THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`v':real^3`; `w':real^3`]) THEN
ASM_REWRITE_TAC[
AZIM_REFL] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THEN
CONJ_TAC THEN MAP_EVERY EXISTS_TAC [`&1 - t`; `t:real`; `&0`] THEN ASM_REWRITE_TAC[
REAL_LE_REFL] THENL
[
CONJ_TAC THENL [ REAL_ARITH_TAC; VECTOR_ARITH_TAC ];
CONJ_TAC THENL [ REAL_ARITH_TAC; VECTOR_ARITH_TAC ]
];
ALL_TAC
] THEN
SUBGOAL_THEN `!v w:real^3. v
IN V /\ w
IN V ==> aff_ge {
vec 0} {p, v}
INTER aff_ge {
vec 0} {w} = if (v = w) then aff_ge {
vec 0} {v} else aff_ge {
vec 0} {}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
COND_CASES_TAC THENL
[
ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[
INTER_ACI] THEN
REWRITE_TAC[GSYM
SUBSET_INTER_ABSORPTION] THEN
MATCH_MP_TAC
AFF_GE_MONO_RIGHT THEN
ASM_SIMP_TAC[
SUBSET;
IN_INSERT;
NOT_IN_EMPTY];
ALL_TAC
] THEN
REWRITE_TAC[
AFF_GE_EQ_AFFINE_HULL;
AFFINE_HULL_SING] THEN
ASM_SIMP_TAC[Fan.AFF_GE_1_2] THEN
REWRITE_TAC[
EXTENSION;
IN_SING;
HALFLINE;
IN_INTER;
IN_ELIM_THM;
VECTOR_MUL_RZERO;
VECTOR_ADD_LID] THEN
GEN_TAC THEN EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `t = &0` THENL
[
UNDISCH_TAC `x = t % w':real^3` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO];
ALL_TAC
] THEN
ASM_CASES_TAC `t3 = &0` THENL
[
UNDISCH_TAC `x = t % w':real^3` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO;
VECTOR_ADD_RID] THEN
REMOVE_THEN "c" (MP_TAC o SPEC `w':real^3`) THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA_ALT;
NOT_EXISTS_THM] THEN
DISCH_TAC THEN
DISCH_THEN (MP_TAC o AP_TERM `\v:real^3. inv t % v`) THEN
ASM_SIMP_TAC[
VECTOR_MUL_ASSOC; REAL_MUL_LINV;
VECTOR_MUL_LID];
ALL_TAC
] THEN
SUBGOAL_THEN `
azim (
vec 0) p x x =
azim (
vec 0) p v' w'` MP_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
MAP_EVERY EXISTS_TAC [`t2:real`; `t3:real`; `&0`; `t:real`] THEN
ASM_REWRITE_TAC[
REAL_LT_LE;
VECTOR_MUL_LZERO;
VECTOR_ADD_LID] THEN
UNDISCH_TAC `x = t % w':real^3` THEN DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
REPLICATE_TAC 2 (FIRST_X_ASSUM (MP_TAC o SPECL [`v':real^3`; `w':real^3`])) THEN
ASM_REWRITE_TAC[
AZIM_REFL] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
DISCH_TAC THEN
CONJ_TAC THENL
[
MAP_EVERY EXISTS_TAC [`&1`; `&0`; `&0`] THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO;
VECTOR_ADD_LID] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
EXISTS_TAC `&0` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO;
REAL_LE_REFL];
ALL_TAC
] THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==> aff_ge {
vec 0} {p, v}
INTER aff_ge {
vec 0} {p} = aff_ge {
vec 0} {p}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[
INTER_ACI] THEN
REWRITE_TAC[GSYM
SUBSET_INTER_ABSORPTION] THEN
MATCH_MP_TAC
AFF_GE_MONO_RIGHT THEN
ASM_SIMP_TAC[
SUBSET;
IN_INSERT;
NOT_IN_EMPTY];
ALL_TAC
] THEN
SUBGOAL_THEN `!v w:real^3. {p, v}
INTER {p, w} = if (v = w) then {p, v} else {p}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
COND_CASES_TAC THENL
[
ASM_REWRITE_TAC[
INTER_ACI];
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION;
IN_INTER;
IN_INSERT;
NOT_IN_EMPTY] THEN
GEN_TAC THEN EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
UNDISCH_TAC `~(v' = w':real^3)` THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
SIMP_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!v:real^3. {p, v}
INTER {p} = {p} /\ {p}
INTER {p, v} = {p}` ASSUME_TAC THENL
[
REWRITE_TAC[
INTER_ACI; GSYM
SUBSET_INTER_ABSORPTION;
SUBSET;
IN_INSERT;
NOT_IN_EMPTY] THEN
SIMP_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!v w:real^3. w
IN V ==> {p, v}
INTER {w} = if (v = w) then {v} else {}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
COND_CASES_TAC THENL
[
ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[
INTER_ACI] THEN
ASM_REWRITE_TAC[GSYM
SUBSET_INTER_ABSORPTION;
SUBSET] THEN
SIMP_TAC[
IN_INSERT];
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION;
IN_INTER;
NOT_IN_EMPTY;
IN_INSERT; DE_MORGAN_THM] THEN
GEN_TAC THEN ASM_CASES_TAC `~(x = w':real^3)` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[NOT_CLAUSES] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!v w:real^3. v
IN V /\ w
IN V ==> aff_ge {
vec 0} {v}
INTER aff_ge {
vec 0} {w} = if (v = w) then aff_ge {
vec 0} {v} else aff_ge {
vec 0} {}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
INTER_ACI] THEN
REWRITE_TAC[
AFF_GE_EQ_AFFINE_HULL;
AFFINE_HULL_SING] THEN
REWRITE_TAC[
EXTENSION;
IN_SING;
HALFLINE;
IN_INTER;
IN_ELIM_THM;
VECTOR_MUL_RZERO;
VECTOR_ADD_LID] THEN
GEN_TAC THEN EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `t = &0` THENL
[
UNDISCH_TAC `x = t % v':real^3` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO];
ALL_TAC
] THEN
ASM_CASES_TAC `t' = &0` THENL
[
UNDISCH_TAC `x = t' % w':real^3` THEN
ASM_REWRITE_TAC[
VECTOR_MUL_LZERO];
ALL_TAC
] THEN
SUBGOAL_THEN `
azim (
vec 0) p x x =
azim (
vec 0) p v' w'` MP_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
MAP_EVERY EXISTS_TAC [`&0`; `t:real`; `&0`; `t':real`] THEN
ASM_REWRITE_TAC[
REAL_LT_LE;
VECTOR_MUL_LZERO;
VECTOR_ADD_LID] THEN
UNDISCH_TAC `x = t % v':real^3` THEN DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
SUBGOAL_THEN `&0 <
azim (
vec 0) p v' w'` MP_TAC THENL
[
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
AZIM_REFL] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
DISCH_TAC THEN
CONJ_TAC THEN EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[
REAL_LE_REFL;
VECTOR_MUL_LZERO];
ALL_TAC
] THEN
SUBGOAL_THEN `!v w:real^3. {v}
INTER {w} = if (v = w) then {v} else {}` ASSUME_TAC THENL
[
REPEAT GEN_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
INTER_ACI] THEN
REWRITE_TAC[
EXTENSION;
IN_INTER;
NOT_IN_EMPTY;
IN_SING] THEN
GEN_TAC THEN REWRITE_TAC[DE_MORGAN_THM] THEN
ASM_CASES_TAC `x = v':real^3` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==> aff_ge {
vec 0} {v}
INTER aff_ge {
vec 0} {p} = aff_ge {
vec 0} {}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
HALFLINE;
AFF_GE_EQ_AFFINE_HULL;
AFFINE_HULL_SING;
IN_INTER;
VECTOR_MUL_RZERO;
VECTOR_ADD_LID] THEN
REWRITE_TAC[
EXTENSION;
IN_INTER;
IN_SING;
IN_ELIM_THM] THEN
GEN_TAC THEN EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
REMOVE_THEN "c" (MP_TAC o SPEC `v':real^3`) THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA_ALT;
NOT_EXISTS_THM] THEN
DISCH_TAC THEN
ASM_CASES_TAC `t = &0` THEN ASM_REWRITE_TAC[
VECTOR_MUL_LZERO] THEN
UNDISCH_TAC `x = t % v':real^3` THEN
DISCH_THEN (MP_TAC o AP_TERM `\v:real^3. inv t % v`) THEN
ASM_SIMP_TAC[
VECTOR_MUL_ASSOC; REAL_MUL_LINV;
VECTOR_MUL_LID];
ALL_TAC
] THEN
DISCH_TAC THEN
CONJ_TAC THEN EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[
REAL_LE_REFL;
VECTOR_MUL_LZERO];
ALL_TAC
] THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==> {v}
INTER {p} = {}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
IN_INTER;
EXTENSION;
NOT_IN_EMPTY;
IN_SING; DE_MORGAN_THM] THEN
GEN_TAC THEN ASM_CASES_TAC `x = p:real^3` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN (ASSUME_TAC o SYM) THEN
UNDISCH_TAC `v':real^3
IN V` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN
ONCE_REWRITE_TAC[
INTER_ACI] THEN ASM_SIMP_TAC[]
THEN TRY (COND_CASES_TAC THEN ASM_REWRITE_TAC[])
);;
let STRICT_CYCLIC_FAN_PROPERTIES = prove(`!V p. let W = V
UNION {p} in let E = {{p,v} | v | v
IN V} in
FAN (
vec 0, W, E) ==>
(!v. v
IN V ==> (p, v)
IN dart1_of_fan (W,E)) /\
set_of_edge p W E = V /\
(!v. v
IN V ==> node (
hypermap_of_fan (W,E)) (p,v) = {(p,w) | w | w
IN V})`,
REPEAT GEN_TAC THEN CONV_TAC (DEPTH_CONV let_CONV) THEN
ABBREV_TAC `W = V
UNION {p:real^3}` THEN
ABBREV_TAC `E = {{p, v:real^3} | v | v
IN V}` THEN
DISCH_TAC THEN
SUBGOAL_THEN `!v:real^3. v
IN V ==> (p, v)
IN dart1_of_fan (W,E)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
EXPAND_TAC "E" THEN
REWRITE_TAC[Fan_defs.dart1_of_fan;
IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`p:real^3`; `v:real^3`] THEN
REWRITE_TAC[] THEN
EXISTS_TAC `v:real^3` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `
set_of_edge p W E = V:real^3->bool` ASSUME_TAC THENL
[
SUBGOAL_THEN `~(p:real^3
IN V)` MP_TAC THENL
[
DISCH_TAC THEN
UNDISCH_TAC `
FAN (
vec 0:real^3,W,E)` THEN
REWRITE_TAC[Fan_defs.FAN; Fan_defs.graph] THEN
DISCH_THEN (CONJUNCTS_THEN2 (fun
th -> ALL_TAC) MP_TAC) THEN
DISCH_THEN (CONJUNCTS_THEN2 MP_TAC (fun
th -> ALL_TAC)) THEN
DISCH_THEN (MP_TAC o SPEC `{p:real^3, p}`) THEN
EXPAND_TAC "E" THEN
REWRITE_TAC[
IN_ELIM_THM; NOT_IMP] THEN
SUBGOAL_THEN `{p:real^3,p} = {p}` ASSUME_TAC THENL [ REWRITE_TAC[
EXTENSION;
IN_INSERT;
NOT_IN_EMPTY]; ALL_TAC ] THEN
ASM_SIMP_TAC[
HAS_SIZE;
CARD_CLAUSES;
FINITE_EMPTY;
NOT_IN_EMPTY] THEN
REWRITE_TAC[DE_MORGAN_THM; ARITH_RULE `~(SUC 0 = 2)`] THEN
EXISTS_TAC `p:real^3` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[Fan_defs.set_of_edge] THEN
EXPAND_TAC "E" THEN EXPAND_TAC "W" THEN
REWRITE_TAC[
EXTENSION;
IN_UNION;
IN_ELIM_THM;
IN_SING] THEN
REPEAT STRIP_TAC THEN
EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `v:real^3`) THEN
ASM_REWRITE_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN
DISCH_TAC THEN UNDISCH_TAC `v:real^3
IN V` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
EXISTS_TAC `x:real^3` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[Hypermap_and_fan.NODE_HYPERMAP_OF_FAN_ALT]);;
let ANGLE_SUM_lemma = prove(`!V p. FINITE V /\ 2 <=
CARD V /\
(!v w. v
IN V /\ w
IN V /\ ~(v = w) ==> &0 <
azim (
vec 0) p v w)
==> ?f. (!x. x
IN V ==> f x
IN V /\ ~(x = f x)) /\
sum V (\x.
azim (
vec 0) p x (f x)) = &2 *
pi`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o MATCH_MP
STRICT_CYCLIC_IMP_FAN) THEN DISCH_TAC THEN
ABBREV_TAC `W = V
UNION {p:real^3}` THEN
ABBREV_TAC `E = {{p, v:real^3} | v | v
IN V}` THEN
MP_TAC (SPEC_ALL
STRICT_CYCLIC_FAN_PROPERTIES) THEN
CONV_TAC (DEPTH_CONV let_CONV) THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
EXISTS_TAC `
sigma_fan (
vec 0) W E p` THEN
CONJ_TAC THENL
[
GEN_TAC THEN DISCH_TAC THEN
MP_TAC (SPECL [`
vec 0:real^3`; `W:real^3->bool`; `E:(real^3->bool)->bool`; `p:real^3`; `x:real^3`] Fan.SIGMA_FAN) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o check (is_conj o concl)) THEN
ASM_SIMP_TAC[
CARD_CLAUSES;
FINITE_EMPTY;
NOT_IN_EMPTY] THEN
ARITH_TAC;
ALL_TAC
] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `?v:real^3. v
IN V` CHOOSE_TAC THENL
[
REWRITE_TAC[
MEMBER_NOT_EMPTY] THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o check (is_conj o concl)) THEN
ASM_REWRITE_TAC[
CARD_CLAUSES] THEN
ARITH_TAC;
ALL_TAC
] THEN
MP_TAC (SPECL [`W:real^3->bool`; `E:(real^3->bool)->bool`; `(p:real^3,v:real^3)`] Hypermap_and_fan.SUM_AZIM_DART) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
dart1_of_fan (W:real^3->bool,E)` THEN
ASM_SIMP_TAC[Hypermap_and_fan.DART1_OF_FAN_SUBSET_DART_OF_FAN];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_SIMP_TAC[] THEN
SUBGOAL_THEN `{p:real^3, w:real^3 | w
IN V} =
IMAGE (\v. p, v) V` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
IMAGE_LEMMA];
ALL_TAC
] THEN
MP_TAC (ISPECL [`\x.
azim (
vec 0) p x (
sigma_fan (
vec 0) W E p x)`; `V:real^3->bool`]
SUM_RESTRICT) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(\x. if x
IN V then
azim (
vec 0) p x (
sigma_fan (
vec 0) W E p x) else &0) = (\x. if x
IN V then ((
azim_dart (W,E)) o (\v. p, v)) x else &0)` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
FUN_EQ_THM] THEN GEN_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
o_THM] THEN
REWRITE_TAC[Fan_defs.azim_dart; Fan_defs.azim_fan] THEN
SUBGOAL_THEN `~(p = x:real^3)` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC Hypermap_and_fan.PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ THEN
MAP_EVERY EXISTS_TAC [`W:real^3->bool`; `E:(real^3->bool)->bool`] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[ARITH_RULE `2 <= a ==> a > 1`];
ALL_TAC
] THEN
MP_TAC (ISPECL [`
azim_dart (W,E) o (\v. p,v)`; `V:real^3->bool`]
SUM_RESTRICT) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC (GSYM
SUM_IMAGE) THEN
SIMP_TAC[
PAIR_EQ]);;
let ANGLE_SUM_BOUND = prove(`!V (p:real^3) a. &0 <= a /\ FINITE V /\ 2 <=
CARD V /\
(!v w. v
IN V /\ w
IN V /\ ~(v = w) ==> a <
azim (
vec 0) p v w)
==> a * &(
CARD V) < &2 *
pi`,
REPEAT STRIP_TAC THEN
MP_TAC (SPEC_ALL
ANGLE_SUM_lemma) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `a:real` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `
sum V (\x.
azim (
vec 0) p x (f x))` THEN
CONJ_TAC THENL
[
REMOVE_ASSUM THEN
SUBGOAL_THEN `!x.
