1 (* ========================================================================= *)
2 (* Convex sets, functions and related things. *)
4 (* (c) Copyright, John Harrison 1998-2008 *)
5 (* (c) Copyright, Lars Schewe 2007 *)
6 (* (c) Copyright, Valentina Bruno 2010 *)
7 (* ========================================================================= *)
9 needs "Multivariate/topology.ml";;
11 (* ------------------------------------------------------------------------- *)
12 (* Some miscelleneous things that are convenient to prove here. *)
13 (* ------------------------------------------------------------------------- *)
15 let TRANSLATION_GALOIS = prove
16 (`!s t a:real^N. s = IMAGE (\x. a + x) t <=> t = IMAGE (\x. --a + x) s`,
17 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN
18 ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN
19 REWRITE_TAC[VECTOR_ARITH `--a + a + x:real^N = x`;
20 VECTOR_ARITH `a + --a + x:real^N = x`] THEN
21 REWRITE_TAC[IMAGE_ID]);;
23 let TRANSLATION_EQ_IMP = prove
24 (`!P:(real^N->bool)->bool.
25 (!a s. P(IMAGE (\x. a + x) s) <=> P s) <=>
26 (!a s. P s ==> P (IMAGE (\x. a + x) s))`,
27 REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
28 MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN
29 EQ_TAC THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM
30 (MP_TAC o SPECL [`--a:real^N`; `IMAGE (\x:real^N. a + x) s`]) THEN
31 ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
32 VECTOR_ARITH `--a + a + x:real^N = x`]);;
34 let DIM_HYPERPLANE = prove
35 (`!a:real^N. ~(a = vec 0) ==> dim {x | a dot x = &0} = dimindex(:N) - 1`,
36 GEOM_BASIS_MULTIPLE_TAC 1 `a:real^N` THEN
37 SIMP_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM; DOT_LMUL; DOT_BASIS;
38 DIMINDEX_GE_1; LE_REFL; REAL_ENTIRE; DIM_SPECIAL_HYPERPLANE]);;
40 let LOWDIM_EQ_HYPERPLANE = prove
41 (`!s. dim s = dimindex(:N) - 1
42 ==> ?a:real^N. ~(a = vec 0) /\ span s = {x | a dot x = &0}`,
44 MP_TAC(ISPEC `s:real^N->bool` LOWDIM_SUBSET_HYPERPLANE) THEN
45 ASM_SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= a ==> a - 1 < a`] THEN
46 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN
47 ASM_REWRITE_TAC[] THEN
48 MP_TAC(ISPEC `a:real^N` SUBSPACE_HYPERPLANE) THEN
49 ONCE_REWRITE_TAC[GSYM SPAN_EQ_SELF] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
50 MATCH_MP_TAC DIM_EQ_SPAN THEN
51 ASM_SIMP_TAC[DIM_HYPERPLANE; LE_REFL] THEN
52 ASM_MESON_TAC[SUBSET_TRANS; SPAN_INC]);;
54 let DIM_EQ_HYPERPLANE = prove
55 (`!s. dim s = dimindex(:N) - 1 <=>
56 ?a:real^N. ~(a = vec 0) /\ span s = {x | a dot x = &0}`,
57 MESON_TAC[DIM_HYPERPLANE; LOWDIM_EQ_HYPERPLANE; DIM_SPAN]);;
59 (* ------------------------------------------------------------------------- *)
60 (* Affine set and affine hull. *)
61 (* ------------------------------------------------------------------------- *)
63 let affine = new_definition
64 `affine s <=> !x y u v. x IN s /\ y IN s /\ (u + v = &1)
65 ==> (u % x + v % y) IN s`;;
67 let AFFINE_ALT = prove
68 (`affine s <=> !x y u. x IN s /\ y IN s ==> ((&1 - u) % x + u % y) IN s`,
69 REWRITE_TAC[affine] THEN
70 MESON_TAC[REAL_ARITH `(u + v = &1) <=> (u = &1 - v)`]);;
72 let AFFINE_SCALING = prove
73 (`!s c. affine s ==> affine (IMAGE (\x. c % x) s)`,
74 REWRITE_TAC[affine; IN_IMAGE] THEN REPEAT STRIP_TAC THEN
75 ASM_REWRITE_TAC[VECTOR_ARITH
76 `u % c % x + v % c % y = c % (u % x + v % y)`] THEN
79 let AFFINE_SCALING_EQ = prove
80 (`!s c. ~(c = &0) ==> (affine (IMAGE (\x. c % x) s) <=> affine s)`,
81 REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[AFFINE_SCALING] THEN
82 DISCH_THEN(MP_TAC o SPEC `inv c` o MATCH_MP AFFINE_SCALING) THEN
83 ASM_SIMP_TAC[GSYM IMAGE_o; o_DEF; VECTOR_MUL_ASSOC;
84 REAL_MUL_LINV; VECTOR_MUL_LID; IMAGE_ID]);;
86 let AFFINE_NEGATIONS = prove
87 (`!s. affine s ==> affine (IMAGE (--) s)`,
88 REWRITE_TAC[affine; IN_IMAGE] THEN REPEAT STRIP_TAC THEN
89 ASM_REWRITE_TAC[VECTOR_ARITH
90 `u % --x + v % --y = --(u % x + v % y)`] THEN
93 let AFFINE_SUMS = prove
94 (`!s t. affine s /\ affine t ==> affine {x + y | x IN s /\ y IN t}`,
95 REWRITE_TAC[affine; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
96 ASM_REWRITE_TAC[VECTOR_ARITH
97 `u % (a + b) + v % (c + d) = (u % a + v % c) + (u % b + v % d)`] THEN
100 let AFFINE_DIFFERENCES = prove
101 (`!s t. affine s /\ affine t ==> affine {x - y | x IN s /\ y IN t}`,
102 REWRITE_TAC[affine; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
103 ASM_REWRITE_TAC[VECTOR_ARITH
104 `u % (a - b) + v % (c - d) = (u % a + v % c) - (u % b + v % d)`] THEN
107 let AFFINE_TRANSLATION_EQ = prove
108 (`!a:real^N s. affine (IMAGE (\x. a + x) s) <=> affine s`,
109 REWRITE_TAC[AFFINE_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
110 REWRITE_TAC[IN_IMAGE; UNWIND_THM1; VECTOR_ARITH
111 `(&1 - u) % (a + x) + u % (a + y) = a + z <=> (&1 - u) % x + u % y = z`]);;
113 add_translation_invariants [AFFINE_TRANSLATION_EQ];;
115 let AFFINE_TRANSLATION = prove
116 (`!s a:real^N. affine s ==> affine (IMAGE (\x. a + x) s)`,
117 REWRITE_TAC[AFFINE_TRANSLATION_EQ]);;
119 let AFFINE_AFFINITY = prove
121 affine s ==> affine (IMAGE (\x. a + c % x) s)`,
122 REPEAT STRIP_TAC THEN
123 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
124 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
125 ASM_SIMP_TAC[IMAGE_o; AFFINE_TRANSLATION; AFFINE_SCALING]);;
127 let AFFINE_LINEAR_IMAGE = prove
128 (`!f s. affine s /\ linear f ==> affine(IMAGE f s)`,
129 REWRITE_TAC[affine; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
130 REWRITE_TAC[IN_IMAGE; linear] THEN MESON_TAC[]);;
132 let AFFINE_LINEAR_IMAGE_EQ = prove
133 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
134 ==> (affine (IMAGE f s) <=> affine s)`,
135 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE AFFINE_LINEAR_IMAGE));;
137 add_linear_invariants [AFFINE_LINEAR_IMAGE_EQ];;
139 let AFFINE_EMPTY = prove
141 REWRITE_TAC[affine; NOT_IN_EMPTY]);;
143 let AFFINE_SING = prove
145 SIMP_TAC[AFFINE_ALT; IN_SING] THEN
146 REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB] THEN
147 REWRITE_TAC[REAL_SUB_ADD; VECTOR_MUL_LID]);;
149 let AFFINE_UNIV = prove
150 (`affine(UNIV:real^N->bool)`,
151 REWRITE_TAC[affine; IN_UNIV]);;
153 let AFFINE_HYPERPLANE = prove
154 (`!a b. affine {x | a dot x = b}`,
155 REWRITE_TAC[affine; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN
156 CONV_TAC REAL_RING);;
158 let AFFINE_STANDARD_HYPERPLANE = prove
159 (`!a b k. affine {x:real^N | x$k = b}`,
161 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
163 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
164 MP_TAC(ISPECL [`basis i:real^N`; `b:real`] AFFINE_HYPERPLANE) THEN
165 ASM_SIMP_TAC[DOT_BASIS]);;
167 let AFFINE_INTERS = prove
168 (`(!s. s IN f ==> affine s) ==> affine(INTERS f)`,
169 REWRITE_TAC[affine; IN_INTERS] THEN MESON_TAC[]);;
171 let AFFINE_INTER = prove
172 (`!s t. affine s /\ affine t ==> affine(s INTER t)`,
173 REWRITE_TAC[affine; IN_INTER] THEN MESON_TAC[]);;
175 let AFFINE_AFFINE_HULL = prove
176 (`!s. affine(affine hull s)`,
177 SIMP_TAC[P_HULL; AFFINE_INTERS]);;
179 let AFFINE_HULL_EQ = prove
180 (`!s. (affine hull s = s) <=> affine s`,
181 SIMP_TAC[HULL_EQ; AFFINE_INTERS]);;
183 let IS_AFFINE_HULL = prove
184 (`!s. affine s <=> ?t. s = affine hull t`,
185 GEN_TAC THEN MATCH_MP_TAC IS_HULL THEN SIMP_TAC[AFFINE_INTERS]);;
187 let AFFINE_HULL_UNIV = prove
188 (`affine hull (:real^N) = (:real^N)`,
189 REWRITE_TAC[AFFINE_HULL_EQ; AFFINE_UNIV]);;
191 let AFFINE_HULLS_EQ = prove
192 (`!s t. s SUBSET affine hull t /\ t SUBSET affine hull s
193 ==> affine hull s = affine hull t`,
194 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULLS_EQ THEN
195 ASM_SIMP_TAC[AFFINE_INTERS]);;
197 let AFFINE_HULL_TRANSLATION = prove
198 (`!a s. affine hull (IMAGE (\x. a + x) s) =
199 IMAGE (\x. a + x) (affine hull s)`,
200 REWRITE_TAC[hull] THEN GEOM_TRANSLATE_TAC[]);;
202 add_translation_invariants [AFFINE_HULL_TRANSLATION];;
204 let AFFINE_HULL_LINEAR_IMAGE = prove
206 ==> affine hull (IMAGE f s) = IMAGE f (affine hull s)`,
207 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
208 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
209 CONJ_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN
210 REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[FUN_IN_IMAGE; HULL_INC] THEN
211 REWRITE_TAC[affine; IN_ELIM_THM] THEN
212 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THENL
213 [FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN
214 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN
215 REWRITE_TAC[IN_IMAGE] THEN
216 MESON_TAC[REWRITE_RULE[affine] AFFINE_AFFINE_HULL];
217 ASM_SIMP_TAC[LINEAR_ADD; LINEAR_CMUL] THEN
218 MESON_TAC[REWRITE_RULE[affine] AFFINE_AFFINE_HULL]]);;
220 add_linear_invariants [AFFINE_HULL_LINEAR_IMAGE];;
222 let IN_AFFINE_HULL_LINEAR_IMAGE = prove
223 (`!f:real^M->real^N s x.
224 linear f /\ x IN affine hull s ==> (f x) IN affine hull (IMAGE f s)`,
225 SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN SET_TAC[]);;
227 let SAME_DISTANCES_TO_AFFINE_HULL = prove
229 (!x. x IN s ==> dist(x,a) = dist(x,b))
230 ==> (!x. x IN affine hull s ==> dist(x,a) = dist(x,b))`,
231 REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN
232 ASM_REWRITE_TAC[AFFINE_ALT; IN_ELIM_THM] THEN
233 REWRITE_TAC[dist; NORM_EQ_SQUARE; NORM_POS_LE; VECTOR_ARITH
234 `((&1 - u) % x + u % y) - a:real^N = (&1 - u) % (x - a) + u % (y - a)`] THEN
235 REWRITE_TAC[NORM_POW_2; DOT_LMUL; DOT_RMUL; VECTOR_ARITH
236 `(x + y) dot (x + y):real^N = (x dot x + y dot y) + &2 * x dot y`] THEN
237 SIMP_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN CONV_TAC REAL_RING);;
239 (* ------------------------------------------------------------------------- *)
240 (* Some convenient lemmas about common affine combinations. *)
241 (* ------------------------------------------------------------------------- *)
243 let IN_AFFINE_ADD_MUL = prove
244 (`!s a x:real^N d. affine s /\ a IN s /\ (a + x) IN s ==> (a + d % x) IN s`,
245 REWRITE_TAC[affine] THEN REPEAT STRIP_TAC THEN
246 SUBST1_TAC(VECTOR_ARITH `a + d % x:real^N = (&1 - d) % a + d % (a + x)`) THEN
247 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
249 let IN_AFFINE_ADD_MUL_DIFF = prove
251 affine s /\ x IN s /\ y IN s /\ z IN s ==> (x + a % (y - z)) IN s`,
252 REWRITE_TAC[affine] THEN REPEAT STRIP_TAC THEN
253 REWRITE_TAC[VECTOR_ARITH
254 `x + a % (y - z):real^N =
255 &1 / &2 % ((&1 - &2 * a) % x + (&2 * a) % y) +
256 &1 / &2 % ((&1 + &2 * a) % x + (-- &2 * a) % z)`] THEN
257 FIRST_ASSUM MATCH_MP_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
258 CONJ_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
261 let IN_AFFINE_MUL_DIFF_ADD = prove
263 affine s /\ x IN s /\ y IN s /\ z IN s ==> a % (x - y) + z IN s`,
264 ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN
265 SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF]);;
267 let IN_AFFINE_SUB_MUL_DIFF = prove
269 affine s /\ x IN s /\ y IN s /\ z IN s ==> x - a % (y - z) IN s`,
270 REWRITE_TAC[VECTOR_ARITH `x - a % (y - z):real^N = x + a % (z - y)`] THEN
271 SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF]);;
273 let AFFINE_DIFFS_SUBSPACE = prove
275 affine s /\ a IN s ==> subspace {x - a | x IN s}`,
276 REWRITE_TAC[subspace; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
277 REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN
278 REWRITE_TAC[VECTOR_ARITH `vec 0:real^N = x - a <=> x = a`;
279 VECTOR_ARITH `x - a + y - a:real^N = z - a <=>
280 z = (a + &1 % (x - a)) + &1 % (y - a)`;
281 VECTOR_ARITH `c % (x - a):real^N = y - a <=>
282 y = a + c % (x - a)`] THEN
283 MESON_TAC[IN_AFFINE_ADD_MUL_DIFF]);;
285 (* ------------------------------------------------------------------------- *)
286 (* Explicit formulations for affine combinations. *)
287 (* ------------------------------------------------------------------------- *)
289 let AFFINE_VSUM = prove
290 (`!s k u x:A->real^N.
291 FINITE k /\ affine s /\ sum k u = &1 /\ (!i. i IN k ==> x i IN s)
292 ==> vsum k (\i. u i % x i) IN s`,
293 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
294 [ASM_REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM; MEMBER_NOT_EMPTY] THEN
295 ASM_CASES_TAC `k:A->bool = {}` THEN ASM_REWRITE_TAC[SUM_CLAUSES] THEN
299 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
300 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
301 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] AFFINE_DIFFS_SUBSPACE) THEN
302 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
303 MP_TAC(ISPECL [`{x - a:real^N | x IN s}`;
304 `(\i. u i % (x i - a)):A->real^N`;
305 `k:A->bool`] SUBSPACE_VSUM) THEN
307 [ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
308 MATCH_MP_TAC SUBSPACE_MUL THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
309 ASM_SIMP_TAC[VSUM_SUB; IN_ELIM_THM; VECTOR_SUB_LDISTRIB; VSUM_RMUL] THEN
310 REWRITE_TAC[VECTOR_ARITH `x - &1 % a:real^N = y - a <=> x = y`] THEN
313 let AFFINE_INDEXED = prove
316 !k u x. (!i:num. 1 <= i /\ i <= k ==> x(i) IN s) /\
318 ==> vsum (1..k) (\i. u(i) % x(i)) IN s`,
319 REPEAT GEN_TAC THEN EQ_TAC THENL
320 [REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN
321 ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG];
322 DISCH_TAC THEN REWRITE_TAC[affine] THEN
323 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
324 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `2`) THEN
325 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then u else v:real`) THEN
326 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then x else y:real^N`) THEN
327 REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG; VSUM_CLAUSES_NUMSEG;
328 NUMSEG_SING; VSUM_SING; SUM_SING] THEN REWRITE_TAC[ARITH] THEN
331 let AFFINE_HULL_INDEXED = prove
332 (`!s. affine hull s =
333 {y:real^N | ?k u x. (!i. 1 <= i /\ i <= k ==> x i IN s) /\
334 (sum (1..k) u = &1) /\
335 (vsum (1..k) (\i. u i % x i) = y)}`,
336 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL
337 [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
338 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
339 MAP_EVERY EXISTS_TAC [`1`; `\i:num. &1`; `\i:num. x:real^N`] THEN
340 ASM_SIMP_TAC[FINITE_RULES; IN_SING; SUM_SING; VECTOR_MUL_LID; VSUM_SING;
341 REAL_POS; NUMSEG_SING];
343 REWRITE_TAC[AFFINE_INDEXED; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
345 REWRITE_TAC[affine; IN_ELIM_THM] THEN
346 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
347 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
348 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
349 [`k1:num`; `u1:num->real`; `x1:num->real^N`;
350 `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN
351 STRIP_TAC THEN EXISTS_TAC `k1 + k2:num` THEN
352 EXISTS_TAC `\i:num. if i <= k1 then u * u1(i) else v * u2(i - k1):real` THEN
353 EXISTS_TAC `\i:num. if i <= k1 then x1(i) else x2(i - k1):real^N` THEN
354 ASM_SIMP_TAC[NUMSEG_ADD_SPLIT; ARITH_RULE `1 <= x + 1 /\ x < x + 1`;
355 IN_NUMSEG; SUM_UNION; VSUM_UNION; FINITE_NUMSEG; DISJOINT_NUMSEG;
356 ARITH_RULE `k1 + 1 <= i ==> ~(i <= k1)`] THEN
357 REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] NUMSEG_OFFSET_IMAGE] THEN
358 ASM_SIMP_TAC[SUM_IMAGE; VSUM_IMAGE; EQ_ADD_LCANCEL; FINITE_NUMSEG] THEN
359 ASM_SIMP_TAC[o_DEF; ADD_SUB2; SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC;
360 FINITE_NUMSEG; REAL_MUL_RID] THEN
361 ASM_MESON_TAC[REAL_LE_MUL; ARITH_RULE
362 `i <= k1 + k2 /\ ~(i <= k1) ==> 1 <= i - k1 /\ i - k1 <= k2`]);;
367 !(s:real^N->bool) (u:real^N->real).
368 FINITE s /\ ~(s = {}) /\ s SUBSET V /\ sum s u = &1
369 ==> vsum s (\x. u x % x) IN V`,
370 GEN_TAC THEN EQ_TAC THENL
371 [REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN
372 ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
373 REWRITE_TAC[affine] THEN DISCH_TAC THEN
374 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
375 STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THENL
376 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
377 ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB;VECTOR_MUL_LID];ALL_TAC] THEN
378 FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N,y}`) THEN
379 DISCH_THEN(MP_TAC o SPEC `\w. if w = x:real^N then u else v:real`) THEN
380 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_RULES; NUMSEG_SING;
381 VSUM_SING; SUM_SING;SUBSET;IN_INSERT;NOT_IN_EMPTY] THEN
384 let AFFINE_EXPLICIT = prove
387 !t u. FINITE t /\ t SUBSET s /\ sum t u = &1
388 ==> vsum t (\x. u(x) % x) IN s`,
389 GEN_TAC THEN REWRITE_TAC[AFFINE] THEN
390 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
391 X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN
392 AP_TERM_TAC THEN ABS_TAC THEN
393 ASM_CASES_TAC `t:real^N->bool = {}` THEN
394 ASM_REWRITE_TAC[SUM_CLAUSES] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
396 let AFFINE_HULL_EXPLICIT = prove
397 (`!(p:real^N -> bool).
399 {y | ?s u. FINITE s /\ ~(s = {}) /\ s SUBSET p /\
400 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
401 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL
402 [REWRITE_TAC[SUBSET;IN_ELIM_THM] THEN
403 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
404 MAP_EVERY EXISTS_TAC [`{x:real^N}`;`\v:real^N. &1:real`] THEN
405 ASM_SIMP_TAC[FINITE_RULES;IN_SING;SUM_SING;VSUM_SING;VECTOR_MUL_LID] THEN
407 REWRITE_TAC[affine;IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
408 EXISTS_TAC `(s UNION s'):real^N->bool` THEN
410 `\a:real^N. (\b:real^N.if (b IN s) then (u * (u' b)) else &0) a +
411 (\b:real^N.if (b IN s') then v * (u'' b) else &0) a` THEN
412 REPEAT CONJ_TAC THENL
413 [ASM_REWRITE_TAC[FINITE_UNION];
415 ASM_REWRITE_TAC[UNION_SUBSET];
416 ASM_SIMP_TAC[REWRITE_RULE[REAL_ARITH `a + b = c + d <=> c = a + b - d`]
417 SUM_INCL_EXCL; GSYM SUM_RESTRICT_SET;
418 SET_RULE `{a | a IN (s:A->bool) /\ a IN s'} = s INTER s'`;
419 SUM_ADD;SUM_LMUL;REAL_MUL_RID;
420 FINITE_INTER;INTER_IDEMPOT] THEN
421 ASM_REWRITE_TAC[SET_RULE `(a INTER b) INTER a = a INTER b`;
422 SET_RULE `(a INTER b) INTER b = a INTER b`;
423 REAL_ARITH `(a + b) + (c + d) - (e + b) = (a + d) + c - e`;
424 REAL_ARITH `a + b - c = a <=> b = c`] THEN
425 AP_TERM_TAC THEN REWRITE_TAC[INTER_COMM];
426 ASM_SIMP_TAC[REWRITE_RULE
427 [VECTOR_ARITH `(a:real^N) + b = c + d <=> c = a + b - d`]
428 VSUM_INCL_EXCL;GSYM VSUM_RESTRICT_SET;
429 SET_RULE `{a | a IN (s:A->bool) /\ a IN s'} = s INTER s'`;
430 VSUM_ADD;FINITE_INTER;INTER_IDEMPOT;VECTOR_ADD_RDISTRIB;
431 GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;
432 MESON[] `(if P then a else b) % (x:real^N) =
433 (if P then a % x else b % x)`;
434 VECTOR_MUL_LZERO;GSYM VSUM_RESTRICT_SET] THEN
435 ASM_REWRITE_TAC[SET_RULE `(a INTER b) INTER a = a INTER b`;
436 SET_RULE `(a INTER b) INTER b = a INTER b`;
438 `((a:real^N) + b) + (c + d) - (e + b) = (a + d) + c - e`;
439 VECTOR_ARITH `(a:real^N) + b - c = a <=> b = c`] THEN
440 AP_TERM_TAC THEN REWRITE_TAC[INTER_COMM]];
441 ASM_CASES_TAC `(p:real^N->bool) = {}` THENL
442 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
443 REWRITE_TAC[SUBSET_EMPTY;EMPTY_SUBSET] THEN ASM SET_TAC[];
445 REWRITE_TAC[AFFINE; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
448 let AFFINE_HULL_EXPLICIT_ALT = prove
449 (`!(p:real^N -> bool).
451 {y | ?s u. FINITE s /\ s SUBSET p /\
452 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
453 GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT] THEN
454 GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM] THEN
455 GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
456 EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
457 POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
458 SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ]);;
460 let AFFINE_HULL_FINITE = prove
462 affine hull s = {y | ?u. sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
463 GEN_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
464 REWRITE_TAC[AFFINE_HULL_EXPLICIT; IN_ELIM_THM] THEN
465 X_GEN_TAC `x:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
466 [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `f:real^N->real`] THEN
468 EXISTS_TAC `\x:real^N. if x IN t then f x else &0` THEN
469 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
470 REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
471 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`];
472 X_GEN_TAC `f:real^N->real` THEN
473 ASM_CASES_TAC `s:real^N->bool = {}` THEN
474 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN STRIP_TAC THEN
475 EXISTS_TAC `support (+) (f:real^N->real) s` THEN
476 EXISTS_TAC `f:real^N->real` THEN
477 MP_TAC(ASSUME `sum s (f:real^N->real) = &1`) THEN
478 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN
479 REWRITE_TAC[iterate] THEN COND_CASES_TAC THEN
480 ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; REAL_OF_NUM_EQ; ARITH] THEN
481 DISCH_THEN(K ALL_TAC) THEN
482 UNDISCH_TAC `sum s (f:real^N->real) = &1` THEN
483 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM SUM_SUPPORT] THEN
484 ASM_CASES_TAC `support (+) (f:real^N->real) s = {}` THEN
485 ASM_SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN
486 DISCH_TAC THEN REWRITE_TAC[SUPPORT_SUBSET] THEN
487 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
488 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
489 REWRITE_TAC[SUPPORT_SUBSET] THEN
490 REWRITE_TAC[support; IN_ELIM_THM; NEUTRAL_REAL_ADD] THEN
491 MESON_TAC[VECTOR_MUL_LZERO]]);;
493 (* ------------------------------------------------------------------------- *)
494 (* Stepping theorems and hence small special cases. *)
495 (* ------------------------------------------------------------------------- *)
497 let AFFINE_HULL_EMPTY = prove
498 (`affine hull {} = {}`,
499 MATCH_MP_TAC HULL_UNIQUE THEN
500 REWRITE_TAC[SUBSET_REFL; AFFINE_EMPTY; EMPTY_SUBSET]);;
502 let AFFINE_HULL_EQ_EMPTY = prove
503 (`!s. (affine hull s = {}) <=> (s = {})`,
504 GEN_TAC THEN EQ_TAC THEN
505 MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; AFFINE_HULL_EMPTY]);;
507 let AFFINE_HULL_FINITE_STEP_GEN = prove
508 (`!P:real^N->real->bool.
509 ((?u. (!x. x IN {} ==> P x (u x)) /\
510 sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=>
511 w = &0 /\ y = vec 0) /\
512 (FINITE(s:real^N->bool) /\
513 (!y. a IN s /\ P a y ==> P a (y / &2)) /\
514 (!x y. a IN s /\ P a x /\ P a y ==> P a (x + y))
515 ==> ((?u. (!x. x IN (a INSERT s) ==> P x (u x)) /\
516 sum (a INSERT s) u = w /\
517 vsum (a INSERT s) (\x. u(x) % x) = y) <=>
518 ?v u. P a v /\ (!x. x IN s ==> P x (u x)) /\
520 vsum s (\x. u(x) % x) = y - v % a))`,
521 GEN_TAC THEN SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; NOT_IN_EMPTY] THEN
522 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_TAC THEN
523 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THENL
524 [ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN EQ_TAC THEN
525 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
526 [X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN
527 EXISTS_TAC `(u:real^N->real) a / &2` THEN
528 EXISTS_TAC `\x:real^N. if x = a then u x / &2 else u x`;
529 MAP_EVERY X_GEN_TAC [`v:real`; `u:real^N->real`] THEN
531 EXISTS_TAC `\x:real^N. if x = a then u x + v else u x`] THEN
532 ASM_SIMP_TAC[] THEN (CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN
533 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
534 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES] THEN
535 ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; VSUM_DELETE] THEN
536 ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`] THEN
537 REWRITE_TAC[SUM_SING; VSUM_SING] THEN
538 (CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);
539 EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
540 [X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN
541 EXISTS_TAC `(u:real^N->real) a` THEN
542 EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[IN_INSERT] THEN
543 REPEAT(FIRST_X_ASSUM(SUBST1_TAC o SYM)) THEN
544 CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC];
545 MAP_EVERY X_GEN_TAC [`v:real`; `u:real^N->real`] THEN
547 EXISTS_TAC `\x:real^N. if x = a then v:real else u x` THEN
548 ASM_SIMP_TAC[IN_INSERT] THEN CONJ_TAC THENL
549 [ASM_MESON_TAC[]; ALL_TAC] THEN
550 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
551 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES] THEN
552 ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; VSUM_DELETE] THEN
553 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ x = a} = {}`] THEN
554 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> s DELETE a = s`] THEN
555 REWRITE_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
556 CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]]]);;
558 let AFFINE_HULL_FINITE_STEP = prove
559 (`((?u. sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=>
560 w = &0 /\ y = vec 0) /\
561 (FINITE(s:real^N->bool)
562 ==> ((?u. sum (a INSERT s) u = w /\
563 vsum (a INSERT s) (\x. u(x) % x) = y) <=>
564 ?v u. sum s u = w - v /\
565 vsum s (\x. u(x) % x) = y - v % a))`,
566 MATCH_ACCEPT_TAC (REWRITE_RULE[]
567 (ISPEC `\x:real^N y:real. T` AFFINE_HULL_FINITE_STEP_GEN)));;
569 let AFFINE_HULL_2 = prove
570 (`!a b. affine hull {a,b} =
571 {u % a + v % b | u + v = &1}`,
572 SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
573 SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
574 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
575 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
576 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
578 let AFFINE_HULL_2_ALT = prove
579 (`!a b. affine hull {a,b} = {a + u % (b - a) | u IN (:real)}`,
580 REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_2] THEN
581 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
582 REWRITE_TAC[IN_ELIM_THM; IN_UNIV; ARITH_RULE `u + v = &1 <=> v = &1 - u`;
583 FORALL_UNWIND_THM2; UNWIND_THM2] THEN
584 CONJ_TAC THEN X_GEN_TAC `u:real` THEN EXISTS_TAC `&1 - u` THEN
587 let AFFINE_HULL_3 = prove
588 (`affine hull {a,b,c} =
589 { u % a + v % b + w % c | u + v + w = &1}`,
590 SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
591 SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
592 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
593 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
594 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
596 (* ------------------------------------------------------------------------- *)
597 (* Some relations between affine hull and subspaces. *)
598 (* ------------------------------------------------------------------------- *)
600 let AFFINE_HULL_INSERT_SUBSET_SPAN = prove
602 affine hull (a INSERT s) SUBSET {a + v | v | v IN span {x - a | x IN s}}`,
603 REPEAT GEN_TAC THEN GEN_REWRITE_TAC I [SUBSET] THEN
604 REWRITE_TAC[AFFINE_HULL_EXPLICIT; SPAN_EXPLICIT; IN_ELIM_THM] THEN
605 REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
607 `(?s u. (?t. P t /\ s = f t) /\ Q s u) <=>
608 (?t u. P t /\ Q (f t) u)`] THEN
610 `(?v. (?s u. P s /\ f s u = v) /\ (x = g a v)) <=>
611 (?s u. ~(P s ==> ~(g a (f s u) = x)))`] THEN
612 SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
613 REWRITE_TAC[o_DEF] THEN X_GEN_TAC `y:real^N` THEN
614 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
615 MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
616 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST1_TAC o SYM)) THEN
618 [`t DELETE (a:real^N)`; `\x. (u:real^N->real)(x + a)`] THEN
619 ASM_SIMP_TAC[FINITE_DELETE; VECTOR_SUB_ADD; SET_RULE
620 `t SUBSET (a INSERT s) ==> t DELETE a SUBSET s`] THEN
621 MATCH_MP_TAC EQ_TRANS THEN
622 EXISTS_TAC `a + vsum t (\x. u x % (x - a)):real^N` THEN CONJ_TAC THENL
623 [AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
624 REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN SET_TAC[];
625 ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; FINITE_DELETE; VSUM_SUB] THEN
626 ASM_REWRITE_TAC[VSUM_RMUL] THEN
627 REWRITE_TAC[VECTOR_ARITH `a + x - &1 % a:real^N = x`]]);;
629 let AFFINE_HULL_INSERT_SPAN = prove
632 ==> affine hull (a INSERT s) =
633 {a + v | v | v IN span {x - a | x IN s}}`,
634 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
635 REWRITE_TAC[AFFINE_HULL_INSERT_SUBSET_SPAN] THEN REWRITE_TAC[SUBSET] THEN
636 REWRITE_TAC[AFFINE_HULL_EXPLICIT; SPAN_EXPLICIT; IN_ELIM_THM] THEN
637 REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
639 `(?s u. (?t. P t /\ s = f t) /\ Q s u) <=>
640 (?t u. P t /\ Q (f t) u)`] THEN
642 `(?v. (?s u. P s /\ f s u = v) /\ (x = g a v)) <=>
643 (?s u. ~(P s ==> ~(g a (f s u) = x)))`] THEN
644 SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
645 REWRITE_TAC[o_DEF] THEN X_GEN_TAC `y:real^N` THEN
646 REWRITE_TAC[NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
647 MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
648 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST1_TAC o SYM)) THEN
650 [`(a:real^N) INSERT t`;
651 `\x. if x = a then &1 - sum t (\x. u(x - a))
652 else (u:real^N->real)(x - a)`] THEN
653 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
654 ASM_CASES_TAC `(a:real^N) IN t` THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
655 ASM_SIMP_TAC[FINITE_INSERT; NOT_INSERT_EMPTY;
656 SET_RULE `s SUBSET t ==> (a INSERT s) SUBSET (a INSERT t)`] THEN
657 SUBGOAL_THEN `!x:real^N. x IN t ==> ~(x = a)` MP_TAC THENL
658 [ASM SET_TAC[]; SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC)] THEN
659 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
660 ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; FINITE_DELETE; VSUM_SUB] THEN
661 ASM_REWRITE_TAC[VSUM_RMUL] THEN VECTOR_ARITH_TAC);;
663 let AFFINE_HULL_SPAN = prove
667 {a + v | v | v IN span {x - a | x | x IN (s DELETE a)}})`,
668 REPEAT STRIP_TAC THEN
669 MP_TAC(ISPECL [`a:real^N`; `s DELETE (a:real^N)`]
670 AFFINE_HULL_INSERT_SPAN) THEN
671 ASM_REWRITE_TAC[IN_DELETE] THEN
672 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
674 let DIFFS_AFFINE_HULL_SPAN = prove
676 a IN s ==> {x - a | x IN affine hull s} = span {x - a | x IN s}`,
677 REPEAT STRIP_TAC THEN
678 FIRST_ASSUM(SUBST1_TAC o MATCH_MP AFFINE_HULL_SPAN) THEN
679 REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID] THEN
680 SIMP_TAC[IMAGE_DELETE_INJ;
681 VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
682 REWRITE_TAC[VECTOR_SUB_REFL; SPAN_DELETE_0]);;
684 let AFFINE_HULL_SING = prove
685 (`!a. affine hull {a} = {a}`,
686 SIMP_TAC[AFFINE_HULL_INSERT_SPAN; NOT_IN_EMPTY] THEN
687 REWRITE_TAC[SET_RULE `{f x | x | F} = {}`; SPAN_EMPTY] THEN
688 REWRITE_TAC[SET_RULE `{f x | x IN {a}} = {f a}`; VECTOR_ADD_RID]);;
690 let AFFINE_HULL_EQ_SING = prove
691 (`!s a:real^N. affine hull s = {a} <=> s = {a}`,
692 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
693 ASM_REWRITE_TAC[AFFINE_HULL_EMPTY] THEN
694 EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[AFFINE_HULL_SING] THEN
695 MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET {a} ==> s = {a}`) THEN
696 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
697 REWRITE_TAC[HULL_SUBSET]);;
699 (* ------------------------------------------------------------------------- *)
701 (* ------------------------------------------------------------------------- *)
703 let convex = new_definition
705 !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
706 ==> (u % x + v % y) IN s`;;
708 let CONVEX_ALT = prove
709 (`convex s <=> !x y u. x IN s /\ y IN s /\ &0 <= u /\ u <= &1
710 ==> ((&1 - u) % x + u % y) IN s`,
711 REWRITE_TAC[convex] THEN
712 MESON_TAC[REAL_ARITH `&0 <= u /\ &0 <= v /\ (u + v = &1)
713 ==> v <= &1 /\ (u = &1 - v)`;
714 REAL_ARITH `u <= &1 ==> &0 <= &1 - u /\ ((&1 - u) + u = &1)`]);;
716 let IN_CONVEX_SET = prove
718 convex s /\ a IN s /\ b IN s /\ &0 <= u /\ u <= &1
719 ==> ((&1 - u) % a + u % b) IN s`,
720 MESON_TAC[CONVEX_ALT]);;
722 let CONVEX_EMPTY = prove
724 REWRITE_TAC[convex; NOT_IN_EMPTY]);;
726 let CONVEX_SING = prove
728 SIMP_TAC[convex; IN_SING; GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]);;
730 let CONVEX_UNIV = prove
731 (`convex(UNIV:real^N->bool)`,
732 REWRITE_TAC[convex; IN_UNIV]);;
734 let CONVEX_INTERS = prove
735 (`(!s. s IN f ==> convex s) ==> convex(INTERS f)`,
736 REWRITE_TAC[convex; IN_INTERS] THEN MESON_TAC[]);;
738 let CONVEX_INTER = prove
739 (`!s t. convex s /\ convex t ==> convex(s INTER t)`,
740 REWRITE_TAC[convex; IN_INTER] THEN MESON_TAC[]);;
742 let CONVEX_HULLS_EQ = prove
743 (`!s t. s SUBSET convex hull t /\ t SUBSET convex hull s
744 ==> convex hull s = convex hull t`,
745 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULLS_EQ THEN
746 ASM_SIMP_TAC[CONVEX_INTERS]);;
748 let CONVEX_HALFSPACE_LE = prove
749 (`!a b. convex {x | a dot x <= b}`,
750 REWRITE_TAC[convex; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN
751 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
752 EXISTS_TAC `(u + v) * b` THEN CONJ_TAC THENL
753 [ASM_MESON_TAC[REAL_ADD_RDISTRIB; REAL_LE_ADD2; REAL_LE_LMUL];
754 ASM_MESON_TAC[REAL_MUL_LID; REAL_LE_REFL]]);;
756 let CONVEX_HALFSPACE_COMPONENT_LE = prove
757 (`!a k. convex {x:real^N | x$k <= a}`,
759 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
761 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
762 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_LE) THEN
763 ASM_SIMP_TAC[DOT_BASIS]);;
765 let CONVEX_HALFSPACE_GE = prove
766 (`!a b. convex {x:real^N | a dot x >= b}`,
768 SUBGOAL_THEN `{x:real^N | a dot x >= b} = {x | --a dot x <= --b}`
769 (fun th -> REWRITE_TAC[th; CONVEX_HALFSPACE_LE]) THEN
770 REWRITE_TAC[EXTENSION; IN_ELIM_THM; DOT_LNEG] THEN REAL_ARITH_TAC);;
772 let CONVEX_HALFSPACE_COMPONENT_GE = prove
773 (`!a k. convex {x:real^N | x$k >= a}`,
775 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
777 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
778 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_GE) THEN
779 ASM_SIMP_TAC[DOT_BASIS]);;
781 let CONVEX_HYPERPLANE = prove
782 (`!a b. convex {x:real^N | a dot x = b}`,
785 `{x:real^N | a dot x = b} = {x | a dot x <= b} INTER {x | a dot x >= b}`
786 (fun th -> SIMP_TAC[th; CONVEX_INTER;
787 CONVEX_HALFSPACE_LE; CONVEX_HALFSPACE_GE]) THEN
788 REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC);;
790 let CONVEX_STANDARD_HYPERPLANE = prove
791 (`!k a. convex {x:real^N | x$k = a}`,
793 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
795 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
796 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HYPERPLANE) THEN
797 ASM_SIMP_TAC[DOT_BASIS]);;
799 let CONVEX_HALFSPACE_LT = prove
800 (`!a b. convex {x | a dot x < b}`,
801 REWRITE_TAC[convex; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN
802 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN
805 let CONVEX_HALFSPACE_COMPONENT_LT = prove
806 (`!a k. convex {x:real^N | x$k < a}`,
808 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
810 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
811 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_LT) THEN
812 ASM_SIMP_TAC[DOT_BASIS]);;
814 let CONVEX_HALFSPACE_GT = prove
815 (`!a b. convex {x | a dot x > b}`,
816 REWRITE_TAC[REAL_ARITH `ax > b <=> --ax < --b`] THEN
817 REWRITE_TAC[GSYM DOT_LNEG; CONVEX_HALFSPACE_LT]);;
819 let CONVEX_HALFSPACE_COMPONENT_GT = prove
820 (`!a k. convex {x:real^N | x$k > a}`,
822 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
824 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
825 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_GT) THEN
826 ASM_SIMP_TAC[DOT_BASIS]);;
828 let CONVEX_POSITIVE_ORTHANT = prove
829 (`convex {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
831 SIMP_TAC[convex; IN_ELIM_THM; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
832 REAL_LE_MUL; REAL_LE_ADD]);;
834 let LIMPT_OF_CONVEX = prove
836 convex s /\ x IN s ==> (x limit_point_of s <=> ~(s = {x}))`,
837 REPEAT STRIP_TAC THEN
838 ASM_CASES_TAC `s = {x:real^N}` THEN ASM_REWRITE_TAC[LIMPT_SING] THEN
839 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL
840 [ASM SET_TAC[]; ALL_TAC] THEN
841 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
842 ABBREV_TAC `u = min (&1 / &2) (e / &2 / norm(y - x:real^N))` THEN
843 SUBGOAL_THEN `&0 < u /\ u < &1` STRIP_ASSUME_TAC THENL
844 [EXPAND_TAC "u" THEN REWRITE_TAC[REAL_LT_MIN; REAL_MIN_LT] THEN
845 CONV_TAC REAL_RAT_REDUCE_CONV THEN
846 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ];
848 EXISTS_TAC `(&1 - u) % x + u % y:real^N` THEN REPEAT CONJ_TAC THENL
849 [FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
850 ASM_SIMP_TAC[REAL_LT_IMP_LE];
851 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH
852 `(&1 - u) % x + u % y:real^N = x <=> u % (y - x) = vec 0`] THEN
854 REWRITE_TAC[dist; NORM_MUL; VECTOR_ARITH
855 `((&1 - u) % x + u % y) - x:real^N = u % (y - x)`] THEN
856 ASM_SIMP_TAC[REAL_ARITH `&0 < u ==> abs u = u`] THEN
857 MATCH_MP_TAC(REAL_ARITH `x <= e / &2 /\ &0 < e ==> x < e`) THEN
858 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
859 ASM_REAL_ARITH_TAC]);;
861 let TRIVIAL_LIMIT_WITHIN_CONVEX = prove
863 convex s /\ x IN s ==> (trivial_limit(at x within s) <=> s = {x})`,
864 SIMP_TAC[TRIVIAL_LIMIT_WITHIN; LIMPT_OF_CONVEX]);;
866 (* ------------------------------------------------------------------------- *)
867 (* Some invariance theorems for convex sets. *)
868 (* ------------------------------------------------------------------------- *)
870 let CONVEX_TRANSLATION_EQ = prove
871 (`!a:real^N s. convex (IMAGE (\x. a + x) s) <=> convex s`,
872 REWRITE_TAC[CONVEX_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
873 REWRITE_TAC[IN_IMAGE; UNWIND_THM1; VECTOR_ARITH
874 `(&1 - u) % (a + x) + u % (a + y) = a + z <=> (&1 - u) % x + u % y = z`]);;
876 add_translation_invariants [CONVEX_TRANSLATION_EQ];;
878 let CONVEX_TRANSLATION = prove
879 (`!s a:real^N. convex s ==> convex (IMAGE (\x. a + x) s)`,
880 REWRITE_TAC[CONVEX_TRANSLATION_EQ]);;
882 let CONVEX_LINEAR_IMAGE = prove
883 (`!f s. convex s /\ linear f ==> convex(IMAGE f s)`,
884 REWRITE_TAC[convex; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
885 REWRITE_TAC[IN_IMAGE; linear] THEN MESON_TAC[]);;
887 let CONVEX_LINEAR_IMAGE_EQ = prove
888 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
889 ==> (convex (IMAGE f s) <=> convex s)`,
890 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONVEX_LINEAR_IMAGE));;
892 add_linear_invariants [CONVEX_LINEAR_IMAGE_EQ];;
894 (* ------------------------------------------------------------------------- *)
895 (* Explicit expressions for convexity in terms of arbitrary sums. *)
896 (* ------------------------------------------------------------------------- *)
898 let CONVEX_VSUM = prove
899 (`!s k u x:A->real^N.
900 FINITE k /\ convex s /\ sum k u = &1 /\
901 (!i. i IN k ==> &0 <= u i /\ x i IN s)
902 ==> vsum k (\i. u i % x i) IN s`,
903 GEN_TAC THEN ASM_CASES_TAC `convex(s:real^N->bool)` THEN
904 ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
905 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
906 SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FORALL_IN_INSERT] THEN
907 CONV_TAC REAL_RAT_REDUCE_CONV THEN
908 MAP_EVERY X_GEN_TAC [`i:A`; `k:A->bool`] THEN
909 GEN_REWRITE_TAC (BINOP_CONV o DEPTH_CONV) [RIGHT_IMP_FORALL_THM] THEN
910 REWRITE_TAC[IMP_IMP] THEN STRIP_TAC THEN
911 MAP_EVERY X_GEN_TAC [`u:A->real`; `x:A->real^N`] THEN
912 ASM_CASES_TAC `(u:A->real) i = &1` THENL
913 [ASM_REWRITE_TAC[REAL_ARITH `&1 + a = &1 <=> a = &0`] THEN
915 SUBGOAL_THEN `vsum k (\i:A. u i % x(i):real^N) = vec 0`
916 (fun th -> ASM_SIMP_TAC[th; VECTOR_ADD_RID; VECTOR_MUL_LID]) THEN
917 MATCH_MP_TAC VSUM_EQ_0 THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
918 REPEAT STRIP_TAC THEN DISJ1_TAC THEN
919 ASM_MESON_TAC[SUM_POS_EQ_0];
921 FIRST_X_ASSUM(MP_TAC o SPEC `\j:A. u(j) / (&1 - u(i))`) THEN
922 ASM_REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
923 ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
924 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN
925 SUBGOAL_THEN `&0 < &1 - u(i:A)` ASSUME_TAC THENL
926 [ASM_MESON_TAC[SUM_POS_LE; REAL_ADD_SYM; REAL_ARITH
927 `&0 <= a /\ &0 <= b /\ b + a = &1 /\ ~(a = &1) ==> &0 < &1 - a`];
929 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE] THEN
930 ASM_SIMP_TAC[REAL_EQ_LDIV_EQ; REAL_MUL_LID; REAL_EQ_SUB_LADD] THEN
931 DISCH_TAC THEN ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN
932 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
933 DISCH_THEN(MP_TAC o SPECL
934 [`vsum k (\j. (u j / (&1 - u(i:A))) % x(j) :real^N)`;
935 `x(i:A):real^N`; `&1 - u(i:A)`; `u(i:A):real`]) THEN
936 REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
937 ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
938 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN
939 REWRITE_TAC[VECTOR_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN
940 ASM_SIMP_TAC[REAL_LT_IMP_LE; VSUM_LMUL] THEN
941 CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; REAL_ARITH_TAC] THEN
942 ASM_MESON_TAC[REAL_ADD_SYM]]);;
944 let CONVEX_VSUM_STRONG = prove
945 (`!s k u x:A->real^N.
949 (!i. i IN k ==> &0 <= u i /\ (u i = &0 \/ x i IN s))
950 ==> vsum k (\i. u i % x i) IN s`,
951 REPEAT STRIP_TAC THEN
953 `vsum k (\i. u i % (x:A->real^N) i) =
954 vsum {i | i IN k /\ ~(u i = &0)} (\i. u i % x i)`
956 [MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
958 MATCH_MP_TAC CONVEX_VSUM THEN
959 ASM_SIMP_TAC[FINITE_RESTRICT; IN_ELIM_THM] THEN
960 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
961 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
962 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM SET_TAC[]]);;
964 let CONVEX_INDEXED = prove
967 !k u x. (!i:num. 1 <= i /\ i <= k ==> &0 <= u(i) /\ x(i) IN s) /\
969 ==> vsum (1..k) (\i. u(i) % x(i)) IN s`,
970 REPEAT GEN_TAC THEN EQ_TAC THENL
971 [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN
972 ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG];
973 DISCH_TAC THEN REWRITE_TAC[convex] THEN
974 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
975 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `2`) THEN
976 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then u else v:real`) THEN
977 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then x else y:real^N`) THEN
978 REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG; VSUM_CLAUSES_NUMSEG;
979 NUMSEG_SING; VSUM_SING; SUM_SING] THEN REWRITE_TAC[ARITH] THEN
982 let CONVEX_EXPLICIT = prove
985 !t u. FINITE t /\ t SUBSET s /\ (!x. x IN t ==> &0 <= u x) /\
987 ==> vsum t (\x. u(x) % x) IN s`,
988 REPEAT GEN_TAC THEN EQ_TAC THENL
989 [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN
990 ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
991 DISCH_TAC THEN REWRITE_TAC[convex] THEN
992 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
993 ASM_CASES_TAC `x:real^N = y` THENL
994 [ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]; ALL_TAC] THEN
995 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N,y}`) THEN
996 DISCH_THEN(MP_TAC o SPEC `\z:real^N. if z = x then u else v:real`) THEN
997 ASM_SIMP_TAC[FINITE_INSERT; FINITE_RULES; SUM_CLAUSES; VSUM_CLAUSES;
999 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; REAL_ADD_RID; SUBSET] THEN
1000 REWRITE_TAC[VECTOR_ADD_RID] THEN ASM_MESON_TAC[]]);;
1005 !(s:real^N->bool) (u:real^N->real).
1006 FINITE s /\ ~(s = {}) /\ s SUBSET V /\
1007 (!x. x IN s ==> &0 <= u x) /\ sum s u = &1
1008 ==> vsum s (\x. u x % x) IN V`,
1009 GEN_TAC THEN REWRITE_TAC[CONVEX_EXPLICIT] THEN
1010 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
1011 X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN
1012 AP_TERM_TAC THEN ABS_TAC THEN
1013 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1014 ASM_REWRITE_TAC[SUM_CLAUSES] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
1016 let CONVEX_FINITE = prove
1020 !u. (!x. x IN s ==> &0 <= u x) /\
1022 ==> vsum s (\x. u(x) % x) IN s)`,
1023 REPEAT STRIP_TAC THEN REWRITE_TAC[CONVEX_EXPLICIT] THEN
1024 EQ_TAC THENL [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN
1025 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
1027 FIRST_X_ASSUM(MP_TAC o SPEC `\x:real^N. if x IN t then u x else &0`) THEN
1028 ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET] THEN
1029 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
1030 ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL; GSYM VSUM_RESTRICT_SET] THEN
1031 ASM_SIMP_TAC[COND_ID; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`]);;
1033 let AFFINE_PCROSS = prove
1034 (`!s:real^M->bool t:real^N->bool.
1035 affine s /\ affine t ==> affine(s PCROSS t)`,
1036 REWRITE_TAC[affine; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1037 SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN
1038 SIMP_TAC[PASTECART_IN_PCROSS]);;
1040 let AFFINE_PCROSS_EQ = prove
1041 (`!s:real^M->bool t:real^N->bool.
1042 affine(s PCROSS t) <=> s = {} \/ t = {} \/ affine s /\ affine t`,
1044 ASM_CASES_TAC `s:real^M->bool = {}` THEN
1045 ASM_REWRITE_TAC[PCROSS_EMPTY; AFFINE_EMPTY] THEN
1046 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1047 ASM_REWRITE_TAC[PCROSS_EMPTY; AFFINE_EMPTY] THEN
1048 EQ_TAC THEN REWRITE_TAC[AFFINE_PCROSS] THEN REPEAT STRIP_TAC THENL
1049 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
1050 `(s:real^M->bool) PCROSS (t:real^N->bool)`] AFFINE_LINEAR_IMAGE) THEN
1051 ASM_REWRITE_TAC[LINEAR_FSTCART];
1052 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
1053 `(s:real^M->bool) PCROSS (t:real^N->bool)`] AFFINE_LINEAR_IMAGE) THEN
1054 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
1055 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1056 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
1057 FSTCART_PASTECART; SNDCART_PASTECART] THEN
1060 let CONVEX_PCROSS = prove
1061 (`!s:real^M->bool t:real^N->bool.
1062 convex s /\ convex t ==> convex(s PCROSS t)`,
1063 REWRITE_TAC[convex; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1064 SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN
1065 SIMP_TAC[PASTECART_IN_PCROSS]);;
1067 let CONVEX_PCROSS_EQ = prove
1068 (`!s:real^M->bool t:real^N->bool.
1069 convex(s PCROSS t) <=> s = {} \/ t = {} \/ convex s /\ convex t`,
1071 ASM_CASES_TAC `s:real^M->bool = {}` THEN
1072 ASM_REWRITE_TAC[PCROSS_EMPTY; CONVEX_EMPTY] THEN
1073 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1074 ASM_REWRITE_TAC[PCROSS_EMPTY; CONVEX_EMPTY] THEN
1075 EQ_TAC THEN REWRITE_TAC[CONVEX_PCROSS] THEN REPEAT STRIP_TAC THENL
1076 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
1077 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_LINEAR_IMAGE) THEN
1078 ASM_REWRITE_TAC[LINEAR_FSTCART];
1079 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
1080 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_LINEAR_IMAGE) THEN
1081 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
1082 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1083 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
1084 FSTCART_PASTECART; SNDCART_PASTECART] THEN
1087 (* ------------------------------------------------------------------------- *)
1088 (* Conic sets and conic hull. *)
1089 (* ------------------------------------------------------------------------- *)
1091 let conic = new_definition
1092 `conic s <=> !x c. x IN s /\ &0 <= c ==> (c % x) IN s`;;
1094 let SUBSPACE_IMP_CONIC = prove
1095 (`!s. subspace s ==> conic s`,
1096 SIMP_TAC[subspace; conic]);;
1098 let CONIC_EMPTY = prove
1100 REWRITE_TAC[conic; NOT_IN_EMPTY]);;
1102 let CONIC_UNIV = prove
1103 (`conic (UNIV:real^N->bool)`,
1104 REWRITE_TAC[conic; IN_UNIV]);;
1106 let CONIC_INTERS = prove
1107 (`(!s. s IN f ==> conic s) ==> conic(INTERS f)`,
1108 REWRITE_TAC[conic; IN_INTERS] THEN MESON_TAC[]);;
1110 let CONIC_LINEAR_IMAGE = prove
1111 (`!f s. conic s /\ linear f ==> conic(IMAGE f s)`,
1112 REWRITE_TAC[conic; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
1113 REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[LINEAR_CMUL]);;
1115 let CONIC_LINEAR_IMAGE_EQ = prove
1116 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
1117 ==> (conic (IMAGE f s) <=> conic s)`,
1118 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONIC_LINEAR_IMAGE));;
1120 add_linear_invariants [CONIC_LINEAR_IMAGE_EQ];;
1122 let CONIC_CONIC_HULL = prove
1123 (`!s. conic(conic hull s)`,
1124 SIMP_TAC[P_HULL; CONIC_INTERS]);;
1126 let CONIC_HULL_EQ = prove
1127 (`!s. (conic hull s = s) <=> conic s`,
1128 SIMP_TAC[HULL_EQ; CONIC_INTERS]);;
1130 let CONIC_NEGATIONS = prove
1131 (`!s. conic s ==> conic (IMAGE (--) s)`,
1132 REWRITE_TAC[conic; RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN
1133 REWRITE_TAC[IN_IMAGE; VECTOR_MUL_RNEG] THEN MESON_TAC[]);;
1135 let CONIC_SPAN = prove
1136 (`!s. conic(span s)`,
1137 SIMP_TAC[SUBSPACE_IMP_CONIC; SUBSPACE_SPAN]);;
1139 let CONIC_HULL_EXPLICIT = prove
1140 (`!s:real^N->bool. conic hull s = {c % x | &0 <= c /\ x IN s}`,
1141 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN
1142 REWRITE_TAC[conic; SUBSET; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
1143 REWRITE_TAC[FORALL_IN_GSPEC] THEN
1144 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; IN_ELIM_THM] THEN
1145 REPEAT CONJ_TAC THENL
1146 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
1147 MAP_EVERY EXISTS_TAC [`&1`; `x:real^N`] THEN
1148 ASM_SIMP_TAC[REAL_POS; VECTOR_MUL_LID];
1149 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MESON_TAC[REAL_LE_MUL];
1152 let CONIC_HULL_LINEAR_IMAGE = prove
1153 (`!f s. linear f ==> conic hull (IMAGE f s) = IMAGE f (conic hull s)`,
1154 REPEAT GEN_TAC THEN REWRITE_TAC[CONIC_HULL_EXPLICIT] THEN
1155 REWRITE_TAC[SET_RULE `IMAGE f {c % x | P c x} = {f(c % x) | P c x}`] THEN
1156 REWRITE_TAC[SET_RULE `{c % x | &0 <= c /\ x IN IMAGE f s} =
1157 {c % f(x) | &0 <= c /\ x IN s}`] THEN
1158 DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]));;
1160 add_linear_invariants [CONIC_HULL_LINEAR_IMAGE];;
1162 let CONVEX_CONIC_HULL = prove
1163 (`!s:real^N->bool. convex s ==> convex (conic hull s)`,
1164 REWRITE_TAC[CONIC_HULL_EXPLICIT] THEN
1165 REWRITE_TAC[CONVEX_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1166 REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM; IMP_IMP] THEN
1167 X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN
1168 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN
1169 MAP_EVERY X_GEN_TAC [`d:real`; `y:real^N`] THEN STRIP_TAC THEN
1170 X_GEN_TAC `u:real` THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
1171 ASM_CASES_TAC `(&1 - u) * c = &0` THENL
1172 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
1173 ASM_MESON_TAC[REAL_LE_MUL];
1175 SUBGOAL_THEN `&0 < (&1 - u) * c + u * d` ASSUME_TAC THENL
1176 [MATCH_MP_TAC REAL_LTE_ADD THEN ASM_REWRITE_TAC[REAL_LT_LE] THEN
1177 CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC;
1179 EXISTS_TAC `(&1 - u) * c + u * d:real` THEN
1180 EXISTS_TAC `((&1 - u) * c) / ((&1 - u) * c + u * d) % x +
1181 (u * d) / ((&1 - u) * c + u * d) % y:real^N` THEN
1182 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN
1183 ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
1184 ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL; REAL_SUB_LE] THEN
1185 ASM_SIMP_TAC[REAL_FIELD
1186 `&0 < u + v ==> u / (u + v) = &1 - (v / (u + v))`] THEN
1187 RULE_ASSUM_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]) THEN
1188 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
1189 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ] THEN
1190 ASM_SIMP_TAC[REAL_MUL_LZERO; REAL_LE_MUL; REAL_MUL_LID; REAL_LE_ADDL;
1193 let CONIC_HALFSPACE_LE = prove
1194 (`!a. conic {x | a dot x <= &0}`,
1195 REWRITE_TAC[conic; IN_ELIM_THM; DOT_RMUL] THEN
1196 REWRITE_TAC[REAL_ARITH `a <= &0 <=> &0 <= --a`] THEN
1197 SIMP_TAC[GSYM REAL_MUL_RNEG; REAL_LE_MUL]);;
1199 let CONIC_HALFSPACE_GE = prove
1200 (`!a. conic {x | a dot x >= &0}`,
1201 SIMP_TAC[conic; IN_ELIM_THM; DOT_RMUL; real_ge; REAL_LE_MUL]);;
1203 let CONIC_HULL_EMPTY = prove
1204 (`conic hull {} = {}`,
1205 MATCH_MP_TAC HULL_UNIQUE THEN
1206 REWRITE_TAC[SUBSET_REFL; CONIC_EMPTY; EMPTY_SUBSET]);;
1208 let CONIC_CONTAINS_0 = prove
1209 (`!s:real^N->bool. conic s ==> (vec 0 IN s <=> ~(s = {}))`,
1210 REPEAT STRIP_TAC THEN EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
1211 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
1212 DISCH_THEN(X_CHOOSE_TAC `x:real^N`) THEN
1213 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [conic]) THEN
1214 DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `&0`]) THEN
1215 ASM_REWRITE_TAC[REAL_POS; VECTOR_MUL_LZERO]);;
1217 let CONIC_HULL_EQ_EMPTY = prove
1218 (`!s. (conic hull s = {}) <=> (s = {})`,
1219 GEN_TAC THEN EQ_TAC THEN
1220 MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; CONIC_HULL_EMPTY]);;
1222 let CONIC_SUMS = prove
1223 (`!s t. conic s /\ conic t ==> conic {x + y:real^N | x IN s /\ y IN t}`,
1224 REWRITE_TAC[conic; IN_ELIM_THM] THEN
1225 MESON_TAC[VECTOR_ADD_LDISTRIB]);;
1227 let CONIC_PCROSS = prove
1228 (`!s:real^M->bool t:real^N->bool.
1229 conic s /\ conic t ==> conic(s PCROSS t)`,
1230 REWRITE_TAC[conic; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1231 SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN
1232 SIMP_TAC[PASTECART_IN_PCROSS]);;
1234 let CONIC_PCROSS_EQ = prove
1235 (`!s:real^M->bool t:real^N->bool.
1236 conic(s PCROSS t) <=> s = {} \/ t = {} \/ conic s /\ conic t`,
1238 ASM_CASES_TAC `s:real^M->bool = {}` THEN
1239 ASM_REWRITE_TAC[PCROSS_EMPTY; CONIC_EMPTY] THEN
1240 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1241 ASM_REWRITE_TAC[PCROSS_EMPTY; CONIC_EMPTY] THEN
1242 EQ_TAC THEN REWRITE_TAC[CONIC_PCROSS] THEN REPEAT STRIP_TAC THENL
1243 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
1244 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONIC_LINEAR_IMAGE) THEN
1245 ASM_REWRITE_TAC[LINEAR_FSTCART];
1246 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
1247 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONIC_LINEAR_IMAGE) THEN
1248 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
1249 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1250 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
1251 FSTCART_PASTECART; SNDCART_PASTECART] THEN
1254 let CONIC_POSITIVE_ORTHANT = prove
1255 (`conic {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i}`,
1256 SIMP_TAC[conic; IN_ELIM_THM; REAL_LE_MUL; VECTOR_MUL_COMPONENT]);;
1258 let SEPARATE_CLOSED_CONES = prove
1259 (`!c d:real^N->bool.
1260 conic c /\ closed c /\ conic d /\ closed d /\ c INTER d SUBSET {vec 0}
1262 !x y. x IN c /\ y IN d
1263 ==> dist(x,y) >= e * max (norm x) (norm y)`,
1266 conic c /\ closed c /\ conic d /\ closed d /\ c INTER d SUBSET {vec 0}
1268 !x y. x IN c /\ y IN d ==> dist(x,y)
1271 [REPEAT STRIP_TAC THEN REWRITE_TAC[real_ge] THEN
1272 MP_TAC(ISPECL [`c INTER sphere(vec 0:real^N,&1)`; `d:real^N->bool`]
1273 SEPARATE_COMPACT_CLOSED) THEN
1274 ASM_SIMP_TAC[CLOSED_INTER_COMPACT; COMPACT_SPHERE] THEN ANTS_TAC THENL
1275 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
1276 `c INTER d SUBSET {a} ==> ~(a IN s) ==> (c INTER s) INTER d = {}`)) THEN
1277 REWRITE_TAC[IN_SPHERE_0; NORM_0] THEN REAL_ARITH_TAC;
1278 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
1279 REWRITE_TAC[IN_INTER; IN_SPHERE_0] THEN STRIP_TAC THEN
1280 ASM_REWRITE_TAC[] THEN
1281 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1282 ASM_CASES_TAC `x:real^N = vec 0` THEN
1283 ASM_REWRITE_TAC[DIST_POS_LE; REAL_MUL_RZERO; NORM_0] THEN
1284 FIRST_X_ASSUM(MP_TAC o SPECL
1285 [`inv(norm x) % x:real^N`; `inv(norm(x:real^N)) % y:real^N`]) THEN
1286 REWRITE_TAC[dist; NORM_MUL; GSYM VECTOR_SUB_LDISTRIB] THEN
1287 REWRITE_TAC[REAL_ARITH `abs x * a = a * abs x`] THEN
1288 REWRITE_TAC[REAL_ABS_INV; GSYM real_div; REAL_ABS_NORM] THEN
1289 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
1290 DISCH_THEN MATCH_MP_TAC THEN
1291 ASM_SIMP_TAC[REAL_DIV_REFL; NORM_EQ_0] THEN
1292 RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN
1293 CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
1294 ASM_SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE]];
1295 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
1296 MP_TAC(SPECL [`c:real^N->bool`; `d:real^N->bool`] th) THEN
1297 MP_TAC(SPECL [`d:real^N->bool`; `c:real^N->bool`] th)) THEN
1298 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN
1299 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; real_ge] THEN
1300 X_GEN_TAC `d:real` THEN STRIP_TAC THEN
1301 X_GEN_TAC `e:real` THEN STRIP_TAC THEN
1302 EXISTS_TAC `min d e:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
1303 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1304 REWRITE_TAC[real_max] THEN COND_CASES_TAC THEN
1305 MATCH_MP_TAC REAL_LE_TRANS THENL
1306 [EXISTS_TAC `d * norm(y:real^N)` THEN ONCE_REWRITE_TAC[DIST_SYM];
1307 EXISTS_TAC `e * norm(x:real^N)`] THEN
1308 ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_RMUL THEN NORM_ARITH_TAC]);;
1310 let CONTINUOUS_ON_COMPACT_SURFACE_PROJECTION = prove
1311 (`!s:real^N->bool v d:real^N->real.
1312 compact s /\ s SUBSET (v DELETE (vec 0)) /\ conic v /\
1313 (!x k. x IN v DELETE (vec 0) ==> (&0 < k /\ (k % x) IN s <=> d x = k))
1314 ==> (\x. d x % x) continuous_on (v DELETE (vec 0))`,
1316 (`!s:real^N->real^N p srf:real^N->bool pnc.
1317 compact srf /\ srf SUBSET pnc /\
1318 IMAGE s pnc SUBSET srf /\ (!x. x IN srf ==> s x = x) /\
1319 p continuous_on pnc /\
1320 (!x. x IN pnc ==> s(p x) = s x /\ p(s x) = p x)
1321 ==> s continuous_on pnc`,
1322 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN
1323 EXISTS_TAC `(s:real^N->real^N) o (p:real^N->real^N)` THEN
1324 CONJ_TAC THENL [ASM_SIMP_TAC[o_DEF]; ALL_TAC] THEN
1325 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN
1326 SUBGOAL_THEN `IMAGE (p:real^N->real^N) pnc = IMAGE p srf` SUBST1_TAC THENL
1328 MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[] THEN
1329 CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]]) in
1330 REWRITE_TAC[conic; IN_DELETE; SUBSET] THEN
1331 REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN
1332 MAP_EVERY EXISTS_TAC [`\x:real^N. inv(norm x) % x`; `s:real^N->bool`] THEN
1333 ASM_REWRITE_TAC[] THEN
1334 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
1335 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
1336 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN CONJ_TAC THENL
1337 [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; CONTINUOUS_ON_ID] THEN
1338 MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
1339 SIMP_TAC[IN_DELETE; NORM_EQ_0; SIMP_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM];
1340 REWRITE_TAC[IN_UNIV; IN_DELETE]] THEN
1342 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
1343 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `&1`]) THEN
1344 ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_LT_01; IN_DELETE] THEN
1345 ASM_MESON_TAC[VECTOR_MUL_LID; SUBSET; IN_DELETE];
1347 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN CONJ_TAC THENL
1348 [FIRST_ASSUM(MP_TAC o SPECL
1349 [`inv(norm x) % x:real^N`; `norm x * (d:real^N->real) x`]) THEN
1350 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `(d:real^N->real) x`]) THEN
1351 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0] THEN STRIP_TAC THEN
1352 ASM_SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE; REAL_LT_MUL; NORM_POS_LT] THEN
1353 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; NORM_EQ_0; REAL_FIELD
1354 `~(n = &0) ==> (n * d) * inv n = d`];
1355 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `(d:real^N->real) x`]) THEN
1356 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
1357 ASM_SIMP_TAC[NORM_MUL; VECTOR_MUL_ASSOC; REAL_INV_MUL] THEN
1358 ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN
1359 ASM_SIMP_TAC[REAL_FIELD `&0 < x ==> (inv(x) * y) * x = y`]]);;
1361 (* ------------------------------------------------------------------------- *)
1362 (* Affine dependence and consequential theorems (from Lars Schewe). *)
1363 (* ------------------------------------------------------------------------- *)
1365 let affine_dependent = new_definition
1366 `affine_dependent (s:real^N -> bool) <=>
1367 ?x. x IN s /\ x IN (affine hull (s DELETE x))`;;
1369 let AFFINE_DEPENDENT_EXPLICIT = prove
1370 (`!p. affine_dependent (p:real^N -> bool) <=>
1371 (?s u. FINITE s /\ s SUBSET p /\
1373 (?v. v IN s /\ ~(u v = &0)) /\
1374 vsum s (\v. u v % v) = (vec 0):real^N)`,
1375 X_GEN_TAC `p:real^N->bool` THEN EQ_TAC THENL
1376 [REWRITE_TAC[affine_dependent;AFFINE_HULL_EXPLICIT;
1378 REPEAT STRIP_TAC THEN
1379 EXISTS_TAC `(x:real^N) INSERT s` THEN
1380 EXISTS_TAC `\v:real^N.if v = x then -- &1 else u v` THEN
1381 ASM_SIMP_TAC[FINITE_INSERT;SUM_CLAUSES;VSUM_CLAUSES;INSERT_SUBSET] THEN
1382 REPEAT CONJ_TAC THENL
1384 COND_CASES_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN
1385 ASM_SIMP_TAC[SUM_CASES; SUM_CLAUSES; SET_RULE
1386 `~((x:real^N) IN s) ==> {v | v IN s /\ v = x} = {} /\
1387 {v | v IN s /\ ~(v = x)} = s`] THEN
1389 SET_TAC[REAL_ARITH `~(-- &1 = &0)`];
1390 MP_TAC (SET_RULE `s SUBSET p DELETE (x:real^N) ==> ~(x IN s)`) THEN
1391 ASM_REWRITE_TAC[] THEN
1393 ASM_SIMP_TAC[VECTOR_ARITH
1394 `(-- &1 % (x:real^N)) + a = vec 0 <=> a = x`] THEN
1395 MATCH_MP_TAC EQ_TRANS THEN
1396 EXISTS_TAC `vsum s (\v:real^N. u v % v)` THEN
1398 MATCH_MP_TAC VSUM_EQ THEN
1401 ASM_REWRITE_TAC[]]];
1403 REWRITE_TAC[affine_dependent;AFFINE_HULL_EXPLICIT;IN_ELIM_THM] THEN
1404 REPEAT STRIP_TAC THEN
1405 EXISTS_TAC `v:real^N` THEN
1406 CONJ_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN
1407 EXISTS_TAC `s DELETE (v:real^N)` THEN
1408 EXISTS_TAC `\x:real^N. -- (&1 / (u v)) * u x` THEN
1409 ASM_SIMP_TAC[FINITE_DELETE;SUM_DELETE;VSUM_DELETE_CASES] THEN
1410 ASM_SIMP_TAC[SUM_LMUL;GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;
1411 VECTOR_MUL_RZERO;VECTOR_ARITH `vec 0 - -- a % x = a % x:real^N`;
1412 REAL_MUL_RZERO;REAL_ARITH `&0 - -- a * b = a * b`] THEN
1413 ASM_SIMP_TAC[REAL_FIELD `~(x = &0) ==> &1 / x * x = &1`;
1414 VECTOR_MUL_ASSOC;VECTOR_MUL_LID] THEN
1415 CONJ_TAC THENL [ALL_TAC;ASM SET_TAC[]] THEN
1416 ASM_SIMP_TAC[SET_RULE `v IN s ==> (s DELETE v = {} <=> s = {v})`] THEN
1417 ASM_CASES_TAC `s = {v:real^N}` THEN
1418 ASM_REWRITE_TAC[] THEN
1419 FIRST_X_ASSUM SUBST_ALL_TAC THEN
1420 FIND_ASSUM MP_TAC `sum {v:real^N} u = &0` THEN
1421 REWRITE_TAC[SUM_SING]
1422 THEN ASM_REWRITE_TAC[]);;
1424 let AFFINE_DEPENDENT_EXPLICIT_FINITE = prove
1425 (`!s. FINITE(s:real^N -> bool)
1426 ==> (affine_dependent s <=>
1428 (?v. v IN s /\ ~(u v = &0)) /\
1429 vsum s (\v. u v % v) = vec 0)`,
1430 REPEAT STRIP_TAC THEN REWRITE_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN
1431 EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_REFL]] THEN
1432 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool`
1433 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)) THEN
1434 EXISTS_TAC `\x:real^N. if x IN t then u(x) else &0` THEN
1435 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
1436 ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
1437 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
1440 let AFFINE_DEPENDENT_TRANSLATION_EQ = prove
1441 (`!a s. affine_dependent (IMAGE (\x. a + x) s) <=> affine_dependent s`,
1442 REWRITE_TAC[affine_dependent] THEN GEOM_TRANSLATE_TAC[]);;
1444 add_translation_invariants [AFFINE_DEPENDENT_TRANSLATION_EQ];;
1446 let AFFINE_DEPENDENT_TRANSLATION = prove
1447 (`!s a. affine_dependent s ==> affine_dependent (IMAGE (\x. a + x) s)`,
1448 REWRITE_TAC[AFFINE_DEPENDENT_TRANSLATION_EQ]);;
1450 let AFFINE_DEPENDENT_LINEAR_IMAGE_EQ = prove
1451 (`!f:real^M->real^N s.
1452 linear f /\ (!x y. f x = f y ==> x = y)
1453 ==> (affine_dependent(IMAGE f s) <=> affine_dependent s)`,
1454 REWRITE_TAC[affine_dependent] THEN GEOM_TRANSFORM_TAC[]);;
1456 add_linear_invariants [AFFINE_DEPENDENT_LINEAR_IMAGE_EQ];;
1458 let AFFINE_DEPENDENT_LINEAR_IMAGE = prove
1459 (`!f:real^M->real^N s.
