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_VSUM_STRONG = prove
314 (`!s k u x:A->real^N.
317 (!i. i IN k ==> u i = &0 \/ x i IN s)
318 ==> vsum k (\i. u i % x i) IN s`,
319 REPEAT STRIP_TAC THEN
321 `vsum k (\i. u i % (x:A->real^N) i) =
322 vsum {i | i IN k /\ ~(u i = &0)} (\i. u i % x i)`
324 [MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
326 MATCH_MP_TAC AFFINE_VSUM THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
327 [ASM_MESON_TAC[SUM_DEGENERATE; REAL_ARITH `~(&1 = &0)`];
328 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
329 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM SET_TAC[];
332 let AFFINE_INDEXED = prove
335 !k u x. (!i:num. 1 <= i /\ i <= k ==> x(i) IN s) /\
337 ==> vsum (1..k) (\i. u(i) % x(i)) IN s`,
338 REPEAT GEN_TAC THEN EQ_TAC THENL
339 [REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN
340 ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG];
341 DISCH_TAC THEN REWRITE_TAC[affine] THEN
342 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
343 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `2`) THEN
344 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then u else v:real`) THEN
345 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then x else y:real^N`) THEN
346 REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG; VSUM_CLAUSES_NUMSEG;
347 NUMSEG_SING; VSUM_SING; SUM_SING] THEN REWRITE_TAC[ARITH] THEN
350 let AFFINE_HULL_INDEXED = prove
351 (`!s. affine hull s =
352 {y:real^N | ?k u x. (!i. 1 <= i /\ i <= k ==> x i IN s) /\
353 (sum (1..k) u = &1) /\
354 (vsum (1..k) (\i. u i % x i) = y)}`,
355 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL
356 [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
357 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
358 MAP_EVERY EXISTS_TAC [`1`; `\i:num. &1`; `\i:num. x:real^N`] THEN
359 ASM_SIMP_TAC[FINITE_RULES; IN_SING; SUM_SING; VECTOR_MUL_LID; VSUM_SING;
360 REAL_POS; NUMSEG_SING];
362 REWRITE_TAC[AFFINE_INDEXED; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
364 REWRITE_TAC[affine; IN_ELIM_THM] THEN
365 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
366 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
367 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
368 [`k1:num`; `u1:num->real`; `x1:num->real^N`;
369 `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN
370 STRIP_TAC THEN EXISTS_TAC `k1 + k2:num` THEN
371 EXISTS_TAC `\i:num. if i <= k1 then u * u1(i) else v * u2(i - k1):real` THEN
372 EXISTS_TAC `\i:num. if i <= k1 then x1(i) else x2(i - k1):real^N` THEN
373 ASM_SIMP_TAC[NUMSEG_ADD_SPLIT; ARITH_RULE `1 <= x + 1 /\ x < x + 1`;
374 IN_NUMSEG; SUM_UNION; VSUM_UNION; FINITE_NUMSEG; DISJOINT_NUMSEG;
375 ARITH_RULE `k1 + 1 <= i ==> ~(i <= k1)`] THEN
376 REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] NUMSEG_OFFSET_IMAGE] THEN
377 ASM_SIMP_TAC[SUM_IMAGE; VSUM_IMAGE; EQ_ADD_LCANCEL; FINITE_NUMSEG] THEN
378 ASM_SIMP_TAC[o_DEF; ADD_SUB2; SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC;
379 FINITE_NUMSEG; REAL_MUL_RID] THEN
380 ASM_MESON_TAC[REAL_LE_MUL; ARITH_RULE
381 `i <= k1 + k2 /\ ~(i <= k1) ==> 1 <= i - k1 /\ i - k1 <= k2`]);;
386 !(s:real^N->bool) (u:real^N->real).