azim (
vec 0) p x (f x) = -- (--azim (
vec 0) p x (f x))` (fun
th -> ONCE_REWRITE_TAC[
th]) THENL [ REAL_ARITH_TAC; ALL_TAC ] THEN
ONCE_REWRITE_TAC[
SUM_NEG] THEN
REWRITE_TAC[REAL_ARITH `a * b < --c <=> c < b * (--a)`] THEN
MATCH_MP_TAC
SUM_BOUND_LT THEN
ASM_REWRITE_TAC[
REAL_LE_NEG] THEN
ASM_SIMP_TAC[
REAL_LE_LT] THEN
SUBGOAL_THEN `?x:real^3. x
IN V` CHOOSE_TAC THENL
[
REWRITE_TAC[
MEMBER_NOT_EMPTY] THEN
DISCH_TAC THEN UNDISCH_TAC `2 <=
CARD (V:real^3->bool)` THEN
ASM_REWRITE_TAC[
CARD_CLAUSES; ARITH_RULE `~(2 <= 0)`];
ALL_TAC
] THEN
EXISTS_TAC `x:real^3` THEN
ASM_SIMP_TAC[
REAL_LT_NEG];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
REAL_LE_REFL]);;
let DIHV_LE_AZIM = prove(`!v w x y. ~collinear {v, w, x} /\ ~collinear {v, w, y}
==>
dihV v w x y <=
azim v w x y`,
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`v:real^3`; `w:real^3`; `x:real^3`; `y:real^3`]
AZIM_DIVH) THEN
ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[
REAL_LE_REFL] THEN
DISCH_TAC THEN
REWRITE_TAC[REAL_ARITH `a <= &2 * b - a <=> a <= b`] THEN
REWRITE_TAC[
DIHV_RANGE]);;
let PYTHAGORAS_PROJECTION = prove(`!x y n:real^N. x
dot n = &0 ==>
dist (x, y) pow 2 =
dist (x, projection n y) pow 2 +
dist (projection n y, y) pow 2`,
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`y:real^N`; `projection (n:real^N) y`; `x:real^N`]
PYTHAGORAS) THEN
ANTS_TAC THENL
[
REWRITE_TAC[projection; VECTOR_ARITH `y - (y - a % n) = a % n:real^N`] THEN
REWRITE_TAC[
orthogonal;
DOT_RSUB;
DOT_LMUL;
DOT_RMUL] THEN
POP_ASSUM MP_TAC THEN SIMP_TAC[
DOT_SYM] THEN DISCH_TAC THEN
REWRITE_TAC[
REAL_ENTIRE] THEN
DISJ2_TAC THEN
ASM_CASES_TAC `n
dot (n:real^N) = &0` THENL
[
SUBGOAL_THEN `n =
vec 0:real^N` ASSUME_TAC THENL
[
ASM_REWRITE_TAC[GSYM
DOT_EQ_0];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
DOT_LZERO] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN
CONV_TAC REAL_FIELD;
ALL_TAC
] THEN
REWRITE_TAC[
dist] THEN
REAL_ARITH_TAC);;
let XYOFCGX_3_0 = prove(`!V S. packing V /\ S
SUBSET V /\ ~affine_dependent S /\
circumcenter S =
vec 0 /\ radV S <
sqrt(&2) /\
CARD S = 3 ==>
(!u v. u
IN S /\ v
IN (V
DIFF S) ==>
dist (v,
vec 0) >
dist (u,
vec 0))`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `
dist (v,
vec 0:real^3) >
dist(u,
vec 0:real^3)` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN PURE_REWRITE_TAC[
real_gt;
REAL_NOT_LT;
DIST_SYM] THEN DISCH_TAC THEN
SUBGOAL_THEN `!u:real^3. u
IN S ==>
dist (u,
vec 0) = radV S` (LABEL_TAC "r") THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[
DIST_SYM] THEN
DISCH_THEN (MATCH_MP_TAC o GSYM) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
dist (v:real^3,
vec 0) <= radV (S:real^3->bool)` ASSUME_TAC THENL
[
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `
dist (u:real^3,
vec 0)` THEN
ASM_REWRITE_TAC[
REAL_LE_LT] THEN
DISJ2_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `~(S = {}:real^3->bool)` ASSUME_TAC THENL
[
DISCH_TAC THEN
UNDISCH_TAC `
CARD (S:real^3->bool) = 3` THEN
ASM_REWRITE_TAC[
CARD_CLAUSES; ARITH_RULE `~(0 = 3)`];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `S:real^3->bool`; `
vec 0:real^3`; `v:real^3`]
XYOFCGX_lemma0) THEN
ASM_REWRITE_TAC[
DIST_SYM; ARITH_RULE `1 < 3`;
ARCV_ANGLE] THEN
STRIP_TAC THEN
SUBGOAL_THEN `?n:real^3. ~(n =
vec 0) /\ (!u. u
IN S ==> n
dot u = &0)` MP_TAC THENL
[
MP_TAC (ISPEC `S:real^3->bool`
LOWDIM_SUBSET_HYPERPLANE) THEN
ANTS_TAC THENL
[
MP_TAC (ISPEC `S:real^3->bool`
AFF_DIM_DIM_0) THEN
ANTS_TAC THENL
[
MP_TAC (ISPEC `S:real^3->bool`
OAPVION1) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (ISPEC `S:real^3->bool`
AFF_DIM_LE_CARD) THEN
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE] THEN
REWRITE_TAC[GSYM
INT_OF_NUM_LT; DIMINDEX_3] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[
SUBSET;
IN_ELIM_THM] THEN DISCH_TAC THEN
EXISTS_TAC `a:real^3` THEN
ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[
SPAN_INC];
ALL_TAC
] THEN
STRIP_TAC THEN
ABBREV_TAC `w:real^3 = projection n v` THEN
ABBREV_TAC `W = (w:real^3)
INSERT S` THEN
SUBGOAL_THEN `(!u:real^3. u
IN S ==> ~(u =
vec 0)) /\ ~(w:real^3 =
vec 0)` STRIP_ASSUME_TAC THENL
[
CONJ_TAC THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (ISPEC `S:real^3->bool`
CIRCUMCENTER_NOT_EQ) THEN
ASM_REWRITE_TAC[ARITH_RULE `1 < 3`] THEN
DISCH_THEN (MP_TAC o SPEC `u':real^3`) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_TAC THEN
MP_TAC (ISPECL [`u:real^3`; `v:real^3`; `n:real^3`]
PYTHAGORAS_PROJECTION) THEN
ASM_SIMP_TAC[
DOT_SYM] THEN
SUBGOAL_THEN `&4 <=
dist (u:real^3, v) pow 2` MP_TAC THENL
[
REWRITE_TAC[REAL_ARITH `&4 = &2 pow 2`] THEN
REWRITE_TAC[GSYM
REAL_LE_SQUARE_ABS;
dist;
REAL_ABS_NORM; REAL_ARITH `abs(&2) = &2`] THEN
REWRITE_TAC[GSYM
dist] THEN
UNDISCH_TAC `packing (V:real^3->bool)` THEN
REWRITE_TAC[packing] THEN
DISCH_THEN MATCH_MP_TAC THEN
CONJ_TAC THENL
[
ONCE_REWRITE_TAC[GSYM
IN] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
CONJ_TAC THENL
[
ONCE_REWRITE_TAC[GSYM
IN] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `V
DIFF S:real^3->bool` THEN
ASM_REWRITE_TAC[
SUBSET_DIFF];
ALL_TAC
] THEN
DISCH_TAC THEN UNDISCH_TAC `v:real^3
IN V
DIFF S` THEN
ASM_REWRITE_TAC[
IN_DIFF; DE_MORGAN_THM] THEN
DISJ2_TAC THEN
POP_ASSUM (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
SUBGOAL_THEN `radV (S:real^3->bool) pow 2 +
dist (
vec 0, v:real^3) pow 2 < &2 + &2` MP_TAC THENL
[
MATCH_MP_TAC
REAL_LT_ADD2 THEN
MP_TAC (SPEC `&2`
SQRT_WORKS) THEN
REWRITE_TAC[REAL_ARITH `&0 <= &2`] THEN
DISCH_TAC THEN
FIRST_ASSUM (fun
th -> ONCE_REWRITE_TAC[GSYM
th]) THEN
REWRITE_TAC[GSYM
REAL_LT_SQUARE_ABS] THEN
CONJ_TAC THENL
[
SUBGOAL_THEN `abs (radV (S:real^3->bool)) = radV S` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
REAL_ABS_REFL] THEN
REMOVE_THEN "r" (MP_TAC o SPEC `u:real^3`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
REWRITE_TAC[
dist;
NORM_POS_LE];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN
UNDISCH_TAC `radV (S:real^3->bool) <
sqrt (&2)` THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
dist;
REAL_ABS_NORM] THEN REWRITE_TAC[GSYM
dist] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `radV (S:real^3->bool)` THEN
ASM_REWRITE_TAC[
DIST_SYM] THEN
POP_ASSUM MP_TAC THEN UNDISCH_TAC `radV (S:real^3->bool) <
sqrt (&2)` THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
REPEAT (FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (free_in `u:real^3` o concl))) THEN
SUBGOAL_THEN `!u:real^3. u
IN S ==>
pi / &2 <
angle (u,
vec 0, w)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
EXPAND_TAC "w" THEN
MATCH_MP_TAC
OBTUSE_ANGLE_PROJECTION THEN
CONJ_TAC THENL
[
FIRST_X_ASSUM ((fun
th -> ALL_TAC) o SPECL [`v:real^3`; `w:real^3`]) THEN
ONCE_REWRITE_TAC[
ANGLE_SYM] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
DOT_SYM] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `FINITE (W:real^3->bool) /\
CARD W = 4` ASSUME_TAC THENL
[
UNDISCH_TAC `w:real^3
INSERT S = W` THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_SIMP_TAC[
FINITE_INSERT;
AFFINE_INDEPENDENT_IMP_FINITE;
CARD_CLAUSES] THEN
SUBGOAL_THEN `~(w:real^3
IN S)` (fun
th -> REWRITE_TAC[
th]) THENL
[
DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `w:real^3`) THEN
ASM_SIMP_TAC[
ANGLE_REFL_MID] THEN
MP_TAC
PI_POS THEN REAL_ARITH_TAC;
ALL_TAC
] THEN
ARITH_TAC;
ALL_TAC
] THEN
MP_TAC (SPECL [`W:real^3->bool`; `n:real^3`; `
pi / &2`]
ANGLE_SUM_BOUND) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[ARITH_RULE `2 <= 4`] THEN
CONJ_TAC THENL [ MP_TAC
PI_POS THEN REAL_ARITH_TAC; ALL_TAC ] THEN
EXPAND_TAC "W" THEN
REWRITE_TAC[
IN_INSERT] THEN
SUBGOAL_THEN `n
dot (w:real^3) = &0` ASSUME_TAC THENL
[
EXPAND_TAC "w" THEN
MP_TAC (ISPECL [`n:real^3`; `v:real^3`]
PROJECTION_ORTHOGONAL) THEN
SIMP_TAC[
DOT_SYM];
ALL_TAC
] THEN
REPEAT STRIP_TAC THENL
[
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `
angle (w':real^3,
vec 0, w)` THEN
ASM_SIMP_TAC[] THEN
ONCE_REWRITE_TAC[
ANGLE_SYM] THEN
ASM_SIMP_TAC[
ANGLE_EQ_DIHV] THEN
ASM_SIMP_TAC[
IN_PLANE_NOT_COLLINEAR;
DIHV_LE_AZIM];
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `
angle (v':real^3,
vec 0, w)` THEN
ASM_SIMP_TAC[
ANGLE_EQ_DIHV] THEN
ASM_SIMP_TAC[
IN_PLANE_NOT_COLLINEAR;
DIHV_LE_AZIM];
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `
angle (v':real^3,
vec 0, w':real^3)` THEN
ASM_SIMP_TAC[
ANGLE_EQ_DIHV] THEN
ASM_SIMP_TAC[
IN_PLANE_NOT_COLLINEAR;
DIHV_LE_AZIM]
];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN MP_TAC
PI_POS THEN
REAL_ARITH_TAC);;
let XYOFCGX_4_0 = prove(`!V S. packing V /\ S
SUBSET V /\ ~affine_dependent S /\
circumcenter S =
vec 0 /\ radV S <
sqrt (&2) /\
CARD S = 4
==> (!u v. u
IN S /\ v
IN V
DIFF S ==>
dist (v,
vec 0) >
dist (u,
vec 0))`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `
dist (v,
vec 0:real^3) >
dist(u,
vec 0:real^3)` THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN PURE_REWRITE_TAC[
real_gt;
REAL_NOT_LT;
DIST_SYM] THEN DISCH_TAC THEN
SUBGOAL_THEN `!u:real^3. u
IN S ==>
dist (u,
vec 0) = radV S` (LABEL_TAC "r") THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[
DIST_SYM] THEN
DISCH_THEN (MATCH_MP_TAC o GSYM) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
dist (v:real^3,
vec 0) <= radV (S:real^3->bool)` ASSUME_TAC THENL
[
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `
dist (u:real^3,
vec 0)` THEN
ASM_REWRITE_TAC[
REAL_LE_LT] THEN
DISJ2_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `~(S = {}:real^3->bool)` ASSUME_TAC THENL
[
DISCH_TAC THEN
UNDISCH_TAC `
CARD (S:real^3->bool) = 4` THEN
ASM_REWRITE_TAC[
CARD_CLAUSES; ARITH_RULE `~(0 = 4)`];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `S:real^3->bool`; `
vec 0:real^3`; `v:real^3`]
XYOFCGX_lemma0) THEN
ASM_REWRITE_TAC[
DIST_SYM; ARITH_RULE `1 < 4`] THEN
STRIP_TAC THEN
REPEAT (FIRST_X_ASSUM ((fun
th -> ALL_TAC) o check (free_in `u:real^3` o concl))) THEN
SUBGOAL_THEN `!u:real^3. u
IN S ==> ~collinear {
vec 0, v, u}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
POP_ASSUM MP_TAC THEN
PURE_REWRITE_TAC[CONJUNCT1 Collect_geom.PER_SET3] THEN
REWRITE_TAC[
COLLINEAR_LEMMA_ALT; DE_MORGAN_THM] THEN
CONJ_TAC THENL
[
MP_TAC (ISPEC `S:real^3->bool`
CIRCUMCENTER_NOT_EQ) THEN
ASM_REWRITE_TAC[ARITH_RULE `1 < 4`] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
NOT_EXISTS_THM] THEN GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `?w:real^3. w
IN S /\ ~(u = w)` CHOOSE_TAC THENL
[
SUBGOAL_THEN `~(S
DELETE u:real^3 = {})` MP_TAC THENL
[
DISCH_THEN (MP_TAC o AP_TERM `\s:real^3->bool.
CARD s`) THEN
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE;
FINITE_DELETE;
CARD_DELETE;
CARD_CLAUSES] THEN
ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY;
IN_DELETE] THEN
STRIP_TAC THEN
EXISTS_TAC `x:real^3` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`u:real^3`; `w:real^3`]) THEN
FIRST_ASSUM (MP_TAC o SPEC `w:real^3`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `u:real^3`) THEN
ASM_REWRITE_TAC[
ARCV_ANGLE;
ANGLE_GT_PI2;
VECTOR_SUB_RZERO;
DOT_LMUL] THEN
DISCH_TAC THEN
SUBGOAL_THEN `c < &0` MP_TAC THENL
[
POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[
REAL_NOT_LT] THEN DISCH_TAC THEN
MATCH_MP_TAC
REAL_LE_MUL THEN
ASM_REWRITE_TAC[
DOT_POS_LE];
ALL_TAC
] THEN
DISCH_TAC THEN
ASM_CASES_TAC `u
dot w:real^3 < &0` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
REAL_NOT_LT] THEN
ONCE_REWRITE_TAC[REAL_ARITH `a * b = (--a) * (--b)`] THEN
MATCH_MP_TAC
REAL_LE_MUL THEN
ASM_REWRITE_TAC[
REAL_NEG_GE0;
REAL_LE_LT];
ALL_TAC
] THEN
MP_TAC (SPECL [`S:real^3->bool`; `v:real^3`; `
pi / &2`]
ANGLE_SUM_BOUND) THEN
ANTS_TAC THENL
[
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE; ARITH_RULE `2 <= 4`] THEN
CONJ_TAC THENL [ MP_TAC
PI_POS THEN REAL_ARITH_TAC; ALL_TAC ] THEN
X_GEN_TAC `u:real^3` THEN X_GEN_TAC `w:real^3` THEN
STRIP_TAC THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `
dihV (
vec 0:real^3) v u w` THEN
CONJ_TAC THENL
[
REWRITE_TAC[
DIHV_GT_PI2] THEN
MP_TAC (ISPECL [`
vec 0:real^3`; `u:real^3`; `w:real^3`; `v:real^3`] Trigonometry.RLXWSTK) THEN
ASM_SIMP_TAC[] THEN
CONV_TAC (DEPTH_CONV let_CONV) THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[REAL_ARITH `(a - b) / c < &0 <=> &0 < (b + --a) / c`] THEN
MATCH_MP_TAC
REAL_LT_DIV THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
REAL_LT_ADD THEN
CONJ_TAC THENL
[
ONCE_REWRITE_TAC[REAL_ARITH `a * b = (--a) * (--b)`] THEN
MATCH_MP_TAC
REAL_LT_MUL THEN
REWRITE_TAC[
REAL_NEG_GT0; GSYM
ARCV_GT_PI2] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
REAL_NEG_GT0; GSYM
ARCV_GT_PI2] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
REAL_LT_MUL THEN
REWRITE_TAC[
ARCV_ANGLE;
REAL_LT_LE;
SIN_ANGLE_POS] THEN
CONJ_TAC THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `w:real^3`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`v:real^3`; `
vec 0:real^3`; `w:real^3`]
COLLINEAR_SIN_ANGLE) THEN
ANTS_TAC THENL
[
POP_ASSUM MP_TAC THEN SIMP_TAC[
COLLINEAR_LEMMA; DE_MORGAN_THM];
ALL_TAC
] THEN
DISCH_TAC THEN
DISCH_THEN (MP_TAC o SYM) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[Collect_geom.PER_SET3];
FIRST_X_ASSUM (MP_TAC o SPEC `u:real^3`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`v:real^3`; `
vec 0:real^3`; `u:real^3`]
COLLINEAR_SIN_ANGLE) THEN
ANTS_TAC THENL
[
POP_ASSUM MP_TAC THEN SIMP_TAC[
COLLINEAR_LEMMA; DE_MORGAN_THM];
ALL_TAC
] THEN
DISCH_TAC THEN
DISCH_THEN (MP_TAC o SYM) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[Collect_geom.PER_SET3]
];
ALL_TAC
] THEN
MATCH_MP_TAC
DIHV_LE_AZIM THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN MP_TAC
PI_POS THEN
REAL_ARITH_TAC);;
let XYOFCGX = prove(`!V S (p:real^3). packing V /\ S
SUBSET V /\ ~affine_dependent S /\
(p = circumcenter S) /\ (radV S <
sqrt(&2)) ==>
(!u v. u
IN S /\ v
IN (V
DIFF S) ==>
dist(v,p) >
dist(u,p))`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC `S = {}:real^3->bool` THENL
[
ASM_REWRITE_TAC[
NOT_IN_EMPTY];
ALL_TAC
] THEN
ABBREV_TAC `W =
IMAGE (\x:real^3. --p + x) V` THEN
ABBREV_TAC `K =
IMAGE (\x:real^3. --p + x) S` THEN
SUBGOAL_THEN `(!u v:real^3. u
IN K /\ v
IN W
DIFF K ==>
dist (v,
vec 0) >
dist(u,
vec 0)) ==> (!u v:real^3. u
IN S /\ v
IN V
DIFF S ==>
dist (v,p) >
dist(u,p))` MP_TAC THENL
[
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`--p + u:real^3`; `--p + v:real^3`]) THEN
REWRITE_TAC[
dist;
VECTOR_SUB_RZERO; VECTOR_ARITH `--a + b = b - a:real^3`] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[
IN_IMAGE;
IN_DIFF] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[
IN_DIFF] THEN DISCH_TAC THEN
REPEAT CONJ_TAC THENL
[
EXISTS_TAC `u:real^3` THEN ASM_REWRITE_TAC[];
EXISTS_TAC `v:real^3` THEN ASM_REWRITE_TAC[];
REWRITE_TAC[VECTOR_ARITH `v - p = x - p <=> x = v:real^3`;
NOT_EXISTS_THM; DE_MORGAN_THM] THEN
GEN_TAC THEN
ASM_CASES_TAC `x = v:real^3` THEN ASM_REWRITE_TAC[]
];
ALL_TAC
] THEN
DISCH_THEN MATCH_MP_TAC THEN
SUBGOAL_THEN `packing W /\ K
SUBSET W /\ ~affine_dependent K /\ circumcenter K =
vec 0:real^3 /\ radV K <
sqrt (&2)` STRIP_ASSUME_TAC THENL
[
REPEAT CONJ_TAC THENL
[
UNDISCH_TAC `packing (V:real^3->bool)` THEN
EXPAND_TAC "W" THEN
REWRITE_TAC[packing] THEN
REWRITE_TAC[
IMAGE;
IN_ELIM_THM;
IN] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`x:real^3`; `x':real^3`]) THEN
POP_ASSUM MP_TAC THEN
ASM_REWRITE_TAC[VECTOR_ARITH `--p + x = --p + x' <=> x = x':real^3`] THEN
SIMP_TAC[
dist; VECTOR_ARITH `(--p + x) - (--p + x') = x - x':real^3`];
EXPAND_TAC "K" THEN EXPAND_TAC "W" THEN
ASM_SIMP_TAC[
IMAGE_SUBSET];
EXPAND_TAC "K" THEN
ASM_REWRITE_TAC[
AFFINE_DEPENDENT_TRANSLATION_EQ];
EXPAND_TAC "K" THEN
MP_TAC (ISPECL [`S:real^3->bool`; `--p:real^3`]
CIRCUMCENTER_TRANSLATION) THEN
ASM_REWRITE_TAC[VECTOR_ARITH `--p + p =
vec 0:real^3`];
MP_TAC (ISPECL [`S:real^3->bool`; `--p:real^3`]
RADV_TRANSLATION) THEN
ASM_SIMP_TAC[]
];
ALL_TAC
] THEN
REPLICATE_TAC 5 (POP_ASSUM MP_TAC) THEN
REPEAT REMOVE_ASSUM THEN REPEAT DISCH_TAC THEN
MP_TAC (ISPEC `K:real^3->bool`
AFFINE_INDEPENDENT_CARD_LE) THEN
ASM_REWRITE_TAC[DIMINDEX_3; ARITH_RULE `a <= 3 + 1 <=> a < 5`; Hypermap_and_fan.gen_NUM_CASES 5] THEN
DISCH_THEN STRIP_ASSUME_TAC THENL
[
POP_ASSUM MP_TAC THEN
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE;
CARD_EQ_0;
NOT_IN_EMPTY];
MATCH_MP_TAC
XYOFCGX_1 THEN ASM_REWRITE_TAC[
LE_REFL];
MATCH_MP_TAC
XYOFCGX_2 THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC
XYOFCGX_3_0 THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC
XYOFCGX_4_0 THEN ASM_REWRITE_TAC[]
]);;
let AFFINE_HULL_PROJECTION_EXISTS = prove(`!S p:real^N. ~(S = {}) ==> ?x n. p = x + n /\ x
IN affine hull S /\
(!v w. v
IN S /\ w
IN S ==> (v - w)
dot n = &0)`,
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
REPEAT STRIP_TAC THEN
ABBREV_TAC `V =
IMAGE (\v:real^N. --x + v) S` THEN
ABBREV_TAC `p0:real^N = --x + p` THEN
SUBGOAL_THEN `?y n:real^N. p0 = y + n /\ y
IN affine hull V /\ (!z. z
IN V ==> z
dot n = &0)` MP_TAC THENL
[
MP_TAC (ISPECL [`V:real^N->bool`; `p0:real^N`]
ORTHOGONAL_SUBSPACE_DECOMP_EXISTS) THEN
REWRITE_TAC[
orthogonal] THEN
STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`y:real^N`; `z:real^N`] THEN
ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[
MP_TAC (ISPEC `V:real^N->bool`
AFFINE_HULL_EQ_SPAN) THEN
ANTS_TAC THENL
[
EXPAND_TAC "V" THEN
REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
EXISTS_TAC `x:real^N` THEN
REWRITE_TAC[VECTOR_ARITH `
vec 0 = --x + x:real^N`] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^N->bool` THEN
ASM_REWRITE_TAC[
HULL_SUBSET];
ALL_TAC
] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `z':real^N`) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `V:real^N->bool` THEN
ASM_REWRITE_TAC[
SPAN_INC];
ALL_TAC
] THEN
SIMP_TAC[
DOT_SYM];
ALL_TAC
] THEN
STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`x + y:real^N`; `n:real^N`] THEN
CONJ_TAC THENL
[
UNDISCH_TAC `--x + p = p0:real^N` THEN
ASM_REWRITE_TAC[] THEN
VECTOR_ARITH_TAC;
ALL_TAC
] THEN
CONJ_TAC THENL
[
UNDISCH_TAC `y:real^N
IN affine hull V` THEN
EXPAND_TAC "V" THEN
REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[VECTOR_ARITH `x + --x + x' = x':real^N`];
ALL_TAC
] THEN
SUBGOAL_THEN `!v:real^N. v
IN S ==> (--x + v)
dot n = &0` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXPAND_TAC "V" THEN REWRITE_TAC[
IN_IMAGE] THEN
EXISTS_TAC `v:real^N` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `v - w = (--x + v) - (--x + w):real^N`] THEN
ONCE_REWRITE_TAC[
DOT_LSUB] THEN
ASM_SIMP_TAC[
REAL_SUB_RZERO]);;
let AFFINE_HULL_PROJECTION_DIST_EQ = prove(`!S p v w x n:real^N. v
IN S /\ w
IN S /\
dist (p, v) =
dist (p, w) /\
p = x + n /\ (!v w. v
IN S /\ w
IN S ==> (v - w)
dot n = &0)
==>
dist (x, v) =
dist (x, w)`,
REPEAT STRIP_TAC THEN
UNDISCH_TAC `p = x + n:real^N` THEN
REWRITE_TAC[VECTOR_ARITH `p = x + n <=> x = p - n:real^N`] THEN
DISCH_TAC THEN
UNDISCH_TAC `
dist (p,v) =
dist(p,w:real^N)` THEN
ASM_REWRITE_TAC[
DIST_EQ;
dist;
NORM_POW_2] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `p - n - v = (p - v) - n:real^N`] THEN
ABBREV_TAC `a = p - v:real^N` THEN
ABBREV_TAC `b = p - w:real^N` THEN
DISCH_TAC THEN
ASM_REWRITE_TAC[
DOT_LSUB;
DOT_RSUB;
DOT_SYM] THEN
REWRITE_TAC[REAL_ARITH `bb - an - (an - nn) = bb - bn - (bn - nn) <=> an - bn = &0`] THEN
ONCE_REWRITE_TAC[GSYM
DOT_LSUB] THEN
EXPAND_TAC "a" THEN EXPAND_TAC "b" THEN
REWRITE_TAC[VECTOR_ARITH `p - v - (p - w) = w - v:real^N`] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[]);;
let ORTHOGONAL_TO_AFFINE_HULL_EQ = prove(`!S n:real^N. (!v w. v
IN S /\ w
IN S ==> (v - w)
dot n = &0) <=>
(!v w. v
IN affine hull S /\ w
IN affine hull S ==> (v - w)
dot n = &0)`,
REPEAT GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
[
ASM_CASES_TAC `S = {}:real^N->bool` THENL
[
UNDISCH_TAC `w:real^N
IN affine hull S` THEN
ASM_REWRITE_TAC[
AFFINE_HULL_EMPTY;
NOT_IN_EMPTY];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
STRIP_TAC THEN
MP_TAC (ISPEC `(
IMAGE (\v:real^N. --x + v) S)`
ORTHOGONAL_TO_SPAN_EQ) THEN
DISCH_THEN (MP_TAC o SPEC `n:real^N`) THEN
REWRITE_TAC[
orthogonal] THEN
SUBGOAL_THEN `span (
IMAGE (\v:real^N. --x + v) S) =
affine hull (
IMAGE (\v:real^N. --x + v) S)` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC (GSYM
AFFINE_HULL_EQ_SPAN) THEN
REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
EXISTS_TAC `x:real^N` THEN
REWRITE_TAC[VECTOR_ARITH `
vec 0 = --x + x:real^N`] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^N->bool` THEN
ASM_REWRITE_TAC[
HULL_SUBSET];
ALL_TAC
] THEN
ONCE_REWRITE_TAC[TAUT `(A <=> B) <=> ((B ==> A) /\ (A ==> B))`] THEN
DISCH_THEN (CONJUNCTS_THEN2 MP_TAC (fun
th -> ALL_TAC)) THEN
REWRITE_TAC[
AFFINE_HULL_TRANSLATION;
IN_IMAGE] THEN
ANTS_TAC THENL
[
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[
DOT_SYM] THEN
ASM_REWRITE_TAC[VECTOR_ARITH `--x + x' = x' - x:real^N`] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_TAC THEN
SUBGOAL_THEN `!v:real^N. v
IN affine hull S ==> (v - x)
dot n = &0` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
DOT_SYM] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `v':real^N` THEN
ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
ALL_TAC
] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `v - w = (v - x) - (w - x):real^N`] THEN
ONCE_REWRITE_TAC[
DOT_LSUB] THEN
ASM_SIMP_TAC[
REAL_SUB_RZERO];
ALL_TAC
] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
CONJ_TAC THEN MATCH_MP_TAC
IN_TRANS THEN EXISTS_TAC `S:real^N->bool` THEN ASM_REWRITE_TAC[
HULL_SUBSET]);;
let AFFINE_HULL_CIRCUMCENTER_PROJECTION = prove(`!s t:real^N->bool. ~affine_dependent s /\ t
SUBSET s /\ ~(t = {})
==> ?n. circumcenter s = circumcenter t + n /\
(!v w. v
IN t /\ w
IN t ==> (v - w)
dot n = &0)`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `p0:real^N = circumcenter t` THEN
ABBREV_TAC `p:real^N = circumcenter s` THEN
MP_TAC (SPECL [`t:real^N->bool`; `p:real^N`]
AFFINE_HULL_PROJECTION_EXISTS) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
EXISTS_TAC `n:real^N` THEN
ASM_REWRITE_TAC[VECTOR_ARITH `x + n = p0 + n <=> x = p0:real^N`] THEN
EXPAND_TAC "p0" THEN
MP_TAC (SPEC `t:real^N->bool`
OAPVION3) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `s:real^N->bool` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(t = {}:real^N->bool)` THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
STRIP_TAC THEN
EXISTS_TAC `
dist (x, x':real^N)` THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
AFFINE_HULL_PROJECTION_DIST_EQ THEN
MAP_EVERY EXISTS_TAC [`t:real^N->bool`; `p:real^N`; `n:real^N`] THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SPEC `s:real^N->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o SPEC `x':real^N`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `w:real^N`) THEN
UNDISCH_TAC `t
SUBSET s:real^N->bool` THEN
REWRITE_TAC[
SUBSET] THEN DISCH_TAC THEN
ASM_SIMP_TAC[]);;
let AFFINE_HULL_CIRCUMCENTER_EQ = prove(`!s t:real^N->bool. ~affine_dependent s /\ t
SUBSET s /\ ~(t = {}) /\ circumcenter s
IN affine hull t
==> circumcenter s = circumcenter t`,
REWRITE_TAC[
SUBSET] THEN REPEAT STRIP_TAC THEN
MP_TAC (SPEC `t:real^N->bool`
OAPVION3) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `s:real^N->bool` THEN
ASM_REWRITE_TAC[
SUBSET];
ALL_TAC
] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `radV (s:real^N->bool)` THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPEC `s:real^N->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `w:real^N`) THEN
ASM_SIMP_TAC[]);;
let AFFINE_HULL_RADV = prove(`!s t:real^N->bool. ~affine_dependent s /\ t
SUBSET s /\ ~(t = {})
==> radV s pow 2 = radV t pow 2 +
dist (circumcenter t, circumcenter s) pow 2 /\
&0 <= radV s /\ &0 <= radV t`,
REWRITE_TAC[
SUBSET] THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN
FIRST_ASSUM MP_TAC THEN REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN STRIP_TAC THEN
SUBGOAL_THEN `~affine_dependent (t:real^N->bool)` ASSUME_TAC THENL
[
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `s:real^N->bool` THEN
ASM_REWRITE_TAC[
SUBSET];
ALL_TAC
] THEN
MP_TAC (SPEC_ALL
AFFINE_HULL_CIRCUMCENTER_PROJECTION) THEN
ASM_REWRITE_TAC[
SUBSET] THEN
STRIP_TAC THEN
POP_ASSUM (LABEL_TAC "vw") THEN
MP_TAC (SPEC `s:real^N->bool`
OAPVION2) THEN
MP_TAC (SPEC `t:real^N->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
DISCH_THEN (MP_TAC o SPEC `x:real^N`) THEN
ASM_SIMP_TAC[] THEN DISCH_TAC THEN
REWRITE_TAC[
DIST_POS_LE] THEN
REWRITE_TAC[
dist;
NORM_POW_2; VECTOR_ARITH `a - (a + n) = --n:real^N`; VECTOR_ARITH `(a + n) - x = (a - x) + n:real^N`] THEN
REWRITE_TAC[
DOT_LADD;
DOT_RADD;
DOT_LNEG;
DOT_RNEG;
DOT_SYM;
REAL_NEG_NEG] THEN
REWRITE_TAC[REAL_ARITH `(a + b) + b + c = a + c <=> b = &0`] THEN
ONCE_REWRITE_TAC[
DOT_SYM] THEN
REMOVE_THEN "vw" MP_TAC THEN ONCE_REWRITE_TAC[
ORTHOGONAL_TO_AFFINE_HULL_EQ] THEN
DISCH_THEN MATCH_MP_TAC THEN
CONJ_TAC THENL
[
MP_TAC (SPEC `t:real^N->bool`
OAPVION1) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
IN_TRANS THEN EXISTS_TAC `t:real^N->bool` THEN
ASM_REWRITE_TAC[
HULL_SUBSET]);;
let RADV_MONO = prove(`!s t:real^N->bool. ~affine_dependent s /\ t
SUBSET s /\ ~(t = {})
==> radV t <= radV s`,
REPEAT GEN_TAC THEN DISCH_THEN (MP_TAC o MATCH_MP
AFFINE_HULL_RADV) THEN
STRIP_TAC THEN
SUBGOAL_THEN `radV (t:real^N->bool) = abs(radV t) /\ radV (s:real^N->bool) = abs(radV s)` (fun
th -> ONCE_REWRITE_TAC[
th]) THENL
[
ONCE_REWRITE_TAC[
EQ_SYM_EQ] THEN
ASM_REWRITE_TAC[
REAL_ABS_REFL];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
REAL_LE_SQUARE_ABS] THEN
REWRITE_TAC[REAL_ARITH `a <= a + b <=> &0 <= b`] THEN
REWRITE_TAC[REAL_POW_2;
REAL_LE_SQUARE]);;
let BARV_CIRCUMCENTER_PROJECTION = prove(`!V ul k i. packing V /\ ul
IN barV V k /\ i <= k
==> let S =
set_of_list (truncate_simplex i ul) in
(?n. circumcenter (
set_of_list ul) = circumcenter S + n /\
(!v w. v
IN S /\ w
IN S ==> (v - w)
dot n = &0))`,
REPEAT STRIP_TAC THEN
UNDISCH_TAC `ul
IN barV V k` THEN GEN_REWRITE_TAC LAND_CONV [
IN] THEN DISCH_TAC THEN
CONV_TAC let_CONV THEN
ABBREV_TAC `S:real^3->bool =
set_of_list (truncate_simplex i ul)` THEN
MATCH_MP_TAC
AFFINE_HULL_CIRCUMCENTER_PROJECTION THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
CONJ_TAC THENL
[
EXPAND_TAC "S" THEN
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`];
ALL_TAC
] THEN
EXPAND_TAC "S" THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `
HD (truncate_simplex i ul):real^3` THEN
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
MP_TAC (ISPECL [`i:num`; `ul:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`; ARITH_RULE `1 <= i + 1`]);;
let HL_DECREASE = prove(`!V ul k i. packing V /\ ul
IN barV V k /\ i <= k
==> hl (truncate_simplex i ul) <= hl ul`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
MP_TAC (SPEC_ALL
HL_PROPERTIES) THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SPEC_ALL
HL_EQ_DIST0) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `i:num`; `truncate_simplex i (ul:(real^3)list)`]
HL_EQ_DIST0) THEN
SUBGOAL_THEN `barV V i (truncate_simplex i ul)` ASSUME_TAC THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ABBREV_TAC `S:real^3->bool =
set_of_list (truncate_simplex i ul)` THEN
ABBREV_TAC `p0:real^3 = circumcenter S` THEN
ABBREV_TAC `p:real^3 = circumcenter (
set_of_list ul)` THEN
SUBGOAL_THEN `
HD (truncate_simplex i ul):real^3 =
HD ul` ASSUME_TAC THENL
[
MATCH_MP_TAC
HD_TRUNCATE_SIMPLEX THEN
MATCH_MP_TAC
LE_TRANS THEN
EXISTS_TAC `k + 1` THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`] THEN
UNDISCH_TAC `barV V k ul` THEN
SIMP_TAC[
BARV;
LE_REFL];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
AFFINE_HULL_PROJECTION_DIST_LE THEN
EXISTS_TAC `S:real^3->bool` THEN
MP_TAC (SPEC_ALL
BARV_CIRCUMCENTER_PROJECTION) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
CONV_TAC (DEPTH_CONV let_CONV) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
EXISTS_TAC `n:real^3` THEN
MP_TAC (SPECL [`V:real^3->bool`; `i:num`; `truncate_simplex i ul:(real^3)list`]
BARV_IMP_HD_IN_SET_OF_LIST) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "p0" THEN
EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_CIRCUMCENTER_EXISTS THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `i:num`] THEN
ASM_REWRITE_TAC[]);;
let XNHPWAB1 = prove(`!V ul k. packing V /\ (ul
IN barV V k) /\ (hl ul <
sqrt(&2)) ==>
(omega_list V ul = circumcenter (
set_of_list ul))`,
REWRITE_TAC[
IN] THEN
REPEAT GEN_TAC THEN SPEC_TAC (`ul:(real^3)list`, `ul:(real^3)list`) THEN
SPEC_TAC (`k:num`, `k:num`) THEN
INDUCT_TAC THENL
[
GEN_TAC THEN
REWRITE_TAC[
BARV; ARITH] THEN
REPEAT STRIP_TAC THEN
MP_TAC (ISPEC `ul:(real^3)list`
LENGTH_1_LEMMA) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> ONCE_REWRITE_TAC[
th]) THEN
REWRITE_TAC[
set_of_list;
CIRCUMCENTER_1;
OMEGA_LIST;
LENGTH; ARITH;
OMEGA_LIST_N;
HD];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `SUC k <= 3` ASSUME_TAC THENL
[
MATCH_MP_TAC
BARV_IMP_K_LE_3 THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `ul:(real^3)list`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ABBREV_TAC `vl:(real^3)list = truncate_simplex k ul` THEN
FIRST_X_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC k <= 3 ==> k <= 3`] THEN
SUBGOAL_THEN `barV V k vl` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = k + 2 /\
LENGTH (vl:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V (SUC k) ul` THEN UNDISCH_TAC `barV V k vl` THEN
SIMP_TAC[
BARV] THEN
ARITH_TAC;
ALL_TAC
] THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (ul:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `SUC k`] THEN
ASM_REWRITE_TAC[
IN] THEN
ARITH_TAC;
ALL_TAC
] THEN
ABBREV_TAC `p0 = circumcenter (
set_of_list vl):real^3` THEN
POP_ASSUM (LABEL_TAC "p0") THEN
DISCH_THEN (LABEL_TAC "po") THEN
SUBGOAL_THEN `p0:real^3
IN affine hull (
set_of_list ul)` ASSUME_TAC THENL
[
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
affine hull (
set_of_list vl):real^3->bool` THEN
CONJ_TAC THENL
[
REMOVE_THEN "p0" (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC
BARV_CIRCUMCENTER_EXISTS THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
HULL_MONO THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
ASM_REWRITE_TAC[] THEN
ARITH_TAC;
ALL_TAC
] THEN
ABBREV_TAC `A:real^3->bool =
affine hull (voronoi_list V ul)` THEN
ABBREV_TAC `p:real^3 =
closest_point A p0` THEN
MP_TAC (ISPECL [`A:real^3->bool`; `p0:real^3`]
CLOSEST_POINT_EXISTS) THEN
ANTS_TAC THENL
[
EXPAND_TAC "A" THEN
REWRITE_TAC[
CLOSED_AFFINE_HULL] THEN
REWRITE_TAC[
AFFINE_HULL_EQ_EMPTY] THEN
DISCH_TAC THEN
UNDISCH_TAC `barV V (SUC k) ul` THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 2`; DE_MORGAN_THM] THEN
DISJ2_TAC THEN DISJ2_TAC THEN
REWRITE_TAC[
AFF_DIM_EMPTY] THEN
UNDISCH_TAC `SUC k <= 3` THEN
REWRITE_TAC[
ADD1; GSYM
INT_OF_NUM_ADD; GSYM
INT_OF_NUM_LE] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
SUBGOAL_THEN `p:real^3
IN affine hull set_of_list ul` ASSUME_TAC THENL
[
ASM_CASES_TAC `p:real^3
IN affine hull set_of_list ul` THEN ASM_REWRITE_TAC[] THEN
MP_TAC (ISPECL [`
set_of_list ul:real^3->bool`; `p:real^3`]
AFFINE_HULL_PROJECTION_EXISTS) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[
SET_OF_LIST_EQ_EMPTY; GSYM
LENGTH_EQ_NIL] THEN
ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THEN
SUBGOAL_THEN `
dist (x, p0) <
dist (p, p0:real^3)` ASSUME_TAC THENL
[
MATCH_MP_TAC
AFFINE_HULL_PROJECTION_DIST_LT THEN
MAP_EVERY EXISTS_TAC [`
affine hull (
set_of_list ul:real^3->bool)`; `n:real^3`] THEN
ASM_REWRITE_TAC[
HULL_HULL; GSYM
ORTHOGONAL_TO_AFFINE_HULL_EQ] THEN
UNDISCH_TAC `p = x + n:real^3` THEN DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
SUBGOAL_THEN `x:real^3
IN A` ASSUME_TAC THENL
[
UNDISCH_TAC `p:real^3
IN A` THEN
EXPAND_TAC "A" THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`]
MHFTTZN2) THEN
ANTS_TAC THENL [ ASM_REWRITE_TAC[];ALL_TAC ] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[bis;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
AFFINE_HULL_PROJECTION_DIST_EQ THEN
MAP_EVERY EXISTS_TAC [`
set_of_list ul:real^3->bool`; `p:real^3`; `n:real^3`] THEN
ASM_SIMP_TAC[] THEN
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
ASM_REWRITE_TAC[ARITH_RULE `1 <= k + 2`];
ALL_TAC
] THEN
UNDISCH_TAC `
dist (x, p0) <
dist (p,p0:real^3)` THEN
POP_ASSUM MP_TAC THEN REMOVE_ASSUM THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `x:real^3`) THEN
ASM_REWRITE_TAC[
DIST_SYM;
REAL_NOT_LT];
ALL_TAC
] THEN
SUBGOAL_THEN `p = circumcenter (
set_of_list ul):real^3` (LABEL_TAC "c") THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`]
MHFTTZN3) THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
SUBGOAL_THEN `p:real^3
IN A
INTER affine hull set_of_list ul` MP_TAC THENL
[
ASM_REWRITE_TAC[
IN_INTER];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
IN_SING];
ALL_TAC
] THEN
SUBGOAL_THEN `p:real^3
IN voronoi_list V ul` ASSUME_TAC THENL
[
REWRITE_TAC[
VORONOI_LIST;
VORONOI_SET;
IN_INTERS;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
voronoi_closed;
IN_ELIM_THM] THEN
SUBGOAL_THEN `!w:real^3. V (w:real^3) <=> w
IN V` (fun
th -> REWRITE_TAC[
th]) THENL [ REWRITE_TAC[
IN]; ALL_TAC ] THEN
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `w:real^3
IN set_of_list ul` THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`]
HL_PROPERTIES) THEN
ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[
REAL_LE_REFL];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `
set_of_list ul:real^3->bool`; `p:real^3`]
XYOFCGX) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[GSYM
HL] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
EXISTS_TAC `V:real^3->bool` THEN EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o SPECL [`v:real^3`; `w:real^3`]) THEN
ASM_REWRITE_TAC[
IN_DIFF;
DIST_SYM;
real_gt;
REAL_LT_LE] THEN
SIMP_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
OMEGA_LIST;
OMEGA_LIST_N; ARITH_RULE `(k + 2) - 1 = SUC k`] THEN
SUBGOAL_THEN `omega_list_n V ul k = p0:real^3` (fun
th -> REWRITE_TAC[
th]) THENL
[
EXPAND_TAC "p0" THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC (GSYM
OMEGA_LIST_LEMMA) THEN
ASM_REWRITE_TAC[] THEN
ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `truncate_simplex (SUC k) ul = ul:(real^3)list` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPECL [`SUC k`; `ul:(real^3)list`; `ul:(real^3)list`]
TRUNCATE_SIMPLEX_INITIAL_SUBLIST) THEN
ASM_REWRITE_TAC[ARITH_RULE `SUC k + 1 = k + 2`;
LE_REFL;
INITIAL_SUBLIST_REFL];
ALL_TAC
] THEN
REMOVE_THEN "c" (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC (GSYM
CLOSEST_POINT_UNIQUE) THEN
ASM_REWRITE_TAC[
CONVEX_VORONOI_LIST;
CLOSED_VORONOI_LIST] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXPAND_TAC "A" THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `voronoi_list V ul:real^3->bool` THEN
ASM_REWRITE_TAC[
HULL_SUBSET]);;
let XNHPWAB2 = prove(`!V ul k. packing V /\ (ul
IN barV V k) /\ (hl ul <
sqrt(&2)) ==>
(omega_list V ul
IN convex hull (
set_of_list ul))`,
REPEAT STRIP_TAC THEN
UNDISCH_TAC `ul
IN barV V k` THEN DISCH_THEN (ASSUME_TAC o REWRITE_RULE[
IN]) THEN
ABBREV_TAC `p:real^3 = omega_list V ul` THEN
ABBREV_TAC `S:real^3->bool =
set_of_list ul` THEN
ASM_CASES_TAC `p:real^3
IN convex hull S` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `circumcenter S = p:real^3` (LABEL_TAC "c") THENL
[
EXPAND_TAC "S" THEN EXPAND_TAC "p" THEN
MATCH_MP_TAC (GSYM
XNHPWAB1) THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