1460 linear f /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
1462 ==> affine_dependent(IMAGE f s)`,
1464 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
1465 REWRITE_TAC[affine_dependent; EXISTS_IN_IMAGE] THEN
1466 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN
1467 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1468 SUBGOAL_THEN `IMAGE (f:real^M->real^N) s DELETE f a = IMAGE f (s DELETE a)`
1469 (fun t -> ASM_SIMP_TAC[FUN_IN_IMAGE; AFFINE_HULL_LINEAR_IMAGE; t]) THEN
1472 let AFFINE_DEPENDENT_MONO = prove
1473 (`!s t:real^N->bool. affine_dependent s /\ s SUBSET t ==> affine_dependent t`,
1474 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
1475 REWRITE_TAC[affine_dependent] THEN MATCH_MP_TAC MONO_EXISTS THEN
1476 X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
1477 FIRST_ASSUM(MP_TAC o MATCH_MP HULL_MONO o SPEC `x:real^N` o MATCH_MP
1478 (SET_RULE `!x. s SUBSET t ==> (s DELETE x) SUBSET (t DELETE x)`)) THEN
1481 let AFFINE_INDEPENDENT_EMPTY = prove
1482 (`~(affine_dependent {})`,
1483 REWRITE_TAC[affine_dependent; NOT_IN_EMPTY]);;
1485 let AFFINE_INDEPENDENT_1 = prove
1486 (`!a:real^N. ~(affine_dependent {a})`,
1487 REWRITE_TAC[affine_dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
1488 REWRITE_TAC[SET_RULE `{a} DELETE a = {}`; AFFINE_HULL_EMPTY; NOT_IN_EMPTY]);;
1490 let AFFINE_INDEPENDENT_2 = prove
1491 (`!a b:real^N. ~(affine_dependent {a,b})`,
1492 REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
1493 [ASM_REWRITE_TAC[INSERT_AC; AFFINE_INDEPENDENT_1];
1494 REWRITE_TAC[affine_dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
1495 ASM_SIMP_TAC[SET_RULE
1496 `~(a = b) ==> {a,b} DELETE a = {b} /\ {a,b} DELETE b = {a}`] THEN
1497 ASM_REWRITE_TAC[AFFINE_HULL_SING; IN_SING]]);;
1499 let AFFINE_INDEPENDENT_SUBSET = prove
1500 (`!s t. ~affine_dependent t /\ s SUBSET t ==> ~affine_dependent s`,
1501 REWRITE_TAC[IMP_CONJ_ALT; CONTRAPOS_THM] THEN
1502 REWRITE_TAC[GSYM IMP_CONJ_ALT; AFFINE_DEPENDENT_MONO]);;
1504 let AFFINE_INDEPENDENT_DELETE = prove
1505 (`!s a. ~affine_dependent s ==> ~affine_dependent(s DELETE a)`,
1507 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_INDEPENDENT_SUBSET) THEN
1510 (* ------------------------------------------------------------------------- *)
1511 (* Coplanarity, and collinearity in terms of affine hull. *)
1512 (* ------------------------------------------------------------------------- *)
1514 let coplanar = new_definition
1515 `coplanar s <=> ?u v w. s SUBSET affine hull {u,v,w}`;;
1517 let COLLINEAR_AFFINE_HULL = prove
1518 (`!s:real^N->bool. collinear s <=> ?u v. s SUBSET affine hull {u,v}`,
1519 GEN_TAC THEN REWRITE_TAC[collinear; AFFINE_HULL_2] THEN EQ_TAC THEN
1520 REWRITE_TAC[SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
1521 REWRITE_TAC[REAL_ARITH `u + v = &1 <=> &1 - u = v`; UNWIND_THM1] THENL
1522 [X_GEN_TAC `u:real^N` THEN DISCH_TAC THEN
1523 ASM_CASES_TAC `s:real^N->bool = {}` THEN
1524 ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
1525 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
1526 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN
1527 DISCH_TAC THEN EXISTS_TAC `x + u:real^N` THEN X_GEN_TAC `y:real^N` THEN
1529 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN
1530 ASM_REWRITE_TAC[VECTOR_ARITH `x - y:real^N = z <=> x = y + z`] THEN
1531 DISCH_THEN(X_CHOOSE_THEN `c:real` SUBST1_TAC) THEN
1532 EXISTS_TAC `&1 + c` THEN VECTOR_ARITH_TAC;
1533 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN
1534 EXISTS_TAC `b - a:real^N` THEN
1535 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1536 FIRST_X_ASSUM(fun th ->
1537 MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN
1538 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
1539 X_GEN_TAC `r:real` THEN DISCH_THEN SUBST1_TAC THEN
1540 X_GEN_TAC `s:real` THEN DISCH_THEN SUBST1_TAC THEN
1541 EXISTS_TAC `s - r:real` THEN VECTOR_ARITH_TAC]);;
1543 let COLLINEAR_IMP_COPLANAR = prove
1544 (`!s. collinear s ==> coplanar s`,
1545 REWRITE_TAC[coplanar; COLLINEAR_AFFINE_HULL] THEN MESON_TAC[INSERT_AC]);;
1547 let COPLANAR_SMALL = prove
1548 (`!s. FINITE s /\ CARD s <= 3 ==> coplanar s`,
1549 GEN_TAC THEN REWRITE_TAC[ARITH_RULE `s <= 3 <=> s <= 2 \/ s = 3`] THEN
1550 REWRITE_TAC[LEFT_OR_DISTRIB; GSYM HAS_SIZE] THEN
1551 DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN
1552 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_SMALL] THEN
1553 CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN REWRITE_TAC[coplanar] THEN
1554 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
1555 STRIP_TAC THEN ASM_REWRITE_TAC[HULL_INC; SUBSET]);;
1557 let COPLANAR_EMPTY = prove
1559 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_EMPTY]);;
1561 let COPLANAR_SING = prove
1562 (`!a. coplanar {a}`,
1563 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_SING]);;
1565 let COPLANAR_2 = prove
1566 (`!a b. coplanar {a,b}`,
1567 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_2]);;
1569 let COPLANAR_3 = prove
1570 (`!a b c. coplanar {a,b,c}`,
1571 REPEAT GEN_TAC THEN MATCH_MP_TAC COPLANAR_SMALL THEN
1572 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_RULES] THEN ARITH_TAC);;
1574 let COLLINEAR_AFFINE_HULL_COLLINEAR = prove
1575 (`!s. collinear(affine hull s) <=> collinear s`,
1576 REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN
1577 MESON_TAC[HULL_HULL; HULL_MONO; HULL_INC; SUBSET]);;
1579 let COPLANAR_AFFINE_HULL_COPLANAR = prove
1580 (`!s. coplanar(affine hull s) <=> coplanar s`,
1581 REWRITE_TAC[coplanar] THEN
1582 MESON_TAC[HULL_HULL; HULL_MONO; HULL_INC; SUBSET]);;
1584 let COPLANAR_TRANSLATION_EQ = prove
1585 (`!a:real^N s. coplanar(IMAGE (\x. a + x) s) <=> coplanar s`,
1586 REWRITE_TAC[coplanar] THEN GEOM_TRANSLATE_TAC[]);;
1588 let COPLANAR_TRANSLATION = prove
1589 (`!a:real^N s. coplanar s ==> coplanar(IMAGE (\x. a + x) s)`,
1590 REWRITE_TAC[COPLANAR_TRANSLATION_EQ]);;
1592 add_translation_invariants [COPLANAR_TRANSLATION_EQ];;
1594 let COPLANAR_LINEAR_IMAGE = prove
1595 (`!f:real^M->real^N s. coplanar s /\ linear f ==> coplanar(IMAGE f s)`,
1596 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
1597 REWRITE_TAC[coplanar; LEFT_IMP_EXISTS_THM] THEN
1598 MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real^M`] THEN STRIP_TAC THEN
1599 MAP_EVERY EXISTS_TAC
1600 [`(f:real^M->real^N) a`; `(f:real^M->real^N) b`; `(f:real^M->real^N) c`] THEN
1601 REWRITE_TAC[SET_RULE `{f a,f b,f c} = IMAGE f {a,b,c}`] THEN
1602 ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE; IMAGE_SUBSET]);;
1604 let COPLANAR_LINEAR_IMAGE_EQ = prove
1605 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
1606 ==> (coplanar (IMAGE f s) <=> coplanar s)`,
1607 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
1609 add_linear_invariants [COPLANAR_LINEAR_IMAGE_EQ];;
1611 let COPLANAR_SUBSET = prove
1612 (`!s t. coplanar t /\ s SUBSET t ==> coplanar s`,
1613 REWRITE_TAC[coplanar] THEN SET_TAC[]);;
1615 let AFFINE_HULL_3_IMP_COLLINEAR = prove
1616 (`!a b c. c IN affine hull {a,b} ==> collinear {a,b,c}`,
1617 ONCE_REWRITE_TAC[GSYM COLLINEAR_AFFINE_HULL_COLLINEAR] THEN
1618 SIMP_TAC[HULL_REDUNDANT_EQ; INSERT_AC] THEN
1619 REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2]);;
1621 let COLLINEAR_3_AFFINE_HULL = prove
1623 ~(a = b) ==> (collinear {a,b,c} <=> c IN affine hull {a,b})`,
1624 REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[AFFINE_HULL_3_IMP_COLLINEAR] THEN
1625 REWRITE_TAC[collinear] THEN
1626 DISCH_THEN(X_CHOOSE_THEN `u:real^N` STRIP_ASSUME_TAC) THEN
1627 FIRST_ASSUM(fun th -> MP_TAC(SPECL [`b:real^N`; `a:real^N`] th) THEN
1628 MP_TAC(SPECL [`c:real^N`; `a:real^N`] th)) THEN
1629 REWRITE_TAC[IN_INSERT; AFFINE_HULL_2; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
1630 REWRITE_TAC[VECTOR_ARITH `a - b:real^N = c <=> a = b + c`] THEN
1631 X_GEN_TAC `x:real` THEN DISCH_TAC THEN X_GEN_TAC `y:real` THEN
1632 ASM_CASES_TAC `y = &0` THEN
1633 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN DISCH_TAC THEN
1634 ASM_REWRITE_TAC[] THEN
1635 MAP_EVERY EXISTS_TAC [`&1 - x / y`; `x / y:real`] THEN
1636 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
1637 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN
1638 ASM_SIMP_TAC[REAL_DIV_RMUL] THEN VECTOR_ARITH_TAC);;
1640 let COLLINEAR_3_EQ_AFFINE_DEPENDENT = prove
1642 collinear{a,b,c} <=>
1643 a = b \/ a = c \/ b = c \/ affine_dependent {a,b,c}`,
1646 ASM_CASES_TAC t THENL [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC])
1647 [`a:real^N = b`; `a:real^N = c`; `b:real^N = c`] THEN
1648 ASM_REWRITE_TAC[affine_dependent] THEN EQ_TAC THENL
1649 [ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN DISCH_TAC THEN
1650 EXISTS_TAC `c:real^N` THEN REWRITE_TAC[IN_INSERT];
1651 REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THENL
1652 [ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`];
1653 ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,a,b}`];
1655 ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL]] THEN
1656 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
1657 `x IN s ==> s SUBSET t ==> x IN t`)) THEN
1658 MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]);;
1660 let AFFINE_DEPENDENT_IMP_COLLINEAR_3 = prove
1661 (`!a b c:real^N. affine_dependent {a,b,c} ==> collinear{a,b,c}`,
1662 REPEAT GEN_TAC THEN REWRITE_TAC[affine_dependent] THEN
1663 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; RIGHT_OR_DISTRIB] THEN
1664 REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2; COLLINEAR_AFFINE_HULL] THEN
1666 [MAP_EVERY EXISTS_TAC [`b:real^N`; `c:real^N`];
1667 MAP_EVERY EXISTS_TAC [`a:real^N`; `c:real^N`];
1668 MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`]] THEN
1669 SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; HULL_INC; IN_INSERT] THEN
1670 POP_ASSUM MP_TAC THEN
1671 MATCH_MP_TAC(SET_RULE `s SUBSET t ==> a IN s ==> a IN t`) THEN
1672 MATCH_MP_TAC HULL_MONO THEN SET_TAC[]);;
1674 let COLLINEAR_3_IN_AFFINE_HULL = prove
1677 ==> (collinear {v0,v1,x} <=> x IN affine hull {v0,v1})`,
1678 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `v0:real^N` THEN
1679 REWRITE_TAC[COLLINEAR_LEMMA; AFFINE_HULL_2] THEN REPEAT STRIP_TAC THEN
1680 ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; IN_ELIM_THM] THEN
1681 ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[] THENL
1682 [MAP_EVERY EXISTS_TAC [`&1`; `&0`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
1684 MESON_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`]]);;
1686 (* ------------------------------------------------------------------------- *)
1687 (* A general lemma. *)
1688 (* ------------------------------------------------------------------------- *)
1690 let CONVEX_CONNECTED = prove
1691 (`!s:real^N->bool. convex s ==> connected s`,
1692 REWRITE_TAC[CONVEX_ALT; connected; SUBSET; EXTENSION; IN_INTER;
1693 IN_UNION; NOT_IN_EMPTY; NOT_FORALL_THM; NOT_EXISTS_THM] THEN
1694 GEN_TAC THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
1695 MAP_EVERY (K(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))) (1--4) THEN
1696 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `x1:real^N` STRIP_ASSUME_TAC)
1697 (X_CHOOSE_THEN `x2:real^N` STRIP_ASSUME_TAC)) THEN
1698 MP_TAC(ISPECL [`\u. (&1 - u) % x1 + u % (x2:real^N)`;
1699 `&0`; `&1`; `e1:real^N->bool`; `e2:real^N->bool`]
1700 (REWRITE_RULE[GSYM open_def] CONNECTED_REAL_LEMMA)) THEN
1701 ASM_REWRITE_TAC[NOT_IMP; REAL_SUB_RZERO; VECTOR_MUL_LID; VECTOR_MUL_LZERO;
1702 REAL_SUB_REFL; VECTOR_ADD_RID; VECTOR_ADD_LID; REAL_POS] THEN
1703 REPEAT(CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]]) THEN
1704 REPEAT STRIP_TAC THEN REWRITE_TAC[dist] THEN
1705 REWRITE_TAC[NORM_MUL; VECTOR_ARITH
1706 `((&1 - a) % x + a % y) - ((&1 - b) % x + b % y) = (a - b) % (y - x)`] THEN
1707 MP_TAC(ISPEC `(x2 - x1):real^N` NORM_POS_LE) THEN
1708 REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THENL
1709 [ALL_TAC; ASM_MESON_TAC[REAL_MUL_RZERO; REAL_LT_01]] THEN
1710 EXISTS_TAC `e / norm((x2 - x1):real^N)` THEN
1711 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_DIV]);;
1713 (* ------------------------------------------------------------------------- *)
1714 (* Various topological facts are queued up here, just because they rely on *)
1715 (* CONNECTED_UNIV, which is a trivial consequence of CONVEX_UNIV. It would *)
1716 (* be fairly easy to prove it earlier and move these back to the topology.ml *)
1717 (* file, which is a bit tidier intellectually. *)
1718 (* ------------------------------------------------------------------------- *)
1720 let CONNECTED_UNIV = prove
1721 (`connected (UNIV:real^N->bool)`,
1722 SIMP_TAC[CONVEX_CONNECTED; CONVEX_UNIV]);;
1724 let CONNECTED_COMPONENT_UNIV = prove
1725 (`!x. connected_component(:real^N) x = (:real^N)`,
1726 MESON_TAC[CONNECTED_CONNECTED_COMPONENT_SET; CONNECTED_UNIV; IN_UNIV]);;
1728 let CONNECTED_COMPONENT_EQ_UNIV = prove
1729 (`!s x. connected_component s x = (:real^N) <=> s = (:real^N)`,
1730 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONNECTED_COMPONENT_UNIV] THEN
1731 MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s = UNIV ==> t = UNIV`) THEN
1732 REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]);;
1735 (`!s. closed s /\ open s <=> s = {} \/ s = (:real^N)`,
1736 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
1737 ASM_REWRITE_TAC[CLOSED_EMPTY; OPEN_EMPTY; CLOSED_UNIV; OPEN_UNIV] THEN
1738 MATCH_MP_TAC(REWRITE_RULE[CONNECTED_CLOPEN] CONNECTED_UNIV) THEN
1739 ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; GSYM CLOSED_IN]);;
1741 let COMPACT_OPEN = prove
1742 (`!s:real^N->bool. compact s /\ open s <=> s = {}`,
1743 MESON_TAC[COMPACT_EMPTY; OPEN_EMPTY; COMPACT_IMP_CLOSED; CLOPEN;
1744 COMPACT_IMP_BOUNDED; NOT_BOUNDED_UNIV]);;
1746 let FRONTIER_NOT_EMPTY = prove
1747 (`!s. ~(s = {}) /\ ~(s = (:real^N)) ==> ~(frontier s = {})`,
1748 REPEAT STRIP_TAC THEN
1749 MP_TAC(ISPECL [`(:real^N)`; `s:real^N->bool`] CONNECTED_INTER_FRONTIER) THEN
1750 REWRITE_TAC[CONNECTED_UNIV] THEN ASM SET_TAC[]);;
1752 let FRONTIER_EQ_EMPTY = prove
1753 (`!s. frontier s = {} <=> s = {} \/ s = (:real^N)`,
1754 MESON_TAC[FRONTIER_NOT_EMPTY; FRONTIER_EMPTY; FRONTIER_UNIV]);;
1756 let EQ_INTERVAL = prove
1758 interval[a,b] = interval[c,d] <=>
1759 interval[a,b] = {} /\ interval[c,d] = {} \/ a = c /\ b = d) /\
1761 interval[a,b] = interval(c,d) <=>
1762 interval[a,b] = {} /\ interval(c,d) = {}) /\
1764 interval(a,b) = interval[c,d] <=>
1765 interval(a,b) = {} /\ interval[c,d] = {}) /\
1767 interval(a,b) = interval(c,d) <=>
1768 interval(a,b) = {} /\ interval(c,d) = {} \/ a = c /\ b = d)`,
1769 REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN
1770 (EQ_TAC THENL [ALL_TAC; STRIP_TAC THEN ASM_REWRITE_TAC[]]) THEN
1771 MATCH_MP_TAC(MESON[]
1772 `(p = {} /\ q = {} ==> r) /\ (~(p = {}) /\ ~(q = {}) ==> p = q ==> r)
1773 ==> p = q ==> r`) THEN
1775 [REWRITE_TAC[INTERVAL_NE_EMPTY; CART_EQ] THEN
1776 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
1777 SIMP_TAC[SUBSET_INTERVAL; GSYM REAL_LE_ANTISYM];
1778 STRIP_TAC THEN MATCH_MP_TAC(MESON[CLOPEN]
1779 `closed s /\ open t /\ ~(s = {}) /\ ~(s = UNIV) ==> ~(s = t)`) THEN
1780 ASM_REWRITE_TAC[CLOSED_INTERVAL; OPEN_INTERVAL; NOT_INTERVAL_UNIV];
1781 STRIP_TAC THEN MATCH_MP_TAC(MESON[CLOPEN]
1782 `closed s /\ open t /\ ~(s = {}) /\ ~(s = UNIV) ==> ~(t = s)`) THEN
1783 ASM_REWRITE_TAC[CLOSED_INTERVAL; OPEN_INTERVAL; NOT_INTERVAL_UNIV];
1784 REWRITE_TAC[INTERVAL_NE_EMPTY; CART_EQ] THEN
1785 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
1786 SIMP_TAC[SUBSET_INTERVAL; GSYM REAL_LE_ANTISYM]]);;
1788 let CLOSED_INTERVAL_EQ = prove
1789 (`(!a b:real^N. closed(interval[a,b])) /\
1790 (!a b:real^N. closed(interval(a,b)) <=> interval(a,b) = {})`,
1791 REWRITE_TAC[CLOSED_INTERVAL] THEN
1792 REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
1793 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN
1794 MP_TAC(ISPEC `interval(a:real^N,b)` CLOPEN) THEN
1795 ASM_REWRITE_TAC[OPEN_INTERVAL] THEN
1796 MESON_TAC[BOUNDED_INTERVAL; NOT_BOUNDED_UNIV]);;
1798 let OPEN_INTERVAL_EQ = prove
1799 (`(!a b:real^N. open(interval[a,b]) <=> interval[a,b] = {}) /\
1800 (!a b:real^N. open(interval(a,b)))`,
1801 REWRITE_TAC[OPEN_INTERVAL] THEN
1802 REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
1803 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN
1804 MP_TAC(ISPEC `interval[a:real^N,b]` CLOPEN) THEN
1805 ASM_REWRITE_TAC[CLOSED_INTERVAL] THEN
1806 MESON_TAC[BOUNDED_INTERVAL; NOT_BOUNDED_UNIV]);;
1808 let COMPACT_INTERVAL_EQ = prove
1809 (`(!a b:real^N. compact(interval[a,b])) /\
1810 (!a b:real^N. compact(interval(a,b)) <=> interval(a,b) = {})`,
1811 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_INTERVAL] THEN
1812 REWRITE_TAC[CLOSED_INTERVAL_EQ]);;
1814 let CONNECTED_CHAIN = prove
1815 (`!f:(real^N->bool)->bool.
1816 (!s. s IN f ==> compact s /\ connected s) /\
1817 (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s)
1818 ==> connected(INTERS f)`,
1819 REPEAT STRIP_TAC THEN
1820 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
1821 ASM_REWRITE_TAC[INTERS_0; CONNECTED_UNIV] THEN
1822 ABBREV_TAC `c:real^N->bool = INTERS f` THEN
1823 SUBGOAL_THEN `compact(c:real^N->bool)` ASSUME_TAC THENL
1824 [EXPAND_TAC "c" THEN MATCH_MP_TAC COMPACT_INTERS THEN ASM SET_TAC[];
1826 ASM_SIMP_TAC[CONNECTED_CLOSED_SET; COMPACT_IMP_CLOSED; NOT_EXISTS_THM] THEN
1827 MAP_EVERY X_GEN_TAC [`a:real^N->bool`; `b:real^N->bool`] THEN STRIP_TAC THEN
1828 MP_TAC(ISPECL [`a:real^N->bool`; `b:real^N->bool`] SEPARATION_NORMAL) THEN
1829 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
1830 MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
1832 SUBGOAL_THEN `?k:real^N->bool. k IN f` STRIP_ASSUME_TAC THENL
1833 [ASM SET_TAC[]; ALL_TAC] THEN
1834 SUBGOAL_THEN `?n:real^N->bool. open n /\ k SUBSET n` MP_TAC THENL
1835 [ASM_MESON_TAC[BOUNDED_SUBSET_BALL; COMPACT_IMP_BOUNDED; OPEN_BALL];
1836 REWRITE_TAC[UNIONS_SUBSET] THEN STRIP_TAC] THEN
1837 MP_TAC(ISPEC `k:real^N->bool` COMPACT_IMP_HEINE_BOREL) THEN
1838 ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o SPEC
1839 `(u UNION v:real^N->bool) INSERT {n DIFF s | s IN f}`) THEN
1840 REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_INSERT; FORALL_IN_IMAGE] THEN
1841 ASM_SIMP_TAC[OPEN_UNION; OPEN_DIFF; COMPACT_IMP_CLOSED; NOT_IMP] THEN
1843 [REWRITE_TAC[UNIONS_INSERT] THEN REWRITE_TAC[SUBSET] THEN
1844 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[IN_UNION] THEN
1845 ASM_CASES_TAC `(x:real^N) IN c` THENL [ASM SET_TAC[]; DISJ2_TAC] THEN
1846 REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN
1847 UNDISCH_TAC `~((x:real^N) IN c)` THEN
1848 SUBST1_TAC(SYM(ASSUME `INTERS f:real^N->bool = c`)) THEN
1849 REWRITE_TAC[IN_INTERS; NOT_FORALL_THM] THEN
1850 MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[];
1852 DISCH_THEN(X_CHOOSE_THEN `g:(real^N->bool)->bool` MP_TAC) THEN
1853 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
1854 REWRITE_TAC[SUBSET_INSERT_DELETE] THEN
1855 SUBGOAL_THEN `FINITE(g DELETE (u UNION v:real^N->bool))` MP_TAC THENL
1856 [ASM_REWRITE_TAC[FINITE_DELETE];
1857 REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`]] THEN
1858 REWRITE_TAC[FINITE_SUBSET_IMAGE] THEN
1859 DISCH_THEN(X_CHOOSE_THEN `f':(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
1861 `?j:real^N->bool. j IN f /\
1862 UNIONS(IMAGE (\s. n DIFF s) f') SUBSET (n DIFF j)`
1863 STRIP_ASSUME_TAC THENL
1864 [ASM_CASES_TAC `f':(real^N->bool)->bool = {}` THEN
1865 ASM_REWRITE_TAC[IMAGE_CLAUSES; UNIONS_0; EMPTY_SUBSET] THENL
1866 [ASM SET_TAC[]; ALL_TAC] THEN
1868 `?j:real^N->bool. j IN f' /\
1869 UNIONS(IMAGE (\s. n DIFF s) f') SUBSET (n DIFF j)`
1870 MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET]] THEN
1872 `!s t:real^N->bool. s IN f' /\ t IN f' ==> s SUBSET t \/ t SUBSET s`
1873 MP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
1874 UNDISCH_TAC `~(f':(real^N->bool)->bool = {})` THEN
1875 UNDISCH_TAC `FINITE(f':(real^N->bool)->bool)` THEN
1876 SPEC_TAC(`f':(real^N->bool)->bool`,`f':(real^N->bool)->bool`) THEN
1877 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[] THEN
1878 REWRITE_TAC[EXISTS_IN_INSERT; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1879 REWRITE_TAC[FORALL_IN_INSERT] THEN POP_ASSUM_LIST(K ALL_TAC) THEN
1880 MAP_EVERY X_GEN_TAC [`i:real^N->bool`; `f:(real^N->bool)->bool`] THEN
1881 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
1882 ASM_REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT; NOT_IN_EMPTY;
1883 UNIONS_0; UNION_EMPTY; SUBSET_REFL] THEN
1884 DISCH_THEN(fun th -> REPEAT DISCH_TAC THEN MP_TAC th) THEN
1885 ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
1886 DISCH_THEN(X_CHOOSE_THEN `j:real^N->bool` STRIP_ASSUME_TAC) THEN
1887 SUBGOAL_THEN `(n DIFF j) SUBSET (n DIFF i) \/
1888 (n DIFF i:real^N->bool) SUBSET (n DIFF j)`
1889 STRIP_ASSUME_TAC THENL
1890 [FIRST_X_ASSUM(MP_TAC o SPEC `j:real^N->bool` o CONJUNCT2) THEN
1892 DISJ1_TAC THEN ASM SET_TAC[];
1893 DISJ2_TAC THEN EXISTS_TAC `j:real^N->bool` THEN ASM SET_TAC[]];
1895 SUBGOAL_THEN `(j INTER k:real^N->bool) SUBSET (u UNION v)` ASSUME_TAC THENL
1896 [MATCH_MP_TAC(SET_RULE
1897 `k SUBSET (u UNION v) UNION (n DIFF j)
1898 ==> (j INTER k) SUBSET (u UNION v)`) THEN
1899 MATCH_MP_TAC SUBSET_TRANS THEN
1900 EXISTS_TAC `UNIONS g :real^N->bool` THEN ASM_REWRITE_TAC[] THEN
1901 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
1902 `UNIONS((u UNION v:real^N->bool) INSERT (g DELETE (u UNION v)))` THEN
1903 CONJ_TAC THENL [MATCH_MP_TAC SUBSET_UNIONS THEN SET_TAC[]; ALL_TAC] THEN
1904 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[UNIONS_INSERT] THEN
1907 SUBGOAL_THEN `connected(j INTER k:real^N->bool)` MP_TAC THENL
1908 [ASM_MESON_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`; INTER_COMM];
1909 REWRITE_TAC[connected] THEN
1910 MAP_EVERY EXISTS_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
1911 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
1913 let CONNECTED_CHAIN_GEN = prove
1914 (`!f:(real^N->bool)->bool.
1915 (!s. s IN f ==> closed s /\ connected s) /\
1916 (?s. s IN f /\ compact s) /\
1917 (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s)
1918 ==> connected(INTERS f)`,
1919 GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
1920 FIRST_X_ASSUM(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
1922 `INTERS f = INTERS(IMAGE (\t:real^N->bool. s INTER t) f)`
1924 [REWRITE_TAC[EXTENSION; INTERS_IMAGE] THEN ASM SET_TAC[];
1925 MATCH_MP_TAC CONNECTED_CHAIN THEN
1926 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
1927 ASM_SIMP_TAC[COMPACT_INTER_CLOSED] THEN
1928 CONJ_TAC THENL [X_GEN_TAC `t:real^N->bool`; ASM SET_TAC[]] THEN
1930 SUBGOAL_THEN `s INTER t:real^N->bool = s \/ s INTER t = t`
1931 (DISJ_CASES_THEN SUBST1_TAC) THEN
1934 let CONNECTED_NEST = prove
1935 (`!s. (!n. compact(s n) /\ connected(s n)) /\
1936 (!m n. m <= n ==> s n SUBSET s m)
1937 ==> connected(INTERS {s n | n IN (:num)})`,
1938 GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONNECTED_CHAIN THEN
1939 ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1940 MATCH_MP_TAC WLOG_LE THEN ASM_MESON_TAC[]);;
1942 let CONNECTED_NEST_GEN = prove
1943 (`!s. (!n. closed(s n) /\ connected(s n)) /\ (?n. compact(s n)) /\
1944 (!m n. m <= n ==> s n SUBSET s m)
1945 ==> connected(INTERS {s n | n IN (:num)})`,
1947 DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN
1948 MATCH_MP_TAC CONNECTED_CHAIN_GEN THEN
1949 ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV; IMP_CONJ; RIGHT_FORALL_IMP_THM;
1950 EXISTS_IN_GSPEC] THEN
1951 MATCH_MP_TAC WLOG_LE THEN ASM_MESON_TAC[]);;
1953 let EQ_BALLS = prove
1954 (`(!a a':real^N r r'.
1955 ball(a,r) = ball(a',r') <=> a = a' /\ r = r' \/ r <= &0 /\ r' <= &0) /\
1957 ball(a,r) = cball(a',r') <=> r <= &0 /\ r' < &0) /\
1959 cball(a,r) = ball(a',r') <=> r < &0 /\ r' <= &0) /\
1961 cball(a,r) = cball(a',r') <=> a = a' /\ r = r' \/ r < &0 /\ r' < &0)`,
1962 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT STRIP_TAC THEN
1964 [ALL_TAC; REWRITE_TAC[EXTENSION; IN_BALL; IN_CBALL] THEN NORM_ARITH_TAC])
1966 [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_BALLS] THEN NORM_ARITH_TAC;
1967 ONCE_REWRITE_TAC[EQ_SYM_EQ];
1969 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_BALLS] THEN NORM_ARITH_TAC] THEN
1970 DISCH_THEN(MP_TAC o MATCH_MP (MESON[CLOPEN; BOUNDED_BALL; NOT_BOUNDED_UNIV]
1971 `s = t ==> closed s /\ open t /\ bounded t ==> s = {} /\ t = {}`)) THEN
1972 REWRITE_TAC[OPEN_BALL; CLOSED_CBALL; BOUNDED_BALL;
1973 BALL_EQ_EMPTY; CBALL_EQ_EMPTY] THEN
1976 let FINITE_CBALL = prove
1977 (`!a:real^N r. FINITE(cball(a,r)) <=> r <= &0`,
1978 REPEAT STRIP_TAC THEN ASM_CASES_TAC `r < &0` THEN
1979 ASM_SIMP_TAC[CBALL_EMPTY; REAL_LT_IMP_LE; FINITE_EMPTY] THEN
1980 ASM_CASES_TAC `r = &0` THEN
1981 ASM_REWRITE_TAC[CBALL_TRIVIAL; FINITE_SING; REAL_LE_REFL] THEN
1982 EQ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
1983 DISCH_THEN(MP_TAC o MATCH_MP EMPTY_INTERIOR_FINITE) THEN
1984 REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC);;
1986 let FINITE_BALL = prove
1987 (`!a:real^N r. FINITE(ball(a,r)) <=> r <= &0`,
1988 REPEAT STRIP_TAC THEN ASM_CASES_TAC `r <= &0` THEN
1989 ASM_SIMP_TAC[BALL_EMPTY; REAL_LT_IMP_LE; FINITE_EMPTY] THEN
1990 DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ]
1991 FINITE_IMP_NOT_OPEN)) THEN
1992 REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC);;
1994 (* ------------------------------------------------------------------------- *)
1995 (* Convex functions into the reals. *)
1996 (* ------------------------------------------------------------------------- *)
1998 parse_as_infix ("convex_on",(12,"right"));;
2000 let convex_on = new_definition
2002 !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
2003 ==> f(u % x + v % y) <= u * f(x) + v * f(y)`;;
2005 let CONVEX_ON_SUBSET = prove
2006 (`!f s t. f convex_on t /\ s SUBSET t ==> f convex_on s`,
2007 REWRITE_TAC[convex_on; SUBSET] THEN MESON_TAC[]);;
2009 let CONVEX_ADD = prove
2010 (`!s f g. f convex_on s /\ g convex_on s ==> (\x. f(x) + g(x)) convex_on s`,
2011 REWRITE_TAC[convex_on; AND_FORALL_THM] THEN
2012 REPEAT(MATCH_MP_TAC MONO_FORALL ORELSE GEN_TAC) THEN
2014 `(b /\ c ==> d) ==> (a ==> b) /\ (a ==> c) ==> a ==> d`) THEN
2017 let CONVEX_CMUL = prove
2018 (`!s c f. &0 <= c /\ f convex_on s ==> (\x. c * f(x)) convex_on s`,
2019 SIMP_TAC[convex_on; REAL_LE_LMUL;
2020 REAL_ARITH `u * c * fx + v * c * fy = c * (u * fx + v * fy)`]);;
2022 let CONVEX_MAX = prove
2023 (`!f g s. f convex_on s /\ g convex_on s
2024 ==> (\x. max (f x) (g x)) convex_on s`,
2025 REWRITE_TAC[convex_on; REAL_MAX_LE] THEN REPEAT STRIP_TAC THEN
2026 FIRST_X_ASSUM(fun th ->
2027 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
2028 ASM_REWRITE_TAC[] THEN
2029 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
2030 MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN
2031 MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REAL_ARITH_TAC);;
2033 let CONVEX_LOWER = prove
2034 (`!f s x y. f convex_on s /\
2035 x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
2036 ==> f(u % x + v % y) <= max (f(x)) (f(y))`,
2037 REWRITE_TAC[convex_on] THEN REPEAT STRIP_TAC THEN
2038 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN
2039 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [SYM th]) THEN
2040 REWRITE_TAC[REAL_ADD_RDISTRIB] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
2041 ASM_MESON_TAC[REAL_LE_ADD2; REAL_LE_LMUL; REAL_MAX_MAX]);;
2043 let CONVEX_LOWER_SEGMENT = prove
2044 (`!f s a b x:real^N.
2045 f convex_on s /\ a IN s /\ b IN s /\ x IN segment[a,b]
2046 ==> f(x) <= max (f a) (f b)`,
2047 REWRITE_TAC[IN_SEGMENT] THEN REPEAT STRIP_TAC THEN
2048 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_LOWER THEN
2049 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
2051 let CONVEX_LOCAL_GLOBAL_MINIMUM = prove
2053 f convex_on s /\ x IN t /\ open t /\ t SUBSET s /\
2054 (!y. y IN t ==> f(x) <= f(y))
2055 ==> !y. y IN s ==> f(x) <= f(y)`,
2056 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
2057 SUBGOAL_THEN `&0 < dist(x:real^N,y)` ASSUME_TAC THENL
2058 [ASM_MESON_TAC[DIST_POS_LT; REAL_LT_REFL]; ALL_TAC] THEN
2059 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
2060 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
2061 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
2062 MP_TAC(SPECL [`&1`; `e / dist(x:real^N,y)`] REAL_DOWN2) THEN
2063 ANTS_TAC THENL [ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_01]; ALL_TAC] THEN
2064 DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
2065 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
2067 SPECL [`x:real^N`; `y:real^N`; `&1 - u`; `u:real`]) THEN
2069 [ASM_SIMP_TAC[REAL_SUB_ADD; REAL_SUB_LE; REAL_LT_IMP_LE] THEN
2070 ASM_MESON_TAC[CENTRE_IN_BALL; SUBSET];
2072 REWRITE_TAC[REAL_NOT_LE] THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
2073 EXISTS_TAC `(&1 - u) * f(x) + u * f(x:real^N):real` THEN
2074 ASM_SIMP_TAC[REAL_LT_LADD; REAL_LT_LMUL] THEN
2075 REWRITE_TAC[REAL_ARITH `(&1 - x) * a + x * a = a`] THEN
2076 FIRST_X_ASSUM MATCH_MP_TAC THEN
2077 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
2078 REWRITE_TAC[IN_BALL; dist] THEN
2079 REWRITE_TAC[VECTOR_ARITH `x - ((&1 - u) % x + u % y):real^N =
2081 REWRITE_TAC[NORM_MUL; GSYM dist] THEN
2082 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ;
2083 REAL_ARITH `&0 < x /\ x < b ==> abs x < b`]);;
2085 let CONVEX_DISTANCE = prove
2086 (`!s a. (\x. dist(a,x)) convex_on s`,
2087 REWRITE_TAC[convex_on; dist] THEN REPEAT STRIP_TAC THEN
2088 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV)
2089 [GSYM VECTOR_MUL_LID] THEN
2090 FIRST_ASSUM(SUBST1_TAC o SYM) THEN
2091 REWRITE_TAC[VECTOR_ARITH
2092 `(u + v) % z - (u % x + v % y) = u % (z - x) + v % (z - y)`] THEN
2093 ASM_MESON_TAC[NORM_TRIANGLE; NORM_MUL; REAL_ABS_REFL]);;
2095 let CONVEX_NORM = prove
2096 (`!s:real^N->bool. norm convex_on s`,
2098 MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`] CONVEX_DISTANCE) THEN
2099 REWRITE_TAC[DIST_0; ETA_AX]);;
2101 let CONVEX_ON_COMPOSE_LINEAR = prove
2102 (`!f g:real^M->real^N s.
2103 f convex_on (IMAGE g s) /\ linear g ==> (f o g) convex_on s`,
2104 REWRITE_TAC[convex_on; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
2105 REWRITE_TAC[FORALL_IN_IMAGE; o_THM] THEN
2106 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN
2107 REPEAT STRIP_TAC THEN
2108 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_ADD th]) THEN
2109 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN
2112 let CONVEX_ON_TRANSLATION = prove
2114 f convex_on (IMAGE (\x. a + x) s) <=> (\x. f(a + x)) convex_on s`,
2115 REWRITE_TAC[convex_on; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
2116 REWRITE_TAC[FORALL_IN_IMAGE; o_THM] THEN
2117 REWRITE_TAC[VECTOR_ARITH
2118 `u % (a + x) + v % (a + y):real^N = (u + v) % a + u % x + v % y`] THEN
2119 SIMP_TAC[VECTOR_MUL_LID]);;
2121 (* ------------------------------------------------------------------------- *)
2122 (* Open and closed balls are convex and hence connected. *)
2123 (* ------------------------------------------------------------------------- *)
2125 let CONVEX_BALL = prove
2126 (`!x:real^N e. convex(ball(x,e))`,
2127 let lemma = REWRITE_RULE[convex_on; IN_UNIV]
2128 (ISPEC `(:real^N)` CONVEX_DISTANCE) in
2129 REWRITE_TAC[convex; IN_BALL] THEN REPEAT STRIP_TAC THEN
2130 W(MP_TAC o PART_MATCH (lhand o rand) lemma o lhand o snd) THEN
2131 ASM_MESON_TAC[REAL_LET_TRANS; REAL_CONVEX_BOUND_LT]);;
2133 let CONNECTED_BALL = prove
2134 (`!x:real^N e. connected(ball(x,e))`,
2135 SIMP_TAC[CONVEX_CONNECTED; CONVEX_BALL]);;
2137 let CONVEX_CBALL = prove
2138 (`!x:real^N e. convex(cball(x,e))`,
2139 REWRITE_TAC[convex; IN_CBALL; dist] THEN MAP_EVERY X_GEN_TAC
2140 [`x:real^N`; `e:real`; `y:real^N`; `z:real^N`; `u:real`; `v:real`] THEN
2141 STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a - b = &1 % a - b`] THEN
2142 FIRST_ASSUM(SUBST1_TAC o SYM) THEN
2143 REWRITE_TAC[VECTOR_ARITH
2144 `(a + b) % x - (a % y + b % z) = a % (x - y) + b % (x - z)`] THEN
2145 MATCH_MP_TAC REAL_LE_TRANS THEN
2146 EXISTS_TAC `norm(u % (x - y)) + norm(v % (x - z):real^N)` THEN
2147 REWRITE_TAC[NORM_TRIANGLE; NORM_MUL] THEN
2148 MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[REAL_ABS_POS] THEN
2149 ASM_SIMP_TAC[REAL_ARITH
2150 `&0 <= u /\ &0 <= v /\ (u + v = &1) ==> (abs(u) + abs(v) = &1)`]);;
2152 let CONNECTED_CBALL = prove
2153 (`!x:real^N e. connected(cball(x,e))`,
2154 SIMP_TAC[CONVEX_CONNECTED; CONVEX_CBALL]);;
2156 let FRONTIER_OF_CONNECTED_COMPONENT_SUBSET = prove
2157 (`!s c x:real^N. frontier(connected_component s x) SUBSET frontier s`,
2158 REPEAT GEN_TAC THEN REWRITE_TAC[frontier; SUBSET; IN_DIFF] THEN
2159 X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THENL
2160 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
2161 `y IN s ==> s SUBSET t ==> y IN t`)) THEN
2162 MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET];
2163 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN
2164 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
2165 SUBGOAL_THEN `ball(y:real^N,e) SUBSET connected_component s y`
2167 [MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
2168 ASM_REWRITE_TAC[CONNECTED_BALL; CENTRE_IN_BALL];
2169 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSURE_APPROACHABLE]) THEN
2170 DISCH_THEN(MP_TAC o SPEC `e:real`) THEN
2171 ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM IN_BALL)] THEN
2172 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
2173 REWRITE_TAC[IN_INTERIOR] THEN EXISTS_TAC `e:real` THEN
2174 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `y:real^N`]
2175 CONNECTED_COMPONENT_OVERLAP) THEN
2176 MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN
2177 ASM_SIMP_TAC[] THEN ASM SET_TAC[]]]);;
2179 let FRONTIER_OF_COMPONENTS_SUBSET = prove
2180 (`!s c:real^N->bool.
2181 c IN components s ==> frontier c SUBSET frontier s`,
2182 SIMP_TAC[components; FORALL_IN_GSPEC;
2183 FRONTIER_OF_CONNECTED_COMPONENT_SUBSET]);;
2185 let FRONTIER_OF_COMPONENTS_CLOSED_COMPLEMENT = prove
2186 (`!s c. closed s /\ c IN components ((:real^N) DIFF s)
2187 ==> frontier c SUBSET s`,
2188 REPEAT STRIP_TAC THEN
2189 FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN
2190 REWRITE_TAC[FRONTIER_COMPLEMENT] THEN
2191 ASM_MESON_TAC[FRONTIER_SUBSET_EQ; SUBSET_TRANS]);;
2193 (* ------------------------------------------------------------------------- *)
2194 (* A couple of lemmas about components (see Newman IV, 3.3 and 3.4). *)
2195 (* ------------------------------------------------------------------------- *)
2197 let CONNECTED_UNION_CLOPEN_IN_COMPLEMENT = prove
2198 (`!s t u:real^N->bool.
2199 connected s /\ connected u /\ s SUBSET u /\
2200 open_in (subtopology euclidean (u DIFF s)) t /\
2201 closed_in (subtopology euclidean (u DIFF s)) t
2202 ==> connected (s UNION t)`,
2204 [`c:real^N->bool`; `h:real^N->bool`; `s:real^N->bool`] THEN
2206 REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN
2207 MATCH_MP_TAC(MESON[]
2208 `!Q. (!x y. P x y <=> P y x) /\
2209 (!x y. P x y ==> Q x \/ Q y) /\
2210 (!x y. P x y /\ Q x ==> F)
2211 ==> (!x y. ~(P x y))`) THEN
2212 EXISTS_TAC `\x:real^N->bool. c SUBSET x` THEN
2213 CONJ_TAC THENL [MESON_TAC[INTER_COMM; UNION_COMM]; ALL_TAC] THEN
2214 REWRITE_TAC[] THEN CONJ_TAC THEN
2215 MAP_EVERY X_GEN_TAC [`h1:real^N->bool`; `h2:real^N->bool`] THENL
2216 [STRIP_TAC THEN UNDISCH_TAC `connected(c:real^N->bool)` THEN
2217 REWRITE_TAC[CONNECTED_CLOSED_IN; NOT_EXISTS_THM] THEN
2219 SPECL [`c INTER h1:real^N->bool`; `c INTER h2:real^N->bool`]) THEN
2221 `(p /\ q) /\ (~r ==> s) ==> ~(p /\ q /\ r) ==> s`) THEN
2222 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL
2224 `closed_in(subtopology euclidean (c UNION h)) (h1:real^N->bool)`;
2226 `closed_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)`] THEN
2227 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2230 FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN
2231 SUBGOAL_THEN `(h2:real^N->bool) SUBSET h` ASSUME_TAC THENL
2232 [ASM SET_TAC[]; ALL_TAC] THEN
2233 UNDISCH_TAC `connected(s:real^N->bool)` THEN
2234 REWRITE_TAC[CONNECTED_CLOPEN] THEN
2235 DISCH_THEN(MP_TAC o SPEC `h2:real^N->bool`) THEN REWRITE_TAC[NOT_IMP] THEN
2236 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2237 SUBGOAL_THEN `s:real^N->bool = (s DIFF c) UNION (c UNION h)`
2238 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL
2239 [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_UNION THEN
2240 MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL
2241 [REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
2242 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2243 SUBGOAL_THEN `(c UNION h) DIFF h2:real^N->bool = h1`
2244 (fun th -> ASM_REWRITE_TAC[th]) THEN ASM SET_TAC[];
2245 DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_TRANS THEN
2246 EXISTS_TAC `h:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2248 `open_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)` THEN
2249 REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN
2251 MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_UNION THEN ASM_REWRITE_TAC[] THEN
2252 MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `h:real^N->bool` THEN
2253 ASM_REWRITE_TAC[] THEN
2255 `closed_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)` THEN
2256 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2259 let COMPONENT_COMPLEMENT_CONNECTED = prove
2260 (`!s u c:real^N->bool.
2261 connected s /\ connected u /\ s SUBSET u /\ c IN components (u DIFF s)
2262 ==> connected(u DIFF c)`,
2264 [`a:real^N->bool`; `s:real^N->bool`; `c:real^N->bool`] THEN
2265 STRIP_TAC THEN UNDISCH_TAC `connected(a:real^N->bool)` THEN
2266 REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN
2267 DISCH_TAC THEN MAP_EVERY X_GEN_TAC
2268 [`h3:real^N->bool`; `h4:real^N->bool`] THEN STRIP_TAC THEN
2269 FIRST_X_ASSUM(MP_TAC o SPECL
2270 [`a INTER h3:real^N->bool`; `a INTER h4:real^N->bool`]) THEN
2271 FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN
2272 FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN
2273 EVERY_ASSUM(fun th -> try
2274 MP_TAC(CONJUNCT1(GEN_REWRITE_RULE I [closed_in] th))
2275 with Failure _ -> ALL_TAC) THEN
2276 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REPEAT DISCH_TAC THEN
2277 REPEAT CONJ_TAC THENL
2278 [UNDISCH_TAC `closed_in (subtopology euclidean (s DIFF c))
2279 (h3:real^N->bool)` THEN
2280 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2282 UNDISCH_TAC `closed_in (subtopology euclidean (s DIFF c))
2283 (h4:real^N->bool)` THEN
2284 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2289 MP_TAC(ISPECL [`s DIFF a:real^N->bool`; `c UNION h3:real^N->bool`;
2290 `c:real^N->bool`] COMPONENTS_MAXIMAL) THEN
2291 ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN
2292 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2293 MATCH_MP_TAC CONNECTED_UNION_CLOPEN_IN_COMPLEMENT THEN
2294 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2295 REPEAT CONJ_TAC THENL
2296 [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED];
2298 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
2299 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2300 SUBGOAL_THEN `s DIFF c DIFF h3:real^N->bool = h4` SUBST1_TAC THEN
2303 MP_TAC(ISPECL [`s DIFF a:real^N->bool`; `c UNION h4:real^N->bool`;
2304 `c:real^N->bool`] COMPONENTS_MAXIMAL) THEN
2305 ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN
2306 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2307 MATCH_MP_TAC CONNECTED_UNION_CLOPEN_IN_COMPLEMENT THEN
2308 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2309 REPEAT CONJ_TAC THENL
2310 [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED];
2312 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
2313 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2314 SUBGOAL_THEN `s DIFF c DIFF h4:real^N->bool = h3` SUBST1_TAC THEN
2317 (* ------------------------------------------------------------------------- *)
2318 (* Sura-Bura's result about components of closed sets. *)
2319 (* ------------------------------------------------------------------------- *)
2321 let SURA_BURA_COMPACT = prove
2322 (`!s c:real^N->bool.
2323 compact s /\ c IN components s
2324 ==> c = INTERS {t | c SUBSET t /\
2325 open_in (subtopology euclidean s) t /\
2326 closed_in (subtopology euclidean s) t}`,
2327 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
2328 CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
2329 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [components]) THEN
2330 REWRITE_TAC[IN_ELIM_THM] THEN
2331 DISCH_THEN(X_CHOOSE_THEN `x:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2332 DISCH_THEN(fun th -> SUBST1_TAC th THEN ASSUME_TAC(SYM th)) THEN
2333 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
2334 SUBGOAL_THEN `(x:real^N) IN c` ASSUME_TAC THENL
2335 [ASM_MESON_TAC[CONNECTED_COMPONENT_REFL; IN]; ALL_TAC] THEN
2336 SUBGOAL_THEN `(c:real^N->bool) SUBSET s` ASSUME_TAC THENL
2337 [ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET]; ALL_TAC] THEN
2338 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL
2340 MATCH_MP_TAC(SET_RULE `s IN t ==> INTERS t SUBSET s`) THEN
2341 REWRITE_TAC[IN_ELIM_THM; CONNECTED_COMPONENT_SUBSET;
2342 OPEN_IN_SUBTOPOLOGY_REFL; CLOSED_IN_SUBTOPOLOGY_REFL] THEN
2343 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]] THEN
2344 W(fun (asl,w) -> ABBREV_TAC(mk_eq(`k:real^N->bool`,rand w))) THEN
2345 SUBGOAL_THEN `closed(k:real^N->bool)` ASSUME_TAC THENL
2346 [EXPAND_TAC "k" THEN MATCH_MP_TAC CLOSED_INTERS THEN
2347 REWRITE_TAC[IN_ELIM_THM] THEN
2348 ASM_MESON_TAC[CLOSED_IN_CLOSED_TRANS; COMPACT_IMP_CLOSED];
2350 REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN
2351 MAP_EVERY X_GEN_TAC [`k1:real^N->bool`; `k2:real^N->bool`] THEN
2353 MP_TAC(ISPECL [`k1:real^N->bool`; `k2:real^N->bool`] SEPARATION_NORMAL) THEN
2354 ASM_REWRITE_TAC[NOT_EXISTS_THM; NOT_IMP] THEN CONJ_TAC THENL
2355 [ASM_MESON_TAC[CLOSED_IN_CLOSED_TRANS; COMPACT_IMP_CLOSED]; ALL_TAC] THEN
2356 MAP_EVERY X_GEN_TAC [`v1:real^N->bool`; `v2:real^N->bool`] THEN
2358 MP_TAC(ISPECL [`s DIFF (v1 UNION v2):real^N->bool`;
2359 `{t:real^N->bool | connected_component s x SUBSET t /\
2360 open_in (subtopology euclidean s) t /\
2361 closed_in (subtopology euclidean s) t}`]
2362 COMPACT_IMP_FIP) THEN
2363 ASM_SIMP_TAC[NOT_IMP; COMPACT_DIFF; OPEN_UNION; IN_ELIM_THM] THEN
2364 REPEAT CONJ_TAC THENL
2365 [ASM_MESON_TAC[CLOSED_IN_CLOSED_TRANS; COMPACT_IMP_CLOSED];
2366 ONCE_REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM];
2368 X_GEN_TAC `f:(real^N->bool)->bool` THEN REPEAT STRIP_TAC THEN
2371 c SUBSET c0 /\ c0 SUBSET (v1 UNION v2) /\
2372 open_in (subtopology euclidean s) c0 /\
2373 closed_in (subtopology euclidean s) c0`
2374 STRIP_ASSUME_TAC THENL
2375 [ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL
2376 [EXISTS_TAC `s:real^N->bool` THEN
2377 ASM_REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV;
2378 OPEN_IN_SUBTOPOLOGY_REFL; CLOSED_IN_SUBTOPOLOGY_REFL] THEN
2380 `(s DIFF (v1 UNION v2)) INTER INTERS f :real^N->bool = {}` THEN
2381 ASM_REWRITE_TAC[INTERS_0; INTER_UNIV] THEN SET_TAC[];
2382 EXISTS_TAC `INTERS f :real^N->bool` THEN REPEAT CONJ_TAC THENL
2384 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
2385 `(s DIFF u) INTER t = {}
2387 ==> t SUBSET u`)) THEN
2388 MATCH_MP_TAC(SET_RULE
2389 `~(f = {}) /\ (!s. s IN f ==> s SUBSET t) ==> INTERS f SUBSET t`) THEN
2390 ASM_MESON_TAC[CLOSED_IN_SUBSET; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY];
2391 MATCH_MP_TAC OPEN_IN_INTERS THEN ASM_SIMP_TAC[];
2392 MATCH_MP_TAC CLOSED_IN_INTERS THEN ASM_SIMP_TAC[]]];
2394 SUBGOAL_THEN `connected(c:real^N->bool)` MP_TAC THENL
2395 [ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT]; ALL_TAC] THEN
2397 `closed_in (subtopology euclidean c0) (c0 INTER v1 :real^N->bool) /\
2398 closed_in (subtopology euclidean c0) (c0 INTER v2 :real^N->bool)`
2401 MATCH_MP_TAC(MESON[]
2402 `closed_in top (c INTER closure v) /\
2403 c INTER closure v = c INTER v
2404 ==> closed_in top (c INTER v)`) THEN
2406 [MESON_TAC[CLOSED_IN_CLOSED; CLOSED_CLOSURE]; ALL_TAC]) THEN
2407 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
2408 `c0 SUBSET vv ==> c0 INTER (vv INTER v') = c0 INTER v
2409 ==> c0 INTER v' = c0 INTER v`)) THEN
2410 REWRITE_TAC[ONCE_REWRITE_RULE[INTER_COMM] UNION_OVER_INTER;
2411 UNION_OVER_INTER] THEN
2412 SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`; CLOSURE_SUBSET] THENL
2413 [ALL_TAC; ONCE_REWRITE_TAC[UNION_COMM]] THEN
2414 MATCH_MP_TAC(SET_RULE `t = {} ==> s UNION (u INTER t) = s`) THEN
2415 ASM_SIMP_TAC[OPEN_INTER_CLOSURE_EQ_EMPTY] THEN ASM SET_TAC[];
2417 REWRITE_TAC[CLOSED_IN_CLOSED] THEN DISCH_THEN(CONJUNCTS_THEN2
2418 (X_CHOOSE_THEN `u1:real^N->bool` STRIP_ASSUME_TAC)
2419 (X_CHOOSE_THEN `u2:real^N->bool` STRIP_ASSUME_TAC)) THEN
2420 SUBGOAL_THEN `closed(c0:real^N->bool)` ASSUME_TAC THENL
2421 [ASM_MESON_TAC[CLOSED_IN_CLOSED_TRANS; COMPACT_IMP_CLOSED]; ALL_TAC] THEN
2422 REWRITE_TAC[CONNECTED_CLOSED] THEN MAP_EVERY EXISTS_TAC
2423 [`c0 INTER u1:real^N->bool`; `c0 INTER u2:real^N->bool`] THEN
2424 ASM_SIMP_TAC[CLOSED_INTER] THEN
2425 REPLICATE_TAC 2 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN CONJ_TAC THENL
2427 SUBGOAL_THEN `c SUBSET (c0 INTER v2 :real^N->bool)` MP_TAC THENL
2428 [ASM SET_TAC[]; ALL_TAC] THEN
2429 SUBGOAL_THEN `k SUBSET (c0 INTER v2 :real^N->bool)` ASSUME_TAC THENL
2430 [ALL_TAC; ASM SET_TAC[]];
2432 SUBGOAL_THEN `c SUBSET (c0 INTER v1 :real^N->bool)` ASSUME_TAC THENL
2433 [ASM SET_TAC[]; ALL_TAC] THEN
2434 SUBGOAL_THEN `k SUBSET (c0 INTER v1 :real^N->bool)` ASSUME_TAC THENL
2435 [ALL_TAC; ASM SET_TAC[]]] THEN
2436 (UNDISCH_THEN `k1 UNION k2 :real^N->bool = k` (K ALL_TAC) THEN
2438 MATCH_MP_TAC(SET_RULE `s IN t ==> INTERS t SUBSET s`) THEN
2439 REWRITE_TAC[IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL
2441 MATCH_MP_TAC OPEN_IN_INTER_OPEN THEN ASM_REWRITE_TAC[];
2442 ASM_REWRITE_TAC[] THEN
2443 MATCH_MP_TAC CLOSED_IN_INTER_CLOSED THEN ASM_REWRITE_TAC[]]));;
2445 let SURA_BURA_CLOSED = prove
2446 (`!s c:real^N->bool.
2447 closed s /\ c IN components s /\ compact c
2448 ==> c = INTERS {k | c SUBSET k /\ compact k /\
2449 open_in (subtopology euclidean s) k}`,
2450 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
2451 [SET_TAC[]; ALL_TAC] THEN
2454 open u /\ c SUBSET u
2455 ==> ?k. c SUBSET k /\ k SUBSET u /\ compact k /\
2456 open_in (subtopology euclidean s) k`
2459 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN
2460 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
2461 MP_TAC(ISPECL [`{x:real^N}`; `c:real^N->bool`] SEPARATION_NORMAL) THEN
2462 ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_SING] THEN
2463 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
2464 MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
2465 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `v:real^N->bool`) THEN
2466 ASM_REWRITE_TAC[IN_INTERS; NOT_FORALL_THM; IN_ELIM_THM; NOT_IMP] THEN
2467 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM SET_TAC[]] THEN
2468 REPEAT STRIP_TAC THEN
2470 `?f. FINITE f /\ c SUBSET UNIONS f /\
2471 (!d:real^N->bool. d IN f ==> open d) /\
2472 (!d:real^N->bool. d IN f ==> bounded d) /\
2473 (!d. d IN f ==> closure d SUBSET u)`
2474 STRIP_ASSUME_TAC THENL
2475 [FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_HEINE_BOREL) THEN
2476 DISCH_THEN(MP_TAC o SPEC
2477 `{ ball(x:real^N,e) | x IN c /\ &0 < e /\ cball(x,e) SUBSET u}`) THEN
2479 [REWRITE_TAC[FORALL_IN_GSPEC; UNIONS_GSPEC; OPEN_BALL] THEN
2480 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN
2481 DISCH_TAC THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN
2482 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
2483 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
2484 ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN
2485 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN
2487 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN
2488 ASM_SIMP_TAC[] THEN REPEAT CONJ_TAC THEN
2489 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
2490 (SET_RULE `t SUBSET s ==> (!x. x IN s ==> P x)
2491 ==> (!x. x IN t ==> P x)`)) THEN
2492 SIMP_TAC[FORALL_IN_GSPEC; OPEN_BALL; BOUNDED_BALL; CLOSURE_BALL]];
2494 ABBREV_TAC `v:real^N->bool = UNIONS f` THEN
2495 SUBGOAL_THEN `bounded(v:real^N->bool)` ASSUME_TAC THENL
2496 [EXPAND_TAC "v" THEN MATCH_MP_TAC BOUNDED_UNIONS THEN
2499 SUBGOAL_THEN `compact(closure v:real^N->bool)` ASSUME_TAC THENL
2500 [ASM_REWRITE_TAC[COMPACT_CLOSURE]; ALL_TAC] THEN
2501 SUBGOAL_THEN `(closure v:real^N->bool) SUBSET u` ASSUME_TAC THENL
2502 [EXPAND_TAC "v" THEN ASM_SIMP_TAC[CLOSURE_UNIONS] THEN
2503 ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC];
2505 SUBGOAL_THEN `open(v:real^N->bool)` ASSUME_TAC THENL
2506 [EXPAND_TAC "v" THEN MATCH_MP_TAC OPEN_UNIONS THEN
2510 [`closure v INTER s:real^N->bool`; `c:real^N->bool`]
2511 SURA_BURA_COMPACT) THEN
2513 [ASM_SIMP_TAC[COMPACT_INTER_CLOSED] THEN
2514 REWRITE_TAC[IN_COMPONENTS_MAXIMAL] THEN
2515 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_COMPONENTS_MAXIMAL]) THEN
2516 ASM_MESON_TAC[SUBSET_INTER; SUBSET_TRANS; CLOSURE_SUBSET];
2521 open_in (subtopology euclidean (closure v INTER s)) t /\
2522 closed_in (subtopology euclidean (closure v INTER s)) t <=>
2523 c SUBSET t /\ t SUBSET (closure v INTER s) /\ compact t /\
2524 open_in (subtopology euclidean (closure v INTER s)) t`
2525 (fun th -> REWRITE_TAC[th])
2527 [GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
2528 [MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
2529 [ASM_MESON_TAC[CLOSED_IN_SUBSET; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY];
2530 STRIP_TAC THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN
2532 [ASM_MESON_TAC[BOUNDED_SUBSET; COMPACT_IMP_BOUNDED;
2533 COMPACT_INTER_CLOSED];
2534 MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN
2535 EXISTS_TAC `closure v INTER s:real^N->bool` THEN
2536 ASM_MESON_TAC[COMPACT_IMP_CLOSED; CLOSED_INTER]]];
2537 MATCH_MP_TAC CLOSED_CLOSED_IN_TRANS THEN
2538 ASM_SIMP_TAC[COMPACT_IMP_CLOSED; CLOSED_INTER]];
2539 DISCH_THEN(ASSUME_TAC o SYM)] THEN
2541 [`(closure v INTER s) DIFF v:real^N->bool`;
2542 `{t:real^N->bool | c SUBSET t /\
2543 t SUBSET (closure v INTER s) /\ compact t /\
2544 open_in (subtopology euclidean (closure v INTER s)) t}`]
2545 COMPACT_IMP_FIP) THEN
2546 ASM_SIMP_TAC[COMPACT_DIFF; COMPACT_INTER_CLOSED] THEN
2548 `p /\ r /\ (~q ==> s) ==> (p /\ q ==> ~r) ==> s`) THEN
2549 REWRITE_TAC[IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL
2550 [MESON_TAC[COMPACT_IMP_CLOSED];
2552 REWRITE_TAC[NOT_FORALL_THM; LEFT_IMP_EXISTS_THM]] THEN
2553 X_GEN_TAC `g:(real^N->bool)->bool` THEN
2554 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SUBSET] THEN
2555 REWRITE_TAC[IN_ELIM_THM; NOT_IMP] THEN
2556 ASM_CASES_TAC `g:(real^N->bool)->bool = {}` THENL
2557 [ASM_REWRITE_TAC[FINITE_EMPTY; NOT_IN_EMPTY; INTERS_0; INTER_UNIV] THEN
2558 REWRITE_TAC[SET_RULE `s DIFF t = {} <=> s SUBSET t`] THEN
2559 STRIP_TAC THEN EXISTS_TAC `closure v INTER s :real^N->bool` THEN
2560 REPEAT CONJ_TAC THENL
2561 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_COMPONENTS_MAXIMAL]) THEN
2562 MP_TAC(ISPEC `v:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[];
2564 ASM_SIMP_TAC[COMPACT_INTER_CLOSED];
2565 SUBGOAL_THEN `closure v INTER s :real^N->bool = s INTER v`
2567 [MP_TAC(ISPEC `v:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[];
2568 ASM_SIMP_TAC[OPEN_IN_OPEN_INTER]]];
2570 EXISTS_TAC `INTERS g :real^N->bool` THEN REPEAT CONJ_TAC THENL
2572 MP_TAC(ISPEC `v:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[];
2573 MATCH_MP_TAC COMPACT_INTERS THEN ASM_MESON_TAC[];
2575 `open_in (subtopology euclidean (closure v INTER s))
2576 (INTERS g:real^N->bool)`
2578 [MATCH_MP_TAC OPEN_IN_INTERS THEN ASM_MESON_TAC[]; ALL_TAC] THEN
2579 FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE
2580 `(s DIFF t) INTER u = {} ==> u SUBSET s ==> u SUBSET t`)) THEN
2581 ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN
2582 REWRITE_TAC[OPEN_IN_OPEN] THEN
2583 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
2584 ASM_REWRITE_TAC[] THEN
2585 EXISTS_TAC `(v:real^N->bool) INTER t` THEN ASM_SIMP_TAC[OPEN_INTER] THEN
2586 MP_TAC(ISPEC `v:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]]]);;
2588 (* ------------------------------------------------------------------------- *)
2589 (* Condition for an open map's image to contain a ball. *)
2590 (* ------------------------------------------------------------------------- *)
2592 let BALL_SUBSET_OPEN_MAP_IMAGE = prove
2593 (`!f:real^M->real^N s a r.
2594 bounded s /\ f continuous_on closure s /\ open(IMAGE f (interior s)) /\
2595 a IN s /\ &0 < r /\ (!z. z IN frontier s ==> r <= norm(f z - f a))
2596 ==> ball(f(a),r) SUBSET IMAGE f s`,
2597 REPEAT STRIP_TAC THEN
2598 MP_TAC(ISPECL [`ball((f:real^M->real^N) a,r)`;
2599 `(:real^N) DIFF IMAGE (f:real^M->real^N) s`]
2600 CONNECTED_INTER_FRONTIER) THEN
2601 REWRITE_TAC[CONNECTED_BALL] THEN MATCH_MP_TAC(SET_RULE
2602 `~(b INTER s = {}) /\ b INTER f = {} ==>
2603 (~(b INTER (UNIV DIFF s) = {}) /\ ~(b DIFF (UNIV DIFF s) = {})
2604 ==> ~(b INTER f = {}))
2605 ==> b SUBSET s`) THEN
2606 REWRITE_TAC[FRONTIER_COMPLEMENT] THEN CONJ_TAC THENL
2607 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
2608 EXISTS_TAC `(f:real^M->real^N) a` THEN
2609 ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ASM SET_TAC[];
2610 REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN t ==> ~(x IN s)`] THEN
2611 REWRITE_TAC[IN_BALL; REAL_NOT_LT]] THEN
2612 MP_TAC(ISPECL[`frontier(IMAGE (f:real^M->real^N) s)`; `(f:real^M->real^N) a`]
2613 DISTANCE_ATTAINS_INF) THEN
2614 REWRITE_TAC[FRONTIER_CLOSED; FRONTIER_EQ_EMPTY] THEN ANTS_TAC THENL
2615 [SIMP_TAC[DE_MORGAN_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2616 MATCH_MP_TAC(MESON[NOT_BOUNDED_UNIV] `bounded s ==> ~(s = UNIV)`) THEN
2617 MATCH_MP_TAC BOUNDED_SUBSET THEN
2618 EXISTS_TAC `IMAGE (f:real^M->real^N) (closure s)` THEN
2619 SIMP_TAC[IMAGE_SUBSET; CLOSURE_SUBSET] THEN
2620 MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
2621 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
2622 ASM_REWRITE_TAC[COMPACT_CLOSURE];
2623 DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC)] THEN
2624 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [frontier]) THEN
2625 REWRITE_TAC[IN_DIFF] THEN
2626 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
2627 REWRITE_TAC[CLOSURE_SEQUENTIAL] THEN
2628 DISCH_THEN(X_CHOOSE_THEN `y:num->real^N`
2629 (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
2630 REWRITE_TAC[IN_IMAGE; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
2631 X_GEN_TAC `z:num->real^M` THEN REWRITE_TAC[FORALL_AND_THM] THEN
2632 ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN
2633 DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
2634 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM COMPACT_CLOSURE]) THEN
2635 REWRITE_TAC[compact] THEN
2636 DISCH_THEN(MP_TAC o SPEC `z:num->real^M`) THEN
2637 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; LEFT_IMP_EXISTS_THM] THEN
2638 MAP_EVERY X_GEN_TAC [`y:real^M`; `r:num->num`] THEN STRIP_TAC THEN
2640 `(((\n. (f:real^M->real^N)(z n)) o (r:num->num)) --> w) sequentially`
2642 [MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[];
2643 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[GSYM o_ASSOC]] THEN
2645 SUBGOAL_THEN `!n. ((z:num->real^M) o (r:num->num)) n IN s` MP_TAC THENL
2646 [ASM_REWRITE_TAC[o_THM];
2647 UNDISCH_THEN `((\n. (f:real^M->real^N) ((z:num->real^M) n)) --> w)
2648 sequentially` (K ALL_TAC) THEN
2649 UNDISCH_THEN `!n. (z:num->real^M) n IN s` (K ALL_TAC)] THEN
2650 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
2651 SPEC_TAC(`(z:num->real^M) o (r:num->num)`, `z:num->real^M`) THEN
2652 REPEAT STRIP_TAC THEN
2653 SUBGOAL_THEN `w = (f:real^M->real^N) y` SUBST_ALL_TAC THENL
2654 [MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
2655 EXISTS_TAC `(f:real^M->real^N) o (z:num->real^M)` THEN
2656 ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
2657 ASM_MESON_TAC[CONTINUOUS_ON_CLOSURE_SEQUENTIALLY];
2659 MATCH_MP_TAC REAL_LE_TRANS THEN
2660 EXISTS_TAC `norm(f y - (f:real^M->real^N) a)` THEN CONJ_TAC THENL
2661 [FIRST_X_ASSUM MATCH_MP_TAC; ASM_MESON_TAC[dist; NORM_SUB]] THEN
2662 ASM_REWRITE_TAC[frontier; IN_DIFF] THEN DISCH_TAC THEN
2663 FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
2664 REWRITE_TAC[interior; IN_ELIM_THM] THEN
2665 EXISTS_TAC `IMAGE (f:real^M->real^N) (interior s)` THEN
2666 ASM_SIMP_TAC[IMAGE_SUBSET; INTERIOR_SUBSET] THEN ASM SET_TAC[]);;
2668 (* ------------------------------------------------------------------------- *)
2669 (* Arithmetic operations on sets preserve convexity. *)
2670 (* ------------------------------------------------------------------------- *)
2672 let CONVEX_SCALING = prove
2673 (`!s c. convex s ==> convex (IMAGE (\x. c % x) s)`,
2674 REWRITE_TAC[convex; IN_IMAGE] THEN REPEAT STRIP_TAC THEN
2675 ASM_REWRITE_TAC[VECTOR_ARITH
2676 `u % c % x + v % c % y = c % (u % x + v % y)`] THEN
2679 let CONVEX_SCALING_EQ = prove
2680 (`!s c. ~(c = &0) ==> (convex (IMAGE (\x. c % x) s) <=> convex s)`,
2681 REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[CONVEX_SCALING] THEN
2682 DISCH_THEN(MP_TAC o SPEC `inv c` o MATCH_MP CONVEX_SCALING) THEN
2683 ASM_SIMP_TAC[GSYM IMAGE_o; o_DEF; VECTOR_MUL_ASSOC;
2684 REAL_MUL_LINV; VECTOR_MUL_LID; IMAGE_ID]);;
2686 let CONVEX_NEGATIONS = prove
2687 (`!s. convex s ==> convex (IMAGE (--) s)`,
2688 REWRITE_TAC[convex; IN_IMAGE] THEN REPEAT STRIP_TAC THEN
2689 ASM_REWRITE_TAC[VECTOR_ARITH
2690 `u % --x + v % --y = --(u % x + v % y)`] THEN
2693 let CONVEX_SUMS = prove
2694 (`!s t. convex s /\ convex t ==> convex {x + y | x IN s /\ y IN t}`,
2695 REWRITE_TAC[convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
2696 ASM_REWRITE_TAC[VECTOR_ARITH
2697 `u % (a + b) + v % (c + d) = (u % a + v % c) + (u % b + v % d)`] THEN
2700 let CONVEX_DIFFERENCES = prove
2701 (`!s t. convex s /\ convex t ==> convex {x - y | x IN s /\ y IN t}`,
2702 REWRITE_TAC[convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
2703 ASM_REWRITE_TAC[VECTOR_ARITH
2704 `u % (a - b) + v % (c - d) = (u % a + v % c) - (u % b + v % d)`] THEN
2707 let CONVEX_AFFINITY = prove
2709 convex s ==> convex (IMAGE (\x. a + c % x) s)`,
2710 REPEAT STRIP_TAC THEN
2711 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
2712 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2713 ASM_SIMP_TAC[IMAGE_o; CONVEX_TRANSLATION; CONVEX_SCALING]);;
2715 let CONVEX_LINEAR_PREIMAGE = prove
2716 (`!f:real^M->real^N.
2717 linear f /\ convex s ==> convex {x | f(x) IN s}`,
2718 REWRITE_TAC[CONVEX_ALT; IN_ELIM_THM] THEN
2719 SIMP_TAC[LINEAR_ADD; LINEAR_CMUL]);;
2721 (* ------------------------------------------------------------------------- *)
2723 (* ------------------------------------------------------------------------- *)
2725 let CONVEX_CONVEX_HULL = prove
2726 (`!s. convex(convex hull s)`,
2727 SIMP_TAC[P_HULL; CONVEX_INTERS]);;
2729 let CONVEX_HULL_EQ = prove
2730 (`!s. (convex hull s = s) <=> convex s`,
2731 SIMP_TAC[HULL_EQ; CONVEX_INTERS]);;
2733 let IS_CONVEX_HULL = prove
2734 (`!s. convex s <=> ?t. s = convex hull t`,
2735 GEN_TAC THEN MATCH_MP_TAC IS_HULL THEN SIMP_TAC[CONVEX_INTERS]);;
2737 let CONVEX_HULL_UNIV = prove
2738 (`convex hull (:real^N) = (:real^N)`,
2739 REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_UNIV]);;
2741 let BOUNDED_CONVEX_HULL = prove
2742 (`!s:real^N->bool. bounded s ==> bounded(convex hull s)`,
2743 GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [bounded] THEN
2744 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
2745 MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(vec 0:real^N,B)` THEN
2746 SIMP_TAC[BOUNDED_CBALL; SUBSET_HULL; CONVEX_CBALL] THEN
2747 ASM_REWRITE_TAC[IN_CBALL; SUBSET; dist; VECTOR_SUB_LZERO; NORM_NEG]);;
2749 let BOUNDED_CONVEX_HULL_EQ = prove
2750 (`!s. bounded(convex hull s) <=> bounded s`,
2751 MESON_TAC[BOUNDED_CONVEX_HULL; HULL_SUBSET; BOUNDED_SUBSET]);;
2753 let FINITE_IMP_BOUNDED_CONVEX_HULL = prove
2754 (`!s. FINITE s ==> bounded(convex hull s)`,
2755 SIMP_TAC[BOUNDED_CONVEX_HULL; FINITE_IMP_BOUNDED]);;
2757 (* ------------------------------------------------------------------------- *)
2758 (* Stepping theorems for convex hulls of finite sets. *)
2759 (* ------------------------------------------------------------------------- *)
2761 let CONVEX_HULL_EMPTY = prove
2762 (`convex hull {} = {}`,
2763 MATCH_MP_TAC HULL_UNIQUE THEN
2764 REWRITE_TAC[SUBSET_REFL; CONVEX_EMPTY; EMPTY_SUBSET]);;
2766 let CONVEX_HULL_EQ_EMPTY = prove
2767 (`!s. (convex hull s = {}) <=> (s = {})`,
2768 GEN_TAC THEN EQ_TAC THEN
2769 MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; CONVEX_HULL_EMPTY]);;
2771 let CONVEX_HULL_SING = prove
2772 (`!a. convex hull {a} = {a}`,
2773 REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_SING]);;
2775 let CONVEX_HULL_EQ_SING = prove
2776 (`!s a:real^N. convex hull s = {a} <=> s = {a}`,
2777 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
2778 ASM_REWRITE_TAC[CONVEX_HULL_EMPTY] THEN
2779 EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[CONVEX_HULL_SING] THEN
2780 MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET {a} ==> s = {a}`) THEN
2781 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
2782 REWRITE_TAC[HULL_SUBSET]);;
2784 let CONVEX_HULL_INSERT = prove
2786 ==> (convex hull (a INSERT s) =
2787 {x:real^N | ?u v b. &0 <= u /\ &0 <= v /\ (u + v = &1) /\
2788 b IN (convex hull s) /\
2789 (x = u % a + v % b)})`,
2790 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
2791 [MATCH_MP_TAC HULL_MINIMAL THEN CONJ_TAC THENL
2792 [REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INSERT] THEN
2793 X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL
2794 [MAP_EVERY EXISTS_TAC [`&1`; `&0`];
2795 MAP_EVERY EXISTS_TAC [`&0`; `&1`]] THEN
2796 ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO] THEN
2797 ASM_REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID] THEN
2798 CONV_TAC REAL_RAT_REDUCE_CONV THEN
2799 ASM_MESON_TAC[MEMBER_NOT_EMPTY; HULL_SUBSET; SUBSET];
2801 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
2802 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2803 MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN
2804 ASM_REWRITE_TAC[] THEN
2805 ASM_MESON_TAC[HULL_SUBSET; SUBSET; IN_INSERT; HULL_MONO]] THEN
2806 REWRITE_TAC[convex; IN_ELIM_THM] THEN
2807 REWRITE_TAC[LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
2808 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
2810 [`x:real^N`; `y:real^N`; `u:real`; `v:real`; `u1:real`; `v1:real`;
2811 `b1:real^N`; `u2:real`; `v2:real`; `b2:real^N`] THEN
2812 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2813 MAP_EVERY EXISTS_TAC [`u * u1 + v * u2`; `u * v1 + v * v2`] THEN
2814 REWRITE_TAC[VECTOR_ARITH
2815 `u % (u1 % a + v1 % b1) + v % (u2 % a + v2 % b2) =
2816 (u * u1 + v * u2) % a + (u * v1) % b1 + (v * v2) % b2`] THEN
2817 ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL] THEN
2818 ASM_REWRITE_TAC[REAL_MUL_RID; REAL_ARITH
2819 `(u * u1 + v * u2) + (u * v1 + v * v2) =
2820 u * (u1 + v1) + v * (u2 + v2)`] THEN
2821 ASM_CASES_TAC `u * v1 + v * v2 = &0` THENL
2822 [FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH
2823 `(a + b = &0) ==> &0 <= a /\ &0 <= b ==> (a = &0) /\ (b = &0)`)) THEN
2824 ASM_SIMP_TAC[REAL_LE_MUL; REAL_ADD_LID; VECTOR_MUL_LZERO;
2825 VECTOR_ADD_RID] THEN
2828 EXISTS_TAC `(u * v1) / (u * v1 + v * v2) % b1 +
2829 (v * v2) / (u * v1 + v * v2) % b2 :real^N` THEN
2830 ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL] THEN
2831 MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN
2832 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_LE_ADD] THEN
2833 ASM_SIMP_TAC[real_div; GSYM REAL_ADD_RDISTRIB; REAL_MUL_RINV]);;
2835 let CONVEX_HULL_INSERT_ALT = prove
2837 convex hull (a INSERT s) =
2839 else {(&1 - u) % a + u % x | &0 <= u /\ u <= &1 /\ x IN convex hull s}`,
2840 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
2841 ASM_REWRITE_TAC[CONVEX_HULL_SING] THEN
2842 ASM_SIMP_TAC[CONVEX_HULL_INSERT] THEN
2843 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> b /\ c /\ a /\ d`] THEN
2844 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
2845 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; REAL_SUB_LE;
2846 REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
2849 (* ------------------------------------------------------------------------- *)
2850 (* Explicit expression for convex hull. *)
2851 (* ------------------------------------------------------------------------- *)
2853 let CONVEX_HULL_INDEXED = prove
2854 (`!s. convex hull s =
2855 {y:real^N | ?k u x. (!i. 1 <= i /\ i <= k ==> &0 <= u i /\ x i IN s) /\
2856 (sum (1..k) u = &1) /\
2857 (vsum (1..k) (\i. u i % x i) = y)}`,
2858 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL
2859 [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
2860 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
2861 MAP_EVERY EXISTS_TAC [`1`; `\i:num. &1`; `\i:num. x:real^N`] THEN
2862 ASM_SIMP_TAC[FINITE_RULES; IN_SING; SUM_SING; VECTOR_MUL_LID; VSUM_SING;
2863 REAL_POS; NUMSEG_SING];
2865 REWRITE_TAC[CONVEX_INDEXED; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
2867 REWRITE_TAC[convex; IN_ELIM_THM] THEN
2868 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
2869 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
2870 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
2871 [`k1:num`; `u1:num->real`; `x1:num->real^N`;
2872 `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN
2873 STRIP_TAC THEN EXISTS_TAC `k1 + k2:num` THEN
2874 EXISTS_TAC `\i:num. if i <= k1 then u * u1(i) else v * u2(i - k1):real` THEN
2875 EXISTS_TAC `\i:num. if i <= k1 then x1(i) else x2(i - k1):real^N` THEN
2876 ASM_SIMP_TAC[NUMSEG_ADD_SPLIT; ARITH_RULE `1 <= x + 1 /\ x < x + 1`;
2877 IN_NUMSEG; SUM_UNION; VSUM_UNION; FINITE_NUMSEG; DISJOINT_NUMSEG;
2878 ARITH_RULE `k1 + 1 <= i ==> ~(i <= k1)`] THEN
2879 REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] NUMSEG_OFFSET_IMAGE] THEN
2880 ASM_SIMP_TAC[SUM_IMAGE; VSUM_IMAGE; EQ_ADD_LCANCEL; FINITE_NUMSEG] THEN
2881 ASM_SIMP_TAC[o_DEF; ADD_SUB2; SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC;
2882 FINITE_NUMSEG; REAL_MUL_RID] THEN
2883 ASM_MESON_TAC[REAL_LE_MUL; ARITH_RULE
2884 `i <= k1 + k2 /\ ~(i <= k1) ==> 1 <= i - k1 /\ i - k1 <= k2`]);;
2886 (* ------------------------------------------------------------------------- *)
2887 (* Another formulation from Lars Schewe. *)
2888 (* ------------------------------------------------------------------------- *)
2890 let CONVEX_HULL_EXPLICIT = prove
2891 (`!p. convex hull p =
2892 {y:real^N | ?s u. FINITE s /\ s SUBSET p /\
2893 (!x. x IN s ==> &0 <= u x) /\
2894 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
2895 REWRITE_TAC[CONVEX_HULL_INDEXED;EXTENSION;IN_ELIM_THM] THEN
2896 REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
2897 [MAP_EVERY EXISTS_TAC [`IMAGE (x':num->real^N) (1..k)`;
2898 `\v:real^N.sum {i | i IN (1..k) /\ x' i = v} u`]
2899 THEN ASM_SIMP_TAC[FINITE_IMAGE;FINITE_NUMSEG;IN_IMAGE] THEN
2900 REPEAT STRIP_TAC THENL
2901 [REWRITE_TAC[IMAGE;SUBSET;IN_ELIM_THM;IN_NUMSEG] THEN
2903 MATCH_MP_TAC SUM_POS_LE THEN
2904 ASM_SIMP_TAC[FINITE_NUMSEG;FINITE_RESTRICT;IN_ELIM_THM;IN_NUMSEG];
2905 ASM_SIMP_TAC[GSYM SUM_IMAGE_GEN;FINITE_IMAGE;FINITE_NUMSEG];
2906 FIRST_X_ASSUM (fun th -> REWRITE_TAC[GSYM th]) THEN
2907 ASM_SIMP_TAC[GSYM VSUM_IMAGE_GEN;FINITE_IMAGE;
2908 FINITE_NUMSEG;VSUM_VMUL;FINITE_RESTRICT] THEN
2909 MP_TAC (ISPECL [`x':num->real^N`;`\i:num.u i % (x' i):real^N`;`(1..k)`]
2910 (GSYM VSUM_IMAGE_GEN)) THEN
2911 ASM_SIMP_TAC[FINITE_NUMSEG]];ALL_TAC] THEN
2912 STRIP_ASSUME_TAC (ASM_REWRITE_RULE [ASSUME `FINITE (s:real^N->bool)`]
2913 (ISPEC `s:real^N->bool` FINITE_INDEX_NUMSEG)) THEN
2914 MAP_EVERY EXISTS_TAC [`CARD (s:real^N->bool)`;
2915 `(u:real^N->real) o (f:num->real^N)`;
2916 `(f:num->real^N)`] THEN
2917 REPEAT STRIP_TAC THENL
2918 [REWRITE_TAC[o_DEF] THEN FIRST_ASSUM MATCH_MP_TAC THEN
2919 FIRST_ASSUM SUBST1_TAC THEN
2920 REWRITE_TAC[IN_IMAGE;IN_NUMSEG] THEN
2922 MATCH_MP_TAC (REWRITE_RULE [SUBSET]
2923 (ASSUME `(s:real^N->bool) SUBSET p`)) THEN
2924 FIRST_ASSUM SUBST1_TAC THEN
2925 REWRITE_TAC[IN_IMAGE;IN_NUMSEG] THEN
2927 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum (s:real^N->bool) u` THEN
2928 CONJ_TAC THENL [ALL_TAC;ASM_REWRITE_TAC[]] THEN
2929 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
2930 [ASSUME `(s:real^N->bool) = IMAGE f (1..CARD s)`] THEN
2931 MATCH_MP_TAC (GSYM SUM_IMAGE) THEN
2933 REWRITE_TAC[MESON [o_THM;FUN_EQ_THM]
2934 `(\i:num. (u o f) i % f i) = (\v:real^N. u v % v) o f`] THEN
2935 MATCH_MP_TAC EQ_TRANS THEN
2936 EXISTS_TAC `vsum (s:real^N->bool) (\v. u v % v)` THEN
2937 CONJ_TAC THENL [ALL_TAC;ASM_REWRITE_TAC[]] THEN
2938 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
2939 [ASSUME `(s:real^N->bool) = IMAGE f (1..CARD s)`] THEN
2940 MATCH_MP_TAC (GSYM VSUM_IMAGE) THEN
2941 ASM SET_TAC[FINITE_NUMSEG]]);;
2943 let CONVEX_HULL_FINITE = prove
2946 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
2948 vsum s (\x. u x % x) = y}`,
2949 GEN_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
2950 REWRITE_TAC[CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN
2951 X_GEN_TAC `x:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
2952 [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `f:real^N->real`] THEN
2954 EXISTS_TAC `\x:real^N. if x IN t then f x else &0` THEN
2955 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
2956 REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
2957 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
2958 REWRITE_TAC[REAL_LE_REFL; COND_ID];
2959 X_GEN_TAC `f:real^N->real` THEN
2960 ASM_CASES_TAC `s:real^N->bool = {}` THEN
2961 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN STRIP_TAC THEN
2962 EXISTS_TAC `support (+) (f:real^N->real) s` THEN
2963 EXISTS_TAC `f:real^N->real` THEN
2964 MP_TAC(ASSUME `sum s (f:real^N->real) = &1`) THEN
2965 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN
2966 REWRITE_TAC[iterate] THEN COND_CASES_TAC THEN
2967 ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; REAL_OF_NUM_EQ; ARITH] THEN
2968 DISCH_THEN(K ALL_TAC) THEN
2969 UNDISCH_TAC `sum s (f:real^N->real) = &1` THEN
2970 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM SUM_SUPPORT] THEN
2971 ASM_CASES_TAC `support (+) (f:real^N->real) s = {}` THEN
2972 ASM_SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN
2973 DISCH_TAC THEN REWRITE_TAC[SUPPORT_SUBSET] THEN CONJ_TAC THENL
2974 [ASM_SIMP_TAC[support; IN_ELIM_THM]; ALL_TAC] THEN
2975 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
2976 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
2977 REWRITE_TAC[SUPPORT_SUBSET] THEN
2978 REWRITE_TAC[support; IN_ELIM_THM; NEUTRAL_REAL_ADD] THEN
2979 MESON_TAC[VECTOR_MUL_LZERO]]);;
2981 let CONVEX_HULL_UNION_EXPLICIT = prove
2982 (`!s t:real^N->bool.