387 FINITE s /\ ~(s = {}) /\ s SUBSET V /\ sum s u = &1
388 ==> vsum s (\x. u x % x) IN V`,
389 GEN_TAC THEN EQ_TAC THENL
390 [REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_VSUM THEN
391 ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
392 REWRITE_TAC[affine] THEN DISCH_TAC THEN
393 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
394 STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THENL
395 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
396 ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB;VECTOR_MUL_LID];ALL_TAC] THEN
397 FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N,y}`) THEN
398 DISCH_THEN(MP_TAC o SPEC `\w. if w = x:real^N then u else v:real`) THEN
399 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_RULES; NUMSEG_SING;
400 VSUM_SING; SUM_SING;SUBSET;IN_INSERT;NOT_IN_EMPTY] THEN
403 let AFFINE_EXPLICIT = prove
406 !t u. FINITE t /\ t SUBSET s /\ sum t u = &1
407 ==> vsum t (\x. u(x) % x) IN s`,
408 GEN_TAC THEN REWRITE_TAC[AFFINE] THEN
409 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
410 X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN
411 AP_TERM_TAC THEN ABS_TAC THEN
412 ASM_CASES_TAC `t:real^N->bool = {}` THEN
413 ASM_REWRITE_TAC[SUM_CLAUSES] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
415 let AFFINE_HULL_EXPLICIT = prove
416 (`!(p:real^N -> bool).
418 {y | ?s u. FINITE s /\ ~(s = {}) /\ s SUBSET p /\
419 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
420 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL
421 [REWRITE_TAC[SUBSET;IN_ELIM_THM] THEN
422 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
423 MAP_EVERY EXISTS_TAC [`{x:real^N}`;`\v:real^N. &1:real`] THEN
424 ASM_SIMP_TAC[FINITE_RULES;IN_SING;SUM_SING;VSUM_SING;VECTOR_MUL_LID] THEN
426 REWRITE_TAC[affine;IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
427 EXISTS_TAC `(s UNION s'):real^N->bool` THEN
429 `\a:real^N. (\b:real^N.if (b IN s) then (u * (u' b)) else &0) a +
430 (\b:real^N.if (b IN s') then v * (u'' b) else &0) a` THEN
431 REPEAT CONJ_TAC THENL
432 [ASM_REWRITE_TAC[FINITE_UNION];
434 ASM_REWRITE_TAC[UNION_SUBSET];
435 ASM_SIMP_TAC[REWRITE_RULE[REAL_ARITH `a + b = c + d <=> c = a + b - d`]
436 SUM_INCL_EXCL; GSYM SUM_RESTRICT_SET;
437 SET_RULE `{a | a IN (s:A->bool) /\ a IN s'} = s INTER s'`;
438 SUM_ADD;SUM_LMUL;REAL_MUL_RID;
439 FINITE_INTER;INTER_IDEMPOT] THEN
440 ASM_REWRITE_TAC[SET_RULE `(a INTER b) INTER a = a INTER b`;
441 SET_RULE `(a INTER b) INTER b = a INTER b`;
442 REAL_ARITH `(a + b) + (c + d) - (e + b) = (a + d) + c - e`;
443 REAL_ARITH `a + b - c = a <=> b = c`] THEN
444 AP_TERM_TAC THEN REWRITE_TAC[INTER_COMM];
445 ASM_SIMP_TAC[REWRITE_RULE
446 [VECTOR_ARITH `(a:real^N) + b = c + d <=> c = a + b - d`]
447 VSUM_INCL_EXCL;GSYM VSUM_RESTRICT_SET;
448 SET_RULE `{a | a IN (s:A->bool) /\ a IN s'} = s INTER s'`;
449 VSUM_ADD;FINITE_INTER;INTER_IDEMPOT;VECTOR_ADD_RDISTRIB;
450 GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;
451 MESON[] `(if P then a else b) % (x:real^N) =
452 (if P then a % x else b % x)`;
453 VECTOR_MUL_LZERO;GSYM VSUM_RESTRICT_SET] THEN
454 ASM_REWRITE_TAC[SET_RULE `(a INTER b) INTER a = a INTER b`;
455 SET_RULE `(a INTER b) INTER b = a INTER b`;
457 `((a:real^N) + b) + (c + d) - (e + b) = (a + d) + c - e`;
458 VECTOR_ARITH `(a:real^N) + b - c = a <=> b = c`] THEN
459 AP_TERM_TAC THEN REWRITE_TAC[INTER_COMM]];
460 ASM_CASES_TAC `(p:real^N->bool) = {}` THENL
461 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
462 REWRITE_TAC[SUBSET_EMPTY;EMPTY_SUBSET] THEN ASM SET_TAC[];
464 REWRITE_TAC[AFFINE; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
467 let AFFINE_HULL_EXPLICIT_ALT = prove
468 (`!(p:real^N -> bool).