SUBGOAL_THEN `~affine_dependent (S:real^3->bool)` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (ISPECL [`S:real^3->bool`; `p:real^3`]
AFFINE_HULL_PROJECTION_SEPARATES) THEN
ANTS_TAC THENL
[
ASM_SIMP_TAC[
AFFINE_INDEPENDENT_IMP_FINITE] THEN
EXPAND_TAC "p" THEN EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_CIRCUMCENTER_EXISTS THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
ASM_CASES_TAC `S
DELETE u:real^3 = {}` THENL
[
SUBGOAL_THEN `S = {u:real^3}` ASSUME_TAC THENL
[
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
EXTENSION;
IN_DELETE;
NOT_IN_EMPTY;
IN_SING; DE_MORGAN_THM] THEN
REPEAT STRIP_TAC THEN
EQ_TAC THEN DISCH_TAC THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `x:real^3`) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REMOVE_THEN "c" MP_TAC THEN
ASM_REWRITE_TAC[
CIRCUMCENTER_1] THEN DISCH_TAC THEN
UNDISCH_TAC `~(p:real^3
IN convex hull S)` THEN
ASM_REWRITE_TAC[
CONVEX_HULL_SING;
IN_SING];
ALL_TAC
] THEN
MP_TAC (ISPECL [`S:real^3->bool`; `S
DELETE u:real^3`]
AFFINE_HULL_CIRCUMCENTER_PROJECTION) THEN
ASM_REWRITE_TAC[
DELETE_SUBSET] THEN
ABBREV_TAC `p0:real^3 = circumcenter (S
DELETE u)` THEN
STRIP_TAC THEN
POP_ASSUM (LABEL_TAC "vw") THEN
SUBGOAL_THEN `p0:real^3
IN affine hull (S
DELETE u)` ASSUME_TAC THENL
[
EXPAND_TAC "p0" THEN
MP_TAC (ISPEC `S
DELETE u:real^3`
OAPVION1) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[
DELETE_SUBSET];
ALL_TAC
] THEN
SIMP_TAC[];
ALL_TAC
] THEN
UNDISCH_TAC `~(S
DELETE u:real^3 = {})` THEN
PURE_REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
DISCH_THEN (X_CHOOSE_THEN `v:real^3` ASSUME_TAC) THEN
SUBGOAL_THEN `
dist (p, v) pow 2 =
dist (p0, v) pow 2 +
dist (p0, p:real^3) pow 2` ASSUME_TAC THENL
[
ASM_REWRITE_TAC[
dist;
NORM_POW_2; VECTOR_ARITH `p0 - (p0 + n) = --n:real^3`; VECTOR_ARITH `(p0 + n) - v = (p0 - v) + n:real^3`] THEN
REWRITE_TAC[
DOT_LADD;
DOT_RADD;
DOT_LNEG;
DOT_RNEG;
REAL_NEG_NEG;
DOT_SYM] THEN
REWRITE_TAC[REAL_ARITH `(p0v +nv) + nv + nn = p0v + nn <=> nv = &0`] THEN
REMOVE_THEN "vw" MP_TAC THEN
ONCE_REWRITE_TAC[
ORTHOGONAL_TO_AFFINE_HULL_EQ] THEN
REWRITE_TAC[
DOT_SYM] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S
DELETE u:real^3` THEN
ASM_REWRITE_TAC[
HULL_SUBSET];
ALL_TAC
] THEN
SUBGOAL_THEN `
dist (p,u:real^3) pow 2 =
dist (p0,u) pow 2 +
dist (p0,p) pow 2 + &2 * (p - p0)
dot (p0 - u)` ASSUME_TAC THENL
[
ASM_REWRITE_TAC[
dist;
NORM_POW_2; VECTOR_ARITH `(p0 + n) - u = (p0 - u) + n:real^3`; VECTOR_ARITH `p0 - (p0 + n) = --n:real^3`] THEN
REWRITE_TAC[
DOT_LADD;
DOT_RADD;
DOT_LNEG;
DOT_RNEG;
REAL_NEG_NEG;
DOT_SYM] THEN
REWRITE_TAC[REAL_ARITH `(a + b) + b + c = a + c + &2 * d <=> b = d`] THEN
REWRITE_TAC[
VECTOR_SUB_REFL;
DOT_LZERO; REAL_ADD_LID];
ALL_TAC
] THEN
SUBGOAL_THEN `
dist (p0, u) <=
dist (p0, v:real^3)` MP_TAC THENL
[
REWRITE_TAC[
dist] THEN ONCE_REWRITE_TAC[GSYM
REAL_ABS_NORM] THEN
REWRITE_TAC[
REAL_LE_SQUARE_ABS; GSYM
dist] THEN
SUBGOAL_THEN `
dist (p0,v:real^3) pow 2 =
dist (p,v) pow 2 -
dist (p0,p) pow 2` (fun
th -> REWRITE_TAC[
th]) THENL
[
REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
dist (p0,u:real^3) pow 2 =
dist (p,u) pow 2 -
dist (p0,p) pow 2 - &2 * ((p - p0)
dot (p0 - u))` (fun
th -> REWRITE_TAC[
th]) THENL
[
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
dist (p, u) =
dist (p, v:real^3)` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o SPEC `u:real^3`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `v:real^3`) THEN
UNDISCH_TAC `v
IN S
DELETE u:real^3` THEN
ASM_SIMP_TAC[
IN_DELETE];
ALL_TAC
] THEN
REWRITE_TAC[REAL_ARITH `a - b - &2 * c <= a - b <=> --c <= &0`] THEN
REWRITE_TAC[GSYM
DOT_RNEG;
VECTOR_NEG_SUB] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `n:real^3` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
REAL_NOT_LE] THEN
MP_TAC (SPECL [`V:real^3->bool`; `S
DELETE u:real^3`; `p0:real^3`]
XYOFCGX) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `S:real^3->bool` THEN
REWRITE_TAC[
DELETE_SUBSET] THEN
EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[
DELETE_SUBSET];
ALL_TAC
] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `radV (S:real^3->bool)` THEN
EXPAND_TAC "S" THEN
ASM_REWRITE_TAC[GSYM
HL] THEN
ASM_REWRITE_TAC[
HL] THEN
MATCH_MP_TAC
RADV_MONO THEN
ASM_REWRITE_TAC[
DELETE_SUBSET; GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `v:real^3` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o SPECL [`v:real^3`; `u:real^3`]) THEN
ASM_REWRITE_TAC[
DIST_SYM;
real_gt] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[
IN_DIFF;
IN_DELETE] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[]);;
let XNHPWAB4 = prove(`!V ul k. packing V /\ (ul
IN barV V k) /\ (hl ul <
sqrt(&2)) ==>
(!i j. (i < j) /\ (j <= k) ==> hl(truncate_simplex i ul) < hl(truncate_simplex j ul))`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
ABBREV_TAC `xl:(real^3)list = truncate_simplex j ul` THEN
ABBREV_TAC `yl:(real^3)list = truncate_simplex i ul` THEN
SUBGOAL_THEN `barV V i yl /\ barV V j xl` ASSUME_TAC THENL
[
EXPAND_TAC "xl" THEN EXPAND_TAC "yl" THEN
CONJ_TAC THEN MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
LE_TRANS THEN
EXISTS_TAC `j:num` THEN
ASM_SIMP_TAC[
LT_IMP_LE];
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (xl:(real^3)list) = j + 1 /\
LENGTH (yl:(real^3)list) = i + 1 /\ i + 1 <= j + 1` ASSUME_TAC THENL
[
POP_ASSUM MP_TAC THEN
SIMP_TAC[
BARV; ARITH_RULE `i + 1 <= j + 1 <=> i <= j`] THEN
ASM_SIMP_TAC[
LT_IMP_LE];
ALL_TAC
] THEN
SUBGOAL_THEN `hl (xl:(real^3)list) <
sqrt (&2) /\ hl (yl:(real^3)list) <
sqrt (&2)` ASSUME_TAC THENL
[
CONJ_TAC THEN MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `hl (ul:(real^3)list)` THEN ASM_REWRITE_TAC[] THENL
[
EXPAND_TAC "xl";
let OMEGA_LIST_N_IN_CONVEX_HULL = prove(`!V ul k i. packing V /\ barV V k ul /\ i <= k /\ hl ul <
sqrt (&2)
==> omega_list_n V ul i
IN convex hull set_of_list ul`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `vl:(real^3)list = truncate_simplex i ul` THEN
SUBGOAL_THEN `barV V i vl` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = i + 1 /\
LENGTH (ul:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k ul` THEN
POP_ASSUM MP_TAC THEN
SIMP_TAC[
BARV];
ALL_TAC
] THEN
SUBGOAL_THEN `omega_list_n V ul i = omega_list V vl` (fun
th -> REWRITE_TAC[
th]) THENL
[
ASM_REWRITE_TAC[
OMEGA_LIST; ARITH_RULE `(i + 1) - 1 = i`] THEN
EXPAND_TAC "vl" THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `i:num`; `0`]
OMEGA_LIST_N_LEMMA) THEN
REWRITE_TAC[ARITH_RULE `i + 0 = i`] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 0 + 1 <= k + 1 <=> i <= k`];
ALL_TAC
] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
convex hull set_of_list vl:real^3->bool` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
XNHPWAB2 THEN
EXISTS_TAC `i:num` THEN
ASM_REWRITE_TAC[
IN] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (ul:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
MATCH_MP_TAC
HULL_MONO THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`]);;
let XNHPWAB3 = prove(`!V ul k. packing V /\ (ul
IN barV V k) /\ (hl ul <
sqrt(&2)) ==>
(
aff_dim { omega_list_n V ul j | j
IN (0..k)} = &k)`,
REWRITE_TAC[
IN_NUMSEG;
IN] THEN
REPEAT GEN_TAC THEN
SPEC_TAC (`ul:(real^3)list`, `ul:(real^3)list`) THEN
SPEC_TAC (`k:num`, `k:num`) THEN
INDUCT_TAC THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
IN_NUMSEG;
LE_ANTISYM;
EQ_SYM_EQ] THEN
SUBGOAL_THEN `{omega_list_n V ul j | j = 0} = {omega_list_n V ul 0}` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
IN_SING] THEN
GEN_TAC THEN EQ_TAC THENL
[
STRIP_TAC THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_TAC THEN
EXISTS_TAC `0` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
AFF_DIM_SING];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
ABBREV_TAC `vl:(real^3)list = truncate_simplex k ul` THEN
FIRST_X_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
SUBGOAL_THEN `barV V k vl` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[ARITH_RULE `k <= SUC k`];
ALL_TAC
] THEN
SUBGOAL_THEN `hl (vl:(real^3)list) <
sqrt (&2)` ASSUME_TAC THENL
[
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (ul:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `SUC k`] THEN
ASM_REWRITE_TAC[ARITH_RULE `k <= SUC k`;
IN];
ALL_TAC
] THEN
ASM_SIMP_TAC[ARITH_RULE `SUC k <= 3 ==> k <= 3`] THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = k + 2 /\
LENGTH (vl:(real^3)list) = k + 1` ASSUME_TAC THENL
[
REPEAT (FIRST_X_ASSUM (MP_TAC o check (free_in `barV` o concl))) THEN
SIMP_TAC[
BARV;
ADD1; ARITH_RULE `(k + 1) + 1 = k + 2`];
ALL_TAC
] THEN
SUBGOAL_THEN `{omega_list_n V ul j | 0 <= j /\ j <= SUC k} = omega_list_n V ul (SUC k)
INSERT {omega_list_n V vl j | 0 <= j /\ j <= k}` MP_TAC THENL
[
SUBGOAL_THEN `!j. 0 <= j /\ j <= k ==> omega_list_n V vl j = omega_list_n V ul j` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
EXPAND_TAC "vl" THEN
SUBGOAL_THEN `k = j + (k - j:num)` MP_TAC THENL
[
POP_ASSUM MP_TAC THEN ARITH_TAC;
ALL_TAC
] THEN
DISCH_THEN (fun
th -> ONCE_REWRITE_TAC[
th]) THEN
MATCH_MP_TAC (GSYM
OMEGA_LIST_N_LEMMA) THEN
ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN
ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
IN_INSERT] THEN
GEN_TAC THEN EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `j = SUC k` THEN ASM_REWRITE_TAC[] THEN
DISJ2_TAC THEN
EXISTS_TAC `j:num` THEN
MP_TAC (ARITH_RULE `j <= SUC k /\ ~(j = SUC k) ==> j <= k`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REPEAT STRIP_TAC THENL
[
EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[
LE_0;
LE_REFL];
ALL_TAC
] THEN
EXISTS_TAC `j:num` THEN
ASM_SIMP_TAC[ARITH_RULE `j <= k ==> j <= SUC k`];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_TAC THEN
ASM_REWRITE_TAC[
AFF_DIM_INSERT] THEN
SUBGOAL_THEN `~(omega_list_n V ul (SUC k)
IN affine hull {omega_list_n V vl j | 0 <= j /\ j <= k})` (fun
th -> REWRITE_TAC[
th;
ADD1;
INT_OF_NUM_ADD]) THEN
SUBGOAL_THEN `omega_list_n V ul (SUC k) = circumcenter (
set_of_list ul)` (fun
th -> REWRITE_TAC[
th]) THENL
[
SUBGOAL_THEN `omega_list_n V ul (SUC k) = omega_list V ul` (fun
th -> REWRITE_TAC[
th]) THENL
[
ASM_REWRITE_TAC[
OMEGA_LIST; ARITH_RULE `(k + 2) - 1 = SUC k`];
ALL_TAC
] THEN
MATCH_MP_TAC
XNHPWAB1 THEN
EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
DISCH_TAC THEN
SUBGOAL_THEN `~affine_dependent ((
set_of_list ul):real^3->bool)` ASSUME_TAC THENL
[
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `SUC k`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
set_of_list vl
SUBSET ((
set_of_list ul):real^3->bool) /\ ~(
set_of_list vl = {})` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `
HD (vl:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
ASM_REWRITE_TAC[] THEN
ARITH_TAC
];
ALL_TAC
] THEN
SUBGOAL_THEN `circumcenter (
set_of_list ul) = circumcenter (
set_of_list vl):real^3` ASSUME_TAC THENL
[
MATCH_MP_TAC
AFFINE_HULL_CIRCUMCENTER_EQ THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
affine hull {omega_list_n V vl j | 0 <= j /\ j <= k}` THEN
ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM
HULL_HULL] THEN
MATCH_MP_TAC
HULL_MONO THEN
REWRITE_TAC[
SUBSET;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
convex hull set_of_list (vl:(real^3)list)` THEN
ASM_REWRITE_TAC[
CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
MATCH_MP_TAC
OMEGA_LIST_N_IN_CONVEX_HULL THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`]
XNHPWAB4) THEN
ASM_REWRITE_TAC[
IN] THEN
DISCH_THEN (MP_TAC o SPECL [`k:num`; `SUC k`]) THEN
ANTS_TAC THENL [ ARITH_TAC; ALL_TAC ] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC (REAL_ARITH `a = b ==> ~(a < b:real)`) THEN
SUBGOAL_THEN `truncate_simplex (SUC k) ul = ul:(real^3)list` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPECL [`SUC k`; `ul:(real^3)list`; `ul:(real^3)list`]
TRUNCATE_SIMPLEX_INITIAL_SUBLIST) THEN
ASM_REWRITE_TAC[ARITH_RULE `SUC k + 1 <= k + 2`; ARITH_RULE `k + 2 = SUC k + 1`;
INITIAL_SUBLIST_REFL] THEN
SIMP_TAC[];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`]
HL_PROPERTIES) THEN
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `k:num`]
HL_PROPERTIES) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `
HD vl:real^3`) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
DISCH_THEN (MP_TAC o SPEC `
HD vl:real^3`) THEN
ANTS_TAC THENL
[
SUBGOAL_THEN `
HD vl =
HD ul:real^3` (fun
th -> REWRITE_TAC[
th]) THENL
[
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HD_TRUNCATE_SIMPLEX THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
SIMP_TAC[]);;
let IN_VORONOI_LIST_IMP_IN_BIS = prove(`!V ul k x. barV V k ul /\ x
IN voronoi_list V ul ==>
(!u. u
IN set_of_list ul ==> x
IN bis (
HD ul) u)`,
REPEAT STRIP_TAC THEN
MP_TAC (ISPEC `ul:(real^3)list`
LENGTH_IMP_CONS) THEN
ANTS_TAC THENL
[
UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV; ARITH_RULE `1 <= k + 1`];
ALL_TAC
] THEN
STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `h:real^3`; `t:(real^3)list`]
VORONOI_LIST_BIS) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN
POP_ASSUM (fun
th -> ASM_REWRITE_TAC[SYM
th]);
ALL_TAC
] THEN
DISCH_TAC THEN
UNDISCH_TAC `x:real^3
IN voronoi_list V ul` THEN
ASM_REWRITE_TAC[
IN_INTER;
IN_INTERS;
IN_ELIM_THM;
HD] THEN
STRIP_TAC THEN
UNDISCH_TAC `u:real^3
IN set_of_list ul` THEN
ASM_REWRITE_TAC[
IN_SET_OF_LIST;
MEM] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
ASM_REWRITE_TAC[bis;
IN_ELIM_THM];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `bis h (u:real^3)`) THEN
DISCH_THEN MATCH_MP_TAC THEN
EXISTS_TAC `u:real^3` THEN
ASM_REWRITE_TAC[
IN_SET_OF_LIST]);;
let WAUFCHE1 = prove(`!V ul k. packing V /\ ul
IN barV V k ==> hl ul <=
dist(omega_list V ul,
HD ul)`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
ABBREV_TAC `p = omega_list V ul` THEN
MP_TAC (ISPECL [`
set_of_list ul:real^3->bool`; `p:real^3`]
AFFINE_HULL_PROJECTION_EXISTS) THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
SUBGOAL_THEN `
HD (ul:(real^3)list)
IN set_of_list ul` ASSUME_TAC THENL
[
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[
SET_OF_LIST_EQ_EMPTY; GSYM
LENGTH_EQ_NIL; ARITH_RULE `~(k + 1 = 0)`] THEN
STRIP_TAC THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `
dist (x:real^3,
HD ul)` THEN
CONJ_TAC THENL
[
MP_TAC (SPEC_ALL
HL_EQ_DIST0) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
REAL_LE_LT] THEN
DISJ2_TAC THEN
AP_TERM_TAC THEN REWRITE_TAC[
PAIR_EQ] THEN
MP_TAC (ISPEC `
set_of_list ul:real^3->bool`
OAPVION3) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (MATCH_MP_TAC o GSYM) THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `
dist (x:real^3,
HD ul)` THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
AFFINE_HULL_PROJECTION_DIST_EQ THEN
MAP_EVERY EXISTS_TAC [`
set_of_list ul:real^3->bool`; `p:real^3`; `n:real^3`] THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `k:num`; `p:real^3`]
IN_VORONOI_LIST_IMP_IN_BIS) THEN
ANTS_TAC THENL
[
EXPAND_TAC "p" THEN
CONJ_TAC THENL [ ASM_REWRITE_TAC[]; ALL_TAC ] THEN
MATCH_MP_TAC
OMEGA_LIST_IN_VORONOI_LIST THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o SPEC `w:real^3`) THEN
ASM_REWRITE_TAC[bis;
IN_ELIM_THM];
ALL_TAC
] THEN
MATCH_MP_TAC
AFFINE_HULL_PROJECTION_DIST_LE THEN
MAP_EVERY EXISTS_TAC [`
set_of_list ul:real^3->bool`; `n:real^3`] THEN
ASM_REWRITE_TAC[]);;
let WAUFCHE2 = prove(`!V ul k. packing V /\ ul
IN barV V k /\ hl ul <
sqrt(&2) ==>
(hl ul =
dist(omega_list V ul,
HD ul))`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`]
HL_EQ_DIST0) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
AP_TERM_TAC THEN
REWRITE_TAC[
PAIR_EQ] THEN
MATCH_MP_TAC (GSYM
XNHPWAB1) THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[
IN] THEN
MATCH_MP_TAC
BARV_IMP_K_LE_3 THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `ul:(real^3)list`] THEN
ASM_REWRITE_TAC[]);;
let CIRCUMCENTER_IN_VORONOI_SET = prove(`!V (S:real^3->bool). packing V /\ S
SUBSET V /\ ~affine_dependent S /\ radV S <
sqrt (&2)
==> circumcenter S
IN voronoi_set V S`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `p:real^3 = circumcenter S` THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
MP_TAC (ISPECL [`V:real^3->bool`; `S:real^3->bool`; `p:real^3`]
XYOFCGX) THEN
ASM_REWRITE_TAC[] THEN
REPEAT DISCH_TAC THEN
REWRITE_TAC[
VORONOI_SET;
IN_INTERS;
IN_ELIM_THM;
voronoi_closed] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
POP_ASSUM (ASSUME_TAC o ONCE_REWRITE_RULE[GSYM
IN]) THEN
ASM_CASES_TAC `w:real^3
IN S` THENL
[
FIRST_ASSUM (MP_TAC o SPEC `w:real^3`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `v:real^3`) THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[
REAL_LE_REFL];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`v:real^3`; `w:real^3`]) THEN
ASM_REWRITE_TAC[
IN_DIFF;
DIST_SYM] THEN
REAL_ARITH_TAC);;
let NEIGHBORHOOD_lemma = prove(`!V S p. packing V /\ S
SUBSET V /\ (!u v. u
IN S /\ v
IN V
DIFF S ==>
dist (v, p) >
dist (u, p))
==> ?r. &0 < r /\ (!x u v. x
IN ball(p, r) /\ u
IN S /\ v
IN V
DIFF S ==>
dist (v, x) >
dist (u, x))`,
REWRITE_TAC[
real_gt] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `V
DIFF S = {}:real^3->bool` THEN ASM_REWRITE_TAC[
NOT_IN_EMPTY] THENL
[
EXISTS_TAC `&1` THEN REWRITE_TAC[
REAL_LT_01];
ALL_TAC
] THEN
SUBGOAL_THEN `?R. S
SUBSET ball(p:real^3, R)` CHOOSE_TAC THENL
[
POP_ASSUM MP_TAC THEN REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN STRIP_TAC THEN
EXISTS_TAC `
dist (x:real^3, p)` THEN
REWRITE_TAC[
ball;
SUBSET;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`x':real^3`; `x:real^3`]) THEN
ASM_REWRITE_TAC[
DIST_SYM];
ALL_TAC
] THEN
SUBGOAL_THEN `?d. &0 < d /\ (!u v. u
IN S /\ v
IN (V
DIFF S)
INTER ball (p:real^3, R + &2) ==>
dist (v, p) >=
dist (u, p) + d)` STRIP_ASSUME_TAC THENL
[
ABBREV_TAC `K =
IMAGE (\(u,v).