2983 convex s /\ convex t
2984 ==> convex hull (s UNION t) =
2986 {(&1 - u) % x + u % y | x IN s /\ y IN t /\ &0 <= u /\ u <= &1}`,
2987 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
2988 [REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN GEN_REWRITE_TAC I [SUBSET] THEN
2989 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
2990 MAP_EVERY X_GEN_TAC [`y:real^N`; `u:real^N->bool`; `f:real^N->real`] THEN
2991 REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2992 SUBST1_TAC(SET_RULE `u:real^N->bool = (u INTER s) UNION (u DIFF s)`) THEN
2993 ASM_SIMP_TAC[SUM_UNION; VSUM_UNION; FINITE_INTER; FINITE_DIFF;
2994 SET_RULE `DISJOINT (u INTER s) (u DIFF s)`] THEN
2995 ASM_CASES_TAC `sum (u INTER s) (f:real^N->real) = &0` THENL
2996 [SUBGOAL_THEN `!x. x IN (u INTER s) ==> (f:real^N->real) x = &0`
2998 [ASM_MESON_TAC[SUM_POS_EQ_0; FINITE_INTER; IN_INTER];
2999 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0] THEN
3000 REWRITE_TAC[VECTOR_ADD_LID; REAL_ADD_LID] THEN
3001 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
3002 REWRITE_TAC[IN_UNION] THEN DISJ2_TAC THEN DISJ1_TAC THEN
3003 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
3004 ASM_SIMP_TAC[FINITE_DIFF; IN_DIFF] THEN ASM SET_TAC[]];
3006 ASM_CASES_TAC `sum (u DIFF s) (f:real^N->real) = &0` THENL
3007 [SUBGOAL_THEN `!x. x IN (u DIFF s) ==> (f:real^N->real) x = &0`
3009 [ASM_MESON_TAC[SUM_POS_EQ_0; FINITE_DIFF; IN_DIFF];
3010 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0] THEN
3011 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN
3012 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
3013 REWRITE_TAC[IN_UNION] THEN DISJ1_TAC THEN
3014 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
3015 ASM_SIMP_TAC[FINITE_INTER; IN_INTER] THEN ASM SET_TAC[]];
3017 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
3018 REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN DISJ2_TAC THEN
3019 MAP_EVERY EXISTS_TAC
3020 [`vsum(u INTER s) (\v:real^N. (f v / sum(u INTER s) f) % v)`;
3021 `sum(u DIFF s) (f:real^N->real)`;
3022 `vsum(u DIFF s) (\v:real^N. (f v / sum(u DIFF s) f) % v)`] THEN
3023 REPEAT CONJ_TAC THENL
3024 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
3025 ASM_SIMP_TAC[INTER_SUBSET; FINITE_INTER; SUM_POS_LE; REAL_LE_DIV;
3026 IN_INTER; real_div; SUM_RMUL; REAL_MUL_RINV];
3027 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
3028 ASM_SIMP_TAC[SUBSET_DIFF; FINITE_DIFF; SUM_POS_LE; REAL_LE_DIV;
3029 IN_DIFF; real_div; SUM_RMUL; REAL_MUL_RINV] THEN
3031 ASM_SIMP_TAC[SUM_POS_LE; IN_DIFF; FINITE_DIFF];
3032 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
3033 `a + b = &1 ==> &0 <= a ==> b <= &1`)) THEN
3034 ASM_SIMP_TAC[SUM_POS_LE; IN_INTER; FINITE_INTER];
3035 ASM_SIMP_TAC[GSYM VSUM_LMUL; FINITE_INTER; FINITE_DIFF] THEN
3036 SIMP_TAC[VECTOR_MUL_ASSOC; REAL_ARITH `a * b / c:real = a / c * b`] THEN
3037 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (REAL_ARITH
3038 `a + b = &1 ==> &1 - b = a`)) THEN
3039 ASM_SIMP_TAC[REAL_DIV_REFL; REAL_MUL_LID]];
3040 REWRITE_TAC[GSYM UNION_ASSOC] THEN ONCE_REWRITE_TAC[UNION_SUBSET] THEN
3041 REWRITE_TAC[HULL_SUBSET] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3042 MAP_EVERY X_GEN_TAC [`x:real^N`; `u:real`; `y:real^N`] THEN STRIP_TAC THEN
3043 MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] CONVEX_CONVEX_HULL) THEN
3044 ASM_SIMP_TAC[HULL_INC; IN_UNION]]);;
3046 let CONVEX_HULL_UNION_NONEMPTY_EXPLICIT = prove
3047 (`!s t:real^N->bool.
3048 convex s /\ ~(s = {}) /\ convex t /\ ~(t = {})
3049 ==> convex hull (s UNION t) =
3050 {(&1 - u) % x + u % y | x IN s /\ y IN t /\ &0 <= u /\ u <= &1}`,
3051 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_UNION_EXPLICIT] THEN
3052 SIMP_TAC[SET_RULE `s UNION t UNION u = u <=> s SUBSET u /\ t SUBSET u`] THEN
3053 CONJ_TAC THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `z:real^N` THEN
3055 [MAP_EVERY EXISTS_TAC [`z:real^N`; `&0`] THEN
3056 REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LID; REAL_POS; VECTOR_MUL_LZERO;
3057 VECTOR_ADD_RID] THEN
3059 SUBGOAL_THEN `?a:real^N. a IN s` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3060 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN
3061 MAP_EVERY EXISTS_TAC [`&1`; `z:real^N`] THEN
3062 ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN VECTOR_ARITH_TAC]);;
3064 let CONVEX_HULL_UNION_UNIONS = prove
3065 (`!f s:real^N->bool.
3066 convex(UNIONS f) /\ ~(f = {})
3067 ==> convex hull (s UNION UNIONS f) =
3068 UNIONS {convex hull (s UNION t) | t IN f}`,
3069 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3071 REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN
3072 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]] THEN
3073 ASM_CASES_TAC `s:real^N->bool = {}` THENL
3074 [ASM_SIMP_TAC[UNION_EMPTY; HULL_P; UNIONS_SUBSET] THEN
3075 X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN
3076 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `convex hull u:real^N->bool` THEN
3077 REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[];
3079 ASM_CASES_TAC `UNIONS f :real^N->bool = {}` THENL
3080 [ASM_REWRITE_TAC[UNION_EMPTY] THEN
3081 SUBGOAL_THEN `?u:real^N->bool. u IN f` CHOOSE_TAC THENL
3082 [ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]; ALL_TAC] THEN
3083 MATCH_MP_TAC SUBSET_TRANS THEN
3084 EXISTS_TAC `convex hull (s UNION u:real^N->bool)` THEN
3085 ASM_SIMP_TAC[HULL_MONO; SUBSET_UNION] THEN ASM SET_TAC[];
3087 GEN_REWRITE_TAC LAND_CONV [HULL_UNION_LEFT] THEN
3088 ASM_SIMP_TAC[CONVEX_HULL_UNION_NONEMPTY_EXPLICIT; CONVEX_HULL_EQ_EMPTY;
3089 CONVEX_CONVEX_HULL] THEN
3090 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3091 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_UNIONS] THEN
3092 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
3093 MAP_EVERY X_GEN_TAC [`a:real`; `u:real^N->bool`] THEN DISCH_TAC THEN
3094 X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THEN
3095 REWRITE_TAC[IN_UNIONS; EXISTS_IN_GSPEC] THEN
3096 EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
3097 MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] CONVEX_CONVEX_HULL) THEN
3098 ASM_MESON_TAC[HULL_MONO; IN_UNION; SUBSET; HULL_INC]);;
3100 (* ------------------------------------------------------------------------- *)
3101 (* A stepping theorem for that expansion. *)
3102 (* ------------------------------------------------------------------------- *)
3104 let CONVEX_HULL_FINITE_STEP = prove
3105 (`((?u. (!x. x IN {} ==> &0 <= u x) /\
3107 vsum {} (\x. u(x) % x) = y) <=> w = &0 /\ y = vec 0) /\
3108 (FINITE(s:real^N->bool)
3109 ==> ((?u. (!x. x IN (a INSERT s) ==> &0 <= u x) /\
3110 sum (a INSERT s) u = w /\
3111 vsum (a INSERT s) (\x. u(x) % x) = y) <=>
3113 ?u. (!x. x IN s ==> &0 <= u x) /\
3115 vsum s (\x. u(x) % x) = y - v % a))`,
3116 MP_TAC(ISPEC `\x:real^N y:real. &0 <= y` AFFINE_HULL_FINITE_STEP_GEN) THEN
3117 SIMP_TAC[REAL_ARITH `&0 <= x / &2 <=> &0 <= x`; REAL_LE_ADD] THEN
3118 REWRITE_TAC[RIGHT_AND_EXISTS_THM]);;
3120 (* ------------------------------------------------------------------------- *)
3121 (* Hence some special cases. *)
3122 (* ------------------------------------------------------------------------- *)
3124 let CONVEX_HULL_2 = prove
3125 (`!a b. convex hull {a,b} =
3126 {u % a + v % b | &0 <= u /\ &0 <= v /\ u + v = &1}`,
3127 SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
3128 SIMP_TAC[CONVEX_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
3129 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
3130 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
3131 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
3133 let CONVEX_HULL_2_ALT = prove
3134 (`!a b. convex hull {a,b} = {a + u % (b - a) | &0 <= u /\ u <= &1}`,
3135 ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN
3136 REWRITE_TAC[CONVEX_HULL_2; EXTENSION; IN_ELIM_THM] THEN
3137 REWRITE_TAC[REAL_ADD_ASSOC; CONJ_ASSOC] THEN
3138 REWRITE_TAC[TAUT `(a /\ x + y = &1) /\ b <=> x + y = &1 /\ a /\ b`] THEN
3139 REWRITE_TAC[REAL_ARITH `x + y = &1 <=> y = &1 - x`; UNWIND_THM2] THEN
3140 REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
3141 BINOP_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);;
3143 let CONVEX_HULL_3 = prove
3144 (`convex hull {a,b,c} =
3145 { u % a + v % b + w % c |
3146 &0 <= u /\ &0 <= v /\ &0 <= w /\ u + v + w = &1}`,
3147 SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
3148 SIMP_TAC[CONVEX_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
3149 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
3150 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
3151 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
3153 let CONVEX_HULL_3_ALT = prove
3154 (`!a b c. convex hull {a,b,c} =
3155 {a + u % (b - a) + v % (c - a) |
3156 &0 <= u /\ &0 <= v /\ u + v <= &1}`,
3157 ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`] THEN
3158 REWRITE_TAC[CONVEX_HULL_3; EXTENSION; IN_ELIM_THM] THEN
3159 REWRITE_TAC[REAL_ADD_ASSOC; CONJ_ASSOC] THEN
3160 REWRITE_TAC[TAUT `(a /\ x + y = &1) /\ b <=> x + y = &1 /\ a /\ b`] THEN
3161 REWRITE_TAC[REAL_ARITH `x + y = &1 <=> y = &1 - x`; UNWIND_THM2] THEN
3162 REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
3163 BINOP_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);;
3165 let CONVEX_HULL_SUMS = prove
3166 (`!s t:real^N->bool.
3167 convex hull {x + y | x IN s /\ y IN t} =
3168 {x + y | x IN convex hull s /\ y IN convex hull t}`,
3169 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3170 [MATCH_MP_TAC HULL_MINIMAL THEN
3171 SIMP_TAC[CONVEX_SUMS; CONVEX_CONVEX_HULL] THEN
3172 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3173 REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[HULL_INC];
3174 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3175 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
3176 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [CONVEX_HULL_INDEXED] THEN
3177 REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
3178 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
3180 [`k1:num`; `u1:num->real`; `x1:num->real^N`;
3181 `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN
3185 vsum(1..k1) (\i. vsum(1..k2) (\j. u1 i % u2 j % (x1 i + x2 j)))`
3187 [REWRITE_TAC[VECTOR_ADD_LDISTRIB; VSUM_ADD_NUMSEG] THEN
3188 ASM_SIMP_TAC[VSUM_LMUL; VSUM_RMUL; VECTOR_MUL_LID];
3189 REWRITE_TAC[VSUM_LMUL] THEN MATCH_MP_TAC CONVEX_VSUM THEN
3190 ASM_SIMP_TAC[FINITE_NUMSEG; CONVEX_CONVEX_HULL; IN_NUMSEG] THEN
3191 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN
3192 ASM_SIMP_TAC[FINITE_NUMSEG; CONVEX_CONVEX_HULL; IN_NUMSEG] THEN
3193 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);;
3195 let AFFINE_HULL_PCROSS,CONVEX_HULL_PCROSS = (CONJ_PAIR o prove)
3196 (`(!s:real^M->bool t:real^N->bool.
3197 affine hull (s PCROSS t) =
3198 (affine hull s) PCROSS (affine hull t)) /\
3199 (!s:real^M->bool t:real^N->bool.
3200 convex hull (s PCROSS t) =
3201 (convex hull s) PCROSS (convex hull t))`,
3203 (`!u v x y:real^M z:real^N.
3205 ==> pastecart z (u % x + v % y) =
3206 u % pastecart z x + v % pastecart z y /\
3207 pastecart (u % x + v % y) z =
3208 u % pastecart x z + v % pastecart y z`,
3209 REWRITE_TAC[PASTECART_ADD; GSYM PASTECART_CMUL] THEN
3210 SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID])
3212 (`INTERS {{x | pastecart x y IN u} | y IN t} =
3213 {x | !y. y IN t ==> pastecart x y IN u}`,
3214 REWRITE_TAC[INTERS_GSPEC; EXTENSION; IN_ELIM_THM] THEN SET_TAC[]) in
3216 [REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3217 [MATCH_MP_TAC HULL_MINIMAL THEN
3218 SIMP_TAC[AFFINE_PCROSS; AFFINE_AFFINE_HULL; HULL_SUBSET; PCROSS_MONO];
3219 REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN
3220 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3221 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3222 [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
3223 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3224 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
3225 SUBGOAL_THEN `pastecart (x:real^M) (y:real^N) IN s PCROSS t` MP_TAC
3226 THENL [ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN
3227 REWRITE_TAC[HULL_INC];
3229 REWRITE_TAC[GSYM lemma2] THEN MATCH_MP_TAC AFFINE_INTERS THEN
3230 REWRITE_TAC[FORALL_IN_GSPEC]] THEN
3231 SIMP_TAC[affine; IN_ELIM_THM; lemma1;
3232 ONCE_REWRITE_RULE[affine] AFFINE_AFFINE_HULL]];
3233 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3234 [MATCH_MP_TAC HULL_MINIMAL THEN
3235 SIMP_TAC[CONVEX_PCROSS; CONVEX_CONVEX_HULL; HULL_SUBSET; PCROSS_MONO];
3236 REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN
3237 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3238 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3239 [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
3240 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3241 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
3242 SUBGOAL_THEN `pastecart (x:real^M) (y:real^N) IN s PCROSS t` MP_TAC
3243 THENL [ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN
3244 REWRITE_TAC[HULL_INC];
3246 REWRITE_TAC[GSYM lemma2] THEN MATCH_MP_TAC CONVEX_INTERS THEN
3247 REWRITE_TAC[FORALL_IN_GSPEC]] THEN
3248 SIMP_TAC[convex; IN_ELIM_THM; lemma1;
3249 ONCE_REWRITE_RULE[convex] CONVEX_CONVEX_HULL]]]);;
3251 (* ------------------------------------------------------------------------- *)
3252 (* Relations among closure notions and corresponding hulls. *)
3253 (* ------------------------------------------------------------------------- *)
3255 let SUBSPACE_IMP_AFFINE = prove
3256 (`!s. subspace s ==> affine s`,
3257 REWRITE_TAC[subspace; affine] THEN MESON_TAC[]);;
3259 let AFFINE_IMP_CONVEX = prove
3260 (`!s. affine s ==> convex s`,
3261 REWRITE_TAC[affine; convex] THEN MESON_TAC[]);;
3263 let SUBSPACE_IMP_CONVEX = prove
3264 (`!s. subspace s ==> convex s`,
3265 MESON_TAC[SUBSPACE_IMP_AFFINE; AFFINE_IMP_CONVEX]);;
3267 let AFFINE_HULL_SUBSET_SPAN = prove
3268 (`!s. (affine hull s) SUBSET (span s)`,
3269 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN
3270 REWRITE_TAC[SUBSET; IN; SUBSPACE_IMP_AFFINE]);;
3272 let CONVEX_HULL_SUBSET_SPAN = prove
3273 (`!s. (convex hull s) SUBSET (span s)`,
3274 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN
3275 REWRITE_TAC[SUBSET; IN; SUBSPACE_IMP_CONVEX]);;
3277 let CONVEX_HULL_SUBSET_AFFINE_HULL = prove
3278 (`!s. (convex hull s) SUBSET (affine hull s)`,
3279 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN
3280 REWRITE_TAC[SUBSET; IN; AFFINE_IMP_CONVEX]);;
3282 let COLLINEAR_CONVEX_HULL_COLLINEAR = prove
3283 (`!s:real^N->bool. collinear(convex hull s) <=> collinear s`,
3284 MESON_TAC[COLLINEAR_SUBSET; HULL_SUBSET; SUBSET_TRANS;
3285 COLLINEAR_AFFINE_HULL_COLLINEAR; CONVEX_HULL_SUBSET_AFFINE_HULL]);;
3287 let AFFINE_SPAN = prove
3288 (`!s. affine(span s)`,
3289 SIMP_TAC[SUBSPACE_IMP_AFFINE; SUBSPACE_SPAN]);;
3291 let CONVEX_SPAN = prove
3292 (`!s. convex(span s)`,
3293 SIMP_TAC[SUBSPACE_IMP_CONVEX; SUBSPACE_SPAN]);;
3295 let AFFINE_EQ_SUBSPACE = prove
3296 (`!s:real^N->bool. vec 0 IN s ==> (affine s <=> subspace s)`,
3297 REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[subspace; affine] THEN
3298 DISCH_TAC THEN MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN
3300 [MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN
3301 SUBST1_TAC(VECTOR_ARITH `c % x:real^N = c % x + (&1 - c) % vec 0`) THEN
3302 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
3303 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
3304 STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH
3305 `x + y:real^N = &2 % (&1 / &2 % x + &1 / &2 % y)`) THEN
3306 FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
3307 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);;
3309 let AFFINE_IMP_SUBSPACE = prove
3310 (`!s. affine s /\ vec 0 IN s ==> subspace s`,
3311 SIMP_TAC[GSYM AFFINE_EQ_SUBSPACE]);;
3313 let AFFINE_HULL_EQ_SPAN = prove
3314 (`!s:real^N->bool. (vec 0) IN affine hull s ==> affine hull s = span s`,
3315 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
3316 REWRITE_TAC[AFFINE_HULL_SUBSET_SPAN] THEN
3317 REWRITE_TAC[SUBSET] THEN MATCH_MP_TAC SPAN_INDUCT THEN
3318 ASM_REWRITE_TAC[SUBSET; subspace; IN_ELIM_THM; HULL_INC] THEN
3319 REPEAT STRIP_TAC THENL
3320 [SUBST1_TAC(VECTOR_ARITH
3321 `x + y:real^N = &2 % (&1 / &2 % x + &1 / &2 % y) + --(&1) % vec 0`) THEN
3322 MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN
3323 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN
3324 MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN
3325 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[];
3326 SUBST1_TAC(VECTOR_ARITH
3327 `c % x:real^N = c % x + (&1 - c) % vec 0`) THEN
3328 MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN
3329 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);;
3331 let CLOSED_AFFINE = prove
3332 (`!s:real^N->bool. affine s ==> closed s`,
3333 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
3334 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN
3335 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3336 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
3337 SUBGOAL_THEN `affine (IMAGE (\x:real^N. --a + x) s)
3338 ==> closed (IMAGE (\x:real^N. --a + x) s)`
3340 [DISCH_THEN(fun th -> MATCH_MP_TAC CLOSED_SUBSPACE THEN MP_TAC th) THEN
3341 MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC AFFINE_EQ_SUBSPACE THEN
3342 REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `a:real^N` THEN
3343 ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
3344 REWRITE_TAC[AFFINE_TRANSLATION_EQ; CLOSED_TRANSLATION_EQ]]);;
3346 let CLOSED_AFFINE_HULL = prove
3347 (`!s. closed(affine hull s)`,
3348 SIMP_TAC[CLOSED_AFFINE; AFFINE_AFFINE_HULL]);;
3350 let CLOSURE_SUBSET_AFFINE_HULL = prove
3351 (`!s. closure s SUBSET affine hull s`,
3352 GEN_TAC THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN
3353 REWRITE_TAC[CLOSED_AFFINE_HULL; HULL_SUBSET]);;
3355 let AFFINE_HULL_CLOSURE = prove
3356 (`!s:real^N->bool. affine hull (closure s) = affine hull s`,
3357 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN
3358 REWRITE_TAC[CLOSURE_SUBSET_AFFINE_HULL; AFFINE_AFFINE_HULL] THEN
3359 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
3360 MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[] THEN
3361 ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]);;
3363 let AFFINE_HULL_EQ_SPAN_EQ = prove
3364 (`!s:real^N->bool. (affine hull s = span s) <=> (vec 0) IN affine hull s`,
3365 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[SPAN_0; AFFINE_HULL_EQ_SPAN]);;
3367 let AFFINE_DEPENDENT_IMP_DEPENDENT = prove
3368 (`!s. affine_dependent s ==> dependent s`,
3369 REWRITE_TAC[affine_dependent; dependent] THEN
3370 MESON_TAC[SUBSET; AFFINE_HULL_SUBSET_SPAN]);;
3372 let DEPENDENT_AFFINE_DEPENDENT_CASES = prove
3374 dependent s <=> affine_dependent s \/ (vec 0) IN affine hull s`,
3375 REWRITE_TAC[DEPENDENT_EXPLICIT; AFFINE_DEPENDENT_EXPLICIT;
3376 AFFINE_HULL_EXPLICIT_ALT; IN_ELIM_THM] THEN
3377 GEN_TAC THEN ONCE_REWRITE_TAC[OR_EXISTS_THM] THEN
3378 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
3379 X_GEN_TAC `t:real^N->bool` THEN
3380 ASM_CASES_TAC `FINITE(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN
3381 EQ_TAC THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN
3382 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC))
3384 [ASM_CASES_TAC `sum t (u:real^N->real) = &0` THENL
3385 [ASM_MESON_TAC[]; ALL_TAC] THEN
3386 DISJ2_TAC THEN EXISTS_TAC `\v:real^N. inv(sum t u) * u v` THEN
3387 ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
3388 ASM_SIMP_TAC[VECTOR_MUL_RZERO; REAL_MUL_LINV];
3389 EXISTS_TAC `u:real^N->real` THEN ASM_MESON_TAC[];
3390 EXISTS_TAC `u:real^N->real` THEN
3391 ASM_REWRITE_TAC[SET_RULE
3392 `(?v. v IN t /\ ~p v) <=> ~(!v. v IN t ==> p v)`] THEN
3394 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
3395 `x = &1 ==> x = &0 ==> F`)) THEN
3396 ASM_MESON_TAC[SUM_EQ_0]]);;
3398 let DEPENDENT_IMP_AFFINE_DEPENDENT = prove
3399 (`!a:real^N s. dependent {x - a | x IN s} /\ ~(a IN s)
3400 ==> affine_dependent(a INSERT s)`,
3401 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
3402 REWRITE_TAC[DEPENDENT_EXPLICIT; AFFINE_DEPENDENT_EXPLICIT] THEN
3403 REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
3404 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN
3405 GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN
3406 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
3407 REWRITE_TAC[TAUT `a /\ x = IMAGE f s /\ b <=> x = IMAGE f s /\ a /\ b`] THEN
3408 REWRITE_TAC[UNWIND_THM2; EXISTS_IN_IMAGE] THEN
3409 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` (X_CHOOSE_THEN `t:real^N->bool`
3410 STRIP_ASSUME_TAC)) THEN
3411 FIRST_X_ASSUM(MP_TAC o check (is_eq o concl)) THEN
3412 ASM_SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
3413 ASM_SIMP_TAC[o_DEF; VECTOR_SUB_LDISTRIB; VSUM_SUB; VSUM_RMUL] THEN
3415 MAP_EVERY EXISTS_TAC
3416 [`(a:real^N) INSERT t`;
3417 `\x. if x = a then --sum t (\x. u (x - a))
3418 else (u:real^N->real) (x - a)`] THEN
3419 ASM_REWRITE_TAC[FINITE_INSERT; SUBSET_REFL] THEN
3420 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
3421 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3422 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; ALL_TAC] THEN
3423 REPEAT CONJ_TAC THENL
3424 [MATCH_MP_TAC(REAL_ARITH `x = y ==> --x + y = &0`) THEN
3425 MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[];
3426 EXISTS_TAC `x:real^N` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3427 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
3428 MATCH_MP_TAC(VECTOR_ARITH
3429 `!s. s - t % a = vec 0 /\ s = u ==> --t % a + u = vec 0`) THEN
3430 EXISTS_TAC `vsum t (\x:real^N. u(x - a) % x)` THEN
3431 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC VSUM_EQ THEN
3432 REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN
3433 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]]);;
3435 let AFFINE_DEPENDENT_BIGGERSET = prove
3437 (FINITE s ==> CARD s >= dimindex(:N) + 2) ==> affine_dependent s`,
3438 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
3439 ASM_SIMP_TAC[CARD_CLAUSES; ARITH_RULE `~(0 >= n + 2)`; FINITE_RULES] THEN
3440 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3441 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
3442 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
3443 `x IN s ==> s = x INSERT (s DELETE x)`)) THEN
3444 SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; IN_DELETE] THEN
3445 REWRITE_TAC[ARITH_RULE `SUC x >= n + 2 <=> x > n`] THEN DISCH_TAC THEN
3446 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3447 REWRITE_TAC[IN_DELETE] THEN MATCH_MP_TAC DEPENDENT_BIGGERSET THEN
3448 REWRITE_TAC[SET_RULE `{x - a:real^N | x | x IN s /\ ~(x = a)} =
3449 IMAGE (\x. x - a) (s DELETE a)`] THEN
3450 ASM_SIMP_TAC[FINITE_IMAGE_INJ_EQ;
3451 VECTOR_ARITH `x - a = y - a <=> x:real^N = y`;
3454 let AFFINE_DEPENDENT_BIGGERSET_GENERAL = prove
3455 (`!s:real^N->bool. (FINITE s ==> CARD s >= dim s + 2) ==> affine_dependent s`,
3456 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
3457 ASM_SIMP_TAC[CARD_CLAUSES; ARITH_RULE `~(0 >= n + 2)`; FINITE_RULES] THEN
3458 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3459 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
3460 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
3461 `x IN s ==> s = x INSERT (s DELETE x)`)) THEN
3462 SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; IN_DELETE] THEN
3463 REWRITE_TAC[ARITH_RULE `SUC x >= n + 2 <=> x > n`] THEN DISCH_TAC THEN
3464 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3465 REWRITE_TAC[IN_DELETE] THEN
3466 MATCH_MP_TAC DEPENDENT_BIGGERSET_GENERAL THEN
3467 REWRITE_TAC[SET_RULE `{x - a:real^N | x | x IN s /\ ~(x = a)} =
3468 IMAGE (\x. x - a) (s DELETE a)`] THEN
3469 ASM_SIMP_TAC[FINITE_IMAGE_INJ_EQ; FINITE_DELETE;
3470 VECTOR_ARITH `x - a = y - a <=> x:real^N = y`;
3471 CARD_IMAGE_INJ] THEN
3472 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check(is_imp o concl)) THEN
3473 ASM_REWRITE_TAC[FINITE_DELETE] THEN
3474 MATCH_MP_TAC(ARITH_RULE `c:num <= b ==> (a > b ==> a > c)`) THEN
3475 MATCH_MP_TAC SUBSET_LE_DIM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
3476 SIMP_TAC[SPAN_SUB; SPAN_SUPERSET; IN_INSERT]);;
3478 let AFFINE_INDEPENDENT_IMP_FINITE = prove
3479 (`!s:real^N->bool. ~(affine_dependent s) ==> FINITE s`,
3480 MESON_TAC[AFFINE_DEPENDENT_BIGGERSET]);;
3482 let AFFINE_INDEPENDENT_CARD_LE = prove
3483 (`!s:real^N->bool. ~(affine_dependent s) ==> CARD s <= dimindex(:N) + 1`,
3484 REWRITE_TAC[ARITH_RULE `s <= n + 1 <=> ~(n + 2 <= s)`; CONTRAPOS_THM] THEN
3485 REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_DEPENDENT_BIGGERSET THEN
3486 ASM_REWRITE_TAC[GE]);;
3488 let AFFINE_INDEPENDENT_CONVEX_AFFINE_HULL = prove
3489 (`!s t:real^N->bool.
3490 ~affine_dependent s /\ t SUBSET s
3491 ==> convex hull t = affine hull t INTER convex hull s`,
3492 REPEAT STRIP_TAC THEN
3493 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3494 SUBGOAL_THEN `FINITE(t:real^N->bool)` ASSUME_TAC THENL
3495 [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
3496 MATCH_MP_TAC(SET_RULE
3497 `ct SUBSET a /\ ct SUBSET cs /\ a INTER cs SUBSET ct
3498 ==> ct = a INTER cs`) THEN
3499 ASM_SIMP_TAC[HULL_MONO; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
3500 REWRITE_TAC[SUBSET; IN_INTER; CONVEX_HULL_FINITE; AFFINE_HULL_FINITE] THEN
3501 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
3502 DISCH_THEN(CONJUNCTS_THEN2
3503 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)
3504 (X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC)) THEN
3505 EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[] THEN
3506 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
3507 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
3508 [AFFINE_DEPENDENT_EXPLICIT]) THEN
3509 REWRITE_TAC[NOT_EXISTS_THM] THEN
3510 DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`;
3511 `\x:real^N. if x IN t then v x - u x:real else v x`]) THEN
3512 ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[MESON[]
3513 `(if p then a else b) % x = if p then a % x else b % x`] THEN
3514 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES; SET_RULE
3515 `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
3516 ASM_SIMP_TAC[GSYM DIFF; SUM_DIFF; VSUM_DIFF; VECTOR_SUB_RDISTRIB;
3517 SUM_SUB; VSUM_SUB] THEN
3518 REWRITE_TAC[REAL_ARITH `a - b + b - a = &0`; NOT_EXISTS_THM;
3519 VECTOR_ARITH `a - b + b - a:real^N = vec 0`] THEN
3520 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
3521 ASM_REWRITE_TAC[REAL_SUB_0] THEN ASM SET_TAC[]);;
3523 let DISJOINT_AFFINE_HULL = prove
3524 (`!s t u:real^N->bool.
3525 ~affine_dependent s /\ t SUBSET s /\ u SUBSET s /\ DISJOINT t u
3526 ==> DISJOINT (affine hull t) (affine hull u)`,
3527 REPEAT STRIP_TAC THEN
3528 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3529 SUBGOAL_THEN `FINITE(t:real^N->bool) /\ FINITE (u:real^N->bool)` ASSUME_TAC
3530 THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
3531 REWRITE_TAC[IN_DISJOINT; AFFINE_HULL_FINITE; IN_ELIM_THM] THEN
3532 DISCH_THEN(X_CHOOSE_THEN `y:real^N` MP_TAC) THEN
3533 DISCH_THEN(CONJUNCTS_THEN2
3534 (X_CHOOSE_THEN `a:real^N->real` STRIP_ASSUME_TAC)
3535 (X_CHOOSE_THEN `b:real^N->real` STRIP_ASSUME_TAC)) THEN
3536 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
3537 [AFFINE_DEPENDENT_EXPLICIT]) THEN
3538 REWRITE_TAC[NOT_EXISTS_THM] THEN
3539 MAP_EVERY EXISTS_TAC
3541 `\x:real^N. if x IN t then a x else if x IN u then --(b x) else &0`] THEN
3542 ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[MESON[]
3543 `(if p then a else b) % x = if p then a % x else b % x`] THEN
3544 ASM_SIMP_TAC[SUM_CASES; SUBSET_REFL; VSUM_CASES; GSYM DIFF; SUM_DIFF;
3545 VSUM_DIFF; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
3546 ASM_SIMP_TAC[SUM_0; VSUM_0; VECTOR_MUL_LZERO; SUM_NEG; VSUM_NEG;
3547 VECTOR_MUL_LNEG; SET_RULE `DISJOINT t u ==> ~(x IN t /\ x IN u)`] THEN
3548 REWRITE_TAC[EMPTY_GSPEC; SUM_CLAUSES; VSUM_CLAUSES] THEN
3549 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
3550 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
3551 UNDISCH_TAC `sum t (a:real^N->real) = &1` THEN
3552 ASM_CASES_TAC `!x:real^N. x IN t ==> a x = &0` THEN
3553 ASM_SIMP_TAC[SUM_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN
3554 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
3555 MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);;
3557 let AFFINE_INDEPENDENT_SPAN_EQ = prove
3558 (`!s. ~(affine_dependent s) /\ CARD s = dimindex(:N) + 1
3559 ==> affine hull s = (:real^N)`,
3560 MATCH_MP_TAC SET_PROVE_CASES THEN
3561 REWRITE_TAC[CARD_CLAUSES; ARITH_RULE `~(0 = n + 1)`] THEN
3562 SIMP_TAC[IMP_CONJ; AFFINE_INDEPENDENT_IMP_FINITE; MESON[HAS_SIZE]
3563 `FINITE s ==> (CARD s = n <=> s HAS_SIZE n)`] THEN
3564 X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN
3565 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN
3566 SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; IMP_CONJ] THEN
3567 REWRITE_TAC[ARITH_RULE `SUC n = m + 1 <=> n = m`; GSYM UNIV_SUBSET] THEN
3568 REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
3569 ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV; LE_REFL; independent] THEN
3570 UNDISCH_TAC `~affine_dependent((vec 0:real^N) INSERT s)` THEN
3571 REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_TAC THEN
3572 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3573 ASM_REWRITE_TAC[VECTOR_SUB_RZERO; SET_RULE `{x | x IN s} = s`]);;
3575 let AFFINE_INDEPENDENT_SPAN_GT = prove
3577 ~(affine_dependent s) /\ dimindex(:N) < CARD s
3578 ==> affine hull s = (:real^N)`,
3579 REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_EQ THEN
3580 ASM_REWRITE_TAC[] THEN
3581 MP_TAC(SPEC `s:real^N->bool` AFFINE_DEPENDENT_BIGGERSET) THEN
3582 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE] THEN ASM_ARITH_TAC);;
3584 let EMPTY_INTERIOR_AFFINE_HULL = prove
3586 FINITE s /\ CARD(s) <= dimindex(:N)
3587 ==> interior(affine hull s) = {}`,
3588 REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3589 REWRITE_TAC[AFFINE_HULL_EMPTY; INTERIOR_EMPTY] THEN
3591 `!x s:real^N->bool n.
3592 ~(x IN s) /\ (x INSERT s) HAS_SIZE n /\ n <= dimindex(:N)
3593 ==> interior(affine hull(x INSERT s)) = {}`
3594 (fun th -> MESON_TAC[th; HAS_SIZE; FINITE_INSERT]) THEN
3595 X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN
3596 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN
3597 REWRITE_TAC[HAS_SIZE; FINITE_INSERT; IMP_CONJ] THEN
3598 SIMP_TAC[CARD_CLAUSES] THEN
3599 REPEAT STRIP_TAC THEN MATCH_MP_TAC EMPTY_INTERIOR_LOWDIM THEN
3600 MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN
3601 ASM_SIMP_TAC[DIM_LE_CARD; DIM_SPAN] THEN ASM_ARITH_TAC);;
3603 let EMPTY_INTERIOR_CONVEX_HULL = prove
3605 FINITE s /\ CARD(s) <= dimindex(:N)
3606 ==> interior(convex hull s) = {}`,
3607 REPEAT STRIP_TAC THEN
3608 MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
3609 EXISTS_TAC `interior(affine hull s):real^N->bool` THEN
3610 SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
3611 ASM_SIMP_TAC[EMPTY_INTERIOR_AFFINE_HULL]);;
3613 let AFFINE_DEPENDENT_CHOOSE = prove
3615 ~(affine_dependent s)
3616 ==> (affine_dependent(a INSERT s) <=> ~(a IN s) /\ a IN affine hull s)`,
3617 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THEN
3618 ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN
3619 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3621 [UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN
3622 ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE; AFFINE_HULL_FINITE;
3623 FINITE_INSERT; IN_ELIM_THM; SUM_CLAUSES; VSUM_CLAUSES] THEN
3624 DISCH_TAC THEN REWRITE_TAC[EXISTS_IN_INSERT] THEN
3625 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` MP_TAC) THEN
3626 ASM_CASES_TAC `(u:real^N->real) a = &0` THEN ASM_REWRITE_TAC[] THENL
3627 [REWRITE_TAC[REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
3629 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
3630 DISCH_THEN(MP_TAC o SPEC `u:real^N->real`) THEN ASM_REWRITE_TAC[];
3631 ONCE_REWRITE_TAC[REAL_ARITH `ua + sa = &0 <=> sa = --ua`;
3632 VECTOR_ARITH `va + sa:real^N = vec 0 <=> sa = --va`] THEN
3633 STRIP_TAC THEN EXISTS_TAC `(\x. --(inv(u a)) * u x):real^N->real` THEN
3634 ASM_SIMP_TAC[SUM_LMUL; GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
3635 ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM VECTOR_MUL_LNEG] THEN
3636 REWRITE_TAC[REAL_ARITH `--a * --b:real = a * b`] THEN
3637 ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID]];
3638 DISCH_TAC THEN REWRITE_TAC[affine_dependent] THEN
3639 EXISTS_TAC `a:real^N` THEN
3640 ASM_SIMP_TAC[IN_INSERT; SET_RULE
3641 `~(a IN s) ==> (a INSERT s) DELETE a = s`]]);;
3643 let AFFINE_INDEPENDENT_INSERT = prove
3645 ~(affine_dependent s) /\ ~(a IN affine hull s)
3646 ==> ~(affine_dependent(a INSERT s))`,
3647 SIMP_TAC[AFFINE_DEPENDENT_CHOOSE]);;
3649 let AFFINE_HULL_EXPLICIT_UNIQUE = prove
3650 (`!s:real^N->bool u u'.
3651 ~(affine_dependent s) /\
3652 sum s u = &1 /\ sum s u' = &1 /\
3653 vsum s (\x. u x % x) = vsum s (\x. u' x % x)
3654 ==> !x. x IN s ==> u x = u' x`,
3655 REPEAT GEN_TAC THEN STRIP_TAC THEN
3656 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3657 FIRST_ASSUM(MP_TAC o MATCH_MP AFFINE_DEPENDENT_EXPLICIT_FINITE) THEN
3658 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
3659 DISCH_THEN(MP_TAC o SPEC `(\x. u x - u' x):real^N->real`) THEN
3660 ASM_SIMP_TAC[VSUM_SUB; SUM_SUB; REAL_SUB_REFL; VECTOR_SUB_RDISTRIB;
3661 VECTOR_SUB_REFL; VECTOR_SUB_EQ; REAL_SUB_0] THEN
3664 let INDEPENDENT_IMP_AFFINE_DEPENDENT_0 = prove
3665 (`!s. independent s ==> ~(affine_dependent(vec 0 INSERT s))`,
3666 REWRITE_TAC[independent; DEPENDENT_AFFINE_DEPENDENT_CASES] THEN
3667 SIMP_TAC[DE_MORGAN_THM; AFFINE_INDEPENDENT_INSERT]);;
3669 let AFFINE_INDEPENDENT_STDBASIS = prove
3670 (`~(affine_dependent
3671 ((vec 0:real^N) INSERT {basis i | 1 <= i /\ i <= dimindex (:N)}))`,
3672 SIMP_TAC[INDEPENDENT_IMP_AFFINE_DEPENDENT_0; INDEPENDENT_STDBASIS]);;
3674 (* ------------------------------------------------------------------------- *)
3675 (* Nonempty affine sets are translates of (unique) subspaces. *)
3676 (* ------------------------------------------------------------------------- *)
3678 let AFFINE_TRANSLATION_SUBSPACE = prove
3680 affine t /\ ~(t = {}) <=> ?a s. subspace s /\ t = IMAGE (\x. a + x) s`,
3681 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
3682 ASM_SIMP_TAC[SUBSPACE_IMP_NONEMPTY; IMAGE_EQ_EMPTY;
3683 AFFINE_TRANSLATION; SUBSPACE_IMP_AFFINE] THEN
3684 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3685 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
3686 ONCE_REWRITE_TAC[TRANSLATION_GALOIS] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
3687 REWRITE_TAC[UNWIND_THM2] THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN
3688 ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ; IN_IMAGE] THEN
3689 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);;
3691 let AFFINE_TRANSLATION_UNIQUE_SUBSPACE = prove
3693 affine t /\ ~(t = {}) <=>
3694 ?!s. ?a. subspace s /\ t = IMAGE (\x. a + x) s`,
3695 GEN_TAC THEN REWRITE_TAC[AFFINE_TRANSLATION_SUBSPACE] THEN
3696 MATCH_MP_TAC(MESON[]
3697 `(!a a' s s'. P s a /\ P s' a' ==> s = s')
3698 ==> ((?a s. P s a) <=> (?!s. ?a. P s a))`) THEN
3700 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
3701 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
3702 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TRANSLATION_GALOIS] THEN
3703 DISCH_THEN SUBST1_TAC THEN CONV_TAC SYM_CONV THEN
3704 REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ADD_ASSOC] THEN
3705 MATCH_MP_TAC SUBSPACE_TRANSLATION_SELF THEN ASM_REWRITE_TAC[] THEN
3706 ONCE_REWRITE_TAC[VECTOR_ARITH `--a' + a:real^N = --(a' - a)`] THEN
3707 MATCH_MP_TAC SUBSPACE_NEG THEN ASM_REWRITE_TAC[] THEN
3708 UNDISCH_TAC `t = IMAGE (\x:real^N. a' + x) s'` THEN
3709 DISCH_THEN(MP_TAC o AP_TERM `\s. (a':real^N) IN s`) THEN
3710 REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a:real^N = a + x <=> x = vec 0`] THEN
3711 ASM_SIMP_TAC[UNWIND_THM2; SUBSPACE_0] THEN
3712 REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a':real^N = a + x <=> x = a' - a`] THEN
3713 REWRITE_TAC[UNWIND_THM2]);;
3715 let AFFINE_TRANSLATION_SUBSPACE_EXPLICIT = prove
3716 (`!t:real^N->bool a.
3718 ==> subspace {x - a | x IN t} /\
3719 t = IMAGE (\x. a + x) {x - a | x IN t}`,
3720 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFFINE_DIFFS_SUBSPACE] THEN
3721 ASM_REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN
3722 REWRITE_TAC[o_DEF; VECTOR_SUB_ADD2; IMAGE_ID]);;
3724 (* ------------------------------------------------------------------------- *)
3725 (* If we take a slice out of a set, we can do it perpendicularly, *)
3726 (* with the normal vector to the slice parallel to the affine hull. *)
3727 (* ------------------------------------------------------------------------- *)
3729 let AFFINE_PARALLEL_SLICE = prove
3732 ==> s INTER {x | a dot x <= b} = {} \/ s SUBSET {x | a dot x <= b} \/
3733 ?a' b'. ~(a' = vec 0) /\
3735 s INTER {x | a' dot x <= b'} = s INTER {x | a dot x <= b} /\
3736 s INTER {x | a' dot x = b'} = s INTER {x | a dot x = b} /\
3737 !w. w IN s ==> (w + a') IN s`,
3738 REPEAT STRIP_TAC THEN
3739 ASM_CASES_TAC `s INTER {x:real^N | a dot x = b} = {}` THENL
3740 [MATCH_MP_TAC(TAUT `~(~p /\ ~q) ==> p \/ q \/ r`) THEN
3741 REPEAT STRIP_TAC THEN SUBGOAL_THEN
3742 `?u v:real^N. u IN s /\ v IN s /\
3743 a dot u <= b /\ ~(a dot v <= b)`
3744 STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3745 SUBGOAL_THEN `(a:real^N) dot u < b` ASSUME_TAC THENL
3746 [ASM_REWRITE_TAC[REAL_LT_LE] THEN ASM SET_TAC[]; ALL_TAC] THEN
3747 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
3748 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
3749 REWRITE_TAC[NOT_IN_EMPTY; IN_INTER; NOT_FORALL_THM; IN_ELIM_THM] THEN
3751 `u + (b - a dot u) / (a dot v - a dot u) % (v - u):real^N` THEN
3752 ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF] THEN
3753 REWRITE_TAC[DOT_RADD; DOT_RMUL; DOT_RSUB] THEN
3754 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
3755 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3756 DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN
3757 REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN POP_ASSUM MP_TAC THEN
3758 GEN_GEOM_ORIGIN_TAC `z:real^N` ["a"; "a'"; "b'"; "w"] THEN
3759 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
3760 REWRITE_TAC[VECTOR_ADD_RID; FORALL_IN_IMAGE] THEN
3761 REWRITE_TAC[DOT_RADD; REAL_ARITH `a + x <= a <=> x <= &0`] THEN
3762 SUBGOAL_THEN `subspace(s:real^N->bool) /\ span s = s`
3763 STRIP_ASSUME_TAC THENL
3764 [ASM_MESON_TAC[AFFINE_IMP_SUBSPACE; SPAN_EQ_SELF]; ALL_TAC] THEN
3765 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`]
3766 ORTHOGONAL_SUBSPACE_DECOMP_EXISTS) THEN
3767 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; orthogonal] THEN
3768 MAP_EVERY X_GEN_TAC [`a':real^N`; `a'':real^N`] THEN
3769 ASM_CASES_TAC `a':real^N = vec 0` THENL
3770 [ASM_REWRITE_TAC[VECTOR_ADD_LID] THEN
3771 ASM_CASES_TAC `a'':real^N = a` THEN ASM_REWRITE_TAC[] THEN
3772 SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_LE_REFL];
3774 STRIP_TAC THEN REPEAT DISJ2_TAC THEN
3775 EXISTS_TAC `a':real^N` THEN ASM_REWRITE_TAC[] THEN
3776 EXISTS_TAC `(a':real^N) dot z` THEN
3777 REPEAT(CONJ_TAC THENL
3778 [MATCH_MP_TAC(SET_RULE
3779 `(!x. x IN s ==> (p x <=> q x))
3780 ==> s INTER {x | p x} = s INTER {x | q x}`) THEN
3781 ASM_SIMP_TAC[DOT_LADD] THEN REAL_ARITH_TAC;
3783 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
3784 EXISTS_TAC `x + a':real^N` THEN
3785 ASM_SIMP_TAC[SUBSPACE_ADD; VECTOR_ADD_ASSOC]]);;
3787 (* ------------------------------------------------------------------------- *)
3788 (* Affine dimension. *)
3789 (* ------------------------------------------------------------------------- *)
3791 let MAXIMAL_AFFINE_INDEPENDENT_SUBSET = prove
3792 (`!s b:real^N->bool.
3793 b SUBSET s /\ ~(affine_dependent b) /\
3794 (!b'. b SUBSET b' /\ b' SUBSET s /\ ~(affine_dependent b') ==> b' = b)
3795 ==> s SUBSET (affine hull b)`,
3796 REPEAT STRIP_TAC THEN
3797 MATCH_MP_TAC(SET_RULE `(!a. a IN t /\ ~(a IN s) ==> F) ==> t SUBSET s`) THEN
3798 X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN
3799 FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N) INSERT b`) THEN
3800 FIRST_ASSUM(MP_TAC o MATCH_MP
3801 (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] HULL_INC)) THEN
3802 ASM_SIMP_TAC[AFFINE_INDEPENDENT_INSERT; INSERT_SUBSET] THEN
3805 let MAXIMAL_AFFINE_INDEPENDENT_SUBSET_AFFINE = prove
3806 (`!s b:real^N->bool.
3807 affine s /\ b SUBSET s /\ ~(affine_dependent b) /\
3808 (!b'. b SUBSET b' /\ b' SUBSET s /\ ~(affine_dependent b') ==> b' = b)
3809 ==> affine hull b = s`,
3810 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3811 [ASM_MESON_TAC[HULL_MONO; HULL_P];
3812 ASM_MESON_TAC[MAXIMAL_AFFINE_INDEPENDENT_SUBSET]]);;
3814 let EXTEND_TO_AFFINE_BASIS = prove
3815 (`!s u:real^N->bool.
3816 ~(affine_dependent s) /\ s SUBSET u
3817 ==> ?t. ~(affine_dependent t) /\ s SUBSET t /\ t SUBSET u /\
3818 affine hull t = affine hull u`,
3819 REPEAT STRIP_TAC THEN
3820 MP_TAC(SPEC `\n. ?t:real^N->bool. ~(affine_dependent t) /\ s SUBSET t /\
3821 t SUBSET u /\ CARD t = n`
3823 DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN REWRITE_TAC[] THEN ANTS_TAC THENL
3824 [ASM_MESON_TAC[SUBSET_REFL; AFFINE_INDEPENDENT_CARD_LE]; ALL_TAC] THEN
3825 DISCH_THEN(X_CHOOSE_THEN `n:num` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
3826 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
3827 ASM_REWRITE_TAC[] THEN
3828 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3829 [ASM_MESON_TAC[HULL_MONO; HULL_P]; ALL_TAC] THEN
3830 MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN
3831 MATCH_MP_TAC MAXIMAL_AFFINE_INDEPENDENT_SUBSET THEN ASM_REWRITE_TAC[] THEN
3832 X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN
3833 FIRST_X_ASSUM(MP_TAC o SPEC `CARD(c:real^N->bool)`) THEN
3834 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
3835 DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN
3836 ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN
3837 CONV_TAC SYM_CONV THEN MATCH_MP_TAC CARD_SUBSET_LE THEN
3838 ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]);;
3840 let AFFINE_BASIS_EXISTS = prove
3842 ?b. ~(affine_dependent b) /\ b SUBSET s /\
3843 affine hull b = affine hull s`,
3845 MP_TAC(ISPECL [`{}:real^N->bool`; `s:real^N->bool`]
3846 EXTEND_TO_AFFINE_BASIS) THEN
3847 REWRITE_TAC[AFFINE_INDEPENDENT_EMPTY; EMPTY_SUBSET]);;
3849 let aff_dim = new_definition
3851 @d:int. ?b. affine hull b = affine hull s /\ ~(affine_dependent b) /\
3852 &(CARD b) = d + &1`;;
3855 (`!s. ?b. affine hull b = affine hull s /\
3856 ~(affine_dependent b) /\
3857 aff_dim s = &(CARD b) - &1`,
3859 REWRITE_TAC[aff_dim; INT_ARITH `y:int = x + &1 <=> x = y - &1`] THEN
3860 CONV_TAC SELECT_CONV THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
3861 REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN
3862 MESON_TAC[AFFINE_BASIS_EXISTS]);;
3864 let AFF_DIM_EMPTY = prove
3865 (`aff_dim {} = -- &1`,
3866 REWRITE_TAC[aff_dim; AFFINE_HULL_EMPTY; AFFINE_HULL_EQ_EMPTY] THEN
3867 REWRITE_TAC[UNWIND_THM2; AFFINE_INDEPENDENT_EMPTY; CARD_CLAUSES] THEN
3868 REWRITE_TAC[INT_ARITH `&0 = d + &1 <=> d:int = -- &1`; SELECT_REFL]);;
3870 let AFF_DIM_AFFINE_HULL = prove
3871 (`!s. aff_dim(affine hull s) = aff_dim s`,
3872 REWRITE_TAC[aff_dim; HULL_HULL]);;
3874 let AFF_DIM_TRANSLATION_EQ = prove
3875 (`!a:real^N s. aff_dim (IMAGE (\x. a + x) s) = aff_dim s`,
3876 REWRITE_TAC[aff_dim] THEN GEOM_TRANSLATE_TAC[] THEN
3877 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a /\ b ==> ~c)`] THEN
3878 SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; CARD_IMAGE_INJ;
3879 VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]);;
3881 add_translation_invariants [AFF_DIM_TRANSLATION_EQ];;
3883 let AFFINE_INDEPENDENT_CARD_DIM_DIFFS = prove
3885 ~affine_dependent s /\ a IN s
3886 ==> CARD s = dim {x - a | x IN s} + 1`,
3887 REPEAT STRIP_TAC THEN
3888 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3889 MATCH_MP_TAC(ARITH_RULE `~(s = 0) /\ v = s - 1 ==> s = v + 1`) THEN
3890 ASM_SIMP_TAC[CARD_EQ_0] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3891 MATCH_MP_TAC DIM_UNIQUE THEN
3892 EXISTS_TAC `{b - a:real^N |b| b IN (s DELETE a)}` THEN REPEAT CONJ_TAC THENL
3894 REWRITE_TAC[SIMPLE_IMAGE; SUBSET; FORALL_IN_IMAGE] THEN
3895 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `x:real^N = a` THENL
3896 [ASM_REWRITE_TAC[VECTOR_SUB_REFL; SPAN_0];
3897 MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]];
3898 UNDISCH_TAC `~affine_dependent(s:real^N->bool)` THEN
3899 REWRITE_TAC[independent; CONTRAPOS_THM] THEN DISCH_TAC THEN
3900 SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL
3901 [ASM SET_TAC[]; ALL_TAC] THEN
3902 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3903 ASM_REWRITE_TAC[IN_DELETE];
3904 REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN
3905 SIMP_TAC[VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
3906 ASM_SIMP_TAC[HAS_SIZE; FINITE_DELETE; CARD_DELETE]]);;
3908 let AFF_DIM_DIM_AFFINE_DIFFS = prove
3909 (`!a:real^N s. affine s /\ a IN s ==> aff_dim s = &(dim {x - a | x IN s})`,
3910 REPEAT STRIP_TAC THEN
3911 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN
3912 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` MP_TAC) THEN
3913 ASM_CASES_TAC `b:real^N->bool = {}` THENL
3914 [ASM_MESON_TAC[AFFINE_HULL_EQ_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN
3916 ASM_REWRITE_TAC[INT_EQ_SUB_RADD; INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN
3917 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3918 DISCH_THEN(X_CHOOSE_TAC `c:real^N`) THEN MATCH_MP_TAC EQ_TRANS THEN
3919 EXISTS_TAC `dim {x - c:real^N | x IN b} + 1` THEN CONJ_TAC THENL
3920 [MATCH_MP_TAC AFFINE_INDEPENDENT_CARD_DIM_DIFFS THEN ASM_REWRITE_TAC[];
3922 MATCH_MP_TAC EQ_TRANS THEN
3923 EXISTS_TAC `dim {x - c:real^N | x IN affine hull b} + 1` THEN CONJ_TAC THENL
3924 [ASM_SIMP_TAC[DIFFS_AFFINE_HULL_SPAN; DIM_SPAN]; ALL_TAC] THEN
3925 ASM_REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
3926 SUBGOAL_THEN `affine hull s:real^N->bool = s` SUBST1_TAC THENL
3927 [ASM_MESON_TAC[AFFINE_HULL_EQ]; ALL_TAC] THEN
3928 SUBGOAL_THEN `(c:real^N) IN s` ASSUME_TAC THENL
3929 [ASM_MESON_TAC[AFFINE_HULL_EQ; HULL_INC]; ALL_TAC] THEN
3930 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3931 REWRITE_TAC[IN_ELIM_THM] THEN
3932 SIMP_TAC[VECTOR_ARITH `x - c:real^N = y - a <=> y = x + &1 % (a - c)`] THEN
3933 ASM_MESON_TAC[IN_AFFINE_ADD_MUL_DIFF]);;
3935 let AFF_DIM_DIM_0 = prove
3936 (`!s:real^N->bool. vec 0 IN affine hull s ==> aff_dim s = &(dim s)`,
3937 REPEAT STRIP_TAC THEN
3938 MP_TAC(ISPECL [`vec 0:real^N`; `affine hull s:real^N->bool`]
3939 AFF_DIM_DIM_AFFINE_DIFFS) THEN
3940 ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; VECTOR_SUB_RZERO] THEN
3941 REWRITE_TAC[AFF_DIM_AFFINE_HULL; SET_RULE `{x | x IN s} = s`] THEN
3942 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; DIM_SPAN]);;
3944 let AFF_DIM_DIM_SUBSPACE = prove
3945 (`!s:real^N->bool. subspace s ==> aff_dim s = &(dim s)`,
3946 MESON_TAC[AFF_DIM_DIM_0; SUBSPACE_0; HULL_INC]);;
3948 let AFF_DIM_LINEAR_IMAGE_LE = prove
3949 (`!f:real^M->real^N s. linear f ==> aff_dim(IMAGE f s) <= aff_dim s`,
3950 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
3951 ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN
3952 MP_TAC(ISPEC `s:real^M->bool` AFFINE_AFFINE_HULL) THEN
3953 SPEC_TAC(`affine hull s:real^M->bool`,`s:real^M->bool`) THEN
3954 GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN
3955 ASM_REWRITE_TAC[IMAGE_CLAUSES; AFF_DIM_EMPTY; INT_LE_REFL] THEN
3956 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3957 DISCH_THEN(X_CHOOSE_TAC `a:real^M`) THEN
3958 SUBGOAL_THEN `dim {x - f(a) |x| x IN IMAGE (f:real^M->real^N) s} <=
3959 dim {x - a | x IN s}`
3961 [REWRITE_TAC[SET_RULE `{f x | x IN IMAGE g s} = {f (g x) | x IN s}`] THEN
3962 ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN
3963 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN
3964 MATCH_MP_TAC DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[];
3965 MATCH_MP_TAC EQ_IMP THEN REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN
3966 BINOP_TAC THEN CONV_TAC SYM_CONV THEN
3967 MATCH_MP_TAC AFF_DIM_DIM_AFFINE_DIFFS THEN
3968 ASM_SIMP_TAC[AFFINE_LINEAR_IMAGE; FUN_IN_IMAGE]]);;
3970 let AFF_DIM_INJECTIVE_LINEAR_IMAGE = prove
3971 (`!f:real^M->real^N s.
3972 linear f /\ (!x y. f x = f y ==> x = y)
3973 ==> aff_dim(IMAGE f s) = aff_dim s`,
3974 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN
3975 CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_LINEAR_IMAGE_LE]; ALL_TAC] THEN
3976 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
3977 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
3978 DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
3979 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC
3980 `aff_dim(IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) s))` THEN
3982 [ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; INT_LE_REFL];
3983 MATCH_MP_TAC AFF_DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[]]);;
3985 add_linear_invariants [AFF_DIM_INJECTIVE_LINEAR_IMAGE];;
3987 let AFF_DIM_AFFINE_INDEPENDENT = prove
3989 ~(affine_dependent b) ==> aff_dim b = &(CARD b) - &1`,
3990 GEN_TAC THEN ASM_CASES_TAC `b:real^N->bool = {}` THENL
3991 [ASM_REWRITE_TAC[CARD_CLAUSES; AFF_DIM_EMPTY] THEN INT_ARITH_TAC;
3993 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3994 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN DISCH_TAC THEN
3995 MP_TAC(ISPECL [`b:real^N->bool`; `a:real^N`]
3996 AFFINE_INDEPENDENT_CARD_DIM_DIFFS) THEN
3997 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
3998 REWRITE_TAC[GSYM INT_OF_NUM_ADD; INT_ARITH `(a + b) - b:int = a`] THEN
3999 MP_TAC(ISPECL [`a:real^N`; `affine hull b:real^N->bool`]
4000 AFF_DIM_DIM_AFFINE_DIFFS) THEN
4001 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC; AFF_DIM_AFFINE_HULL] THEN
4002 DISCH_THEN(K ALL_TAC) THEN AP_TERM_TAC THEN
4003 ASM_MESON_TAC[DIFFS_AFFINE_HULL_SPAN; DIM_SPAN]);;
4005 let AFF_DIM_UNIQUE = prove
4006 (`!s b:real^N->bool.
4007 affine hull b = affine hull s /\ ~(affine_dependent b)
4008 ==> aff_dim s = &(CARD b) - &1`,
4009 MESON_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_AFFINE_INDEPENDENT]);;
4011 let AFF_DIM_SING = prove
4012 (`!a:real^N. aff_dim {a} = &0`,
4013 GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
4014 EXISTS_TAC `&(CARD {a:real^N}) - &1:int` THEN CONJ_TAC THENL
4015 [MATCH_MP_TAC AFF_DIM_AFFINE_INDEPENDENT THEN
4016 REWRITE_TAC[AFFINE_INDEPENDENT_1];
4017 SIMP_TAC[CARD_CLAUSES; FINITE_RULES; ARITH; NOT_IN_EMPTY; INT_SUB_REFL]]);;
4019 let AFF_DIM_LE_CARD = prove
4020 (`!s:real^N->bool. FINITE s ==> aff_dim s <= &(CARD s) - &1`,
4021 MATCH_MP_TAC SET_PROVE_CASES THEN
4022 SIMP_TAC[AFF_DIM_EMPTY; CARD_CLAUSES] THEN CONV_TAC INT_REDUCE_CONV THEN
4023 GEOM_ORIGIN_TAC `a:real^N` THEN
4024 SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC] THEN
4025 SIMP_TAC[CARD_IMAGE_INJ; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
4026 SIMP_TAC[DIM_INSERT_0; INT_LE_SUB_LADD; CARD_CLAUSES; FINITE_INSERT] THEN
4027 REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_LE; ADD1; LE_ADD_RCANCEL] THEN
4028 SIMP_TAC[DIM_LE_CARD]);;
4030 let AFF_DIM_GE = prove
4031 (`!s:real^N->bool. -- &1 <= aff_dim s`,
4032 GEN_TAC THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN
4033 STRIP_TAC THEN ASM_REWRITE_TAC[INT_LE_SUB_LADD; INT_ADD_LINV; INT_POS]);;
4035 let AFF_DIM_SUBSET = prove
4036 (`!s t:real^N->bool. s SUBSET t ==> aff_dim s <= aff_dim t`,
4037 MATCH_MP_TAC SET_PROVE_CASES THEN REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EMPTY] THEN
4038 GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN
4039 SUBGOAL_THEN `(vec 0:real^N) IN t` ASSUME_TAC THENL
4040 [ASM SET_TAC[]; ALL_TAC] THEN
4041 ASM_SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC; INT_OF_NUM_LE; DIM_SUBSET]);;
4043 let AFF_DIM_LE_DIM = prove
4044 (`!s:real^N->bool. aff_dim s <= &(dim s)`,
4045 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
4046 ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN
4047 MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[SPAN_INC]);;
4049 let AFF_DIM_CONVEX_HULL = prove
4050 (`!s:real^N->bool. aff_dim(convex hull s) = aff_dim s`,
4051 GEN_TAC THEN MATCH_MP_TAC(INT_ARITH
4052 `!c:int. c = a /\ a <= b /\ b <= c ==> b = a`) THEN
4053 EXISTS_TAC `aff_dim(affine hull s:real^N->bool)` THEN
4054 SIMP_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_SUBSET; HULL_SUBSET;
4055 CONVEX_HULL_SUBSET_AFFINE_HULL]);;
4057 let AFF_DIM_CLOSURE = prove
4058 (`!s:real^N->bool. aff_dim(closure s) = aff_dim s`,
4059 GEN_TAC THEN MATCH_MP_TAC(INT_ARITH
4060 `!h. h = s /\ s <= c /\ c <= h ==> c:int = s`) THEN
4061 EXISTS_TAC `aff_dim(affine hull s:real^N->bool)` THEN
4062 REPEAT CONJ_TAC THENL
4063 [REWRITE_TAC[AFF_DIM_AFFINE_HULL];
4064 MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET];
4065 MATCH_MP_TAC AFF_DIM_SUBSET THEN
4066 MATCH_MP_TAC CLOSURE_MINIMAL THEN
4067 REWRITE_TAC[CLOSED_AFFINE_HULL; HULL_SUBSET]]);;
4069 let AFF_DIM_2 = prove
4070 (`!a b:real^N. aff_dim {a,b} = if a = b then &0 else &1`,
4071 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
4072 [ASM_REWRITE_TAC[INSERT_AC; AFF_DIM_SING]; ALL_TAC] THEN
4073 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `&(CARD {a:real^N,b}) - &1:int` THEN
4074 ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; AFFINE_INDEPENDENT_2] THEN
4075 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_INSERT; NOT_IN_EMPTY] THEN
4076 CONV_TAC NUM_REDUCE_CONV THEN INT_ARITH_TAC);;
4078 let AFF_DIM_EQ_MINUS1 = prove
4079 (`!s:real^N->bool. aff_dim s = -- &1 <=> s = {}`,
4080 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_EMPTY] THEN
4081 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
4082 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN
4083 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
4084 MATCH_MP_TAC(INT_ARITH `&0:int <= n ==> ~(n = -- &1)`) THEN
4085 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim {a:real^N}` THEN
4086 ASM_SIMP_TAC[AFF_DIM_SUBSET; SING_SUBSET] THEN
4087 REWRITE_TAC[AFF_DIM_SING; INT_LE_REFL]);;
4089 let AFF_DIM_POS_LE = prove
4090 (`!s:real^N->bool. &0 <= aff_dim s <=> ~(s = {})`,
4091 GEN_TAC THEN REWRITE_TAC[GSYM AFF_DIM_EQ_MINUS1] THEN
4092 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_GE) THEN INT_ARITH_TAC);;
4094 let AFF_DIM_EQ_0 = prove
4095 (`!s:real^N->bool. aff_dim s = &0 <=> ?a. s = {a}`,
4096 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_SING; LEFT_IMP_EXISTS_THM] THEN
4097 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN
4098 CONV_TAC INT_REDUCE_CONV THEN DISCH_TAC THEN
4099 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
4100 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
4101 MATCH_MP_TAC(SET_RULE
4102 `(!b. ~(b = a) /\ {a,b} SUBSET s ==> F) ==> a IN s ==> s = {a}`) THEN
4103 X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN
4104 FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN
4105 MP_TAC(ISPECL [`a:real^N`; `b:real^N`] AFF_DIM_2) THEN
4106 ASM_SIMP_TAC[] THEN INT_ARITH_TAC);;
4108 let CONNECTED_IMP_PERFECT_AFF_DIM = prove
4110 connected s /\ ~(aff_dim s = &0) /\ x IN s ==> x limit_point_of s`,
4111 REWRITE_TAC[AFF_DIM_EQ_0; CONNECTED_IMP_PERFECT]);;
4113 let AFF_DIM_UNIV = prove
4114 (`aff_dim(:real^N) = &(dimindex(:N))`,
4115 SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_UNIV; DIM_UNIV]);;
4117 let AFF_DIM_EQ_AFFINE_HULL = prove
4118 (`!s t:real^N->bool.
4119 s SUBSET t /\ aff_dim t <= aff_dim s
4120 ==> affine hull s = affine hull t`,
4121 MATCH_MP_TAC SET_PROVE_CASES THEN
4122 SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; AFF_DIM_GE;
4123 INT_ARITH `a:int <= x ==> (x <= a <=> x = a)`] THEN
4124 X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN
4125 SIMP_TAC[INSERT_SUBSET; IMP_CONJ; AFF_DIM_DIM_0; IN_INSERT; DIM_EQ_SPAN;
4126 HULL_INC; AFFINE_HULL_EQ_SPAN; INT_OF_NUM_LE]);;
4128 let AFF_DIM_SUMS_INTER = prove
4129 (`!s t:real^N->bool.
4130 affine s /\ affine t /\ ~(s INTER t = {})
4131 ==> aff_dim {x + y | x IN s /\ y IN t} =
4132 (aff_dim s + aff_dim t) - aff_dim(s INTER t)`,
4133 REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN
4134 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN
4135 GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN
4136 GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN X_GEN_TAC `a:real^N` THEN
4137 GEOM_ORIGIN_TAC `a:real^N` THEN
4138 REWRITE_TAC[VECTOR_ARITH `(a + x) + (a + y):real^N = &2 % a + (x + y)`] THEN
4139 ONCE_REWRITE_TAC[SET_RULE `{a + x + y:real^N | x IN s /\ y IN t} =
4140 IMAGE (\x. a + x) {x + y | x IN s /\ y IN t}`] THEN
4141 REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; IN_INTER] THEN
4142 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN STRIP_TAC THEN
4144 SUBGOAL_THEN `(vec 0:real^N) IN {x + y | x IN s /\ y IN t}` ASSUME_TAC THENL
4145 [REWRITE_TAC[IN_ELIM_THM] THEN REPEAT(EXISTS_TAC `vec 0:real^N`) THEN
4146 ASM_REWRITE_TAC[VECTOR_ADD_LID];
4148 ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; IN_INTER] THEN
4149 REWRITE_TAC[INT_EQ_SUB_LADD; INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN
4150 MATCH_MP_TAC DIM_SUMS_INTER THEN ASM_SIMP_TAC[AFFINE_IMP_SUBSPACE]);;
4152 let AFF_DIM_PSUBSET = prove
4153 (`!s t. (affine hull s) PSUBSET (affine hull t) ==> aff_dim s < aff_dim t`,
4154 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4155 SIMP_TAC[PSUBSET; AFF_DIM_SUBSET; INT_LT_LE] THEN
4156 MESON_TAC[INT_EQ_IMP_LE; AFF_DIM_EQ_AFFINE_HULL; HULL_HULL]);;
4158 let AFF_DIM_EQ_FULL = prove
4159 (`!s. aff_dim s = &(dimindex(:N)) <=> affine hull s = (:real^N)`,
4160 GEN_TAC THEN EQ_TAC THENL
4161 [DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM AFFINE_HULL_UNIV] THEN
4162 MATCH_MP_TAC AFF_DIM_EQ_AFFINE_HULL THEN
4163 ASM_REWRITE_TAC[SUBSET_UNIV; AFF_DIM_UNIV; INT_LE_REFL];
4164 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4165 SIMP_TAC[AFF_DIM_UNIV]]);;
4167 let AFF_DIM_LE_UNIV = prove
4168 (`!s:real^N->bool. aff_dim s <= &(dimindex(:N))`,
4169 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN
4170 MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[SUBSET_UNIV]);;
4172 let AFFINE_INDEPENDENT_IFF_CARD = prove
4174 ~affine_dependent s <=> FINITE s /\ aff_dim s = &(CARD s) - &1`,
4175 GEN_TAC THEN EQ_TAC THEN
4176 SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; AFFINE_INDEPENDENT_IMP_FINITE] THEN
4177 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4178 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN DISCH_TAC THEN
4179 X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC
4180 (ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN
4181 MATCH_MP_TAC(ARITH_RULE `!b:int. a <= b - &1 /\ b < s ==> ~(a = s - &1)`) THEN
4182 EXISTS_TAC `&(CARD(b:real^N->bool)):int` THEN CONJ_TAC THENL
4183 [ASM_MESON_TAC[AFF_DIM_LE_CARD; FINITE_SUBSET; AFF_DIM_AFFINE_HULL];
4184 REWRITE_TAC[INT_OF_NUM_LT] THEN MATCH_MP_TAC CARD_PSUBSET THEN
4185 ASM_REWRITE_TAC[PSUBSET] THEN ASM_MESON_TAC[]]);;
4187 let AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR = prove
4188 (`!s t:real^N->bool.
4189 convex s /\ ~(s INTER interior t = {})
4190 ==> affine hull (s INTER t) = affine hull s`,
4191 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM;
4192 LEFT_IMP_EXISTS_THM] THEN
4193 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`; `a:real^N`] THEN
4194 GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN
4195 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
4196 SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
4197 SIMP_TAC[SUBSET_HULL; AFFINE_AFFINE_HULL] THEN
4198 FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SIMP_RULE[SUBSET] INTERIOR_SUBSET)) THEN
4199 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INTER] THEN
4200 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN
4201 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_CBALL_0] THEN
4202 X_GEN_TAC `e:real` THEN STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_UNIV] THEN
4203 X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
4204 ASM_SIMP_TAC[SPAN_SUPERSET; IN_INTER] THEN DISCH_TAC THEN
4205 ABBREV_TAC `k = min (&1 / &2) (e / norm(x:real^N))` THEN
4206 SUBGOAL_THEN `&0 < k /\ k < &1` STRIP_ASSUME_TAC THENL
4207 [EXPAND_TAC "k" THEN
4208 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; NORM_POS_LT; REAL_MIN_LT] THEN
4209 CONV_TAC REAL_RAT_REDUCE_CONV;
4211 SUBGOAL_THEN `x:real^N = inv k % k % x` SUBST1_TAC THENL
4212 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID;
4215 MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN
4216 REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL
4217 [ONCE_REWRITE_TAC[VECTOR_ARITH
4218 `k % x:real^N = (&1 - k) % vec 0 + k % x`] THEN
4219 MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE];
4220 FIRST_X_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "k" THEN
4221 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
4222 ASM_REAL_ARITH_TAC]);;
4224 let AFFINE_HULL_CONVEX_INTER_OPEN = prove
4225 (`!s t:real^N->bool.
4226 convex s /\ open t /\ ~(s INTER t = {})
4227 ==> affine hull (s INTER t) = affine hull s`,
4228 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR; INTERIOR_OPEN]);;
4230 let AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR = prove
4231 (`!s t:real^N->bool.
4232 affine s /\ ~(s INTER interior t = {})
4233 ==> affine hull (s INTER t) = s`,
4234 SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR; AFFINE_IMP_CONVEX;
4237 let AFFINE_HULL_AFFINE_INTER_OPEN = prove
4238 (`!s t:real^N->bool.
4239 affine s /\ open t /\ ~(s INTER t = {})
4240 ==> affine hull (s INTER t) = s`,
4241 SIMP_TAC[AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR; INTERIOR_OPEN]);;
4243 let CONVEX_AND_AFFINE_INTER_OPEN = prove
4244 (`!s t u:real^N->bool.
4245 convex s /\ affine t /\ open u /\
4246 s INTER u = t INTER u /\ ~(s INTER u = {})
4247 ==> affine hull s = t`,
4248 REPEAT STRIP_TAC THEN
4249 MATCH_MP_TAC(MESON[] `!u v. x = u /\ u = v /\ v = y ==> x = y`) THEN
4250 MAP_EVERY EXISTS_TAC
4251 [`affine hull (s INTER u:real^N->bool)`;
4252 `affine hull t:real^N->bool`] THEN
4253 REPEAT CONJ_TAC THENL
4254 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN
4256 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN
4257 ASM_SIMP_TAC[AFFINE_IMP_CONVEX] THEN ASM SET_TAC[];
4258 ASM_REWRITE_TAC[AFFINE_HULL_EQ]]);;
4260 let AFFINE_HULL_CONVEX_INTER_OPEN_IN = prove
4261 (`!s t:real^N->bool.
4262 convex s /\ open_in (subtopology euclidean (affine hull s)) t /\
4264 ==> affine hull (s INTER t) = affine hull s`,
4265 REPEAT STRIP_TAC THEN
4266 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4267 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4268 ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t INTER u = s INTER u`;
4270 MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM SET_TAC[]);;
4272 let AFFINE_HULL_AFFINE_INTER_OPEN_IN = prove
4273 (`!s t:real^N->bool.
4274 affine s /\ open_in (subtopology euclidean s) t /\ ~(s INTER t = {})
4275 ==> affine hull (s INTER t) = s`,
4276 REPEAT STRIP_TAC THEN
4277 MP_TAC(ISPECL [`affine hull s:real^N->bool`; `t:real^N->bool`]
4278 AFFINE_HULL_CONVEX_INTER_OPEN_IN) THEN
4279 ASM_SIMP_TAC[HULL_HULL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; HULL_P]);;
4281 let AFFINE_HULL_CONVEX_INTER_OPEN_IN = prove
4282 (`!s t:real^N->bool.
4283 convex s /\ open_in (subtopology euclidean (affine hull s)) t /\
4285 ==> affine hull (s INTER t) = affine hull s`,
4286 REPEAT STRIP_TAC THEN
4287 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4288 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4289 ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t INTER u = s INTER u`;
4291 MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM SET_TAC[]);;
4293 let AFFINE_HULL_AFFINE_INTER_OPEN_IN = prove
4294 (`!s t:real^N->bool.
4295 affine s /\ open_in (subtopology euclidean s) t /\ ~(s INTER t = {})
4296 ==> affine hull (s INTER t) = s`,
4297 REPEAT STRIP_TAC THEN
4298 MP_TAC(ISPECL [`affine hull s:real^N->bool`; `t:real^N->bool`]
4299 AFFINE_HULL_CONVEX_INTER_OPEN_IN) THEN
4300 ASM_SIMP_TAC[HULL_HULL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; HULL_P]);;
4302 let AFFINE_HULL_OPEN_IN = prove
4303 (`!s t:real^N->bool.
4304 open_in (subtopology euclidean (affine hull t)) s /\ ~(s = {})
4305 ==> affine hull s = affine hull t`,
4306 REPEAT STRIP_TAC THEN
4307 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4308 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4309 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC AFFINE_HULL_AFFINE_INTER_OPEN THEN
4310 REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM SET_TAC[]);;
4312 let AFFINE_HULL_OPEN = prove
4313 (`!s. open s /\ ~(s = {}) ==> affine hull s = (:real^N)`,
4314 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4315 SUBST1_TAC(SET_RULE `s = (:real^N) INTER s`) THEN
4316 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN; CONVEX_UNIV] THEN
4317 REWRITE_TAC[AFFINE_HULL_UNIV]);;
4319 let AFFINE_HULL_NONEMPTY_INTERIOR = prove
4320 (`!s. ~(interior s = {}) ==> affine hull s = (:real^N)`,
4321 REPEAT STRIP_TAC THEN
4322 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
4323 EXISTS_TAC `affine hull (interior s:real^N->bool)` THEN
4324 SIMP_TAC[HULL_MONO; INTERIOR_SUBSET] THEN
4325 ASM_SIMP_TAC[AFFINE_HULL_OPEN; OPEN_INTERIOR]);;
4327 let AFF_DIM_OPEN = prove
4328 (`!s:real^N->bool. open s /\ ~(s = {}) ==> aff_dim s = &(dimindex(:N))`,
4329 SIMP_TAC[AFF_DIM_EQ_FULL; AFFINE_HULL_OPEN]);;
4331 let AFF_DIM_NONEMPTY_INTERIOR = prove
4332 (`!s:real^N->bool. ~(interior s = {}) ==> aff_dim s = &(dimindex(:N))`,
4333 SIMP_TAC[AFF_DIM_EQ_FULL; AFFINE_HULL_NONEMPTY_INTERIOR]);;
4335 let SPAN_OPEN = prove
4336 (`!s. open s /\ ~(s = {}) ==> span s = (:real^N)`,
4337 REPEAT STRIP_TAC THEN
4338 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
4339 EXISTS_TAC `affine hull s:real^N->bool` THEN
4340 ASM_SIMP_TAC[AFFINE_HULL_OPEN; AFFINE_HULL_SUBSET_SPAN]);;
4342 let DIM_OPEN = prove
4343 (`!s:real^N->bool. open s /\ ~(s = {}) ==> dim s = dimindex(:N)`,
4344 SIMP_TAC[DIM_EQ_FULL; SPAN_OPEN]);;
4346 let AFF_DIM_INSERT = prove
4348 aff_dim (a INSERT s) =
4349 if a IN affine hull s then aff_dim s else aff_dim s + &1`,
4350 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC SET_PROVE_CASES THEN
4351 SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_SING; AFFINE_HULL_EMPTY; NOT_IN_EMPTY] THEN
4352 CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN
4353 MAP_EVERY X_GEN_TAC [`b:real^N`; `s:real^N->bool`; `a:real^N`] THEN
4354 GEOM_ORIGIN_TAC `b:real^N` THEN
4355 SIMP_TAC[AFFINE_HULL_EQ_SPAN; AFF_DIM_DIM_0; HULL_INC; IN_INSERT] THEN
4356 MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN
4357 DISCH_THEN(K ALL_TAC) THEN
4358 SPEC_TAC(`(vec 0:real^N) INSERT s`,`s:real^N->bool`) THEN
4359 SIMP_TAC[DIM_INSERT; INT_OF_NUM_ADD] THEN MESON_TAC[]);;
4361 let AFFINE_BOUNDED_EQ_TRIVIAL = prove
4363 affine s ==> (bounded s <=> s = {} \/ ?a. s = {a})`,
4364 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
4365 ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN
4366 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
4367 DISCH_THEN(X_CHOOSE_THEN `b:real^N` MP_TAC) THEN
4368 GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN
4369 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SUBSPACE_BOUNDED_EQ_TRIVIAL] THEN
4370 FIRST_ASSUM(MP_TAC o MATCH_MP SUBSPACE_0) THEN SET_TAC[]);;
4372 let AFFINE_BOUNDED_EQ_LOWDIM = prove
4374 affine s ==> (bounded s <=> aff_dim s <= &0)`,
4375 SIMP_TAC[AFF_DIM_GE; INT_ARITH
4376 `--(&1):int <= x ==> (x <= &0 <=> x = --(&1) \/ x = &0)`] THEN
4377 SIMP_TAC[AFF_DIM_EQ_0; AFF_DIM_EQ_MINUS1; AFFINE_BOUNDED_EQ_TRIVIAL]);;
4379 let COLLINEAR_AFF_DIM = prove
4380 (`!s:real^N->bool. collinear s <=> aff_dim s <= &1`,
4381 GEN_TAC THEN EQ_TAC THENL
4382 [REWRITE_TAC[COLLINEAR_AFFINE_HULL; LEFT_IMP_EXISTS_THM] THEN
4383 MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN STRIP_TAC THEN
4384 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim{u:real^N,v}` THEN
4386 [ASM_MESON_TAC[AFF_DIM_SUBSET; AFF_DIM_AFFINE_HULL];
4387 MATCH_MP_TAC INT_LE_TRANS THEN
4388 EXISTS_TAC `&(CARD{u:real^N,v}) - &1:int` THEN
4389 SIMP_TAC[AFF_DIM_LE_CARD; FINITE_INSERT; FINITE_EMPTY] THEN
4390 REWRITE_TAC[INT_ARITH `x - &1:int <= &1 <=> x <= &2`; INT_OF_NUM_LE] THEN
4391 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ARITH_TAC];
4392 ONCE_REWRITE_TAC[GSYM COLLINEAR_AFFINE_HULL_COLLINEAR;
4393 GSYM AFF_DIM_AFFINE_HULL] THEN
4394 MP_TAC(ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN
4395 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) THEN
4396 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I
4397 [AFFINE_INDEPENDENT_IFF_CARD]) THEN
4398 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4399 ASM_REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR;
4400 AFF_DIM_AFFINE_HULL] THEN
4401 REWRITE_TAC[INT_ARITH `x - &1:int <= &1 <=> x <= &2`; INT_OF_NUM_LE] THEN
4402 ASM_SIMP_TAC[COLLINEAR_SMALL]]);;
4404 let HOMEOMORPHIC_AFFINE_SETS = prove
4405 (`!s:real^M->bool t:real^N->bool.