470 {y | ?s u. FINITE s /\ s SUBSET p /\
471 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
472 GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_EXPLICIT] THEN
473 GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_ELIM_THM] THEN
474 GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
475 EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
476 POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
477 SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ]);;
479 let AFFINE_HULL_FINITE = prove
481 affine hull s = {y | ?u. sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
482 GEN_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
483 REWRITE_TAC[AFFINE_HULL_EXPLICIT; IN_ELIM_THM] THEN
484 X_GEN_TAC `x:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
485 [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `f:real^N->real`] THEN
487 EXISTS_TAC `\x:real^N. if x IN t then f x else &0` THEN
488 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
489 REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
490 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`];
491 X_GEN_TAC `f:real^N->real` THEN
492 ASM_CASES_TAC `s:real^N->bool = {}` THEN
493 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN STRIP_TAC THEN
494 EXISTS_TAC `support (+) (f:real^N->real) s` THEN
495 EXISTS_TAC `f:real^N->real` THEN
496 MP_TAC(ASSUME `sum s (f:real^N->real) = &1`) THEN
497 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN
498 REWRITE_TAC[iterate] THEN COND_CASES_TAC THEN
499 ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; REAL_OF_NUM_EQ; ARITH] THEN
500 DISCH_THEN(K ALL_TAC) THEN
501 UNDISCH_TAC `sum s (f:real^N->real) = &1` THEN
502 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM SUM_SUPPORT] THEN
503 ASM_CASES_TAC `support (+) (f:real^N->real) s = {}` THEN
504 ASM_SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN
505 DISCH_TAC THEN REWRITE_TAC[SUPPORT_SUBSET] THEN
506 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
507 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
508 REWRITE_TAC[SUPPORT_SUBSET] THEN
509 REWRITE_TAC[support; IN_ELIM_THM; NEUTRAL_REAL_ADD] THEN
510 MESON_TAC[VECTOR_MUL_LZERO]]);;
512 (* ------------------------------------------------------------------------- *)
513 (* Stepping theorems and hence small special cases. *)
514 (* ------------------------------------------------------------------------- *)
516 let AFFINE_HULL_EMPTY = prove
517 (`affine hull {} = {}`,
518 MATCH_MP_TAC HULL_UNIQUE THEN
519 REWRITE_TAC[SUBSET_REFL; AFFINE_EMPTY; EMPTY_SUBSET]);;
521 let AFFINE_HULL_EQ_EMPTY = prove
522 (`!s. (affine hull s = {}) <=> (s = {})`,
523 GEN_TAC THEN EQ_TAC THEN
524 MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; AFFINE_HULL_EMPTY]);;
526 let AFFINE_HULL_FINITE_STEP_GEN = prove
527 (`!P:real^N->real->bool.