dist (v,p:real^3) -
dist (u,p)) {u,v | u
IN S /\ v
IN (V
DIFF S)
INTER ball (p:real^3, R + &2)}` THEN
SUBGOAL_THEN `FINITE (K:real->bool)` ASSUME_TAC THENL
[
EXPAND_TAC "K" THEN
MATCH_MP_TAC
FINITE_IMAGE THEN
MATCH_MP_TAC
FINITE_PRODUCT THEN
CONJ_TAC THEN MATCH_MP_TAC
FINITE_SUBSET THENL
[
EXISTS_TAC `V
INTER ball (p:real^3, R)` THEN
ASM_SIMP_TAC[
KIUMVTC;
SUBSET_INTER];
EXISTS_TAC `V
INTER (
ball (p:real^3, R + &2))` THEN
ASM_SIMP_TAC[
KIUMVTC;
SUBSET_INTER;
INTER_SUBSET] THEN
SIMP_TAC[
SUBSET;
IN_INTER;
IN_DIFF]
];
ALL_TAC
] THEN
ASM_CASES_TAC `K = {}:real->bool` THENL
[
POP_ASSUM MP_TAC THEN EXPAND_TAC "K" THEN
REWRITE_TAC[
IMAGE_EQ_EMPTY;
EXTENSION;
NOT_IN_EMPTY;
IN_ELIM_THM;
NOT_EXISTS_THM] THEN
DISCH_TAC THEN
EXISTS_TAC `&1` THEN
REWRITE_TAC[
REAL_LT_01] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`u:real^3,v:real^3`; `u:real^3`; `v:real^3`]) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
EXISTS_TAC `
inf (K:real->bool)` THEN
MP_TAC (SPEC `K:real->bool`
INF_FINITE) THEN
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
ABBREV_TAC `d =
inf K` THEN
CONJ_TAC THENL
[
UNDISCH_TAC `d:real
IN K` THEN
EXPAND_TAC "K" THEN
REWRITE_TAC[
IN_IMAGE;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`u:real^3`; `v:real^3`]) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN
SIMP_TAC[
IN_INTER];
ALL_TAC
] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `
dist (v:real^3,p) -
dist (u,p)`) THEN
ANTS_TAC THENL
[
EXPAND_TAC "K" THEN
REWRITE_TAC[
IN_IMAGE;
IN_ELIM_THM] THEN
EXISTS_TAC `u:real^3,v:real^3` THEN
REWRITE_TAC[] THEN
MAP_EVERY EXISTS_TAC [`u:real^3`; `v:real^3`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
ABBREV_TAC `r = min (&1) (d / &2)` THEN
SUBGOAL_THEN `&0 < r /\ r <= &1 /\ r <= d / &2` ASSUME_TAC THENL
[
EXPAND_TAC "r" THEN
REWRITE_TAC[
REAL_MIN_MIN;
REAL_LT_MIN] THEN
UNDISCH_TAC `&0 < d` THEN REAL_ARITH_TAC;
ALL_TAC
] THEN
EXISTS_TAC `r:real` THEN
ASM_REWRITE_TAC[
ball;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[REAL_ARITH `a < b <=> &0 < b - a`] THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `(
dist (v, p) -
dist (u, p)) - &2 *
dist (p:real^3, x)` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `
dist (v, p) -
dist (u, p:real^3) - &2 * r` THEN
CONJ_TAC THENL
[
ASM_CASES_TAC `v
IN ball (p:real^3,R + &2)` THENL
[
FIRST_X_ASSUM (MP_TAC o SPECL [`u:real^3`; `v:real^3`]) THEN
ASM_REWRITE_TAC[
IN_INTER] THEN
FIRST_X_ASSUM (MP_TAC o check (is_conj o concl)) THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN
SUBGOAL_THEN `u
IN ball(p:real^3,R)` MP_TAC THENL
[
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `S:real^3->bool` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
ball;
IN_ELIM_THM;
REAL_NOT_LT;
DIST_SYM] THEN
FIRST_X_ASSUM (MP_TAC o check (is_conj o concl)) THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
UNDISCH_TAC `
dist (p:real^3,x) < r` THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
MP_TAC (ISPECL [`u:real^3`; `p:real^3`; `x:real^3`]
DIST_TRIANGLE) THEN
MP_TAC (ISPECL [`p:real^3`; `x:real^3`; `v:real^3`]
DIST_TRIANGLE) THEN
REWRITE_TAC[
DIST_SYM] THEN
REAL_ARITH_TAC);;
let SUBSPACES_INTER_BALL_EQ_IMP_EQ = prove(`!s t r.
subspace s /\
subspace t /\
&0 < r /\ s
INTER ball (
vec 0:real^N, r) = t
INTER ball (
vec 0, r)
==> s = t`,
REWRITE_TAC[
subspace] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `!v:real^N. ?d. d % v
IN ball (
vec 0,r) /\ v = (inv d) % (d % v)` ASSUME_TAC THENL
[
GEN_TAC THEN
EXISTS_TAC `(r / &2) * inv(
norm (v:real^N))` THEN
ASM_CASES_TAC `v =
vec 0:real^N` THENL
[
ASM_REWRITE_TAC[
VECTOR_MUL_RZERO;
CENTRE_IN_BALL];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[GSYM
NORM_EQ_0] THEN DISCH_TAC THEN
CONJ_TAC THENL
[
REWRITE_TAC[
ball;
IN_ELIM_THM;
dist;
VECTOR_SUB_LZERO;
NORM_NEG;
NORM_MUL] THEN
REWRITE_TAC[
REAL_ABS_MUL;
REAL_ABS_INV;
REAL_ABS_NORM; GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_MUL_LINV] THEN
UNDISCH_TAC `&0 < r` THEN REAL_ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
VECTOR_MUL_ASSOC;
REAL_INV_MUL;
REAL_INV_INV] THEN
REWRITE_TAC[REAL_ARITH `(ia * b) * a * ib = (ia * a) * (ib * b)`] THEN
ASM_SIMP_TAC[REAL_MUL_LINV; REAL_ARITH `&0 < r ==> ~(r / &2 = &0)`] THEN
REWRITE_TAC[REAL_MUL_LID;
VECTOR_MUL_LID];
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION] THEN GEN_TAC THEN
EQ_TAC THEN DISCH_TAC THENL
[
FIRST_X_ASSUM (MP_TAC o SPEC `x:real^N`) THEN
STRIP_TAC THEN
ABBREV_TAC `v:real^N = d % x` THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`inv d`; `v:real^N`]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN MATCH_MP_TAC THEN
SUBGOAL_THEN `v
IN s
INTER ball (
vec 0:real^N,r)` MP_TAC THENL
[
ASM_REWRITE_TAC[
IN_INTER] THEN
EXPAND_TAC "v" THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[
IN_INTER];
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `x:real^N`) THEN
STRIP_TAC THEN
ABBREV_TAC `v:real^N = d % x` THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
SUBGOAL_THEN `v
IN t
INTER ball (
vec 0:real^N,r)` MP_TAC THENL
[
ASM_REWRITE_TAC[
IN_INTER] THEN
EXPAND_TAC "v" THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REPLICATE_TAC 4 REMOVE_ASSUM THEN
POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
SIMP_TAC[
IN_INTER]);;
let YIFVQDV_1 = prove(`!V ul k p. packing V /\ ul
IN barV V k /\
hl ul <
sqrt (&2) /\ p
permutes (0..k) ==>
left_action_list p ul
IN barV V k`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
ABBREV_TAC `vl =
left_action_list p (ul:(real^3)list)` THEN
REWRITE_TAC[
BARV] THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = k + 1 /\
LENGTH (ul:(real^3)list) = k + 1` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST] THEN
UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
SUBGOAL_THEN `
set_of_list vl =
set_of_list ul:real^3->bool` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
MP_TAC (ISPECL [`ul:(real^3)list`; `p:num->num`]
SET_OF_LIST_LEFT_ACTION_LIST) THEN
ASM_REWRITE_TAC[ARITH_RULE `(k + 1) - 1 = k`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?j. vl':(real^3)list = truncate_simplex j vl /\ j + 1 <= k + 1 /\
LENGTH vl' = j + 1` CHOOSE_TAC THENL
[
EXISTS_TAC `
LENGTH (vl':(real^3)list) - 1` THEN
ABBREV_TAC `n =
LENGTH (vl':(real^3)list)` THEN
MP_TAC (ISPECL [`vl:(real^3)list`; `vl':(real^3)list`]
INITIAL_SUBLIST_IMP_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[ARITH_RULE `1 <= n <=> 0 < n`] THEN
ASM_SIMP_TAC[ARITH_RULE `0 < n ==> n - 1 + 1 = n`];
ALL_TAC
] THEN
REWRITE_TAC[
VORONOI_NONDG] THEN
ASM_REWRITE_TAC[] THEN
REPEAT CONJ_TAC THENL
[
MATCH_MP_TAC
LET_TRANS THEN
EXISTS_TAC `k + 1` THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `barV V k ul` THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_SIMP_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 1`];
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `
set_of_list (vl:(real^3)list)` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ABBREV_TAC `a = \j. if j = 0 then &4 else
aff_dim (
INTERS {bis (
HD vl) (v:real^3) | v | v
IN set_of_list (truncate_simplex (j - 1) vl)})` THEN
SUBGOAL_THEN `!i. i <= k ==>
aff_dim (voronoi_list (V:real^3->bool) (truncate_simplex i vl)) = a (i + 1)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
EXPAND_TAC "a" THEN
REWRITE_TAC[ARITH_RULE `~(i + 1 = 0) /\ (i + 1) - 1 = i`] THEN
SUBGOAL_THEN `
HD vl =
HD (truncate_simplex i vl):real^3` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC (GSYM
HD_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`];
ALL_TAC
] THEN
MATCH_MP_TAC
YIFVQDV_lemma_aff_dim THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `
set_of_list (truncate_simplex i vl)
SUBSET set_of_list vl:real^3->bool` ASSUME_TAC THENL
[
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`];
ALL_TAC
] THEN
REPEAT CONJ_TAC THENL
[
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `
set_of_list vl:real^3->bool` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `
set_of_list vl:real^3->bool` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (vl:(real^3)list)` THEN
CONJ_TAC THENL
[
REWRITE_TAC[
HL] THEN
MATCH_MP_TAC
RADV_MONO THEN
ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `
HD (truncate_simplex i vl):real^3` THEN
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
MP_TAC (ISPECL [`i:num`; `vl:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[ARITH_RULE `i + 1 <= k + 1 <=> i <= k`] THEN
SIMP_TAC[ARITH_RULE `1 <= i + 1`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
HL] THEN
ASM_REWRITE_TAC[GSYM
HL];
ALL_TAC
] THEN
SUBGOAL_THEN `a 0 =
int_of_num 4 /\ a 1 = &3` ASSUME_TAC THENL
[
EXPAND_TAC "a" THEN
REWRITE_TAC[ARITH_RULE `~(1 = 0) /\ 1 - 1 = 0`] THEN
ASM_SIMP_TAC[
TRUNCATE_0_EQ_HEAD; ARITH_RULE `1 <= k + 1`] THEN
REWRITE_TAC[
set_of_list;
IN_SING] THEN
SUBGOAL_THEN `
INTERS {bis (
HD vl) v | v | v =
HD vl:real^3} = bis (
HD vl) (
HD vl)` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
EXTENSION;
IN_INTERS;
IN_ELIM_THM;
IN_SING] THEN
GEN_TAC THEN REWRITE_TAC[GSYM
EXTENSION] THEN
EQ_TAC THEN REPEAT STRIP_TAC THENL
[
POP_ASSUM (MP_TAC o SPEC `bis (
HD vl) (
HD vl):real^3->bool`) THEN
ANTS_TAC THENL [ EXISTS_TAC `
HD vl:real^3` THEN REWRITE_TAC[]; ALL_TAC ] THEN
REWRITE_TAC[];
ASM_REWRITE_TAC[]
];
ALL_TAC
] THEN
REWRITE_TAC[bis] THEN
SUBGOAL_THEN `{x:real^3 | T} =
UNIV` (fun
th -> REWRITE_TAC[
th]) THENL [ REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
IN_UNIV]; ALL_TAC ] THEN
REWRITE_TAC[
AFF_DIM_UNIV; DIMINDEX_3];
ALL_TAC
] THEN
SUBGOAL_THEN `!j. j <= k ==>
int_of_num 3 - &j <= a j - &1 /\ a j - &1 <= a (j + 1) /\ (a (j + 1) = &3 - &j <=> (!i. i <= j ==> a (i + 1) = &3 - &i))` ASSUME_TAC THENL
[
INDUCT_TAC THENL
[
DISCH_TAC THEN
ASM_SIMP_TAC[ARITH; ARITH_RULE `i <= 0 <=> i = 0`] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
REWRITE_TAC[
ADD1] THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o check (is_imp o concl)) THEN
ASM_SIMP_TAC[ARITH_RULE `j' + 1 <= k ==> j' <= k`] THEN
STRIP_TAC THEN
CONJ_TAC THENL
[
REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `a (j' + 1) -
int_of_num 1 <= a ((j' + 1) + 1)` ASSUME_TAC THENL
[
EXPAND_TAC "a" THEN
REWRITE_TAC[ARITH_RULE `~(j' + 1 = 0) /\ ~((j' + 1) + 1 = 0) /\ (j' + 1) - 1 = j' /\ ((j' + 1) + 1) - 1 = j' + 1`] THEN
ABBREV_TAC `s:real^3->bool =
INTERS {bis (
HD vl) v | v | v
IN set_of_list (truncate_simplex j' vl)}` THEN
ABBREV_TAC `t:real^3->bool =
INTERS {bis (
HD vl) v | v | v
IN set_of_list (truncate_simplex (j' + 1) vl)}` THEN
SUBGOAL_THEN `~(t = {}:real^3->bool)` ASSUME_TAC THENL
[
EXPAND_TAC "t" THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
ABBREV_TAC `S:real^3->bool =
set_of_list (truncate_simplex (j' + 1) vl)` THEN
ABBREV_TAC `c:real^3 = circumcenter S` THEN
EXISTS_TAC `c:real^3` THEN
REWRITE_TAC[
IN_INTERS;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[bis;
IN_ELIM_THM] THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
AFFINE_INDEPENDENT_SUBSET THEN
EXISTS_TAC `
set_of_list vl:real^3->bool` THEN
CONJ_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
EXPAND_TAC "S" THEN
MATCH_MP_TAC
SET_OF_LIST_TRUNCATE_SIMPLEX_SUBSET THEN
MATCH_MP_TAC
LE_TRANS THEN
EXISTS_TAC `k + 1` THEN
ASM_REWRITE_TAC[
LE_REFL; ARITH_RULE `a + 1 <= b + 1 <=> a <= b`];
ALL_TAC
] THEN
DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o SPEC `v:real^3`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `
HD vl:real^3`) THEN
ANTS_TAC THENL
[
EXPAND_TAC "S" THEN
SUBGOAL_THEN `
HD vl:real^3 =
HD (truncate_simplex (j' + 1) vl)` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC (GSYM
HD_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[ARITH_RULE `j' + 1 <= k ==> (j' + 1) + 1 <= k + 1`];
ALL_TAC
] THEN
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
MP_TAC (ISPECL [`j' + 1`; `vl:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[ARITH_RULE `j' + 1 <= k ==> (j' + 1) + 1 <= k + 1`; ARITH_RULE `1 <= a + 1`];
ALL_TAC
] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `?w:real^3. t = s
INTER bis (
HD vl) w` CHOOSE_TAC THENL
[
ABBREV_TAC `w:real^3 =
LAST (truncate_simplex (j' + 1) vl)` THEN
EXISTS_TAC `w:real^3` THEN
EXPAND_TAC "t" THEN EXPAND_TAC "s" THEN
MP_TAC (ISPECL [`vl:(real^3)list`; `j':num`]
TRUNCATE_SIMPLEX_ADD1) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `j' + 1 <= k` THEN
ARITH_TAC;
ALL_TAC
] THEN
ABBREV_TAC `yl = truncate_simplex (j' + 1) vl:(real^3)list` THEN
DISCH_THEN (fun
th -> ONCE_REWRITE_TAC[
th]) THEN
ASM_REWRITE_TAC[
SET_OF_LIST_APPEND;
set_of_list;
IN_UNION;
IN_SING] THEN
EXPAND_TAC "s" THEN
SET_TAC[];
ALL_TAC
] THEN
POP_ASSUM (ASSUME_TAC o REWRITE_RULE[
BIS_EQ_HYPERPLANE]) THEN
ABBREV_TAC `aa = &2 % (w -
HD vl:real^3)` THEN
ABBREV_TAC `bb = w
dot (w:real^3) -
HD vl
dot (
HD vl:real^3)` THEN
MP_TAC (ISPECL [`aa:real^3`; `bb:real`; `s:real^3->bool`]
AFF_DIM_AFFINE_INTER_HYPERPLANE) THEN
ANTS_TAC THENL
[
EXPAND_TAC "s" THEN
REWRITE_TAC[GSYM
AFFINE_HULL_EQ] THEN
MATCH_MP_TAC
HULL_INTERS_EQ_INTERS THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[
BIS_EQ_HYPERPLANE;
AFFINE_HYPERPLANE];
ALL_TAC
] THEN
REMOVE_ASSUM THEN REMOVE_ASSUM THEN POP_ASSUM (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
COND_CASES_TAC THEN INT_ARITH_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
EQ_TAC THENL
[
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `a (j' + 1) =
int_of_num 3 - &j'` MP_TAC THENL
[
REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[GSYM
INT_OF_NUM_ADD] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN
ASM_CASES_TAC `i = j' + 1` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN
ARITH_TAC;
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o SPEC `j' + 1`) THEN
REWRITE_TAC[
LE_REFL];
ALL_TAC
] THEN
SUBGOAL_THEN `a (k + 1) =
int_of_num 3 - &k` MP_TAC THENL
[
REMOVE_ASSUM THEN FIRST_X_ASSUM (MP_TAC o SPEC `k:num`) THEN
REWRITE_TAC[
LE_REFL] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
SUBGOAL_THEN `truncate_simplex k vl = vl:(real^3)list` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPECL [`k:num`; `vl:(real^3)list`; `vl:(real^3)list`]
TRUNCATE_SIMPLEX_INITIAL_SUBLIST) THEN
ASM_REWRITE_TAC[
LE_REFL;
INITIAL_SUBLIST_REFL];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
VORONOI_LIST] THEN
REWRITE_TAC[GSYM
VORONOI_LIST] THEN
UNDISCH_TAC `barV V k ul` THEN
REWRITE_TAC[
BARV;
VORONOI_NONDG] THEN
STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `ul:(real^3)list`) THEN
ASM_REWRITE_TAC[
INITIAL_SUBLIST_REFL; ARITH_RULE `0 < k + 1`; GSYM
INT_OF_NUM_ADD] THEN
INT_ARITH_TAC;
ALL_TAC
] THEN
FIRST_X_ASSUM (MP_TAC o SPEC `k:num`) THEN
REWRITE_TAC[
LE_REFL] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_THEN (MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[ARITH_RULE `j <= k <=> j + 1 <= k + 1`] THEN
DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[ARITH_RULE `j <= k <=> j + 1 <= k + 1`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th; GSYM
INT_OF_NUM_ADD]) THEN
INT_ARITH_TAC);;
let YIFVQDV = prove(`!V ul k p. packing V /\ ul
IN barV V k /\
hl ul <
sqrt(&2) /\ p
permutes (0..k) ==>
(
left_action_list p ul
IN barV V k) /\ (omega_list V (
left_action_list p ul) = omega_list V ul)`,
REWRITE_TAC[
IN] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
MP_TAC (SPEC_ALL
YIFVQDV_1) THEN
ASM_REWRITE_TAC[
IN] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
ABBREV_TAC `vl:(real^3)list =
left_action_list p ul` THEN
MP_TAC (SPEC_ALL
XNHPWAB1) THEN
ASM_REWRITE_TAC[
IN] THEN
DISCH_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `k:num`]
XNHPWAB1) THEN
ASM_REWRITE_TAC[