4406 affine s /\ affine t /\ aff_dim s = aff_dim t ==> s homeomorphic t`,
4408 ASM_CASES_TAC `t:real^N->bool = {}` THEN
4409 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN
4410 POP_ASSUM MP_TAC THEN
4411 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN
4412 ASM_CASES_TAC `s:real^M->bool = {}` THEN
4413 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN
4414 POP_ASSUM MP_TAC THEN REWRITE_TAC
4415 [GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN
4416 MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`] THEN
4417 GEOM_ORIGIN_TAC `a:real^M` THEN GEOM_ORIGIN_TAC `b:real^N` THEN
4418 SIMP_TAC[AFFINE_EQ_SUBSPACE; AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_EQ] THEN
4419 MESON_TAC[HOMEOMORPHIC_SUBSPACES]);;
4421 let AFF_DIM_OPEN_IN = prove
4422 (`!s t:real^N->bool.
4423 ~(s = {}) /\ open_in (subtopology euclidean t) s /\ affine t
4424 ==> aff_dim s = aff_dim t`,
4426 REWRITE_TAC[IMP_CONJ; GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN
4427 X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN
4428 REPEAT GEN_TAC THEN STRIP_TAC THEN DISCH_TAC THEN
4429 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
4430 SUBGOAL_THEN `(vec 0:real^N) IN t` ASSUME_TAC THENL
4431 [ASM SET_TAC[]; ALL_TAC] THEN
4432 ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; AFFINE_EQ_SUBSPACE] THEN
4433 DISCH_TAC THEN AP_TERM_TAC THEN
4434 ASM_SIMP_TAC[GSYM LE_ANTISYM; DIM_SUBSET] THEN
4435 SUBGOAL_THEN `?e. &0 < e /\ cball(vec 0:real^N,e) INTER t SUBSET s`
4437 [FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4438 FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o
4439 GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
4441 REWRITE_TAC[SUBSET; IN_INTER; IN_CBALL_0] THEN
4442 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)] THEN
4443 FIRST_ASSUM(MP_TAC o MATCH_MP ORTHONORMAL_BASIS_SUBSPACE) THEN
4444 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4445 MP_TAC(ISPECL [`s:real^N->bool`; `IMAGE (\x:real^N. e % x) b`]
4446 INDEPENDENT_CARD_LE_DIM) THEN
4447 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
4448 ASM_SIMP_TAC[CARD_IMAGE_INJ; VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ] THEN
4449 ANTS_TAC THENL [REWRITE_TAC[SUBSET]; MESON_TAC[]] THEN CONJ_TAC THENL
4450 [REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN
4451 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NORM_MUL] THEN
4452 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC SUBSPACE_MUL] THEN
4454 MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE THEN
4455 ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; LINEAR_SCALING]]);;
4457 let DIM_OPEN_IN = prove
4458 (`!s t:real^N->bool.
4459 ~(s = {}) /\ open_in (subtopology euclidean t) s /\ subspace t
4461 REPEAT STRIP_TAC THEN
4462 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
4463 ASM_SIMP_TAC[GSYM LE_ANTISYM; DIM_SUBSET] THEN
4464 REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN
4465 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim(s:real^N->bool)` THEN
4466 REWRITE_TAC[AFF_DIM_LE_DIM] THEN ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE] THEN
4467 MATCH_MP_TAC INT_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN
4468 MATCH_MP_TAC AFF_DIM_OPEN_IN THEN ASM_SIMP_TAC[SUBSPACE_IMP_AFFINE]);;
4470 let AFF_DIM_CONVEX_INTER_NONEMPTY_INTERIOR = prove
4471 (`!s t:real^N->bool.
4472 convex s /\ ~(s INTER interior t = {})
4473 ==> aff_dim(s INTER t) = aff_dim s`,
4474 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4475 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR] THEN
4476 REWRITE_TAC[AFF_DIM_AFFINE_HULL]);;
4478 let AFF_DIM_CONVEX_INTER_OPEN = prove
4479 (`!s t:real^N->bool.
4480 convex s /\ open t /\ ~(s INTER t = {})
4481 ==> aff_dim(s INTER t) = aff_dim s`,
4482 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4483 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN] THEN
4484 REWRITE_TAC[AFF_DIM_AFFINE_HULL]);;
4486 let AFFINE_HULL_HALFSPACE_LT = prove
4487 (`!a b. affine hull {x | a dot x < b} =
4488 if a = vec 0 /\ b <= &0 then {} else (:real^N)`,
4489 REPEAT GEN_TAC THEN COND_CASES_TAC THEN
4490 ASM_SIMP_TAC[AFFINE_HULL_EQ_EMPTY; HALFSPACE_EQ_EMPTY_LT;
4491 AFFINE_HULL_OPEN; OPEN_HALFSPACE_LT]);;
4493 let AFFINE_HULL_HALFSPACE_LE = prove
4494 (`!a b. affine hull {x | a dot x <= b} =
4495 if a = vec 0 /\ b < &0 then {} else (:real^N)`,
4496 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL
4497 [ASM_SIMP_TAC[DOT_LZERO; SET_RULE `{x | p} = if p then UNIV else {}`] THEN
4498 COND_CASES_TAC THEN ASM_SIMP_TAC[AFFINE_HULL_EMPTY; AFFINE_HULL_UNIV] THEN
4499 COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
4500 ASM_SIMP_TAC[GSYM CLOSURE_HALFSPACE_LT; AFFINE_HULL_CLOSURE] THEN
4501 ASM_REWRITE_TAC[AFFINE_HULL_HALFSPACE_LT]]);;
4503 let AFFINE_HULL_HALFSPACE_GT = prove
4504 (`!a b. affine hull {x | a dot x > b} =
4505 if a = vec 0 /\ b >= &0 then {} else (:real^N)`,
4506 REPEAT GEN_TAC THEN COND_CASES_TAC THEN
4507 ASM_SIMP_TAC[AFFINE_HULL_EQ_EMPTY; HALFSPACE_EQ_EMPTY_GT;
4508 AFFINE_HULL_OPEN; OPEN_HALFSPACE_GT]);;
4510 let AFFINE_HULL_HALFSPACE_GE = prove
4511 (`!a b. affine hull {x | a dot x >= b} =
4512 if a = vec 0 /\ b > &0 then {} else (:real^N)`,
4514 MP_TAC(ISPECL [`--a:real^N`; `--b:real`] AFFINE_HULL_HALFSPACE_LE) THEN
4515 SIMP_TAC[real_ge; DOT_LNEG; REAL_LE_NEG2; VECTOR_NEG_EQ_0] THEN
4516 REWRITE_TAC[REAL_ARITH `--b < &0 <=> b > &0`]);;
4518 let AFF_DIM_HALFSPACE_LT = prove
4520 aff_dim {x | a dot x < b} =
4521 if a = vec 0 /\ b <= &0 then --(&1) else &(dimindex(:N))`,
4522 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4523 SIMP_TAC[AFFINE_HULL_HALFSPACE_LT] THEN COND_CASES_TAC THEN
4524 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4526 let AFF_DIM_HALFSPACE_LE = prove
4528 aff_dim {x | a dot x <= b} =
4529 if a = vec 0 /\ b < &0 then --(&1) else &(dimindex(:N))`,
4530 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4531 SIMP_TAC[AFFINE_HULL_HALFSPACE_LE] THEN COND_CASES_TAC THEN
4532 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4534 let AFF_DIM_HALFSPACE_GT = prove
4536 aff_dim {x | a dot x > b} =
4537 if a = vec 0 /\ b >= &0 then --(&1) else &(dimindex(:N))`,
4538 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4539 SIMP_TAC[AFFINE_HULL_HALFSPACE_GT] THEN COND_CASES_TAC THEN
4540 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4542 let AFF_DIM_HALFSPACE_GE = prove
4544 aff_dim {x | a dot x >= b} =
4545 if a = vec 0 /\ b > &0 then --(&1) else &(dimindex(:N))`,
4546 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4547 SIMP_TAC[AFFINE_HULL_HALFSPACE_GE] THEN COND_CASES_TAC THEN
4548 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4550 let CHOOSE_AFFINE_SUBSET = prove
4551 (`!s:real^N->bool d.
4552 affine s /\ --(&1) <= d /\ d <= aff_dim s
4553 ==> ?t. affine t /\ t SUBSET s /\ aff_dim t = d`,
4554 REPEAT GEN_TAC THEN ASM_CASES_TAC `d:int = --(&1)` THENL
4555 [STRIP_TAC THEN EXISTS_TAC `{}:real^N->bool` THEN
4556 ASM_REWRITE_TAC[EMPTY_SUBSET; AFFINE_EMPTY; AFF_DIM_EMPTY];
4557 ASM_SIMP_TAC[INT_ARITH
4558 `~(d:int = --(&1)) ==> (--(&1) <= d <=> &0 <= d)`] THEN
4559 POP_ASSUM(K ALL_TAC)] THEN
4560 ASM_CASES_TAC `s:real^N->bool = {}` THENL
4561 [ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN INT_ARITH_TAC;
4562 POP_ASSUM MP_TAC] THEN
4563 GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN
4564 DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN
4565 GEOM_ORIGIN_TAC `a:real^N` THEN
4566 SIMP_TAC[IMP_CONJ; AFF_DIM_DIM_SUBSPACE; AFFINE_EQ_SUBSPACE] THEN
4567 REPEAT GEN_TAC THEN DISCH_TAC THEN DISCH_TAC THEN
4568 REWRITE_TAC[GSYM INT_OF_NUM_EXISTS] THEN
4569 DISCH_THEN(X_CHOOSE_THEN `n:num` SUBST1_TAC) THEN
4570 REWRITE_TAC[INT_OF_NUM_LE] THEN DISCH_TAC THEN
4571 MP_TAC(ISPECL [`s:real^N->bool`; `n:num`]
4572 CHOOSE_SUBSPACE_OF_SUBSPACE) THEN
4573 ASM_SIMP_TAC[SPAN_OF_SUBSPACE] THEN
4574 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN
4575 ASM_SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_IMP_AFFINE]);;
4577 (* ------------------------------------------------------------------------- *)
4578 (* Existence of a rigid transform between congruent sets. *)
4579 (* ------------------------------------------------------------------------- *)
4581 let RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS = prove
4582 (`!x:A->real^N y:A->real^N s.
4583 (!i j. i IN s /\ j IN s ==> dist(x i,x j) = dist(y i,y j))
4584 ==> ?a f. orthogonal_transformation f /\
4585 !i. i IN s ==> y i = a + f(x i)`,
4587 (`!x:(real^N)^M y:(real^N)^M.
4588 (!i j. 1 <= i /\ i <= dimindex(:M) /\
4589 1 <= j /\ j <= dimindex(:M)
4590 ==> dist(x$i,x$j) = dist(y$i,y$j))
4591 ==> ?a f. orthogonal_transformation f /\
4592 !i. 1 <= i /\ i <= dimindex(:M)
4593 ==> y$i = a + f(x$i)`,
4594 REPEAT STRIP_TAC THEN
4595 ABBREV_TAC `(X:real^M^N) = lambda i j. (x:real^N^M)$j$i - x$1$i` THEN
4596 ABBREV_TAC `(Y:real^M^N) = lambda i j. (y:real^N^M)$j$i - y$1$i` THEN
4597 SUBGOAL_THEN `transp(X:real^M^N) ** X = transp(Y:real^M^N) ** Y`
4599 [REWRITE_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN] THEN
4600 MAP_EVERY EXPAND_TAC ["X"; "Y"] THEN
4601 SIMP_TAC[CART_EQ; column; LAMBDA_BETA; dot] THEN
4602 REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT; GSYM dot] THEN
4603 REWRITE_TAC[DOT_NORM_SUB; VECTOR_ARITH
4604 `(x - a) - (y - a):real^N = x - y`] THEN
4605 ASM_SIMP_TAC[GSYM dist; DIMINDEX_GE_1; LE_REFL];
4608 `?M:real^N^N. orthogonal_matrix M /\ (Y:real^M^N) = M ** (X:real^M^N)`
4609 (CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL
4611 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [CART_EQ] THEN
4612 MAP_EVERY EXPAND_TAC ["X"; "Y"] THEN
4613 SIMP_TAC[LAMBDA_BETA; matrix_mul] THEN
4614 REWRITE_TAC[REAL_ARITH `x - y:real = z <=> x = y + z`] THEN STRIP_TAC THEN
4615 EXISTS_TAC `(y:real^N^M)$1 - (M:real^N^N) ** (x:real^N^M)$1` THEN
4616 EXISTS_TAC `\x:real^N. (M:real^N^N) ** x` THEN
4617 ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX;
4618 MATRIX_OF_MATRIX_VECTOR_MUL; MATRIX_VECTOR_MUL_LINEAR] THEN
4619 SIMP_TAC[CART_EQ; matrix_vector_mul; LAMBDA_BETA;
4620 VECTOR_ADD_COMPONENT] THEN
4621 ASM_SIMP_TAC[REAL_SUB_LDISTRIB; SUM_SUB_NUMSEG] THEN
4622 REWRITE_TAC[VECTOR_SUB_COMPONENT; REAL_ARITH
4623 `a + y - b:real = a - z + y <=> z = b`] THEN
4624 SIMP_TAC[LAMBDA_BETA]] THEN
4625 MP_TAC(ISPEC `transp(X:real^M^N) ** X`
4626 SYMMETRIC_MATRIX_DIAGONALIZABLE_EXPLICIT) THEN
4627 REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_TRANSP; LEFT_IMP_EXISTS_THM] THEN
4628 MAP_EVERY X_GEN_TAC [`P:real^M^M`; `d:num->real`] THEN
4629 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4630 DISCH_THEN(fun th -> MP_TAC th THEN ASM_REWRITE_TAC[] THEN MP_TAC th) THEN
4631 REWRITE_TAC[MATRIX_MUL_ASSOC; GSYM MATRIX_TRANSP_MUL] THEN
4632 REWRITE_TAC[GSYM MATRIX_MUL_ASSOC; LEFT_IMP_EXISTS_THM] THEN
4633 REWRITE_TAC[IMP_IMP] THEN
4634 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [CART_EQ] THEN
4635 SIMP_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN; LAMBDA_BETA] THEN STRIP_TAC THEN
4636 MP_TAC(ISPECL [`\i. column i ((X:real^M^N) ** (P:real^M^M))`;
4637 `\i. column i ((Y:real^M^N) ** (P:real^M^M))`;
4639 ORTHOGONAL_TRANSFORMATION_BETWEEN_ORTHOGONAL_SETS) THEN
4640 REWRITE_TAC[IN_NUMSEG] THEN ANTS_TAC THENL
4641 [ASM_SIMP_TAC[pairwise; IN_NUMSEG; NORM_EQ; orthogonal]; ALL_TAC] THEN
4642 DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
4643 EXISTS_TAC `matrix(f:real^N->real^N)` THEN CONJ_TAC THENL
4644 [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX]; ALL_TAC] THEN
4646 `!M:real^M^N. M = M ** (P:real^M^M) ** transp P`
4647 (fun th -> GEN_REWRITE_TAC BINOP_CONV [th])
4649 [ASM_MESON_TAC[orthogonal_matrix; MATRIX_MUL_RID];
4650 REWRITE_TAC[MATRIX_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC] THEN
4651 REWRITE_TAC[GSYM MATRIX_MUL_ASSOC] THEN
4652 ASM_SIMP_TAC[MATRIX_EQUAL_COLUMNS] THEN
4653 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
4654 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthogonal_transformation]) THEN
4655 DISCH_THEN(ASSUME_TAC o GSYM o MATCH_MP MATRIX_WORKS o CONJUNCT1) THEN
4656 ASM_REWRITE_TAC[] THEN
4657 SIMP_TAC[CART_EQ; matrix_vector_mul; column; LAMBDA_BETA] THEN
4658 X_GEN_TAC `j:num` THEN STRIP_TAC THEN
4659 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [matrix_mul] THEN
4660 ASM_SIMP_TAC[LAMBDA_BETA]) in
4661 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:A->bool = {}` THENL
4662 [REPEAT STRIP_TAC THEN
4663 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `\x:real^N. x`] THEN
4664 ASM_REWRITE_TAC[NOT_IN_EMPTY; ORTHOGONAL_TRANSFORMATION_ID];
4665 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
4666 DISCH_THEN(X_CHOOSE_TAC `m:A`) THEN DISCH_TAC] THEN
4668 `?r. IMAGE r (1..dimindex(:(N,1)finite_sum)) SUBSET s /\
4669 affine hull (IMAGE (y o r) (1..dimindex(:(N,1)finite_sum))) =
4670 affine hull (IMAGE (y:A->real^N) s)`
4672 [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
4673 SIMP_TAC[IMAGE_o; TAUT `p /\ q <=> ~(p ==> ~q)`;
4674 HULL_MONO; IMAGE_SUBSET] THEN REWRITE_TAC[NOT_IMP] THEN
4675 MP_TAC(ISPEC `IMAGE (y:A->real^N) s` AFFINE_BASIS_EXISTS) THEN
4676 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4677 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFFINE_INDEPENDENT_IFF_CARD]) THEN
4679 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN
4680 DISCH_THEN(X_CHOOSE_THEN `f:num->real^N` STRIP_ASSUME_TAC) THEN
4681 SUBGOAL_THEN `CARD(b:real^N->bool) <= dimindex(:(N,1)finite_sum)`
4683 [REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN
4684 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH
4685 `a:int = c - &1 ==> a + &1 <= n ==> c <= n`)) THEN
4686 REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN
4687 REWRITE_TAC[INT_LE_RADD; AFF_DIM_LE_UNIV];
4689 UNDISCH_TAC `b SUBSET IMAGE (y:A->real^N) s` THEN
4690 ONCE_ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
4691 REWRITE_TAC[IN_IMAGE] THEN
4692 GEN_REWRITE_TAC (LAND_CONV o DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
4693 REWRITE_TAC[SKOLEM_THM; IN_NUMSEG] THEN
4694 DISCH_THEN(X_CHOOSE_THEN `r:num->A` STRIP_ASSUME_TAC) THEN
4695 EXISTS_TAC `\i. if i <= CARD(b:real^N->bool) then r i else (m:A)` THEN
4696 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
4697 UNDISCH_THEN `affine hull b:real^N->bool = affine hull IMAGE y (s:A->bool)`
4698 (SUBST1_TAC o SYM) THEN
4699 REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC HULL_MONO THEN
4700 ONCE_ASM_REWRITE_TAC[] THEN
4701 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG] THEN
4702 X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM IMAGE_o] THEN
4703 REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN EXISTS_TAC `i:num` THEN
4704 ASM_REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[LE_TRANS];
4705 REWRITE_TAC[SUBSET; IN_NUMSEG; FORALL_IN_IMAGE] THEN
4706 STRIP_TAC THEN MP_TAC(ISPECL
4707 [`(lambda i. x(r i:A)):real^N^(N,1)finite_sum`;
4708 `(lambda i. y(r i:A)):real^N^(N,1)finite_sum`] lemma) THEN
4709 ASM_SIMP_TAC[LAMBDA_BETA] THEN
4710 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
4711 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN
4712 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4713 X_GEN_TAC `k:A` THEN STRIP_TAC THEN
4716 affine hull IMAGE (y o (r:num->A)) (1..dimindex(:(N,1)finite_sum))
4717 ==> dist(z,y k) = dist(z,a + (f:real^N->real^N)(x k))`
4719 [MATCH_MP_TAC SAME_DISTANCES_TO_AFFINE_HULL THEN
4720 REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_NUMSEG] THEN
4721 X_GEN_TAC `j:num` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
4722 EXISTS_TAC `dist(x(r(j:num)),(x:A->real^N) k)` THEN CONJ_TAC THENL
4723 [CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[];
4724 REWRITE_TAC[dist] THEN ASM_SIMP_TAC[NORM_ARITH
4725 `(a + x) - (a + y):real^N = x - y`] THEN
4726 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION; LINEAR_SUB]];
4727 ASM_SIMP_TAC[NORM_ARITH
4728 `a:real^N = b <=> dist(a:real^N,a) = dist(a,b)`] THEN
4729 DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HULL_INC THEN
4730 REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[]]]);;
4732 let RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS_STRONG = prove
4733 (`!x:A->real^N y:A->real^N s t.
4734 t SUBSET s /\ affine hull (IMAGE y t) = affine hull (IMAGE y s) /\
4735 (!i j. i IN s /\ j IN t ==> dist(x i,x j) = dist(y i,y j))
4736 ==> ?a f. orthogonal_transformation f /\
4737 !i. i IN s ==> y i = a + f(x i)`,
4738 REPEAT STRIP_TAC THEN
4739 MP_TAC(ISPECL [`x:A->real^N`; `y:A->real^N`; `t:A->bool`]
4740 RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS) THEN
4741 ANTS_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
4742 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
4743 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN
4744 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:A` THEN DISCH_TAC THEN
4746 `!z. z IN affine hull (IMAGE (y:A->real^N) t)
4747 ==> dist(z,y i) = dist(z,a + (f:real^N->real^N)(x i))`
4749 [MATCH_MP_TAC SAME_DISTANCES_TO_AFFINE_HULL THEN
4750 REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_NUMSEG] THEN
4751 X_GEN_TAC `j:A` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
4752 EXISTS_TAC `dist(a + f(x(j:A):real^N):real^N,a + f(x i))` THEN
4753 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
4754 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = dist(x,y)`] THEN
4755 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_ISOMETRY; DIST_SYM];
4756 ASM_SIMP_TAC[NORM_ARITH
4757 `a:real^N = b <=> dist(a:real^N,a) = dist(a,b)`] THEN
4758 DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HULL_INC THEN
4759 REWRITE_TAC[IN_IMAGE] THEN ASM_MESON_TAC[]]);;
4761 let RIGID_TRANSFORMATION_BETWEEN_3 = prove
4762 (`!a b c a' b' c':real^N.
4763 dist(a,b) = dist(a',b') /\
4764 dist(b,c) = dist(b',c') /\
4765 dist(c,a) = dist(c',a')
4766 ==> ?k f. orthogonal_transformation f /\
4767 a' = k + f a /\ b' = k + f b /\ c' = k + f c`,
4768 REPEAT STRIP_TAC THEN
4770 [`FST:real^N#real^N->real^N`; `SND:real^N#real^N->real^N`;
4771 `{(a:real^N,a':real^N), (b,b'), (c,c')}`]
4772 RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS) THEN
4773 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_INSERT] THEN
4774 REWRITE_TAC[NOT_IN_EMPTY; IMP_IMP] THEN DISCH_THEN MATCH_MP_TAC THEN
4775 ASM_MESON_TAC[DIST_REFL; DIST_SYM]);;
4777 let RIGID_TRANSFORMATION_BETWEEN_2 = prove
4778 (`!a b a' b':real^N.
4779 dist(a,b) = dist(a',b')
4780 ==> ?k f. orthogonal_transformation f /\
4781 a' = k + f a /\ b' = k + f b`,
4782 REPEAT STRIP_TAC THEN
4783 MP_TAC(ISPECL [`a:real^N`; `b:real^N`; `a:real^N`;
4784 `a':real^N`; `b':real^N`; `a':real^N`]
4785 RIGID_TRANSFORMATION_BETWEEN_3) THEN
4786 ASM_MESON_TAC[DIST_EQ_0; DIST_SYM]);;
4788 (* ------------------------------------------------------------------------- *)
4789 (* Caratheodory's theorem. *)
4790 (* ------------------------------------------------------------------------- *)
4792 let CONVEX_HULL_CARATHEODORY_AFF_DIM = prove
4793 (`!p. convex hull p =
4794 {y:real^N | ?s u. FINITE s /\ s SUBSET p /\
4795 &(CARD s) <= aff_dim p + &1 /\
4796 (!x. x IN s ==> &0 <= u x) /\
4797 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
4798 GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN
4799 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN
4800 EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
4801 MATCH_MP_TAC(TAUT `!q. (p ==> q) /\ (q ==> r) ==> (p ==> r)`) THEN
4802 EXISTS_TAC `?n s u. CARD s = n /\
4803 FINITE s /\ s SUBSET p /\
4804 (!x. x IN s ==> &0 <= u x) /\
4805 sum s u = &1 /\ vsum s (\v. u v % v) = (y:real^N)` THEN
4806 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
4807 GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
4808 DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN
4809 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4810 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
4811 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4812 GEN_REWRITE_TAC I [GSYM INT_NOT_LT] THEN DISCH_TAC THEN
4813 FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN REWRITE_TAC[NOT_IMP] THEN
4815 [MATCH_MP_TAC(ARITH_RULE `~(n = 0) ==> n - 1 < n`) THEN
4816 DISCH_THEN SUBST_ALL_TAC THEN
4817 UNDISCH_TAC `aff_dim(p:real^N->bool) + &1 < &0` THEN
4818 REWRITE_TAC[INT_ARITH `p + &1:int < &0 <=> ~(-- &1 <= p)`] THEN
4819 REWRITE_TAC[AFF_DIM_GE];
4821 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_AFFINE_INDEPENDENT) THEN
4822 ASM_REWRITE_TAC[] THEN
4823 SUBGOAL_THEN `~(aff_dim(s:real^N->bool) = &n - &1)` ASSUME_TAC THENL
4824 [FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN
4826 UNDISCH_TAC `aff_dim(p:real^N->bool) + &1 < &n` THEN
4829 ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE] THEN
4830 DISCH_THEN(X_CHOOSE_THEN `w:real^N->real` STRIP_ASSUME_TAC) THEN
4832 `?t. (!v:real^N. v IN s ==> u(v) + t * w(v) >= &0) /\
4833 ?a. a IN s /\ u(a) + t * w(a) = &0`
4834 STRIP_ASSUME_TAC THENL
4836 `i = IMAGE (\v. u(v) / --w(v)) {v:real^N | v IN s /\ w v < &0}` THEN
4837 EXISTS_TAC `inf i` THEN MP_TAC(SPEC `i:real->bool` INF_FINITE) THEN
4839 [EXPAND_TAC "i" THEN
4840 ASM_SIMP_TAC[FINITE_IMAGE; FINITE_RESTRICT; IMAGE_EQ_EMPTY] THEN
4841 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
4842 MP_TAC(ISPECL [`w:real^N->real`; `s:real^N->bool`] SUM_ZERO_EXISTS) THEN
4843 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
4845 ABBREV_TAC `t = inf i` THEN
4846 EXPAND_TAC "i" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
4847 REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN
4848 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `a:real^N`
4849 STRIP_ASSUME_TAC) MP_TAC) THEN
4850 SIMP_TAC[REAL_LE_RDIV_EQ; REAL_ARITH `x < &0 ==> &0 < --x`; real_ge] THEN
4851 REWRITE_TAC[REAL_ARITH `t * --w <= u <=> &0 <= u + t * w`] THEN
4852 STRIP_TAC THEN CONJ_TAC THENL
4853 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
4854 DISJ_CASES_TAC(REAL_ARITH `(w:real^N->real) x < &0 \/ &0 <= w x`) THEN
4855 ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_ADD THEN ASM_SIMP_TAC[] THEN
4856 MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[] THEN
4857 MATCH_MP_TAC REAL_LE_DIV THEN ASM_SIMP_TAC[] THEN
4858 MATCH_MP_TAC(REAL_ARITH `w < &0 ==> &0 <= --w`) THEN ASM_REWRITE_TAC[];
4859 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN
4860 UNDISCH_TAC `w(a:real^N) < &0` THEN CONV_TAC REAL_FIELD];
4862 MAP_EVERY EXISTS_TAC
4863 [`s DELETE (a:real^N)`; `(\v. u(v) + t * w(v)):real^N->real`] THEN
4864 ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; CARD_DELETE; FINITE_DELETE] THEN
4865 ASM_SIMP_TAC[SUM_ADD; VECTOR_ADD_RDISTRIB; VSUM_ADD] THEN
4866 ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL; VSUM_LMUL] THEN
4867 REPEAT CONJ_TAC THENL
4868 [ASM SET_TAC[]; ASM SET_TAC[real_ge]; REAL_ARITH_TAC; VECTOR_ARITH_TAC]);;
4870 let CARATHEODORY_AFF_DIM = prove
4871 (`!p. convex hull p =
4872 {x:real^N | ?s. FINITE s /\ s SUBSET p /\
4873 &(CARD s) <= aff_dim p + &1 /\
4874 x IN convex hull s}`,
4875 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL
4876 [GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
4877 [CONVEX_HULL_CARATHEODORY_AFF_DIM] THEN
4878 REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
4879 ASM_MESON_TAC[HULL_SUBSET; CONVEX_EXPLICIT; CONVEX_CONVEX_HULL];
4880 MESON_TAC[SUBSET; HULL_MONO]]);;
4882 let CONVEX_HULL_CARATHEODORY = prove
4883 (`!p. convex hull p =
4884 {y:real^N | ?s u. FINITE s /\ s SUBSET p /\
4885 CARD(s) <= dimindex(:N) + 1 /\
4886 (!x. x IN s ==> &0 <= u x) /\
4887 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
4889 GEN_TAC THEN REWRITE_TAC[EXTENSION] THEN X_GEN_TAC `y:real^N` THEN
4891 [REWRITE_TAC[CONVEX_HULL_CARATHEODORY_AFF_DIM; IN_ELIM_THM] THEN
4892 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
4894 ASM_REWRITE_TAC[GSYM INT_OF_NUM_LE; GSYM INT_OF_NUM_ADD] THEN
4895 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH
4896 `a:int <= x + &1 ==> x <= y ==> a <= y + &1`)) THEN
4897 REWRITE_TAC[AFF_DIM_LE_UNIV];
4898 REWRITE_TAC[CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN MESON_TAC[]]);;
4900 let CARATHEODORY = prove
4901 (`!p. convex hull p =
4902 {x:real^N | ?s. FINITE s /\ s SUBSET p /\
4903 CARD(s) <= dimindex(:N) + 1 /\
4904 x IN convex hull s}`,
4905 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL
4906 [GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
4907 [CONVEX_HULL_CARATHEODORY] THEN
4908 REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
4909 ASM_MESON_TAC[HULL_SUBSET; CONVEX_EXPLICIT; CONVEX_CONVEX_HULL];
4910 MESON_TAC[SUBSET; HULL_MONO]]);;
4912 (* ------------------------------------------------------------------------- *)
4913 (* Some results on decomposing convex hulls, e.g. simplicial subdivision. *)
4914 (* ------------------------------------------------------------------------- *)
4916 let AFFINE_HULL_INTER,CONVEX_HULL_INTER = (CONJ_PAIR o prove)
4917 (`(!s t:real^N->bool.
4918 ~(affine_dependent(s UNION t))
4919 ==> affine hull s INTER affine hull t = affine hull (s INTER t)) /\
4921 ~(affine_dependent (s UNION t))
4922 ==> convex hull s INTER convex hull t = convex hull (s INTER t))`,
4924 (REPEAT STRIP_TAC THEN
4925 FIRST_ASSUM(MP_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
4926 REWRITE_TAC[FINITE_UNION] THEN STRIP_TAC THEN
4927 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET_INTER] THEN
4928 SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
4929 REWRITE_TAC[SUBSET; AFFINE_HULL_FINITE; CONVEX_HULL_FINITE;
4930 IN_ELIM_THM; IN_INTER] THEN
4931 X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2
4932 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)
4933 (X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC)) THEN
4934 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
4935 [AFFINE_DEPENDENT_EXPLICIT]) THEN
4936 REWRITE_TAC[NOT_EXISTS_THM] THEN
4937 DISCH_THEN(MP_TAC o SPEC `(s UNION t):real^N->bool`) THEN
4938 ASM_REWRITE_TAC[SUBSET_REFL] THEN
4939 DISCH_THEN(MP_TAC o SPEC
4940 `\x:real^N. (if x IN s then u x else &0) -
4941 (if x IN t then v x else &0)`) THEN
4942 ASM_SIMP_TAC[SUM_SUB; FINITE_UNION; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN
4944 `(if p then a else b) % x = (if p then a % x else b % x)`] THEN
4945 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO; FINITE_UNION] THEN
4946 ASM_REWRITE_TAC[SUM_0; VSUM_0;
4947 SET_RULE `{x | x IN (s UNION t) /\ x IN s} = s`;
4948 SET_RULE `{x | x IN (s UNION t) /\ x IN t} = t`] THEN
4949 MATCH_MP_TAC(TAUT `a /\ c /\ (~b ==> d) ==> ~(a /\ b /\ c) ==> d`) THEN
4950 REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC; ALL_TAC] THEN
4951 DISCH_TAC THEN EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[] THEN
4952 CONJ_TAC THEN MATCH_MP_TAC EQ_TRANS THENL
4953 [EXISTS_TAC `sum s (u:real^N->real)`;
4954 EXISTS_TAC `vsum s (\x. (u:real^N->real) x % x)`] THEN
4955 (CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM ACCEPT_TAC]) THEN
4956 CONV_TAC SYM_CONV THENL
4957 [MATCH_MP_TAC SUM_EQ_SUPERSET; MATCH_MP_TAC VSUM_EQ_SUPERSET] THEN
4958 ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET; IN_INTER] THEN
4959 X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4960 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISCH_TAC THEN
4961 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
4962 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
4963 ASM_REWRITE_TAC[REAL_SUB_RZERO] THEN ASM SET_TAC[]));;
4965 let AFFINE_HULL_INTERS = prove
4966 (`!s:(real^N->bool)->bool.
4967 ~(affine_dependent(UNIONS s))
4968 ==> affine hull (INTERS s) = INTERS {affine hull t | t IN s}`,
4969 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
4970 MP_TAC th THEN MP_TAC(MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE th)) THEN
4971 SPEC_TAC(`s:(real^N->bool)->bool`,`s:(real^N->bool)->bool`) THEN
4972 REWRITE_TAC[FINITE_UNIONS; IMP_CONJ] THEN
4973 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
4974 REWRITE_TAC[UNIONS_0; INTERS_0; UNIONS_INSERT; INTERS_INSERT;
4975 SET_RULE `{f x | x IN {}} = {}`; AFFINE_HULL_UNIV] THEN
4976 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN
4977 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
4978 REWRITE_TAC[FORALL_IN_INSERT] THEN
4979 DISCH_THEN(fun th -> STRIP_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
4980 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
4981 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
4982 REWRITE_TAC[CONTRAPOS_THM] THEN
4983 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
4986 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL
4987 [ASM_REWRITE_TAC[INTERS_0; INTER_UNIV; IN_SING] THEN
4988 REWRITE_TAC[SET_RULE `{f x | x = a} = {f a}`; INTERS_1];
4990 W(MP_TAC o PART_MATCH (rhs o rand) AFFINE_HULL_INTER o lhand o snd) THEN
4992 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
4993 REWRITE_TAC[CONTRAPOS_THM] THEN
4994 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
4995 UNDISCH_TAC `~(f:(real^N->bool)->bool = {})` THEN SET_TAC[];
4996 DISCH_THEN(SUBST1_TAC o SYM)] THEN
4997 REWRITE_TAC[SET_RULE
4998 `{f x | x IN (a INSERT s)} = (f a) INSERT {f x | x IN s}`] THEN
4999 ASM_REWRITE_TAC[INTERS_INSERT]);;
5001 let CONVEX_HULL_INTERS = prove
5002 (`!s:(real^N->bool)->bool.
5003 ~(affine_dependent(UNIONS s))
5004 ==> convex hull (INTERS s) = INTERS {convex hull t | t IN s}`,
5005 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
5006 MP_TAC th THEN MP_TAC(MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE th)) THEN
5007 SPEC_TAC(`s:(real^N->bool)->bool`,`s:(real^N->bool)->bool`) THEN
5008 REWRITE_TAC[FINITE_UNIONS; IMP_CONJ] THEN
5009 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
5010 REWRITE_TAC[UNIONS_0; INTERS_0; UNIONS_INSERT; INTERS_INSERT;
5011 SET_RULE `{f x | x IN {}} = {}`; CONVEX_HULL_UNIV] THEN
5012 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN
5013 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
5014 REWRITE_TAC[FORALL_IN_INSERT] THEN
5015 DISCH_THEN(fun th -> STRIP_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
5016 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
5017 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
5018 REWRITE_TAC[CONTRAPOS_THM] THEN
5019 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
5022 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL
5023 [ASM_REWRITE_TAC[INTERS_0; INTER_UNIV; IN_SING] THEN
5024 REWRITE_TAC[SET_RULE `{f x | x = a} = {f a}`; INTERS_1];
5026 W(MP_TAC o PART_MATCH (rhs o rand) CONVEX_HULL_INTER o lhand o snd) THEN
5028 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
5029 REWRITE_TAC[CONTRAPOS_THM] THEN
5030 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
5031 UNDISCH_TAC `~(f:(real^N->bool)->bool = {})` THEN SET_TAC[];
5032 DISCH_THEN(SUBST1_TAC o SYM)] THEN
5033 REWRITE_TAC[SET_RULE
5034 `{f x | x IN (a INSERT s)} = (f a) INSERT {f x | x IN s}`] THEN
5035 ASM_REWRITE_TAC[INTERS_INSERT]);;
5037 let IN_CONVEX_HULL_EXCHANGE = prove
5039 a IN convex hull s /\ x IN convex hull s
5040 ==> ?b. b IN s /\ x IN convex hull (a INSERT (s DELETE b))`,
5041 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL
5042 [EXISTS_TAC `a:real^N` THEN ASM_SIMP_TAC[INSERT_DELETE]; ALL_TAC] THEN
5043 ASM_CASES_TAC `FINITE(s:real^N->bool) /\ CARD s <= dimindex(:N) + 1` THENL
5045 UNDISCH_TAC `(x:real^N) IN convex hull s` THEN
5046 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [CARATHEODORY] THEN
5047 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
5048 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
5049 ASM_CASES_TAC `t:real^N->bool = s` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
5050 SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b IN t)` MP_TAC THENL
5051 [ASM SET_TAC[]; ALL_TAC] THEN
5052 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN
5053 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5054 UNDISCH_TAC `(x:real^N) IN convex hull t` THEN
5055 SPEC_TAC(`x:real^N`,`x:real^N`) THEN REWRITE_TAC[GSYM SUBSET] THEN
5056 MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]] THEN
5057 MP_TAC(ASSUME `(a:real^N) IN convex hull s`) THEN
5058 MP_TAC(ASSUME `(x:real^N) IN convex hull s`) THEN
5059 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
5060 X_GEN_TAC `v:real^N->real` THEN STRIP_TAC THEN
5061 X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN
5062 ASM_CASES_TAC `?b. b IN s /\ (v:real^N->real) b = &0` THENL
5063 [FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN
5064 X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5065 EXISTS_TAC `\x:real^N. if x = a then &0 else v x` THEN
5066 ASM_SIMP_TAC[FORALL_IN_INSERT; REAL_LE_REFL] THEN
5067 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_DELETE] THEN
5068 ASM_REWRITE_TAC[IN_DELETE] THEN
5069 ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; COND_ID] THEN
5070 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
5071 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; REAL_LE_REFL; COND_ID] THEN
5072 REWRITE_TAC[VECTOR_MUL_LZERO; SUM_0; VSUM_0] THEN
5073 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ ~(x = a)} = s`] THEN
5074 CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC];
5076 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
5077 REWRITE_TAC[TAUT `~(a /\ b) <=> a ==> ~b`] THEN DISCH_TAC THEN
5078 MP_TAC(ISPEC `IMAGE (\b. (u:real^N->real) b / v b) s` SUP_FINITE) THEN
5079 ASM_CASES_TAC `s:real^N->bool = {}` THENL
5080 [ASM_MESON_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN
5081 ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
5082 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
5083 REWRITE_TAC[IN_IMAGE] THEN MATCH_MP_TAC MONO_EXISTS THEN
5084 X_GEN_TAC `b:real^N` THEN
5085 DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
5086 ASM_REWRITE_TAC[] THEN
5087 SUBGOAL_THEN `!b. b IN s ==> &0 < (v:real^N->real) b` ASSUME_TAC THENL
5088 [ASM_SIMP_TAC[REAL_LT_LE]; ALL_TAC] THEN
5089 SUBGOAL_THEN `&0 < (u:real^N->real) b /\ &0 < v b` STRIP_ASSUME_TAC THENL
5090 [ASM_SIMP_TAC[REAL_LT_LE] THEN
5091 UNDISCH_TAC `!x. x IN s ==> (u:real^N->real) x / v x <= u b / v b` THEN
5092 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN
5093 DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN
5094 REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN
5095 ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (x <= &0 <=> x = &0)`] THEN
5096 DISCH_TAC THEN UNDISCH_TAC `sum s (u:real^N->real) = &1` THEN
5097 MATCH_MP_TAC(REAL_ARITH `x = &0 ==> x = &1 ==> F`) THEN
5098 ASM_SIMP_TAC[SUM_EQ_0];
5100 EXISTS_TAC `(\x. if x = a then v b / u b else v x - (v b / u b) * u x):
5102 ASM_SIMP_TAC[FORALL_IN_INSERT; REAL_LE_DIV; REAL_LT_IMP_LE] THEN
5103 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
5104 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_DELETE] THEN
5105 ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; IN_DELETE] THEN
5106 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; FINITE_DELETE] THEN
5107 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ ~(x = a)} = s`;
5108 SET_RULE `~(a IN s) ==> {x | x IN s /\ x = a} = {}`] THEN
5109 REWRITE_TAC[VSUM_CLAUSES; SUM_CLAUSES] THEN
5110 ASM_CASES_TAC `b:real^N = a` THENL [ASM_MESON_TAC[]; ASM_REWRITE_TAC[]] THEN
5111 ASM_SIMP_TAC[VECTOR_SUB_RDISTRIB; VSUM_SUB; SUM_SUB] THEN
5112 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VECTOR_ADD_LID; REAL_ADD_LID] THEN
5113 ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
5114 ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN REPEAT CONJ_TAC THENL
5115 [ALL_TAC; REAL_ARITH_TAC; VECTOR_ARITH_TAC] THEN
5116 X_GEN_TAC `c:real^N` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
5117 STRIP_TAC THEN ASM_CASES_TAC `(u:real^N->real) c = &0` THENL
5118 [ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO]; ALL_TAC] THEN
5119 REWRITE_TAC[REAL_SUB_LE] THEN
5120 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_LT_LE] THEN
5121 ONCE_REWRITE_TAC[GSYM REAL_INV_DIV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
5122 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_LE]);;
5124 let IN_CONVEX_HULL_EXCHANGE_UNIQUE = prove
5125 (`!s t t' a x:real^N.
5126 ~(affine_dependent s) /\
5127 a IN convex hull s /\
5128 t SUBSET s /\ t' SUBSET s /\
5129 x IN convex hull (a INSERT t) /\
5130 x IN convex hull (a INSERT t')
5131 ==> x IN convex hull (a INSERT (t INTER t'))`,
5132 REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL
5133 [REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE
5134 `a INSERT (s INTER t) = (a INSERT s) INTER (a INSERT t)`] THEN
5135 W(MP_TAC o PART_MATCH (rand o rand) CONVEX_HULL_INTER o rand o snd) THEN
5137 [UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN
5138 REWRITE_TAC[CONTRAPOS_THM] THEN
5139 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO);
5140 DISCH_THEN(SUBST1_TAC o SYM)] THEN
5143 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5144 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
5145 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN
5146 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `b:real^N->real` STRIP_ASSUME_TAC)
5148 REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5149 SUBGOAL_THEN `~((a:real^N) IN t) /\ ~(a IN t')` STRIP_ASSUME_TAC THENL
5150 [ASM SET_TAC[]; ALL_TAC] THEN
5151 SUBGOAL_THEN `FINITE(t:real^N->bool) /\ FINITE(t':real^N->bool)`
5152 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
5153 ASM_SIMP_TAC[AFFINE_HULL_FINITE_STEP_GEN; REAL_LE_ADD;
5154 REAL_ARITH `&0 <= a / &2 <=> &0 <= a`] THEN
5155 REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
5156 MAP_EVERY X_GEN_TAC [`u':real`; `u:real^N->real`] THEN REPEAT DISCH_TAC THEN
5157 MAP_EVERY X_GEN_TAC [`v':real`; `v:real^N->real`] THEN REPEAT DISCH_TAC THEN
5158 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
5159 [AFFINE_DEPENDENT_EXPLICIT]) THEN
5160 REWRITE_TAC[NOT_EXISTS_THM] THEN
5161 DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN
5162 ASM_REWRITE_TAC[SUBSET_REFL] THEN
5163 DISCH_THEN(MP_TAC o SPEC
5164 `\x:real^N. (if x IN t then u x else &0) - (if x IN t' then v x else &0) +
5165 (u' - v') * b x`) THEN
5166 ASM_SIMP_TAC[SUM_ADD; VSUM_ADD; SUM_LMUL; VSUM_LMUL; VECTOR_ADD_RDISTRIB] THEN
5167 ASM_SIMP_TAC[SUM_SUB; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN
5169 `(if p then a else b) % x = (if p then a % x else b % x)`] THEN
5170 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO; SUM_0; VSUM_0] THEN
5171 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
5172 ASM_SIMP_TAC[SUM_ADD; SUM_LMUL; VSUM_ADD; VSUM_LMUL; VECTOR_ADD_RDISTRIB;
5173 GSYM VECTOR_MUL_ASSOC] THEN
5174 MATCH_MP_TAC(TAUT `a /\ c /\ (~b ==> d) ==> ~(a /\ b /\ c) ==> d`) THEN
5175 REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC; ALL_TAC] THEN
5178 `(!x. x IN s ==> (if x IN t then u x else &0) <=
5179 (if x IN t' then v x else &0)) \/
5180 (!x:real^N. x IN s ==> (if x IN t' then v x else &0) <=
5181 (if x IN t then u x else &0))`
5182 (DISJ_CASES_THEN(LABEL_TAC "*")) THENL
5183 [MP_TAC(REAL_ARITH `&0 <= (u' - v') \/ &0 <= (v' - u')`) THEN
5184 MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN
5185 DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
5186 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
5187 DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THENL
5188 [MATCH_MP_TAC(REAL_ARITH `&0 <= c ==> a - b + c = &0 ==> a <= b`);
5189 MATCH_MP_TAC(REAL_ARITH `&0 <= --c ==> a - b + c = &0 ==> b <= a`)] THEN
5190 ASM_SIMP_TAC[REAL_LE_MUL; GSYM REAL_MUL_LNEG; REAL_NEG_SUB];
5191 EXISTS_TAC `(\x. if x = a then u' else u x):real^N->real`;
5192 EXISTS_TAC `(\x. if x = a then v' else v x):real^N->real`] THEN
5193 ASM_SIMP_TAC[FORALL_IN_INSERT] THEN
5194 (CONJ_TAC THENL [ASM_MESON_TAC[IN_INTER]; ALL_TAC]) THEN
5195 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_INTER] THEN
5196 ASM_REWRITE_TAC[IN_INTER] THEN
5197 REWRITE_TAC[REAL_ARITH `u' + u = &1 <=> u = &1 - u'`;
5198 VECTOR_ARITH `u' + u:real^N = y <=> u = y - u'`] THEN
5200 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
5201 CONV_TAC SYM_CONV THENL
5202 [MATCH_MP_TAC SUM_EQ_SUPERSET; MATCH_MP_TAC VSUM_EQ_SUPERSET]) THEN
5203 ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET; IN_INTER] THEN
5204 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
5205 X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5206 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISCH_TAC THEN
5207 REMOVE_THEN "*" (MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN
5208 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN ASM SET_TAC[]);;
5210 let CONVEX_HULL_EXCHANGE_UNION = prove
5214 UNIONS {convex hull (a INSERT (s DELETE b)) |b| b IN s}`,
5215 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
5216 [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[UNIONS_IMAGE] THEN
5217 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
5218 ASM_MESON_TAC[IN_CONVEX_HULL_EXCHANGE];
5219 REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_GSPEC;
5220 IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5221 GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[GSYM SUBSET] THEN
5222 ASM_SIMP_TAC[SUBSET_HULL; CONVEX_CONVEX_HULL] THEN
5223 ASM_REWRITE_TAC[INSERT_SUBSET] THEN
5224 MESON_TAC[HULL_INC; SUBSET; IN_DELETE]]);;
5226 let CONVEX_HULL_EXCHANGE_INTER = prove
5228 ~affine_dependent s /\
5229 a IN convex hull s /\
5232 ==> (convex hull (a INSERT t)) INTER (convex hull (a INSERT t')) =
5233 convex hull (a INSERT (t INTER t'))`,
5234 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
5235 [REWRITE_TAC[SUBSET; IN_INTER] THEN REPEAT STRIP_TAC THEN
5236 MATCH_MP_TAC IN_CONVEX_HULL_EXCHANGE_UNIQUE THEN
5237 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[];
5238 REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THEN
5239 MATCH_MP_TAC HULL_MONO THEN SET_TAC[]]);;
5241 (* ------------------------------------------------------------------------- *)
5242 (* Representing affine hull as hyperplane or finite intersection of them. *)
5243 (* ------------------------------------------------------------------------- *)
5245 let AFF_DIM_EQ_HYPERPLANE = prove
5246 (`!s. aff_dim s = &(dimindex(:N)) - &1 <=>
5247 ?a b. ~(a = vec 0) /\ affine hull s = {x:real^N | a dot x = b}`,
5248 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
5249 [ASM_REWRITE_TAC[AFF_DIM_EMPTY; INT_ARITH `--a:int = b - a <=> b = &0`] THEN
5250 SIMP_TAC[INT_OF_NUM_EQ; LE_1; DIMINDEX_GE_1; AFFINE_HULL_EMPTY] THEN
5251 REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; NOT_EXISTS_THM] THEN
5252 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN
5253 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5254 DISCH_THEN(MP_TAC o SPEC `(b / (a dot a)) % a:real^N`) THEN
5255 ASM_SIMP_TAC[DOT_RMUL; REAL_DIV_RMUL; DOT_EQ_0];
5256 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
5257 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:real^N` THEN
5258 GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN
5259 SIMP_TAC[AFF_DIM_DIM_0; HULL_INC] THEN
5260 SIMP_TAC[INT_OF_NUM_SUB; DIMINDEX_GE_1; INT_OF_NUM_EQ] THEN
5261 SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; DIM_EQ_HYPERPLANE] THEN
5262 REPEAT STRIP_TAC THEN AP_TERM_TAC THEN
5263 GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `a:real^N` THEN
5264 REWRITE_TAC[] THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
5265 ASM_REWRITE_TAC[DOT_RADD; REAL_ARITH `a + b:real = c <=> b = c - a`] THEN
5266 EQ_TAC THEN STRIP_TAC THENL
5267 [EXISTS_TAC `(a:real^N) dot c` THEN ASM_REWRITE_TAC[REAL_SUB_REFL];
5268 ASM_REWRITE_TAC[] THEN
5269 FIRST_X_ASSUM(MP_TAC o AP_TERM `\s. (vec 0:real^N) IN s`) THEN
5270 ASM_SIMP_TAC[SPAN_SUPERSET; IN_ELIM_THM; DOT_RZERO]]]);;
5272 let AFF_DIM_HYPERPLANE = prove
5273 (`!a b. ~(a = vec 0)
5274 ==> aff_dim {x:real^N | a dot x = b} = &(dimindex(:N)) - &1`,
5275 REPEAT STRIP_TAC THEN REWRITE_TAC[AFF_DIM_EQ_HYPERPLANE] THEN
5276 MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real`] THEN
5277 ASM_REWRITE_TAC[AFFINE_HULL_EQ; AFFINE_HYPERPLANE]);;
5279 let BOUNDED_HYPERPLANE_EQ_TRIVIAL = prove
5280 (`!a b. bounded {x:real^N | a dot x = b} <=>
5281 if a = vec 0 then ~(b = &0) else dimindex(:N) = 1`,
5282 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
5283 ASM_REWRITE_TAC[DOT_LZERO] THENL
5284 [ASM_CASES_TAC `b = &0` THEN
5285 ASM_REWRITE_TAC[EMPTY_GSPEC; BOUNDED_EMPTY] THEN
5286 REWRITE_TAC[NOT_BOUNDED_UNIV; SET_RULE `{x | T} = UNIV`];
5287 ASM_SIMP_TAC[AFFINE_BOUNDED_EQ_LOWDIM; AFF_DIM_HYPERPLANE;
5288 AFFINE_HYPERPLANE] THEN
5289 REWRITE_TAC[INT_ARITH `a - &1:int <= &0 <=> a <= &1`; INT_OF_NUM_LE] THEN
5290 MATCH_MP_TAC(ARITH_RULE `1 <= n ==> (n <= 1 <=> n = 1)`) THEN
5291 REWRITE_TAC[DIMINDEX_GE_1]]);;
5293 let AFFINE_HULL_FINITE_INTERSECTION_HYPERPLANES = prove
5296 &(CARD f) + aff_dim s = &(dimindex(:N)) /\
5297 affine hull s = INTERS f /\
5298 (!h. h IN f ==> ?a b. ~(a = vec 0) /\ h = {x | a dot x = b})`,
5299 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
5300 MP_TAC(ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN
5301 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
5302 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
5303 MP_TAC(ISPECL [`b:real^N->bool`; `(:real^N)`] EXTEND_TO_AFFINE_BASIS) THEN
5304 ASM_REWRITE_TAC[SUBSET_UNIV; AFFINE_HULL_UNIV] THEN
5305 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
5306 SUBGOAL_THEN `FINITE(c:real^N->bool)` ASSUME_TAC THENL
5307 [ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]; ALL_TAC] THEN
5308 REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
5309 REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN
5310 ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; CARD_DIFF] THEN
5311 REWRITE_TAC[INT_ARITH `f + b - &1:int = c - &1 <=> f = c - b`] THEN
5312 ASM_SIMP_TAC[INT_OF_NUM_SUB; CARD_SUBSET; GSYM CARD_DIFF; INT_OF_NUM_EQ] THEN
5313 ASM_CASES_TAC `c:real^N->bool = b` THENL
5314 [EXISTS_TAC `{}:(real^N->bool)->bool` THEN
5315 ASM_REWRITE_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY; INTERS_0;
5319 EXISTS_TAC `{affine hull (c DELETE a) |a| (a:real^N) IN (c DIFF b)}` THEN
5320 REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN
5321 ASM_SIMP_TAC[FINITE_IMAGE; FINITE_DIFF] THEN REPEAT CONJ_TAC THENL
5322 [MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_SIMP_TAC[FINITE_DIFF] THEN
5323 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF] THEN
5324 STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN ASM_REWRITE_TAC[] THEN
5325 UNDISCH_TAC `~affine_dependent(c:real^N->bool)` THEN
5326 REWRITE_TAC[affine_dependent] THEN EXISTS_TAC `x:real^N` THEN
5327 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[];
5328 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN
5329 ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN
5330 W(MP_TAC o PART_MATCH (rhs o rand) AFFINE_HULL_INTERS o rand o snd) THEN
5332 [MATCH_MP_TAC AFFINE_INDEPENDENT_SUBSET THEN
5333 EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
5334 REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE;
5335 IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SET_TAC[];
5336 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN
5337 GEN_REWRITE_TAC I [EXTENSION] THEN
5338 REWRITE_TAC[IN_INTERS; FORALL_IN_IMAGE] THEN ASM SET_TAC[]];
5339 REWRITE_TAC[GSYM AFF_DIM_EQ_HYPERPLANE] THEN
5340 ASM_SIMP_TAC[IN_DIFF; AFFINE_INDEPENDENT_DELETE;
5341 AFF_DIM_AFFINE_INDEPENDENT; CARD_DELETE] THEN
5342 REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
5343 REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN
5344 ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; CARD_DIFF] THEN
5345 REPEAT STRIP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
5346 MATCH_MP_TAC(GSYM INT_OF_NUM_SUB) THEN
5347 MATCH_MP_TAC(ARITH_RULE `~(c = 0) ==> 1 <= c`) THEN
5348 ASM_SIMP_TAC[CARD_EQ_0] THEN ASM SET_TAC[]]);;
5350 let AFFINE_HYPERPLANE_SUMS_EQ_UNIV = prove
5353 ~(s INTER {v | a dot v = b} = {}) /\
5354 ~(s DIFF {v | a dot v = b} = {})
5355 ==> {x + y | x IN s /\ y IN {v | a dot v = b}} = (:real^N)`,
5356 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL
5357 [ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[]; ALL_TAC] THEN
5358 REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> b ==> a /\ c ==> d`] THEN
5359 GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN
5360 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN X_GEN_TAC `c:real^N` THEN
5361 ONCE_REWRITE_TAC[SET_RULE
5362 `{x + y:real^N | x IN s /\ P y} =
5363 {z | ?x y. x IN s /\ P y /\ z = x + y}`] THEN
5364 GEOM_ORIGIN_TAC `c:real^N` THEN REPEAT GEN_TAC THEN
5365 REWRITE_TAC[DOT_RADD; REAL_ARITH `b dot c + a = d <=> a = d - b dot c`] THEN
5366 REWRITE_TAC[IN_INTER; IN_ELIM_THM; DOT_RZERO] THEN
5367 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
5368 ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE; HULL_INC] THEN STRIP_TAC THEN
5369 REWRITE_TAC[VECTOR_ARITH `c + z:real^N = (c + x) + (c + y) <=>
5370 z = c + x + y`] THEN
5371 REWRITE_TAC[SET_RULE
5372 `{z | ?x y. x IN s /\ P y /\ z = c + x + y} =
5373 IMAGE (\x. c + x) {x + y:real^N | x IN s /\ y IN {v | P v}}`] THEN
5374 MATCH_MP_TAC(SET_RULE
5375 `!f. (!x. g(f x) = x) /\ s = UNIV ==> IMAGE g s = UNIV`) THEN
5376 EXISTS_TAC `\x:real^N. x - c` THEN
5377 REWRITE_TAC[VECTOR_ARITH `c + x - c:real^N = x`] THEN
5378 MATCH_MP_TAC(MESON[SPAN_EQ_SELF] `subspace s /\ span s = t ==> s = t`) THEN
5380 [ASM_SIMP_TAC[SUBSPACE_SUMS; SUBSPACE_HYPERPLANE];
5382 REWRITE_TAC[GSYM DIM_EQ_FULL] THEN
5383 REWRITE_TAC[GSYM LE_ANTISYM; DIM_SUBSET_UNIV] THEN
5384 MATCH_MP_TAC(ARITH_RULE `m - 1 < n ==> m <= n`) THEN
5385 MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `dim {x:real^N | a dot x = &0}` THEN
5386 CONJ_TAC THENL [ASM_SIMP_TAC[DIM_HYPERPLANE; LE_REFL]; ALL_TAC] THEN
5387 MATCH_MP_TAC DIM_PSUBSET THEN
5388 ASM_SIMP_TAC[snd(EQ_IMP_RULE(SPEC_ALL SPAN_EQ_SELF));
5389 SUBSPACE_SUMS; SUBSPACE_HYPERPLANE] THEN
5390 REWRITE_TAC[PSUBSET; SUBSET; FORALL_IN_GSPEC] THEN
5391 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN CONJ_TAC THENL
5392 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
5393 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN
5394 ASM_SIMP_TAC[SUBSPACE_0; VECTOR_ADD_LID];
5395 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
5396 REWRITE_TAC[NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
5397 X_GEN_TAC `x:real^N` THEN SIMP_TAC[IN_DIFF; IN_ELIM_THM] THEN
5398 DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
5399 ASM_REWRITE_TAC[DOT_RZERO; VECTOR_ADD_RID]]);;
5401 let AFF_DIM_AFFINE_INTER_HYPERPLANE = prove
5402 (`!a b s:real^N->bool.
5404 ==> aff_dim(s INTER {x | a dot x = b}) =
5405 if s INTER {v | a dot v = b} = {} then -- &1
5406 else if s SUBSET {v | a dot v = b} then aff_dim s
5407 else aff_dim s - &1`,
5408 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL
5409 [ASM_REWRITE_TAC[DOT_LZERO] THEN ASM_CASES_TAC `b = &0` THEN
5410 ASM_REWRITE_TAC[EMPTY_GSPEC; INTER_EMPTY; AFF_DIM_EMPTY] THEN
5411 SIMP_TAC[SET_RULE `{x | T} = UNIV`; IN_UNIV; INTER_UNIV; SUBSET_UNIV] THEN
5412 COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY];
5413 STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN
5414 COND_CASES_TAC THENL [AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN
5415 MP_TAC(ISPECL [`s:real^N->bool`; `{x:real^N | a dot x = b}`]
5416 AFF_DIM_SUMS_INTER) THEN
5417 ASM_SIMP_TAC[AFFINE_HYPERPLANE; AFF_DIM_HYPERPLANE] THEN
5418 ASM_SIMP_TAC[AFFINE_HYPERPLANE_SUMS_EQ_UNIV; AFF_DIM_UNIV;
5419 SET_RULE `~(s SUBSET t) ==> ~(s DIFF t = {})`] THEN
5420 SPEC_TAC(`aff_dim (s INTER {x:real^N | a dot x = b})`,`i:int`) THEN
5423 let AFF_DIM_LT_FULL = prove
5424 (`!s. aff_dim s < &(dimindex(:N)) <=> ~(affine hull s = (:real^N))`,
5425 GEN_TAC THEN REWRITE_TAC[GSYM AFF_DIM_EQ_FULL] THEN
5426 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_LE_UNIV) THEN ARITH_TAC);;
5428 let AFF_LOWDIM_SUBSET_HYPERPLANE = prove
5430 aff_dim s < &(dimindex(:N))
5431 ==> ?a b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`,
5432 MATCH_MP_TAC SET_PROVE_CASES THEN CONJ_TAC THENL
5433 [DISCH_TAC THEN EXISTS_TAC `basis 1:real^N` THEN
5434 SIMP_TAC[EMPTY_SUBSET; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1];
5435 MAP_EVERY X_GEN_TAC [`c:real^N`; `s:real^N->bool`] THEN
5436 CONV_TAC(ONCE_DEPTH_CONV(GEN_ALPHA_CONV `a:real^N`)) THEN
5437 GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN
5438 SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; IN_INSERT; INT_OF_NUM_LT] THEN
5439 REPEAT GEN_TAC THEN DISCH_TAC THEN
5440 DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN
5441 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N` THEN
5442 STRIP_TAC THEN EXISTS_TAC `(u:real^N) dot c` THEN
5443 ASM_REWRITE_TAC[DOT_RADD; REAL_EQ_ADD_LCANCEL_0] THEN
5444 ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS]]);;
5446 (* ------------------------------------------------------------------------- *)
5447 (* An additional lemma about hyperplanes. *)
5448 (* ------------------------------------------------------------------------- *)
5450 let SUBSET_HYPERPLANES = prove
5452 {x | a dot x = b} SUBSET {x | a' dot x = b'} <=>
5453 {x | a dot x = b} = {} \/ {x | a' dot x = b'} = (:real^N) \/
5454 {x | a dot x = b} = {x | a' dot x = b'}`,
5456 ASM_CASES_TAC `{x:real^N | a dot x = b} = {}` THEN
5457 ASM_REWRITE_TAC[EMPTY_SUBSET] THEN
5458 ASM_CASES_TAC `{x | a' dot x = b'} = (:real^N)` THEN
5459 ASM_REWRITE_TAC[SUBSET_UNIV] THEN
5460 RULE_ASSUM_TAC(REWRITE_RULE
5461 [HYPERPLANE_EQ_EMPTY; HYPERPLANE_EQ_UNIV]) THEN
5462 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
5463 ASM_CASES_TAC `{x:real^N | a dot x = b} SUBSET {x | a' dot x = b'}` THEN
5464 ASM_REWRITE_TAC[] THEN
5465 MP_TAC(ISPECL [`{x:real^N | a dot x = b}`; `{x:real^N | a' dot x = b'}`]
5466 AFF_DIM_PSUBSET) THEN
5467 ASM_SIMP_TAC[PSUBSET;
5468 REWRITE_RULE[GSYM AFFINE_HULL_EQ] AFFINE_HYPERPLANE] THEN
5469 ASM_CASES_TAC `{x:real^N | a dot x = b} = {x | a' dot x = b'}` THEN
5470 ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_CASES_TAC `a':real^N = vec 0` THENL
5471 [ASM_CASES_TAC `b' = &0` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
5472 ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[];
5474 ASM_CASES_TAC `a:real^N = vec 0` THENL
5475 [ASM_CASES_TAC `b = &0` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
5476 ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[];
5478 ASM_SIMP_TAC[AFF_DIM_HYPERPLANE; INT_LT_REFL]);;
5480 (* ------------------------------------------------------------------------- *)
5481 (* Openness and compactness are preserved by convex hull operation. *)
5482 (* ------------------------------------------------------------------------- *)
5484 let OPEN_CONVEX_HULL = prove
5485 (`!s:real^N->bool. open s ==> open(convex hull s)`,
5487 REWRITE_TAC[CONVEX_HULL_EXPLICIT; OPEN_CONTAINS_CBALL] THEN
5488 REWRITE_TAC[IN_ELIM_THM; SUBSET; LEFT_IMP_EXISTS_THM] THEN DISCH_TAC THEN
5489 MAP_EVERY X_GEN_TAC [`a:real^N`; `t:real^N->bool`; `u:real^N->real`] THEN
5491 SUBGOAL_THEN `?b. !x:real^N. x IN t ==> &0 < b(x) /\ cball(x,b(x)) SUBSET s`
5492 STRIP_ASSUME_TAC THENL
5493 [REWRITE_TAC[GSYM SKOLEM_THM] THEN ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
5494 ABBREV_TAC `i = IMAGE (b:real^N->real) t` THEN
5495 EXISTS_TAC `inf i` THEN MP_TAC(SPEC `i:real->bool` INF_FINITE) THEN
5496 EXPAND_TAC "i" THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_IMAGE] THEN
5498 [EXPAND_TAC "i" THEN CONJ_TAC THENL
5499 [ASM_SIMP_TAC[FINITE_IMAGE]; ALL_TAC] THEN
5500 REWRITE_TAC[IMAGE_EQ_EMPTY] THEN
5501 ASM_MESON_TAC[SUM_CLAUSES; REAL_ARITH `~(&1 = &0)`];
5503 STRIP_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
5504 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_CBALL; dist] THEN
5505 DISCH_TAC THEN EXISTS_TAC `IMAGE (\v:real^N. v + (y - a)) t` THEN
5506 EXISTS_TAC `\v. (u:real^N->real)(v - (y - a))` THEN
5507 ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; SUM_IMAGE; VSUM_IMAGE;
5508 VECTOR_ARITH `v + a:real^N = w + a <=> v = w`] THEN
5509 ASM_REWRITE_TAC[o_DEF; VECTOR_ARITH `(v + a) - a:real^N = v`] THEN
5510 ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB; ETA_AX] THEN
5511 ASM_SIMP_TAC[VSUM_ADD; VSUM_RMUL] THEN
5512 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
5513 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
5514 SUBGOAL_THEN `z + (y - a) IN cball(z:real^N,b z)`
5515 (fun th -> ASM_MESON_TAC[th; SUBSET]) THEN
5516 REWRITE_TAC[IN_CBALL; dist; NORM_ARITH
5517 `norm(z - (z + a - y)) = norm(y - a)`] THEN
5518 ASM_MESON_TAC[REAL_LE_TRANS]);;
5520 let COMPACT_CONVEX_COMBINATIONS = prove
5521 (`!s t. compact s /\ compact t
5522 ==> compact { (&1 - u) % x + u % y :real^N |
5523 &0 <= u /\ u <= &1 /\ x IN s /\ y IN t}`,
5524 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5525 `{ (&1 - u) % x + u % y :real^N | &0 <= u /\ u <= &1 /\ x IN s /\ y IN t} =
5526 IMAGE (\z. (&1 - drop(fstcart z)) % fstcart(sndcart z) +
5527 drop(fstcart z) % sndcart(sndcart z))
5528 { pastecart u w | u IN interval[vec 0,vec 1] /\
5529 w IN { pastecart x y | x IN s /\ y IN t} }`
5531 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN
5532 X_GEN_TAC `x:real^N` THEN
5533 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN
5534 CONV_TAC(ONCE_DEPTH_CONV UNWIND_CONV) THEN
5535 REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN
5536 REWRITE_TAC[IN_INTERVAL_1; GSYM EXISTS_DROP; DROP_VEC] THEN MESON_TAC[];
5538 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
5539 ASM_SIMP_TAC[COMPACT_PCROSS; GSYM PCROSS; COMPACT_INTERVAL] THEN
5540 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
5541 X_GEN_TAC `z:real^(1,(N,N)finite_sum)finite_sum` THEN
5542 DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[PCROSS] THEN
5543 MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC THEN
5544 MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN
5545 CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_SUB) THEN
5546 REWRITE_TAC[CONTINUOUS_CONST] THEN
5547 MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
5548 REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART; ETA_AX] THEN
5549 GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN
5550 MATCH_MP_TAC LINEAR_COMPOSE THEN
5551 REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);;
5553 let COMPACT_CONVEX_HULL = prove
5554 (`!s:real^N->bool. compact s ==> compact(convex hull s)`,
5555 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CARATHEODORY] THEN
5556 SPEC_TAC(`dimindex(:N) + 1`,`n:num`) THEN
5557 ASM_CASES_TAC `s:real^N->bool = {}` THENL
5558 [ASM_REWRITE_TAC[SUBSET_EMPTY] THEN
5559 CONV_TAC(ONCE_DEPTH_CONV UNWIND_CONV) THEN
5560 REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
5561 REWRITE_TAC[SET_RULE `{x | F} = {}`; COMPACT_EMPTY];
5563 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
5564 DISCH_THEN(X_CHOOSE_TAC `w:real^N`) THEN INDUCT_TAC THENL
5566 `{x:real^N | ?t. FINITE t /\ t SUBSET s /\ CARD t <= 0 /\
5567 x IN convex hull t} = {}`
5568 (fun th -> REWRITE_TAC[th; COMPACT_EMPTY]) THEN
5569 REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; LE; IN_ELIM_THM] THEN
5570 MESON_TAC[CARD_EQ_0; CONVEX_HULL_EMPTY; NOT_IN_EMPTY];
5572 ASM_CASES_TAC `n = 0` THENL
5573 [ASM_REWRITE_TAC[ARITH_RULE `s <= SUC 0 <=> s = 0 \/ s = 1`] THEN
5574 UNDISCH_TAC `compact(s:real^N->bool)` THEN
5575 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
5576 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
5577 REWRITE_TAC[TAUT `a /\ b /\ (c \/ d) /\ e <=>
5578 (a /\ c) /\ (b /\ e) \/ (a /\ d) /\ (b /\ e)`] THEN
5579 REWRITE_TAC[GSYM HAS_SIZE; num_CONV `1`; HAS_SIZE_CLAUSES] THEN
5580 REWRITE_TAC[EXISTS_OR_THM; LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN
5581 CONV_TAC(TOP_DEPTH_CONV UNWIND_CONV) THEN
5582 REWRITE_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY] THEN
5583 REWRITE_TAC[CONVEX_HULL_SING] THEN SET_TAC[];
5586 `{x:real^N | ?t. FINITE t /\ t SUBSET s /\ CARD t <= SUC n /\
5587 x IN convex hull t} =
5588 { (&1 - u) % x + u % y :real^N |
5589 &0 <= u /\ u <= &1 /\ x IN s /\
5590 y IN {x | ?t. FINITE t /\ t SUBSET s /\
5591 CARD t <= n /\ x IN convex hull t}}`
5592 (fun th -> ASM_SIMP_TAC[th; COMPACT_CONVEX_COMBINATIONS]) THEN
5593 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
5594 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
5596 REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM;
5597 LEFT_AND_EXISTS_THM] THEN
5598 MAP_EVERY X_GEN_TAC [`u:real^N`; `c:real`; `v:real^N`;
5599 `t:real^N->bool`] THEN
5600 STRIP_TAC THEN EXISTS_TAC `(u:real^N) INSERT t` THEN
5601 ASM_REWRITE_TAC[FINITE_INSERT; INSERT_SUBSET] THEN
5602 ASM_SIMP_TAC[CARD_CLAUSES] THEN CONJ_TAC THENL
5603 [ASM_ARITH_TAC; ALL_TAC] THEN
5604 MATCH_MP_TAC IN_CONVEX_SET THEN
5605 ASM_REWRITE_TAC[CONVEX_CONVEX_HULL] THEN CONJ_TAC THEN
5606 ASM_MESON_TAC[HULL_SUBSET; SUBSET; IN_INSERT; HULL_MONO]] THEN
5607 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
5608 ASM_CASES_TAC `CARD(t:real^N->bool) <= n` THENL
5609 [MAP_EVERY EXISTS_TAC [`w:real^N`; `&1`; `x:real^N`] THEN
5610 ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN
5611 CONJ_TAC THENL [ASM_MESON_TAC[]; VECTOR_ARITH_TAC];
5613 SUBGOAL_THEN `(t:real^N->bool) HAS_SIZE (SUC n)` MP_TAC THENL
5614 [ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_ARITH_TAC;
5616 REWRITE_TAC[HAS_SIZE_CLAUSES] THEN
5617 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `u:real^N->bool`
5618 STRIP_ASSUME_TAC)) THEN
5619 FIRST_X_ASSUM SUBST_ALL_TAC THEN
5620 UNDISCH_TAC `(x:real^N) IN convex hull (a INSERT u)` THEN
5621 RULE_ASSUM_TAC(REWRITE_RULE[FINITE_INSERT]) THEN
5622 ASM_CASES_TAC `(u:real^N->bool) = {}` THENL
5623 [ASM_REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN
5624 DISCH_THEN SUBST_ALL_TAC THEN
5625 MAP_EVERY EXISTS_TAC [`a:real^N`; `&1`; `a:real^N`] THEN
5626 ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN
5627 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
5628 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5629 EXISTS_TAC `{a:real^N}` THEN SIMP_TAC[FINITE_RULES] THEN
5630 REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN
5631 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5632 SIMP_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY] THEN
5633 UNDISCH_TAC `~(n = 0)` THEN ARITH_TAC;
5635 ASM_SIMP_TAC[CONVEX_HULL_INSERT; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
5636 MAP_EVERY X_GEN_TAC [`c:real`; `d:real`; `z:real^N`] THEN STRIP_TAC THEN
5637 MAP_EVERY EXISTS_TAC [`a:real^N`; `d:real`; `z:real^N`] THEN
5638 FIRST_X_ASSUM(SUBST_ALL_TAC o MATCH_MP (REAL_ARITH
5639 `c + d = &1 ==> c = (&1 - d)`)) THEN
5640 ASM_REWRITE_TAC[REAL_ARITH `d <= &1 <=> &0 <= &1 - d`] THEN
5641 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5642 EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
5643 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5644 UNDISCH_TAC `CARD ((a:real^N) INSERT u) <= SUC n` THEN
5645 ASM_SIMP_TAC[CARD_CLAUSES; LE_SUC]);;
5647 let FINITE_IMP_COMPACT_CONVEX_HULL = prove
5648 (`!s:real^N->bool. FINITE s ==> compact(convex hull s)`,
5649 SIMP_TAC[FINITE_IMP_COMPACT; COMPACT_CONVEX_HULL]);;
5651 (* ------------------------------------------------------------------------- *)
5652 (* Extremal points of a simplex are some vertices. *)
5653 (* ------------------------------------------------------------------------- *)
5655 let DIST_INCREASES_ONLINE = prove
5656 (`!a b d. ~(d = vec 0)
5657 ==> dist(a,b + d) > dist(a,b) \/ dist(a,b - d) > dist(a,b)`,
5658 REWRITE_TAC[dist; vector_norm; real_gt; GSYM NORM_POS_LT] THEN
5659 SIMP_TAC[SQRT_MONO_LT_EQ; DOT_POS_LE; SQRT_LT_0] THEN
5660 REWRITE_TAC[DOT_RSUB; DOT_RADD; DOT_LSUB; DOT_LADD] THEN REAL_ARITH_TAC);;
5662 let NORM_INCREASES_ONLINE = prove
5663 (`!a:real^N d. ~(d = vec 0)
5664 ==> norm(a + d) > norm(a) \/ norm(a - d) > norm(a)`,
5665 MP_TAC(ISPEC `vec 0 :real^N` DIST_INCREASES_ONLINE) THEN
5666 REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG]);;
5668 let SIMPLEX_FURTHEST_LT = prove
5671 ==> !x. x IN (convex hull s) /\ ~(x IN s)
5672 ==> ?y. y IN (convex hull s) /\ norm(x - a) < norm(y - a)`,
5673 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
5674 REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
5675 MAP_EVERY X_GEN_TAC [`x:real^N`; `s:real^N->bool`] THEN
5676 ASM_CASES_TAC `s:real^N->bool = {}` THENL
5677 [ASM_REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN MESON_TAC[];
5679 ASM_SIMP_TAC[CONVEX_HULL_INSERT] THEN
5680 STRIP_TAC THEN X_GEN_TAC `y:real^N` THEN
5681 REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
5682 MAP_EVERY X_GEN_TAC [`u:real`; `v:real`; `b:real^N`] THEN
5683 ASM_CASES_TAC `y:real^N IN (convex hull s)` THENL
5684 [REWRITE_TAC[IN_INSERT; DE_MORGAN_THM] THEN STRIP_TAC THEN
5685 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN
5686 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
5687 X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN
5688 MAP_EVERY EXISTS_TAC [`&0`; `&1`; `c:real^N`] THEN
5689 ASM_REWRITE_TAC[REAL_ADD_LID; REAL_POS] THEN VECTOR_ARITH_TAC;
5691 ASM_CASES_TAC `u = &0` THENL
5692 [ASM_SIMP_TAC[REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
5693 ASM_MESON_TAC[VECTOR_MUL_LID];
5695 ASM_CASES_TAC `v = &0` THENL
5696 [ASM_SIMP_TAC[REAL_ADD_RID; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN
5697 ASM_CASES_TAC `u = &1` THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN
5698 ASM_CASES_TAC `y = a:real^N` THEN ASM_REWRITE_TAC[IN_INSERT] THEN
5701 REWRITE_TAC[IN_INSERT; DE_MORGAN_THM] THEN STRIP_TAC THEN
5702 MP_TAC(SPECL [`u:real`; `v:real`] REAL_DOWN2) THEN ANTS_TAC THENL
5703 [ASM_REWRITE_TAC[REAL_LT_LE]; ALL_TAC] THEN
5704 DISCH_THEN(X_CHOOSE_THEN `w:real` STRIP_ASSUME_TAC) THEN
5705 MP_TAC(ISPECL [`a:real^N`; `y:real^N`; `w % (x - b):real^N`]
5706 DIST_INCREASES_ONLINE) THEN
5708 [ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN
5709 REWRITE_TAC[VECTOR_ARITH `(x - y = vec 0) <=> (x = y)`] THEN
5710 DISCH_THEN SUBST_ALL_TAC THEN
5711 UNDISCH_TAC `~(y:real^N IN convex hull s)` THEN
5712 ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID];
5714 ASM_REWRITE_TAC[dist; real_gt] THEN
5715 REWRITE_TAC[VECTOR_ARITH
5716 `((u % x + v % b) + w % (x - b) = (u + w) % x + (v - w) % b) /\
5717 ((u % x + v % b) - w % (x - b) = (u - w) % x + (v + w) % b)`] THEN
5719 [MAP_EVERY EXISTS_TAC
5720 [`(u + w) % x + (v - w) % b:real^N`; `u + w`; `v - w`; `b:real^N`];
5721 MAP_EVERY EXISTS_TAC
5722 [`(u - w) % x + (v + w) % b:real^N`; `u - w`; `v + w`; `b:real^N`]] THEN
5723 ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[] THEN
5724 ASM_SIMP_TAC[REAL_LE_ADD; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
5725 UNDISCH_TAC `u + v = &1` THEN REAL_ARITH_TAC);;
5727 let SIMPLEX_FURTHEST_LE = prove
5729 FINITE s /\ ~(s = {})
5731 !x. x IN (convex hull s) ==> norm(x - a) <= norm(y - a)`,
5732 REPEAT GEN_TAC THEN DISCH_TAC THEN
5733 MP_TAC(ISPEC `convex hull (s:real^N->bool)` DISTANCE_ATTAINS_SUP) THEN
5734 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ANTS_TAC THENL
5735 [ASM_SIMP_TAC[FINITE_IMP_COMPACT_CONVEX_HULL] THEN
5736 ASM_MESON_TAC[SUBSET_EMPTY; HULL_SUBSET];
5738 ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[dist] THEN
5739 ASM_MESON_TAC[SIMPLEX_FURTHEST_LT; REAL_NOT_LE]);;
5741 let SIMPLEX_FURTHEST_LE_EXISTS = prove
5744 ==> !x. x IN (convex hull s)
5745 ==> ?y. y IN s /\ norm(x - a) <= norm(y - a)`,
5746 MESON_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY; SIMPLEX_FURTHEST_LE]);;
5748 let SIMPLEX_EXTREMAL_LE = prove
5750 FINITE s /\ ~(s = {})
5751 ==> ?u v. u IN s /\ v IN s /\
5752 !x y. x IN convex hull s /\ y IN convex hull s
5753 ==> norm(x - y) <= norm(u - v)`,
5754 REPEAT STRIP_TAC THEN
5755 MP_TAC(ISPEC `convex hull (s:real^N->bool)` COMPACT_SUP_MAXDISTANCE) THEN
5757 [ASM_SIMP_TAC[FINITE_IMP_COMPACT_CONVEX_HULL] THEN
5758 ASM_MESON_TAC[SUBSET_EMPTY; HULL_SUBSET];
5760 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
5761 SIMP_TAC[] THEN ASM_MESON_TAC[SIMPLEX_FURTHEST_LT; REAL_NOT_LE; NORM_SUB]);;
5763 let SIMPLEX_EXTREMAL_LE_EXISTS = prove
5764 (`!s:real^N->bool x y. FINITE s /\ x IN convex hull s /\ y IN convex hull s
5765 ==> ?u v. u IN s /\ v IN s /\
5766 norm(x - y) <= norm(u - v)`,
5767 MESON_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY; SIMPLEX_EXTREMAL_LE]);;
5769 let DIAMETER_CONVEX_HULL = prove
5770 (`!s:real^N->bool. diameter(convex hull s) = diameter s`,
5772 (`!a b s. (!x. x IN s ==> dist(a,x) <= b)
5773 ==> (!x. x IN convex hull s ==> dist(a,x) <= b)`,
5774 REPEAT GEN_TAC THEN DISCH_TAC THEN
5775 MATCH_MP_TAC HULL_INDUCT THEN ASM_REWRITE_TAC[GSYM cball; CONVEX_CBALL]) in
5776 GEN_TAC THEN REWRITE_TAC[diameter; CONVEX_HULL_EQ_EMPTY] THEN
5777 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUP_EQ THEN
5778 REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `b:real` THEN
5779 EQ_TAC THENL [MESON_TAC[SUBSET; HULL_SUBSET]; ALL_TAC] THEN
5780 MATCH_MP_TAC(TAUT `!b. (a ==> b) /\ (b ==> c) ==> a ==> c`) THEN
5781 EXISTS_TAC `!x:real^N y. x IN s /\ y IN convex hull s ==> norm(x - y) <= b`
5783 [MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THEN
5784 ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
5785 REWRITE_TAC[GSYM dist; lemma];
5786 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
5787 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^N` THEN
5788 ASM_CASES_TAC `(y:real^N) IN convex hull s` THEN ASM_REWRITE_TAC[] THEN
5789 REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[DIST_SYM] dist); lemma]]);;
5791 let DIAMETER_SIMPLEX = prove
5794 ==> diameter(convex hull s) = sup { dist(x,y) | x IN s /\ y IN s}`,
5795 REWRITE_TAC[DIAMETER_CONVEX_HULL] THEN SIMP_TAC[diameter; dist]);;
5797 (* ------------------------------------------------------------------------- *)
5798 (* Closest point of a convex set is unique, with a continuous projection. *)
5799 (* ------------------------------------------------------------------------- *)
5801 let CLOSER_POINTS_LEMMA = prove
5805 !v. &0 < v /\ v <= u ==> norm(v % z - y) < norm y`,
5806 REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB; DOT_LMUL; DOT_RMUL;
5807 REAL_SUB_LDISTRIB; real_gt] THEN REPEAT GEN_TAC THEN
5808 REWRITE_TAC[REAL_ARITH `(a - b) - (c - d) < d <=> a < b + c`] THEN
5809 STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `(z:real^N) dot y = y dot z`) THEN
5810 SIMP_TAC[GSYM REAL_ADD_LDISTRIB; REAL_LT_LMUL_EQ] THEN
5811 EXISTS_TAC `(y dot (z:real^N)) / (z dot z)` THEN
5812 SUBGOAL_THEN `&0 < z dot (z:real^N)` ASSUME_TAC THENL
5813 [ASM_MESON_TAC[DOT_POS_LT; DOT_RZERO; REAL_LT_REFL]; ALL_TAC] THEN
5814 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LE_RDIV_EQ] THEN
5815 ASM_SIMP_TAC[REAL_ARITH `&0 < y /\ x <= y ==> x < y + y`; REAL_LT_MUL]);;
5817 let CLOSER_POINT_LEMMA = prove
5818 (`!x y z. (y - x) dot (z - x) > &0
5819 ==> ?u. &0 < u /\ u <= &1 /\ dist(x + u % (z - x),y) < dist(x,y)`,
5820 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSER_POINTS_LEMMA) THEN
5821 ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[dist; NORM_LT] THEN
5822 REWRITE_TAC[VECTOR_ARITH
5823 `(y - (x + z)) dot (y - (x + z)) = (z - (y - x)) dot (z - (y - x))`] THEN
5824 DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
5825 EXISTS_TAC `min u (&1)` THEN
5826 ASM_SIMP_TAC[REAL_LT_MIN; REAL_MIN_LE; REAL_LT_01; REAL_LE_REFL]);;
5828 let ANY_CLOSEST_POINT_DOT = prove
5830 convex s /\ closed s /\ x IN s /\ y IN s /\
5831 (!z. z IN s ==> dist(a,x) <= dist(a,z))
5832 ==> (a - x) dot (y - x) <= &0`,
5833 REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN
5834 DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN
5835 DISCH_THEN(X_CHOOSE_THEN `u:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5836 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5837 REWRITE_TAC[REAL_NOT_LT] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
5838 FIRST_X_ASSUM MATCH_MP_TAC THEN
5839 REWRITE_TAC[VECTOR_ARITH `x + u % (y - x) = (&1 - u) % x + u % y`] THEN
5840 MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]);;
5842 let ANY_CLOSEST_POINT_UNIQUE = prove
5844 convex s /\ closed s /\ x IN s /\ y IN s /\
5845 (!z. z IN s ==> dist(a,x) <= dist(a,z)) /\
5846 (!z. z IN s ==> dist(a,y) <= dist(a,z))
5848 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
5849 REWRITE_TAC[GSYM NORM_LE_0; NORM_LE_SQUARE] THEN
5850 SUBGOAL_THEN `(a - x:real^N) dot (y - x) <= &0 /\ (a - y) dot (x - y) <= &0`
5851 MP_TAC THENL [ASM_MESON_TAC[ANY_CLOSEST_POINT_DOT]; ALL_TAC] THEN
5852 REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB] THEN REAL_ARITH_TAC);;
5854 let CLOSEST_POINT_UNIQUE = prove
5856 convex s /\ closed s /\ x IN s /\
5857 (!z. z IN s ==> dist(a,x) <= dist(a,z))
5858 ==> x = closest_point s a`,
5859 REPEAT STRIP_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_UNIQUE THEN
5860 MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `a:real^N`] THEN
5861 ASM_MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);;
5863 let CLOSEST_POINT_DOT = prove
5865 convex s /\ closed s /\ x IN s
5866 ==> (a - closest_point s a) dot (x - closest_point s a) <= &0`,
5867 REPEAT STRIP_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_DOT THEN
5868 EXISTS_TAC `s:real^N->bool` THEN
5869 ASM_MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);;
5871 let CLOSEST_POINT_LT = prove
5872 (`!s a x. convex s /\ closed s /\ x IN s /\ ~(x = closest_point s a)
5873 ==> dist(a,closest_point s a) < dist(a,x)`,
5875 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5876 REWRITE_TAC[GSYM REAL_NOT_LE; CONTRAPOS_THM] THEN
5877 DISCH_TAC THEN MATCH_MP_TAC CLOSEST_POINT_UNIQUE THEN
5878 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSEST_POINT_LE; REAL_LE_TRANS]);;
5880 let CLOSEST_POINT_LIPSCHITZ = prove
5882 convex s /\ closed s /\ ~(s = {})
5883 ==> dist(closest_point s x,closest_point s y) <= dist(x,y)`,
5884 REPEAT STRIP_TAC THEN REWRITE_TAC[dist; NORM_LE] THEN
5886 `(x - closest_point s x :real^N) dot
5887 (closest_point s y - closest_point s x) <= &0 /\
5888 (y - closest_point s y) dot
5889 (closest_point s x - closest_point s y) <= &0`
5891 [CONJ_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_DOT THEN
5892 EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[CLOSEST_POINT_EXISTS];
5893 MP_TAC(ISPEC `(x - closest_point s x :real^N) - (y - closest_point s y)`
5895 REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB; DOT_SYM] THEN REAL_ARITH_TAC]);;
5897 let CONTINUOUS_AT_CLOSEST_POINT = prove
5898 (`!s x. convex s /\ closed s /\ ~(s = {})
5899 ==> (closest_point s) continuous (at x)`,
5900 REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at] THEN
5901 ASM_MESON_TAC[CLOSEST_POINT_LIPSCHITZ; REAL_LET_TRANS]);;
5903 let CONTINUOUS_ON_CLOSEST_POINT = prove
5904 (`!s t. convex s /\ closed s /\ ~(s = {})
5905 ==> (closest_point s) continuous_on t`,
5906 MESON_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON; CONTINUOUS_AT_CLOSEST_POINT]);;
5908 (* ------------------------------------------------------------------------- *)
5909 (* Relating closest points and orthogonality. *)
5910 (* ------------------------------------------------------------------------- *)
5912 let ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL = prove
5914 affine s /\ b IN s /\ (!x. x IN s ==> dist(a,b) <= dist(a,x))
5915 ==> (!x. x IN s ==> orthogonal (x - b) (a - b))`,
5916 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `b:real^N` THEN
5917 REWRITE_TAC[DIST_0; VECTOR_SUB_RZERO; orthogonal; dist; NORM_LE] THEN
5918 REWRITE_TAC[DOT_LSUB] THEN REWRITE_TAC[DOT_RSUB] THEN
5919 REWRITE_TAC[DOT_SYM; REAL_ARITH `a <= a - y - (y - x) <=> &2 * y <= x`] THEN
5920 REPEAT STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
5921 ASM_REWRITE_TAC[DOT_RZERO] THEN FIRST_X_ASSUM(fun th ->
5922 MP_TAC(SPEC `vec 0 + --((a dot x) / (x dot x)) % (x - vec 0:real^N)` th) THEN
5923 MP_TAC(SPEC `vec 0 + (a dot x) / (x dot x) % (x - vec 0:real^N)` th)) THEN
5924 ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF] THEN
5925 REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ADD_LID; DOT_RMUL] THEN
5926 REWRITE_TAC[DOT_LMUL; IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
5927 `&2 * x * a <= b * c * z /\ &2 * --x * a <= --b * --c * z
5928 ==> &2 * abs(x * a) <= b * c * z`)) THEN
5929 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
5930 ASM_SIMP_TAC[REAL_NOT_LE; REAL_DIV_RMUL; DOT_EQ_0] THEN
5931 MATCH_MP_TAC(REAL_ARITH `~(x = &0) ==> x < &2 * abs x`) THEN
5932 RULE_ASSUM_TAC(REWRITE_RULE[GSYM DOT_EQ_0]) THEN
5933 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD);;
5935 let ORTHOGONAL_ANY_CLOSEST_POINT = prove
5937 b IN s /\ (!x. x IN s ==> orthogonal (x - b) (a - b))
5938 ==> (!x. x IN s ==> dist(a,b) <= dist(a,x))`,
5939 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `b:real^N` THEN
5940 REWRITE_TAC[dist; NORM_LE; orthogonal; VECTOR_SUB_RZERO] THEN
5941 SIMP_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN
5942 REWRITE_TAC[DOT_POS_LE; REAL_ARITH `a <= a - &0 - (&0 - x) <=> &0 <= x`]);;
5944 let CLOSEST_POINT_AFFINE_ORTHOGONAL = prove
5946 affine s /\ ~(s = {}) /\ x IN s
5947 ==> orthogonal (x - closest_point s a) (a - closest_point s a)`,
5948 GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5949 DISCH_TAC THEN DISCH_TAC THEN GEN_TAC THEN
5950 MATCH_MP_TAC ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL THEN
5951 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSEST_POINT_EXISTS THEN
5952 ASM_SIMP_TAC[CLOSED_AFFINE]);;
5954 let CLOSEST_POINT_AFFINE_ORTHOGONAL_EQ = prove
5957 ==> (closest_point s a = b <=>
5958 !x. x IN s ==> orthogonal (x - b) (a - b))`,
5959 REPEAT STRIP_TAC THEN EQ_TAC THENL
5960 [ASM_MESON_TAC[CLOSEST_POINT_AFFINE_ORTHOGONAL; MEMBER_NOT_EMPTY];
5961 DISCH_TAC THEN CONV_TAC SYM_CONV THEN
5962 MATCH_MP_TAC CLOSEST_POINT_UNIQUE THEN
5963 ASM_SIMP_TAC[CLOSED_AFFINE; AFFINE_IMP_CONVEX] THEN
5964 MATCH_MP_TAC ORTHOGONAL_ANY_CLOSEST_POINT THEN ASM_REWRITE_TAC[]]);;
5966 (* ------------------------------------------------------------------------- *)
5967 (* Various point-to-set separating/supporting hyperplane theorems. *)
5968 (* ------------------------------------------------------------------------- *)
5970 let SUPPORTING_HYPERPLANE_COMPACT_POINT_SUP = prove
5971 (`!a c s:real^N->bool.