528 ((?u. (!x. x IN {} ==> P x (u x)) /\
529 sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=>
530 w = &0 /\ y = vec 0) /\
531 (FINITE(s:real^N->bool) /\
532 (!y. a IN s /\ P a y ==> P a (y / &2)) /\
533 (!x y. a IN s /\ P a x /\ P a y ==> P a (x + y))
534 ==> ((?u. (!x. x IN (a INSERT s) ==> P x (u x)) /\
535 sum (a INSERT s) u = w /\
536 vsum (a INSERT s) (\x. u(x) % x) = y) <=>
537 ?v u. P a v /\ (!x. x IN s ==> P x (u x)) /\
539 vsum s (\x. u(x) % x) = y - v % a))`,
540 GEN_TAC THEN SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; NOT_IN_EMPTY] THEN
541 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_TAC THEN
542 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THENL
543 [ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN EQ_TAC THEN
544 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
545 [X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN
546 EXISTS_TAC `(u:real^N->real) a / &2` THEN
547 EXISTS_TAC `\x:real^N. if x = a then u x / &2 else u x`;
548 MAP_EVERY X_GEN_TAC [`v:real`; `u:real^N->real`] THEN
550 EXISTS_TAC `\x:real^N. if x = a then u x + v else u x`] THEN
551 ASM_SIMP_TAC[] THEN (CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN
552 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
553 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES] THEN
554 ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; VSUM_DELETE] THEN
555 ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`] THEN
556 REWRITE_TAC[SUM_SING; VSUM_SING] THEN
557 (CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);
558 EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
559 [X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN
560 EXISTS_TAC `(u:real^N->real) a` THEN
561 EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[IN_INSERT] THEN
562 REPEAT(FIRST_X_ASSUM(SUBST1_TAC o SYM)) THEN
563 CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC];
564 MAP_EVERY X_GEN_TAC [`v:real`; `u:real^N->real`] THEN
566 EXISTS_TAC `\x:real^N. if x = a then v:real else u x` THEN
567 ASM_SIMP_TAC[IN_INSERT] THEN CONJ_TAC THENL
568 [ASM_MESON_TAC[]; ALL_TAC] THEN
569 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
570 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES] THEN
571 ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; VSUM_DELETE] THEN
572 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ x = a} = {}`] THEN
573 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> s DELETE a = s`] THEN
574 REWRITE_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
575 CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]]]);;
577 let AFFINE_HULL_FINITE_STEP = prove
578 (`((?u. sum {} u = w /\ vsum {} (\x. u(x) % x) = y) <=>
579 w = &0 /\ y = vec 0) /\
580 (FINITE(s:real^N->bool)
581 ==> ((?u. sum (a INSERT s) u = w /\
582 vsum (a INSERT s) (\x. u(x) % x) = y) <=>
583 ?v u. sum s u = w - v /\
584 vsum s (\x. u(x) % x) = y - v % a))`,
585 MATCH_ACCEPT_TAC (REWRITE_RULE[]
586 (ISPEC `\x:real^N y:real. T` AFFINE_HULL_FINITE_STEP_GEN)));;
588 let AFFINE_HULL_2 = prove
589 (`!a b. affine hull {a,b} =
590 {u % a + v % b | u + v = &1}`,
591 SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
592 SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
593 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
594 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
595 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
597 let AFFINE_HULL_2_ALT = prove