HL;
IN] THEN
SUBGOAL_THEN `
set_of_list vl =
set_of_list ul:real^3->bool` (fun
th -> REWRITE_TAC[
th]) THENL
[
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
SET_OF_LIST_LEFT_ACTION_LIST THEN
UNDISCH_TAC `barV V k ul` THEN
ASM_SIMP_TAC[
BARV; ARITH_RULE `(k + 1) - 1 = k`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[GSYM
HL]);;
let HL_TRUNCATE_SIMPLEX_OMEGA_N = prove(`!V k ul j. packing V /\ barV V k ul /\ j <= k /\ hl ul <
sqrt (&2)
==> hl (truncate_simplex j ul) =
dist (omega_list_n V ul j,
HD ul)`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `vl:(real^3)list = truncate_simplex j ul` THEN
SUBGOAL_THEN `barV V j vl` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `j:num`; `vl:(real^3)list`]
HL_EQ_DIST0) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
AP_TERM_TAC THEN
REWRITE_TAC[
PAIR_EQ] THEN
CONJ_TAC THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `j:num`]
OMEGA_LIST_LEMMA) THEN
ANTS_TAC THENL
[
UNDISCH_TAC `barV V k ul` THEN UNDISCH_TAC `j <= k:num` THEN
SIMP_TAC[
BARV] THEN ARITH_TAC;
ALL_TAC
] THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC (GSYM
XNHPWAB1) THEN
EXISTS_TAC `j:num` THEN
ASM_REWRITE_TAC[
IN] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (ul:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
HD_TRUNCATE_SIMPLEX THEN
UNDISCH_TAC `barV V k ul` THEN UNDISCH_TAC `j <= k:num` THEN
SIMP_TAC[
BARV] THEN ARITH_TAC);;
let KSOQKWL_lemma0 = prove(`!V ul vl k. packing V /\ barV V k ul /\ barV V k vl /\
~(
HD ul =
HD vl)
==> ~({omega_list_n V ul i | i <= k} = {omega_list_n V vl i | i <= k})`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[
EXTENSION;
NOT_FORALL_THM;
IN_ELIM_THM] THEN
EXISTS_TAC `
HD ul:real^3` THEN
MATCH_MP_TAC (TAUT `~(A ==> B) ==> ~(A <=> B)`) THEN
REWRITE_TAC[NOT_IMP] THEN
CONJ_TAC THENL [ EXISTS_TAC `0` THEN REWRITE_TAC[
OMEGA_LIST_N;
LE_0]; ALL_TAC ] THEN
REWRITE_TAC[
NOT_EXISTS_THM; TAUT `~(A /\ B) <=> A ==> ~B`] THEN
REPEAT STRIP_TAC THEN
POP_ASSUM (ASSUME_TAC o SYM) THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = k + 1 /\
LENGTH (ul:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k vl` THEN UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
SUBGOAL_THEN `i + 1 <= k + 1` ASSUME_TAC THENL [ UNDISCH_TAC `i <= k:num` THEN ARITH_TAC; ALL_TAC ] THEN
SUBGOAL_THEN `omega_list_n V vl i
IN voronoi_closed V (
HD vl)` MP_TAC THENL
[
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `voronoi_list V (truncate_simplex i vl)` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
OMEGA_LIST_N_IN_VORONOI_LIST THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
HD vl =
HD (truncate_simplex i vl):real^3` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC (GSYM
HD_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
VORONOI_LIST_SUBSET_VORONOI_CLOSED THEN
MP_TAC (ISPECL [`i:num`; `vl:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
ASM_REWRITE_TAC[
voronoi_closed;
IN_ELIM_THM;
NOT_FORALL_THM] THEN
EXISTS_TAC `
HD ul:real^3` THEN
REWRITE_TAC[NOT_IMP] THEN
CONJ_TAC THENL
[
ONCE_REWRITE_TAC[GSYM
IN] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `
set_of_list ul:real^3->bool` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
HD_IN_SET_OF_LIST THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
DIST_REFL;
DIST_LE_0]);;
let KSOQKWL_lemma1 = prove(`!V ul vl k j. packing V /\ barV V k ul /\ barV V k vl /\
hl ul <
sqrt (&2) /\ hl vl <
sqrt (&2) /\
0 < j /\ j <= k /\
truncate_simplex (j - 1) ul = truncate_simplex (j - 1) vl /\
hl (truncate_simplex j ul) <= hl (truncate_simplex j vl) /\
~(omega_list_n V ul j = omega_list_n V vl j)
==> ~({omega_list_n V ul i | i <= k} = {omega_list_n V vl i | i <= k})`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[
EXTENSION;
NOT_FORALL_THM;
IN_ELIM_THM] THEN
EXISTS_TAC `omega_list_n V ul j` THEN
MATCH_MP_TAC (TAUT `~(A ==> B) ==> ~(A <=> B)`) THEN
REWRITE_TAC[NOT_IMP;
NOT_EXISTS_THM] THEN
CONJ_TAC THENL
[
EXISTS_TAC `j:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (ul:(real^3)list) = k + 1 /\
LENGTH (vl:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k ul` THEN UNDISCH_TAC `barV V k vl` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
SUBGOAL_THEN `j + 1 <= k + 1` ASSUME_TAC THENL [ ASM_REWRITE_TAC[ARITH_RULE `j + 1 <= k + 1 <=> j <= k`]; ALL_TAC ] THEN
MP_TAC (ARITH_RULE `0 < j /\ j <= k ==> j - 1 + 1 = j /\ j <= k + 1`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
REWRITE_TAC[TAUT `~(A /\ B) <=> (A ==> ~B)`] THEN
REPEAT STRIP_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`; `j:num`]
HL_TRUNCATE_SIMPLEX_OMEGA_N) THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `vl:(real^3)list`; `i:num`]
HL_TRUNCATE_SIMPLEX_OMEGA_N) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `
HD vl =
HD ul:real^3` ASSUME_TAC THENL
[
SUBGOAL_THEN `
HD vl =
HD (truncate_simplex (j - 1) ul):real^3` (fun
th -> REWRITE_TAC[
th]) THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC (GSYM
HD_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
HD_TRUNCATE_SIMPLEX THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_CASES_TAC `i < j:num` THENL
[
SUBGOAL_THEN `truncate_simplex i vl:(real^3)list = truncate_simplex i ul` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPECL [`ul:(real^3)list`; `i:num`; `j - 1`]
TRUNCATE_TRUNCATE_SIMPLEX) THEN
ANTS_TAC THENL [ ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN ARITH_TAC; ALL_TAC ] THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC (GSYM
TRUNCATE_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
ALL_TAC
] THEN
MATCH_MP_TAC (REAL_ARITH `a < b ==> ~(b = a:real)`) THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `k:num`]
XNHPWAB4) THEN
ASM_REWRITE_TAC[
IN] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[
NOT_LT;
LE_LT] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
MATCH_MP_TAC (REAL_ARITH `a < b ==> ~(a = b:real)`) THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (truncate_simplex j vl:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SPECL [`V:real^3->bool`; `vl:(real^3)list`; `k:num`]
XNHPWAB4) THEN
ASM_REWRITE_TAC[
IN] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
UNDISCH_TAC `omega_list_n V ul j = omega_list_n V vl i` THEN
POP_ASSUM (fun
th -> ASM_REWRITE_TAC[SYM
th]));;
let AFFINE_INDEPENDENT_OMEGA_LIST_N = prove(`!V ul k. packing V /\ barV V k ul /\ hl ul <
sqrt (&2)
==> ~affine_dependent {omega_list_n V ul i | i <= k}`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[
AFFINE_INDEPENDENT_IFF_CARD] THEN
ABBREV_TAC `A = {omega_list_n V ul i | i <= k}` THEN
SUBGOAL_THEN `FINITE (A:real^3->bool) /\
CARD A <= k + 1` ASSUME_TAC THENL
[
EXPAND_TAC "A" THEN
SUBGOAL_THEN `!i. i <= k <=> i
IN 0..k` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
IN_NUMSEG;
LE_0];
ALL_TAC
] THEN
ONCE_REWRITE_TAC[GSYM
IMAGE_LEMMA] THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
FINITE_IMAGE THEN REWRITE_TAC[
FINITE_NUMSEG];
ALL_TAC
] THEN
MP_TAC (ISPECL [`\i. omega_list_n V ul i`; `0..k`]
CARD_IMAGE_LE) THEN
REWRITE_TAC[
FINITE_NUMSEG;
CARD_NUMSEG; ARITH_RULE `(k + 1) - 0 = k + 1`];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `k:num`]
XNHPWAB3) THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
IN;
LE_0] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC (ISPEC `A:real^3->bool`
AFF_DIM_LE_CARD) THEN
ASM_REWRITE_TAC[] THEN
REMOVE_ASSUM THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[GSYM
INT_OF_NUM_LE; GSYM
INT_OF_NUM_ADD] THEN
INT_ARITH_TAC);;
let ROGERS_EQ = prove(`!V ul vl k. packing V /\ barV V k ul /\ barV V k vl /\
hl ul <
sqrt (&2) /\ hl vl <
sqrt (&2)
==> (rogers V ul = rogers V vl <=>
{omega_list_n V ul i | i <= k} = {omega_list_n V vl i | i <= k})`,
let NUM_FINITE_IMP_MAX_EXISTS = prove(`!K:num->bool. FINITE K /\ ~(K = {}) ==> ?m. m
IN K /\ (!j. j
IN K ==> j <= m)`,
REWRITE_TAC[GSYM IMP_IMP] THEN
MATCH_MP_TAC FINITE_INDUCT THEN REWRITE_TAC[
NOT_INSERT_EMPTY] THEN
REPEAT STRIP_TAC THEN POP_ASSUM MP_TAC THEN
ASM_CASES_TAC `s:num->bool = {}` THEN ASM_REWRITE_TAC[] THENL
[
ASM_REWRITE_TAC[
IN_SING] THEN
EXISTS_TAC `x:num` THEN
SIMP_TAC[
LE_REFL];
ALL_TAC
] THEN
STRIP_TAC THEN
ASM_CASES_TAC `m <= x:num` THENL
[
EXISTS_TAC `x:num` THEN
REWRITE_TAC[
IN_INSERT] THEN
GEN_TAC THEN
ASM_CASES_TAC `j = x:num` THEN ASM_REWRITE_TAC[
LE_REFL] THEN
DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[] THEN UNDISCH_TAC `m <= x:num` THEN
ARITH_TAC;
ALL_TAC
] THEN
EXISTS_TAC `m:num` THEN
ASM_REWRITE_TAC[
IN_INSERT] THEN
GEN_TAC THEN
ASM_CASES_TAC `j = x:num` THENL
[
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(m <= x:num)` THEN ARITH_TAC;
ALL_TAC
] THEN
ASM_SIMP_TAC[]);;
let NOT_ID_IMP_LISTS_NOT_EQ = prove(`!ul:(A)list p k.
LENGTH ul = k + 1 /\
CARD (
set_of_list ul) = k + 1 /\
p
permutes (0..k) /\ ~(p = I)
==> ~(ul =
left_action_list p ul)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?j:num. j < k + 1 /\ ~(p j = j)` CHOOSE_TAC THENL
[
UNDISCH_TAC `~(p = I:num->num)` THEN
REWRITE_TAC[
FUN_EQ_THM;
I_THM;
NOT_FORALL_THM] THEN
STRIP_TAC THEN
EXISTS_TAC `x:num` THEN
ASM_REWRITE_TAC[] THEN
DISJ_CASES_TAC (ARITH_RULE `x < k + 1 \/ k < x:num`) THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `p
permutes 0..k` THEN
REWRITE_TAC[
permutes] THEN
DISCH_THEN (CONJUNCTS_THEN2 (MP_TAC o SPEC `x:num`) (fun
th -> ALL_TAC)) THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
LE_0] THEN ARITH_TAC;
ALL_TAC
] THEN
UNDISCH_TAC `ul:(A)list =
left_action_list p ul` THEN
DISCH_THEN (MP_TAC o AP_TERM `\l:(A)list.
EL ((p:num->num) j) l`) THEN
MP_TAC (SPECL [`ul:(A)list`; `p:num->num`; `j:num`]
EL_LEFT_ACTION_LIST) THEN
ASM_REWRITE_TAC[ARITH_RULE `(k + 1) - 1 = k`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
MP_TAC (SPEC `ul:(A)list`
CARD_SET_OF_LIST_EQ_LENGTH_IMP_ALL_DISTINCT) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
MP_TAC (ISPECL [`p:num->num`; `0..k`] Hypermap_and_fan.PERMUTES_IMP_INSIDE) THEN
ASM_REWRITE_TAC[
IN_NUMSEG] THEN
DISCH_THEN (MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[
LE_0; ARITH_RULE `a <= b <=> a < b + 1`]);;
let KSOQKWL = prove(`!V ul p k. packing V /\ ul
IN barV V k /\ hl ul <
sqrt(&2) /\
p
permutes (0..k) /\ (rogers V ul = rogers V (
left_action_list p ul)) ==> (p = I)`,
REWRITE_TAC[
IN] THEN REPEAT STRIP_TAC THEN
POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
ABBREV_TAC `vl:(real^3)list =
left_action_list p ul` THEN
SUBGOAL_THEN `barV V k vl` ASSUME_TAC THENL
[
EXPAND_TAC "vl" THEN
MP_TAC (SPEC_ALL
YIFVQDV) THEN
ASM_SIMP_TAC[
IN];
ALL_TAC
] THEN
MP_TAC (SPEC_ALL
BARV_IMP_LENGTH_EQ_CARD) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
SUBGOAL_THEN `hl (vl:(real^3)list) <
sqrt (&2)` ASSUME_TAC THENL
[
SUBGOAL_THEN `hl (vl:(real^3)list) = hl (ul:(real^3)list)` (fun
th -> ASM_REWRITE_TAC[
th]) THEN
REWRITE_TAC[
HL] THEN
AP_TERM_TAC THEN
EXPAND_TAC "vl" THEN
MATCH_MP_TAC
SET_OF_LIST_LEFT_ACTION_LIST THEN
UNDISCH_TAC `barV V k ul` THEN SIMP_TAC[
BARV] THEN
ASM_REWRITE_TAC[ARITH_RULE `(k + 1) - 1 = k`];
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (vl:(real^3)list) = k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `barV V k vl` THEN SIMP_TAC[
BARV];
ALL_TAC
] THEN
ASM_SIMP_TAC[
ROGERS_EQ] THEN
MP_TAC (ISPECL [`ul:(real^3)list`; `p:num->num`; `k:num`]
NOT_ID_IMP_EXISTS_MAX_EQ_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `~(
HD ul =
HD vl:real^3)` THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
KSOQKWL_lemma0 THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
SUBGOAL_THEN `~(omega_list_n V ul (j + 1) = omega_list_n V vl (j + 1))` ASSUME_TAC THENL
[
DISCH_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `ul:(real^3)list`; `j + 1`]
HL_TRUNCATE_SIMPLEX_OMEGA_N) THEN
MP_TAC (SPECL [`V:real^3->bool`; `k:num`; `vl:(real^3)list`; `j + 1`]
HL_TRUNCATE_SIMPLEX_OMEGA_N) THEN
MP_TAC (ARITH_RULE `j < k ==> j + 1 <= k`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ABBREV_TAC `S =
set_of_list (truncate_simplex (j + 1) ul:(real^3)list)` THEN
ABBREV_TAC `c:real^3 = circumcenter S` THEN
MP_TAC (SPECL [`V:real^3->bool`; `S:real^3->bool`; `c:real^3`]
XYOFCGX) THEN
SUBGOAL_THEN `barV V (j + 1) (truncate_simplex (j + 1) ul)` ASSUME_TAC THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `~affine_dependent (S:real^3->bool)` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `j + 1`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `S
SUBSET V:real^3->bool` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `j + 1` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "S" THEN
REWRITE_TAC[GSYM
HL] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (ul:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
ABBREV_TAC `x =
EL (j + 1) ul:real^3` THEN
ABBREV_TAC `y =
EL (j + 1) vl:real^3` THEN
DISCH_THEN (MP_TAC o SPECL [`x:real^3`; `y:real^3`]) THEN
ABBREV_TAC `xl:(real^3)list = truncate_simplex (j + 1) ul` THEN
ABBREV_TAC `yl:(real^3)list = truncate_simplex (j + 1) vl` THEN
SUBGOAL_THEN `barV V (j + 1) yl` ASSUME_TAC THENL
[
EXPAND_TAC "yl" THEN
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MP_TAC (SPECL [`V:real^3->bool`; `xl:(real^3)list`; `j + 1`]
BARV_IMP_LENGTH_EQ_CARD) THEN
MP_TAC (SPECL [`V:real^3->bool`; `yl:(real^3)list`; `j + 1`]
BARV_IMP_LENGTH_EQ_CARD) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN DISCH_TAC THEN
MP_TAC (ARITH_RULE `j + 1 <= k ==> (j + 1) + 1 <= k + 1`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN `!n. n < (j + 1) + 1 ==> (
EL n xl):real^3 = if (n = j + 1) then x else
EL n (truncate_simplex j vl)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`ul:(real^3)list`; `j + 1`]
EL_TRUNCATE_SIMPLEX) THEN
MP_TAC (ISPECL [`ul:(real^3)list`; `j:num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THENL
[
DISCH_TAC THEN
DISCH_THEN (MP_TAC o SPEC `j + 1`) THEN
ASM_REWRITE_TAC[
LE_REFL];
ALL_TAC
] THEN
DISCH_TAC THEN DISCH_TAC THEN
POP_ASSUM (MP_TAC o SPEC `n:num`) THEN
POP_ASSUM (MP_TAC o SPEC `n:num`) THEN
ASM_SIMP_TAC[ARITH_RULE `j + 1 <= k ==> j + 1 <= k + 1`; ARITH_RULE `n < (j + 1) + 1 /\ ~(n = j + 1) ==> n <= j /\ n <= j + 1`];
ALL_TAC
] THEN
SUBGOAL_THEN `!n. n < (j + 1) + 1 ==> (
EL n yl):real^3 = if (n = j + 1) then y else
EL n (truncate_simplex j vl)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`vl:(real^3)list`; `j + 1`]
EL_TRUNCATE_SIMPLEX) THEN
MP_TAC (ISPECL [`vl:(real^3)list`; `j:num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THENL
[
DISCH_TAC THEN
DISCH_THEN (MP_TAC o SPEC `j + 1`) THEN
ASM_REWRITE_TAC[
LE_REFL];
ALL_TAC
] THEN
DISCH_TAC THEN DISCH_TAC THEN
POP_ASSUM (MP_TAC o SPEC `n:num`) THEN
POP_ASSUM (MP_TAC o SPEC `n:num`) THEN
ASM_SIMP_TAC[ARITH_RULE `j + 1 <= k ==> j + 1 <= k + 1`; ARITH_RULE `n < (j + 1) + 1 /\ ~(n = j + 1) ==> n <= j /\ n <= j + 1`];
ALL_TAC
] THEN
SUBGOAL_THEN `x:real^3
IN S` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN
REWRITE_TAC[
IN_SET_OF_LIST] THEN
REMOVE_ASSUM THEN POP_ASSUM (MP_TAC o SPEC `j + 1`) THEN
REWRITE_TAC[ARITH_RULE `j + 1 < (j + 1) + 1`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
MATCH_MP_TAC
MEM_EL THEN
ASM_REWRITE_TAC[ARITH_RULE `j + 1 < (j + 1) + 1`];
ALL_TAC
] THEN
SUBGOAL_THEN `y:real^3
IN (
set_of_list yl)
DIFF S` ASSUME_TAC THENL
[
EXPAND_TAC "S" THEN
REWRITE_TAC[
IN_DIFF;
IN_SET_OF_LIST;
MEM_EXISTS_EL] THEN
CONJ_TAC THENL
[
EXISTS_TAC `j + 1` THEN
ASM_SIMP_TAC[ARITH_RULE `j + 1 < (j + 1) + 1`];
ALL_TAC
] THEN
REWRITE_TAC[
NOT_EXISTS_THM; TAUT `~(A /\ B) <=> (A ==> ~B)`] THEN GEN_TAC THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_ASSUM (MP_TAC o SPEC `j + 1:num`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `i:num`) THEN