5972 compact s /\ ~(s = {})
5973 ==> ?b y. y IN s /\ a dot (y - c) = b /\
5974 (!x. x IN s ==> a dot (x - c) <= b)`,
5975 REPEAT STRIP_TAC THEN
5976 MP_TAC(ISPECL [`\x:real^N. a dot (x - c)`; `s:real^N->bool`]
5977 CONTINUOUS_ATTAINS_SUP) THEN
5978 ASM_REWRITE_TAC[] THEN
5979 ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
5980 SUBGOAL_THEN `(\x:real^N. a dot (x - c)) = (\x. a dot x) o (\x. x - c)`
5981 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
5982 REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
5983 SIMP_TAC[CONTINUOUS_ON_LIFT_DOT; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST;
5984 CONTINUOUS_ON_ID]);;
5986 let SUPPORTING_HYPERPLANE_COMPACT_POINT_INF = prove
5987 (`!a c s:real^N->bool.
5988 compact s /\ ~(s = {})
5989 ==> ?b y. y IN s /\ a dot (y - c) = b /\
5990 (!x. x IN s ==> a dot (x - c) >= b)`,
5991 REPEAT STRIP_TAC THEN
5992 MP_TAC(ISPECL [`--a:real^N`; `c:real^N`; `s:real^N->bool`]
5993 SUPPORTING_HYPERPLANE_COMPACT_POINT_SUP) THEN
5994 ASM_REWRITE_TAC[] THEN
5995 DISCH_THEN(X_CHOOSE_THEN `b:real`
5996 (fun th -> EXISTS_TAC `--b:real` THEN MP_TAC th)) THEN
5997 REWRITE_TAC[DOT_LNEG; REAL_ARITH `x >= -- b <=> --x <= b`] THEN
5998 REWRITE_TAC[REAL_NEG_EQ]);;
6000 let SUPPORTING_HYPERPLANE_CLOSED_POINT = prove
6001 (`!s z:real^N. convex s /\ closed s /\ ~(s = {}) /\ ~(z IN s)
6002 ==> ?a b y. a dot z < b /\ y IN s /\ (a dot y = b) /\
6003 (!x. x IN s ==> a dot x >= b)`,
6004 REPEAT STRIP_TAC THEN
6005 MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN
6006 ASM_REWRITE_TAC[] THEN
6007 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
6008 EXISTS_TAC `y - z:real^N` THEN EXISTS_TAC `(y - z:real^N) dot y` THEN
6009 EXISTS_TAC `y:real^N` THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
6010 ASM_REWRITE_TAC[GSYM DOT_RSUB; DOT_POS_LT; VECTOR_SUB_EQ] THEN
6011 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN
6012 DISCH_TAC THEN SUBGOAL_THEN
6013 `!u. &0 <= u /\ u <= &1 ==> dist(z:real^N,y) <= dist(z,(&1 - u) % y + u % x)`
6014 MP_TAC THENL [ASM_MESON_TAC[CONVEX_ALT]; ALL_TAC] THEN
6015 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
6016 REWRITE_TAC[real_ge; REAL_NOT_LE; NOT_FORALL_THM; NOT_IMP] THEN
6017 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `x < y <=> y - x > &0`] THEN
6018 REWRITE_TAC[VECTOR_ARITH
6019 `(a - b) dot x - (a - b) dot y = (b - a) dot (y - x)`] THEN
6020 DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN
6021 REWRITE_TAC[VECTOR_ARITH `y + u % (x - y) = (&1 - u) % y + u % x`] THEN
6022 MESON_TAC[REAL_LT_IMP_LE]);;
6024 let SEPARATING_HYPERPLANE_CLOSED_POINT_INSET = prove
6025 (`!s z:real^N. convex s /\ closed s /\ ~(s = {}) /\ ~(z IN s)
6027 (a - z) dot z < b /\
6028 (!x. x IN s ==> (a - z) dot x > b)`,
6029 REPEAT STRIP_TAC THEN
6030 MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN
6031 ASM_REWRITE_TAC[] THEN
6032 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
6033 EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN
6034 EXISTS_TAC `(y - z:real^N) dot z + norm(y - z) pow 2 / &2` THEN
6035 SUBGOAL_THEN `&0 < norm(y - z:real^N)` ASSUME_TAC THENL
6036 [ASM_MESON_TAC[NORM_POS_LT; VECTOR_SUB_EQ]; ALL_TAC] THEN
6037 ASM_SIMP_TAC[REAL_LT_ADDR; REAL_LT_DIV; REAL_POW_LT;
6038 REAL_OF_NUM_LT; ARITH] THEN
6039 REWRITE_TAC[NORM_POW_2; REAL_ARITH `a > b + c <=> c < a - b`] THEN
6040 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
6041 SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
6042 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
6043 REWRITE_TAC[VECTOR_ARITH
6044 `((y - z) dot x - (y - z) dot z) * &2 - (y - z) dot (y - z) =
6045 &2 * ((y - z) dot (x - y)) + (y - z) dot (y - z)`] THEN
6046 MATCH_MP_TAC(REAL_ARITH `~(--x > &0) /\ &0 < y ==> &0 < &2 * x + y`) THEN
6047 ASM_SIMP_TAC[GSYM NORM_POW_2; REAL_POW_LT] THEN
6048 REWRITE_TAC[GSYM DOT_LNEG; VECTOR_NEG_SUB] THEN
6049 DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN
6050 REWRITE_TAC[NOT_EXISTS_THM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
6051 GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
6052 REWRITE_TAC[REAL_NOT_LT] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
6053 REWRITE_TAC[VECTOR_ARITH `y + u % (x - y) = (&1 - u) % y + u % x`] THEN
6054 ASM_MESON_TAC[CONVEX_ALT; REAL_LT_IMP_LE]);;
6056 let SEPARATING_HYPERPLANE_CLOSED_0_INSET = prove
6058 convex s /\ closed s /\ ~(s = {}) /\ ~(vec 0 IN s)
6059 ==> ?a b. a IN s /\ ~(a = vec 0) /\ &0 < b /\
6060 (!x. x IN s ==> a dot x > b)`,
6061 REPEAT GEN_TAC THEN DISCH_TAC THEN
6062 FIRST_ASSUM(MP_TAC o MATCH_MP SEPARATING_HYPERPLANE_CLOSED_POINT_INSET) THEN
6063 REWRITE_TAC[DOT_RZERO; real_gt] THEN
6064 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
6065 SIMP_TAC[VECTOR_SUB_RZERO] THEN ASM_MESON_TAC[]);;
6067 let SEPARATING_HYPERPLANE_CLOSED_POINT = prove
6068 (`!s z:real^N. convex s /\ closed s /\ ~(z IN s)
6069 ==> ?a b. a dot z < b /\ (!x. x IN s ==> a dot x > b)`,
6070 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6071 [MAP_EVERY EXISTS_TAC [`--z:real^N`; `&1`] THEN
6072 SIMP_TAC[DOT_LNEG; REAL_ARITH `&0 <= x ==> --x < &1`; DOT_POS_LE] THEN
6073 ASM_MESON_TAC[NOT_IN_EMPTY];
6075 ASM_MESON_TAC[SEPARATING_HYPERPLANE_CLOSED_POINT_INSET]);;
6077 let SEPARATING_HYPERPLANE_CLOSED_0 = prove
6079 convex s /\ closed s /\ ~(vec 0 IN s)
6080 ==> ?a b. ~(a = vec 0) /\ &0 < b /\ (!x. x IN s ==> a dot x > b)`,
6081 REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6082 [EXISTS_TAC `basis 1:real^N` THEN EXISTS_TAC `&1` THEN
6083 ASM_REWRITE_TAC[NOT_IN_EMPTY; REAL_LT_01; GSYM NORM_POS_LT] THEN
6084 ASM_SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_LT_01];
6085 FIRST_X_ASSUM(MP_TAC o MATCH_MP SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6086 REWRITE_TAC[DOT_RZERO; real_gt] THEN
6087 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
6088 ASM_MESON_TAC[MEMBER_NOT_EMPTY; DOT_LZERO; REAL_LT_ANTISYM]]);;
6090 (* ------------------------------------------------------------------------- *)
6091 (* Now set-to-set for closed/compact sets. *)
6092 (* ------------------------------------------------------------------------- *)
6094 let SEPARATING_HYPERPLANE_CLOSED_COMPACT = prove
6095 (`!s t. convex s /\ closed s /\
6096 convex t /\ compact t /\ ~(t = {}) /\ DISJOINT s t
6097 ==> ?a:real^N b. (!x. x IN s ==> a dot x < b) /\
6098 (!x. x IN t ==> a dot x > b)`,
6099 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6100 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
6101 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
6102 REWRITE_TAC[BOUNDED_POS] THEN
6103 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN
6104 SUBGOAL_THEN `?z:real^N. norm(z) = b + &1` CHOOSE_TAC THENL
6105 [ASM_SIMP_TAC[VECTOR_CHOOSE_SIZE; REAL_ARITH `&0 < b ==> &0 <= b + &1`];
6107 MP_TAC(SPECL [`t:real^N->bool`; `z:real^N`]
6108 SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6109 ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
6110 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN
6111 ASM_MESON_TAC[REAL_ARITH `~(b + &1 <= b)`];
6113 MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0 :real^N`]
6114 SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6115 ASM_SIMP_TAC[CLOSED_COMPACT_DIFFERENCES; CONVEX_DIFFERENCES] THEN
6117 [REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
6118 REWRITE_TAC[VECTOR_SUB_EQ] THEN
6119 ASM_MESON_TAC[DISJOINT; NOT_IN_EMPTY; IN_INTER; EXTENSION];
6121 SIMP_TAC[DOT_RZERO; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
6122 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN
6123 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6124 GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN
6125 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
6126 ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
6127 REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL; DOT_RSUB] THEN
6128 REWRITE_TAC[real_gt; REAL_LT_SUB_LADD] THEN DISCH_TAC THEN
6129 EXISTS_TAC `--a:real^N` THEN
6130 MP_TAC(SPEC `IMAGE (\x:real^N. a dot x) t` SUP) THEN
6131 ABBREV_TAC `k = sup (IMAGE (\x:real^N. a dot x) t)` THEN
6132 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN ANTS_TAC THENL
6133 [ASM_MESON_TAC[REAL_ARITH `b + x < y ==> x <= y - b`; MEMBER_NOT_EMPTY];
6135 STRIP_TAC THEN EXISTS_TAC `--(k + b / &2)` THEN
6136 REWRITE_TAC[DOT_LNEG; REAL_LT_NEG2] THEN REPEAT STRIP_TAC THEN
6137 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH;
6138 REAL_ARITH `&0 < b /\ x <= k ==> x < k + b`] THEN
6139 FIRST_X_ASSUM(MP_TAC o SPEC `k - b / &2`) THEN
6140 ASM_SIMP_TAC[REAL_ARITH `k <= k - b2 <=> ~(&0 < b2)`; REAL_LT_DIV;
6141 REAL_OF_NUM_LT; ARITH; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP] THEN
6142 X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6143 MATCH_MP_TAC(REAL_ARITH
6144 `!b. (b2 + b2 = b) /\ b + ay < ax ==> ~(ay <= k - b2) ==> k + b2 < ax`) THEN
6145 ASM_MESON_TAC[REAL_HALF]);;
6147 let SEPARATING_HYPERPLANE_COMPACT_CLOSED = prove
6148 (`!s t. convex s /\ compact s /\ ~(s = {}) /\
6149 convex t /\ closed t /\ DISJOINT s t
6150 ==> ?a:real^N b. (!x. x IN s ==> a dot x < b) /\
6151 (!x. x IN t ==> a dot x > b)`,
6152 REPEAT STRIP_TAC THEN
6153 MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`]
6154 SEPARATING_HYPERPLANE_CLOSED_COMPACT) THEN
6155 ANTS_TAC THENL [ASM_MESON_TAC[DISJOINT_SYM]; ALL_TAC] THEN
6156 REWRITE_TAC[real_gt] THEN
6157 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real`
6158 STRIP_ASSUME_TAC)) THEN
6159 MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`] THEN
6160 ASM_REWRITE_TAC[REAL_LT_NEG2; DOT_LNEG]);;
6162 let SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO = prove
6163 (`!s t:real^N->bool.
6164 convex s /\ compact s /\ ~(s = {}) /\
6165 convex t /\ closed t /\ DISJOINT s t
6166 ==> ?a b. ~(a = vec 0) /\
6167 (!x. x IN s ==> a dot x < b) /\
6168 (!x. x IN t ==> a dot x > b)`,
6169 REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL
6170 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN
6171 EXISTS_TAC `basis 1:real^N` THEN
6173 `bounded(IMAGE (\x:real^N. lift(basis 1 dot x)) s)`
6175 [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
6176 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
6177 ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DOT];
6178 REWRITE_TAC[BOUNDED_POS_LT; FORALL_IN_IMAGE; NORM_LIFT] THEN
6179 SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN
6180 MESON_TAC[REAL_ARITH `abs x < b ==> x < b`]];
6182 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`]
6183 SEPARATING_HYPERPLANE_COMPACT_CLOSED) THEN
6184 ASM_REWRITE_TAC[] THEN
6185 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
6186 ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_REWRITE_TAC[] THEN
6187 REWRITE_TAC[DOT_LZERO; real_gt] THEN
6188 ASM_MESON_TAC[REAL_LT_ANTISYM; MEMBER_NOT_EMPTY]]);;
6190 let SEPARATING_HYPERPLANE_COMPACT_COMPACT = prove
6191 (`!s t:real^N->bool.
6192 convex s /\ compact s /\ convex t /\ compact t /\ DISJOINT s t
6193 ==> ?a b. ~(a = vec 0) /\
6194 (!x. x IN s ==> a dot x < b) /\
6195 (!x. x IN t ==> a dot x > b)`,
6196 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6197 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN
6198 EXISTS_TAC `--basis 1:real^N` THEN
6200 `bounded(IMAGE (\x:real^N. lift(basis 1 dot x)) t)`
6202 [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
6203 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
6204 ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DOT];
6205 REWRITE_TAC[BOUNDED_POS_LT; FORALL_IN_IMAGE; NORM_LIFT] THEN
6206 SIMP_TAC[VECTOR_NEG_EQ_0; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN
6207 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN
6208 EXISTS_TAC `--b:real` THEN REWRITE_TAC[DOT_LNEG] THEN
6209 REWRITE_TAC[REAL_ARITH `--x > --y <=> x < y`] THEN
6210 ASM_MESON_TAC[REAL_ARITH `abs x < b ==> x < b`]];
6212 MATCH_MP_TAC SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO THEN
6213 ASM_SIMP_TAC[COMPACT_IMP_CLOSED]]);;
6215 (* ------------------------------------------------------------------------- *)
6216 (* General case without assuming closure and getting non-strict separation. *)
6217 (* ------------------------------------------------------------------------- *)
6219 let SEPARATING_HYPERPLANE_SET_0_INSPAN = prove
6221 convex s /\ ~(s = {}) /\ ~(vec 0 IN s)
6222 ==> ?a b. a IN span s /\ ~(a = vec 0) /\
6223 !x. x IN s ==> &0 <= a dot x`,
6224 REPEAT STRIP_TAC THEN
6225 ABBREV_TAC `k = \c:real^N. {x | &0 <= c dot x}` THEN
6227 `~((span s INTER frontier(cball(vec 0:real^N,&1))) INTER
6228 (INTERS (IMAGE k (s:real^N->bool))) = {})`
6231 SIMP_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_INTERS; NOT_FORALL_THM;
6232 FORALL_IN_IMAGE; FRONTIER_CBALL; REAL_LT_01] THEN
6233 EXPAND_TAC "k" THEN REWRITE_TAC[IN_SPHERE_0; IN_ELIM_THM; NORM_NEG] THEN
6234 MESON_TAC[NORM_EQ_0; REAL_ARITH `~(&1 = &0)`; DOT_SYM]] THEN
6235 MATCH_MP_TAC COMPACT_IMP_FIP THEN
6236 SIMP_TAC[COMPACT_CBALL; COMPACT_FRONTIER; FORALL_IN_IMAGE;
6237 CLOSED_INTER_COMPACT; CLOSED_SPAN] THEN
6239 [EXPAND_TAC "k" THEN REWRITE_TAC[GSYM real_ge; CLOSED_HALFSPACE_GE];
6241 REWRITE_TAC[FINITE_SUBSET_IMAGE] THEN GEN_TAC THEN
6242 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` MP_TAC) THEN
6243 ASM_CASES_TAC `c:real^N->bool = {}` THENL
6244 [ASM_SIMP_TAC[INTERS_0; INTER_UNIV; IMAGE_CLAUSES] THEN
6245 DISCH_THEN(K ALL_TAC) THEN
6246 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
6247 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
6248 SUBGOAL_THEN `~(a:real^N = vec 0)` ASSUME_TAC THENL
6249 [ASM_MESON_TAC[]; ALL_TAC] THEN
6250 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
6251 EXISTS_TAC `inv(norm a) % a:real^N` THEN
6252 ASM_SIMP_TAC[IN_INTER; FRONTIER_CBALL; SPAN_CLAUSES; IN_SPHERE_0] THEN
6253 REWRITE_TAC[DIST_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN
6254 ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0];
6256 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
6257 MP_TAC(ISPEC `convex hull (c:real^N->bool)`
6258 SEPARATING_HYPERPLANE_CLOSED_0_INSET) THEN
6260 [ASM_REWRITE_TAC[CONVEX_HULL_EQ_EMPTY] THEN
6261 ASM_MESON_TAC[CONVEX_CONVEX_HULL; SUBSET; SUBSET_HULL; HULL_SUBSET;
6262 FINITE_IMP_COMPACT_CONVEX_HULL; COMPACT_IMP_CLOSED];
6264 REWRITE_TAC[DOT_RZERO; real_gt] THEN
6265 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real`
6266 STRIP_ASSUME_TAC)) THEN
6267 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_INTERS; FORALL_IN_IMAGE] THEN
6268 EXPAND_TAC "k" THEN SIMP_TAC[IN_ELIM_THM; FRONTIER_CBALL; REAL_LT_01] THEN
6269 REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG] THEN
6270 EXISTS_TAC `inv(norm(a)) % a:real^N` THEN REWRITE_TAC[DOT_RMUL] THEN
6271 SUBGOAL_THEN `(a:real^N) IN s` ASSUME_TAC THENL
6272 [ASM_MESON_TAC[SUBSET; HULL_MINIMAL]; ASM_SIMP_TAC[SPAN_CLAUSES]] THEN
6273 REWRITE_TAC[IN_SPHERE_0; VECTOR_SUB_LZERO; NORM_NEG; NORM_MUL] THEN
6274 REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM] THEN
6275 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN
6276 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_EQ_LDIV_EQ; NORM_POS_LT] THEN
6277 REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN
6278 ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS; HULL_SUBSET; SUBSET; DOT_SYM]);;
6280 let SEPARATING_HYPERPLANE_SET_POINT_INAFF = prove
6282 convex s /\ ~(s = {}) /\ ~(z IN s)
6283 ==> ?a b. (z + a) IN affine hull (z INSERT s) /\ ~(a = vec 0) /\
6284 a dot z <= b /\ (!x. x IN s ==> a dot x >= b)`,
6285 REPEAT STRIP_TAC THEN
6286 MP_TAC(ISPEC `IMAGE (\x:real^N. --z + x) s`
6287 SEPARATING_HYPERPLANE_SET_0_INSPAN) THEN
6288 ASM_SIMP_TAC[FORALL_IN_IMAGE; CONVEX_TRANSLATION; IMAGE_EQ_EMPTY] THEN
6289 REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --z + x <=> x = z`] THEN
6290 ASM_SIMP_TAC[UNWIND_THM2; AFFINE_HULL_INSERT_SPAN; IN_ELIM_THM] THEN
6291 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
6292 REWRITE_TAC[GSYM SIMPLE_IMAGE; VECTOR_ARITH `--x + y:real^N = y - x`] THEN
6293 STRIP_TAC THEN ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
6294 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
6295 EXISTS_TAC `(a:real^N) dot z` THEN REWRITE_TAC[REAL_LE_REFL] THEN
6296 ASM_REWRITE_TAC[REAL_ARITH `x >= y <=> &0 <= x - y`; GSYM DOT_RSUB]);;
6298 let SEPARATING_HYPERPLANE_SET_0 = prove
6300 convex s /\ ~(vec 0 IN s)
6301 ==> ?a b. ~(a = vec 0) /\ !x. x IN s ==> &0 <= a dot x`,
6302 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6303 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
6304 MESON_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1];
6305 ASM_MESON_TAC[SEPARATING_HYPERPLANE_SET_0_INSPAN]]);;
6307 let SEPARATING_HYPERPLANE_SETS = prove
6308 (`!s t. convex s /\ convex t /\ ~(s = {}) /\ ~(t = {}) /\ DISJOINT s t
6309 ==> ?a:real^N b. ~(a = vec 0) /\
6310 (!x. x IN s ==> a dot x <= b) /\
6311 (!x. x IN t ==> a dot x >= b)`,
6312 REPEAT STRIP_TAC THEN
6313 MP_TAC(ISPEC `{y - x:real^N | y IN t /\ x IN s}`
6314 SEPARATING_HYPERPLANE_SET_0) THEN
6315 ASM_SIMP_TAC[CONVEX_DIFFERENCES] THEN ANTS_TAC THENL
6316 [REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
6317 REWRITE_TAC[VECTOR_SUB_EQ] THEN
6318 ASM_MESON_TAC[DISJOINT; NOT_IN_EMPTY; IN_INTER; EXTENSION];
6320 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
6321 SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
6322 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6323 GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN
6324 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
6325 ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
6326 REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL; DOT_RSUB; REAL_SUB_LE] THEN
6328 MP_TAC(SPEC `IMAGE (\x:real^N. a dot x) s` SUP) THEN
6329 ABBREV_TAC `k = sup (IMAGE (\x:real^N. a dot x) s)` THEN
6330 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY; real_ge] THEN ANTS_TAC THENL
6331 [ASM_MESON_TAC[MEMBER_NOT_EMPTY]; ASM_MESON_TAC[]]);;
6333 (* ------------------------------------------------------------------------- *)
6334 (* More convexity generalities. *)
6335 (* ------------------------------------------------------------------------- *)
6337 let CONVEX_CLOSURE = prove
6338 (`!s:real^N->bool. convex s ==> convex(closure s)`,
6339 REWRITE_TAC[convex; CLOSURE_SEQUENTIAL] THEN
6340 GEN_TAC THEN DISCH_TAC THEN
6341 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
6342 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:num->real^N`) MP_TAC) THEN
6343 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `b:num->real^N`) MP_TAC) THEN
6344 STRIP_TAC THEN EXISTS_TAC `\n:num. u % a(n) + v % b(n) :real^N` THEN
6345 ASM_SIMP_TAC[LIM_ADD; LIM_CMUL]);;
6347 let CONVEX_INTERIOR = prove
6348 (`!s:real^N->bool. convex s ==> convex(interior s)`,
6349 REWRITE_TAC[CONVEX_ALT; IN_INTERIOR; SUBSET; IN_BALL; dist] THEN
6350 REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT GEN_TAC THEN
6351 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `d:real`) MP_TAC) THEN
6352 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `e:real`) STRIP_ASSUME_TAC) THEN
6353 EXISTS_TAC `min d e` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
6354 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
6355 SUBST1_TAC(VECTOR_ARITH `z:real^N =
6356 (&1 - u) % (z - u % (y - x)) + u % (z + (&1 - u) % (y - x))`) THEN
6357 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
6358 ASM_MESON_TAC[VECTOR_ARITH `x - (z - u % (y - x)) =
6359 ((&1 - u) % x + u % y) - z:real^N`;
6360 VECTOR_ARITH `y - (z + (&1 - u) % (y - x)) =
6361 ((&1 - u) % x + u % y) - z:real^N`]);;
6363 (* ------------------------------------------------------------------------- *)
6364 (* Moving and scaling convex hulls. *)
6365 (* ------------------------------------------------------------------------- *)
6367 let CONVEX_HULL_TRANSLATION = prove
6369 convex hull (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (convex hull s)`,
6370 REPEAT GEN_TAC THEN MATCH_MP_TAC HULL_IMAGE THEN
6371 REWRITE_TAC[CONVEX_TRANSLATION_EQ; CONVEX_CONVEX_HULL] THEN
6372 REWRITE_TAC[VECTOR_ARITH `a + x:real^N = y <=> x = y - a`; EXISTS_REFL] THEN
6375 add_translation_invariants [CONVEX_HULL_TRANSLATION];;
6377 let CONVEX_HULL_SCALING = prove
6378 (`!s:real^N->bool c.
6379 convex hull (IMAGE (\x. c % x) s) = IMAGE (\x. c % x) (convex hull s)`,
6380 REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THENL
6381 [ASM_SIMP_TAC[IMAGE_CONST; VECTOR_MUL_LZERO; CONVEX_HULL_EQ_EMPTY] THEN
6382 COND_CASES_TAC THEN REWRITE_TAC[CONVEX_HULL_EMPTY; CONVEX_HULL_SING];
6384 MATCH_MP_TAC HULL_IMAGE THEN
6385 ASM_SIMP_TAC[CONVEX_SCALING_EQ; CONVEX_CONVEX_HULL] THEN
6386 REWRITE_TAC[VECTOR_ARITH `c % x = c % y <=> c % (x - y) = vec 0`] THEN
6387 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
6388 X_GEN_TAC `x:real^N` THEN EXISTS_TAC `inv c % x:real^N` THEN
6389 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]);;
6391 let CONVEX_HULL_AFFINITY = prove
6393 convex hull (IMAGE (\x. a + c % x) s) =
6394 IMAGE (\x. a + c % x) (convex hull s)`,
6396 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
6397 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
6398 ASM_SIMP_TAC[IMAGE_o; CONVEX_HULL_TRANSLATION; CONVEX_HULL_SCALING]);;
6400 (* ------------------------------------------------------------------------- *)
6401 (* Convex set as intersection of halfspaces. *)
6402 (* ------------------------------------------------------------------------- *)
6404 let CONVEX_HALFSPACE_INTERSECTION = prove
6405 (`!s. closed(s:real^N->bool) /\ convex s
6406 ==> s = INTERS {h | s SUBSET h /\ ?a b. h = {x | a dot x <= b}}`,
6407 REPEAT STRIP_TAC THEN
6408 GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_INTERS] THEN
6409 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
6410 REWRITE_TAC[MESON[] `(!t. (P t /\ ?a b. t = x a b) ==> Q t) <=>
6411 (!a b. P(x a b) ==> Q(x a b))`] THEN
6412 EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN STRIP_TAC THEN
6413 MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
6414 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`]
6415 SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6416 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
6417 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN
6418 FIRST_X_ASSUM(MP_TAC o SPECL [`--a:real^N`; `--b:real`]) THEN
6419 ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; DOT_LNEG; NOT_IMP] THEN
6420 ASM_SIMP_TAC[REAL_LE_NEG2; REAL_LT_NEG2; REAL_NOT_LE;
6421 REAL_ARITH `a > b ==> b <= a`]);;
6423 (* ------------------------------------------------------------------------- *)
6424 (* Radon's theorem (from Lars Schewe). *)
6425 (* ------------------------------------------------------------------------- *)
6427 let RADON_EX_LEMMA = prove
6428 (`!(c:real^N->bool).
6429 FINITE c /\ affine_dependent c
6430 ==> (?u. sum c u = &0 /\ (?v. v IN c /\ ~(u v = &0)) /\
6431 vsum c (\v. u v % v) = (vec 0):real^N)`,
6432 REWRITE_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN
6433 REPEAT STRIP_TAC THEN
6434 EXISTS_TAC `\v:real^N. if v IN s then u v else &0` THEN
6435 ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET] THEN
6436 ASM_SIMP_TAC[COND_RAND;COND_RATOR;
6437 VECTOR_MUL_LZERO;GSYM VSUM_RESTRICT_SET] THEN
6438 ASM_SIMP_TAC[SET_RULE `s SUBSET c ==> {x | x IN c /\ x IN s} = s`] THEN
6439 EXISTS_TAC `v:real^N` THEN
6440 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
6442 let RADON_S_LEMMA = prove
6444 FINITE s /\ sum s f = &0
6445 ==> sum {x | x IN s /\ &0 < f x} f =
6446 -- sum {x | x IN s /\ f x < &0} f`,
6447 REWRITE_TAC[REAL_ARITH `a = --b <=> a + b = &0`] THEN
6448 REPEAT STRIP_TAC THEN
6449 ASM_SIMP_TAC[FINITE_RESTRICT;GSYM SUM_UNION;
6450 REWRITE_RULE [REAL_ARITH `&0 < f x ==> ~(f x < &0)`]
6451 (SET_RULE `(!x:A. &0 < f x ==> ~(f x < &0))
6452 ==> DISJOINT {x | x IN s /\ &0 < f x}
6453 {x | x IN s /\ f x < &0}`)] THEN
6454 MATCH_MP_TAC (REAL_ARITH `!a b.a = &0 /\ a + b = &0 ==> b = &0`) THEN
6455 EXISTS_TAC `sum {x:A | x IN s /\ f x = &0} f` THEN
6457 [ASM_SIMP_TAC[SUM_RESTRICT_SET] THEN REWRITE_TAC[COND_ID;SUM_0];
6459 SUBGOAL_THEN `DISJOINT {x:A | x IN s /\ f x = &0}
6460 ({x | x IN s /\ &0 < f x} UNION
6461 {x | x IN s /\ f x < &0})` ASSUME_TAC THENL
6462 [REWRITE_TAC[DISJOINT;UNION;INTER;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN
6465 ASM_SIMP_TAC[FINITE_UNION;FINITE_RESTRICT;GSYM SUM_UNION] THEN
6466 FIRST_X_ASSUM (SUBST1_TAC o GSYM) THEN
6467 MATCH_MP_TAC (MESON[] `a = b ==> sum a f = sum b f`) THEN
6468 REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION] THEN
6469 MESON_TAC[REAL_LT_TOTAL]);;
6471 let RADON_V_LEMMA = prove
6473 FINITE s /\ vsum s f = vec 0 /\ (!x. g x = &0 ==> f x = vec 0)
6474 ==> (vsum {x | x IN s /\ &0 < g x} f) :real^N =
6475 -- vsum {x | x IN s /\ g x < &0} f`,
6476 REWRITE_TAC[VECTOR_ARITH `a:real^N = --b <=> a + b = vec 0`] THEN
6477 REPEAT STRIP_TAC THEN
6478 ASM_SIMP_TAC[FINITE_RESTRICT;GSYM VSUM_UNION;
6479 REWRITE_RULE [REAL_ARITH `&0 < f x ==> ~(f x < &0)`]
6480 (SET_RULE `(!x:A. &0 < f x ==> ~(f x < &0))
6481 ==> DISJOINT {x | x IN s /\ &0 < f x}
6482 {x | x IN s /\ f x < &0}`)] THEN
6483 MATCH_MP_TAC (VECTOR_ARITH
6484 `!a b. (a:real^N) = vec 0 /\ a + b = vec 0 ==> b = vec 0`) THEN
6485 EXISTS_TAC `(vsum {x:A | x IN s /\ g x = &0} f):real^N` THEN
6487 [ASM_SIMP_TAC[VSUM_RESTRICT_SET;COND_ID;VSUM_0];ALL_TAC] THEN
6488 SUBGOAL_THEN `DISJOINT {x:A | x IN s /\ g x = &0}
6489 ({x | x IN s /\ &0 < g x} UNION
6490 {x | x IN s /\ g x < &0})` ASSUME_TAC THENL
6491 [REWRITE_TAC[DISJOINT;UNION;INTER;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN
6494 ASM_SIMP_TAC[FINITE_UNION;FINITE_RESTRICT;GSYM VSUM_UNION] THEN
6495 FIRST_X_ASSUM (SUBST1_TAC o GSYM) THEN
6496 MATCH_MP_TAC (MESON[] `a = b ==> vsum a f = vsum b f`) THEN
6497 REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION] THEN
6498 MESON_TAC[REAL_LT_TOTAL]);;
6500 let RADON_PARTITION = prove
6501 (`!(c:real^N->bool).
6502 FINITE c /\ affine_dependent c
6503 ==> ?(m:real^N->bool) (p:real^N->bool).
6506 ~(DISJOINT (convex hull m) (convex hull p))`,
6507 REPEAT STRIP_TAC THEN
6508 MP_TAC (ISPEC `c:real^N->bool` RADON_EX_LEMMA) THEN
6509 ASM_REWRITE_TAC[] THEN
6510 REPEAT STRIP_TAC THEN
6511 MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v <= &0}`;
6512 `{v:real^N | v IN c /\ u v > &0}`] THEN
6513 REPEAT CONJ_TAC THENL
6514 [REWRITE_TAC[DISJOINT;INTER;
6515 IN_ELIM_THM;REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN
6517 REWRITE_TAC[UNION;IN_ELIM_THM;REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN
6520 SUBGOAL_THEN `~(sum {x:real^N | x IN c /\ u x > &0} u = &0)` ASSUME_TAC THENL
6521 [MATCH_MP_TAC (REAL_ARITH `a > &0 ==> ~(a = &0)`) THEN
6522 REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN
6523 MATCH_MP_TAC (REWRITE_RULE[SUM_0] (ISPEC `\x. &0` SUM_LT_ALL)) THEN
6524 ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN
6525 REWRITE_TAC[MESON[]`~(!x. ~(P x /\ Q x)) = ?x. P x /\ Q x`] THEN
6526 ASM_CASES_TAC `&0 < u (v:real^N)` THENL
6527 [ASM SET_TAC[];ALL_TAC] THEN
6528 POP_ASSUM MP_TAC THEN POP_ASSUM (K ALL_TAC) THEN POP_ASSUM MP_TAC THEN
6529 REWRITE_TAC[IMP_IMP;REAL_ARITH `~(a = &0) /\ ~(&0 < a) <=> a < &0`] THEN
6531 REWRITE_TAC[MESON[REAL_NOT_LT]
6532 `(?x:real^N. P x /\ &0 < u x) <=> (!x. P x ==> u x <= &0) ==> F`] THEN
6534 MP_TAC (ISPECL [`u:real^N->real`;`\x:real^N. &0`;`c:real^N->bool`]
6536 ASM_REWRITE_TAC[SUM_0;REAL_ARITH `~(&0 < &0)`] THEN
6537 ASM_MESON_TAC[];ALL_TAC] THEN
6538 REWRITE_TAC[SET_RULE `~DISJOINT a b <=> ?y. y IN a /\ y IN b`] THEN
6539 EXISTS_TAC `&1 / (sum {x:real^N | x IN c /\ u x > &0} u) %
6540 vsum {x:real^N | x IN c /\ u x > &0} (\x. u x % x)` THEN
6541 REWRITE_TAC[CONVEX_HULL_EXPLICIT;IN_ELIM_THM] THEN
6543 [MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v < &0}`;
6545 &1 / (sum {x:real^N | x IN c /\ u x > &0} u) *
6547 ASM_SIMP_TAC[FINITE_RESTRICT;SUBSET;IN_ELIM_THM] THEN
6548 REPEAT CONJ_TAC THENL
6550 REPEAT STRIP_TAC THEN
6551 MATCH_MP_TAC REAL_LE_MUL THEN
6552 CONJ_TAC THENL [ALL_TAC;
6553 ASM_REWRITE_TAC[REAL_NEG_GE0;REAL_LE_LT]] THEN
6554 MATCH_MP_TAC REAL_LE_DIV THEN
6555 REWRITE_TAC[REAL_LE_01] THEN
6556 MATCH_MP_TAC SUM_POS_LE THEN
6557 ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM] THEN
6559 ASM_SIMP_TAC[FINITE_RESTRICT;SUM_LMUL] THEN
6560 MATCH_MP_TAC (REAL_FIELD `!a. ~(a = &0) /\ a * b = a * c ==> b = c`) THEN
6561 EXISTS_TAC `sum {x:real^N | x IN c /\ u x > &0} u` THEN
6562 REWRITE_TAC[SUM_LMUL] THEN
6563 ASM_SIMP_TAC[REAL_FIELD `~(a = &0) ==> a * &1 / a * b = b`] THEN
6564 REWRITE_TAC[SUM_NEG;REAL_MUL_RID] THEN
6565 REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN
6566 MATCH_MP_TAC (GSYM RADON_S_LEMMA) THEN
6569 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;VECTOR_MUL_LCANCEL] THEN
6570 REWRITE_TAC[VECTOR_MUL_LNEG;VSUM_NEG] THEN
6572 MATCH_MP_TAC (REWRITE_RULE[REAL_ARITH `&0 < a <=> a > &0`]
6573 (GSYM RADON_V_LEMMA)) THEN
6574 ASM_REWRITE_TAC[] THEN
6575 MESON_TAC[VECTOR_MUL_LZERO];ALL_TAC] THEN
6576 MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v > &0}`;
6578 &1 / (sum {x:real^N | x IN c /\ u x > &0} u) *
6580 ASM_SIMP_TAC[FINITE_RESTRICT;SUBSET;IN_ELIM_THM] THEN
6581 REPEAT CONJ_TAC THENL
6582 [REPEAT STRIP_TAC THEN
6583 MATCH_MP_TAC REAL_LE_MUL THEN
6584 CONJ_TAC THENL [ALL_TAC;
6585 ASM_SIMP_TAC[REAL_ARITH `a > &0 ==> &0 <= a`]] THEN
6586 MATCH_MP_TAC REAL_LE_DIV THEN
6587 REWRITE_TAC[REAL_LE_01] THEN
6588 MATCH_MP_TAC SUM_POS_LE THEN
6589 ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM] THEN
6591 ASM_SIMP_TAC[FINITE_RESTRICT;SUM_LMUL] THEN
6592 MATCH_MP_TAC (REAL_FIELD `!a. ~(a = &0) /\ a * b = a * c ==> b = c`) THEN
6593 EXISTS_TAC `sum {x:real^N | x IN c /\ u x > &0} u` THEN
6594 REWRITE_TAC[SUM_LMUL] THEN
6595 ASM_SIMP_TAC[REAL_FIELD `~(a = &0) ==> a * &1 / a * b = b`] THEN
6596 REWRITE_TAC[SUM_NEG;REAL_MUL_RID] THEN
6597 REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN
6598 MATCH_MP_TAC (GSYM RADON_S_LEMMA) THEN
6601 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;VECTOR_MUL_LCANCEL] THEN
6602 REWRITE_TAC[VECTOR_MUL_LNEG;VSUM_NEG] THEN
6604 MATCH_MP_TAC (REWRITE_RULE[REAL_ARITH `&0 < a <=> a > &0`]
6605 (GSYM RADON_V_LEMMA)) THEN
6606 ASM_REWRITE_TAC[] THEN
6607 MESON_TAC[VECTOR_MUL_LZERO]);;
6610 (`!(c:real^N->bool).
6612 ==> ?(m:real^N->bool) (p:real^N->bool).
6616 ~(DISJOINT (convex hull m) (convex hull p))`,
6617 REPEAT STRIP_TAC THEN MP_TAC
6618 (ISPEC `c:real^N->bool` AFFINE_DEPENDENT_EXPLICIT) THEN
6619 ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN MP_TAC
6620 (ISPEC `s:real^N->bool` RADON_PARTITION) THEN
6622 [ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN
6623 MAP_EVERY EXISTS_TAC [`s:real^N->bool`;`u:real^N->real`] THEN
6624 ASM SET_TAC[];ALL_TAC] THEN
6625 DISCH_THEN STRIP_ASSUME_TAC THEN
6626 MAP_EVERY EXISTS_TAC [`m:real^N->bool`;`p:real^N->bool`] THEN
6629 (* ------------------------------------------------------------------------- *)
6630 (* Helly's theorem. *)
6631 (* ------------------------------------------------------------------------- *)
6633 let HELLY_INDUCT = prove
6634 (`!n f. f HAS_SIZE n /\ n >= dimindex(:N) + 1 /\
6635 (!s:real^N->bool. s IN f ==> convex s) /\
6636 (!t. t SUBSET f /\ CARD(t) = dimindex(:N) + 1
6637 ==> ~(INTERS t = {}))
6638 ==> ~(INTERS f = {})`,
6639 INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `~(0 >= n + 1)`] THEN GEN_TAC THEN
6640 POP_ASSUM(LABEL_TAC "*") THEN STRIP_TAC THEN
6641 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_SIZE_SUC]) THEN
6642 STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
6643 FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE
6644 `SUC n >= m + 1 ==> m = n \/ n >= m + 1`))
6646 [FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
6647 ASM_SIMP_TAC[CARD_CLAUSES; SUBSET_REFL] THEN ARITH_TAC;
6650 `?X. !s:real^N->bool. s IN f ==> X(s) IN INTERS (f DELETE s)`
6651 STRIP_ASSUME_TAC THENL
6652 [REWRITE_TAC[GSYM SKOLEM_THM; MEMBER_NOT_EMPTY; RIGHT_EXISTS_IMP_THM] THEN
6653 GEN_TAC THEN STRIP_TAC THEN REMOVE_THEN "*" MATCH_MP_TAC THEN
6654 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM SET_TAC[];
6657 `?s t:real^N->bool. s IN f /\ t IN f /\ ~(s = t) /\ X s:real^N = X t`
6659 [FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC) THEN
6660 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
6661 EXISTS_TAC `(X:(real^N->bool)->real^N) t` THEN
6662 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC ONCE_DEPTH_CONV
6665 ==> INTERS f = INTERS(f DELETE s) INTER INTERS(f DELETE t)`)
6667 REWRITE_TAC[IN_INTER] THEN ASM_MESON_TAC[];
6669 MP_TAC(ISPEC `IMAGE (X:(real^N->bool)->real^N) f` RADON_PARTITION) THEN
6671 [ASM_SIMP_TAC[FINITE_IMAGE] THEN
6672 MATCH_MP_TAC AFFINE_DEPENDENT_BIGGERSET THEN
6673 ASM_SIMP_TAC[FINITE_IMAGE] THEN
6674 MATCH_MP_TAC(ARITH_RULE
6675 `!f n. n >= d + 1 /\ f = SUC n /\ c = f ==> c >= d + 2`) THEN
6676 MAP_EVERY EXISTS_TAC [`CARD(f:(real^N->bool)->bool)`; `n:num`] THEN
6677 REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN
6678 MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_MESON_TAC[];
6680 ONCE_REWRITE_TAC[SET_RULE
6681 `P /\ m UNION p = s /\ Q <=>
6682 m SUBSET s /\ p SUBSET s /\ m UNION p = s /\ P /\ Q`] THEN
6683 REWRITE_TAC[SUBSET_IMAGE; DISJOINT] THEN
6685 `(?m p. (?u. P u /\ m = t u) /\ (?u. P u /\ p = t u) /\ Q m p) ==> r <=>
6686 (!u v. P u /\ P v /\ Q (t u) (t v) ==> r)`] THEN
6687 MAP_EVERY X_GEN_TAC [`g:(real^N->bool)->bool`; `h:(real^N->bool)->bool`] THEN
6688 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
6689 SUBGOAL_THEN `(f:(real^N->bool)->bool) = h UNION g` SUBST1_TAC THENL
6690 [MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[UNION_SUBSET] THEN
6691 REWRITE_TAC[SUBSET; IN_UNION] THEN X_GEN_TAC `s:real^N->bool` THEN
6692 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
6693 DISCH_THEN(MP_TAC o ISPEC `X:(real^N->bool)->real^N` o
6694 MATCH_MP FUN_IN_IMAGE) THEN
6695 FIRST_X_ASSUM(fun th ->
6696 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN
6697 ONCE_REWRITE_TAC[DISJ_SYM] THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN
6698 MATCH_MP_TAC MONO_OR THEN ASM_MESON_TAC[SUBSET];
6700 MATCH_MP_TAC(SET_RULE
6701 `g SUBSET INTERS g' /\ h SUBSET INTERS h'
6702 ==> ~(g INTER h = {}) ==> ~(INTERS(g' UNION h') = {})`) THEN
6703 FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE
6704 `IMAGE X s INTER IMAGE X t = {} ==> s INTER t = {}`)) THEN
6705 CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN
6706 (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET; CONVEX_INTERS]]) THEN
6707 REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE] THEN ASM SET_TAC[]);;
6710 (`!f:(real^N->bool)->bool.
6711 FINITE f /\ CARD(f) >= dimindex(:N) + 1 /\
6712 (!s. s IN f ==> convex s) /\
6713 (!t. t SUBSET f /\ CARD(t) = dimindex(:N) + 1 ==> ~(INTERS t = {}))
6714 ==> ~(INTERS f = {})`,
6715 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC HELLY_INDUCT THEN
6716 ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_MESON_TAC[]);;
6718 let HELLY_ALT = prove
6719 (`!f:(real^N->bool)->bool.
6721 (!s. s IN f ==> convex s) /\
6722 (!t. t SUBSET f /\ CARD(t) <= dimindex(:N) + 1 ==> ~(INTERS t = {}))
6723 ==> ~(INTERS f = {})`,
6724 GEN_TAC THEN STRIP_TAC THEN
6725 ASM_CASES_TAC `CARD(f:(real^N->bool)->bool) < dimindex(:N) + 1` THEN
6726 ASM_SIMP_TAC[SUBSET_REFL; LT_IMP_LE] THEN MATCH_MP_TAC HELLY THEN
6727 ASM_SIMP_TAC[GE; GSYM NOT_LT] THEN ASM_MESON_TAC[LE_REFL]);;
6729 let HELLY_CLOSED_ALT = prove
6730 (`!f:(real^N->bool)->bool.
6731 (!s. s IN f ==> convex s /\ closed s) /\ (?s. s IN f /\ bounded s) /\
6732 (!t. t SUBSET f /\ FINITE t /\ CARD(t) <= dimindex(:N) + 1
6733 ==> ~(INTERS t = {}))
6734 ==> ~(INTERS f = {})`,
6736 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
6737 MATCH_MP_TAC CLOSED_FIP THEN ASM_SIMP_TAC[] THEN
6738 X_GEN_TAC `g:(real^N->bool)->bool` THEN STRIP_TAC THEN
6739 MATCH_MP_TAC HELLY_ALT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
6741 ASM_MESON_TAC[SUBSET_TRANS; FINITE_SUBSET]]);;
6743 let HELLY_COMPACT_ALT = prove
6744 (`!f:(real^N->bool)->bool.
6745 (!s. s IN f ==> convex s /\ compact s) /\
6746 (!t. t SUBSET f /\ FINITE t /\ CARD(t) <= dimindex(:N) + 1
6747 ==> ~(INTERS t = {}))
6748 ==> ~(INTERS f = {})`,
6749 GEN_TAC THEN STRIP_TAC THEN
6750 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
6751 ASM_REWRITE_TAC[INTERS_0; UNIV_NOT_EMPTY] THEN
6752 MATCH_MP_TAC HELLY_CLOSED_ALT THEN
6753 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN
6754 ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPACT_IMP_BOUNDED]);;
6756 let HELLY_CLOSED = prove
6757 (`!f:(real^N->bool)->bool.
6758 (FINITE f ==> CARD f >= dimindex (:N) + 1) /\
6759 (!s. s IN f ==> convex s /\ closed s) /\ (?s. s IN f /\ bounded s) /\
6760 (!t. t SUBSET f /\ FINITE t /\ CARD(t) = dimindex(:N) + 1
6761 ==> ~(INTERS t = {}))
6762 ==> ~(INTERS f = {})`,
6763 GEN_TAC THEN REWRITE_TAC[GE] THEN
6764 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
6765 MATCH_MP_TAC HELLY_CLOSED_ALT THEN ASM_REWRITE_TAC[] THEN
6766 X_GEN_TAC `g:(real^N->bool)->bool` THEN STRIP_TAC THEN
6767 MP_TAC(ISPECL [`dimindex(:N) + 1`; `g:(real^N->bool)->bool`;
6768 `f:(real^N->bool)->bool`] CHOOSE_SUBSET_BETWEEN) THEN
6769 ASM_REWRITE_TAC[] THEN
6770 DISCH_THEN(X_CHOOSE_THEN `h:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
6771 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ ~(s = {}) ==> ~(t = {})`) THEN
6772 EXISTS_TAC `INTERS h: real^N->bool` THEN
6773 CONJ_TAC THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC] THEN
6774 ASM_MESON_TAC[HAS_SIZE]);;
6776 let HELLY_COMPACT = prove
6777 (`!f:(real^N->bool)->bool.
6778 (FINITE f ==> CARD f >= dimindex (:N) + 1) /\
6779 (!s. s IN f ==> convex s /\ compact s) /\
6780 (!t. t SUBSET f /\ FINITE t /\ CARD(t) = dimindex(:N) + 1
6781 ==> ~(INTERS t = {}))
6782 ==> ~(INTERS f = {})`,
6783 GEN_TAC THEN STRIP_TAC THEN
6784 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
6785 ASM_REWRITE_TAC[INTERS_0; UNIV_NOT_EMPTY] THEN
6786 MATCH_MP_TAC HELLY_CLOSED THEN
6787 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN
6788 ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPACT_IMP_BOUNDED]);;
6790 (* ------------------------------------------------------------------------- *)
6791 (* Kirchberger's theorem *)
6792 (* ------------------------------------------------------------------------- *)
6794 let KIRCHBERGER = prove
6795 (`!s t:real^N->bool.
6796 compact s /\ compact t /\
6797 (!s' t'. s' SUBSET s /\ t' SUBSET t /\ FINITE s' /\ FINITE t' /\
6798 CARD(s') + CARD(t') <= dimindex(:N) + 2
6799 ==> ?a b. (!x. x IN s' ==> a dot x < b) /\
6800 (!x. x IN t' ==> a dot x > b))
6801 ==> ?a b. ~(a = vec 0) /\
6802 (!x. x IN s ==> a dot x < b) /\
6803 (!x. x IN t ==> a dot x > b)`,
6805 (`(!x. x IN convex hull s ==> a dot x < b) /\
6806 (!x. x IN convex hull t ==> a dot x > b) <=>
6807 (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`,
6808 REWRITE_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN
6809 SIMP_TAC[SUBSET_HULL; CONVEX_HALFSPACE_LT; CONVEX_HALFSPACE_GT])
6810 and KIRCH_LEMMA = prove
6811 (`!s t:real^N->bool.
6812 FINITE s /\ FINITE t /\
6813 (!s' t'. s' SUBSET s /\ t' SUBSET t /\
6814 CARD(s') + CARD(t') <= dimindex(:N) + 2
6815 ==> ?a b. (!x. x IN s' ==> a dot x < b) /\
6816 (!x. x IN t' ==> a dot x > b))
6817 ==> ?a b. (!x. x IN s ==> a dot x < b) /\
6818 (!x. x IN t ==> a dot x > b)`,
6819 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
6820 [`IMAGE (\r. {z:real^(N,1)finite_sum |
6821 fstcart z dot r < drop(sndcart z)}) s UNION
6822 IMAGE (\r. {z:real^(N,1)finite_sum |
6823 fstcart z dot r > drop(sndcart z)}) t`]
6825 REWRITE_TAC[FORALL_SUBSET_UNION; IN_UNION; IMP_CONJ] THEN
6826 REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_SUBSET_IMAGE] THEN
6827 ASM_SIMP_TAC[FINITE_UNION; FINITE_IMAGE; INTERS_UNION] THEN
6828 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE; IN_INTER;
6829 EXISTS_PASTECART; IN_ELIM_PASTECART_THM;
6830 FSTCART_PASTECART; SNDCART_PASTECART] THEN
6831 REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
6832 REWRITE_TAC[FORALL_AND_THM; FORALL_IN_IMAGE; RIGHT_IMP_FORALL_THM] THEN
6833 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; GSYM EXISTS_DROP] THEN
6834 DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
6835 [REWRITE_TAC[REAL_ARITH `a > b <=> --a < --b`; GSYM DOT_RNEG] THEN
6836 REWRITE_TAC[convex; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN
6837 SIMP_TAC[PASTECART_ADD; GSYM PASTECART_CMUL; IN_ELIM_PASTECART_THM] THEN
6838 SIMP_TAC[DOT_LADD; DOT_LMUL; DROP_ADD; DROP_CMUL; GSYM FORALL_DROP] THEN
6839 REWRITE_TAC[REAL_ARITH `--(a * x + b * y):real = a * --x + b * --y`] THEN
6840 REPEAT STRIP_TAC THEN
6841 FIRST_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH
6843 ==> &0 <= u /\ &0 <= v
6844 ==> u = &0 /\ v = &1 \/ u = &1 /\ v = &0 \/ &0 < u /\ &0 < v`)) THEN
6845 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
6846 ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID;
6847 REAL_ADD_LID; REAL_ADD_RID] THEN
6848 MATCH_MP_TAC REAL_LT_ADD2 THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ];
6849 REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1;
6850 ARITH_RULE `(n + 1) + 1 = n + 2`] THEN
6851 MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
6852 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
6853 SUBGOAL_THEN `FINITE(u:real^N->bool) /\ FINITE(v:real^N->bool)`
6854 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
6855 W(MP_TAC o PART_MATCH (lhs o rand) CARD_UNION o lhand o lhand o snd) THEN
6856 ASM_SIMP_TAC[FINITE_IMAGE] THEN ANTS_TAC THENL
6857 [REWRITE_TAC[SET_RULE `IMAGE f s INTER IMAGE g t = {} <=>
6858 !x y. x IN s /\ y IN t ==> ~(f x = g y)`] THEN
6859 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
6860 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
6861 DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN
6862 REWRITE_TAC[GSYM FORALL_DROP; DOT_LZERO] THEN
6863 DISCH_THEN(MP_TAC o SPEC `&1`) THEN REAL_ARITH_TAC;
6864 DISCH_THEN SUBST1_TAC] THEN
6865 DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN
6866 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(ARITH_RULE
6867 `a = a' /\ b = b' ==> a + b <= n + 2 ==> a' + b' <= n + 2`) THEN
6868 CONJ_TAC THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN
6869 ASM_REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
6870 SIMP_TAC[GSYM FORALL_DROP; real_gt; VECTOR_EQ_LDOT;
6871 MESON[REAL_LT_TOTAL; REAL_LT_REFL]
6872 `((!y:real. a < y <=> b < y) <=> a = b) /\
6873 ((!y:real. y < a <=> y < b) <=> a = b)`]]) in
6874 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM lemma] THEN
6875 MATCH_MP_TAC SEPARATING_HYPERPLANE_COMPACT_COMPACT THEN
6876 ASM_SIMP_TAC[CONVEX_CONVEX_HULL; COMPACT_CONVEX_HULL;
6877 CONVEX_HULL_EQ_EMPTY] THEN
6879 `!s' t'. (s':real^N->bool) SUBSET s /\ t' SUBSET t /\
6880 FINITE s' /\ CARD(s') <= dimindex(:N) + 1 /\
6881 FINITE t' /\ CARD(t') <= dimindex(:N) + 1
6882 ==> DISJOINT (convex hull s') (convex hull t')`
6884 [REPEAT STRIP_TAC THEN
6885 MP_TAC(ISPECL [`s':real^N->bool`; `t':real^N->bool`] KIRCH_LEMMA) THEN
6887 [ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
6888 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; FINITE_SUBSET];
6889 ONCE_REWRITE_TAC[GSYM lemma] THEN SET_TAC[REAL_LT_ANTISYM; real_gt]];
6890 POP_ASSUM_LIST(K ALL_TAC) THEN STRIP_TAC THEN
6891 REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s /\ x IN t ==> F`] THEN
6892 X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[CARATHEODORY] THEN
6893 REWRITE_TAC[IN_ELIM_THM] THEN
6894 DISCH_THEN(CONJUNCTS_THEN2
6895 (X_CHOOSE_THEN `s':real^N->bool` STRIP_ASSUME_TAC)
6896 (X_CHOOSE_THEN `t':real^N->bool` STRIP_ASSUME_TAC)) THEN
6897 FIRST_X_ASSUM(MP_TAC o SPECL [`s':real^N->bool`; `t':real^N->bool`]) THEN
6898 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
6900 (* ------------------------------------------------------------------------- *)
6901 (* Convex hull is "preserved" by a linear function. *)
6902 (* ------------------------------------------------------------------------- *)
6904 let CONVEX_HULL_LINEAR_IMAGE = prove
6905 (`!f s. linear f ==> convex hull (IMAGE f s) = IMAGE f (convex hull s)`,
6906 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
6907 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
6908 CONJ_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN
6909 REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[FUN_IN_IMAGE; HULL_INC] THEN
6910 REWRITE_TAC[convex; IN_ELIM_THM] THEN
6911 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THENL
6912 [FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN
6913 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN
6914 REWRITE_TAC[IN_IMAGE] THEN
6915 MESON_TAC[REWRITE_RULE[convex] CONVEX_CONVEX_HULL];
6916 ASM_SIMP_TAC[LINEAR_ADD; LINEAR_CMUL] THEN
6917 MESON_TAC[REWRITE_RULE[convex] CONVEX_CONVEX_HULL]]);;
6919 add_linear_invariants [CONVEX_HULL_LINEAR_IMAGE];;
6921 let IN_CONVEX_HULL_LINEAR_IMAGE = prove
6922 (`!f:real^M->real^N s x.
6923 linear f /\ x IN convex hull s ==> (f x) IN convex hull (IMAGE f s)`,
6924 SIMP_TAC[CONVEX_HULL_LINEAR_IMAGE] THEN SET_TAC[]);;
6926 (* ------------------------------------------------------------------------- *)
6927 (* Convexity of general and special intervals. *)
6928 (* ------------------------------------------------------------------------- *)
6930 let IS_INTERVAL_CONVEX = prove
6931 (`!s:real^N->bool. is_interval s ==> convex s`,
6932 REWRITE_TAC[is_interval; convex] THEN
6933 REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
6934 MAP_EVERY EXISTS_TAC [`x:real^N`; `y:real^N`] THEN
6935 ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
6936 GEN_TAC THEN STRIP_TAC THEN
6937 DISJ_CASES_TAC(SPECL [`(x:real^N)$i`; `(y:real^N)$i`] REAL_LE_TOTAL) THENL
6938 [DISJ1_TAC; DISJ2_TAC] THEN
6939 MATCH_MP_TAC(REAL_ARITH
6940 `&1 * a <= b /\ b <= &1 * c ==> a <= b /\ b <= c`) THEN
6941 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
6942 ASM_SIMP_TAC[GSYM VECTOR_MUL_COMPONENT;
6943 VECTOR_ADD_RDISTRIB; VECTOR_ADD_COMPONENT] THEN
6944 ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_LE_LMUL;
6945 REAL_LE_LADD; REAL_LE_RADD]);;
6947 let IS_INTERVAL_CONNECTED = prove
6948 (`!s:real^N->bool. is_interval s ==> connected s`,
6949 MESON_TAC[IS_INTERVAL_CONVEX; CONVEX_CONNECTED]);;
6951 let IS_INTERVAL_CONNECTED_1 = prove
6952 (`!s:real^1->bool. is_interval s <=> connected s`,
6953 GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[IS_INTERVAL_CONNECTED] THEN
6954 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
6955 REWRITE_TAC[IS_INTERVAL_1; connected; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM;
6956 NOT_IMP; FORALL_LIFT; LIFT_DROP] THEN
6957 MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `x:real`] THEN STRIP_TAC THEN
6958 MAP_EVERY EXISTS_TAC
6959 [`{z:real^1 | basis 1 dot z < x}`; `{z:real^1 | basis 1 dot z > x}`] THEN
6960 REWRITE_TAC[OPEN_HALFSPACE_LT; OPEN_HALFSPACE_GT] THEN
6961 SIMP_TAC[SUBSET; EXTENSION; IN_UNION; IN_INTER; GSYM drop; NOT_FORALL_THM;
6962 real_gt; NOT_IN_EMPTY; IN_ELIM_THM; DOT_BASIS; DIMINDEX_1; ARITH] THEN
6963 REPEAT CONJ_TAC THENL
6964 [ASM_MESON_TAC[REAL_LT_TOTAL; LIFT_DROP];
6966 EXISTS_TAC `lift a`;
6967 EXISTS_TAC `lift b`] THEN
6968 ASM_REWRITE_TAC[REAL_LT_LE; LIFT_DROP] THEN ASM_MESON_TAC[]);;
6970 let CONVEX_INTERVAL = prove
6971 (`!a b:real^N. convex(interval [a,b]) /\ convex(interval (a,b))`,
6972 SIMP_TAC[IS_INTERVAL_CONVEX; IS_INTERVAL_INTERVAL]);;
6974 let CONNECTED_INTERVAL = prove
6975 (`(!a b:real^N. connected(interval[a,b])) /\
6976 (!a b:real^N. connected(interval(a,b)))`,
6977 SIMP_TAC[CONVEX_CONNECTED; CONVEX_INTERVAL]);;
6979 (* ------------------------------------------------------------------------- *)
6980 (* On real^1, is_interval, convex and connected are all equivalent. *)
6981 (* ------------------------------------------------------------------------- *)
6983 let IS_INTERVAL_CONVEX_1 = prove
6984 (`!s:real^1->bool. is_interval s <=> convex s`,
6985 MESON_TAC[IS_INTERVAL_CONVEX; CONVEX_CONNECTED; IS_INTERVAL_CONNECTED_1]);;
6987 let CONVEX_CONNECTED_1 = prove
6988 (`!s:real^1->bool. convex s <=> connected s`,
6989 REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; GSYM IS_INTERVAL_CONNECTED_1]);;
6991 let CONNECTED_CONVEX_1 = prove
6992 (`!s:real^1->bool. connected s <=> convex s`,
6993 REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; GSYM IS_INTERVAL_CONNECTED_1]);;
6995 let CONNECTED_COMPACT_INTERVAL_1 = prove
6996 (`!s:real^1->bool. connected s /\ compact s <=> ?a b. s = interval[a,b]`,
6997 REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_COMPACT]);;
6999 let CONVEX_CONNECTED_1_GEN = prove
7001 dimindex(:N) = 1 ==> (convex s <=> connected s)`,
7002 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
7003 REWRITE_TAC[GSYM DIMINDEX_1] THEN
7004 DISCH_THEN(ACCEPT_TAC o C GEOM_EQUAL_DIMENSION_RULE CONVEX_CONNECTED_1));;
7006 let CONNECTED_CONVEX_1_GEN = prove
7008 dimindex(:N) = 1 ==> (convex s <=> connected s)`,
7009 SIMP_TAC[CONVEX_CONNECTED_1_GEN]);;
7011 (* ------------------------------------------------------------------------- *)
7012 (* Jung's theorem. *)
7013 (* Proof taken from http://cstheory.wordpress.com/2010/08/07/jungs-theorem/ *)
7014 (* ------------------------------------------------------------------------- *)
7017 (`!s:real^N->bool r.