598 (`!a b. affine hull {a,b} = {a + u % (b - a) | u IN (:real)}`,
599 REPEAT GEN_TAC THEN REWRITE_TAC[AFFINE_HULL_2] THEN
600 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
601 REWRITE_TAC[IN_ELIM_THM; IN_UNIV; ARITH_RULE `u + v = &1 <=> v = &1 - u`;
602 FORALL_UNWIND_THM2; UNWIND_THM2] THEN
603 CONJ_TAC THEN X_GEN_TAC `u:real` THEN EXISTS_TAC `&1 - u` THEN
606 let AFFINE_HULL_3 = prove
607 (`affine hull {a,b,c} =
608 { u % a + v % b + w % c | u + v + w = &1}`,
609 SIMP_TAC[AFFINE_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
610 SIMP_TAC[AFFINE_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
611 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
612 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
613 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
615 (* ------------------------------------------------------------------------- *)
616 (* Some relations between affine hull and subspaces. *)
617 (* ------------------------------------------------------------------------- *)
619 let AFFINE_HULL_INSERT_SUBSET_SPAN = prove
621 affine hull (a INSERT s) SUBSET {a + v | v | v IN span {x - a | x IN s}}`,
622 REPEAT GEN_TAC THEN GEN_REWRITE_TAC I [SUBSET] THEN
623 REWRITE_TAC[AFFINE_HULL_EXPLICIT; SPAN_EXPLICIT; IN_ELIM_THM] THEN
624 REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
626 `(?s u. (?t. P t /\ s = f t) /\ Q s u) <=>
627 (?t u. P t /\ Q (f t) u)`] THEN
629 `(?v. (?s u. P s /\ f s u = v) /\ (x = g a v)) <=>
630 (?s u. ~(P s ==> ~(g a (f s u) = x)))`] THEN
631 SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
632 REWRITE_TAC[o_DEF] THEN X_GEN_TAC `y:real^N` THEN
633 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
634 MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
635 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST1_TAC o SYM)) THEN
637 [`t DELETE (a:real^N)`; `\x. (u:real^N->real)(x + a)`] THEN
638 ASM_SIMP_TAC[FINITE_DELETE; VECTOR_SUB_ADD; SET_RULE
639 `t SUBSET (a INSERT s) ==> t DELETE a SUBSET s`] THEN
640 MATCH_MP_TAC EQ_TRANS THEN
641 EXISTS_TAC `a + vsum t (\x. u x % (x - a)):real^N` THEN CONJ_TAC THENL
642 [AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
643 REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN SET_TAC[];
644 ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; FINITE_DELETE; VSUM_SUB] THEN
645 ASM_REWRITE_TAC[VSUM_RMUL] THEN
646 REWRITE_TAC[VECTOR_ARITH `a + x - &1 % a:real^N = x`]]);;
648 let AFFINE_HULL_INSERT_SPAN = prove
651 ==> affine hull (a INSERT s) =
652 {a + v | v | v IN span {x - a | x IN s}}`,
653 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
654 REWRITE_TAC[AFFINE_HULL_INSERT_SUBSET_SPAN] THEN REWRITE_TAC[SUBSET] THEN
655 REWRITE_TAC[AFFINE_HULL_EXPLICIT; SPAN_EXPLICIT; IN_ELIM_THM] THEN
656 REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
658 `(?s u. (?t. P t /\ s = f t) /\ Q s u) <=>
659 (?t u. P t /\ Q (f t) u)`] THEN
661 `(?v. (?s u. P s /\ f s u = v) /\ (x = g a v)) <=>
662 (?s u. ~(P s ==> ~(g a (f s u) = x)))`] THEN
663 SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
664 REWRITE_TAC[o_DEF] THEN X_GEN_TAC `y:real^N` THEN
665 REWRITE_TAC[NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
666 MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
667 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (SUBST1_TAC o SYM)) THEN
669 [`(a:real^N) INSERT t`;
670 `\x. if x = a then &1 - sum t (\x. u(x - a))
671 else (u:real^N->real)(x - a)`] THEN