FIRST_X_ASSUM (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THENL
[
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[ARITH_RULE `j + 1 < (j + 1) + 1`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
MP_TAC (ISPECL [`yl:(real^3)list`]
CARD_SET_OF_LIST_EQ_LENGTH_IMP_ALL_DISTINCT) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN MATCH_MP_TAC THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
ALL_TAC
] THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `(
set_of_list yl:real^3->bool)
DIFF S` THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `
set_of_list yl:real^3->bool
SUBSET V` MP_TAC THENL
[
MATCH_MP_TAC
BARV_SUBSET THEN
EXISTS_TAC `j + 1` THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SIMP_TAC[
SUBSET;
IN_DIFF];
ALL_TAC
] THEN
SUBGOAL_THEN `omega_list_n V ul (j + 1) = c:real^3` ASSUME_TAC THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `xl:(real^3)list`; `j + 1`]
XNHPWAB1) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[
IN] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (ul:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "xl" THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
EXPAND_TAC "c" THEN EXPAND_TAC "S" THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
EXPAND_TAC "xl" THEN
MATCH_MP_TAC (GSYM
OMEGA_LIST_LEMMA) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `hl (xl:(real^3)list) =
dist (x:real^3,c)` (fun
th -> REWRITE_TAC[SYM
th]) THENL
[
ASM_REWRITE_TAC[
HL] THEN
MP_TAC (ISPEC `S:real^3->bool`
OAPVION2) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `x:real^3`) THEN
ASM_REWRITE_TAC[
DIST_SYM];
ALL_TAC
] THEN
SUBGOAL_THEN `hl (yl:(real^3)list) =
dist (y:real^3,c)` (fun
th -> REWRITE_TAC[SYM
th]) THENL
[
ASM_REWRITE_TAC[
HL] THEN
MP_TAC (SPECL [`V:real^3->bool`; `yl:(real^3)list`; `j + 1`]
XNHPWAB1) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[
IN] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (vl:(real^3)list)` THEN
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "yl" THEN
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `k:num`] THEN
ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
DISCH_THEN (ASSUME_TAC o SYM) THEN
MP_TAC (ISPEC `
set_of_list yl:real^3->bool`
OAPVION2) THEN
ANTS_TAC THENL
[
MATCH_MP_TAC
BARV_AFFINE_INDEPENDENT THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `j + 1`] THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o SPEC `y:real^3`) THEN
ANTS_TAC THENL
[
UNDISCH_TAC `y:real^3
IN set_of_list yl
DIFF S` THEN SIMP_TAC[
IN_DIFF];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[
th]) THEN
SUBGOAL_THEN `omega_list V yl = omega_list_n V ul (j + 1)` (fun
th -> REWRITE_TAC[
th]) THENL
[
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "yl" THEN
MATCH_MP_TAC
OMEGA_LIST_LEMMA THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
REMOVE_ASSUM THEN POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
DIST_SYM];
ALL_TAC
] THEN
REAL_ARITH_TAC;
ALL_TAC
] THEN
ASM_CASES_TAC `hl (truncate_simplex (j + 1) ul:(real^3)list) <= hl (truncate_simplex (j + 1) vl:(real^3)list)` THENL
[
MATCH_MP_TAC
KSOQKWL_lemma1 THEN
EXISTS_TAC `j + 1` THEN
ASM_REWRITE_TAC[ARITH_RULE `0 < j + 1`; ARITH_RULE `j + 1 <= k <=> j < k`; ARITH_RULE `(j + 1) - 1 = j`];
ALL_TAC
] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[
REAL_NOT_LE;
REAL_LT_LE] THEN DISCH_TAC THEN
MATCH_MP_TAC (GSYM
KSOQKWL_lemma1) THEN
EXISTS_TAC `j + 1` THEN
ASM_REWRITE_TAC[ARITH_RULE `0 < j + 1`; ARITH_RULE `j + 1 <= k <=> j < k`; ARITH_RULE `(j + 1) - 1 = j`]);;
let IVFICRK = prove(`!k. ?g. (
BIJ g { (i,
sigma ) | i
IN 0..(k+1) /\
sigma permutes (0..k) } { p | p
permutes (0..(k+1)) })
/\ (!(ul:(A)list) i
sigma j. (
LENGTH ul = k+2) /\ j <= k /\ i
IN 0..(k+1) /\
sigma permutes (0..k) ==>
(
EL j (
left_action_list (g(i,
sigma)) ul) =
EL j (
left_action_list sigma (
DROP ul i) )))`,
GEN_TAC THEN
ABBREV_TAC `f = (\i j. if j = k + 1 then i else (if (i <= j /\ j < k + 1) then j + 1 else j))` THEN
ABBREV_TAC `fi = (\i j. if j = i then k + 1 else (if (i < j /\ j <= k + 1) then j - 1 else j))` THEN
SUBGOAL_THEN `!i. i <= k + 1 ==> (f:num->num->num) i o fi i = I:num->num /\ fi i o f i = I` ASSUME_TAC THENL
[
REWRITE_TAC[
IN_NUMSEG;
FUN_EQ_THM;
I_THM;
o_THM] THEN
REPEAT STRIP_TAC THENL
[
EXPAND_TAC "fi" THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "f" THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "f" THEN
ASM_SIMP_TAC[ARITH_RULE `x <= k + 1 ==> ~(x - 1 = k + 1)`] THEN
MP_TAC (ARITH_RULE `i < x /\ x <= k + 1 ==> i <= x - 1 /\ x - 1 < k + 1`) THEN
ASM_SIMP_TAC[] THEN
POP_ASSUM MP_TAC THEN ARITH_TAC;
ALL_TAC
] THEN
EXPAND_TAC "f" THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[DE_MORGAN_THM;
NOT_LT;
NOT_LE] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
MP_TAC (ARITH_RULE `x <= i /\ ~(x = i) /\ i <= k + 1 ==> ~(x = k + 1)`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC (ARITH_RULE `x <= i /\ i <= x ==> x = i:num`) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[ARITH_RULE `k + 1 < x ==> ~(x = k + 1) /\ ~(x < k + 1)`];
ALL_TAC
] THEN
EXPAND_TAC "f" THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "fi" THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "fi" THEN
ASM_SIMP_TAC[ARITH_RULE `i <= x ==> ~(x + 1 = i)`] THEN
MP_TAC (ARITH_RULE `i <= x /\ x < k + 1 ==> i < x + 1 /\ x + 1 <= k + 1`) THEN
ASM_SIMP_TAC[ARITH_RULE `(x + 1) - 1 = x`];
ALL_TAC
] THEN
EXPAND_TAC "fi" THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[DE_MORGAN_THM;
NOT_LE;
NOT_LT] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
ASM_SIMP_TAC[ARITH_RULE `x < i ==> ~(x = i:num) /\ ~(i < x)`];
ALL_TAC
] THEN
MP_TAC (ARITH_RULE `k + 1 <= x /\ ~(x = k + 1) ==> ~(x <= k + 1)`) THEN
ASM_SIMP_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!i j. i <= k + 1 ==> (f:num->num->num) i (fi i j) = j /\ fi i (f i j) = j` ASSUME_TAC THENL
[
GEN_TAC THEN GEN_TAC THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[
FUN_EQ_THM;
o_THM;
I_THM] THEN
DISCH_THEN (CONJUNCTS_THEN (ASSUME_TAC o SPEC `j:num`)) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!i. i <= k + 1 ==> f i
permutes (0..k+1) /\ fi i
permutes (0..k+1)` ASSUME_TAC THENL
[
GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `(f:num->num->num) i
permutes (0..k+1)` ASSUME_TAC THENL
[
REWRITE_TAC[
permutes;
IN_NUMSEG; DE_MORGAN_THM;
NOT_LE; ARITH_RULE `~(x < 0)`] THEN
REPEAT STRIP_TAC THENL
[
EXPAND_TAC "f" THEN
ASM_SIMP_TAC[ARITH_RULE `k + 1 < x ==> ~(x = k + 1) /\ ~(x < k + 1)`];
ALL_TAC
] THEN
REWRITE_TAC[
EXISTS_UNIQUE] THEN
EXISTS_TAC `(fi:num->num->num) i y` THEN
CONJ_TAC THENL
[
ASM_SIMP_TAC[];
ALL_TAC
] THEN
GEN_TAC THEN DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC Hypermap_and_fan.INVERSE_PERMUTES THEN
EXISTS_TAC `(f:num->num->num) i` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ABBREV_TAC `g = \(i:num,sigma:num->num).
sigma o (fi:num->num->num) i` THEN
EXISTS_TAC `g:(num#(num->num))->num->num` THEN
SUBGOAL_THEN `!i sigma. i <= k + 1 /\
sigma permutes 0..k ==> g (i,
sigma)
permutes (0..k+1)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
EXPAND_TAC "g" THEN
REWRITE_TAC[] THEN
MATCH_MP_TAC
PERMUTES_COMPOSE THEN
ASM_SIMP_TAC[] THEN
MATCH_MP_TAC
PERMUTES_SUBSET THEN
EXISTS_TAC `0..k` THEN
ASM_REWRITE_TAC[
SUBSET_NUMSEG] THEN
ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `!i sigma. i <= k + 1 /\
sigma permutes 0..k ==>
inverse (g (i,
sigma)) (k + 1) = i` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `g (i:num, sigma:num->num) i = k + 1` MP_TAC THENL
[
EXPAND_TAC "g" THEN REWRITE_TAC[
o_THM] THEN
EXPAND_TAC "fi" THEN REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
permutes;
IN_NUMSEG; DE_MORGAN_THM;
NOT_LE; ARITH_RULE `~(x < 0)`] THEN
DISCH_THEN (CONJUNCTS_THEN2 (MP_TAC o SPEC `k + 1`) (fun
th -> ALL_TAC)) THEN
SIMP_TAC[ARITH_RULE `k < k + 1`];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o AP_TERM `\x:num. (
inverse (g (i:num, sigma:num->num)) x):num`) THEN BETA_TAC THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
MP_TAC (ISPECL [`(g (i:num, sigma:num->num)):num->num`; `0..k+1`]
PERMUTES_INVERSES) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
CONJ_TAC THENL
[
REWRITE_TAC[
BIJ;
INJ;
SURJ;
IN_ELIM_THM;
IN_NUMSEG; ARITH_RULE `0 <= i`] THEN
REPEAT STRIP_TAC THENL
[
ASM_SIMP_TAC[];
ASM_REWRITE_TAC[
PAIR_EQ] THEN
FIRST_ASSUM (MP_TAC o SPECL [`i:num`; `sigma:num->num`]) THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i':num`; `sigma':num->num`]) THEN
ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o AP_TERM `(\x:num->num. x o (f:num->num->num) i)`) THEN
ASM_SIMP_TAC[GSYM
o_ASSOC;
I_O_ID];
ASM_SIMP_TAC[];
ABBREV_TAC `r =
inverse (x:num->num) (k + 1)` THEN
EXISTS_TAC `r:num, (x:num->num) o (f:num->num->num) r` THEN
SUBGOAL_THEN `r <= k + 1` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`x:num->num`; `0..k+1`]
PERMUTES_INVERSE) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`
inverse (x:num->num)`; `0..k+1`] Hypermap_and_fan.PERMUTES_IMP_INSIDE) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `k + 1`) THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
LE_0;
LE_REFL];
ALL_TAC
] THEN
CONJ_TAC THENL
[
EXISTS_TAC `r:num` THEN
EXISTS_TAC `(x:num->num) o (f:num->num->num) r` THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(x:num->num) o (f:num->num->num) r
permutes 0..k+1` MP_TAC THENL
[
MATCH_MP_TAC
PERMUTES_COMPOSE THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
SIMP_TAC[
permutes;
IN_NUMSEG; DE_MORGAN_THM;
NOT_LE; ARITH_RULE `~(x < 0)`] THEN
DISCH_THEN (CONJUNCTS_THEN2 ASSUME_TAC (fun
th -> ALL_TAC)) THEN
X_GEN_TAC `j:num` THEN
ONCE_REWRITE_TAC[ARITH_RULE `k < j <=> j = k + 1 \/ k + 1 < j`] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "r" THEN
REWRITE_TAC[
o_THM] THEN
EXPAND_TAC "f" THEN
REWRITE_TAC[] THEN
MP_TAC (ISPECL [`x:num->num`; `0..k+1`]
PERMUTES_INVERSES) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
EXPAND_TAC "g" THEN REWRITE_TAC[GSYM
o_ASSOC] THEN
ASM_SIMP_TAC[
I_O_ID]
];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `j < k + 2 /\ j < k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `j <= k:num` THEN ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `i < k + 2` ASSUME_TAC THENL
[
UNDISCH_TAC `i
IN 0..k+1` THEN REWRITE_TAC[
IN_NUMSEG] THEN
ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (
DROP (ul:(A)list) i) = k + 1` ASSUME_TAC THENL
[
MP_TAC (SPECL [`i:num`; `ul:(A)list`]
LENGTH_DROP) THEN
ASM_REWRITE_TAC[] THEN
ARITH_TAC;
ALL_TAC
] THEN
ASM_SIMP_TAC[
left_action_list;
EL_TABLE] THEN
ABBREV_TAC `r =
inverse (sigma:num->num) j` THEN
MP_TAC (SPECL [`i:num`; `r:num`; `ul:(A)list`]
EL_DROP) THEN
SUBGOAL_THEN `r <= k:num` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`sigma:num->num`; `0..k`]
PERMUTES_INVERSE) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`
inverse (sigma:num->num)`; `0..k`] Hypermap_and_fan.PERMUTES_IMP_INSIDE) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
LE_0];
ALL_TAC
] THEN
ASM_REWRITE_TAC[ARITH_RULE `r < (k + 2) - 1 <=> r <= k`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
SUBGOAL_THEN `
inverse ((sigma:num->num) o (fi:num->num->num) i) = (f:num->num->num) i o
inverse sigma` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC
INVERSE_UNIQUE_o THEN
SUBGOAL_THEN `!a:num->num b:num->num c:num->num d:num->num. (a o b) o c o d = a o (b o c) o d` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
o_ASSOC];
ALL_TAC
] THEN
UNDISCH_TAC `!i. i <= k + 1 ==> (f:num->num->num) i o fi i = I /\ fi i o f i = I` THEN
DISCH_THEN (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[ARITH_RULE `i <= k + 1 <=> i < k + 2`] THEN
DISCH_TAC THEN
MP_TAC (ISPECL [`sigma:num->num`; `0..k`]
PERMUTES_INVERSES_o) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[
I_O_ID];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
o_THM] THEN
EXPAND_TAC "f" THEN
ASM_SIMP_TAC[ARITH_RULE `r <= k ==> ~(r = k + 1) /\ r < k + 1`] THEN
ONCE_REWRITE_TAC[GSYM
NOT_LE] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[]);;
let IVFICRK_real3 = prove(`!k. ?g. (
BIJ g { (i,
sigma ) | i
IN 0..(k+1) /\
sigma permutes (0..k) } { p | p
permutes (0..(k+1)) })
/\ (!(ul:(real^3)list) i
sigma j. (
LENGTH ul = k+2) /\ j <= k /\ i
IN 0..(k+1) /\
sigma permutes (0..k) ==>
(
EL j (
left_action_list (g(i,
sigma)) ul) =
EL j (
left_action_list sigma (
DROP ul i) )))`,
GEN_TAC THEN
ABBREV_TAC `f = (\i j. if j = k + 1 then i else (if (i <= j /\ j < k + 1) then j + 1 else j))` THEN
ABBREV_TAC `fi = (\i j. if j = i then k + 1 else (if (i < j /\ j <= k + 1) then j - 1 else j))` THEN
SUBGOAL_THEN `!i. i <= k + 1 ==> (f:num->num->num) i o fi i = I:num->num /\ fi i o f i = I` ASSUME_TAC THENL
[
REWRITE_TAC[
IN_NUMSEG;
FUN_EQ_THM;
I_THM;
o_THM] THEN
REPEAT STRIP_TAC THENL
[
EXPAND_TAC "fi" THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "f" THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "f" THEN
ASM_SIMP_TAC[ARITH_RULE `x <= k + 1 ==> ~(x - 1 = k + 1)`] THEN
MP_TAC (ARITH_RULE `i < x /\ x <= k + 1 ==> i <= x - 1 /\ x - 1 < k + 1`) THEN
ASM_SIMP_TAC[] THEN
POP_ASSUM MP_TAC THEN ARITH_TAC;
ALL_TAC
] THEN
EXPAND_TAC "f" THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[DE_MORGAN_THM;
NOT_LT;
NOT_LE] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
MP_TAC (ARITH_RULE `x <= i /\ ~(x = i) /\ i <= k + 1 ==> ~(x = k + 1)`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC (ARITH_RULE `x <= i /\ i <= x ==> x = i:num`) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[ARITH_RULE `k + 1 < x ==> ~(x = k + 1) /\ ~(x < k + 1)`];
ALL_TAC
] THEN
EXPAND_TAC "f" THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "fi" THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
COND_CASES_TAC THENL
[
EXPAND_TAC "fi" THEN
ASM_SIMP_TAC[ARITH_RULE `i <= x ==> ~(x + 1 = i)`] THEN
MP_TAC (ARITH_RULE `i <= x /\ x < k + 1 ==> i < x + 1 /\ x + 1 <= k + 1`) THEN
ASM_SIMP_TAC[ARITH_RULE `(x + 1) - 1 = x`];
ALL_TAC
] THEN
EXPAND_TAC "fi" THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[DE_MORGAN_THM;
NOT_LE;
NOT_LT] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
ASM_SIMP_TAC[ARITH_RULE `x < i ==> ~(x = i:num) /\ ~(i < x)`];
ALL_TAC
] THEN
MP_TAC (ARITH_RULE `k + 1 <= x /\ ~(x = k + 1) ==> ~(x <= k + 1)`) THEN
ASM_SIMP_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!i j. i <= k + 1 ==> (f:num->num->num) i (fi i j) = j /\ fi i (f i j) = j` ASSUME_TAC THENL
[
GEN_TAC THEN GEN_TAC THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[
FUN_EQ_THM;
o_THM;
I_THM] THEN
DISCH_THEN (CONJUNCTS_THEN (ASSUME_TAC o SPEC `j:num`)) THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!i. i <= k + 1 ==> f i
permutes (0..k+1) /\ fi i
permutes (0..k+1)` ASSUME_TAC THENL
[
GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `(f:num->num->num) i
permutes (0..k+1)` ASSUME_TAC THENL
[
REWRITE_TAC[
permutes;
IN_NUMSEG; DE_MORGAN_THM;
NOT_LE; ARITH_RULE `~(x < 0)`] THEN
REPEAT STRIP_TAC THENL
[
EXPAND_TAC "f" THEN
ASM_SIMP_TAC[ARITH_RULE `k + 1 < x ==> ~(x = k + 1) /\ ~(x < k + 1)`];
ALL_TAC
] THEN
REWRITE_TAC[
EXISTS_UNIQUE] THEN
EXISTS_TAC `(fi:num->num->num) i y` THEN
CONJ_TAC THENL
[
ASM_SIMP_TAC[];
ALL_TAC
] THEN
GEN_TAC THEN DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC Hypermap_and_fan.INVERSE_PERMUTES THEN
EXISTS_TAC `(f:num->num->num) i` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ABBREV_TAC `g = \(i:num,sigma:num->num).