7019 sqrt(&(dimindex(:N)) / &(2 * dimindex(:N) + 2)) * diameter s <= r
7020 ==> ?a. s SUBSET cball(a,r)`,
7022 (`&0 < x /\ x <= y ==> (x - &1) / x <= (y - &1) / y`,
7023 SIMP_TAC[REAL_LE_LDIV_EQ] THEN REPEAT STRIP_TAC THEN
7024 ONCE_REWRITE_TAC[REAL_ARITH `x / y * z:real = (x * z) / y`] THEN
7025 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL
7026 [ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_LE_RDIV_EQ]] THEN
7027 ASM_REAL_ARITH_TAC) in
7028 REPEAT STRIP_TAC THEN
7029 SUBGOAL_THEN `&0 <= r` ASSUME_TAC THENL
7030 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
7031 REAL_LE_TRANS)) THEN
7032 MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[DIAMETER_POS_LE] THEN
7033 SIMP_TAC[SQRT_POS_LE; REAL_LE_DIV; REAL_POS];
7035 MP_TAC(ISPEC `IMAGE (\x:real^N. cball(x,r)) s` HELLY_COMPACT_ALT) THEN
7036 REWRITE_TAC[FORALL_IN_IMAGE; COMPACT_CBALL; CONVEX_CBALL] THEN
7037 REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> q /\ p ==> r ==> s`] THEN
7038 REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN
7039 REWRITE_TAC[INTERS_IMAGE; GSYM MEMBER_NOT_EMPTY] THEN
7040 REWRITE_TAC[SUBSET; IN_CBALL; IN_ELIM_THM] THEN
7041 ANTS_TAC THENL [ALL_TAC; MESON_TAC[DIST_SYM]] THEN
7042 X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[GSYM SUBSET] THEN
7044 ASM_SIMP_TAC[CARD_IMAGE_INJ; EQ_BALLS; GSYM REAL_NOT_LE] THEN
7045 UNDISCH_TAC `FINITE(t:real^N->bool)` THEN
7046 SUBGOAL_THEN `bounded(t:real^N->bool)` MP_TAC THENL
7047 [ASM_MESON_TAC[BOUNDED_SUBSET]; ALL_TAC] THEN
7048 UNDISCH_TAC `&0 <= r` THEN
7050 `sqrt(&(dimindex(:N)) / &(2 * dimindex(:N) + 2)) *
7051 diameter(t:real^N->bool) <= r`
7053 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
7054 REAL_LE_TRANS)) THEN
7055 MATCH_MP_TAC REAL_LE_LMUL THEN
7056 ASM_SIMP_TAC[DIAMETER_SUBSET; SQRT_POS_LE; REAL_POS; REAL_LE_DIV];
7057 POP_ASSUM_LIST(K ALL_TAC) THEN
7058 SPEC_TAC(`t:real^N->bool`,`s:real^N->bool`) THEN
7059 REPEAT STRIP_TAC] THEN
7060 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
7061 MP_TAC(ISPEC `{d | &0 <= d /\ ?a:real^N. s SUBSET cball(a,d)}` INF) THEN
7062 ABBREV_TAC `d = inf {d | &0 <= d /\ ?a:real^N. s SUBSET cball(a,d)}` THEN
7063 REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL
7064 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
7065 ASM_MESON_TAC[BOUNDED_SUBSET_CBALL; REAL_LT_IMP_LE];
7066 DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "P") (LABEL_TAC "M"))] THEN
7067 SUBGOAL_THEN `&0 <= d` ASSUME_TAC THENL
7068 [ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN
7069 SUBGOAL_THEN `?a:real^N. s SUBSET cball(a,d)` MP_TAC THENL
7071 `!n. ?a:real^N. s SUBSET cball(a,d + inv(&n + &1))`
7073 [X_GEN_TAC `n:num` THEN
7074 REMOVE_THEN "M" (MP_TAC o SPEC `d + inv(&n + &1)`) THEN
7075 REWRITE_TAC[REAL_ARITH `d + i <= d <=> ~(&0 < i)`] THEN
7076 REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
7077 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LE] THEN
7078 MESON_TAC[SUBSET_CBALL; REAL_LT_IMP_LE; SUBSET_TRANS];
7080 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SKOLEM_THM] THEN
7081 X_GEN_TAC `aa:num->real^N` THEN DISCH_TAC THEN
7082 SUBGOAL_THEN `?t. compact t /\ !n. (aa:num->real^N) n IN t` MP_TAC THENL
7083 [FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o
7084 MATCH_MP BOUNDED_SUBSET_CBALL) THEN
7085 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_CBALL_0] THEN
7086 X_GEN_TAC `B:real` THEN STRIP_TAC THEN
7087 EXISTS_TAC `cball(vec 0:real^N,B + d + &1)` THEN
7088 REWRITE_TAC[COMPACT_CBALL; IN_CBALL_0] THEN X_GEN_TAC `n:num` THEN
7089 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_CBALL]) THEN
7090 MATCH_MP_TAC(NORM_ARITH
7091 `(?x:real^N. norm(x) <= B /\ dist(a,x) <= d) ==> norm(a) <= B + d`) THEN
7092 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7093 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN
7094 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `d + inv(&n + &1)` THEN
7095 ASM_SIMP_TAC[REAL_LE_LADD] THEN
7096 MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC;
7098 REWRITE_TAC[compact; LEFT_IMP_EXISTS_THM] THEN
7099 X_GEN_TAC `t:real^N->bool` THEN
7100 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
7101 DISCH_THEN(MP_TAC o SPEC `aa:num->real^N`) THEN ASM_REWRITE_TAC[] THEN
7102 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
7103 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
7104 REWRITE_TAC[SUBSET; IN_CBALL] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7105 REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
7106 MP_TAC(SPEC `(dist(a:real^N,x) - d) / &2` REAL_ARCH_INV) THEN
7107 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
7108 DISCH_THEN(MP_TAC o SPEC `(dist(a:real^N,x) - d) / &2`) THEN
7109 ASM_SIMP_TAC[REAL_SUB_LT; REAL_HALF; o_THM] THEN
7110 DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
7111 DISCH_THEN(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC) THEN
7112 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [SUBSET]) THEN
7113 DISCH_THEN(MP_TAC o SPECL [`(r:num->num)(N1 + N2)`; `x:real^N`]) THEN
7114 ASM_REWRITE_TAC[IN_CBALL; REAL_NOT_LE] THEN
7115 FIRST_X_ASSUM(MP_TAC o SPEC `N1 + N2:num`) THEN
7116 ASM_REWRITE_TAC[LE_ADD] THEN
7117 SUBGOAL_THEN `inv(&(r (N1 + N2:num)) + &1) < (dist(a:real^N,x) - d) / &2`
7118 MP_TAC THENL [ALL_TAC; NORM_ARITH_TAC] THEN
7119 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N2)` THEN
7120 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
7121 CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_INV_EQ]; ALL_TAC] THEN
7122 REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_ADD] THEN
7123 MATCH_MP_TAC(ARITH_RULE
7124 `N1 + N2 <= r(N1 + N2) ==> N2 <= r(N1 + N2) + 1`) THEN
7125 ASM_MESON_TAC[MONOTONE_BIGGER];
7127 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
7128 ONCE_REWRITE_TAC[DIST_SYM] THEN
7129 REWRITE_TAC[GSYM IN_CBALL; GSYM SUBSET] THEN
7130 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
7131 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN
7132 MATCH_MP_TAC SUBSET_CBALL THEN
7133 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
7134 `a * s <= r ==> d <= a * s ==> d <= r`)) THEN
7135 UNDISCH_THEN `&0 <= r` (K ALL_TAC) THEN REMOVE_THEN "M" (K ALL_TAC) THEN
7136 FIRST_X_ASSUM(K ALL_TAC o SYM) THEN REMOVE_THEN "P" MP_TAC THEN
7137 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
7138 ABBREV_TAC `n = CARD(s:real^N->bool)` THEN
7139 SUBGOAL_THEN `(s:real^N->bool) HAS_SIZE n` MP_TAC THENL
7140 [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN
7141 UNDISCH_THEN `CARD(s:real^N->bool) = n` (K ALL_TAC) THEN
7142 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
7143 SPEC_TAC(`d:real`,`r:real`) THEN GEN_TAC THEN
7144 GEOM_ORIGIN_TAC `a:real^N` THEN SIMP_TAC[HAS_SIZE] THEN
7145 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
7146 ABBREV_TAC `t = {x:real^N | x IN s /\ norm(x) = r}` THEN
7147 SUBGOAL_THEN `FINITE(t:real^N->bool)` ASSUME_TAC THENL
7148 [EXPAND_TAC "t" THEN ASM_SIMP_TAC[FINITE_RESTRICT]; ALL_TAC] THEN
7149 SUBGOAL_THEN `(vec 0:real^N) IN convex hull t` MP_TAC THENL
7150 [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
7151 MP_TAC(ISPEC `convex hull t:real^N->bool`
7152 SEPARATING_HYPERPLANE_CLOSED_0) THEN
7153 ASM_SIMP_TAC[CONVEX_CONVEX_HULL; NOT_IMP; COMPACT_CONVEX_HULL;
7154 FINITE_IMP_COMPACT; COMPACT_IMP_CLOSED] THEN
7155 REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN
7156 X_GEN_TAC `v:real^N` THEN
7157 ABBREV_TAC `k = CARD(s:real^N->bool)` THEN
7158 SUBGOAL_THEN `(s:real^N->bool) HAS_SIZE k` MP_TAC THENL
7159 [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN
7160 UNDISCH_THEN `CARD(s:real^N->bool) = k` (K ALL_TAC) THEN
7161 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
7162 GEOM_BASIS_MULTIPLE_TAC 1 `v:real^N` THEN X_GEN_TAC `m:real` THEN
7163 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
7164 STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
7165 ASM_SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; REAL_LT_IMP_NZ] THEN
7166 REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[HAS_SIZE] THEN
7167 DISCH_THEN(SUBST_ALL_TAC o SYM) THEN X_GEN_TAC `b:real` THEN DISCH_TAC THEN
7168 ASM_SIMP_TAC[DOT_LMUL; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN
7169 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
7170 ASM_SIMP_TAC[real_gt; GSYM REAL_LT_LDIV_EQ] THEN
7171 SUBGOAL_THEN `&0 < b / m` MP_TAC THENL
7172 [ASM_SIMP_TAC[REAL_LT_DIV];
7173 UNDISCH_THEN `&0 < b` (K ALL_TAC) THEN
7174 SPEC_TAC(`b / m:real`,`b:real`)] THEN
7175 X_GEN_TAC `b:real` THEN DISCH_TAC THEN DISCH_TAC THEN
7177 `!x:real^N e. &0 < e /\ e < b /\ x IN t ==> norm(x - e % basis 1) < r`
7179 [MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real`] THEN STRIP_TAC THEN
7180 SUBGOAL_THEN `r = norm(x:real^N)` SUBST1_TAC THENL
7181 [ASM SET_TAC[]; REWRITE_TAC[NORM_LT; dot]] THEN
7182 SIMP_TAC[SUM_CLAUSES_LEFT; DIMINDEX_GE_1] THEN
7183 SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT;
7184 BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL;
7185 ARITH_RULE `2 <= n ==> 1 <= n /\ ~(n = 1)`; ARITH] THEN
7186 REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO; REAL_LT_RADD] THEN
7187 REWRITE_TAC[GSYM REAL_POW_2; GSYM REAL_LT_SQUARE_ABS] THEN
7188 MATCH_MP_TAC(REAL_ARITH
7189 `!b. &0 < e /\ e < b /\ b < x ==> abs(x - e * &1) < abs x`) THEN
7190 EXISTS_TAC `b:real` THEN ASM_REWRITE_TAC[] THEN
7191 ASM_MESON_TAC[HULL_INC];
7195 !x:real^N a. x IN (s DIFF t) /\ norm(a) < d ==> norm(x - a) < r`
7196 STRIP_ASSUME_TAC THENL
7197 [ASM_CASES_TAC `s DIFF t:real^N->bool = {}` THENL
7198 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN MESON_TAC[REAL_LT_01]; ALL_TAC] THEN
7199 EXISTS_TAC `inf (IMAGE (\x:real^N. r - norm x) (s DIFF t))` THEN
7200 SUBGOAL_THEN `FINITE(s DIFF t:real^N->bool)` ASSUME_TAC THENL
7201 [ASM_MESON_TAC[FINITE_DIFF]; ALL_TAC] THEN
7202 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
7203 REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC
7204 [NORM_ARITH `norm a < r - norm x ==> norm(x - a:real^N) < r`] THEN
7205 EXPAND_TAC "t" THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM; REAL_SUB_LT] THEN
7206 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_CBALL_0]) THEN
7207 ASM_MESON_TAC[REAL_LT_LE];
7210 `?a. !x. x IN s ==> norm(x - a:real^N) < r`
7211 STRIP_ASSUME_TAC THENL
7212 [EXISTS_TAC `min (b / &2) (d / &2) % basis 1:real^N` THEN
7213 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7214 ASM_CASES_TAC `(x:real^N) IN t` THENL
7215 [MATCH_MP_TAC(ASSUME
7216 `!x:real^N e. &0 < e /\ e < b /\ x IN t
7217 ==> norm (x - e % basis 1) < r`) THEN
7218 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
7220 `!x:real^N a. x IN s DIFF t /\ norm a < d ==> norm (x - a) < r`) THEN
7221 ASM_SIMP_TAC[IN_DIFF; NORM_MUL; LE_REFL; NORM_BASIS;
7223 ASM_REAL_ARITH_TAC];
7224 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL
7225 [ASM_MESON_TAC[MEMBER_NOT_EMPTY; NORM_ARITH
7226 `norm(x:real^N) < r ==> &0 < r`];
7229 `!x a:real^N. &0 <= x /\ s SUBSET cball (a,x) ==> r <= x` (MP_TAC o
7230 SPECL [`max (&0) (r - inf (IMAGE (\x:real^N. r - norm(x - a)) s))`;
7232 ASM_SIMP_TAC[REAL_ARITH `&0 < r ==> (r <= max (&0) a <=> r <= a)`] THEN
7233 REWRITE_TAC[SUBSET; IN_CBALL; REAL_ARITH `a <= max a b`] THEN
7234 REWRITE_TAC[NOT_IMP; REAL_ARITH `~(r <= r - x) <=> &0 < x`] THEN
7235 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
7236 ASM_REWRITE_TAC[FORALL_IN_IMAGE; REAL_SUB_LT] THEN
7237 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7238 MATCH_MP_TAC(REAL_ARITH `d <= b ==> d <= max a b`) THEN
7239 ONCE_REWRITE_TAC[REAL_ARITH `a <= b - c <=> c <= b - a`] THEN
7240 ASM_SIMP_TAC[REAL_INF_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
7241 REWRITE_TAC[EXISTS_IN_IMAGE; ONCE_REWRITE_RULE[NORM_SUB] dist] THEN
7242 ASM_MESON_TAC[REAL_LE_REFL]];
7244 ASM_CASES_TAC `t:real^N->bool = {}` THEN
7245 ASM_REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
7246 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN
7247 DISCH_THEN(X_CHOOSE_THEN `l:real^N->real` STRIP_ASSUME_TAC) THEN
7248 MATCH_MP_TAC REAL_LE_TRANS THEN
7249 EXISTS_TAC `sqrt((&(dimindex (:N)) / &(2 * dimindex (:N) + 2)) *
7250 diameter(s:real^N->bool) pow 2)` THEN
7252 [MATCH_MP_TAC REAL_LE_RSQRT;
7253 ASM_SIMP_TAC[SQRT_MUL; DIAMETER_POS_LE; REAL_POW_LE; REAL_LE_DIV;
7254 REAL_POS; POW_2_SQRT; REAL_LE_REFL]] THEN
7257 `sum t (\y:real^N. &2 * r pow 2) <=
7258 sum t (\y. (&1 - l y) * diameter(s:real^N->bool) pow 2)`
7260 [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN
7261 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7262 MATCH_MP_TAC REAL_LE_TRANS THEN
7263 EXISTS_TAC `sum (t DELETE x) (\x:real^N. l(x)) *
7264 diameter(s:real^N->bool) pow 2` THEN CONJ_TAC THENL
7265 [ALL_TAC; ASM_SIMP_TAC[SUM_DELETE; ETA_AX; REAL_LE_REFL]] THEN
7266 REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
7267 EXISTS_TAC `sum (t DELETE x) (\y:real^N. l y * norm(y - x) pow 2)` THEN
7270 MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FINITE_DELETE; IN_DELETE] THEN
7271 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN
7272 ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_POW_LE2 THEN
7273 REWRITE_TAC[NORM_POS_LE] THEN
7274 MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM SET_TAC[]] THEN
7275 MATCH_MP_TAC REAL_LE_TRANS THEN
7276 EXISTS_TAC `sum t (\y:real^N. l y * norm (y - x) pow 2)` THEN
7279 MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ_SUPERSET THEN
7280 ASM_REWRITE_TAC[FINITE_DELETE] THEN
7281 CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[IN_DELETE]] THEN
7282 SIMP_TAC[TAUT `p /\ ~(p /\ ~q) <=> p /\ q`] THEN
7283 REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN REAL_ARITH_TAC] THEN
7284 REWRITE_TAC[NORM_POW_2; VECTOR_ARITH
7285 `(y - x:real^N) dot (y - x) = (x dot x + y dot y) - &2 * x dot y`] THEN
7286 MATCH_MP_TAC REAL_LE_TRANS THEN
7287 EXISTS_TAC `sum t (\y:real^N. l y * (&2 * r pow 2 - &2 * (x dot y)))` THEN
7290 MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN
7291 UNDISCH_TAC `(x:real^N) IN t` THEN EXPAND_TAC "t" THEN
7292 REWRITE_TAC[IN_DELETE; IN_ELIM_THM] THEN
7293 SIMP_TAC[NORM_EQ_SQUARE; NORM_POW_2] THEN REAL_ARITH_TAC] THEN
7294 REWRITE_TAC[REAL_ARITH `x * (&2 * y - &2 * z) = &2 * (x * y - x * z)`] THEN
7295 REWRITE_TAC[SUM_LMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN
7296 REWRITE_TAC[REAL_POS] THEN
7297 ASM_SIMP_TAC[SUM_SUB; FINITE_DELETE; SUM_RMUL] THEN
7298 REWRITE_TAC[GSYM DOT_RMUL] THEN
7299 ASM_SIMP_TAC[GSYM DOT_RSUM; DOT_RZERO] THEN REAL_ARITH_TAC;
7300 ASM_SIMP_TAC[SUM_CONST; SUM_RMUL; SUM_SUB] THEN
7301 REWRITE_TAC[REAL_OF_NUM_MUL; MULT_CLAUSES] THEN
7302 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [REAL_MUL_SYM] THEN
7303 SUBGOAL_THEN `&0 < &(CARD(t:real^N->bool) * 2)` ASSUME_TAC THENL
7304 [REWRITE_TAC[REAL_OF_NUM_LT; ARITH_RULE `0 < n * 2 <=> ~(n = 0)`] THEN
7305 ASM_SIMP_TAC[CARD_EQ_0];
7306 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
7307 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
7308 REWRITE_TAC[REAL_ARITH `(a * b) / c:real = a / c * b`] THEN
7309 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_LE_POW_2] THEN
7310 REWRITE_TAC[ARITH_RULE `2 * n + 2 = (n + 1) * 2`; GSYM REAL_OF_NUM_MUL;
7311 real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN
7312 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[GSYM real_div] THEN
7313 CONV_TAC REAL_RAT_REDUCE_CONV THEN
7314 SUBGOAL_THEN `&(dimindex(:N)) = &(dimindex(:N) + 1) - &1`
7316 [REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC;
7317 MATCH_MP_TAC lemma THEN
7318 ASM_SIMP_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; CARD_EQ_0;
7319 ARITH_RULE `0 < n <=> ~(n = 0)`] THEN
7320 MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN
7321 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CARD_SUBSET THEN
7324 (* ------------------------------------------------------------------------- *)
7325 (* Convex cones and corresponding hulls. *)
7326 (* ------------------------------------------------------------------------- *)
7328 let convex_cone = new_definition
7329 `convex_cone s <=> ~(s = {}) /\ convex s /\ conic s`;;
7331 let CONVEX_CONE = prove
7335 (!x y. x IN s /\ y IN s ==> (x + y) IN s) /\
7336 (!x c. x IN s /\ &0 <= c ==> (c % x) IN s)`,
7337 GEN_TAC THEN REWRITE_TAC[convex_cone; GSYM conic] THEN
7338 ASM_CASES_TAC `conic(s:real^N->bool)` THEN
7339 ASM_SIMP_TAC[CONIC_CONTAINS_0] THEN AP_TERM_TAC THEN
7340 RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN
7341 REWRITE_TAC[convex] THEN EQ_TAC THEN
7342 ASM_SIMP_TAC[REAL_SUB_LE] THEN DISCH_TAC THEN
7343 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
7344 FIRST_X_ASSUM(MP_TAC o SPECL
7345 [`&2 % (x:real^N)`; `&2 % (y:real^N)`; `&1 / &2`; `&1 / &2`]) THEN
7346 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
7347 ASM_SIMP_TAC[VECTOR_MUL_LID; REAL_POS]);;
7349 let CONVEX_CONE_LINEAR_IMAGE = prove
7350 (`!f:real^M->real^N s.
7351 convex_cone s /\ linear f ==> convex_cone(IMAGE f s)`,
7352 SIMP_TAC[convex_cone; CONVEX_LINEAR_IMAGE; IMAGE_EQ_EMPTY;
7353 CONIC_LINEAR_IMAGE]);;
7355 let CONVEX_CONE_LINEAR_IMAGE_EQ = prove
7356 (`!f:real^M->real^N s.
7357 linear f /\ (!x y. f x = f y ==> x = y)
7358 ==> (convex_cone(IMAGE f s) <=> convex_cone s)`,
7359 REWRITE_TAC[convex_cone] THEN
7360 MESON_TAC[IMAGE_EQ_EMPTY; CONVEX_LINEAR_IMAGE_EQ; CONIC_LINEAR_IMAGE_EQ]);;
7362 add_linear_invariants [CONVEX_CONE_LINEAR_IMAGE_EQ];;
7364 let CONVEX_CONE_HALFSPACE_GE = prove
7365 (`!a. convex_cone {x | a dot x >= &0}`,
7366 SIMP_TAC[CONVEX_CONE; real_ge; IN_ELIM_THM; DOT_RZERO; DOT_RADD; DOT_RMUL;
7367 REAL_LE_ADD; REAL_LE_MUL; REAL_LE_REFL]);;
7369 let CONVEX_CONE_HALFSPACE_LE = prove
7370 (`!a. convex_cone {x | a dot x <= &0}`,
7371 REWRITE_TAC[REAL_ARITH `x <= &0 <=> &0 <= --x`; GSYM DOT_LNEG] THEN
7372 REWRITE_TAC[GSYM real_ge; CONVEX_CONE_HALFSPACE_GE]);;
7374 let CONVEX_CONE_CONTAINS_0 = prove
7375 (`!s:real^N->bool. convex_cone s ==> vec 0 IN s`,
7376 SIMP_TAC[CONVEX_CONE]);;
7378 let CONVEX_CONE_INTERS = prove
7379 (`!f. (!s:real^N->bool. s IN f ==> convex_cone s) ==> convex_cone(INTERS f)`,
7380 SIMP_TAC[convex_cone; CONIC_INTERS; CONVEX_INTERS] THEN
7381 REWRITE_TAC[GSYM convex_cone] THEN GEN_TAC THEN DISCH_TAC THEN
7382 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN
7383 ASM_SIMP_TAC[IN_INTERS; CONVEX_CONE_CONTAINS_0]);;
7385 let CONVEX_CONE_CONVEX_CONE_HULL = prove
7386 (`!s. convex_cone(convex_cone hull s)`,
7387 SIMP_TAC[P_HULL; CONVEX_CONE_INTERS]);;
7389 let CONVEX_CONVEX_CONE_HULL = prove
7390 (`!s. convex(convex_cone hull s)`,
7391 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);;
7393 let CONIC_CONVEX_CONE_HULL = prove
7394 (`!s. conic(convex_cone hull s)`,
7395 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);;
7397 let CONVEX_CONE_HULL_NONEMPTY = prove
7398 (`!s. ~(convex_cone hull s = {})`,
7399 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);;
7401 let CONVEX_CONE_HULL_CONTAINS_0 = prove
7402 (`!s. vec 0 IN convex_cone hull s`,
7403 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; CONVEX_CONE]);;
7405 let CONVEX_CONE_HULL_ADD = prove
7407 x IN convex_cone hull s /\ y IN convex_cone hull s
7408 ==> x + y IN convex_cone hull s`,
7409 MESON_TAC[CONVEX_CONE; CONVEX_CONE_CONVEX_CONE_HULL]);;
7411 let CONVEX_CONE_HULL_MUL = prove
7413 &0 <= c /\ x IN convex_cone hull s
7414 ==> (c % x) IN convex_cone hull s`,
7415 MESON_TAC[CONVEX_CONE; CONVEX_CONE_CONVEX_CONE_HULL]);;
7417 let CONVEX_CONE_SUMS = prove
7418 (`!s t. convex_cone s /\ convex_cone t
7419 ==> convex_cone {x + y:real^N | x IN s /\ y IN t}`,
7420 SIMP_TAC[convex_cone; CONIC_SUMS; CONVEX_SUMS] THEN SET_TAC[]);;
7422 let CONVEX_CONE_PCROSS = prove
7423 (`!s:real^M->bool t:real^N->bool.
7424 convex_cone s /\ convex_cone t ==> convex_cone(s PCROSS t)`,
7425 SIMP_TAC[convex_cone; CONVEX_PCROSS; CONIC_PCROSS; PCROSS_EQ_EMPTY]);;
7427 let CONVEX_CONE_PCROSS_EQ = prove
7428 (`!s:real^M->bool t:real^N->bool.
7429 convex_cone(s PCROSS t) <=> convex_cone s /\ convex_cone t`,
7431 ASM_CASES_TAC `s:real^M->bool = {}` THENL
7432 [ASM_REWRITE_TAC[PCROSS_EMPTY; convex_cone]; ALL_TAC] THEN
7433 ASM_CASES_TAC `t:real^N->bool = {}` THENL
7434 [ASM_REWRITE_TAC[PCROSS_EMPTY; convex_cone]; ALL_TAC] THEN
7435 EQ_TAC THEN REWRITE_TAC[CONVEX_CONE_PCROSS] THEN REPEAT STRIP_TAC THENL
7436 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
7437 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_CONE_LINEAR_IMAGE) THEN
7438 ASM_REWRITE_TAC[LINEAR_FSTCART];
7439 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
7440 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_CONE_LINEAR_IMAGE) THEN
7441 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
7442 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
7443 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
7444 FSTCART_PASTECART; SNDCART_PASTECART] THEN
7447 let CONVEX_CONE_HULL_UNION = prove
7448 (`!s t. convex_cone hull(s UNION t) =
7449 {x + y:real^N | x IN convex_cone hull s /\ y IN convex_cone hull t}`,
7450 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
7451 [MATCH_MP_TAC HULL_MINIMAL THEN
7452 SIMP_TAC[CONVEX_CONE_SUMS; CONVEX_CONE_CONVEX_CONE_HULL] THEN
7453 REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN
7454 X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL
7455 [MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
7456 ASM_SIMP_TAC[HULL_INC; CONVEX_CONE_HULL_CONTAINS_0; VECTOR_ADD_RID];
7457 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN
7458 ASM_SIMP_TAC[HULL_INC; CONVEX_CONE_HULL_CONTAINS_0; VECTOR_ADD_LID]];
7459 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
7460 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_CONE_HULL_ADD THEN
7461 ASM_MESON_TAC[HULL_MONO; SUBSET_UNION; SUBSET]]);;
7463 let CONVEX_CONE_SING = prove
7464 (`convex_cone {vec 0}`,
7465 SIMP_TAC[CONVEX_CONE; IN_SING; VECTOR_ADD_LID; VECTOR_MUL_RZERO]);;
7467 let CONVEX_HULL_SUBSET_CONVEX_CONE_HULL = prove
7468 (`!s. convex hull s SUBSET convex_cone hull s`,
7469 GEN_TAC THEN MATCH_MP_TAC HULL_ANTIMONO THEN
7470 SIMP_TAC[convex_cone; SUBSET; IN]);;
7472 let CONIC_HULL_SUBSET_CONVEX_CONE_HULL = prove
7473 (`!s. conic hull s SUBSET convex_cone hull s`,
7474 GEN_TAC THEN MATCH_MP_TAC HULL_ANTIMONO THEN
7475 SIMP_TAC[convex_cone; SUBSET; IN]);;
7477 let CONVEX_CONE_HULL_SEPARATE_NONEMPTY = prove
7480 ==> convex_cone hull s = conic hull (convex hull s)`,
7481 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN
7482 MATCH_MP_TAC HULL_MINIMAL THEN
7483 REWRITE_TAC[CONIC_CONVEX_CONE_HULL; CONVEX_HULL_SUBSET_CONVEX_CONE_HULL] THEN
7484 ASM_SIMP_TAC[CONVEX_CONIC_HULL; CONVEX_CONVEX_HULL; CONIC_CONIC_HULL;
7485 convex_cone; CONIC_HULL_EQ_EMPTY; CONVEX_HULL_EQ_EMPTY] THEN
7486 ASM_MESON_TAC[HULL_SUBSET; SUBSET_REFL; SUBSET_TRANS]);;
7488 let CONVEX_CONE_HULL_EMPTY = prove
7489 (`convex_cone hull {} = {vec 0}`,
7490 MATCH_MP_TAC HULL_UNIQUE THEN
7491 REWRITE_TAC[CONVEX_CONE_CONTAINS_0; EMPTY_SUBSET; CONVEX_CONE_SING;
7492 SET_RULE `{a} SUBSET s <=> a IN s`; CONVEX_CONE_CONTAINS_0]);;
7494 let CONVEX_CONE_HULL_SEPARATE = prove
7496 convex_cone hull s = vec 0 INSERT conic hull (convex hull s)`,
7497 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
7498 ASM_SIMP_TAC[CONVEX_CONE_HULL_EMPTY; CONVEX_HULL_EMPTY; CONIC_HULL_EMPTY] THEN
7499 ASM_SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY] THEN
7500 MATCH_MP_TAC(SET_RULE `a IN s ==> s = a INSERT s`) THEN
7501 ASM_SIMP_TAC[CONIC_CONTAINS_0; CONIC_CONIC_HULL] THEN
7502 ASM_REWRITE_TAC[CONIC_HULL_EQ_EMPTY; CONVEX_HULL_EQ_EMPTY]);;
7504 let CONVEX_CONE_HULL_CONVEX_HULL_NONEMPTY = prove
7507 ==> convex_cone hull s = {c % x | &0 <= c /\ x IN convex hull s}`,
7508 SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY; CONIC_HULL_EXPLICIT]);;
7510 let CONVEX_CONE_HULL_CONVEX_HULL = prove
7512 convex_cone hull s =
7513 vec 0 INSERT {c % x | &0 <= c /\ x IN convex hull s}`,
7514 REWRITE_TAC[CONVEX_CONE_HULL_SEPARATE; CONIC_HULL_EXPLICIT]);;
7516 let CONVEX_CONE_HULL_LINEAR_IMAGE = prove
7517 (`!f:real^M->real^N s.
7519 ==> convex_cone hull (IMAGE f s) = IMAGE f (convex_cone hull s)`,
7520 REPEAT STRIP_TAC THEN
7521 ASM_CASES_TAC `s:real^M-> bool = {}` THEN
7522 ASM_SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY; IMAGE_EQ_EMPTY;
7523 CONVEX_HULL_LINEAR_IMAGE; CONIC_HULL_LINEAR_IMAGE] THEN
7524 REWRITE_TAC[IMAGE_CLAUSES; CONVEX_CONE_HULL_EMPTY] THEN
7525 MATCH_MP_TAC(SET_RULE `f x = y ==> {y} = {f x}`) THEN
7526 ASM_MESON_TAC[LINEAR_0]);;
7528 add_linear_invariants [CONVEX_CONE_HULL_LINEAR_IMAGE];;
7530 let SUBSPACE_IMP_CONVEX_CONE = prove
7531 (`!s. subspace s ==> convex_cone s`,
7532 SIMP_TAC[subspace; CONVEX_CONE]);;
7534 let CONVEX_CONE_SPAN = prove
7535 (`!s. convex_cone(span s)`,
7536 SIMP_TAC[convex_cone; CONVEX_SPAN; CONIC_SPAN; GSYM MEMBER_NOT_EMPTY] THEN
7537 MESON_TAC[SPAN_0]);;
7539 let CONVEX_CONE_NEGATIONS = prove
7540 (`!s. convex_cone s ==> convex_cone (IMAGE (--) s)`,
7541 SIMP_TAC[convex_cone; IMAGE_EQ_EMPTY; CONIC_NEGATIONS; CONVEX_NEGATIONS]);;
7543 let SUBSPACE_CONVEX_CONE_SYMMETRIC = prove
7545 subspace s <=> convex_cone s /\ (!x. x IN s ==> --x IN s)`,
7546 GEN_TAC THEN REWRITE_TAC[subspace; CONVEX_CONE] THEN
7547 EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THENL
7548 [ASM_MESON_TAC[VECTOR_ARITH `--x:real^N = -- &1 % x`];
7549 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN DISCH_TAC THEN
7550 DISJ_CASES_TAC(SPEC `c:real` REAL_LE_NEGTOTAL) THEN ASM_SIMP_TAC[] THEN
7551 ASM_MESON_TAC[VECTOR_ARITH `c % x:real^N = --(--c % x)`]]);;
7553 let SPAN_CONVEX_CONE_ALLSIGNS = prove
7554 (`!s:real^N->bool. span s = convex_cone hull (s UNION IMAGE (--) s)`,
7555 GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
7556 [MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN CONJ_TAC THENL
7557 [MESON_TAC[HULL_SUBSET; SUBSET_UNION; SUBSET_TRANS]; ALL_TAC] THEN
7558 REWRITE_TAC[SUBSPACE_CONVEX_CONE_SYMMETRIC;
7559 CONVEX_CONE_CONVEX_CONE_HULL] THEN
7560 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
7561 [X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN
7562 DISCH_TAC THEN MATCH_MP_TAC HULL_INC THEN
7563 REWRITE_TAC[IN_UNION; IN_IMAGE] THEN ASM_MESON_TAC[VECTOR_NEG_NEG];
7564 SUBGOAL_THEN `!s. {x:real^N | (--x) IN s} = IMAGE (--) s`
7565 (fun th -> SIMP_TAC[th; CONVEX_CONE_NEGATIONS;
7566 CONVEX_CONE_CONVEX_CONE_HULL]) THEN
7567 GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
7568 REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[VECTOR_NEG_NEG]];
7569 MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_CONE_SPAN] THEN
7570 REWRITE_TAC[UNION_SUBSET; SPAN_INC] THEN
7571 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
7572 MESON_TAC[SPAN_SUPERSET; SPAN_NEG]]);;
7574 (* ------------------------------------------------------------------------- *)
7575 (* Epigraphs of convex functions. *)
7576 (* ------------------------------------------------------------------------- *)
7578 let epigraph = new_definition
7579 `epigraph s (f:real^N->real) =
7580 {xy:real^((N,1)finite_sum) |
7581 fstcart xy IN s /\ f(fstcart xy) <= drop(sndcart xy)}`;;
7583 let IN_EPIGRAPH = prove
7584 (`!x y. (pastecart x (lift y)) IN epigraph s f <=> x IN s /\ f(x) <= y`,
7585 REWRITE_TAC[epigraph; IN_ELIM_THM; FSTCART_PASTECART; SNDCART_PASTECART;
7588 let CONVEX_EPIGRAPH = prove
7589 (`!f s. f convex_on s /\ convex s <=> convex(epigraph s f)`,
7590 REWRITE_TAC[convex; convex_on; IN_ELIM_THM; SNDCART_ADD; SNDCART_CMUL;
7591 epigraph; FSTCART_ADD; FSTCART_CMUL; FORALL_PASTECART; FSTCART_PASTECART;
7592 SNDCART_PASTECART] THEN
7593 REWRITE_TAC[GSYM FORALL_DROP; DROP_ADD; DROP_CMUL] THEN
7594 MESON_TAC[REAL_LE_REFL; REAL_LE_ADD2; REAL_LE_LMUL; REAL_LE_TRANS]);;
7596 let CONVEX_EPIGRAPH_CONVEX = prove
7597 (`!f s. convex s ==> (f convex_on s <=> convex(epigraph s f))`,
7598 REWRITE_TAC[GSYM CONVEX_EPIGRAPH] THEN CONV_TAC TAUT);;
7600 let CONVEX_ON_EPIGRAPH_SLICE_LE = prove
7601 (`!f:real^N->real s a.
7602 f convex_on s /\ convex s ==> convex {x | x IN s /\ f(x) <= a}`,
7603 SIMP_TAC[convex_on; convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
7604 FIRST_X_ASSUM(fun th ->
7605 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
7606 ASM_REWRITE_TAC[] THEN
7607 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
7608 MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[]);;
7610 let CONVEX_ON_EPIGRAPH_SLICE_LT = prove
7611 (`!f:real^N->real s a.
7612 f convex_on s /\ convex s ==> convex {x | x IN s /\ f(x) < a}`,
7613 SIMP_TAC[convex_on; convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
7614 FIRST_X_ASSUM(fun th ->
7615 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
7616 ASM_REWRITE_TAC[] THEN
7617 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN
7618 MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REWRITE_TAC[]);;
7620 (* ------------------------------------------------------------------------- *)
7621 (* Use this to derive general bound property of convex function. *)
7622 (* ------------------------------------------------------------------------- *)
7624 let FORALL_OF_PASTECART = prove
7625 (`(!p. P (fstcart o p) (sndcart o p)) <=> (!x:A->B^M y:A->B^N. P x y)`,
7626 EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THEN
7627 FIRST_ASSUM(MP_TAC o SPEC `\a:A. pastecart (x a :B^M) (y a :B^N)`) THEN
7628 REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART; ETA_AX]);;
7630 let FORALL_OF_DROP = prove
7631 (`(!v. P (drop o v)) <=> (!x:A->real. P x)`,
7632 EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THEN
7633 FIRST_ASSUM(MP_TAC o SPEC `\a:A. lift(x a)`) THEN
7634 REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]);;
7636 let CONVEX_ON_JENSEN = prove
7637 (`!f:real^N->real s.
7639 ==> (f convex_on s <=>
7641 (!i:num. 1 <= i /\ i <= k ==> &0 <= u(i) /\ x(i) IN s) /\
7643 ==> f(vsum (1..k) (\i. u(i) % x(i)))
7644 <= sum (1..k) (\i. u(i) * f(x(i))))`,
7646 (`(!x. P x ==> (Q x = R x)) ==> (!x. P x) ==> ((!x. Q x) <=> (!x. R x))`,
7648 REPEAT STRIP_TAC THEN FIRST_ASSUM
7649 (fun th -> REWRITE_TAC[MATCH_MP CONVEX_EPIGRAPH_CONVEX th]) THEN
7650 REWRITE_TAC[CONVEX_INDEXED; epigraph] THEN
7651 SIMP_TAC[IN_ELIM_THM; SNDCART_ADD; SNDCART_CMUL; FINITE_NUMSEG;
7652 FSTCART_ADD; FSTCART_CMUL; FORALL_PASTECART; DROP_CMUL;
7653 FSTCART_PASTECART; SNDCART_PASTECART;
7654 FSTCART_VSUM; SNDCART_VSUM; DROP_VSUM; o_DEF] THEN
7655 REWRITE_TAC[GSYM(ISPEC `fstcart` o_THM); GSYM(ISPEC `sndcart` o_THM)] THEN
7656 REWRITE_TAC[GSYM(ISPEC `drop` o_THM)] THEN
7657 REWRITE_TAC[FORALL_OF_PASTECART; FORALL_OF_DROP] THEN
7658 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_INDEXED]) THEN
7659 REPEAT(MATCH_MP_TAC lemma THEN GEN_TAC) THEN SIMP_TAC[] THEN
7660 REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN
7661 REWRITE_TAC[FORALL_AND_THM] THEN DISCH_THEN(K ALL_TAC) THEN
7662 EQ_TAC THEN SIMP_TAC[REAL_LE_REFL] THEN
7663 DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN
7664 ASM_REWRITE_TAC[] THEN
7665 MATCH_MP_TAC(REAL_ARITH `a <= b ==> x <= a ==> x <= b`) THEN
7666 ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LE_LMUL]);;
7668 (* ------------------------------------------------------------------------- *)
7669 (* Another intermediate value theorem formulation. *)
7670 (* ------------------------------------------------------------------------- *)
7672 let IVT_INCREASING_COMPONENT_ON_1 = prove
7673 (`!f:real^1->real^N a b y k.
7674 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
7675 f continuous_on interval[a,b] /\
7676 f(a)$k <= y /\ y <= f(b)$k
7677 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
7678 REPEAT STRIP_TAC THEN
7679 MP_TAC(ISPECL [`IMAGE (f:real^1->real^N) (interval[a,b])`]
7680 CONNECTED_IVT_COMPONENT) THEN
7681 REWRITE_TAC[EXISTS_IN_IMAGE] THEN DISCH_THEN MATCH_MP_TAC THEN
7682 REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_IN_IMAGE] THEN
7683 ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; CONVEX_CONNECTED;
7684 CONVEX_INTERVAL] THEN
7685 EXISTS_TAC `a:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN
7686 EXISTS_TAC `b:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL]);;
7688 let IVT_INCREASING_COMPONENT_1 = prove
7689 (`!f:real^1->real^N a b y k.
7690 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
7691 (!x. x IN interval[a,b] ==> f continuous at x) /\
7692 f(a)$k <= y /\ y <= f(b)$k
7693 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
7694 REPEAT STRIP_TAC THEN MATCH_MP_TAC IVT_INCREASING_COMPONENT_ON_1 THEN
7695 ASM_SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
7697 let IVT_DECREASING_COMPONENT_ON_1 = prove
7698 (`!f:real^1->real^N a b y k.
7699 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
7700 f continuous_on interval[a,b] /\
7701 f(b)$k <= y /\ y <= f(a)$k
7702 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
7703 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EQ_NEG2] THEN
7704 ASM_SIMP_TAC[GSYM VECTOR_NEG_COMPONENT] THEN
7705 MATCH_MP_TAC IVT_INCREASING_COMPONENT_ON_1 THEN
7706 ASM_SIMP_TAC[VECTOR_NEG_COMPONENT; CONTINUOUS_ON_NEG; REAL_LE_NEG2]);;
7708 let IVT_DECREASING_COMPONENT_1 = prove
7709 (`!f:real^1->real^N a b y k.
7710 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
7711 (!x. x IN interval[a,b] ==> f continuous at x) /\
7712 f(b)$k <= y /\ y <= f(a)$k
7713 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
7714 REPEAT STRIP_TAC THEN MATCH_MP_TAC IVT_DECREASING_COMPONENT_ON_1 THEN
7715 ASM_SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
7717 (* ------------------------------------------------------------------------- *)
7718 (* A bound within a convex hull, and so an interval. *)
7719 (* ------------------------------------------------------------------------- *)
7721 let CONVEX_ON_CONVEX_HULL_BOUND = prove
7722 (`!f s b. f convex_on (convex hull s) /\
7723 (!x:real^N. x IN s ==> f(x) <= b)
7724 ==> !x. x IN convex hull s ==> f(x) <= b`,
7725 REPEAT GEN_TAC THEN SIMP_TAC[CONVEX_ON_JENSEN; CONVEX_CONVEX_HULL] THEN
7726 STRIP_TAC THEN GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_INDEXED] THEN
7727 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
7728 MAP_EVERY X_GEN_TAC [`k:num`; `u:num->real`; `v:num->real^N`] THEN
7729 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN
7730 MATCH_MP_TAC REAL_LE_TRANS THEN
7731 EXISTS_TAC `sum(1..k) (\i. u i * f(v i :real^N))` THEN CONJ_TAC THENL
7732 [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET];
7734 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..k) (\i. u i * b)` THEN
7736 [MATCH_MP_TAC SUM_LE_NUMSEG THEN ASM_SIMP_TAC[REAL_LE_LMUL];
7738 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[SUM_LMUL] THEN
7739 ASM_MESON_TAC[REAL_LE_REFL; REAL_MUL_RID]);;
7741 let UNIT_INTERVAL_CONVEX_HULL = prove
7742 (`interval [vec 0,vec 1:real^N] =
7744 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
7745 ==> ((x$i = &0) \/ (x$i = &1))}`,
7747 (`FINITE {i | 1 <= i /\ i <= n /\ P(i)} /\
7748 CARD {i | 1 <= i /\ i <= n /\ P(i)} <= n`,
7750 [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `1..n`;
7751 GEN_REWRITE_TAC RAND_CONV [ARITH_RULE `x = (x + 1) - 1`] THEN
7752 REWRITE_TAC[GSYM CARD_NUMSEG] THEN MATCH_MP_TAC CARD_SUBSET] THEN
7753 SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; SUBSET; IN_ELIM_THM]) in
7754 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
7756 MATCH_MP_TAC HULL_MINIMAL THEN
7757 REWRITE_TAC[CONVEX_INTERVAL; SUBSET; IN_INTERVAL; IN_ELIM_THM] THEN
7758 SIMP_TAC[VEC_COMPONENT] THEN MESON_TAC[REAL_LE_REFL; REAL_POS]] THEN
7761 x IN interval[vec 0,vec 1] /\
7762 n <= dimindex(:N) /\
7763 CARD {i | 1 <= i /\ i <= dimindex(:N) /\ ~(x$i = &0)} <= n
7764 ==> x IN convex hull
7765 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
7766 ==> ((x$i = &0) \/ (x$i = &1))}`
7769 REWRITE_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN
7770 FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `dimindex(:N)` THEN
7771 ASM_REWRITE_TAC[LE_REFL; lemma]] THEN
7772 INDUCT_TAC THEN X_GEN_TAC `x:real^N` THENL
7773 [SIMP_TAC[LE; lemma; CARD_EQ_0] THEN
7774 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
7775 [EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; BETA_THM] THEN
7776 REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN STRIP_TAC THEN
7777 SUBGOAL_THEN `x = vec 0:real^N` SUBST1_TAC THENL
7778 [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT]; ALL_TAC] THEN
7779 MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
7780 SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT];
7783 `{i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)} = {}`
7785 [DISCH_THEN(K ALL_TAC) THEN
7786 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
7787 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
7788 [EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; BETA_THM] THEN
7789 REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN STRIP_TAC THEN
7790 SUBGOAL_THEN `x = vec 0:real^N` SUBST1_TAC THENL
7791 [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT]; ALL_TAC] THEN
7792 MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
7793 SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT];
7797 {i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)}`
7799 ABBREV_TAC `xi = inf
7801 {i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)})` THEN
7802 ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; lemma] THEN
7803 REWRITE_TAC[FORALL_IN_IMAGE] THEN
7804 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [IN_IMAGE; IN_ELIM_THM] THEN
7805 REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
7806 FIRST_X_ASSUM(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC) THEN
7807 FIRST_X_ASSUM SUBST_ALL_TAC THEN REPEAT STRIP_TAC THEN
7808 SUBGOAL_THEN `&0 <= (x:real^N)$i /\ x$i <= &1` STRIP_ASSUME_TAC THENL
7809 [UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN
7810 ASM_SIMP_TAC[IN_INTERVAL; VEC_COMPONENT];
7812 FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
7813 `x <= &1 ==> (x = &1) \/ x < &1`))
7816 `x = lambda i. if (x:real^N)$i = &0 then &0 else &1`
7818 [UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN
7819 ASM_SIMP_TAC[CART_EQ; IN_INTERVAL; VEC_COMPONENT; LAMBDA_BETA] THEN
7820 ASM_MESON_TAC[REAL_LE_ANTISYM];
7822 MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
7823 SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA] THEN MESON_TAC[];
7827 x$i % (lambda j. if x$j = &0 then &0 else &1) +
7829 (lambda j. if x$j = &0 then &0 else (x$j - x$i) / (&1 - x$i))`
7831 [SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
7832 LAMBDA_BETA; VEC_COMPONENT] THEN
7833 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
7834 ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_LID] THEN
7835 ASM_SIMP_TAC[REAL_DIV_LMUL; ARITH_RULE `x < &1 ==> ~(&1 - x = &0)`] THEN
7836 REPEAT STRIP_TAC THEN REAL_ARITH_TAC;
7838 MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN
7839 ASM_SIMP_TAC[REAL_ARITH `x < &1 ==> &0 <= &1 - x`;
7840 REAL_ARITH `x + &1 - x = &1`] THEN
7842 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
7843 SIMP_TAC[LAMBDA_BETA; IN_ELIM_THM] THEN MESON_TAC[];
7845 FIRST_X_ASSUM MATCH_MP_TAC THEN
7846 ASM_SIMP_TAC[ARITH_RULE `SUC k <= n ==> k <= n`] THEN CONJ_TAC THENL
7847 [SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; VEC_COMPONENT] THEN
7848 GEN_TAC THEN STRIP_TAC THEN
7849 COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL; REAL_POS] THEN
7850 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ;
7851 REAL_ARITH `x < &1 ==> &0 < &1 - x`] THEN
7852 ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_LE; REAL_MUL_LID] THEN
7853 ASM_SIMP_TAC[REAL_ARITH `a - b <= &1 - b <=> a <= &1`] THEN
7854 UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN
7855 ASM_SIMP_TAC[CART_EQ; IN_INTERVAL; VEC_COMPONENT; LAMBDA_BETA];
7857 MATCH_MP_TAC LE_TRANS THEN
7859 `CARD({i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)}
7862 [MATCH_MP_TAC CARD_SUBSET THEN REWRITE_TAC[lemma; FINITE_DELETE] THEN
7863 REWRITE_TAC[SUBSET; IN_DELETE; IN_ELIM_THM] THEN
7864 GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
7865 ASM_SIMP_TAC[LAMBDA_BETA] THEN
7866 COND_CASES_TAC THEN ASM_REWRITE_TAC[CONTRAPOS_THM] THEN
7867 SIMP_TAC[real_div; REAL_SUB_REFL; REAL_MUL_LZERO];
7868 SIMP_TAC[lemma; CARD_DELETE] THEN COND_CASES_TAC THEN
7869 ASM_SIMP_TAC[ARITH_RULE `x <= SUC n ==> x - 1 <= n`] THEN
7870 RULE_ASSUM_TAC(REWRITE_RULE[IN_ELIM_THM]) THEN
7873 (* ------------------------------------------------------------------------- *)
7874 (* Representation of any interval as a finite convex hull. *)
7875 (* ------------------------------------------------------------------------- *)
7877 let CLOSED_INTERVAL_AS_CONVEX_HULL = prove
7878 (`!a b:real^N. ?s. FINITE s /\ interval[a,b] = convex hull s`,
7879 REPEAT GEN_TAC THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL
7880 [ASM_MESON_TAC[CONVEX_HULL_EMPTY; FINITE_EMPTY]; ALL_TAC] THEN
7881 ASM_SIMP_TAC[CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL] THEN
7883 `?s:real^N->bool. FINITE s /\ interval[vec 0,vec 1] = convex hull s`
7884 STRIP_ASSUME_TAC THENL
7886 `{x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
7887 ==> ((x$i = &0) \/ (x$i = &1))}` THEN
7888 REWRITE_TAC[UNIT_INTERVAL_CONVEX_HULL] THEN
7889 MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC
7890 `IMAGE (\s. (lambda i. if i IN s then &1 else &0):real^N)
7891 {t | t SUBSET (1..dimindex(:N))}` THEN
7892 ASM_SIMP_TAC[FINITE_POWERSET; FINITE_IMAGE; FINITE_NUMSEG] THEN
7893 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_IMAGE] THEN
7894 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC
7895 `{i | 1 <= i /\ i <= dimindex(:N) /\ ((x:real^N)$i = &1)}` THEN
7896 SIMP_TAC[CART_EQ; IN_ELIM_THM; IN_NUMSEG; LAMBDA_BETA] THEN
7898 EXISTS_TAC `IMAGE (\x:real^N. a + x)
7899 (IMAGE (\x. (lambda i. ((b:real^N)$i - a$i) * x$i))
7900 (s:real^N->bool))` THEN
7901 ASM_SIMP_TAC[FINITE_IMAGE; CONVEX_HULL_TRANSLATION] THEN
7902 AP_TERM_TAC THEN MATCH_MP_TAC(GSYM CONVEX_HULL_LINEAR_IMAGE) THEN
7903 SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
7904 VECTOR_MUL_COMPONENT] THEN
7905 REPEAT STRIP_TAC THEN REAL_ARITH_TAC]);;
7907 (* ------------------------------------------------------------------------- *)
7908 (* Bounded convex function on open set is continuous. *)
7909 (* ------------------------------------------------------------------------- *)
7911 let CONVEX_ON_BOUNDED_CONTINUOUS = prove
7912 (`!f:real^N->real s b.
7913 open s /\ f convex_on s /\ (!x. x IN s ==> abs(f x) <= b)
7914 ==> (lift o f) continuous_on s`,
7915 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
7916 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7917 REWRITE_TAC[CONTINUOUS_AT_LIFT_RANGE] THEN
7918 ABBREV_TAC `B = abs(b) + &1` THEN
7919 SUBGOAL_THEN `&0 < B /\ !x:real^N. x IN s ==> abs(f x) <= B`
7920 STRIP_ASSUME_TAC THENL
7921 [EXPAND_TAC "B" THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
7922 ASM_MESON_TAC[REAL_ARITH `x <= b ==> x <= abs b + &1`];
7924 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
7925 REWRITE_TAC[REAL_ARITH `abs(x - y) < e <=> x - y < e /\ y - x < e`] THEN
7926 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
7927 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[SUBSET; IN_CBALL] THEN
7928 ASM_REWRITE_TAC[] THEN
7929 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
7930 EXISTS_TAC `min (k / &2) (e / (&2 * B) * k)` THEN
7931 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; REAL_LT_MUL;
7932 REAL_OF_NUM_LT; ARITH] THEN
7933 X_GEN_TAC `y:real^N` THEN
7934 ASM_CASES_TAC `y:real^N = x` THEN ASM_REWRITE_TAC[REAL_SUB_REFL] THEN
7936 ABBREV_TAC `t = k / norm(y - x:real^N)` THEN
7937 SUBGOAL_THEN `&2 < t` ASSUME_TAC THENL
7938 [EXPAND_TAC "t" THEN
7939 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
7940 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
7941 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH];
7943 FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP (REAL_ARITH
7944 `&2 < t ==> &0 < t /\ ~(t = &0) /\ &0 < t - &1 /\
7945 &0 < &1 + t /\ ~(&1 + t = &0)`)) THEN
7946 SUBGOAL_THEN `y:real^N IN s` ASSUME_TAC THENL
7947 [FIRST_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[dist] THEN
7948 ONCE_REWRITE_TAC[NORM_SUB] THEN
7949 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
7950 `x < k / &2 ==> k / &2 <= k ==> x <= k`)) THEN
7951 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
7952 UNDISCH_TAC `&0 < k` THEN REAL_ARITH_TAC;
7955 [ABBREV_TAC `w:real^N = x + t % (y - x)` THEN
7956 SUBGOAL_THEN `w:real^N IN s` STRIP_ASSUME_TAC THENL
7957 [FIRST_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "w" THEN
7958 REWRITE_TAC[dist; VECTOR_ARITH `x - (x + t) = --t:real^N`] THEN
7959 EXPAND_TAC "t" THEN REWRITE_TAC[NORM_NEG; NORM_MUL; REAL_ABS_DIV] THEN
7960 REWRITE_TAC[REAL_ABS_NORM; real_div; GSYM REAL_MUL_ASSOC] THEN
7961 ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; NORM_POS_LT; VECTOR_SUB_EQ;
7962 REAL_MUL_RID; REAL_ARITH `&0 < x ==> abs(x) <= x`];
7964 SUBGOAL_THEN `(&1 / t) % w + (t - &1) / t % x = y:real^N` ASSUME_TAC THENL
7965 [EXPAND_TAC "w" THEN
7966 REWRITE_TAC[VECTOR_ARITH
7967 `b % (x + c % (y - x)) + a % x =
7968 (a + b - b * c) % x + (b * c) % y`] THEN
7969 ASM_SIMP_TAC[REAL_DIV_RMUL; VECTOR_MUL_LID] THEN
7970 ASM_SIMP_TAC[real_div; REAL_MUL_RINV; REAL_SUB_REFL;
7971 VECTOR_MUL_LZERO; VECTOR_ADD_LID;
7972 REAL_ARITH `(a - &1) * b + &1 * b - &1 = a * b - &1`];
7974 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
7975 DISCH_THEN(MP_TAC o SPECL
7976 [`w:real^N`; `x:real^N`; `&1 / t`; `(t - &1) / t`]) THEN
7977 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_DIV; REAL_LT_01] THEN
7978 REWRITE_TAC[real_div; GSYM REAL_ADD_RDISTRIB] THEN
7979 ASM_SIMP_TAC[REAL_SUB_ADD2; REAL_MUL_RINV] THEN
7980 MATCH_MP_TAC(REAL_ARITH
7981 `a * fw + (b - &1) * fx < e
7982 ==> fy <= a * fw + b * fx ==> fy - fx < e`) THEN
7983 ASM_SIMP_TAC[real_div; REAL_SUB_RDISTRIB; REAL_MUL_RINV; REAL_MUL_LID;
7984 REAL_ARITH `a * x + y - a * y - y = a * (x - y)`] THEN
7985 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
7986 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
7987 MATCH_MP_TAC(REAL_ARITH
7988 `!b. abs(x) <= b /\ abs(y) <= b /\ &2 * b < z ==> x - y < z`) THEN
7989 EXISTS_TAC `B:real` THEN ASM_SIMP_TAC[] THEN EXPAND_TAC "t" THEN
7990 REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN
7991 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
7992 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
7993 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN
7994 REWRITE_TAC[real_div; REAL_ARITH `(a * b) * inv c = (b * inv c) * a`] THEN
7995 ASM_REWRITE_TAC[GSYM real_div];
7997 ABBREV_TAC `w:real^N = x - t % (y - x)` THEN
7998 SUBGOAL_THEN `w:real^N IN s` STRIP_ASSUME_TAC THENL
7999 [FIRST_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "w" THEN
8000 REWRITE_TAC[dist; VECTOR_ARITH `x - (x - t) = t:real^N`] THEN
8001 EXPAND_TAC "t" THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN
8002 REWRITE_TAC[REAL_ABS_NORM; real_div; GSYM REAL_MUL_ASSOC] THEN
8003 ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; NORM_POS_LT; VECTOR_SUB_EQ;
8004 REAL_MUL_RID; REAL_ARITH `&0 < x ==> abs(x) <= x`];
8006 SUBGOAL_THEN `(&1 / (&1 + t)) % w + t / (&1 + t) % y = x:real^N`
8008 [EXPAND_TAC "w" THEN
8009 REWRITE_TAC[VECTOR_ARITH
8010 `b % (x - c % (y - x)) + a % y =
8011 (b * (&1 + c)) % x + (a - b * c) % y`] THEN
8012 ASM_SIMP_TAC[REAL_DIV_RMUL; VECTOR_MUL_LID] THEN
8013 REWRITE_TAC[real_div; REAL_MUL_AC; REAL_MUL_LID; REAL_MUL_RID] THEN
8014 REWRITE_TAC[REAL_SUB_REFL; VECTOR_MUL_LZERO; VECTOR_ADD_RID];
8016 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
8017 DISCH_THEN(MP_TAC o SPECL
8018 [`w:real^N`; `y:real^N`; `&1 / (&1 + t)`; `t / (&1 + t)`]) THEN
8019 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_DIV; REAL_LT_01] THEN
8020 REWRITE_TAC[real_div; GSYM REAL_ADD_RDISTRIB] THEN
8021 ASM_SIMP_TAC[REAL_SUB_ADD2; REAL_MUL_RINV] THEN
8022 MATCH_MP_TAC(REAL_ARITH
8023 `a * fw + (b - &1) * fx < e
8024 ==> fy <= a * fw + b * fx ==> fy - fx < e`) THEN
8025 SUBGOAL_THEN `t * inv(&1 + t) - &1 = --(inv(&1 + t))` SUBST1_TAC THENL
8026 [REWRITE_TAC[REAL_ARITH `(a * b - &1 = --b) <=> ((&1 + a) * b = &1)`] THEN
8027 ASM_SIMP_TAC[REAL_MUL_RINV];
8029 REWRITE_TAC[REAL_ARITH `(&1 * a) * x + --a * y = a * (x - y)`] THEN
8030 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8031 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
8032 MATCH_MP_TAC(REAL_ARITH
8033 `!b. abs(x) <= b /\ abs(y) <= b /\ &2 * b < z ==> x - y < z`) THEN
8034 EXISTS_TAC `B:real` THEN ASM_SIMP_TAC[] THEN
8035 MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x < e * k ==> x < e * (&1 + k)`) THEN
8036 EXPAND_TAC "t" THEN REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN
8037 REWRITE_TAC[GSYM real_div] THEN
8038 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
8039 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8040 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN
8041 REWRITE_TAC[real_div; REAL_ARITH `(a * b) * inv c = (b * inv c) * a`] THEN
8042 ASM_REWRITE_TAC[GSYM real_div]]);;
8044 (* ------------------------------------------------------------------------- *)
8045 (* Upper bound on a ball implies upper and lower bounds. *)
8046 (* ------------------------------------------------------------------------- *)
8048 let CONVEX_BOUNDS_LEMMA = prove
8050 f convex_on cball(x,e) /\
8051 (!y. y IN cball(x,e) ==> f(y) <= b)
8052 ==> !y. y IN cball(x,e) ==> abs(f(y)) <= b + &2 * abs(f(x))`,
8053 REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 <= e` THENL
8055 REWRITE_TAC[IN_CBALL] THEN ASM_MESON_TAC[DIST_POS_LE; REAL_LE_TRANS]] THEN
8056 REPEAT STRIP_TAC THEN
8057 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
8058 DISCH_THEN(MP_TAC o SPECL
8059 [`y:real^N`; `&2 % x - y:real^N`; `&1 / &2`; `&1 / &2`]) THEN
8060 REWRITE_TAC[GSYM VECTOR_ADD_LDISTRIB; GSYM REAL_ADD_LDISTRIB] THEN
8061 REWRITE_TAC[VECTOR_ARITH `y + x - y = x:real^N`] THEN
8062 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
8063 ABBREV_TAC `z = &2 % x - y:real^N` THEN
8064 SUBGOAL_THEN `z:real^N IN cball(x,e)` ASSUME_TAC THENL
8065 [UNDISCH_TAC `y:real^N IN cball(x,e)` THEN
8066 EXPAND_TAC "z" THEN REWRITE_TAC[dist; IN_CBALL] THEN
8067 REWRITE_TAC[VECTOR_ARITH `x - (&2 % x - y) = y - x`] THEN
8068 REWRITE_TAC[NORM_SUB];
8070 ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8071 REWRITE_TAC[real_div; REAL_MUL_LID] THEN REWRITE_TAC[GSYM real_div] THEN
8072 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
8073 FIRST_X_ASSUM(fun th ->
8074 MAP_EVERY (MP_TAC o C SPEC th) [`y:real^N`; `z:real^N`]) THEN
8075 ASM_REWRITE_TAC[CENTRE_IN_CBALL] THEN REAL_ARITH_TAC);;
8077 (* ------------------------------------------------------------------------- *)
8078 (* Hence a convex function on an open set is continuous. *)
8079 (* ------------------------------------------------------------------------- *)
8081 let CONVEX_ON_CONTINUOUS = prove
8082 (`!f s:real^N->bool. open s /\ f convex_on s ==> lift o f continuous_on s`,
8083 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT] THEN
8084 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
8085 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
8086 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
8087 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
8088 ABBREV_TAC `d = e / &(dimindex(:N))` THEN
8089 SUBGOAL_THEN `&0 < d` ASSUME_TAC THENL
8090 [EXPAND_TAC "d" THEN MATCH_MP_TAC REAL_LT_DIV THEN
8091 ASM_REWRITE_TAC[REAL_OF_NUM_LT; DIMINDEX_GE_1;
8092 ARITH_RULE `0 < d <=> 1 <= d`];
8095 `?b. !y:real^N. y IN interval[(x - lambda i. d),(x + lambda i. d)]
8097 STRIP_ASSUME_TAC THENL
8098 [MP_TAC(ISPECL [`x - (lambda i. d):real^N`; `x + (lambda i. d):real^N`]
8099 CLOSED_INTERVAL_AS_CONVEX_HULL) THEN
8100 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
8101 ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `c = {}:real^N->bool` THEN
8102 ASM_REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
8103 MP_TAC(ISPEC `IMAGE (f:real^N->real) c` SUP_FINITE) THEN
8104 ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
8105 ABBREV_TAC `k = sup(IMAGE (f:real^N->real) c)` THEN
8106 STRIP_TAC THEN EXISTS_TAC `k:real` THEN
8107 MATCH_MP_TAC CONVEX_ON_CONVEX_HULL_BOUND THEN
8108 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_ON_SUBSET THEN
8109 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
8110 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `cball (x:real^N,e)` THEN
8111 ASM_REWRITE_TAC[] THEN
8112 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
8113 REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL] THEN
8114 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; LAMBDA_BETA] THEN
8115 X_GEN_TAC `z:real^N` THEN
8116 REWRITE_TAC[REAL_ARITH `x - d <= z /\ z <= x + d <=> abs(x - z) <= d`] THEN
8117 DISCH_TAC THEN REWRITE_TAC[dist] THEN
8118 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
8119 `sum(1..dimindex(:N)) (\i. abs((x - z:real^N)$i))` THEN
8120 REWRITE_TAC[NORM_LE_L1] THEN
8121 MATCH_MP_TAC SUM_BOUND_GEN THEN
8122 REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; CARD_NUMSEG] THEN
8123 ASM_SIMP_TAC[IN_NUMSEG; NOT_LT; DIMINDEX_GE_1; ADD_SUB;
8124 VECTOR_SUB_COMPONENT];
8126 SUBGOAL_THEN `cball(x:real^N,d) SUBSET cball(x,e)` ASSUME_TAC THENL
8127 [REWRITE_TAC[SUBSET; IN_CBALL] THEN GEN_TAC THEN
8128 MATCH_MP_TAC(REAL_ARITH `d <= e ==> x <= d ==> x <= e`) THEN
8130 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; DIMINDEX_GE_1;
8131 ARITH_RULE `0 < x <=> 1 <= x`] THEN
8132 GEN_REWRITE_TAC LAND_CONV [GSYM REAL_MUL_RID] THEN
8133 ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_OF_NUM_LE; DIMINDEX_GE_1];
8136 `!y:real^N. y IN cball(x,d) ==> abs(f(y)) <= b + &2 * abs(f(x))`
8138 [MATCH_MP_TAC CONVEX_BOUNDS_LEMMA THEN CONJ_TAC THENL
8139 [ASM_MESON_TAC[CONVEX_ON_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN
8140 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
8141 UNDISCH_TAC `y:real^N IN cball(x,d)` THEN REWRITE_TAC[IN_CBALL] THEN
8142 REWRITE_TAC[IN_INTERVAL; IN_CBALL; dist] THEN DISCH_TAC THEN
8143 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; LAMBDA_BETA] THEN
8144 REWRITE_TAC[REAL_ARITH `x - d <= z /\ z <= x + d <=> abs(x - z) <= d`] THEN
8145 SIMP_TAC[GSYM VECTOR_SUB_COMPONENT] THEN REPEAT STRIP_TAC THEN
8146 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(x - y:real^N)` THEN
8147 ASM_SIMP_TAC[COMPONENT_LE_NORM];
8149 SUBGOAL_THEN `(lift o f) continuous_on (ball(x:real^N,d))` MP_TAC THENL
8150 [MATCH_MP_TAC CONVEX_ON_BOUNDED_CONTINUOUS THEN REWRITE_TAC[OPEN_BALL] THEN
8151 EXISTS_TAC `b + &2 * abs(f(x:real^N))` THEN
8152 ASM_MESON_TAC[SUBSET; CONVEX_ON_SUBSET; SUBSET_TRANS; BALL_SUBSET_CBALL];
8154 ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_BALL; CENTRE_IN_BALL]);;
8156 (* ------------------------------------------------------------------------- *)
8157 (* Characterizations of convex functions in terms of sequents. *)
8158 (* ------------------------------------------------------------------------- *)
8160 let CONVEX_ON_LEFT_SECANT_MUL,CONVEX_ON_RIGHT_SECANT_MUL = (CONJ_PAIR o prove)
8161 (`(!f s:real^N->bool.
8163 !a b x. a IN s /\ b IN s /\ x IN segment[a,b]
8164 ==> (f x - f a) * norm(b - a) <= (f b - f a) * norm(x - a)) /\
8167 !a b x. a IN s /\ b IN s /\ x IN segment[a,b]
8168 ==> (f b - f a) * norm(b - x) <= (f b - f x) * norm(b - a))`,
8170 REPEAT GEN_TAC THEN REWRITE_TAC[convex_on] THEN
8171 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8172 X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN
8173 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8174 X_GEN_TAC `b:real^N` THEN REWRITE_TAC[] THEN
8175 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8176 ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8177 REWRITE_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN
8178 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
8179 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8180 X_GEN_TAC `u:real` THEN REWRITE_TAC[] THEN
8181 REWRITE_TAC[TAUT `a /\ x = y <=> x = y /\ a`;
8182 TAUT `a /\ x = y /\ b <=> x = y /\ a /\ b`] THEN
8183 REWRITE_TAC[REAL_ARITH `v + u = &1 <=> v = &1 - u`] THEN
8184 REWRITE_TAC[FORALL_UNWIND_THM2; IMP_CONJ] THEN
8185 REWRITE_TAC[REAL_SUB_LE] THEN
8186 ASM_CASES_TAC `&0 <= u` THEN ASM_REWRITE_TAC[] THEN
8187 ASM_CASES_TAC `u <= &1` THEN ASM_REWRITE_TAC[] THEN
8188 REWRITE_TAC[VECTOR_ARITH `((&1 - u) % a + u % b) - a:real^N = u % (b - a)`;
8189 VECTOR_ARITH `b - ((&1 - u) % a + u % b):real^N = (&1 - u) % (b - a)`] THEN
8190 REWRITE_TAC[NORM_MUL; REAL_MUL_ASSOC] THEN
8191 (ASM_CASES_TAC `b:real^N = a` THENL
8192 [ASM_REWRITE_TAC[VECTOR_SUB_REFL; REAL_SUB_REFL;
8193 VECTOR_ARITH `(&1 - u) % a + u % a:real^N = a`] THEN
8195 ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
8196 ASM_SIMP_TAC[REAL_ARITH
8197 `&0 <= u /\ u <= &1 ==> abs u = u /\ abs(&1 - u) = &1 - u`] THEN
8200 let CONVEX_ON_LEFT_SECANT,CONVEX_ON_RIGHT_SECANT = (CONJ_PAIR o prove)
8201 (`(!f s:real^N->bool.
8203 !a b x. a IN s /\ b IN s /\ x IN segment(a,b)
8204 ==> (f x - f a) / norm(x - a) <= (f b - f a) / norm(b - a)) /\
8207 !a b x. a IN s /\ b IN s /\ x IN segment(a,b)
8208 ==> (f b - f a) / norm(b - a) <= (f b - f x) / norm(b - x))`,
8209 CONJ_TAC THEN REPEAT GEN_TAC THENL
8210 [REWRITE_TAC[CONVEX_ON_LEFT_SECANT_MUL];
8211 REWRITE_TAC[CONVEX_ON_RIGHT_SECANT_MUL]] THEN
8212 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8213 X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN
8214 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8215 X_GEN_TAC `b:real^N` THEN REWRITE_TAC[] THEN
8216 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8217 ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8218 ASM_CASES_TAC `a:real^N = b` THEN
8219 ASM_REWRITE_TAC[SEGMENT_REFL; NOT_IN_EMPTY; REAL_SUB_REFL; VECTOR_SUB_REFL;
8220 NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LE_REFL] THEN
8221 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8222 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[] THEN
8223 REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN
8224 MAP_EVERY ASM_CASES_TAC [`x:real^N = a`; `x:real^N = b`] THEN
8225 ASM_REWRITE_TAC[REAL_LE_REFL; REAL_SUB_REFL; VECTOR_SUB_REFL; NORM_0;
8226 REAL_MUL_LZERO; REAL_MUL_RZERO] THEN
8227 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM REAL_LE_LDIV_EQ; NORM_POS_LT;
8229 AP_TERM_TAC THEN REAL_ARITH_TAC);;
8231 (* ------------------------------------------------------------------------- *)
8232 (* Starlike sets and more stuff about line segments. *)
8233 (* ------------------------------------------------------------------------- *)
8235 let starlike = new_definition
8236 `starlike s <=> ?a. a IN s /\ !x. x IN s ==> segment[a,x] SUBSET s`;;
8238 let CONVEX_CONTAINS_SEGMENT = prove
8239 (`!s. convex s <=> !a b. a IN s /\ b IN s ==> segment[a,b] SUBSET s`,
8240 REWRITE_TAC[CONVEX_ALT; segment; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8242 let CONVEX_CONTAINS_SEGMENT_EQ = prove
8244 convex s <=> !a b. segment[a,b] SUBSET s <=> a IN s /\ b IN s`,
8245 REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET] THEN
8246 MESON_TAC[ENDS_IN_SEGMENT]);;
8248 let CONVEX_CONTAINS_SEGMENT_IMP = prove
8249 (`!s a b. convex s ==> (segment[a,b] SUBSET s <=> a IN s /\ b IN s)`,
8250 SIMP_TAC[CONVEX_CONTAINS_SEGMENT_EQ]);;
8252 let CONVEX_IMP_STARLIKE = prove
8253 (`!s. convex s /\ ~(s = {}) ==> starlike s`,
8254 REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; starlike; GSYM MEMBER_NOT_EMPTY] THEN
8257 let SEGMENT_CONVEX_HULL = prove
8258 (`!a b. segment[a,b] = convex hull {a,b}`,
8260 SIMP_TAC[CONVEX_HULL_INSERT; CONVEX_HULL_SING; NOT_INSERT_EMPTY] THEN
8261 REWRITE_TAC[IN_SING; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
8262 REWRITE_TAC[segment; EXTENSION; IN_ELIM_THM] THEN
8263 REWRITE_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
8264 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
8265 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> c /\ a /\ b /\ d`] THEN
8266 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN
8267 REWRITE_TAC[REAL_LE_SUB_LADD; REAL_ADD_LID] THEN MESON_TAC[]);;
8269 let SEGMENT_FURTHEST_LE = prove
8271 x IN segment[a,b] ==> norm(y - x) <= norm(y - a) \/
8272 norm(y - x) <= norm(y - b)`,
8273 REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN REPEAT STRIP_TAC THEN
8274 MP_TAC(ISPECL [`y:real^N`; `{a:real^N,b}`] SIMPLEX_FURTHEST_LE) THEN
8275 ASM_REWRITE_TAC[FINITE_INSERT; FINITE_RULES; NOT_INSERT_EMPTY] THEN
8276 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THEN
8277 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
8278 ASM_MESON_TAC[NORM_SUB]);;
8280 let SEGMENT_BOUND = prove
8282 x IN segment[a,b] ==> norm(x - a) <= norm(b - a) /\
8283 norm(x - b) <= norm(b - a)`,
8284 REPEAT STRIP_TAC THEN
8285 MP_TAC(SPECL [`a:real^N`; `b:real^N`; `x:real^N`] SEGMENT_FURTHEST_LE) THENL
8286 [DISCH_THEN(MP_TAC o SPEC `a:real^N`);
8287 DISCH_THEN(MP_TAC o SPEC `b:real^N`)] THEN
8288 REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN
8289 ASM_MESON_TAC[NORM_POS_LE; REAL_LE_TRANS; NORM_SUB]);;
8291 let BETWEEN_IN_CONVEX_HULL = prove
8292 (`!x a b:real^N. between x (a,b) <=> x IN convex hull {a,b}`,
8293 REWRITE_TAC[BETWEEN_IN_SEGMENT; SEGMENT_CONVEX_HULL]);;
8295 let STARLIKE_LINEAR_IMAGE = prove
8296 (`!f s. starlike s /\ linear f ==> starlike(IMAGE f s)`,
8297 REWRITE_TAC[starlike; FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN
8298 SIMP_TAC[CLOSED_SEGMENT_LINEAR_IMAGE] THEN SET_TAC[]);;
8300 let STARLIKE_LINEAR_IMAGE_EQ = prove
8301 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
8302 ==> (starlike (IMAGE f s) <=> starlike s)`,
8303 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE STARLIKE_LINEAR_IMAGE));;
8305 add_linear_invariants [STARLIKE_LINEAR_IMAGE_EQ];;
8307 let STARLIKE_TRANSLATION_EQ = prove
8308 (`!a s. starlike (IMAGE (\x. a + x) s) <=> starlike s`,
8309 REWRITE_TAC[starlike] THEN GEOM_TRANSLATE_TAC[]);;
8311 add_translation_invariants [STARLIKE_TRANSLATION_EQ];;
8313 let BETWEEN_LINEAR_IMAGE_EQ = prove
8314 (`!f x y z. linear f /\ (!x y. f x = f y ==> x = y)
8315 ==> (between (f x) (f y,f z) <=> between x (y,z))`,
8316 SIMP_TAC[BETWEEN_IN_SEGMENT; CLOSED_SEGMENT_LINEAR_IMAGE] THEN SET_TAC[]);;
8318 add_linear_invariants [BETWEEN_LINEAR_IMAGE_EQ];;
8320 let BETWEEN_TRANSLATION = prove
8321 (`!a x y. between (a + x) (a + y,a + z) <=> between x (y,z)`,
8322 REWRITE_TAC[between] THEN NORM_ARITH_TAC);;
8324 add_translation_invariants [STARLIKE_TRANSLATION_EQ];;
8326 let STARLIKE_CLOSURE = prove
8327 (`!s:real^N->bool. starlike s ==> starlike(closure s)`,
8328 GEN_TAC THEN REWRITE_TAC[starlike; SUBSET; segment; FORALL_IN_GSPEC] THEN
8329 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
8330 STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN
8331 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE] THEN
8332 DISCH_TAC THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN
8333 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN
8334 ASM_REWRITE_TAC[] THEN
8335 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
8336 EXISTS_TAC `(&1 - u) % a + u % y:real^N` THEN
8337 ASM_SIMP_TAC[dist; NORM_MUL; VECTOR_ARITH
8338 `(v % a + u % y) - (v % a + u % z):real^N = u % (y - z)`] THEN
8339 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
8340 REAL_LET_TRANS)) THEN
8341 REWRITE_TAC[dist; REAL_ARITH `u * n <= n <=> &0 <= n * (&1 - u)`] THEN
8342 MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
8343 ASM_REAL_ARITH_TAC);;
8345 let STARLIKE_UNIV = prove
8346 (`starlike(:real^N)`,
8347 MESON_TAC[CONVEX_IMP_STARLIKE; CONVEX_UNIV;
8348 BOUNDED_EMPTY; NOT_BOUNDED_UNIV]);;
8350 let STARLIKE_PCROSS = prove
8351 (`!s:real^M->bool t:real^N->bool.
8352 starlike s /\ starlike t ==> starlike(s PCROSS t)`,
8353 SIMP_TAC[starlike; EXISTS_IN_PCROSS; SUBSET; IN_SEGMENT] THEN
8354 REPEAT GEN_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
8355 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
8356 REWRITE_TAC[FORALL_IN_PCROSS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
8357 REWRITE_TAC[FORALL_UNWIND_THM2; IMP_IMP] THEN
8358 REWRITE_TAC[GSYM PASTECART_CMUL; PASTECART_ADD] THEN
8359 REWRITE_TAC[PASTECART_IN_PCROSS] THEN MESON_TAC[]);;
8361 let STARLIKE_PCROSS_EQ = prove
8362 (`!s:real^M->bool t:real^N->bool.