672 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
673 ASM_CASES_TAC `(a:real^N) IN t` THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
674 ASM_SIMP_TAC[FINITE_INSERT; NOT_INSERT_EMPTY;
675 SET_RULE `s SUBSET t ==> (a INSERT s) SUBSET (a INSERT t)`] THEN
676 SUBGOAL_THEN `!x:real^N. x IN t ==> ~(x = a)` MP_TAC THENL
677 [ASM SET_TAC[]; SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC)] THEN
678 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
679 ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; FINITE_DELETE; VSUM_SUB] THEN
680 ASM_REWRITE_TAC[VSUM_RMUL] THEN VECTOR_ARITH_TAC);;
682 let AFFINE_HULL_SPAN = prove
686 {a + v | v | v IN span {x - a | x | x IN (s DELETE a)}})`,
687 REPEAT STRIP_TAC THEN
688 MP_TAC(ISPECL [`a:real^N`; `s DELETE (a:real^N)`]
689 AFFINE_HULL_INSERT_SPAN) THEN
690 ASM_REWRITE_TAC[IN_DELETE] THEN
691 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
693 let DIFFS_AFFINE_HULL_SPAN = prove
695 a IN s ==> {x - a | x IN affine hull s} = span {x - a | x IN s}`,
696 REPEAT STRIP_TAC THEN
697 FIRST_ASSUM(SUBST1_TAC o MATCH_MP AFFINE_HULL_SPAN) THEN
698 REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o; o_DEF; VECTOR_ADD_SUB; IMAGE_ID] THEN
699 SIMP_TAC[IMAGE_DELETE_INJ;
700 VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
701 REWRITE_TAC[VECTOR_SUB_REFL; SPAN_DELETE_0]);;
703 let AFFINE_HULL_SING = prove
704 (`!a. affine hull {a} = {a}`,
705 SIMP_TAC[AFFINE_HULL_INSERT_SPAN; NOT_IN_EMPTY] THEN
706 REWRITE_TAC[SET_RULE `{f x | x | F} = {}`; SPAN_EMPTY] THEN
707 REWRITE_TAC[SET_RULE `{f x | x IN {a}} = {f a}`; VECTOR_ADD_RID]);;
709 let AFFINE_HULL_EQ_SING = prove
710 (`!s a:real^N. affine hull s = {a} <=> s = {a}`,
711 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
712 ASM_REWRITE_TAC[AFFINE_HULL_EMPTY] THEN
713 EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[AFFINE_HULL_SING] THEN
714 MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET {a} ==> s = {a}`) THEN
715 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
716 REWRITE_TAC[HULL_SUBSET]);;
718 (* ------------------------------------------------------------------------- *)
720 (* ------------------------------------------------------------------------- *)
722 let convex = new_definition
724 !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
725 ==> (u % x + v % y) IN s`;;
727 let CONVEX_ALT = prove
728 (`convex s <=> !x y u. x IN s /\ y IN s /\ &0 <= u /\ u <= &1
729 ==> ((&1 - u) % x + u % y) IN s`,
730 REWRITE_TAC[convex] THEN
731 MESON_TAC[REAL_ARITH `&0 <= u /\ &0 <= v /\ (u + v = &1)
732 ==> v <= &1 /\ (u = &1 - v)`;
733 REAL_ARITH `u <= &1 ==> &0 <= &1 - u /\ ((&1 - u) + u = &1)`]);;
735 let IN_CONVEX_SET = prove
737 convex s /\ a IN s /\ b IN s /\ &0 <= u /\ u <= &1
738 ==> ((&1 - u) % a + u % b) IN s`,
739 MESON_TAC[CONVEX_ALT]);;
741 let MIDPOINT_IN_CONVEX = prove
743 convex s /\ x IN s /\ y IN s ==> midpoint(x,y) IN s`,
744 REPEAT STRIP_TAC THEN
745 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `y:real^N`; `&1 / &2`]
747 ASM_REWRITE_TAC[midpoint] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
748 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
749 CONV_TAC VECTOR_ARITH);;
751 let CONVEX_CONTAINS_SEGMENT = prove
752 (`!s. convex s <=> !a b. a IN s /\ b IN s ==> segment[a,b] SUBSET s`,
753 REWRITE_TAC[CONVEX_ALT; segment; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
755 let CONVEX_CONTAINS_OPEN_SEGMENT = prove