sigma o (fi:num->num->num) i` THEN
EXISTS_TAC `g:(num#(num->num))->num->num` THEN
SUBGOAL_THEN `!i sigma. i <= k + 1 /\
sigma permutes 0..k ==> g (i,
sigma)
permutes (0..k+1)` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
EXPAND_TAC "g" THEN
REWRITE_TAC[] THEN
MATCH_MP_TAC
PERMUTES_COMPOSE THEN
ASM_SIMP_TAC[] THEN
MATCH_MP_TAC
PERMUTES_SUBSET THEN
EXISTS_TAC `0..k` THEN
ASM_REWRITE_TAC[
SUBSET_NUMSEG] THEN
ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `!i sigma. i <= k + 1 /\
sigma permutes 0..k ==>
inverse (g (i,
sigma)) (k + 1) = i` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `g (i:num, sigma:num->num) i = k + 1` MP_TAC THENL
[
EXPAND_TAC "g" THEN REWRITE_TAC[
o_THM] THEN
EXPAND_TAC "fi" THEN REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
permutes;
IN_NUMSEG; DE_MORGAN_THM;
NOT_LE; ARITH_RULE `~(x < 0)`] THEN
DISCH_THEN (CONJUNCTS_THEN2 (MP_TAC o SPEC `k + 1`) (fun
th -> ALL_TAC)) THEN
SIMP_TAC[ARITH_RULE `k < k + 1`];
ALL_TAC
] THEN
DISCH_THEN (MP_TAC o AP_TERM `\x:num. (
inverse (g (i:num, sigma:num->num)) x):num`) THEN BETA_TAC THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
MP_TAC (ISPECL [`(g (i:num, sigma:num->num)):num->num`; `0..k+1`]
PERMUTES_INVERSES) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
CONJ_TAC THENL
[
REWRITE_TAC[
BIJ;
INJ;
SURJ;
IN_ELIM_THM;
IN_NUMSEG; ARITH_RULE `0 <= i`] THEN
REPEAT STRIP_TAC THENL
[
ASM_SIMP_TAC[];
ASM_REWRITE_TAC[
PAIR_EQ] THEN
FIRST_ASSUM (MP_TAC o SPECL [`i:num`; `sigma:num->num`]) THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i':num`; `sigma':num->num`]) THEN
ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
POP_ASSUM MP_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o AP_TERM `(\x:num->num. x o (f:num->num->num) i)`) THEN
ASM_SIMP_TAC[GSYM
o_ASSOC;
I_O_ID];
ASM_SIMP_TAC[];
ABBREV_TAC `r =
inverse (x:num->num) (k + 1)` THEN
EXISTS_TAC `r:num, (x:num->num) o (f:num->num->num) r` THEN
SUBGOAL_THEN `r <= k + 1` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`x:num->num`; `0..k+1`]
PERMUTES_INVERSE) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`
inverse (x:num->num)`; `0..k+1`] Hypermap_and_fan.PERMUTES_IMP_INSIDE) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (MP_TAC o SPEC `k + 1`) THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
LE_0;
LE_REFL];
ALL_TAC
] THEN
CONJ_TAC THENL
[
EXISTS_TAC `r:num` THEN
EXISTS_TAC `(x:num->num) o (f:num->num->num) r` THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(x:num->num) o (f:num->num->num) r
permutes 0..k+1` MP_TAC THENL
[
MATCH_MP_TAC
PERMUTES_COMPOSE THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
SIMP_TAC[
permutes;
IN_NUMSEG; DE_MORGAN_THM;
NOT_LE; ARITH_RULE `~(x < 0)`] THEN
DISCH_THEN (CONJUNCTS_THEN2 ASSUME_TAC (fun
th -> ALL_TAC)) THEN
X_GEN_TAC `j:num` THEN
ONCE_REWRITE_TAC[ARITH_RULE `k < j <=> j = k + 1 \/ k + 1 < j`] THEN
DISCH_THEN DISJ_CASES_TAC THENL
[
ASM_REWRITE_TAC[] THEN
EXPAND_TAC "r" THEN
REWRITE_TAC[
o_THM] THEN
EXPAND_TAC "f" THEN
REWRITE_TAC[] THEN
MP_TAC (ISPECL [`x:num->num`; `0..k+1`]
PERMUTES_INVERSES) THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
EXPAND_TAC "g" THEN REWRITE_TAC[GSYM
o_ASSOC] THEN
ASM_SIMP_TAC[
I_O_ID]
];
ALL_TAC
] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `j < k + 2 /\ j < k + 1` ASSUME_TAC THENL
[
UNDISCH_TAC `j <= k:num` THEN ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `i < k + 2` ASSUME_TAC THENL
[
UNDISCH_TAC `i
IN 0..k+1` THEN REWRITE_TAC[
IN_NUMSEG] THEN
ARITH_TAC;
ALL_TAC
] THEN
SUBGOAL_THEN `
LENGTH (
DROP (ul:(real^3)list) i) = k + 1` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`i:num`; `ul:(real^3)list`]
LENGTH_DROP) THEN
ASM_REWRITE_TAC[] THEN
ARITH_TAC;
ALL_TAC
] THEN
ASM_SIMP_TAC[
left_action_list;
EL_TABLE] THEN
ABBREV_TAC `r =
inverse (sigma:num->num) j` THEN
MP_TAC (ISPECL [`i:num`; `r:num`; `ul:(real^3)list`]
EL_DROP) THEN
SUBGOAL_THEN `r <= k:num` ASSUME_TAC THENL
[
MP_TAC (ISPECL [`sigma:num->num`; `0..k`]
PERMUTES_INVERSE) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`
inverse (sigma:num->num)`; `0..k`] Hypermap_and_fan.PERMUTES_IMP_INSIDE) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
LE_0];
ALL_TAC
] THEN
ASM_REWRITE_TAC[ARITH_RULE `r < (k + 2) - 1 <=> r <= k`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
SUBGOAL_THEN `
inverse ((sigma:num->num) o (fi:num->num->num) i) = (f:num->num->num) i o
inverse sigma` (fun
th -> REWRITE_TAC[
th]) THENL
[
MATCH_MP_TAC
INVERSE_UNIQUE_o THEN
SUBGOAL_THEN `!a:num->num b:num->num c:num->num d:num->num. (a o b) o c o d = a o (b o c) o d` (fun
th -> REWRITE_TAC[
th]) THENL
[
REWRITE_TAC[
o_ASSOC];
ALL_TAC
] THEN
UNDISCH_TAC `!i. i <= k + 1 ==> (f:num->num->num) i o fi i = I /\ fi i o f i = I` THEN
DISCH_THEN (MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[ARITH_RULE `i <= k + 1 <=> i < k + 2`] THEN
DISCH_TAC THEN
MP_TAC (ISPECL [`sigma:num->num`; `0..k`]
PERMUTES_INVERSES_o) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[
I_O_ID];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
o_THM] THEN
EXPAND_TAC "f" THEN
ASM_SIMP_TAC[ARITH_RULE `r <= k ==> ~(r = k + 1) /\ r < k + 1`] THEN
ONCE_REWRITE_TAC[GSYM
NOT_LE] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[]);;
let WQPRRDY = prove(`!V ul k. packing V /\ ul
IN barV V k /\ hl ul <
sqrt(&2) ==>
(
convex hull (
set_of_list ul) =
UNIONS { rogers V (
left_action_list p ul) | p
permutes (0..k) })`,
GEN_TAC THEN REWRITE_TAC[
IN] THEN ONCE_REWRITE_TAC[
SWAP_FORALL_THM] THEN
INDUCT_TAC THEN REPEAT STRIP_TAC THENL
[
SUBGOAL_THEN `?x:real^3. ul = [x]` CHOOSE_TAC THENL
[
EXISTS_TAC `
HD ul:real^3` THEN
MATCH_MP_TAC
LENGTH_1_LEMMA THEN
UNDISCH_TAC `barV V 0 ul` THEN SIMP_TAC[
BARV; ARITH];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
set_of_list;
CONVEX_HULL_SING;
PERMUTES_TRIVIAL;
SING_GSPEC_APP;
LEFT_ACTION_LIST_I] THEN
REWRITE_TAC[
ROGERS;
LENGTH; ARITH_RULE `j < SUC 0 <=> j = 0`;
SING_GSPEC;
IMAGE_LEMMA;
IN_SING;
SING_GSPEC_APP;
CONVEX_HULL_SING] THEN
REWRITE_TAC[
OMEGA_LIST_N;
HD;
UNIONS_1];
ALL_TAC
] THEN
ABBREV_TAC `S:real^3->bool =
set_of_list ul` THEN
MP_TAC (ISPECL [`S:real^3->bool`; `omega_list V ul`]
CONVEX_HULL_EXCHANGE_UNION) THEN
ANTS_TAC THENL
[
EXPAND_TAC "S" THEN
MATCH_MP_TAC
XNHPWAB2 THEN
EXISTS_TAC `SUC k` THEN ASM_REWRITE_TAC[
IN];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ABBREV_TAC `u = omega_list V ul` THEN
EXPAND_TAC "S" THEN ASM_REWRITE_TAC[
IN_SET_OF_LIST] THEN
MP_TAC (ISPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`]
BARV_IMP_LENGTH_EQ_CARD) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
MP_TAC (SPEC `k:num`
IVFICRK_real3) THEN
DISCH_THEN (CHOOSE_THEN (CONJUNCTS_THEN2 (LABEL_TAC "g0") ASSUME_TAC)) THEN
POP_ASSUM (MP_TAC o ISPEC `ul:(real^3)list`) THEN
ASM_REWRITE_TAC[ARITH_RULE `SUC k + 1 = k + 2`;
IN_NUMSEG;
LE_0] THEN DISCH_THEN (LABEL_TAC "tmp") THEN
SUBGOAL_THEN `!i p. i < SUC k + 1 /\ p
permutes 0..k ==> g (i,p)
permutes 0..SUC k` (LABEL_TAC "g1") THENL
[
REPEAT STRIP_TAC THEN
REMOVE_THEN "g0" MP_TAC THEN
REWRITE_TAC[
BIJ;
INJ; GSYM
CONJ_ASSOC] THEN
DISCH_THEN (CONJUNCTS_THEN2 MP_TAC (fun
th -> ALL_TAC)) THEN
DISCH_THEN (MP_TAC o SPEC `i:num, p:num->num`) THEN
REWRITE_TAC[
IN_ELIM_THM;
ADD1] THEN
ANTS_TAC THENL
[
MAP_EVERY EXISTS_TAC [`i:num`; `p:num->num`] THEN
ASM_REWRITE_TAC[
IN_NUMSEG;
LE_0; ARITH_RULE `i <= k + 1 <=> i < SUC k + 1`];
ALL_TAC
] THEN
REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!s. s
permutes 0..SUC k ==> ?i p. i < SUC k + 1 /\ p
permutes 0..k /\ s = g(i,p)` (LABEL_TAC "g2") THENL
[
REWRITE_TAC[
ADD1; ARITH_RULE `i < (k + 1) + 1 <=> i <= k + 1`] THEN
REPEAT STRIP_TAC THEN
REMOVE_THEN "g0" MP_TAC THEN
REWRITE_TAC[
BIJ;
SURJ] THEN
REPLICATE_TAC 2 (DISCH_THEN (CONJUNCTS_THEN2 (fun
th -> ALL_TAC) MP_TAC)) THEN
DISCH_THEN (MP_TAC o SPEC `s:num->num`) THEN
ASM_REWRITE_TAC[
IN_ELIM_THM;
IN_NUMSEG;
LE_0] THEN
STRIP_TAC THEN
MAP_EVERY EXISTS_TAC [`i:num`; `sigma:num->num`] THEN
POP_ASSUM MP_TAC THEN
ASM_REWRITE_TAC[
EQ_SYM_EQ];
ALL_TAC
] THEN
REMOVE_THEN "g0" (fun
th -> ALL_TAC) THEN
SUBGOAL_THEN `!i p r. i < SUC k + 1 /\ r <= k /\ p
permutes 0..k ==> truncate_simplex r (
left_action_list (g(i,p)) ul:(real^3)list) = truncate_simplex r (
left_action_list p (
DROP ul i))` (LABEL_TAC "tr") THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
LIST_EL_EQ] THEN
ABBREV_TAC `xl:(real^3)list = truncate_simplex r (
left_action_list (g (i:num,p:num->num)) ul)` THEN
ABBREV_TAC `yl:(real^3)list = truncate_simplex r (
left_action_list p (
DROP ul i))` THEN
MP_TAC (ISPECL [`i:num`; `ul:(real^3)list`]
LENGTH_DROP) THEN
ASM_REWRITE_TAC[ARITH_RULE `(SUC k + 1) - 1 = SUC k`] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`r:num`; `
left_action_list (g (i:num,p:num->num)) ul:(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
MP_TAC (ARITH_RULE `r <= k ==> r + 1 <= SUC k + 1`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST] THEN
MP_TAC (ISPECL [`r:num`; `
left_action_list p (
DROP ul i):(real^3)list`]
LENGTH_TRUNCATE_SIMPLEX) THEN
ASM_SIMP_TAC[
LENGTH_LEFT_ACTION_LIST; ARITH_RULE `r <= k ==> r + 1 <= SUC k`] THEN
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`
left_action_list (g(i:num,p:num->num)) ul:(real^3)list`; `r:num`; `j:num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST] THEN
MP_TAC (ARITH_RULE `j < r + 1 ==> j <= r`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC (ISPECL [`
left_action_list p (
DROP ul i):(real^3)list`; `r:num`; `j:num`]
EL_TRUNCATE_SIMPLEX) THEN
ASM_REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST; ARITH_RULE `r + 1 <= SUC k <=> r <= k`] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `r <= k:num` THEN UNDISCH_TAC `j <= r:num` THEN UNDISCH_TAC `i < SUC k + 1` THEN
ARITH_TAC;
ALL_TAC
] THEN
REMOVE_THEN "tmp" (fun
th -> ALL_TAC) THEN
SUBGOAL_THEN `!i b:real^3. i < SUC k + 1 /\ b =
EL i ul ==>
convex hull (u
INSERT (S
DELETE b)) =
UNIONS {
convex hull (u
INSERT rogers V (
left_action_list p (
DROP ul i))) | p
permutes 0..k}` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
MP_TAC (ISPECL [`i:num`; `ul:(real^3)list`]
SET_OF_LIST_DELETE_EQ_DROP) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
ABBREV_TAC `vl:(real^3)list =
DROP ul i` THEN
FIRST_X_ASSUM (MP_TAC o SPEC `vl:(real^3)list`) THEN
ANTS_TAC THENL
[
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i:num`; `I:num->num`; `k:num`]) THEN
ASM_REWRITE_TAC[
PERMUTES_I;
LE_REFL;
LEFT_ACTION_LIST_I] THEN
MP_TAC (ISPECL [`i:num`; `ul:(real^3)list`]
LENGTH_DROP) THEN
ASM_REWRITE_TAC[ARITH_RULE `(SUC k + 1) - 1 = k + 1`] THEN DISCH_TAC THEN
ABBREV_TAC `xl:(real^3)list =
left_action_list (g (i:num, I:num->num)) ul` THEN
SUBGOAL_THEN `truncate_simplex k vl = vl:(real^3)list` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPECL [`k:num`; `vl:(real^3)list`; `vl:(real^3)list`]
TRUNCATE_SIMPLEX_INITIAL_SUBLIST) THEN
ASM_REWRITE_TAC[
LE_REFL;
INITIAL_SUBLIST_REFL];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i:num`; `I:num->num`]) THEN
ASM_REWRITE_TAC[
PERMUTES_I] THEN DISCH_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`; `(g(i:num,I:num->num)):num->num`]
YIFVQDV) THEN
ASM_REWRITE_TAC[
IN] THEN DISCH_TAC THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
TRUNCATE_SIMPLEX_BARV THEN
EXISTS_TAC `SUC k` THEN ASM_REWRITE_TAC[ARITH_RULE `k <= SUC k`];
ALL_TAC
] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `hl (xl:(real^3)list)` THEN
CONJ_TAC THENL
[
MATCH_MP_TAC
HL_DECREASE THEN
MAP_EVERY EXISTS_TAC [`V:real^3->bool`; `SUC k`] THEN
ASM_REWRITE_TAC[
IN; ARITH_RULE `k <= SUC k`];
ALL_TAC
] THEN
EXPAND_TAC "xl" THEN
REWRITE_TAC[
HL] THEN
ASM_SIMP_TAC[ARITH_RULE `(SUC k + 1) - 1 = SUC k`;
SET_OF_LIST_LEFT_ACTION_LIST] THEN
EXPAND_TAC "S" THEN REWRITE_TAC[GSYM
HL] THEN ASM_REWRITE_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `
convex hull (u:real^3
INSERT set_of_list vl) =
convex hull (u
INSERT convex hull set_of_list vl)` (fun
th -> REWRITE_TAC[
th]) THENL
[
MP_TAC (ISPECL [`{u:real^3}`; `
set_of_list vl:real^3->bool`]
CONV_UNION_lemma) THEN
REWRITE_TAC[SET_RULE `!x s. {x:real^3}
UNION s = x
INSERT s`];
ALL_TAC
] THEN
DISCH_THEN (LABEL_TAC "A") THEN ASM_REWRITE_TAC[] THEN
ABBREV_TAC `f = {rogers V (
left_action_list p vl) | p
permutes 0..k}` THEN
ONCE_REWRITE_TAC[SET_RULE `!x s. (x:real^3)
INSERT s = {x}
UNION s`] THEN
MP_TAC (ISPECL [`f:(real^3->bool)->bool`; `{u:real^3}`]
CONVEX_HULL_UNION_UNIONS) THEN
ANTS_TAC THENL
[
REMOVE_THEN "A" (fun
th -> REWRITE_TAC[SYM
th]) THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY;
CONVEX_CONVEX_HULL] THEN
EXISTS_TAC `rogers V vl` THEN
EXPAND_TAC "f" THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
EXISTS_TAC `I:num->num` THEN
REWRITE_TAC[
LEFT_ACTION_LIST_I;
PERMUTES_I];
ALL_TAC
] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
AP_TERM_TAC THEN
ASM SET_TAC[];
ALL_TAC
] THEN
SUBGOAL_THEN `!i p. i < SUC k + 1 /\ p
permutes 0..k ==>
convex hull (u:real^3
INSERT rogers V (
left_action_list p (
DROP ul i))) = rogers V (
left_action_list (g (i, p)) ul)` (LABEL_TAC "r") THENL
[
REPEAT STRIP_TAC THEN
REWRITE_TAC[
ROGERS] THEN
ABBREV_TAC `xl:(real^3)list =
left_action_list p (
DROP ul i)` THEN
ABBREV_TAC `yl:(real^3)list =
left_action_list (g (i:num, p:num->num)) ul` THEN
SUBGOAL_THEN `!s:real^3->bool.
convex hull (u
INSERT convex hull s) =
convex hull (u
INSERT s)` (fun
th -> REWRITE_TAC[
th]) THENL
[
GEN_TAC THEN
MP_TAC (ISPECL [`{u:real^3}`; `s:real^3->bool`]
CONV_UNION_lemma) THEN
SIMP_TAC[SET_RULE `{u:real^3}
UNION s = u
INSERT s`];
ALL_TAC
] THEN
AP_TERM_TAC THEN
EXPAND_TAC "xl" THEN EXPAND_TAC "yl" THEN
ASM_REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST] THEN
MP_TAC (ISPECL [`i:num`; `ul:(real^3)list`]
LENGTH_DROP) THEN
ASM_REWRITE_TAC[ARITH_RULE `(SUC k + 1) - 1 = SUC k`] THEN DISCH_TAC THEN
SUBGOAL_THEN `u = omega_list_n V yl (SUC k)` ASSUME_TAC THENL
[
MP_TAC (SPECL [`V:real^3->bool`; `ul:(real^3)list`; `SUC k`; `(g (i:num, p:num->num)):num->num`]
YIFVQDV) THEN
ASM_SIMP_TAC[
IN] THEN
DISCH_THEN (fun
th -> REWRITE_TAC[GSYM
th]) THEN
MP_TAC (SPECL [`V:real^3->bool`; `yl:(real^3)list`; `SUC k`]
OMEGA_LIST_LEMMA) THEN
EXPAND_TAC "yl" THEN ASM_REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST;
LE_REFL] THEN
MP_TAC (ISPECL [`SUC k`; `yl:(real^3)list`; `yl:(real^3)list`]
TRUNCATE_SIMPLEX_INITIAL_SUBLIST) THEN
EXPAND_TAC "yl" THEN ASM_REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST;
LE_REFL;
INITIAL_SUBLIST_REFL] THEN
DISCH_THEN (fun
th -> ASM_REWRITE_TAC[
th]);
ALL_TAC
] THEN
SUBGOAL_THEN `!j. j < SUC k ==> omega_list_n V xl j = omega_list_n V yl j` ASSUME_TAC THENL
[
REPEAT STRIP_TAC THEN
REMOVE_THEN "tr" (MP_TAC o SPECL [`i:num`; `p:num->num`; `k:num`]) THEN
ASM_REWRITE_TAC[
LE_REFL] THEN
DISCH_TAC THEN
MP_TAC (SPECL [`V:real^3->bool`; `xl:(real^3)list`; `j:num`; `k - j:num`]
OMEGA_LIST_N_LEMMA) THEN
MP_TAC (SPECL [`V:real^3->bool`; `yl:(real^3)list`; `j:num`; `k - j:num`]
OMEGA_LIST_N_LEMMA) THEN
MP_TAC (ARITH_RULE `j < SUC k ==> j + k - j = k /\ j + k - j + 1 = k + 1`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
EXPAND_TAC "yl" THEN EXPAND_TAC "xl" THEN REWRITE_TAC[
LENGTH_LEFT_ACTION_LIST] THEN
ASM_REWRITE_TAC[ARITH_RULE `k + 1 <= SUC k /\ k + 1 <= SUC k + 1`] THEN
SIMP_TAC[];
ALL_TAC
] THEN
ASM_REWRITE_TAC[
EXTENSION;
IN_IMAGE;
IN_INSERT;
IN_ELIM_THM] THEN
GEN_TAC THEN EQ_TAC THENL
[
STRIP_TAC THENL
[
EXISTS_TAC `SUC k` THEN
ASM_REWRITE_TAC[] THEN ARITH_TAC;
ALL_TAC
] THEN
EXISTS_TAC `x':num` THEN
ASM_SIMP_TAC[] THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
ALL_TAC
] THEN
STRIP_TAC THEN
ASM_CASES_TAC `x' = SUC k` THENL
[
DISJ1_TAC THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
DISJ2_TAC THEN
EXISTS_TAC `x':num` THEN
MP_TAC (ARITH_RULE `x' < SUC k + 1 /\ ~(x' = SUC k) ==> x' < SUC k`) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
REWRITE_TAC[
EXTENSION;
IN_UNIONS;
IN_ELIM_THM] THEN GEN_TAC THEN
REWRITE_TAC[GSYM
EXTENSION;
MEM_EXISTS_EL] THEN
EQ_TAC THENL
[
ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
POP_ASSUM MP_TAC THEN
ASM_REWRITE_TAC[] THEN REMOVE_ASSUM THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i:num`; `b:real^3`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM] THEN
STRIP_TAC THEN
POP_ASSUM MP_TAC THEN POP_ASSUM (fun
th -> REWRITE_TAC[
th]) THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i:num`; `p:num->num`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
DISCH_TAC THEN
EXISTS_TAC `rogers V (
left_action_list (g (i:num, p:num->num)) ul)` THEN
ASM_REWRITE_TAC[] THEN
EXISTS_TAC `(g (i:num, p:num->num)):num->num` THEN
ASM_SIMP_TAC[];
ALL_TAC
] THEN
STRIP_TAC THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `p:num->num`) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
EXISTS_TAC `
convex hull (u:real^3
INSERT (S
DELETE (
EL i ul)))` THEN
CONJ_TAC THENL
[
EXISTS_TAC `
EL i ul:real^3` THEN REWRITE_TAC[] THEN
EXISTS_TAC `i:num` THEN
ASM_REWRITE_TAC[];
ALL_TAC
] THEN
MATCH_MP_TAC
IN_TRANS THEN
EXISTS_TAC `rogers V (
left_action_list p ul)` THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i:num`; `p':num->num`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[SYM
th]) THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`i:num`; `
EL i ul:real^3`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN (fun
th -> REWRITE_TAC[
th]) THEN
REWRITE_TAC[
SUBSET;
IN_UNIONS] THEN
REPEAT STRIP_TAC THEN
EXISTS_TAC `
convex hull (u
INSERT rogers V (
left_action_list p' (
DROP ul i)))` THEN
ASM_REWRITE_TAC[
IN_ELIM_THM] THEN
EXISTS_TAC `p':num->num` THEN
ASM_REWRITE_TAC[]);;