8363 starlike(s PCROSS t) <=> starlike s /\ starlike t`,
8365 ASM_CASES_TAC `s:real^M->bool = {}` THENL
8366 [ASM_REWRITE_TAC[PCROSS_EMPTY] THEN MESON_TAC[starlike; NOT_IN_EMPTY];
8368 ASM_CASES_TAC `t:real^N->bool = {}` THENL
8369 [ASM_REWRITE_TAC[PCROSS_EMPTY] THEN MESON_TAC[starlike; NOT_IN_EMPTY];
8371 EQ_TAC THEN REWRITE_TAC[STARLIKE_PCROSS] THEN REPEAT STRIP_TAC THENL
8372 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
8373 `(s:real^M->bool) PCROSS (t:real^N->bool)`] STARLIKE_LINEAR_IMAGE) THEN
8374 ASM_REWRITE_TAC[LINEAR_FSTCART];
8375 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
8376 `(s:real^M->bool) PCROSS (t:real^N->bool)`] STARLIKE_LINEAR_IMAGE) THEN
8377 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
8378 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
8379 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
8380 FSTCART_PASTECART; SNDCART_PASTECART] THEN
8383 let BETWEEN_DIST_LT = prove
8385 dist(c,a) < r /\ dist(c,b) < r /\ between x (a,b) ==> dist(c,x) < r`,
8386 REPEAT STRIP_TAC THEN
8387 SUBGOAL_THEN `convex hull {a,b} SUBSET ball(c:real^N,r)` MP_TAC THENL
8388 [MATCH_MP_TAC HULL_MINIMAL THEN
8389 ASM_REWRITE_TAC[CONVEX_BALL; INSERT_SUBSET; EMPTY_SUBSET; IN_BALL];
8390 ASM_SIMP_TAC[SUBSET; GSYM BETWEEN_IN_CONVEX_HULL; IN_BALL]]);;
8392 let BETWEEN_DIST_LE = prove
8394 dist(c,a) <= r /\ dist(c,b) <= r /\ between x (a,b) ==> dist(c,x) <= r`,
8396 REPEAT STRIP_TAC THEN
8397 SUBGOAL_THEN `convex hull {a,b} SUBSET cball(c:real^N,r)` MP_TAC THENL
8398 [MATCH_MP_TAC HULL_MINIMAL THEN
8399 ASM_REWRITE_TAC[CONVEX_CBALL; INSERT_SUBSET; EMPTY_SUBSET; IN_CBALL];
8400 ASM_SIMP_TAC[SUBSET; GSYM BETWEEN_IN_CONVEX_HULL; IN_CBALL]]);;
8402 let BETWEEN_NORM_LT = prove
8404 norm a < r /\ norm b < r /\ between x (a,b) ==> norm x < r`,
8405 REWRITE_TAC[GSYM(CONJUNCT2(SPEC_ALL DIST_0)); BETWEEN_DIST_LT]);;
8407 let BETWEEN_NORM_LE = prove
8409 norm a <= r /\ norm b <= r /\ between x (a,b) ==> norm x <= r`,
8410 REWRITE_TAC[GSYM(CONJUNCT2(SPEC_ALL DIST_0)); BETWEEN_DIST_LE]);;
8412 (* ------------------------------------------------------------------------- *)
8413 (* Shrinking towards the interior of a convex set. *)
8414 (* ------------------------------------------------------------------------- *)
8416 let IN_INTERIOR_CONVEX_SHRINK = prove
8418 convex s /\ c IN interior s /\
8419 x IN s /\ &0 < e /\ e <= &1
8420 ==> x - e % (x - c) IN interior s`,
8421 REPEAT STRIP_TAC THEN
8422 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN
8423 REWRITE_TAC[IN_INTERIOR; SUBSET; IN_BALL; dist] THEN
8424 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
8425 EXISTS_TAC `e * d:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
8426 X_GEN_TAC `y':real^N` THEN DISCH_TAC THEN
8427 FIRST_X_ASSUM(MP_TAC o SPEC `(&1 / e) % y' - ((&1 - e) / e) % x:real^N`) THEN
8429 [UNDISCH_TAC `norm (x - e % (x - c) - y':real^N) < e * d` THEN
8430 SUBGOAL_THEN `x - e % (x - c) - y':real^N =
8431 e % (c - (&1 / e % y' - (&1 - e) / e % x))`
8433 [ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC;
8434 REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC;
8435 ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL_EQ; real_abs; REAL_LT_IMP_LE]];
8437 SUBGOAL_THEN `y' = (&1 - (&1 - e)) % (&1 / e % y' - (&1 - e) / e % x) +
8438 (&1 - e) % x:real^N`
8440 [ASM_SIMP_TAC[REAL_ARITH `&1 - (&1 - e) = e`; VECTOR_SUB_LDISTRIB;
8441 VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
8443 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
8444 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);;
8446 let IN_INTERIOR_CLOSURE_CONVEX_SHRINK = prove
8448 convex s /\ c IN interior s /\
8449 x IN closure s /\ &0 < e /\ e <= &1
8450 ==> x - e % (x - c) IN interior s`,
8451 REPEAT STRIP_TAC THEN
8452 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN
8453 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
8454 SUBGOAL_THEN `?y:real^N. y IN s /\ norm(y - x) * (&1 - e) < e * d`
8455 STRIP_ASSUME_TAC THENL
8456 [ASM_CASES_TAC `(x:real^N) IN s` THENL
8457 [EXISTS_TAC `x:real^N` THEN
8458 ASM_SIMP_TAC[REAL_LT_MUL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO];
8460 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [closure]) THEN
8461 ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; LIMPT_APPROACHABLE; dist] THEN
8462 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
8463 `e <= &1 ==> e = &1 \/ e < &1`)) THEN
8464 ASM_SIMP_TAC[REAL_SUB_REFL; GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THENL
8465 [DISCH_THEN(MP_TAC o SPEC `&1`) THEN
8466 REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_01];
8467 DISCH_THEN(MP_TAC o SPEC `(e * d) / (&1 - e)`)] THEN
8468 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_SUB_LT; REAL_MUL_LZERO; REAL_LT_MUL;
8470 MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[];
8472 ABBREV_TAC `z:real^N = c + ((&1 - e) / e) % (x - y)` THEN
8473 SUBGOAL_THEN `x - e % (x - c):real^N = y - e % (y - z)` SUBST1_TAC THENL
8474 [EXPAND_TAC "z" THEN
8475 REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB] THEN
8476 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
8479 MATCH_MP_TAC IN_INTERIOR_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN
8480 FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET] o
8481 MATCH_MP SUBSET_INTERIOR) THEN
8482 SIMP_TAC[INTERIOR_OPEN; OPEN_BALL] THEN
8483 REWRITE_TAC[IN_BALL; dist] THEN EXPAND_TAC "z" THEN
8484 REWRITE_TAC[NORM_ARITH `norm(c - (c + x)) = norm(x)`] THEN
8485 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN
8486 ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
8487 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8488 REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN
8489 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
8490 ASM_MESON_TAC[REAL_MUL_SYM; NORM_SUB]);;
8492 let IN_INTERIOR_CLOSURE_CONVEX_SEGMENT = prove
8494 convex s /\ a IN interior s /\ b IN closure s
8495 ==> segment(a,b) SUBSET interior s`,
8496 REWRITE_TAC[SUBSET; IN_SEGMENT] THEN REPEAT STRIP_TAC THEN
8497 ASM_REWRITE_TAC[VECTOR_ARITH
8498 `(&1 - u) % a + u % b:real^N = b - (&1 - u) % (b - a)`] THEN
8499 MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
8500 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
8502 (* ------------------------------------------------------------------------- *)
8503 (* Relative interior of a set. *)
8504 (* ------------------------------------------------------------------------- *)
8506 let relative_interior = new_definition
8507 `relative_interior s =
8508 {x | ?t. open_in (subtopology euclidean (affine hull s)) t /\
8509 x IN t /\ t SUBSET s}`;;
8511 let relative_frontier = new_definition
8512 `relative_frontier s = closure s DIFF relative_interior s`;;
8514 let RELATIVE_INTERIOR = prove
8515 (`!s. relative_interior s =
8517 ?t. open t /\ x IN t /\ t INTER (affine hull s) SUBSET s}`,
8518 REWRITE_TAC[EXTENSION; relative_interior; IN_ELIM_THM] THEN
8519 REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN
8520 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
8521 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> b /\ a /\ c /\ d`] THEN
8522 REWRITE_TAC[UNWIND_THM2; SUBSET; IN_INTER; RIGHT_AND_EXISTS_THM] THEN
8523 AP_TERM_TAC THEN ABS_TAC THEN MESON_TAC[HULL_INC]);;
8525 let RELATIVE_INTERIOR_EQ = prove
8526 (`!s. relative_interior s = s <=>
8527 open_in(subtopology euclidean (affine hull s)) s`,
8528 GEN_TAC THEN REWRITE_TAC[EXTENSION; relative_interior; IN_ELIM_THM] THEN
8529 GEN_REWRITE_TAC RAND_CONV [OPEN_IN_SUBOPEN] THEN MESON_TAC[SUBSET]);;
8531 let RELATIVE_INTERIOR_OPEN_IN = prove
8532 (`!s. open_in(subtopology euclidean (affine hull s)) s
8533 ==> relative_interior s = s`,
8534 REWRITE_TAC[RELATIVE_INTERIOR_EQ]);;
8536 let RELATIVE_INTERIOR_EMPTY = prove
8537 (`relative_interior {} = {}`,
8538 SIMP_TAC[RELATIVE_INTERIOR_OPEN_IN; OPEN_IN_EMPTY]);;
8540 let RELATIVE_FRONTIER_EMPTY = prove
8541 (`relative_frontier {} = {}`,
8542 REWRITE_TAC[relative_frontier; CLOSURE_EMPTY; EMPTY_DIFF]);;
8544 let RELATIVE_INTERIOR_AFFINE = prove
8545 (`!s:real^N->bool. affine s ==> relative_interior s = s`,
8546 SIMP_TAC[RELATIVE_INTERIOR_EQ; OPEN_IN_SUBTOPOLOGY_REFL; HULL_P] THEN
8547 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);;
8549 let RELATIVE_INTERIOR_UNIV = prove
8550 (`!s. relative_interior(affine hull s) = affine hull s`,
8551 REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_OPEN_IN THEN
8552 REWRITE_TAC[HULL_HULL; OPEN_IN_SUBTOPOLOGY_REFL] THEN
8553 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);;
8555 let OPEN_IN_RELATIVE_INTERIOR = prove
8556 (`!s. open_in (subtopology euclidean (affine hull s))
8557 (relative_interior s)`,
8558 GEN_TAC THEN REWRITE_TAC[relative_interior] THEN
8559 GEN_REWRITE_TAC I [OPEN_IN_SUBOPEN] THEN
8560 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8562 let RELATIVE_INTERIOR_SUBSET = prove
8563 (`!s. (relative_interior s) SUBSET s`,
8564 REWRITE_TAC[SUBSET; relative_interior; IN_ELIM_THM] THEN MESON_TAC[]);;
8566 let SUBSET_RELATIVE_INTERIOR = prove
8567 (`!s t. s SUBSET t /\ affine hull s = affine hull t
8568 ==> (relative_interior s) SUBSET (relative_interior t)`,
8569 REWRITE_TAC[relative_interior; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8571 let RELATIVE_INTERIOR_MAXIMAL = prove
8572 (`!s t. t SUBSET s /\
8573 open_in(subtopology euclidean (affine hull s)) t
8574 ==> t SUBSET (relative_interior s)`,
8575 REWRITE_TAC[relative_interior; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8577 let RELATIVE_INTERIOR_UNIQUE = prove
8578 (`!s t. t SUBSET s /\
8579 open_in(subtopology euclidean (affine hull s)) t /\
8580 (!t'. t' SUBSET s /\
8581 open_in(subtopology euclidean (affine hull s)) t'
8583 ==> (relative_interior s = t)`,
8584 MESON_TAC[SUBSET_ANTISYM; RELATIVE_INTERIOR_MAXIMAL; RELATIVE_INTERIOR_SUBSET;
8585 OPEN_IN_RELATIVE_INTERIOR]);;
8587 let IN_RELATIVE_INTERIOR = prove
8589 x IN relative_interior s <=>
8590 x IN s /\ ?e. &0 < e /\ (ball(x,e) INTER (affine hull s)) SUBSET s`,
8591 REPEAT GEN_TAC THEN REWRITE_TAC[relative_interior; IN_ELIM_THM] THEN
8592 REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN
8593 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
8594 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> b /\ a /\ c /\ d`] THEN
8595 REWRITE_TAC[UNWIND_THM2; SUBSET; IN_INTER] THEN EQ_TAC THENL
8596 [ASM_MESON_TAC[SUBSET; OPEN_CONTAINS_BALL];
8597 STRIP_TAC THEN EXISTS_TAC `ball(x:real^N,e)` THEN
8598 ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL; HULL_INC]]);;
8600 let IN_RELATIVE_INTERIOR_CBALL = prove
8602 x IN relative_interior s <=>
8603 x IN s /\ ?e. &0 < e /\ (cball(x,e) INTER affine hull s) SUBSET s`,
8604 REPEAT GEN_TAC THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN
8605 AP_TERM_TAC THEN EQ_TAC THEN
8606 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THENL
8607 [EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN
8608 MATCH_MP_TAC SUBSET_TRANS THEN
8609 EXISTS_TAC `ball(x:real^N,e) INTER affine hull s` THEN
8610 ASM_REWRITE_TAC[] THEN
8611 REWRITE_TAC[SUBSET; IN_INTER; IN_BALL; IN_CBALL] THEN
8612 ASM_SIMP_TAC[REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`];
8613 EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN
8614 MATCH_MP_TAC SUBSET_TRANS THEN
8615 EXISTS_TAC `cball(x:real^N,e) INTER affine hull s` THEN
8616 ASM_REWRITE_TAC[] THEN
8617 SIMP_TAC[SUBSET; IN_INTER; IN_BALL; IN_CBALL; REAL_LT_IMP_LE]]);;
8619 let OPEN_IN_SUBSET_RELATIVE_INTERIOR = prove
8620 (`!s t. open_in(subtopology euclidean (affine hull t)) s
8621 ==> (s SUBSET relative_interior t <=> s SUBSET t)`,
8622 MESON_TAC[RELATIVE_INTERIOR_MAXIMAL; RELATIVE_INTERIOR_SUBSET;
8625 let RELATIVE_INTERIOR_TRANSLATION = prove
8627 relative_interior (IMAGE (\x. a + x) s) =
8628 IMAGE (\x. a + x) (relative_interior s)`,
8629 REWRITE_TAC[relative_interior; OPEN_IN_OPEN] THEN GEOM_TRANSLATE_TAC[]);;
8631 add_translation_invariants [RELATIVE_INTERIOR_TRANSLATION];;
8633 let RELATIVE_FRONTIER_TRANSLATION = prove
8635 relative_frontier (IMAGE (\x. a + x) s) =
8636 IMAGE (\x. a + x) (relative_frontier s)`,
8637 REWRITE_TAC[relative_frontier] THEN GEOM_TRANSLATE_TAC[]);;
8639 add_translation_invariants [RELATIVE_FRONTIER_TRANSLATION];;
8641 let RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE = prove
8642 (`!f:real^M->real^N s.
8643 linear f /\ (!x y. f x = f y ==> x = y)
8644 ==> relative_interior(IMAGE f s) = IMAGE f (relative_interior s)`,
8645 REPEAT GEN_TAC THEN DISCH_TAC THEN
8646 ASM_SIMP_TAC[relative_interior; AFFINE_HULL_LINEAR_IMAGE] THEN
8647 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> c /\ a /\ b`] THEN
8648 REWRITE_TAC[EXISTS_SUBSET_IMAGE] THEN
8649 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_INJECTIVE_LINEAR_IMAGE) THEN
8650 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
8652 add_linear_invariants [RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE];;
8654 let RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE = prove
8655 (`!f:real^M->real^N s.
8656 linear f /\ (!x y. f x = f y ==> x = y)
8657 ==> relative_frontier(IMAGE f s) = IMAGE f (relative_frontier s)`,
8658 REWRITE_TAC[relative_frontier] THEN GEOM_TRANSFORM_TAC[]);;
8660 add_linear_invariants [RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE];;
8662 let RELATIVE_INTERIOR_EQ_EMPTY = prove
8664 convex s ==> (relative_interior s = {} <=> s = {})`,
8667 vec 0 IN s /\ convex s ==> ~(relative_interior s = {})`
8670 GEN_TAC THEN DISCH_TAC THEN
8671 ASM_CASES_TAC `s:real^N->bool = {}` THEN
8672 ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY] THEN
8673 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
8674 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
8675 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\x:real^N. --a + x) s`) THEN
8676 REWRITE_TAC[CONVEX_TRANSLATION_EQ; RELATIVE_INTERIOR_TRANSLATION] THEN
8677 ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; IN_IMAGE] THEN
8678 DISCH_THEN MATCH_MP_TAC THEN EXISTS_TAC `a:real^N` THEN
8679 ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC] THEN
8680 GEN_TAC THEN STRIP_TAC THEN
8681 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_RELATIVE_INTERIOR] THEN
8682 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
8683 X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC
8684 (ISPEC `s:real^N->bool` BASIS_EXISTS) THEN
8685 SUBGOAL_THEN `span(s:real^N->bool) = span b` SUBST_ALL_TAC THENL
8686 [ASM_SIMP_TAC[SPAN_EQ] THEN ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS];
8688 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
8689 ABBREV_TAC `n = dim(s:real^N->bool)` THEN
8691 `!c. (!v. v IN b ==> &0 <= c(v)) /\ sum b c <= &1
8692 ==> vsum b (\v:real^N. c(v) % v) IN s`
8694 [REPEAT STRIP_TAC THEN SUBGOAL_THEN
8695 `vsum (vec 0 INSERT b :real^N->bool)
8696 (\v. (if v = vec 0 then &1 - sum b c else c v) % v) IN s`
8698 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
8699 ASM_SIMP_TAC[INSERT_SUBSET; FINITE_INSERT; SUM_CLAUSES;
8700 INDEPENDENT_NONZERO; IN_INSERT] THEN
8701 CONJ_TAC THENL [ASM_MESON_TAC[REAL_SUB_LE]; ALL_TAC] THEN
8702 REWRITE_TAC[REAL_ARITH `&1 - x + y = &1 <=> x = y`] THEN
8703 MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[INDEPENDENT_NONZERO];
8704 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
8705 ASM_SIMP_TAC[VSUM_CLAUSES; INDEPENDENT_NONZERO] THEN
8706 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
8707 MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[INDEPENDENT_NONZERO]];
8709 ABBREV_TAC `a:real^N = vsum b (\v. inv(&2 * &n + &1) % v)` THEN
8710 EXISTS_TAC `a:real^N` THEN CONJ_TAC THENL
8711 [EXPAND_TAC "a" THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
8712 ASM_SIMP_TAC[SUM_CONST; REAL_LE_INV_EQ; REAL_ARITH `&0 < &2 * &n + &1`;
8713 GSYM real_div; REAL_LT_IMP_LE; REAL_LE_LDIV_EQ] THEN
8716 MP_TAC(ISPECL [`b:real^N->bool`; `inv(&2 * &n + &1)`]
8717 BASIS_COORDINATES_CONTINUOUS) THEN
8718 ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
8719 ANTS_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
8720 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
8721 ASM_SIMP_TAC[SUBSET; IN_INTER; IMP_CONJ_ALT] THEN
8722 ASM_SIMP_TAC[SPAN_FINITE; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
8723 GEN_TAC THEN X_GEN_TAC `u:real^N->real` THEN
8724 DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[IN_BALL; dist] THEN
8725 EXPAND_TAC "a" THEN ASM_SIMP_TAC[GSYM VSUM_SUB] THEN
8726 DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN
8727 REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB] THEN
8728 DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o C MATCH_MP th)) THEN
8729 REWRITE_TAC[REAL_ARITH `abs(x - y) < x <=> &0 < y /\ abs(y) < &2 * x`] THEN
8730 SIMP_TAC[REAL_LT_IMP_LE] THEN DISCH_TAC THEN
8731 MATCH_MP_TAC REAL_LE_TRANS THEN
8732 EXISTS_TAC `&(CARD(b:real^N->bool)) * &2 * inv(&2 * &n + &1)` THEN
8734 [MATCH_MP_TAC SUM_BOUND THEN
8735 ASM_SIMP_TAC[REAL_ARITH `abs x < a ==> x <= a`];
8736 ASM_REWRITE_TAC[REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN
8737 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &2 * &n + &1`] THEN
8740 let RELATIVE_INTERIOR_INTERIOR = prove
8741 (`!s. affine hull s = (:real^N)
8742 ==> relative_interior s = interior s`,
8743 SIMP_TAC[relative_interior; interior; SUBTOPOLOGY_UNIV; OPEN_IN]);;
8745 let RELATIVE_INTERIOR_OPEN = prove
8746 (`!s:real^N->bool. open s ==> relative_interior s = s`,
8747 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
8748 ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY] THEN
8749 ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_OPEN; INTERIOR_EQ]);;
8751 let RELATIVE_INTERIOR_NONEMPTY_INTERIOR = prove
8752 (`!s. ~(interior s = {}) ==> relative_interior s = interior s`,
8753 MESON_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_NONEMPTY_INTERIOR]);;
8755 let RELATIVE_FRONTIER_NONEMPTY_INTERIOR = prove
8756 (`!s. ~(interior s = {}) ==> relative_frontier s = frontier s`,
8757 SIMP_TAC[relative_frontier; frontier; RELATIVE_INTERIOR_NONEMPTY_INTERIOR]);;
8759 let RELATIVE_FRONTIER_FRONTIER = prove
8760 (`!s. affine hull s = (:real^N) ==> relative_frontier s = frontier s`,
8761 SIMP_TAC[relative_frontier; frontier; RELATIVE_INTERIOR_INTERIOR]);;
8763 let AFFINE_HULL_CONVEX_HULL = prove
8764 (`!s. affine hull (convex hull s) = affine hull s`,
8765 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN
8766 REWRITE_TAC[AFFINE_AFFINE_HULL; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
8767 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN
8768 ASM_MESON_TAC[SUBSET_TRANS; HULL_SUBSET]);;
8770 let INTERIOR_SIMPLEX_NONEMPTY = prove
8772 independent s /\ s HAS_SIZE (dimindex(:N))
8773 ==> ?a. a IN interior(convex hull (vec 0 INSERT s))`,
8774 REPEAT STRIP_TAC THEN
8775 MP_TAC(ISPEC `convex hull (vec 0 INSERT s):real^N->bool`
8776 RELATIVE_INTERIOR_EQ_EMPTY) THEN
8777 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_HULL] THEN
8778 REWRITE_TAC[CONVEX_HULL_EQ_EMPTY; CONVEX_CONVEX_HULL; NOT_INSERT_EMPTY] THEN
8779 REWRITE_TAC[MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
8780 AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
8781 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; HULL_INC] THEN
8782 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
8783 EXISTS_TAC `span s:real^N->bool` THEN CONJ_TAC THENL
8784 [MATCH_MP_TAC SPAN_MONO THEN MATCH_MP_TAC(SET_RULE
8785 `(a INSERT s) SUBSET P hull (a INSERT s)
8786 ==> s SUBSET P hull (a INSERT s)`) THEN REWRITE_TAC[HULL_SUBSET];
8787 MATCH_MP_TAC(SET_RULE `UNIV SUBSET s ==> s = UNIV`) THEN
8788 MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
8789 ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV] THEN
8790 ASM_MESON_TAC[LE_REFL;HAS_SIZE]]);;
8792 let INTERIOR_SUBSET_RELATIVE_INTERIOR = prove
8793 (`!s. interior s SUBSET relative_interior s`,
8794 REWRITE_TAC[SUBSET; IN_INTERIOR; IN_RELATIVE_INTERIOR; IN_INTER] THEN
8795 MESON_TAC[CENTRE_IN_BALL]);;
8797 let CONVEX_RELATIVE_INTERIOR = prove
8798 (`!s:real^N->bool. convex s ==> convex(relative_interior s)`,
8799 REWRITE_TAC[CONVEX_ALT; IN_RELATIVE_INTERIOR; IN_INTER;
8800 SUBSET; IN_BALL; dist] THEN
8801 GEN_TAC THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
8802 REWRITE_TAC[TAUT `(a /\ b) /\ (c /\ d) /\ e ==> f <=>
8803 a /\ c /\ e ==> b /\ d ==> f`] THEN
8804 STRIP_TAC THEN ASM_SIMP_TAC[] THEN
8805 MATCH_MP_TAC(MESON[] `(!d e. P d /\ Q e ==> R(min d e))
8806 ==> (?e. P e) /\ (?e. Q e) ==> (?e. R e)`) THEN
8807 REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
8808 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
8809 SUBST1_TAC(VECTOR_ARITH `z:real^N =
8810 (&1 - u) % (z - u % (y - x)) + u % (z + (&1 - u) % (y - x))`) THEN
8811 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
8812 FIRST_X_ASSUM(CONJUNCTS_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
8813 REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN MATCH_MP_TAC MONO_AND THEN
8814 CONJ_TAC THEN DISCH_THEN MATCH_MP_TAC THEN
8816 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
8817 `norm x < e ==> norm x = y ==> y < e`)) THEN
8818 AP_TERM_TAC THEN VECTOR_ARITH_TAC;
8819 REWRITE_TAC[VECTOR_ARITH `a - b % c:real^N = a + --b % c`] THEN
8820 MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
8821 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]]));;
8823 let IN_RELATIVE_INTERIOR_CONVEX_SHRINK = prove
8825 convex s /\ c IN relative_interior s /\
8826 x IN s /\ &0 < e /\ e <= &1
8827 ==> x - e % (x - c) IN relative_interior s`,
8828 REPEAT STRIP_TAC THEN
8829 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN
8830 REWRITE_TAC[IN_RELATIVE_INTERIOR; SUBSET; IN_INTER; IN_BALL; dist] THEN
8831 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
8832 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL
8833 [REWRITE_TAC[VECTOR_ARITH
8834 `x - e % (x - c):real^N = (&1 - e) % x + e % c`] THEN
8835 FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
8836 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
8838 EXISTS_TAC `e * d:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
8839 X_GEN_TAC `y':real^N` THEN STRIP_TAC THEN
8840 FIRST_X_ASSUM(MP_TAC o SPEC `(&1 / e) % y' - ((&1 - e) / e) % x:real^N`) THEN
8843 [UNDISCH_TAC `norm (x - e % (x - c) - y':real^N) < e * d` THEN
8844 SUBGOAL_THEN `x - e % (x - c) - y':real^N =
8845 e % (c - (&1 / e % y' - (&1 - e) / e % x))`
8847 [ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC;
8848 REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC;
8849 ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL_EQ; real_abs; REAL_LT_IMP_LE]];
8850 REWRITE_TAC[real_div; REAL_SUB_RDISTRIB] THEN
8851 ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN
8852 REWRITE_TAC[VECTOR_ARITH `a % y - (b - c) % x:real^N =
8853 (c - b) % x + a % y`] THEN
8854 MATCH_MP_TAC(REWRITE_RULE[AFFINE_ALT] AFFINE_AFFINE_HULL) THEN
8855 ASM_SIMP_TAC[HULL_INC]];
8857 SUBGOAL_THEN `y' = (&1 - (&1 - e)) % (&1 / e % y' - (&1 - e) / e % x) +
8858 (&1 - e) % x:real^N`
8860 [ASM_SIMP_TAC[REAL_ARITH `&1 - (&1 - e) = e`; VECTOR_SUB_LDISTRIB;
8861 VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
8863 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
8864 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);;
8866 let IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK = prove
8868 convex s /\ c IN relative_interior s /\
8869 x IN closure s /\ &0 < e /\ e <= &1
8870 ==> x - e % (x - c) IN relative_interior s`,
8871 REPEAT STRIP_TAC THEN
8872 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN
8873 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
8874 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
8875 SUBGOAL_THEN `?y:real^N. y IN s /\ norm(y - x) * (&1 - e) < e * d`
8876 STRIP_ASSUME_TAC THENL
8877 [ASM_CASES_TAC `(x:real^N) IN s` THENL
8878 [EXISTS_TAC `x:real^N` THEN
8879 ASM_SIMP_TAC[REAL_LT_MUL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO];
8881 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [closure]) THEN
8882 ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; LIMPT_APPROACHABLE; dist] THEN
8883 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
8884 `e <= &1 ==> e = &1 \/ e < &1`)) THEN
8885 ASM_SIMP_TAC[REAL_SUB_REFL; GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THENL
8886 [DISCH_THEN(MP_TAC o SPEC `&1`) THEN
8887 REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_01];
8888 DISCH_THEN(MP_TAC o SPEC `(e * d) / (&1 - e)`)] THEN
8889 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_SUB_LT; REAL_MUL_LZERO; REAL_LT_MUL;
8891 MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[];
8893 ABBREV_TAC `z:real^N = c + ((&1 - e) / e) % (x - y)` THEN
8894 SUBGOAL_THEN `x - e % (x - c):real^N = y - e % (y - z)` SUBST1_TAC THENL
8895 [EXPAND_TAC "z" THEN
8896 REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB] THEN
8897 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
8900 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN
8901 SUBGOAL_THEN `dist(c:real^N,z) < d` ASSUME_TAC THENL
8902 [EXPAND_TAC "z" THEN
8903 REWRITE_TAC[NORM_ARITH `dist(c:real^N,c + x) = norm x`] THEN
8904 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN
8905 REWRITE_TAC[REAL_ARITH `a / b * c:real = (c * a) / b`] THEN
8906 ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_LT_LDIV_EQ] THEN
8909 SUBGOAL_THEN `(z:real^N) IN affine hull s` ASSUME_TAC THENL
8910 [EXPAND_TAC "z" THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
8911 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC] THEN
8912 MATCH_MP_TAC(SET_RULE `!t. x IN t /\ t = s ==> x IN s`) THEN
8913 EXISTS_TAC `closure(affine hull s):real^N->bool` THEN
8914 SIMP_TAC[CLOSURE_EQ; CLOSED_AFFINE_HULL] THEN
8915 ASM_MESON_TAC[SUBSET_CLOSURE; HULL_INC; SUBSET];
8917 ASM_REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN CONJ_TAC THENL
8918 [ASM_MESON_TAC[IN_BALL; IN_INTER; SUBSET]; ALL_TAC] THEN
8919 EXISTS_TAC `d - dist(c:real^N,z)` THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN
8920 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
8921 (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN
8922 REWRITE_TAC[SUBSET; IN_INTER] THEN GEN_TAC THEN
8923 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
8924 UNDISCH_TAC `dist(c:real^N,z) < d` THEN REWRITE_TAC[IN_BALL] THEN
8927 let IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT = prove
8929 convex s /\ a IN relative_interior s /\ b IN closure s
8930 ==> segment(a,b) SUBSET relative_interior s`,
8931 REWRITE_TAC[SUBSET; IN_SEGMENT] THEN REPEAT STRIP_TAC THEN
8932 ASM_REWRITE_TAC[VECTOR_ARITH
8933 `(&1 - u) % a + u % b:real^N = b - (&1 - u) % (b - a)`] THEN
8934 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
8935 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
8937 let RELATIVE_INTERIOR_SING = prove
8938 (`!a. relative_interior {a} = {a}`,
8939 GEN_TAC THEN MATCH_MP_TAC(SET_RULE
8940 `s SUBSET {a} /\ ~(s = {}) ==> s = {a}`) THEN
8941 SIMP_TAC[RELATIVE_INTERIOR_SUBSET; RELATIVE_INTERIOR_EQ_EMPTY;
8945 let RELATIVE_FRONTIER_SING = prove
8946 (`!a:real^N. relative_frontier {a} = {}`,
8947 REWRITE_TAC[relative_frontier; RELATIVE_INTERIOR_SING; CLOSURE_SING] THEN
8950 let RELATIVE_FRONTIER_CBALL = prove
8951 (`!a:real^N r. relative_frontier(cball(a,r)) =
8952 if r = &0 then {} else sphere(a,r)`,
8953 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
8954 ASM_SIMP_TAC[CBALL_SING; RELATIVE_FRONTIER_SING] THEN
8955 ASM_CASES_TAC `r < &0` THEN
8956 ASM_SIMP_TAC[CBALL_EMPTY; SPHERE_EMPTY; RELATIVE_FRONTIER_EMPTY] THEN
8957 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
8958 ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; INTERIOR_CBALL;
8959 BALL_EQ_EMPTY; GSYM REAL_NOT_LT; FRONTIER_CBALL]);;
8961 let RELATIVE_FRONTIER_BALL = prove
8962 (`!a:real^N r. relative_frontier(ball(a,r)) =
8963 if r = &0 then {} else sphere(a,r)`,
8964 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
8965 ASM_SIMP_TAC[BALL_EMPTY; REAL_LE_REFL; RELATIVE_FRONTIER_EMPTY] THEN
8966 ASM_CASES_TAC `r < &0` THEN
8967 ASM_SIMP_TAC[BALL_EMPTY; REAL_LT_IMP_LE; SPHERE_EMPTY;
8968 RELATIVE_FRONTIER_EMPTY] THEN
8969 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
8970 ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; INTERIOR_OPEN; OPEN_BALL;
8971 BALL_EQ_EMPTY; GSYM REAL_NOT_LT; FRONTIER_BALL]);;
8973 let STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS = prove
8974 (`!s t:real^N->bool.
8975 convex s /\ ~(s = {}) /\
8976 relative_interior s SUBSET t /\ t SUBSET closure s
8978 REPEAT STRIP_TAC THEN
8979 SUBGOAL_THEN `~(relative_interior s:real^N->bool = {})` MP_TAC THENL
8980 [ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; REWRITE_TAC[starlike]] THEN
8981 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN
8982 X_GEN_TAC `a:real^N` THEN
8983 REPEAT STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
8984 MATCH_MP_TAC(SET_RULE
8985 `a IN s /\ b IN s /\ segment[a,b] DIFF {a,b} SUBSET s
8986 ==> segment[a:real^N,b] SUBSET s`) THEN
8987 ASM_REWRITE_TAC[GSYM open_segment] THEN
8988 ASM_MESON_TAC[IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT; SUBSET]);;
8990 let RELATIVE_INTERIOR_PROLONG = prove
8992 x IN relative_interior s /\ y IN s
8993 ==> ?t. &1 < t /\ (y + t % (x - y)) IN s`,
8995 REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL; IN_ELIM_THM] THEN
8996 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
8997 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real`
8998 STRIP_ASSUME_TAC)) THEN
8999 ASM_CASES_TAC `y:real^N = x` THENL
9000 [ASM_REWRITE_TAC[VECTOR_ARITH `y + t % (x - x):real^N = y`] THEN
9001 EXISTS_TAC `&2` THEN CONV_TAC REAL_RAT_REDUCE_CONV;
9002 EXISTS_TAC `&1 + e / norm(x - y:real^N)` THEN
9003 ASM_SIMP_TAC[REAL_LT_ADDR; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ] THEN
9004 REWRITE_TAC[VECTOR_ARITH
9005 `y + (&1 + e) % (x - y):real^N = x + e % (x - y)`] THEN
9006 FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
9007 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; IN_INTER; IN_AFFINE_ADD_MUL_DIFF;
9008 HULL_INC; IN_CBALL] THEN
9009 REWRITE_TAC[NORM_ARITH `dist(x:real^N,x + y) = norm y`] THEN
9010 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
9011 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
9012 ASM_REAL_ARITH_TAC]);;
9014 let RELATIVE_INTERIOR_CONVEX_PROLONG = prove
9016 ==> relative_interior s =
9017 {x:real^N | x IN s /\
9018 !y. y IN s ==> ?t. &1 < t /\ (y + t % (x - y)) IN s}`,
9019 REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
9020 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
9021 [SIMP_TAC[RELATIVE_INTERIOR_PROLONG] THEN
9022 MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET];
9024 SUBGOAL_THEN `?y:real^N. y IN relative_interior s` STRIP_ASSUME_TAC THENL
9025 [ASM_SIMP_TAC[MEMBER_NOT_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY] THEN
9028 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ANTS_TAC THENL
9029 [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]; ALL_TAC] THEN
9030 ASM_CASES_TAC `y:real^N = x` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
9031 DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN
9032 MP_TAC(ISPECL [`s:real^N->bool`; `y:real^N`; `y + t % (x - y):real^N`]
9033 IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN
9034 ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; ALL_TAC] THEN
9035 REWRITE_TAC[SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN
9036 REWRITE_TAC[IN_SEGMENT; IN_ELIM_THM] THEN
9037 ASM_REWRITE_TAC[VECTOR_ARITH `y:real^N = y + x <=> x = vec 0`;
9038 VECTOR_ARITH `(&1 - u) % y + u % (y + t % (x - y)):real^N =
9039 y + t % u % (x - y)`] THEN
9040 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
9041 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
9042 EXISTS_TAC `inv t:real` THEN
9043 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_INV_EQ;
9044 REAL_INV_LT_1; REAL_LT_IMP_NZ; REAL_ARITH `&1 < x ==> &0 < x`] THEN
9045 VECTOR_ARITH_TAC]);;
9047 let RELATIVE_INTERIOR_EQ_CLOSURE = prove
9049 relative_interior s = closure s <=> affine s`,
9050 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9051 ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY; CLOSURE_EMPTY; AFFINE_EMPTY] THEN
9053 SIMP_TAC[RELATIVE_INTERIOR_AFFINE; CLOSURE_CLOSED; CLOSED_AFFINE] THEN
9054 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
9055 `relative_interior s = closure s
9056 ==> relative_interior s SUBSET s /\ s SUBSET closure s
9057 ==> relative_interior s = s /\ closure s = s`)) THEN
9058 REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET] THEN
9059 REWRITE_TAC[RELATIVE_INTERIOR_EQ; CLOSURE_EQ; GSYM AFFINE_HULL_EQ] THEN
9060 STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
9061 `~(s = {}) ==> s = {} \/ s = a ==> a = s`)) THEN
9062 MP_TAC(ISPEC `affine hull s:real^N->bool` CONNECTED_CLOPEN) THEN
9063 SIMP_TAC[AFFINE_IMP_CONVEX; CONVEX_CONNECTED; AFFINE_AFFINE_HULL] THEN
9064 DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
9065 MATCH_MP_TAC CLOSED_SUBSET THEN ASM_REWRITE_TAC[HULL_SUBSET]);;
9067 let RAY_TO_RELATIVE_FRONTIER = prove
9069 bounded s /\ a IN relative_interior s /\
9070 (a + l) IN affine hull s /\ ~(l = vec 0)
9072 (a + d % l) IN relative_frontier s /\
9073 !e. &0 <= e /\ e < d ==> (a + e % l) IN relative_interior s`,
9074 REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier] THEN
9076 `{d | &0 < d /\ ~((a + d % l:real^N) IN relative_interior(s))}` INF) THEN
9078 `d = inf {d | &0 < d /\ ~((a + d % l:real^N) IN relative_interior(s))}` THEN
9080 `?e. &0 < e /\ !d. &0 <= d /\ d < e
9081 ==> (a + d % l:real^N) IN relative_interior s`
9082 (X_CHOOSE_THEN `k:real` (LABEL_TAC "0"))
9084 [MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN
9085 REWRITE_TAC[open_in; GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
9086 DISCH_THEN(MP_TAC o SPEC `a:real^N` o CONJUNCT2) THEN
9087 ASM_REWRITE_TAC[] THEN
9088 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9089 EXISTS_TAC `e / norm(l:real^N)` THEN
9090 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT] THEN X_GEN_TAC `x:real` THEN
9091 STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
9092 [MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
9093 ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN
9094 ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET];
9095 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a) = norm x`] THEN
9096 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN
9097 ASM_REAL_ARITH_TAC];
9100 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
9101 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE]] THEN
9102 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N` o
9103 MATCH_MP BOUNDED_SUBSET_BALL) THEN
9104 REWRITE_TAC[SUBSET; IN_BALL] THEN
9105 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
9106 EXISTS_TAC `B / norm(l:real^N)` THEN
9107 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT] THEN
9108 DISCH_THEN(MP_TAC o MATCH_MP
9109 (REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET)) THEN
9110 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV
9111 [GSYM CONTRAPOS_THM]) THEN
9112 REWRITE_TAC[REAL_NOT_LT] THEN DISCH_THEN MATCH_MP_TAC THEN
9113 REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN
9114 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM;
9115 REAL_DIV_RMUL; NORM_EQ_0] THEN
9117 REWRITE_TAC[IN_ELIM_THM] THEN
9118 DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "1") (LABEL_TAC "2")) THEN
9119 EXISTS_TAC `d:real` THEN
9120 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
9121 [MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `k:real` THEN
9122 ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE];
9124 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
9125 [REWRITE_TAC[REAL_LE_LT] THEN
9126 ASM_MESON_TAC[VECTOR_ARITH `a + &0 % l:real^N = a`;
9127 REAL_NOT_LT; REAL_LT_IMP_LE];
9129 REWRITE_TAC[IN_DIFF] THEN CONJ_TAC THENL
9130 [REWRITE_TAC[CLOSURE_APPROACHABLE] THEN
9131 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
9132 EXISTS_TAC `a + (d - min d (x / &2 / norm(l:real^N))) % l` THEN
9134 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET) THEN
9135 FIRST_X_ASSUM MATCH_MP_TAC THEN
9136 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
9137 MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < d ==> d - min d x < d`) THEN
9138 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT];
9139 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = norm(x - y)`] THEN
9140 REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN
9141 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN
9142 MATCH_MP_TAC(REAL_ARITH
9143 `&0 < x /\ x < y /\ &0 < d ==> abs((d - min d x) - d) < y`) THEN
9144 REWRITE_TAC[REAL_ARITH `x / &2 / y < x / y <=> &0 < x / y`] THEN
9145 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT]];
9147 MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN
9148 REWRITE_TAC[open_in; GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
9149 DISCH_THEN(MP_TAC o SPEC `a + d % l:real^N` o CONJUNCT2) THEN
9150 ASM_REWRITE_TAC[] THEN
9151 DISCH_THEN(X_CHOOSE_THEN `e:real`
9152 (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "3"))) THEN
9153 REMOVE_THEN "2" (MP_TAC o SPEC `d + e / norm(l:real^N)`) THEN
9154 ASM_SIMP_TAC[NOT_IMP; REAL_ARITH `~(d + l <= d) <=> &0 < l`;
9155 REAL_LT_DIV; NORM_POS_LT] THEN
9156 X_GEN_TAC `x:real` THEN
9157 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9158 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
9159 REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN
9160 ASM_CASES_TAC `x < d` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
9161 REMOVE_THEN "3" MATCH_MP_TAC THEN CONJ_TAC THENL
9162 [MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
9163 ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN
9164 ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET];
9165 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = norm(x - y)`] THEN
9166 REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN
9167 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN
9168 ASM_REAL_ARITH_TAC]]]);;
9170 let RAY_TO_FRONTIER = prove
9172 bounded s /\ a IN interior s /\ ~(l = vec 0)
9173 ==> ?d. &0 < d /\ (a + d % l) IN frontier s /\
9174 !e. &0 <= e /\ e < d ==> (a + e % l) IN interior s`,
9175 REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN
9176 SUBGOAL_THEN `interior s:real^N->bool = relative_interior s` SUBST1_TAC THENL
9178 REWRITE_TAC[GSYM relative_frontier] THEN
9179 MATCH_MP_TAC RAY_TO_RELATIVE_FRONTIER THEN ASM_REWRITE_TAC[]] THEN
9180 ASM_MESON_TAC[NOT_IN_EMPTY; RELATIVE_INTERIOR_NONEMPTY_INTERIOR; IN_UNIV;
9181 AFFINE_HULL_NONEMPTY_INTERIOR]);;
9183 let RELATIVE_FRONTIER_NOT_SING = prove
9184 (`!s a:real^N. bounded s ==> ~(relative_frontier s = {a})`,
9185 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9186 ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY; NOT_INSERT_EMPTY] THEN
9187 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9188 DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN
9189 ASM_CASES_TAC `s = {z:real^N}` THEN
9190 ASM_REWRITE_TAC[RELATIVE_FRONTIER_SING; NOT_INSERT_EMPTY] THEN
9191 SUBGOAL_THEN `?w:real^N. w IN s /\ ~(w = z)` STRIP_ASSUME_TAC THENL
9192 [ASM SET_TAC[]; REPEAT STRIP_TAC] THEN
9194 `~((w:real^N) IN relative_frontier s /\ z IN relative_frontier s)`
9195 MP_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN
9196 MAP_EVERY UNDISCH_TAC
9197 [`relative_frontier s = {a:real^N}`; `bounded(s:real^N->bool)`;
9198 `~(w:real^N = z)`; `(z:real^N) IN s`; `(w:real^N) IN s`;
9199 `~((w:real^N) IN relative_frontier s /\ z IN relative_frontier s)`] THEN
9200 POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[DE_MORGAN_THM] THEN
9201 MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`z:real^N`; `w:real^N`] THEN
9202 MATCH_MP_TAC(MESON[]
9203 `(!w z. Q w z <=> Q z w) /\ (!w z. P z ==> Q w z)
9204 ==> !w z. P w \/ P z ==> Q w z`) THEN
9205 CONJ_TAC THENL [MESON_TAC[]; REPEAT GEN_TAC] THEN
9206 DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN
9207 REWRITE_TAC[relative_frontier; IN_DIFF] THEN
9208 CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; DISCH_TAC] THEN
9210 (ISPECL [`s:real^N->bool`; `z:real^N`; `d % (w - z):real^N`]
9211 RAY_TO_RELATIVE_FRONTIER)) THEN
9212 ASM_SIMP_TAC[VECTOR_SUB_EQ; IN_AFFINE_ADD_MUL_DIFF; AFFINE_AFFINE_HULL;
9213 HULL_INC; VECTOR_MUL_EQ_0] THEN
9214 DISCH_THEN(fun th -> MP_TAC(SPEC `&1` th) THEN MP_TAC(SPEC `--(&1)` th)) THEN
9215 CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[IN_SING] THEN
9216 DISCH_THEN(X_CHOOSE_THEN `e:real` (STRIP_ASSUME_TAC o GSYM)) THEN
9217 ASM_REWRITE_TAC[VECTOR_MUL_RCANCEL; VECTOR_MUL_ASSOC; VECTOR_SUB_EQ;
9218 VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
9219 ASM_REAL_ARITH_TAC);;
9221 let RELATIVE_INTERIOR_PCROSS = prove
9222 (`!s:real^M->bool t:real^N->bool.
9223 relative_interior(s PCROSS t) =
9224 relative_interior s PCROSS relative_interior t`,
9225 REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC
9226 [`s:real^M->bool = {}`; `t:real^N->bool = {}`] THEN
9227 ASM_REWRITE_TAC[PCROSS_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN
9228 REWRITE_TAC[relative_interior; AFFINE_HULL_PCROSS] THEN
9229 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM;
9230 PASTECART_IN_PCROSS] THEN
9231 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN EQ_TAC THENL
9232 [ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ q /\ p`] THEN
9233 DISCH_THEN(X_CHOOSE_THEN `u:real^(M,N)finite_sum->bool`
9234 (CONJUNCTS_THEN ASSUME_TAC)) THEN
9235 FIRST_ASSUM(MP_TAC o MATCH_MP PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN
9236 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
9237 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
9238 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
9239 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
9240 W(MP_TAC o PART_MATCH (funpow 3 rand) SUBSET_PCROSS o snd) THEN
9242 DISCH_THEN(CONJUNCTS_THEN2
9243 (X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC)
9244 (X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC)) THEN
9245 EXISTS_TAC `(v:real^M->bool) PCROSS (w:real^N->bool)` THEN
9246 ASM_SIMP_TAC[PASTECART_IN_PCROSS; SUBSET_PCROSS; OPEN_IN_PCROSS]]);;
9248 let RELATIVE_FRONTIER_EQ_EMPTY = prove
9249 (`!s:real^N->bool. relative_frontier s = {} <=> affine s`,
9250 GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN
9251 REWRITE_TAC[GSYM RELATIVE_INTERIOR_EQ_CLOSURE] THEN
9252 MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN
9253 MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]);;
9255 let DIAMETER_BOUNDED_BOUND_LT = prove
9257 bounded s /\ x IN relative_interior s /\ y IN closure s /\
9259 ==> norm(x - y) < diameter s`,
9262 bounded s /\ x IN relative_interior s /\ y IN s /\
9264 ==> norm(x - y) < diameter s`,
9265 REPEAT STRIP_TAC THEN FIRST_X_ASSUM
9266 (MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN
9267 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real`
9268 STRIP_ASSUME_TAC)) THEN
9269 ASM_SIMP_TAC[REAL_LT_LE; DIAMETER_BOUNDED_BOUND] THEN
9270 ASM_CASES_TAC `y:real^N = x` THEN
9271 ASM_SIMP_TAC[VECTOR_SUB_REFL; NORM_0] THEN
9273 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
9274 DISCH_THEN(MP_TAC o SPEC `x + e / norm(x - y) % (x - y):real^N`) THEN
9275 REWRITE_TAC[NOT_IMP; IN_INTER] THEN REPEAT CONJ_TAC THENL
9276 [REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(x:real^M,x + y) = norm y`] THEN
9277 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL;
9278 NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC;
9279 MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
9280 ASM_SIMP_TAC[HULL_INC; AFFINE_AFFINE_HULL];
9281 DISCH_TAC THEN MP_TAC(ISPECL
9282 [`s:real^N->bool`; `x + e / norm(x - y) % (x - y):real^N`; `y:real^N`]
9283 DIAMETER_BOUNDED_BOUND) THEN
9284 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
9285 REWRITE_TAC[VECTOR_ARITH
9286 `(x + e % (x - y)) - y:real^N = (&1 + e) % (x - y)`] THEN
9287 SIMP_TAC[NORM_MUL; REAL_ARITH `~(a * n <= n) <=> &0 < n * (a - &1)`] THEN
9288 MATCH_MP_TAC REAL_LT_MUL THEN
9289 ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN
9290 MATCH_MP_TAC(REAL_ARITH `&0 < e ==> &0 < abs(&1 + e) - &1`) THEN
9291 MATCH_MP_TAC REAL_LT_DIV THEN
9292 ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ]]) in
9293 REPEAT STRIP_TAC THEN
9294 MP_TAC(ISPECL [`closure s:real^N->bool`; `x:real^N`; `y:real^N`]
9296 ASM_SIMP_TAC[DIAMETER_CLOSURE; BOUNDED_CLOSURE] THEN
9297 DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
9298 (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN
9299 MATCH_MP_TAC SUBSET_RELATIVE_INTERIOR THEN
9300 REWRITE_TAC[CLOSURE_SUBSET; AFFINE_HULL_CLOSURE]);;
9302 let DIAMETER_ATTAINED_RELATIVE_FRONTIER = prove
9304 bounded s /\ ~(diameter s = &0)
9305 ==> ?x y. x IN relative_frontier s /\
9306 y IN relative_frontier s /\
9307 norm(x - y) = diameter s`,
9308 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9309 ASM_REWRITE_TAC[DIAMETER_EMPTY; relative_frontier] THEN REPEAT STRIP_TAC THEN
9310 MP_TAC(ISPEC `closure s:real^N->bool` DIAMETER_COMPACT_ATTAINED) THEN
9311 ASM_SIMP_TAC[COMPACT_CLOSURE; CLOSURE_EQ_EMPTY; DIAMETER_CLOSURE] THEN
9312 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
9313 STRIP_TAC THEN ASM_REWRITE_TAC[IN_DIFF] THEN REPEAT STRIP_TAC THEN
9314 MP_TAC(ISPEC `s:real^N->bool` DIAMETER_BOUNDED_BOUND_LT) THENL
9315 [DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `y:real^N`]);
9316 DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `x:real^N`])] THEN
9317 ASM_MESON_TAC[REAL_LT_REFL; NORM_SUB]);;
9319 let DIAMETER_RELATIVE_FRONTIER = prove
9321 bounded s /\ ~(?a. s = {a})
9322 ==> diameter(relative_frontier s) = diameter s`,
9323 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9324 ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY] THEN
9325 REWRITE_TAC[relative_frontier] THEN
9326 ASM_SIMP_TAC[GSYM DIAMETER_CLOSURE; GSYM REAL_LE_ANTISYM] THEN
9327 ASM_SIMP_TAC[SUBSET_DIFF; DIAMETER_SUBSET; BOUNDED_CLOSURE] THEN
9328 ASM_SIMP_TAC[DIAMETER_CLOSURE] THEN
9329 MP_TAC(ISPEC `s:real^N->bool` DIAMETER_ATTAINED_RELATIVE_FRONTIER) THEN
9330 ASM_SIMP_TAC[DIAMETER_EQ_0; relative_frontier] THEN STRIP_TAC THEN
9331 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN
9332 ASM_SIMP_TAC[BOUNDED_CLOSURE; BOUNDED_DIFF]);;
9334 let DIAMETER_ATTAINED_FRONTIER = prove
9336 bounded s /\ ~(diameter s = &0)
9337 ==> ?x y. x IN frontier s /\ y IN frontier s /\
9338 norm(x - y) = diameter s`,
9340 DISCH_THEN(MP_TAC o MATCH_MP DIAMETER_ATTAINED_RELATIVE_FRONTIER) THEN
9341 REWRITE_TAC[frontier; relative_frontier; IN_DIFF] THEN
9342 MESON_TAC[REWRITE_RULE[SUBSET] INTERIOR_SUBSET_RELATIVE_INTERIOR]);;
9344 let DIAMETER_FRONTIER = prove
9345 (`!s:real^N->bool. bounded s ==> diameter(frontier s) = diameter s`,
9346 REPEAT STRIP_TAC THEN ASM_CASES_TAC `?a:real^N. s = {a}` THENL
9347 [ASM_MESON_TAC[FRONTIER_SING]; ALL_TAC] THEN
9348 MATCH_MP_TAC(REAL_ARITH
9349 `!r. r <= f /\ f <= s /\ r = s ==> f = s`) THEN
9350 EXISTS_TAC `diameter(closure s DIFF relative_interior s:real^N->bool)` THEN
9351 REPEAT CONJ_TAC THENL
9352 [ASM_SIMP_TAC[GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_SUBSET THEN
9353 ASM_SIMP_TAC[BOUNDED_FRONTIER] THEN REWRITE_TAC[frontier] THEN
9354 MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET_RELATIVE_INTERIOR) THEN
9356 ASM_SIMP_TAC[GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_SUBSET THEN
9357 ASM_SIMP_TAC[BOUNDED_CLOSURE; frontier; SUBSET_DIFF];
9358 ASM_SIMP_TAC[DIAMETER_RELATIVE_FRONTIER; GSYM relative_frontier]]);;
9360 let DIAMETER_SPHERE = prove
9361 (`!a:real^N r. diameter(sphere(a,r)) = if r < &0 then &0 else &2 * r`,
9362 REWRITE_TAC[GSYM FRONTIER_CBALL] THEN
9363 ASM_SIMP_TAC[DIAMETER_FRONTIER; BOUNDED_CBALL; DIAMETER_CBALL]);;
9365 let CLOSEST_POINT_IN_RELATIVE_INTERIOR = prove
9367 closed s /\ ~(s = {}) /\ x IN affine hull s
9368 ==> ((closest_point s x) IN relative_interior s <=>
9369 x IN relative_interior s)`,
9370 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN
9371 ASM_SIMP_TAC[CLOSEST_POINT_SELF] THEN
9372 MATCH_MP_TAC(TAUT `~q /\ ~p ==> (p <=> q)`) THEN CONJ_TAC THENL
9373 [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]; STRIP_TAC] THEN
9374 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN
9375 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9376 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9377 SUBGOAL_THEN `~(closest_point s (x:real^N) = x)` ASSUME_TAC THENL
9378 [ASM_MESON_TAC[]; ALL_TAC] THEN
9379 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`;
9380 `closest_point s x -
9381 (min (&1) (e / norm(closest_point s x - x))) %
9382 (closest_point s x - x):real^N`]
9383 CLOSEST_POINT_LE) THEN
9384 ASM_REWRITE_TAC[dist; NOT_IMP; VECTOR_ARITH
9385 `x - (y - e % (y - x)):real^N = (&1 - e) % (x - y)`] THEN
9387 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
9388 REWRITE_TAC[IN_CBALL; IN_INTER] THEN CONJ_TAC THENL
9389 [REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - x) = norm x`] THEN
9390 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
9391 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
9392 MATCH_MP_TAC(REAL_ARITH `&0 <= a ==> abs(min (&1) a) <= a`) THEN
9393 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_DIV; NORM_POS_LE];
9394 MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN
9395 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]];
9396 REWRITE_TAC[NORM_MUL; REAL_ARITH
9397 `~(n <= a * n) <=> &0 < (&1 - a) * n`] THEN
9398 MATCH_MP_TAC REAL_LT_MUL THEN
9399 ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN
9400 MATCH_MP_TAC(REAL_ARITH
9401 `&0 < e /\ e <= &1 ==> &0 < &1 - abs(&1 - e)`) THEN
9402 REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LT_01; REAL_LE_REFL] THEN
9403 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]);;
9405 let CLOSEST_POINT_IN_RELATIVE_FRONTIER = prove
9407 closed s /\ ~(s = {}) /\ x IN affine hull s DIFF relative_interior s
9408 ==> closest_point s x IN relative_frontier s`,
9409 SIMP_TAC[relative_frontier; IN_DIFF; CLOSEST_POINT_IN_RELATIVE_INTERIOR] THEN
9410 MESON_TAC[CLOSURE_SUBSET; CLOSEST_POINT_IN_SET; SUBSET]);;
9412 (* ------------------------------------------------------------------------- *)
9413 (* Interior, relative interior and closure interrelations. *)
9414 (* ------------------------------------------------------------------------- *)
9416 let CONVEX_CLOSURE_INTERIOR = prove
9418 convex s /\ ~(interior s = {})
9419 ==> closure(interior s) = closure s`,
9420 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
9421 SIMP_TAC[SUBSET_CLOSURE; INTERIOR_SUBSET] THEN
9422 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9423 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN REWRITE_TAC[SUBSET] THEN
9424 X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
9425 [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; ALL_TAC] THEN
9426 REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN
9427 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
9428 EXISTS_TAC `b - min (e / &2 / norm(b - a)) (&1) % (b - a):real^N` THEN
9429 REPEAT CONJ_TAC THENL
9430 [MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
9431 ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LE_REFL; REAL_LT_01];
9432 REWRITE_TAC[VECTOR_ARITH `b - x:real^N = b <=> x = vec 0`] THEN
9433 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
9434 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min x (&1) = &0)`);
9435 REWRITE_TAC[NORM_ARITH `dist(b - x:real^N,b) = norm x`] THEN
9436 REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LET_TRANS THEN
9437 EXISTS_TAC `e / &2 / norm(b - a:real^N) * norm(b - a)` THEN CONJ_TAC THENL
9438 [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
9439 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> abs(min x (&1)) <= x`);
9440 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_POS_LT; REAL_LT_IMP_NZ;
9442 ASM_REAL_ARITH_TAC]] THEN
9443 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_OF_NUM_LT;
9444 VECTOR_SUB_EQ; ARITH]);;
9446 let EMPTY_INTERIOR_SUBSET_HYPERPLANE = prove
9447 (`!s. convex s /\ interior s = {}
9448 ==> ?a:real^N b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`,
9450 (`!s. convex s /\ (vec 0) IN s /\ interior s = {}
9451 ==> ?a:real^N b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`,
9453 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
9454 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
9455 SUBGOAL_THEN `~(relative_interior(s:real^N->bool) = {})` MP_TAC THENL
9456 [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; MEMBER_NOT_EMPTY]; ALL_TAC] THEN
9457 ASM_REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN
9458 AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
9459 MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
9460 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
9461 ONCE_REWRITE_TAC[GSYM SPAN_UNIV] THEN MATCH_MP_TAC DIM_EQ_SPAN THEN
9462 REWRITE_TAC[SUBSET_UNIV; DIM_UNIV; GSYM NOT_LT] THEN
9463 DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN
9464 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
9465 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
9466 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN
9467 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN EXISTS_TAC `&0` THEN
9468 ASM_MESON_TAC[SUBSET_TRANS; SPAN_INC]) in
9469 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
9470 [ASM_MESON_TAC[EMPTY_SUBSET; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1];
9473 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9474 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
9475 MP_TAC(ISPEC `IMAGE (\x:real^N. --a + x) s` lemma) THEN
9476 ASM_REWRITE_TAC[CONVEX_TRANSLATION_EQ; INTERIOR_TRANSLATION;
9477 IMAGE_EQ_EMPTY; IN_IMAGE; UNWIND_THM2;
9478 VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN
9479 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN
9480 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; DOT_RADD] THEN
9481 MESON_TAC[REAL_ARITH `a + x:real = b <=> x = b - a`]);;
9483 let CONVEX_INTERIOR_CLOSURE = prove
9484 (`!s:real^N->bool. convex s ==> interior(closure s) = interior s`,
9485 REPEAT STRIP_TAC THEN
9486 ASM_CASES_TAC `interior(s:real^N->bool) = {}` THENL
9487 [MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN
9488 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
9489 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN
9490 MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
9491 EXISTS_TAC `interior {x:real^N | a dot x = b}` THEN CONJ_TAC THENL
9492 [ALL_TAC; ASM_SIMP_TAC[INTERIOR_HYPERPLANE]] THEN
9493 MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN
9494 ASM_REWRITE_TAC[CLOSED_HYPERPLANE];
9496 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9497 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
9498 MATCH_MP_TAC SUBSET_ANTISYM THEN
9499 SIMP_TAC[SUBSET_INTERIOR; CLOSURE_SUBSET] THEN
9500 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN
9501 MP_TAC(ASSUME `(b:real^N) IN interior(closure s)`) THEN
9502 GEN_REWRITE_TAC LAND_CONV [IN_INTERIOR_CBALL] THEN
9503 REWRITE_TAC[SUBSET; IN_CBALL; LEFT_IMP_EXISTS_THM] THEN
9504 X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9505 ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[] THEN
9506 DISCH_THEN(MP_TAC o SPEC `b + e / norm(b - a) % (b - a):real^N`) THEN
9507 ASM_SIMP_TAC[NORM_ARITH `dist(b:real^N,b + e) = norm e`; NORM_MUL;
9508 REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ;
9509 REAL_ARITH `&0 < e ==> abs e <= e`] THEN
9512 `b = (b + e / norm(b - a) % (b - a)) -
9513 e / norm(b - a) / (&1 + e / norm(b - a)) %
9514 ((b + e / norm(b - a) % (b - a)) - a):real^N`
9516 [REWRITE_TAC[VECTOR_ARITH
9517 `b = (b + e % (b - a)) - d % ((b + e % (b - a)) - a) <=>
9518 (e - d * (&1 + e)) % (b - a) = vec 0`] THEN
9519 ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0];
9520 MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK] THEN
9521 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_DIV; NORM_POS_LT;
9522 VECTOR_SUB_EQ; REAL_ARITH `&0 < x ==> &0 < &1 + x`;
9523 REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`;
9524 REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_DIV_RMUL;
9525 REAL_LT_IMP_NZ; REAL_LE_ADDL; NORM_POS_LE; REAL_SUB_REFL]);;
9527 let FRONTIER_CLOSURE_CONVEX = prove
9528 (`!s:real^N->bool. convex s ==> frontier(closure s) = frontier s`,
9529 SIMP_TAC[frontier; CLOSURE_CLOSURE; CONVEX_INTERIOR_CLOSURE]);;
9531 let CONVEX_CLOSURE_RELATIVE_INTERIOR = prove
9533 convex s ==> closure(relative_interior s) = closure s`,
9534 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
9535 SIMP_TAC[SUBSET_CLOSURE; RELATIVE_INTERIOR_SUBSET] THEN
9536 ASM_CASES_TAC `relative_interior(s:real^N->bool) = {}` THENL
9537 [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; SUBSET_REFL]; ALL_TAC] THEN
9538 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9539 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN REWRITE_TAC[SUBSET] THEN
9540 X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
9541 [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; ALL_TAC] THEN
9542 REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN
9543 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
9544 EXISTS_TAC `b - min (e / &2 / norm(b - a)) (&1) % (b - a):real^N` THEN
9545 REPEAT CONJ_TAC THENL
9546 [MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
9547 ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LE_REFL; REAL_LT_01];
9548 REWRITE_TAC[VECTOR_ARITH `b - x:real^N = b <=> x = vec 0`] THEN
9549 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
9550 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min x (&1) = &0)`);
9551 REWRITE_TAC[NORM_ARITH `dist(b - x:real^N,b) = norm x`] THEN
9552 REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LET_TRANS THEN
9553 EXISTS_TAC `e / &2 / norm(b - a:real^N) * norm(b - a)` THEN CONJ_TAC THENL
9554 [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
9555 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> abs(min x (&1)) <= x`);
9556 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_POS_LT; REAL_LT_IMP_NZ;
9558 ASM_REAL_ARITH_TAC]] THEN
9559 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_OF_NUM_LT;
9560 VECTOR_SUB_EQ; ARITH]);;
9562 let AFFINE_HULL_RELATIVE_INTERIOR = prove
9564 ==> affine hull (relative_interior s) = affine hull s`,
9565 MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; AFFINE_HULL_CLOSURE]);;
9567 let CONVEX_RELATIVE_INTERIOR_CLOSURE = prove
9569 convex s ==> relative_interior(closure s) = relative_interior s`,
9570 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9571 ASM_REWRITE_TAC[CLOSURE_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN
9572 SUBGOAL_THEN `?a:real^N. a IN relative_interior s` STRIP_ASSUME_TAC THENL
9573 [ASM_SIMP_TAC[MEMBER_NOT_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY];
9575 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET] THEN CONJ_TAC THENL
9577 REWRITE_TAC[IN_RELATIVE_INTERIOR; AFFINE_HULL_CLOSURE; SUBSET] THEN
9578 MESON_TAC[CLOSURE_SUBSET; SUBSET]] THEN
9579 X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN
9580 MP_TAC(ASSUME `(b:real^N) IN relative_interior(closure s)`) THEN
9581 GEN_REWRITE_TAC LAND_CONV [IN_RELATIVE_INTERIOR_CBALL] THEN
9582 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9583 REWRITE_TAC[SUBSET; IN_CBALL; IN_INTER; LEFT_IMP_EXISTS_THM;
9584 AFFINE_HULL_CLOSURE] THEN
9585 X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9586 ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[] THEN
9587 DISCH_THEN(MP_TAC o SPEC `b + e / norm(b - a) % (b - a):real^N`) THEN
9588 ASM_SIMP_TAC[NORM_ARITH `dist(b:real^N,b + e) = norm e`; NORM_MUL;
9589 REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ;
9590 REAL_ARITH `&0 < e ==> abs e <= e`] THEN
9592 [MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
9593 ASM_MESON_TAC[SUBSET; AFFINE_AFFINE_HULL; RELATIVE_INTERIOR_SUBSET;
9594 CLOSURE_SUBSET_AFFINE_HULL; HULL_INC];
9598 `b = (b + e / norm(b - a) % (b - a)) -
9599 e / norm(b - a) / (&1 + e / norm(b - a)) %
9600 ((b + e / norm(b - a) % (b - a)) - a):real^N`
9602 [REWRITE_TAC[VECTOR_ARITH
9603 `b = (b + e % (b - a)) - d % ((b + e % (b - a)) - a) <=>
9604 (e - d * (&1 + e)) % (b - a) = vec 0`] THEN
9605 ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0];
9606 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK] THEN
9607 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_DIV; NORM_POS_LT;
9608 VECTOR_SUB_EQ; REAL_ARITH `&0 < x ==> &0 < &1 + x`;
9609 REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`;
9610 REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_DIV_RMUL;
9611 REAL_LT_IMP_NZ; REAL_LE_ADDL; NORM_POS_LE; REAL_SUB_REFL]);;
9613 let RELATIVE_FRONTIER_CLOSURE = prove
9614 (`!s. convex s ==> relative_frontier(closure s) = relative_frontier s`,
9615 SIMP_TAC[relative_frontier; CLOSURE_CLOSURE;
9616 CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
9618 let CONNECTED_INTER_RELATIVE_FRONTIER = prove
9619 (`!s t:real^N->bool.
9620 connected s /\ s SUBSET affine hull t /\
9621 ~(s INTER t = {}) /\ ~(s DIFF t = {})
9622 ==> ~(s INTER relative_frontier t = {})`,
9623 REWRITE_TAC[relative_frontier] THEN REPEAT STRIP_TAC THEN
9624 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_OPEN_IN]) THEN
9625 REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC
9626 [`s INTER relative_interior t:real^N->bool`;
9627 `s DIFF closure t:real^N->bool`] THEN
9628 REPEAT CONJ_TAC THENL
9629 [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN
9630 EXISTS_TAC `affine hull t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
9631 MATCH_MP_TAC OPEN_IN_INTER THEN
9632 REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; OPEN_IN_SUBTOPOLOGY_REFL] THEN
9633 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV];
9634 ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN
9635 MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN
9636 REWRITE_TAC[GSYM closed; CLOSED_CLOSURE];
9638 MATCH_MP_TAC(SET_RULE
9639 `i SUBSET t /\ t SUBSET c ==> (s INTER i) INTER (s DIFF c) = {}`) THEN
9640 REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET];
9641 MP_TAC(ISPEC `t:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[];
9642 MP_TAC(ISPEC `t:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN
9645 let CLOSED_RELATIVE_FRONTIER = prove
9646 (`!s:real^N->bool. closed(relative_frontier s)`,
9647 REPEAT GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN
9648 MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN
9649 EXISTS_TAC `affine hull s:real^N->bool` THEN
9650 REWRITE_TAC[CLOSED_AFFINE_HULL] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN
9651 REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN
9652 MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[CLOSED_CLOSURE] THEN
9653 MATCH_MP_TAC(SET_RULE
9654 `s SUBSET closure t /\ closure t = t ==> s SUBSET t`) THEN
9655 SIMP_TAC[SUBSET_CLOSURE; HULL_SUBSET; CLOSURE_EQ; CLOSED_AFFINE_HULL]);;
9657 let CLOSED_RELATIVE_BOUNDARY = prove
9658 (`!s. closed s ==> closed(s DIFF relative_interior s)`,
9659 MESON_TAC[CLOSED_RELATIVE_FRONTIER; relative_frontier; CLOSURE_CLOSED]);;
9661 let COMPACT_RELATIVE_BOUNDARY = prove
9662 (`!s. compact s ==> compact(s DIFF relative_interior s)`,
9663 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_RELATIVE_BOUNDARY;
9666 let BOUNDED_RELATIVE_FRONTIER = prove
9667 (`!s:real^N->bool. bounded s ==> bounded(relative_frontier s)`,
9668 REWRITE_TAC[relative_frontier] THEN
9669 MESON_TAC[BOUNDED_CLOSURE; BOUNDED_SUBSET; SUBSET_DIFF]);;
9671 let COMPACT_RELATIVE_FRONTIER_BOUNDED = prove
9672 (`!s:real^N->bool. bounded s ==> compact(relative_frontier s)`,
9673 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_RELATIVE_FRONTIER;
9674 BOUNDED_RELATIVE_FRONTIER]);;
9676 let COMPACT_RELATIVE_FRONTIER = prove
9677 (`!s:real^N->bool. compact s ==> compact(relative_frontier s)`,
9678 SIMP_TAC[COMPACT_RELATIVE_FRONTIER_BOUNDED; COMPACT_IMP_BOUNDED]);;
9680 let CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE = prove
9681 (`!s t. convex s /\ convex t
9682 ==> (relative_interior s = relative_interior t <=>
9683 closure s = closure t)`,
9684 MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR;
9685 CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
9687 let CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE_STRADDLE = prove
9688 (`!s t. convex s /\ convex t
9689 ==> (relative_interior s = relative_interior t <=>
9690 relative_interior s SUBSET t /\ t SUBSET closure s)`,
9691 MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR;
9692 CONVEX_RELATIVE_INTERIOR_CLOSURE; SUBSET_CLOSURE;
9693 SUBSET_ANTISYM; RELATIVE_INTERIOR_SUBSET;
9694 CLOSURE_SUBSET; CLOSURE_CLOSURE]);;
9696 let RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX = prove
9697 (`!f:real^M->real^N s.
9698 linear f /\ convex s
9699 ==> relative_interior(IMAGE f s) = IMAGE f (relative_interior s)`,
9700 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
9702 `relative_interior (IMAGE f (relative_interior s)) =
9703 relative_interior (IMAGE (f:real^M->real^N) s)`
9704 (fun th -> REWRITE_TAC[SYM th; RELATIVE_INTERIOR_SUBSET]) THEN
9705 ASM_SIMP_TAC[CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE_STRADDLE;
9706 CONVEX_RELATIVE_INTERIOR; CONVEX_LINEAR_IMAGE] THEN
9708 [MATCH_MP_TAC SUBSET_TRANS THEN
9709 EXISTS_TAC `IMAGE (f:real^M->real^N) (relative_interior s)` THEN
9710 SIMP_TAC[RELATIVE_INTERIOR_SUBSET; IMAGE_SUBSET];
9711 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
9712 `IMAGE (f:real^M->real^N) (closure(relative_interior s))` THEN
9713 ASM_SIMP_TAC[CLOSURE_LINEAR_IMAGE_SUBSET] THEN
9714 ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN
9715 MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET]];
9716 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `z:real^M` THEN
9718 ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_PROLONG; CONVEX_LINEAR_IMAGE] THEN
9719 REWRITE_TAC[IN_ELIM_THM; FORALL_IN_IMAGE] THEN CONJ_TAC THENL
9720 [MATCH_MP_TAC FUN_IN_IMAGE THEN
9721 ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET];
9723 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
9724 MP_TAC(ISPECL [`s:real^M->bool`; `z:real^M`; `x:real^M`]
9725 RELATIVE_INTERIOR_PROLONG) THEN ASM_REWRITE_TAC[] THEN
9726 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN
9727 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
9728 DISCH_THEN(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP FUN_IN_IMAGE) THEN
9729 ASM_MESON_TAC[LINEAR_ADD; LINEAR_SUB; LINEAR_CMUL]]);;
9731 let CLOSURE_INTERS_CONVEX = prove
9732 (`!f:(real^N->bool)->bool.
9733 (!s. s IN f ==> convex s) /\
9734 ~(INTERS(IMAGE relative_interior f) = {})
9735 ==> closure(INTERS f) = INTERS(IMAGE closure f)`,
9736 REPEAT STRIP_TAC THEN
9737 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[CLOSURE_INTERS_SUBSET] THEN
9738 REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE] THEN
9739 X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN
9740 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9741 REWRITE_TAC[INTERS_IMAGE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
9742 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
9743 REWRITE_TAC[CLOSURE_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
9744 ASM_CASES_TAC `b:real^N = a` THENL
9745 [EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[DIST_REFL; IN_INTERS] THEN
9746 ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET];
9748 EXISTS_TAC `b - min (&1 / &2) (e / &2 / norm(b - a)) % (b - a):real^N` THEN
9751 REWRITE_TAC[NORM_ARITH `dist(b - a:real^N,b) = norm a`; NORM_MUL] THEN
9752 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
9753 MATCH_MP_TAC(REAL_ARITH
9754 `&0 < a /\ &0 < x /\ x < y ==> abs(min a x) < y`) THEN
9755 ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_HALF; REAL_LT_DIV; NORM_POS_LT;
9757 ASM_REAL_ARITH_TAC] THEN
9758 REWRITE_TAC[IN_INTERS] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN
9760 (MESON[RELATIVE_INTERIOR_SUBSET; SUBSET]
9761 `!x. x IN relative_interior s ==> x IN s`) THEN
9762 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
9763 ASM_SIMP_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_DIV; NORM_POS_LT;
9767 let CLOSURE_INTERS_CONVEX_OPEN = prove
9768 (`!f:(real^N->bool)->bool.
9769 (!s. s IN f ==> convex s /\ open s)
9770 ==> closure(INTERS f) =
9771 if INTERS f = {} then {}
9772 else INTERS(IMAGE closure f)`,
9773 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CLOSURE_EMPTY] THEN
9774 MATCH_MP_TAC CLOSURE_INTERS_CONVEX THEN ASM_SIMP_TAC[] THEN
9775 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
9776 `~(s = {}) ==> s = t ==> ~(t = {})`)) THEN
9777 AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE
9778 `(!x. x IN s ==> f x = x) ==> s = IMAGE f s`) THEN
9779 ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN; INTERIOR_EQ]);;
9781 let CLOSURE_INTER_CONVEX = prove
9782 (`!s t:real^N->bool.
9783 convex s /\ convex t /\
9784 ~(relative_interior s INTER relative_interior t = {})
9785 ==> closure(s INTER t) = closure(s) INTER closure(t)`,
9786 REPEAT STRIP_TAC THEN
9787 MP_TAC(ISPEC `{s:real^N->bool,t}` CLOSURE_INTERS_CONVEX) THEN
9788 ASM_SIMP_TAC[IMAGE_CLAUSES; INTERS_2] THEN
9789 ASM_REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY]);;
9791 let CLOSURE_INTER_CONVEX_OPEN = prove
9792 (`!s t. convex s /\ open s /\ convex t /\ open t
9793 ==> closure(s INTER t) =
9794 if s INTER t = {} then {} else closure(s) INTER closure(t)`,
9795 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CLOSURE_EMPTY] THEN
9796 MATCH_MP_TAC CLOSURE_INTER_CONVEX THEN
9797 ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN]);;
9799 let CLOSURE_CONVEX_INTER_SUPERSET = prove
9800 (`!s t:real^N->bool.
9801 convex s /\ ~(interior s = {}) /\ interior s SUBSET closure t
9802 ==> closure(s INTER t) = closure s`,
9803 REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
9804 SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET; SUBSET_INTER] THEN
9805 MATCH_MP_TAC SUBSET_TRANS THEN
9806 EXISTS_TAC `closure(interior s):real^N->bool` THEN CONJ_TAC THENL
9807 [ASM_SIMP_TAC[CONVEX_CLOSURE_INTERIOR; SUBSET_REFL];
9808 ASM_SIMP_TAC[GSYM CLOSURE_OPEN_INTER_SUPERSET; OPEN_INTERIOR] THEN
9809 MATCH_MP_TAC SUBSET_CLOSURE THEN
9810 MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN SET_TAC[]]);;
9812 let CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET = prove
9814 convex s /\ ~(interior s = {})
9816 { inv(&2 pow n) % x | n,x |
9817 !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }) =
9819 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_CONVEX_INTER_SUPERSET THEN
9820 ASM_REWRITE_TAC[CLOSURE_DYADIC_RATIONALS; SUBSET_UNIV]);;
9822 let CLOSURE_RATIONALS_IN_CONVEX_SET = prove
9824 convex s /\ ~(interior s = {})
9826 { x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) }) =
9828 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_CONVEX_INTER_SUPERSET THEN
9829 ASM_REWRITE_TAC[CLOSURE_RATIONAL_COORDINATES; SUBSET_UNIV]);;
9831 let RELATIVE_INTERIOR_CONVEX_INTER_AFFINE = prove
9832 (`!s t:real^N->bool.
9833 convex s /\ affine t /\ ~(interior s INTER t = {})
9834 ==> relative_interior(s INTER t) = interior s INTER t`,
9836 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM] THEN
9837 DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN
9838 GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN
9839 REPEAT GEN_TAC THEN ASM_CASES_TAC `(vec 0:real^N) IN t` THEN
9840 ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN STRIP_TAC THEN
9841 GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN
9842 MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`]
9843 (ONCE_REWRITE_RULE[INTER_COMM]
9844 AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR)) THEN
9845 ASM_SIMP_TAC[SUBSPACE_IMP_AFFINE; IN_RELATIVE_INTERIOR_CBALL] THEN
9846 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTER; IN_INTERIOR_CBALL]] THEN
9847 DISCH_THEN SUBST1_TAC THEN
9848 ASM_CASES_TAC `(x:real^N) IN t` THEN ASM_REWRITE_TAC[] THEN
9849 SIMP_TAC[SUBSET; IN_INTER] THEN
9850 ASM_CASES_TAC `(x:real^N) IN s` THENL
9851 [ASM_REWRITE_TAC[]; ASM_MESON_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE]] THEN
9852 EQ_TAC THENL [REWRITE_TAC[IN_CBALL]; MESON_TAC[]] THEN
9853 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9854 ASM_CASES_TAC `x:real^N = vec 0` THENL
9855 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN
9856 ASM_REWRITE_TAC[SUBSET; IN_CBALL];
9859 [`s:real^N->bool`; `vec 0:real^N`; `(&1 + e / norm x) % x:real^N`]
9860 IN_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN
9861 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
9862 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN
9863 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SUBSPACE_MUL] THEN
9864 REWRITE_TAC[VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID;
9865 NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN
9866 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM;
9867 REAL_DIV_RMUL; NORM_EQ_0] THEN
9869 REWRITE_TAC[SUBSET; IN_INTERIOR_CBALL; IN_CBALL] THEN
9870 DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_SEGMENT] THEN
9871 CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
9872 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
9873 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
9874 EXISTS_TAC `inv(&1 + e / norm(x:real^N))` THEN
9875 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_LT_DIV; NORM_POS_LT; VECTOR_MUL_LID;
9876 REAL_LT_INV_EQ; REAL_MUL_LINV; REAL_INV_LT_1; REAL_ARITH
9877 `&0 < x ==> &1 < &1 + x /\ &0 < &1 + x /\ ~(&1 + x = &0)`]]);;
9879 (* ------------------------------------------------------------------------- *)
9880 (* Homeomorphism of all convex compact sets with same affine dimension, and *)
9881 (* in particular all those with nonempty interior. *)
9882 (* ------------------------------------------------------------------------- *)
9884 let COMPACT_FRONTIER_LINE_LEMMA = prove
9885 (`!s x. compact s /\ (vec 0 IN s) /\ ~(x = vec 0 :real^N)
9886 ==> ?u. &0 <= u /\ (u % x) IN frontier s /\
9887 !v. u < v ==> ~((v % x) IN s)`,
9888 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
9889 REWRITE_TAC[BOUNDED_POS] THEN
9890 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN
9892 [`{y:real^N | ?u. &0 <= u /\ u <= b / norm(x) /\ (y = u % x)} INTER s`;
9894 DISTANCE_ATTAINS_SUP) THEN
9898 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN
9899 ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN
9900 EXISTS_TAC `&0` THEN
9901 ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL; REAL_LT_IMP_LE;
9902 REAL_LT_DIV; NORM_POS_LT]] THEN
9903 MATCH_MP_TAC COMPACT_INTER THEN ASM_REWRITE_TAC[] THEN
9905 `{y:real^N | ?u. &0 <= u /\ u <= b / norm(x) /\ (y = u % x)} =
9906 IMAGE (\u. drop u % x) (interval [vec 0,lambda i. b / norm(x:real^N)])`
9908 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; IN_INTERVAL] THEN
9909 SIMP_TAC[LAMBDA_BETA] THEN
9910 SIMP_TAC[DIMINDEX_1; ARITH_RULE `1 <= i /\ i <= 1 <=> (i = 1)`] THEN
9911 REWRITE_TAC[GSYM drop; LEFT_FORALL_IMP_THM; EXISTS_REFL; DROP_VEC] THEN
9912 REWRITE_TAC[EXISTS_LIFT; LIFT_DROP] THEN MESON_TAC[];
9914 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
9915 REWRITE_TAC[COMPACT_INTERVAL] THEN
9916 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REPEAT STRIP_TAC THEN
9917 MATCH_MP_TAC CONTINUOUS_VMUL THEN
9918 REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_AT_ID];
9920 REWRITE_TAC[IN_INTER; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
9921 ONCE_REWRITE_TAC[TAUT `(a /\ b /\ c) /\ d <=> c /\ a /\ b /\ d`] THEN
9922 SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
9923 GEN_REWRITE_TAC (BINDER_CONV o ONCE_DEPTH_CONV) [SWAP_FORALL_THM] THEN
9924 SIMP_TAC[IMP_CONJ] THEN
9925 REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
9926 REWRITE_TAC[IMP_IMP] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN
9927 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real` THEN
9928 REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG; NORM_MUL] THEN
9929 ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT] THEN
9930 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
9931 ASM_SIMP_TAC[real_abs] THEN REPEAT STRIP_TAC THENL
9932 [REWRITE_TAC[FRONTIER_STRADDLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
9934 [EXISTS_TAC `u % x :real^N` THEN ASM_REWRITE_TAC[DIST_REFL];
9936 EXISTS_TAC `(u + (e / &2) / norm(x)) % x :real^N` THEN
9937 REWRITE_TAC[dist; VECTOR_ARITH `u % x - (u + a) % x = --(a % x)`] THEN
9938 ASM_SIMP_TAC[NORM_NEG; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; NORM_EQ_0;
9939 REAL_DIV_RMUL; REAL_ABS_NUM; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT;
9940 ARITH; REAL_ARITH `abs e < e * &2 <=> &0 < e`] THEN
9942 FIRST_X_ASSUM(MP_TAC o SPEC `u + (e / &2) / norm(x:real^N)`) THEN
9943 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
9944 `&0 < e /\ &0 <= u /\ u + e <= b
9945 ==> ~(&0 <= u + e /\ u + e <= b ==> u + e <= u)`) THEN
9946 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NORM_POS_LT] THEN
9947 FIRST_X_ASSUM(MP_TAC o SPEC `(u + (e / &2) / norm(x:real^N)) % x`) THEN
9948 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
9951 FIRST_X_ASSUM(MP_TAC o SPEC `v:real`) THEN
9952 ASM_REWRITE_TAC[GSYM REAL_NOT_LT] THEN ASM_REWRITE_TAC[REAL_NOT_LT] THEN
9953 CONJ_TAC THENL [ASM_MESON_TAC[REAL_LET_TRANS; REAL_LT_IMP_LE]; ALL_TAC] THEN
9954 FIRST_X_ASSUM(MP_TAC o SPEC `v % x:real^N`) THEN
9955 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
9958 let STARLIKE_COMPACT_PROJECTIVE = prove
9959 (`!s:real^N->bool a.
9960 compact s /\ a IN relative_interior s /\
9961 (!x. x IN s ==> segment(a,x) SUBSET relative_interior s)
9962 ==> s DIFF relative_interior s homeomorphic
9963 sphere(a,&1) INTER affine hull s /\
9964 s homeomorphic cball(a,&1) INTER affine hull s`,
9965 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN
9966 REWRITE_TAC[SUBSET; IMP_IMP; RIGHT_IMP_FORALL_THM] THEN
9967 GEN_TAC THEN STRIP_TAC THEN
9969 `!x:real^N u. x IN s /\ &0 <= u /\ u < &1
9970 ==> (u % x) IN relative_interior s`
9972 [REWRITE_TAC[REAL_ARITH `&0 <= u <=> u = &0 \/ &0 < u`] THEN
9973 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
9974 ASM_CASES_TAC `x:real^N = vec 0` THEN
9975 ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
9976 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_SEGMENT] THEN
9977 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN ASM_MESON_TAC[];
9978 FIRST_X_ASSUM(K ALL_TAC o SPECL [`x:real^N`; `x:real^N`])] THEN
9979 FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET]
9980 RELATIVE_INTERIOR_SUBSET)) THEN
9981 ABBREV_TAC `proj = \x:real^N. inv(norm(x)) % x` THEN
9983 `!x:real^N y. (proj(x) = proj(y):real^N) /\ (norm x = norm y) <=> (x = y)`
9985 [REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
9986 ASM_CASES_TAC `y:real^N = vec 0` THEN
9987 ASM_SIMP_TAC[NORM_EQ_0; NORM_0] THEN
9988 ASM_CASES_TAC `x:real^N = vec 0` THENL
9989 [ASM_MESON_TAC[NORM_EQ_0]; ALL_TAC] THEN
9990 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
9991 EXPAND_TAC "proj" THEN REWRITE_TAC[] THEN
9992 ASM_REWRITE_TAC[VECTOR_ARITH
9993 `a % x = a % y <=> a % (x - y):real^N = vec 0`] THEN
9994 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0; VECTOR_SUB_EQ];
9997 `(!x. x IN affine hull s ==> proj x IN affine hull s) /\
9998 (!x. ~(x = vec 0) ==> norm(proj x) = &1) /\
9999 (!x:real^N. proj x = vec 0 <=> x = vec 0)`
10000 STRIP_ASSUME_TAC THENL
10001 [EXPAND_TAC "proj" THEN REWRITE_TAC[NORM_MUL; VECTOR_MUL_EQ_0] THEN
10002 REWRITE_TAC[REAL_INV_EQ_0; NORM_EQ_0; REAL_ABS_INV; REAL_ABS_NORM] THEN
10003 SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN REPEAT STRIP_TAC THEN
10004 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN
10005 MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
10006 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID; HULL_INC];
10008 SUBGOAL_THEN `(proj:real^N->real^N) continuous_on (UNIV DELETE vec 0)`
10010 [MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
10011 REWRITE_TAC[IN_DELETE; IN_UNIV] THEN EXPAND_TAC "proj" THEN
10012 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
10013 ASM_SIMP_TAC[CONTINUOUS_AT_ID] THEN
10014 REWRITE_TAC[GSYM(ISPEC `lift` o_DEF);
10015 GSYM(ISPEC `inv:real->real` o_DEF)] THEN
10016 MATCH_MP_TAC CONTINUOUS_AT_INV THEN
10017 ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ; CONTINUOUS_AT_LIFT_NORM];
10019 ABBREV_TAC `usph = {x:real^N | x IN affine hull s /\ norm x = &1}` THEN
10020 SUBGOAL_THEN ` sphere(vec 0:real^N,&1) INTER affine hull s = usph`
10022 [EXPAND_TAC "usph" THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE_0] THEN
10026 `!x. x IN affine hull s /\ ~(x = vec 0)
10027 ==> (proj:real^N->real^N) x IN usph`
10028 ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
10029 SUBGOAL_THEN `?surf. homeomorphism (s DIFF relative_interior s,usph)
10030 (proj:real^N->real^N,surf)`
10032 [MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN
10033 ASM_SIMP_TAC[COMPACT_RELATIVE_BOUNDARY] THEN REPEAT CONJ_TAC THENL
10034 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10035 CONTINUOUS_ON_SUBSET)) THEN
10037 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
10038 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF] THEN
10039 EXPAND_TAC "usph" THEN REWRITE_TAC[IN_ELIM_THM] THEN
10040 ASM_MESON_TAC[HULL_INC];
10041 MAP_EVERY EXPAND_TAC ["proj"; "usph"] THEN
10042 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN
10043 ASM_CASES_TAC `x:real^N = vec 0` THEN
10044 ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN STRIP_TAC THEN
10045 MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `x:real^N`]
10046 RAY_TO_RELATIVE_FRONTIER) THEN
10047 REWRITE_TAC[relative_frontier] THEN
10048 ASM_SIMP_TAC[COMPACT_IMP_BOUNDED; CLOSURE_CLOSED; COMPACT_IMP_CLOSED;
10049 VECTOR_ADD_LID] THEN
10050 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
10051 EXPAND_TAC "proj" THEN REWRITE_TAC[IN_IMAGE] THEN
10052 EXISTS_TAC `d % x:real^N` THEN ASM_REWRITE_TAC[NORM_MUL] THEN
10053 ASM_SIMP_TAC[REAL_MUL_RID; real_abs; REAL_LT_IMP_LE] THEN
10054 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ;
10056 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
10057 REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN
10058 ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN
10059 ASM_CASES_TAC `y:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN
10060 UNDISCH_TAC `(proj:real^N->real^N) x = proj y` THEN
10061 EXPAND_TAC "proj" THEN
10062 REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH
10063 `norm(x:real^N) = norm(y:real^N) \/
10064 norm x < norm y \/ norm y < norm x`)
10066 [ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL; REAL_INV_EQ_0; NORM_EQ_0];
10067 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
10068 [`y:real^N`; `norm(x:real^N) / norm(y:real^N)`]);
10069 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
10070 [`x:real^N`; `norm(y:real^N) / norm(x:real^N)`])] THEN
10071 ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_LDIV_EQ; NORM_POS_LT;
10073 ASM_REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THENL
10074 [FIRST_X_ASSUM(SUBST1_TAC o SYM); ALL_TAC] THEN
10075 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; NORM_EQ_0] THEN
10076 ASM_REWRITE_TAC[VECTOR_MUL_LID]];
10077 DISCH_THEN(fun th ->
10079 [MESON_TAC[homeomorphic; th];
10080 ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
10081 MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN
10082 SIMP_TAC[COMPACT_INTER_CLOSED; CLOSED_AFFINE_HULL; COMPACT_CBALL] THEN
10084 REWRITE_TAC[HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN
10085 X_GEN_TAC `surf:real^N->real^N` THEN STRIP_TAC THEN
10086 EXISTS_TAC `\x:real^N. norm(x) % (surf:real^N->real^N)(proj(x))` THEN
10087 REWRITE_TAC[]] THEN
10089 `(proj:real^N->real^N) continuous_on s DIFF relative_interior s`
10091 REWRITE_TAC[] THEN CONJ_TAC THENL
10092 [REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; IN_INTER] THEN
10093 X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
10094 ASM_CASES_TAC `x = vec 0:real^N` THENL
10095 [ASM_REWRITE_TAC[CONTINUOUS_WITHIN; VECTOR_MUL_LZERO; NORM_0] THEN
10096 MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN
10097 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
10098 REWRITE_TAC[BOUNDED_POS] THEN MATCH_MP_TAC MONO_EXISTS THEN
10099 REPEAT STRIP_TAC THENL
10100 [REWRITE_TAC[LIM_WITHIN; o_THM; DIST_0; NORM_LIFT; REAL_ABS_NORM] THEN
10102 REWRITE_TAC[EVENTUALLY_WITHIN] THEN EXISTS_TAC `&1` THEN
10103 REWRITE_TAC[REAL_LT_01; IN_INTER; DIST_0; NORM_POS_LT] THEN
10105 MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
10106 EXISTS_TAC `affine hull s:real^N->bool` THEN
10107 REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
10108 SIMP_TAC[CONTINUOUS_LIFT_NORM_COMPOSE; CONTINUOUS_WITHIN_ID; o_DEF] THEN
10110 `((surf:real^N->real^N) o (proj:real^N->real^N)) continuous_on
10111 (affine hull s DELETE vec 0)`
10113 [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
10114 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN
10115 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10116 CONTINUOUS_ON_SUBSET)) THEN
10117 SIMP_TAC[SUBSET; IN_DELETE; IN_UNIV; FORALL_IN_IMAGE] THEN
10118 EXPAND_TAC "usph" THEN ASM_SIMP_TAC[IN_ELIM_THM];
10119 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
10120 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_SIMP_TAC[IN_DELETE] THEN
10121 REWRITE_TAC[CONTINUOUS_WITHIN; o_DEF] THEN MATCH_MP_TAC EQ_IMP THEN
10122 MATCH_MP_TAC LIM_TRANSFORM_WITHIN_SET THEN
10123 REWRITE_TAC[EVENTUALLY_AT] THEN EXISTS_TAC `norm(x:real^N)` THEN
10124 ASM_REWRITE_TAC[IN_DELETE; IN_INTER; IN_CBALL; NORM_POS_LT] THEN
10125 X_GEN_TAC `y:real^N` THEN
10126 ASM_CASES_TAC `(y:real^N) IN affine hull s` THEN ASM_REWRITE_TAC[] THEN
10127 CONV_TAC NORM_ARITH]];
10130 `!a x. &0 < a ==> (proj:real^N->real^N)(a % x) = proj x`
10132 [REPEAT GEN_TAC THEN EXPAND_TAC "proj" THEN
10133 REWRITE_TAC[NORM_MUL; REAL_INV_MUL; VECTOR_MUL_ASSOC] THEN
10134 SIMP_TAC[REAL_FIELD `&0 < a ==> (inv(a) * x) * a = x`; real_abs;
10139 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
10140 ASM_CASES_TAC `y:real^N = vec 0` THENL
10141 [ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_EQ_0; NORM_0; NORM_EQ_0] THEN
10144 ASM_CASES_TAC `x:real^N = vec 0` THENL
10145 [CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
10146 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_EQ_0; NORM_0; NORM_EQ_0] THEN
10149 REWRITE_TAC[IN_INTER; IN_CBALL_0] THEN
10150 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
10151 DISCH_THEN(fun th -> MP_TAC th THEN
10152 MP_TAC(AP_TERM `proj:real^N->real^N` th)) THEN
10153 ASM_SIMP_TAC[NORM_POS_LT; VECTOR_MUL_RCANCEL] THEN ASM SET_TAC[]] THEN
10154 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
10155 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_CBALL_0] THEN
10156 X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
10157 ASM_REWRITE_TAC[NORM_0; VECTOR_MUL_LZERO; IN_INTER] THEN
10158 REWRITE_TAC[IN_CBALL_0; REAL_LE_LT] THEN STRIP_TAC THENL
10159 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET) THEN
10160 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[NORM_POS_LE] THEN
10162 ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ASM SET_TAC[]];
10164 REWRITE_TAC[SUBSET; IN_IMAGE; IN_CBALL_0; IN_INTER] THEN
10165 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
10166 ASM_CASES_TAC `x:real^N = vec 0` THENL
10167 [EXISTS_TAC `vec 0:real^N` THEN
10168 ASM_SIMP_TAC[NORM_0; VECTOR_MUL_LZERO; HULL_INC; REAL_POS];
10171 `!x. x IN usph ==> ~((surf:real^N->real^N) x = vec 0)`
10172 ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
10173 EXISTS_TAC `inv(norm(surf(proj x:real^N):real^N)) % x:real^N` THEN
10174 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN
10175 REWRITE_TAC[GSYM CONJ_ASSOC] THEN
10176 ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
10177 [NORM_POS_LT; REAL_LT_INV_EQ; HULL_INC; REAL_LT_MUL; NORM_MUL;
10178 REAL_ABS_INV; REAL_ABS_NORM] THEN
10179 REPEAT CONJ_TAC THENL
10180 [MATCH_MP_TAC(REAL_FIELD `~(y = &0) ==> x = (inv y * x) * y`) THEN
10181 ASM_SIMP_TAC[NORM_EQ_0; HULL_INC];
10182 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
10183 ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
10184 [GSYM real_div; REAL_LE_LDIV_EQ; NORM_POS_LT; HULL_INC; REAL_MUL_LID] THEN
10185 FIRST_X_ASSUM(MP_TAC o SPECL
10186 [`x:real^N`; `norm(surf(proj x:real^N):real^N) / norm(x:real^N)`]) THEN
10187 ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_LDIV_EQ; NORM_POS_LT] THEN
10188 GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN
10189 REWRITE_TAC[REAL_NOT_LT; REAL_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN
10191 `norm(surf(proj x)) / norm x % x:real^N = surf(proj x:real^N)`
10193 [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC I [GSYM th]) THEN
10194 ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
10195 [NORM_POS_LT; REAL_LT_INV_EQ; HULL_INC; REAL_LT_MUL; NORM_MUL;
10196 REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_DIV; REAL_LT_DIV;
10197 REAL_DIV_RMUL; NORM_EQ_0];
10198 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
10199 `IMAGE f s SUBSET t DIFF u ==> x IN s ==> ~(f x IN u)`)) THEN
10200 ASM_SIMP_TAC[HULL_INC]];
10201 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN
10202 MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
10203 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID; HULL_INC]]);;
10205 let HOMEOMORPHIC_CONVEX_COMPACT_SETS,
10206 HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS = (CONJ_PAIR o prove)
10207 (`(!s:real^M->bool t:real^N->bool.
10208 convex s /\ compact s /\ convex t /\ compact t /\ aff_dim s = aff_dim t
10209 ==> s homeomorphic t) /\
10210 (!s:real^M->bool t:real^N->bool.
10211 convex s /\ bounded s /\ convex t /\ bounded t /\ aff_dim s = aff_dim t
10212 ==> relative_frontier s homeomorphic relative_frontier t)`,
10214 (`!s:real^M->bool t:real^N->bool.
10215 convex s /\ compact s /\ convex t /\ compact t /\
10216 aff_dim s = aff_dim t
10217 ==> (s DIFF relative_interior s) homeomorphic
10218 (t DIFF relative_interior t) /\
10220 REPEAT GEN_TAC THEN
10221 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
10222 ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THENL
10223 [UNDISCH_TAC `relative_interior t:real^N->bool = {}` THEN
10224 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1;
10225 EMPTY_DIFF; HOMEOMORPHIC_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY];
10226 FIRST_X_ASSUM(X_CHOOSE_THEN `b:real^N` MP_TAC o
10227 GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN
10228 CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
10229 ASM_CASES_TAC `relative_interior s:real^M->bool = {}` THENL
10230 [UNDISCH_TAC `relative_interior s:real^M->bool = {}` THEN
10231 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1;
10232 EMPTY_DIFF; HOMEOMORPHIC_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY];
10233 FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^M` MP_TAC o
10234 GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN
10235 REPEAT(POP_ASSUM MP_TAC) THEN
10236 GEOM_ORIGIN_TAC `b:real^N` THEN REPEAT GEN_TAC THEN
10237 GEOM_ORIGIN_TAC `a:real^M` THEN REPEAT GEN_TAC THEN
10238 REPEAT DISCH_TAC THEN
10239 MP_TAC(ISPECL [`s:real^M->bool`; `vec 0:real^M`]
10240 STARLIKE_COMPACT_PROJECTIVE) THEN
10241 MP_TAC(ISPECL [`t:real^N->bool`; `vec 0:real^N`]
10242 STARLIKE_COMPACT_PROJECTIVE) THEN
10243 ASM_SIMP_TAC[IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT;
10244 REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN
10245 DISCH_THEN(fun th -> MATCH_MP_TAC MONO_AND THEN MP_TAC th) THEN
10246 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
10247 DISCH_THEN(fun th ->
10248 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHIC_TRANS) THEN
10249 MP_TAC(ONCE_REWRITE_RULE[HOMEOMORPHIC_SYM] th)) THEN
10250 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_TRANS) THEN
10251 REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET]
10252 RELATIVE_INTERIOR_SUBSET))) THEN
10253 FIRST_X_ASSUM(MP_TAC o SYM) THEN
10254 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; AFF_DIM_DIM_0] THEN
10255 REWRITE_TAC[INT_OF_NUM_EQ] THEN DISCH_TAC THEN
10256 MP_TAC(ISPECL [`span s:real^M->bool`; `span t:real^N->bool`]
10257 ISOMETRIES_SUBSPACES) THEN
10258 ASM_REWRITE_TAC[SUBSPACE_SPAN; DIM_SPAN; homeomorphic; HOMEOMORPHISM] THEN
10259 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN
10260 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN
10261 SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_CBALL_0; IN_SPHERE_0] THEN
10262 SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN ASM SET_TAC[]) in
10263 SIMP_TAC[lemma; relative_frontier] THEN REPEAT STRIP_TAC THEN
10264 MP_TAC(SPECL [`closure s:real^M->bool`; `closure t:real^N->bool`] lemma) THEN
10265 ASM_SIMP_TAC[CONVEX_CLOSURE; COMPACT_CLOSURE; AFF_DIM_CLOSURE] THEN
10266 ASM_SIMP_TAC[CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
10268 let HOMEOMORPHIC_CONVEX_COMPACT = prove
10269 (`!s:real^N->bool t:real^N->bool.
10270 convex s /\ compact s /\ ~(interior s = {}) /\
10271 convex t /\ compact t /\ ~(interior t = {})
10272 ==> s homeomorphic t`,
10273 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT_SETS THEN
10274 ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR]);;
10276 let HOMEOMORPHIC_CONVEX_COMPACT_CBALL = prove
10277 (`!s:real^N->bool b:real^N e.
10278 convex s /\ compact s /\ ~(interior s = {}) /\ &0 < e
10279 ==> s homeomorphic cball(b,e)`,
10280 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN
10281 ASM_REWRITE_TAC[COMPACT_CBALL; INTERIOR_CBALL; CONVEX_CBALL] THEN
10282 ASM_REWRITE_TAC[BALL_EQ_EMPTY; REAL_NOT_LE]);;
10284 let HOMEOMORPHIC_CLOSED_INTERVALS = prove
10285 (`!a b:real^N c d:real^N.
10286 ~(interval(a,b) = {}) /\ ~(interval(c,d) = {})
10287 ==> interval[a,b] homeomorphic interval[c,d]`,
10288 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN
10289 REWRITE_TAC[CONVEX_INTERVAL; COMPACT_INTERVAL] THEN
10290 ASM_REWRITE_TAC[INTERIOR_CLOSED_INTERVAL]);;
10292 (* ------------------------------------------------------------------------- *)
10293 (* More about affine dimension of special sets. *)
10294 (* ------------------------------------------------------------------------- *)
10296 let AFF_DIM_NONEMPTY_INTERIOR_EQ = prove
10298 convex s ==> (aff_dim s = &(dimindex (:N)) <=> ~(interior s = {}))`,
10299 REPEAT STRIP_TAC THEN EQ_TAC THEN
10300 ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR] THEN
10301 REPEAT STRIP_TAC THEN
10302 MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN
10303 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
10304 FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN
10305 ASM_SIMP_TAC[AFF_DIM_HYPERPLANE] THEN INT_ARITH_TAC);;
10307 let AFF_DIM_BALL = prove
10309 aff_dim(ball(a,r)) = if &0 < r then &(dimindex(:N)) else --(&1)`,
10310 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
10311 [MATCH_MP_TAC AFF_DIM_OPEN THEN
10312 ASM_REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE];
10313 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT; GSYM BALL_EQ_EMPTY]) THEN
10314 ASM_REWRITE_TAC[AFF_DIM_EMPTY]]);;
10316 let AFF_DIM_CBALL = prove
10318 aff_dim(cball(a,r)) =
10319 if &0 < r then &(dimindex(:N))
10320 else if r = &0 then &0 else --(&1)`,
10321 REPEAT GEN_TAC THEN REPEAT COND_CASES_TAC THENL
10322 [MATCH_MP_TAC AFF_DIM_NONEMPTY_INTERIOR THEN
10323 ASM_REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC;
10324 ASM_SIMP_TAC[CBALL_SING; AFF_DIM_SING];
10325 MATCH_MP_TAC(MESON[AFF_DIM_EMPTY] `s = {} ==> aff_dim s = --(&1)`) THEN
10326 REWRITE_TAC[CBALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC]);;
10328 let AFF_DIM_INTERVAL = prove
10330 aff_dim(interval[a,b]) =
10331 if interval[a,b] = {} then --(&1)
10332 else &(CARD {i | 1 <= i /\ i <= dimindex(:N) /\ a$i < b$i})) /\
10334 aff_dim(interval(a,b)) =
10335 if interval(a,b) = {} then --(&1)
10336 else &(dimindex(:N)))`,
10337 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
10338 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_OPEN; OPEN_INTERVAL] THEN
10339 POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN
10340 REPEAT STRIP_TAC THEN
10341 REWRITE_TAC[VECTOR_ADD_COMPONENT; VEC_COMPONENT; REAL_LT_LADD] THEN
10342 ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; ENDS_IN_INTERVAL] THEN AP_TERM_TAC THEN
10343 ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN MATCH_MP_TAC DIM_UNIQUE THEN EXISTS_TAC
10344 `{basis i:real^N | 1 <= i /\ i <= dimindex(:N) /\ &0 < (b:real^N)$i}` THEN
10345 RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY; VEC_COMPONENT]) THEN
10346 REPEAT CONJ_TAC THENL
10347 [REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
10348 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
10349 SUBGOAL_THEN `basis i:real^N = inv(b$i) % (b:real^N)$i % basis i`
10351 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN
10352 REWRITE_TAC[VECTOR_MUL_LID];
10353 MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN
10354 SIMP_TAC[IN_INTERVAL; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN
10355 X_GEN_TAC `j:num` THEN REWRITE_TAC[VEC_COMPONENT] THEN
10356 COND_CASES_TAC THEN
10357 ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_MUL_RID; REAL_LE_REFL]];
10358 MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN
10359 REWRITE_TAC[SUBSPACE_SPAN; SUBSET; IN_INTERVAL; VEC_COMPONENT] THEN
10360 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
10361 GEN_REWRITE_TAC LAND_CONV [GSYM BASIS_EXPANSION] THEN
10362 MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG] THEN
10363 X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN
10364 ASM_CASES_TAC `&0 < (b:real^N)$i` THENL
10365 [MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[];
10366 SUBGOAL_THEN `(x:real^N)$i = &0`
10367 (fun th -> REWRITE_TAC[th; VECTOR_MUL_LZERO; SPAN_0]) THEN
10368 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN
10369 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC];
10370 MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN
10371 REWRITE_TAC[SET_RULE `~(a IN {f x | P x}) <=> !x. P x ==> ~(f x = a)`] THEN
10372 SIMP_TAC[BASIS_NONZERO; pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
10373 SIMP_TAC[FORALL_IN_GSPEC; BASIS_INJ_EQ; ORTHOGONAL_BASIS_BASIS];
10374 GEN_REWRITE_TAC LAND_CONV [SIMPLE_IMAGE_GEN] THEN
10375 MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN
10376 SIMP_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC; BASIS_INJ_EQ;
10378 SIMP_TAC[CONJ_ASSOC; GSYM IN_NUMSEG; FINITE_RESTRICT; FINITE_NUMSEG]]);;
10380 (* ------------------------------------------------------------------------- *)
10381 (* Deducing convexity from midpoint convexity in common cases. *)
10382 (* ------------------------------------------------------------------------- *)
10384 let MIDPOINT_CONVEX_DYADIC_RATIONALS = prove
10385 (`!f:real^N->real s.
10386 (!x y. x IN s /\ y IN s
10387 ==> midpoint(x,y) IN s /\
10388 f(midpoint(x,y)) <= (f(x) + f(y)) / &2)
10390 x IN s /\ y IN s /\ m + p = 2 EXP n
10391 ==> (&m / &2 pow n % x + &p / &2 pow n % y) IN s /\
10392 f(&m / &2 pow n % x + &p / &2 pow n % y)
10393 <= &m / &2 pow n * f x + &p / &2 pow n * f y`,
10394 REPEAT GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THENL
10395 [REWRITE_TAC[ARITH_RULE
10396 `m + p = 2 EXP 0 <=> m = 0 /\ p = 1 \/ m = 1 /\ p = 0`] THEN
10397 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
10398 CONV_TAC REAL_RAT_REDUCE_CONV THEN
10399 ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO;
10400 VECTOR_ADD_LID; VECTOR_ADD_RID] THEN
10402 MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL
10403 [REWRITE_TAC[VECTOR_ADD_SYM; REAL_ADD_SYM; ADD_SYM] THEN MESON_TAC[];
10405 MAP_EVERY X_GEN_TAC [`m:num`; `p:num`] THEN DISCH_TAC THEN
10406 REPEAT GEN_TAC THEN REWRITE_TAC[EXP; real_pow] THEN STRIP_TAC THEN
10407 REWRITE_TAC[real_div; REAL_INV_MUL] THEN
10408 ONCE_REWRITE_TAC[REAL_ARITH `x * inv(&2) * y = inv(&2) * x * y`] THEN
10409 ONCE_REWRITE_TAC[GSYM REAL_MUL_ASSOC; GSYM VECTOR_MUL_ASSOC] THEN
10410 REWRITE_TAC[GSYM REAL_ADD_LDISTRIB; GSYM VECTOR_ADD_LDISTRIB] THEN
10411 SUBGOAL_THEN `2 EXP n <= p` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
10412 SUBGOAL_THEN `&p * inv(&2 pow n) = &(p - 2 EXP n) * inv(&2 pow n) + &1`
10414 [ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW] THEN
10415 ASM_SIMP_TAC[REAL_SUB_RDISTRIB; REAL_MUL_RINV; REAL_LT_IMP_NZ;
10416 REAL_LT_POW2] THEN REAL_ARITH_TAC;
10417 REWRITE_TAC[VECTOR_ADD_RDISTRIB; REAL_ADD_RDISTRIB] THEN
10418 REWRITE_TAC[VECTOR_MUL_LID; REAL_MUL_LID] THEN
10419 REWRITE_TAC[VECTOR_ADD_ASSOC; REAL_ADD_ASSOC] THEN
10420 REWRITE_TAC[GSYM midpoint; GSYM real_div] THEN FIRST_X_ASSUM(fun th ->
10421 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
10422 FIRST_X_ASSUM(fun th ->
10423 W(MP_TAC o PART_MATCH (lhand o rand) th o funpow 3 lhand o snd)) THEN
10424 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
10425 [ASM_ARITH_TAC; SIMP_TAC[] THEN REAL_ARITH_TAC]]]);;
10427 let CONTINUOUS_MIDPOINT_CONVEX = prove
10428 (`!f:real^N->real s.
10429 (lift o f) continuous_on s /\ convex s /\
10430 (!x y. x IN s /\ y IN s ==> f(midpoint(x,y)) <= (f(x) + f(y)) / &2)
10431 ==> f convex_on s`,
10432 REWRITE_TAC[midpoint] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[convex_on] THEN
10433 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
10434 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
10435 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
10436 REWRITE_TAC[REAL_ARITH `u + v = &1 <=> v = &1 - u`; IMP_CONJ] THEN
10437 REWRITE_TAC[FORALL_UNWIND_THM2; REAL_SUB_LE] THEN
10438 REWRITE_TAC[FORALL_DROP; GSYM DROP_VEC; IMP_IMP; GSYM IN_INTERVAL_1] THEN
10439 MP_TAC(ISPEC `interval[vec 0:real^1,vec 1]`
10440 CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET) THEN
10441 SIMP_TAC[CONVEX_INTERVAL; INTERIOR_CLOSED_INTERVAL;
10442 CLOSURE_CLOSED; CLOSED_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN
10443 REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop] THEN
10444 DISCH_THEN(fun th -> SUBST1_TAC(SYM th) THEN ASSUME_TAC th) THEN
10445 ONCE_REWRITE_TAC[REAL_ARITH `a <= b <=> a - b <= &0`] THEN
10446 MATCH_MP_TAC CONTINUOUS_LE_ON_CLOSURE THEN
10447 REWRITE_TAC[IN_INTER; IMP_CONJ_ALT; FORALL_IN_GSPEC] THEN
10448 FIRST_X_ASSUM SUBST1_TAC THEN
10449 REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; GSYM FORALL_DROP; DROP_VEC] THEN
10451 [REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_ADD; LIFT_CMUL] THEN
10452 MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THENL
10453 [REPLICATE_TAC 2 (ONCE_REWRITE_TAC[GSYM o_DEF]) THEN
10454 REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
10457 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10458 CONTINUOUS_ON_SUBSET)) THEN
10459 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; GSYM FORALL_DROP;
10460 DROP_VEC] THEN REPEAT STRIP_TAC THEN
10461 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
10462 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC];
10464 MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN
10465 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
10466 SIMP_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST;
10467 LIFT_SUB; CONTINUOUS_ON_SUB];
10468 MAP_EVERY X_GEN_TAC [`n:num`; `i:real`] THEN
10469 ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_LT_INV_EQ; REAL_LT_POW2] THEN
10470 ASM_CASES_TAC `&0 <= i` THEN ASM_SIMP_TAC[INTEGER_POS] THEN
10471 DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST_ALL_TAC) THEN
10472 REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN
10473 SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN
10474 GEN_REWRITE_TAC (LAND_CONV o DEPTH_CONV)
10475 [REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN DISCH_TAC THEN
10476 MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`]
10477 MIDPOINT_CONVEX_DYADIC_RATIONALS) THEN
10479 [ASM_SIMP_TAC[midpoint] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB] THEN
10480 REPEAT STRIP_TAC THEN
10481 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
10482 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
10483 DISCH_THEN(MP_TAC o SPECL
10484 [`n:num`; `m:num`; `2 EXP n - m`; `x:real^N`; `y:real^N`]) THEN
10485 ASM_REWRITE_TAC[] THEN
10486 ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN
10487 ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW] THEN
10488 ASM_SIMP_TAC[REAL_LT_POW2; REAL_FIELD
10489 `&0 < y ==> (y - x) / y = &1 - x / y`] THEN
10490 REAL_ARITH_TAC]]);;
10492 (* ------------------------------------------------------------------------- *)
10493 (* Slightly shaper separating/supporting hyperplane results. *)
10494 (* ------------------------------------------------------------------------- *)
10496 let SEPARATING_HYPERPLANE_RELATIVE_INTERIORS = prove
10497 (`!s t. convex s /\ convex t /\
10498 ~(s = {} /\ t = (:real^N) \/ s = (:real^N) /\ t = {}) /\
10499 DISJOINT (relative_interior s) (relative_interior t)
10500 ==> ?a b. ~(a = vec 0) /\
10501 (!x. x IN s ==> a dot x <= b) /\
10502 (!x. x IN t ==> a dot x >= b)`,
10503 REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC
10504 [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
10505 ASM_REWRITE_TAC[NOT_IN_EMPTY; UNIV_NOT_EMPTY; CONVEX_EMPTY;
10506 RELATIVE_INTERIOR_EMPTY] THEN
10508 [EXISTS_TAC `basis 1:real^N` THEN
10509 SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL];
10510 FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^N` o MATCH_MP (SET_RULE
10511 `~(s = UNIV) ==> ?a. ~(a IN s)`)) THEN
10512 MP_TAC(ISPECL [`t:real^N->bool`; `x:real^N`]
10513 SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN
10515 FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^N` o MATCH_MP (SET_RULE
10516 `~(s = UNIV) ==> ?a. ~(a IN s)`)) THEN
10517 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`]
10518 SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN
10519 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; real_ge] THEN
10520 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN
10521 MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`] THEN
10522 ASM_REWRITE_TAC[VECTOR_NEG_EQ_0; DOT_LNEG; REAL_LE_NEG2];
10523 MP_TAC(ISPECL [`relative_interior s:real^N->bool`;
10524 `relative_interior t:real^N->bool`]
10525 SEPARATING_HYPERPLANE_SETS) THEN
10526 ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY; CONVEX_RELATIVE_INTERIOR] THEN
10527 SIMP_TAC[real_ge] THEN MATCH_MP_TAC MONO_EXISTS THEN
10528 X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN
10529 X_GEN_TAC `b:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
10530 CONJ_TAC THEN MATCH_MP_TAC
10531 (MESON[CONVEX_CLOSURE_RELATIVE_INTERIOR; CLOSURE_SUBSET; SUBSET]
10532 `convex s /\ (!x. x IN closure(relative_interior s) ==> P x)
10533 ==> !x. x IN s ==> P x`) THEN
10534 ASM_REWRITE_TAC[] THENL
10535 [MATCH_MP_TAC CONTINUOUS_LE_ON_CLOSURE;
10536 MATCH_MP_TAC CONTINUOUS_GE_ON_CLOSURE] THEN
10537 ASM_REWRITE_TAC[CONTINUOUS_ON_LIFT_DOT]]);;
10539 let SUPPORTING_HYPERPLANE_RELATIVE_BOUNDARY = prove
10541 convex s /\ x IN s /\ ~(x IN relative_interior s)
10542 ==> ?a. ~(a = vec 0) /\
10543 (!y. y IN s ==> a dot x <= a dot y) /\
10544 (!y. y IN relative_interior s ==> a dot x < a dot y)`,
10545 REPEAT STRIP_TAC THEN
10546 MP_TAC(ISPECL [`relative_interior s:real^N->bool`; `x:real^N`]
10547 SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN
10548 ASM_SIMP_TAC[CONVEX_SING; CONVEX_RELATIVE_INTERIOR;
10549 RELATIVE_INTERIOR_EQ_EMPTY; real_ge] THEN
10550 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
10551 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
10552 REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
10553 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[] THEN
10554 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
10555 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
10556 MP_TAC(ISPECL [`lift o (\x:real^N. a dot x)`;
10557 `relative_interior s:real^N->bool`;
10558 `y:real^N`; `(a:real^N) dot x`; `1`]
10559 CONTINUOUS_ON_CLOSURE_COMPONENT_GE) THEN
10560 REWRITE_TAC[CONTINUOUS_ON_LIFT_DOT; GSYM drop; o_THM; LIFT_DROP] THEN
10561 ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN
10562 ASM_MESON_TAC[CLOSURE_SUBSET; REAL_LE_TRANS; SUBSET];
10564 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
10565 REWRITE_TAC[REAL_LT_LE] THEN CONJ_TAC THENL
10566 [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN
10567 DISCH_TAC THEN UNDISCH_TAC `(y:real^N) IN relative_interior s` THEN
10568 REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL] THEN
10569 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
10570 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_INTER; IN_CBALL] THEN
10571 X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
10572 DISCH_THEN(MP_TAC o SPEC `y + --(e / norm(a)) % ((x + a) - x):real^N`) THEN
10573 REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL
10574 [SIMP_TAC[NORM_ARITH `dist(y:real^N,y + e) = norm e`; VECTOR_ADD_SUB] THEN
10575 REWRITE_TAC[NORM_MUL; REAL_ABS_NEG; REAL_ABS_DIV; REAL_ABS_NORM] THEN
10576 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC;
10577 MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
10578 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC] THEN
10579 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
10580 `x IN s ==> s SUBSET t ==> x IN t`)) THEN
10581 MATCH_MP_TAC HULL_MONO THEN
10582 ASM_REWRITE_TAC[INSERT_SUBSET; RELATIVE_INTERIOR_SUBSET];
10583 REWRITE_TAC[VECTOR_ADD_SUB] THEN DISCH_TAC THEN
10584 UNDISCH_TAC `!y:real^N. y IN s ==> a dot x <= a dot y` THEN
10585 DISCH_THEN(MP_TAC o SPEC `y + --(e / norm(a)) % a:real^N`) THEN
10586 ASM_REWRITE_TAC[DOT_RMUL; DOT_RNEG; DOT_RADD] THEN
10587 MATCH_MP_TAC(REAL_ARITH `&0 < x * y ==> ~(a <= a + --x * y)`) THEN
10588 ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; NORM_POS_LT; DOT_POS_LT]]);;
10590 let SUPPORTING_HYPERPLANE_RELATIVE_FRONTIER = prove
10592 convex s /\ x IN closure s /\ ~(x IN relative_interior s)
10593 ==> ?a. ~(a = vec 0) /\
10594 (!y. y IN closure s ==> a dot x <= a dot y) /\
10595 (!y. y IN relative_interior s ==> a dot x < a dot y)`,
10596 REPEAT STRIP_TAC THEN
10597 MP_TAC(ISPECL [`closure s:real^N->bool`; `x:real^N`]
10598 SUPPORTING_HYPERPLANE_RELATIVE_BOUNDARY) THEN
10599 ASM_SIMP_TAC[CONVEX_CLOSURE; CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
10601 (* ------------------------------------------------------------------------- *)
10602 (* Containment of rays in unbounded convex sets. *)
10603 (* ------------------------------------------------------------------------- *)
10605 let UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY = prove
10607 convex s /\ ~bounded s /\ closed s /\ a IN s
10608 ==> ?l. ~(l = vec 0) /\ !t. &0 <= t ==> (a + t % l) IN s`,
10609 GEN_GEOM_ORIGIN_TAC `a:real^N` ["l"] THEN
10610 REWRITE_TAC[VECTOR_ADD_LID] THEN REPEAT STRIP_TAC THEN
10611 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [BOUNDED_POS]) THEN
10612 REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN
10613 DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&n + &1:real`) THEN
10614 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_ARITH `&0 < &n + &1`] THEN
10615 REWRITE_TAC[REAL_NOT_LE; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
10616 X_GEN_TAC `x:num->real^N` THEN REWRITE_TAC[FORALL_AND_THM] THEN
10618 SUBGOAL_THEN `!n. ~((x:num->real^N) n = vec 0)` ASSUME_TAC THENL
10619 [ASM_MESON_TAC[NORM_ARITH `~(&n + &1 < norm(vec 0:real^N))`]; ALL_TAC] THEN
10620 MP_TAC(ISPEC `sphere(vec 0:real^N,&1)` compact) THEN
10621 REWRITE_TAC[COMPACT_SPHERE] THEN
10622 DISCH_THEN(MP_TAC o SPEC `\n. inv(norm(x n)) % (x:num->real^N) n`) THEN
10623 ASM_SIMP_TAC[IN_SPHERE_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM;
10624 REAL_MUL_LINV; NORM_EQ_0; o_DEF] THEN
10625 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN
10626 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL
10627 [ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`]; ALL_TAC] THEN
10628 X_GEN_TAC `t:real` THEN DISCH_TAC THEN
10629 MATCH_MP_TAC CLOSED_CONTAINS_SEQUENTIAL_LIMIT THEN
10631 `?N:num. !n. N <= n ==> t / norm(x n:real^N) <= &1`
10632 STRIP_ASSUME_TAC THENL
10633 [ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT] THEN
10634 MP_TAC(SPEC `t:real` REAL_ARCH_SIMPLE) THEN
10635 MATCH_MP_TAC MONO_EXISTS THEN
10636 REWRITE_TAC[GSYM REAL_OF_NUM_LE; REAL_MUL_LID] THEN
10637 ASM_MESON_TAC[REAL_ARITH `t <= m /\ m <= n /\ n + &1 < x ==> t <= x`];
10638 EXISTS_TAC `\n:num. t / norm((x:num->real^N)(r(N + n))) % x(r(N + n))` THEN
10639 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
10640 [X_GEN_TAC `n:num` THEN
10641 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
10642 DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN
10643 ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
10644 DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE] THEN
10645 FIRST_X_ASSUM MATCH_MP_TAC THEN
10646 FIRST_ASSUM(MP_TAC o SPEC `N + n:num` o MATCH_MP MONOTONE_BIGGER) THEN
10648 REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THEN
10649 MATCH_MP_TAC LIM_CMUL THEN ONCE_REWRITE_TAC[ADD_SYM] THEN
10650 FIRST_ASSUM(MP_TAC o SPEC `N:num` o MATCH_MP SEQ_OFFSET) THEN
10651 ASM_REWRITE_TAC[]]]);;
10653 let CONVEX_CLOSED_CONTAINS_SAME_RAY = prove
10655 convex s /\ closed s /\ b IN s /\ (!t. &0 <= t ==> (a + t % l) IN s)
10656 ==> !t. &0 <= t ==> (b + t % l) IN s`,
10657 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `&0`) THEN
10658 REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN DISCH_TAC THEN
10659 MATCH_MP_TAC(ISPEC `sequentially` LIM_IN_CLOSED_SET) THEN
10660 EXISTS_TAC `\n. (&1 - t / (&n + &1)) % b +
10661 t / (&n + &1) % (a + (&n + &1) % l):real^N` THEN
10662 ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL
10663 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
10664 MP_TAC(SPEC `t:real` REAL_ARCH_SIMPLE) THEN
10665 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
10666 DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
10667 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
10668 ASM_SIMP_TAC[REAL_LE_DIV; REAL_ARITH `&0 <= &n + &1`] THEN
10669 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
10670 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN ASM_REAL_ARITH_TAC;
10671 REWRITE_TAC[VECTOR_ARITH
10672 `(&1 - u) % b + u % c:real^N = b + u % (c - b)`] THEN
10673 MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
10674 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_SUB_LDISTRIB] THEN
10675 SIMP_TAC[VECTOR_MUL_ASSOC; REAL_FIELD `t / (&n + &1) * (&n + &1) = t`] THEN
10676 SIMP_TAC[VECTOR_ARITH `(v % a + b) - v % c:real^N = b + v % (a - c)`] THEN
10677 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN
10678 MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
10679 REWRITE_TAC[real_div; VECTOR_ARITH `(x * y) % a:real^N = y % x % a`] THEN
10680 MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN
10681 EXISTS_TAC `norm(t % (a - b):real^N)` THEN
10682 REWRITE_TAC[REAL_LE_REFL; EVENTUALLY_TRUE; o_DEF] THEN
10683 MP_TAC(MATCH_MP SEQ_OFFSET SEQ_HARMONIC) THEN
10684 SIMP_TAC[REAL_OF_NUM_ADD]]);;
10686 let UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAYS = prove
10688 convex s /\ ~bounded s /\ closed s
10689 ==> ?l. ~(l = vec 0) /\ !a t. a IN s /\ &0 <= t ==> (a + t % l) IN s`,
10690 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
10691 ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN
10692 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
10693 ASM_MESON_TAC[UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY;
10694 CONVEX_CLOSED_CONTAINS_SAME_RAY]);;
10696 let RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAY = prove
10698 convex s /\ ~bounded s /\ a IN relative_interior s
10699 ==> ?l. ~(l = vec 0) /\
10700 !t. &0 <= t ==> (a + t % l) IN relative_interior s`,
10701 REPEAT STRIP_TAC THEN
10702 MP_TAC(ISPECL [`closure s:real^N->bool`; `a:real^N`]
10703 UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY) THEN
10704 ASM_SIMP_TAC[CONVEX_CLOSURE; CLOSED_CLOSURE] THEN ANTS_TAC THENL
10705 [ASM_MESON_TAC[BOUNDED_SUBSET; SUBSET; CLOSURE_SUBSET;
10706 RELATIVE_INTERIOR_SUBSET];
10707 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN STRIP_TAC THEN
10708 ASM_REWRITE_TAC[] THEN
10709 ONCE_REWRITE_TAC[VECTOR_ARITH
10710 `a + t % l:real^N =
10711 (a + (&2 * t) % l) - inv(&2) % ((a + (&2 * t) % l) - a)`] THEN
10712 REPEAT STRIP_TAC THEN
10713 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
10714 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN
10715 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]);;
10717 let RELATIVE_INTERIOR_CONVEX_CONTAINS_SAME_RAY = prove
10719 convex s /\ b IN relative_interior s /\
10720 (!t. &0 <= t ==> (a + t % l) IN relative_interior s)
10721 ==> !t. &0 <= t ==> (b + t % l) IN relative_interior s`,
10722 REPEAT STRIP_TAC THEN
10723 MP_TAC(ISPECL [`closure s:real^N->bool`; `a:real^N`; `b:real^N`; `l:real^N`]
10724 CONVEX_CLOSED_CONTAINS_SAME_RAY) THEN
10725 ASM_SIMP_TAC[CONVEX_CLOSURE; CLOSED_CLOSURE] THEN ANTS_TAC THENL
10726 [ASM_MESON_TAC[BOUNDED_SUBSET; SUBSET; CLOSURE_SUBSET;
10727 RELATIVE_INTERIOR_SUBSET];
10729 ONCE_REWRITE_TAC[VECTOR_ARITH
10730 `a + t % l:real^N =
10731 (a + (&2 * t) % l) - inv(&2) % ((a + (&2 * t) % l) - a)`] THEN
10732 REPEAT STRIP_TAC THEN
10733 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
10734 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN
10735 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]);;
10737 let RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAYS = prove
10739 convex s /\ ~bounded s
10740 ==> ?l. ~(l = vec 0) /\
10741 !a t. a IN relative_interior s /\ &0 <= t
10742 ==> (a + t % l) IN relative_interior s`,
10743 REPEAT GEN_TAC THEN
10744 ASM_CASES_TAC `relative_interior s:real^N->bool = {}` THENL
10745 [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; BOUNDED_EMPTY]; ALL_TAC] THEN
10746 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
10747 ASM_MESON_TAC[RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAY;
10748 RELATIVE_INTERIOR_CONVEX_CONTAINS_SAME_RAY]);;
10750 (* ------------------------------------------------------------------------- *)
10751 (* Explicit formulas for interior and relative interior of convex hull. *)
10752 (* ------------------------------------------------------------------------- *)
10754 let EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL = prove
10756 ==> {y:real^N | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\
10758 vsum s (\x. u x % x) = y}
10759 SUBSET relative_interior(convex hull s)`,
10760 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
10761 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ] THEN
10762 REWRITE_TAC[EMPTY_GSPEC; EMPTY_SUBSET] THEN
10763 REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_MAXIMAL THEN
10764 REWRITE_TAC[AFFINE_HULL_CONVEX_HULL] THEN CONJ_TAC THENL
10765 [REWRITE_TAC[CONVEX_HULL_FINITE; SUBSET; IN_ELIM_THM] THEN
10766 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_LT_IMP_LE];
10768 REWRITE_TAC[open_in; IN_ELIM_THM] THEN CONJ_TAC THENL
10769 [REWRITE_TAC[AFFINE_HULL_FINITE; SUBSET; IN_ELIM_THM] THEN
10770 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_LT_IMP_LE];
10772 X_GEN_TAC `y:real^N` THEN
10773 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
10774 ABBREV_TAC `e = inf (IMAGE (\x:real^N. min (&1 - u x) (u x)) s)` THEN
10775 SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL
10776 [EXPAND_TAC "e" THEN
10777 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
10778 ASM_SIMP_TAC[REAL_LT_MIN; REAL_SUB_LT; FORALL_IN_IMAGE];
10780 MP_TAC(ISPEC `IMAGE (\z:real^N. z - y) (affine hull s)` BASIS_EXISTS) THEN
10781 REWRITE_TAC[SUBSET_IMAGE] THEN
10782 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool`
10783 (CONJUNCTS_THEN2 (X_CHOOSE_THEN `c:real^N->bool` (STRIP_ASSUME_TAC o GSYM))
10785 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; HAS_SIZE] THEN
10786 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
10787 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
10788 ASM_SIMP_TAC[SPAN_FINITE; IN_ELIM_THM] THEN
10789 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
10790 REWRITE_TAC[SKOLEM_THM] THEN
10791 DISCH_THEN(X_CHOOSE_TAC `compo:real^N->real^N->real`) THEN
10792 FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o
10793 MATCH_MP BASIS_COORDINATES_LIPSCHITZ) THEN
10795 `!i. i IN b ==> ?u. sum s u = &0 /\ vsum s (\x:real^N. u x % x) = i`
10797 [EXPAND_TAC "b" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
10798 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
10799 SUBGOAL_THEN `(x:real^N) IN affine hull s` MP_TAC THENL
10800 [ASM SET_TAC[]; REWRITE_TAC[AFFINE_HULL_FINITE; IN_ELIM_THM]] THEN
10801 DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN
10802 EXISTS_TAC `(\x. v x - u x):real^N->real` THEN
10803 ASM_SIMP_TAC[SUM_SUB; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN
10804 REWRITE_TAC[REAL_SUB_REFL; VECTOR_SUB_RZERO];
10805 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
10806 [RIGHT_IMP_EXISTS_THM; SKOLEM_THM; FORALL_AND_THM;
10807 TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN
10808 DISCH_THEN(X_CHOOSE_THEN `w:real^N->real^N->real` STRIP_ASSUME_TAC)] THEN
10809 EXISTS_TAC `e / B /
10810 (&1 + sum (b:real^N->bool)
10811 (\i. abs(sup(IMAGE (abs o w i) (s:real^N->bool)))))` THEN
10812 ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 <= x ==> &0 < &1 + x`;
10813 SUM_POS_LE; REAL_ABS_POS] THEN
10814 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
10816 `\x:real^N. u x + sum (b:real^N->bool)
10817 (\i. compo (z:real^N) i * w i x)` THEN
10818 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
10820 ASM_SIMP_TAC[SUM_ADD; REAL_ARITH `&1 + x = &1 <=> x = &0`] THEN
10821 W(MP_TAC o PART_MATCH (lhs o rand) SUM_SWAP o lhand o snd) THEN
10822 ASM_REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN
10823 MATCH_MP_TAC SUM_EQ_0 THEN
10824 ASM_SIMP_TAC[SUM_LMUL; ETA_AX; REAL_MUL_RZERO; SUM_0];
10825 ASM_SIMP_TAC[VSUM_ADD; VECTOR_ADD_RDISTRIB] THEN
10826 ONCE_REWRITE_TAC[VECTOR_ARITH `y + w:real^N = z <=> w = z - y`] THEN
10827 ASM_SIMP_TAC[GSYM VSUM_LMUL; GSYM VSUM_RMUL; GSYM VECTOR_MUL_ASSOC] THEN
10828 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_SWAP o lhand o snd) THEN
10829 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
10830 ASM_SIMP_TAC[VSUM_LMUL] THEN MATCH_MP_TAC EQ_TRANS THEN
10831 EXISTS_TAC `vsum b (\v:real^N. compo (z:real^N) v % v)` THEN
10832 CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[]] THEN
10833 MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[]] THEN
10834 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
10835 MATCH_MP_TAC(REAL_ARITH
10836 `abs(x) < min u (&1 - u) ==> &0 < u + x /\ u + x < &1`) THEN
10837 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC
10838 `B * norm(z - y:real^N) * sum (b:real^N->bool)
10839 (\i. abs(sup(IMAGE (abs o w i) (s:real^N->bool))))` THEN
10841 [REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_ABS_LE THEN
10842 ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_MUL_ASSOC] THEN
10843 X_GEN_TAC `i:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
10844 REWRITE_TAC[REAL_ABS_POS] THEN CONJ_TAC THENL
10845 [FIRST_X_ASSUM(MP_TAC o SPECL [`(compo:real^N->real^N->real) z`;
10848 MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs a`) THEN
10849 ASM_SIMP_TAC[REAL_LE_SUP_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
10850 REWRITE_TAC[EXISTS_IN_IMAGE; o_THM] THEN ASM_MESON_TAC[REAL_LE_REFL]];
10852 GEN_REWRITE_TAC LAND_CONV [REAL_MUL_SYM] THEN
10853 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
10854 MATCH_MP_TAC(REAL_ARITH
10855 `&0 <= x /\ x * (&1 + e) < d ==> x * e < d`) THEN
10856 REWRITE_TAC[NORM_POS_LE] THEN
10857 ASM_SIMP_TAC[NORM_POS_LE; GSYM REAL_LT_RDIV_EQ;
10858 REAL_ARITH `&0 <= x ==> &0 < &1 + x`;
10859 SUM_POS_LE; REAL_ABS_POS] THEN
10860 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
10861 `dist(z:real^N,y) < k ==> k <= d ==> norm(z - y) < d`)) THEN
10862 ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_ARITH `&0 <= x ==> &0 < &1 + x`;
10863 SUM_POS_LE; REAL_ABS_POS] THEN
10864 EXPAND_TAC "e" THEN
10865 ASM_SIMP_TAC[REAL_INF_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
10866 REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `x:real^N` THEN
10867 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
10869 let EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL = prove
10871 ==> {y:real^N | ?u. (!x. x IN s ==> &0 < u x) /\
10873 vsum s (\x. u x % x) = y}
10874 SUBSET relative_interior(convex hull s)`,
10875 REPEAT STRIP_TAC THEN
10876 ASM_CASES_TAC `s:real^N->bool = {}` THEN
10877 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ] THEN
10878 REWRITE_TAC[EMPTY_GSPEC; EMPTY_SUBSET] THEN
10879 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
10880 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
10881 ASM_CASES_TAC `s = {a:real^N}` THENL
10882 [ASM_REWRITE_TAC[SUM_SING; VSUM_SING; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
10883 REWRITE_TAC[RELATIVE_INTERIOR_SING; CONVEX_HULL_SING] THEN
10884 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_SING] THEN
10885 MESON_TAC[VECTOR_MUL_LID];
10886 FIRST_ASSUM(MP_TAC o MATCH_MP
10887 EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL) THEN
10888 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) THEN
10889 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `w:real^N` THEN
10890 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->real` THEN
10891 STRIP_TAC THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
10892 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL
10893 [ASM SET_TAC[]; ALL_TAC] THEN
10894 SUBGOAL_THEN `sum {x,y} u <= sum s (u:real^N->real)` MP_TAC THENL
10895 [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
10896 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; REAL_LT_IMP_LE; IN_DIFF] THEN
10898 ASM_SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
10899 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
10900 MATCH_MP_TAC(REAL_ARITH `&0 < y ==> x + y + &0 <= &1 ==> x < &1`) THEN
10901 ASM_SIMP_TAC[]]]);;
10903 let RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT = prove
10904 (`!s. ~(affine_dependent s)
10905 ==> relative_interior(convex hull s) =
10906 {y:real^N | ?u. (!x. x IN s ==> &0 < u x) /\
10908 vsum s (\x. u x % x) = y}`,
10909 REPEAT STRIP_TAC THEN
10910 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
10911 MATCH_MP_TAC SUBSET_ANTISYM THEN
10912 ASM_SIMP_TAC[EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL] THEN
10913 ASM_CASES_TAC `?a:real^N. s = {a}` THENL
10914 [FIRST_X_ASSUM(CHOOSE_THEN SUBST1_TAC) THEN
10915 ASM_REWRITE_TAC[SUM_SING; VSUM_SING; CONVEX_HULL_SING;
10916 RELATIVE_INTERIOR_SING] THEN
10917 REWRITE_TAC[IN_ELIM_THM; SUBSET; IN_SING] THEN
10918 REPEAT STRIP_TAC THEN EXISTS_TAC `\x:real^N. &1` THEN
10919 ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_LT_01];
10921 MATCH_MP_TAC(SET_RULE
10922 `relative_interior s SUBSET s /\
10923 (!x. x IN s /\ ~(x IN t) ==> ~(x IN relative_interior s))
10924 ==> relative_interior s SUBSET t`) THEN
10925 REWRITE_TAC[RELATIVE_INTERIOR_SUBSET] THEN
10926 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN
10927 REWRITE_TAC[AFFINE_HULL_CONVEX_HULL; IN_ELIM_THM; NOT_EXISTS_THM] THEN
10928 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN
10929 DISCH_THEN(CONJUNCTS_THEN2
10930 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)
10931 (MP_TAC o SPEC `u:real^N->real`)) THEN
10932 ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; IN_RELATIVE_INTERIOR; DE_MORGAN_THM;
10933 SUBSET; IN_ELIM_THM; IN_BALL; IN_INTER] THEN
10934 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN DISJ2_TAC THEN
10935 DISCH_THEN(X_CHOOSE_THEN `e:real`
10936 (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))) THEN
10937 SUBGOAL_THEN `(u:real^N->real) a = &0` ASSUME_TAC THENL
10938 [ASM_SIMP_TAC[REAL_ARITH `&0 <= x /\ ~(&0 < x) ==> x = &0`]; ALL_TAC] THEN
10939 SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b = a)` STRIP_ASSUME_TAC THENL
10940 [ASM SET_TAC[];ALL_TAC] THEN
10941 SUBGOAL_THEN `?d. &0 < d /\ norm(d % (a - b):real^N) < e`
10942 STRIP_ASSUME_TAC THENL
10943 [EXISTS_TAC `e / &2 / norm(a - b:real^N)` THEN
10944 ASM_SIMP_TAC[NORM_MUL; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NORM_POS_LT;
10945 REAL_ABS_DIV; REAL_ABS_NORM; REAL_ABS_NUM;
10946 REAL_DIV_RMUL; REAL_LT_IMP_NZ; VECTOR_SUB_EQ] THEN
10947 ASM_REAL_ARITH_TAC;
10949 REMOVE_THEN "*" (MP_TAC o SPEC `y - d % (a - b):real^N`) THEN
10950 ASM_REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - b) = norm b`] THEN
10951 REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL
10952 [MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN
10953 ASM_SIMP_TAC[HULL_INC; AFFINE_AFFINE_HULL] THEN
10954 REWRITE_TAC[AFFINE_HULL_FINITE; IN_ELIM_THM] THEN
10955 EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[];
10957 DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN
10958 UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN
10959 ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE] THEN
10960 EXISTS_TAC `\x:real^N. (v x - u x) -
10961 (if x = a then --d else if x = b then d else &0)` THEN
10962 REWRITE_TAC[VECTOR_SUB_RDISTRIB; MESON[]
10963 `(if p then a else b) % x = (if p then a % x else b % x)`] THEN
10964 ASM_SIMP_TAC[SUM_SUB; VSUM_SUB] THEN
10965 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES; FINITE_RESTRICT; IN_ELIM_THM] THEN
10966 ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`;
10967 SET_RULE `b IN s /\ ~(b = a)
10968 ==> {x | (x IN s /\ ~(x = a)) /\ x = b} = {b}`] THEN
10969 ASM_SIMP_TAC[VECTOR_MUL_LZERO; SUM_0; VSUM_0; SUM_SING; VSUM_SING] THEN
10970 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
10971 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
10972 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN
10973 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN
10974 ASM_REAL_ARITH_TAC);;
10976 let EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL = prove
10977 (`!s. FINITE s /\ affine hull s = (:real^N)
10978 ==> {y | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\
10980 vsum s (\x. u x % x) = y}
10981 SUBSET interior(convex hull s)`,
10982 REPEAT STRIP_TAC THEN
10983 FIRST_ASSUM(MP_TAC o
10984 MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL) THEN
10985 ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_CONVEX_HULL]);;
10987 let EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL_MINIMAL = prove
10988 (`!s. FINITE s /\ affine hull s = (:real^N)
10989 ==> {y | ?u. (!x. x IN s ==> &0 < u x) /\
10991 vsum s (\x. u x % x) = y}
10992 SUBSET interior(convex hull s)`,
10993 REPEAT STRIP_TAC THEN
10994 FIRST_ASSUM(MP_TAC o
10995 MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL) THEN
10996 ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_CONVEX_HULL]);;
10998 let INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL = prove
11000 ~(affine_dependent s)
11001 ==> interior(convex hull s) =
11002 if CARD(s) <= dimindex(:N) then {}
11003 else {y | ?u. (!x. x IN s ==> &0 < u x) /\
11005 vsum s (\x. u x % x) = y}`,
11006 REPEAT STRIP_TAC THEN
11007 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11008 COND_CASES_TAC THEN ASM_SIMP_TAC[EMPTY_INTERIOR_CONVEX_HULL] THEN
11009 MATCH_MP_TAC EQ_TRANS THEN
11010 EXISTS_TAC `relative_interior(convex hull s):real^N->bool` THEN
11012 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
11013 REWRITE_TAC[AFFINE_HULL_CONVEX_HULL] THEN
11014 MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_GT THEN
11015 ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC;
11016 ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT]]);;
11018 let INTERIOR_CONVEX_HULL_EXPLICIT = prove
11020 ~(affine_dependent s)
11021 ==> interior(convex hull s) =
11022 if CARD(s) <= dimindex(:N) then {}
11023 else {y | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\
11025 vsum s (\x. u x % x) = y}`,
11026 REPEAT STRIP_TAC THEN
11027 ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN
11028 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
11029 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `v:real^N` THEN
11030 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `u:real^N->real` THEN
11031 EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN
11032 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
11033 MP_TAC(ISPEC `s:real^N->bool` CHOOSE_SUBSET) THEN
11034 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE] THEN
11035 DISCH_THEN(MP_TAC o SPEC `2`) THEN ANTS_TAC THENL
11036 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE
11037 `~(c <= n) ==> 1 <= n ==> 2 <= c`)) THEN
11038 REWRITE_TAC[DIMINDEX_GE_1];
11040 CONV_TAC(ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN
11041 REWRITE_TAC[SUBSET] THEN
11042 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` (CONJUNCTS_THEN2 ASSUME_TAC
11044 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real^N`
11045 STRIP_ASSUME_TAC)) THEN
11046 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL
11047 [ASM SET_TAC[]; ALL_TAC] THEN
11048 SUBGOAL_THEN `sum {x,y} u <= sum s (u:real^N->real)` MP_TAC THENL
11049 [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
11050 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; REAL_LT_IMP_LE; IN_DIFF] THEN
11053 ASM_SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11054 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
11055 MATCH_MP_TAC(REAL_ARITH `&0 < y ==> x + y + &0 <= &1 ==> x < &1`) THEN
11058 let INTERIOR_CONVEX_HULL_3_MINIMAL = prove
11061 ==> interior(convex hull {a,b,c}) =
11062 {v | ?x y z. &0 < x /\
11066 x % a + y % b + z % c = v}`,
11067 REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT; DE_MORGAN_THM] THEN
11068 REPEAT STRIP_TAC THEN
11069 ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN
11070 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11071 CONV_TAC(LAND_CONV(RATOR_CONV(LAND_CONV(ONCE_DEPTH_CONV(REWRITE_CONV
11072 [IN_INSERT; NOT_IN_EMPTY]))))) THEN
11073 ASM_REWRITE_TAC[DIMINDEX_2; ARITH] THEN
11074 SIMP_TAC[FINITE_INSERT; FINITE_UNION; FINITE_EMPTY; RIGHT_EXISTS_AND_THM;
11075 AFFINE_HULL_FINITE_STEP_GEN; REAL_LT_ADD; REAL_HALF] THEN
11076 REWRITE_TAC[REAL_ARITH `&1 - a - b - c = &0 <=> a + b + c = &1`;
11077 VECTOR_ARITH `y - a - b - c:real^N = vec 0 <=> a + b + c = y`]);;
11079 let INTERIOR_CONVEX_HULL_3 = prove
11082 ==> interior(convex hull {a,b,c}) =
11083 {v | ?x y z. &0 < x /\ x < &1 /\
11084 &0 < y /\ y < &1 /\
11085 &0 < z /\ z < &1 /\
11087 x % a + y % b + z % c = v}`,
11088 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_3_MINIMAL] THEN
11089 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN GEN_TAC THEN
11090 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN EQ_TAC THEN STRIP_TAC THEN
11091 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
11093 (* ------------------------------------------------------------------------- *)
11094 (* Similar results for closure and (relative or absolute) frontier. *)
11095 (* ------------------------------------------------------------------------- *)
11097 let CLOSURE_CONVEX_HULL = prove
11098 (`!s. compact s ==> closure(convex hull s) = convex hull s`,
11099 SIMP_TAC[CLOSURE_CLOSED; COMPACT_IMP_CLOSED; COMPACT_CONVEX_HULL]);;
11101 let RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT = prove
11103 ~(affine_dependent s)
11104 ==> relative_frontier(convex hull s) =
11105 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
11106 (?x. x IN s /\ u x = &0) /\
11108 vsum s (\x. u x % x) = y}`,
11109 REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier; UNIONS_GSPEC] THEN
11110 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11111 ASM_SIMP_TAC[CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11112 ASM_SIMP_TAC[CONVEX_HULL_FINITE; RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT] THEN
11113 GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `y:real^N` THEN
11114 REWRITE_TAC[IN_DIFF; IN_ELIM_THM] THEN EQ_TAC THENL
11115 [DISCH_THEN(CONJUNCTS_THEN2
11116 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) ASSUME_TAC) THEN
11117 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
11118 DISCH_THEN(MP_TAC o SPEC `u:real^N->real`) THEN
11119 ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
11120 DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
11121 ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (~(&0 < x) <=> x = &0)`] THEN
11122 DISCH_TAC THEN EXISTS_TAC `u:real^N->real` THEN
11123 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
11124 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real`
11125 (REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN
11127 [EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[]; ALL_TAC] THEN
11128 DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN
11129 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
11130 [AFFINE_DEPENDENT_EXPLICIT]) THEN
11131 REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC
11132 [`s:real^N->bool`; `(\x. u x - v x):real^N->real`] THEN
11133 ASM_SIMP_TAC[SUBSET_REFL; VECTOR_SUB_RDISTRIB; SUM_SUB; VSUM_SUB] THEN
11134 REWRITE_TAC[REAL_SUB_0; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[REAL_LT_REFL]]);;
11136 let FRONTIER_CONVEX_HULL_EXPLICIT = prove
11138 ~(affine_dependent s)
11139 ==> frontier(convex hull s) =
11140 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
11141 (dimindex(:N) < CARD s ==> ?x. x IN s /\ u x = &0) /\
11143 vsum s (\x. u x % x) = y}`,
11144 REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN
11145 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11147 (ARITH_RULE `CARD(s:real^N->bool) <= dimindex(:N) \/
11148 dimindex(:N) < CARD(s:real^N->bool)`)
11150 [ASM_SIMP_TAC[GSYM NOT_LE; INTERIOR_CONVEX_HULL_EXPLICIT] THEN
11151 ASM_SIMP_TAC[CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT; DIFF_EMPTY] THEN
11152 REWRITE_TAC[CONVEX_HULL_FINITE];
11153 ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT] THEN
11154 REWRITE_TAC[relative_frontier] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
11155 MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
11156 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
11157 EXISTS_TAC `affine hull s:real^N->bool` THEN
11158 ASM_SIMP_TAC[AFFINE_INDEPENDENT_SPAN_GT; HULL_MONO; HULL_SUBSET]]);;
11160 let RELATIVE_FRONTIER_CONVEX_HULL_CASES = prove
11162 ~(affine_dependent s)
11163 ==> relative_frontier(convex hull s) =
11164 UNIONS { convex hull (s DELETE a) |a| a IN s }`,
11165 REPEAT STRIP_TAC THEN REWRITE_TAC[UNIONS_GSPEC] THEN
11166 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11167 ASM_SIMP_TAC[RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT] THEN
11168 REWRITE_TAC[EXTENSION; IN_ELIM_THM; CONVEX_HULL_FINITE] THEN
11169 X_GEN_TAC `y:real^N` THEN EQ_TAC THENL
11170 [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` MP_TAC) THEN
11171 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
11172 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
11173 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
11174 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `u:real^N->real` THEN
11175 ASM_SIMP_TAC[IN_DELETE; SUM_DELETE; VSUM_DELETE; REAL_SUB_RZERO] THEN
11177 REWRITE_TAC[IN_DELETE] THEN
11178 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (CONJUNCTS_THEN2 ASSUME_TAC
11179 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC))) THEN
11180 EXISTS_TAC `(\x. if x = a then &0 else u x):real^N->real` THEN
11181 ASM_SIMP_TAC[COND_RAND; COND_RATOR; REAL_LE_REFL; COND_ID] THEN
11182 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
11183 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO] THEN
11184 ASM_SIMP_TAC[GSYM DELETE; SUM_0; VSUM_0; REAL_ADD_LID; VECTOR_ADD_LID]]);;
11186 let FRONTIER_CONVEX_HULL_CASES = prove
11188 ~(affine_dependent s)
11189 ==> frontier(convex hull s) =
11190 if CARD(s) <= dimindex(:N) then convex hull s
11191 else UNIONS { convex hull (s DELETE a) |a| a IN s }`,
11192 REPEAT STRIP_TAC THEN
11193 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11194 ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11195 COND_CASES_TAC THENL
11196 [ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT; DIFF_EMPTY]; ALL_TAC] THEN
11197 ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_CONVEX_HULL_CASES] THEN
11198 ASM_SIMP_TAC[relative_frontier; frontier;
11199 CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11200 AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
11201 RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE]) THEN
11202 MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
11203 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
11204 EXISTS_TAC `affine hull s:real^N->bool` THEN
11205 ASM_SIMP_TAC[AFFINE_INDEPENDENT_SPAN_GT; HULL_MONO; HULL_SUBSET]);;
11207 let IN_FRONTIER_CONVEX_HULL = prove
11209 FINITE s /\ CARD s <= dimindex(:N) + 1 /\ x IN s
11210 ==> x IN frontier(convex hull s)`,
11211 REPEAT STRIP_TAC THEN ASM_CASES_TAC `affine_dependent(s:real^N->bool)` THENL
11212 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [affine_dependent]) THEN
11213 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
11214 ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11215 ASM_SIMP_TAC[HULL_INC; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE
11216 `!t. s SUBSET t /\ t = {} ==> ~(x IN s)`) THEN
11217 EXISTS_TAC `interior(affine hull s):real^N->bool` THEN
11218 SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
11219 SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL
11220 [ASM SET_TAC[]; ALL_TAC] THEN
11221 ASM_SIMP_TAC[HULL_REDUNDANT] THEN
11222 MATCH_MP_TAC EMPTY_INTERIOR_AFFINE_HULL THEN
11223 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC;
11224 ASM_SIMP_TAC[FRONTIER_CONVEX_HULL_CASES] THEN
11225 COND_CASES_TAC THEN ASM_SIMP_TAC[HULL_INC] THEN
11226 REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM] THEN
11227 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` MP_TAC THENL
11228 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
11229 `x IN s ==> ~(s = {x}) ==> ?y. y IN s /\ ~(y = x)`)) THEN DISCH_TAC THEN
11230 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_LE]) THEN
11231 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11232 REWRITE_TAC[NOT_LT; NOT_IN_EMPTY; ARITH_SUC; DIMINDEX_GE_1];
11233 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN
11234 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);;
11236 let NOT_IN_INTERIOR_CONVEX_HULL = prove
11238 FINITE s /\ CARD s <= dimindex(:N) + 1 /\ x IN s
11239 ==> ~(x IN interior(convex hull s))`,
11240 REPEAT GEN_TAC THEN
11241 DISCH_THEN(MP_TAC o MATCH_MP IN_FRONTIER_CONVEX_HULL) THEN
11242 SIMP_TAC[frontier; IN_DIFF]);;
11244 let INTERIOR_CONVEX_HULL_EQ_EMPTY = prove
11246 s HAS_SIZE (dimindex(:N) + 1)
11247 ==> (interior(convex hull s) = {} <=> affine_dependent s)`,
11248 REPEAT GEN_TAC THEN REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC THEN
11249 ASM_CASES_TAC `affine_dependent(s:real^N->bool)` THENL
11250 [ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
11251 [affine_dependent]) THEN
11252 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
11253 ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11254 ASM_SIMP_TAC[HULL_INC; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE
11255 `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
11256 EXISTS_TAC `interior(affine hull s):real^N->bool` THEN
11257 SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
11258 SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL
11259 [ASM SET_TAC[]; ALL_TAC] THEN
11260 ASM_SIMP_TAC[HULL_REDUNDANT] THEN
11261 MATCH_MP_TAC EMPTY_INTERIOR_AFFINE_HULL THEN
11262 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC;
11263 ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN
11264 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; ARITH_RULE `~(n + 1 <= n)`] THEN
11265 EXISTS_TAC `vsum s (\x:real^N. inv(&(dimindex(:N)) + &1) % x)` THEN
11266 REWRITE_TAC[IN_ELIM_THM] THEN
11267 EXISTS_TAC `\x:real^N. inv(&(dimindex(:N)) + &1)` THEN
11268 ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
11269 ASM_SIMP_TAC[SUM_CONST; GSYM REAL_OF_NUM_ADD] THEN
11270 CONV_TAC REAL_FIELD]);;
11272 (* ------------------------------------------------------------------------- *)
11273 (* Similar things in special case (could use above as lemmas here instead). *)
11274 (* ------------------------------------------------------------------------- *)
11276 let SIMPLEX_EXPLICIT = prove
11278 FINITE s /\ ~(vec 0 IN s)
11279 ==> convex hull (vec 0 INSERT s) =
11280 { y | ?u. (!x. x IN s ==> &0 <= u x) /\
11282 vsum s (\x. u x % x) = y}`,
11283 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT] THEN
11284 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN
11285 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; IN_INSERT] THEN
11286 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
11287 EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THENL
11288 [EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[REAL_LE_REFL] THEN
11289 FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N`) THEN REWRITE_TAC[] THEN
11290 ASM_REAL_ARITH_TAC;
11291 EXISTS_TAC `\x:real^N. if x = vec 0 then &1 - sum (s:real^N->bool) u
11293 ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
11294 [X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
11295 ASM_REWRITE_TAC[REAL_SUB_LE];
11296 MATCH_MP_TAC(REAL_ARITH `s = t ==> &1 - s + t = &1`) THEN
11297 MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[];
11298 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
11299 MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[]]]);;
11301 let STD_SIMPLEX = prove
11302 (`convex hull (vec 0 INSERT { basis i | 1 <= i /\ i <= dimindex(:N)}) =
11303 {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i) /\
11304 sum (1..dimindex(:N)) (\i. x$i) <= &1 }`,
11305 W(MP_TAC o PART_MATCH (lhs o rand) SIMPLEX_EXPLICIT o lhs o snd) THEN ANTS_TAC THENL
11306 [REWRITE_TAC[SIMPLE_IMAGE; GSYM IN_NUMSEG] THEN
11307 SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; IN_IMAGE] THEN
11308 REWRITE_TAC[IN_NUMSEG] THEN MESON_TAC[BASIS_NONZERO];
11310 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION] THEN
11311 ONCE_REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN
11312 REWRITE_TAC[SIMPLE_IMAGE; GSYM IN_NUMSEG] THEN
11313 SUBGOAL_THEN `!u. sum (IMAGE (basis:num->real^N) (1..dimindex(:N))) u =
11314 sum (1..dimindex(:N)) (u o basis)`
11315 (fun th -> REWRITE_TAC[th])
11317 [GEN_TAC THEN MATCH_MP_TAC SUM_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN
11318 REWRITE_TAC[GSYM CONJ_ASSOC; BASIS_INJ];
11320 SUBGOAL_THEN `!u. vsum (IMAGE (basis:num->real^N) (1..dimindex(:N))) u =
11321 vsum (1..dimindex(:N)) ((u:real^N->real^N) o basis)`
11322 (fun th -> REWRITE_TAC[th])
11324 [GEN_TAC THEN MATCH_MP_TAC VSUM_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN
11325 REWRITE_TAC[GSYM CONJ_ASSOC; BASIS_INJ; FINITE_NUMSEG];
11327 REWRITE_TAC[o_DEF; BASIS_EXPANSION_UNIQUE; FORALL_IN_IMAGE] THEN
11328 REWRITE_TAC[IN_NUMSEG] THEN EQ_TAC THENL
11329 [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
11330 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
11331 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
11332 `x <= &1 ==> x = y ==> y <= &1`)) THEN
11333 MATCH_MP_TAC SUM_EQ THEN ASM_SIMP_TAC[IN_NUMSEG];
11334 STRIP_TAC THEN EXISTS_TAC `\y:real^N. y dot x` THEN
11335 ASM_SIMP_TAC[DOT_BASIS]]);;
11337 let INTERIOR_STD_SIMPLEX = prove
11339 (convex hull (vec 0 INSERT { basis i | 1 <= i /\ i <= dimindex(:N)})) =
11340 {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 < x$i) /\
11341 sum (1..dimindex(:N)) (\i. x$i) < &1 }`,
11342 REWRITE_TAC[EXTENSION; IN_INTERIOR; IN_ELIM_THM; STD_SIMPLEX] THEN
11343 REWRITE_TAC[SUBSET; IN_BALL; IN_ELIM_THM] THEN
11344 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
11345 [DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
11346 FIRST_ASSUM(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[DIST_REFL] THEN
11347 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_LT_LE] THEN
11349 [X_GEN_TAC `k:num` THEN STRIP_TAC THEN
11350 FIRST_X_ASSUM(MP_TAC o SPEC `x - (e / &2) % basis k:real^N`) THEN
11351 REWRITE_TAC[NORM_ARITH `dist(x,x - e) = norm(e)`; NORM_MUL] THEN
11352 ASM_SIMP_TAC[NORM_BASIS; REAL_ARITH `&0 < e ==> abs(e / &2) * &1 < e`;
11353 VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN
11354 DISCH_THEN(MP_TAC o SPEC `k:num` o CONJUNCT1) THEN ASM_REWRITE_TAC[] THEN
11355 ASM_SIMP_TAC[BASIS_COMPONENT] THEN UNDISCH_TAC `&0 < e` THEN
11357 FIRST_X_ASSUM(MP_TAC o SPEC `x + (e / &2) % basis 1:real^N`) THEN
11358 REWRITE_TAC[NORM_ARITH `dist(x,x + e) = norm(e)`; NORM_MUL] THEN
11359 ASM_SIMP_TAC[NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN
11360 ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> abs(e / &2) * &1 < e`] THEN
11361 DISCH_THEN(MP_TAC o CONJUNCT2) THEN
11362 MATCH_MP_TAC(REAL_ARITH `x < y ==> y <= &1 ==> ~(x = &1)`) THEN
11363 MATCH_MP_TAC SUM_LT THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
11364 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c <=> ~(a /\ b ==> ~c)`] THEN
11365 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
11366 BASIS_COMPONENT] THEN
11368 [GEN_TAC THEN COND_CASES_TAC;
11369 EXISTS_TAC `1` THEN REWRITE_TAC[LE_REFL; DIMINDEX_GE_1]] THEN
11370 ASM_REAL_ARITH_TAC];
11373 `min (inf(IMAGE (\i. (x:real^N)$i) (1..dimindex(:N))))
11374 ((&1 - sum (1..dimindex(:N)) (\i. x$i)) / &(dimindex(:N)))` THEN
11375 ASM_SIMP_TAC[REAL_LT_MIN] THEN
11376 SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; FINITE_NUMSEG;
11377 IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN
11378 REWRITE_TAC[FORALL_IN_IMAGE] THEN
11379 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT;
11380 ARITH_RULE `0 < x <=> 1 <= x`; DIMINDEX_GE_1] THEN
11381 ASM_REWRITE_TAC[IN_NUMSEG; REAL_MUL_LZERO; REAL_SUB_LT] THEN
11382 REPEAT(POP_ASSUM(K ALL_TAC)) THEN X_GEN_TAC `y:real^N` THEN
11383 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
11384 [MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN
11385 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
11386 ASM_REWRITE_TAC[] THEN
11387 MATCH_MP_TAC(REAL_ARITH `abs(xk - yk) <= d ==> d < xk ==> &0 <= yk`);
11388 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV)
11389 [GSYM CARD_NUMSEG_1] THEN
11390 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
11391 SIMP_TAC[GSYM SUM_CONST; FINITE_NUMSEG] THEN
11392 MATCH_MP_TAC(REAL_ARITH
11393 `s2 <= s0 + s1 ==> s0 < &1 - s1 ==> s2 <= &1`) THEN
11394 REWRITE_TAC[GSYM SUM_ADD_NUMSEG] THEN
11395 MATCH_MP_TAC SUM_LE_NUMSEG THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
11396 MATCH_MP_TAC(REAL_ARITH `abs(y - x) <= z ==> x <= z + y`)] THEN
11397 ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; dist] THEN
11398 MATCH_MP_TAC COMPONENT_LE_NORM THEN ASM_REWRITE_TAC[]]);;
git://colo12-c703.uibk.ac.at / Multivariate Analysis/.git / blob