756 (`!s. convex s <=> !a b. a IN s /\ b IN s ==> segment(a,b) SUBSET s`,
757 ONCE_REWRITE_TAC[segment] THEN REWRITE_TAC[CONVEX_CONTAINS_SEGMENT] THEN
760 let CONVEX_CONTAINS_SEGMENT_EQ = prove
762 convex s <=> !a b. segment[a,b] SUBSET s <=> a IN s /\ b IN s`,
763 REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; SUBSET] THEN
764 MESON_TAC[ENDS_IN_SEGMENT]);;
766 let CONVEX_CONTAINS_SEGMENT_IMP = prove
767 (`!s a b. convex s ==> (segment[a,b] SUBSET s <=> a IN s /\ b IN s)`,
768 SIMP_TAC[CONVEX_CONTAINS_SEGMENT_EQ]);;
770 let CONVEX_EMPTY = prove
772 REWRITE_TAC[convex; NOT_IN_EMPTY]);;
774 let CONVEX_SING = prove
776 SIMP_TAC[convex; IN_SING; GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]);;
778 let CONVEX_UNIV = prove
779 (`convex(UNIV:real^N->bool)`,
780 REWRITE_TAC[convex; IN_UNIV]);;
782 let CONVEX_INTERS = prove
783 (`(!s. s IN f ==> convex s) ==> convex(INTERS f)`,
784 REWRITE_TAC[convex; IN_INTERS] THEN MESON_TAC[]);;
786 let CONVEX_INTER = prove
787 (`!s t. convex s /\ convex t ==> convex(s INTER t)`,
788 REWRITE_TAC[convex; IN_INTER] THEN MESON_TAC[]);;
790 let CONVEX_HALFSPACE_LE = prove
791 (`!a b. convex {x | a dot x <= b}`,
792 REWRITE_TAC[convex; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN
793 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
794 EXISTS_TAC `(u + v) * b` THEN CONJ_TAC THENL
795 [ASM_MESON_TAC[REAL_ADD_RDISTRIB; REAL_LE_ADD2; REAL_LE_LMUL];
796 ASM_MESON_TAC[REAL_MUL_LID; REAL_LE_REFL]]);;
798 let CONVEX_HALFSPACE_COMPONENT_LE = prove
799 (`!a k. convex {x:real^N | x$k <= a}`,
801 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
803 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
804 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_LE) THEN
805 ASM_SIMP_TAC[DOT_BASIS]);;
807 let CONVEX_HALFSPACE_GE = prove
808 (`!a b. convex {x:real^N | a dot x >= b}`,
810 SUBGOAL_THEN `{x:real^N | a dot x >= b} = {x | --a dot x <= --b}`
811 (fun th -> REWRITE_TAC[th; CONVEX_HALFSPACE_LE]) THEN
812 REWRITE_TAC[EXTENSION; IN_ELIM_THM; DOT_LNEG] THEN REAL_ARITH_TAC);;
814 let CONVEX_HALFSPACE_COMPONENT_GE = prove
815 (`!a k. convex {x:real^N | x$k >= a}`,
817 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
819 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
820 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_GE) THEN
821 ASM_SIMP_TAC[DOT_BASIS]);;
823 let CONVEX_HYPERPLANE = prove
824 (`!a b. convex {x:real^N | a dot x = b}`,
827 `{x:real^N | a dot x = b} = {x | a dot x <= b} INTER {x | a dot x >= b}`
828 (fun th -> SIMP_TAC[th; CONVEX_INTER;
829 CONVEX_HALFSPACE_LE; CONVEX_HALFSPACE_GE]) THEN
830 REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC);;
832 let CONVEX_STANDARD_HYPERPLANE = prove
833 (`!k a. convex {x:real^N | x$k = a}`,
835 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
837 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
838 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HYPERPLANE) THEN
839 ASM_SIMP_TAC[DOT_BASIS]);;
841 let CONVEX_HALFSPACE_LT = prove
842 (`!a b. convex {x | a dot x < b}`,
843 REWRITE_TAC[convex; IN_ELIM_THM; DOT_RADD; DOT_RMUL] THEN
844 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN
847 let CONVEX_HALFSPACE_COMPONENT_LT = prove
848 (`!a k. convex {x:real^N | x$k < a}`,
850 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
852 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
853 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_LT) THEN
854 ASM_SIMP_TAC[DOT_BASIS]);;
856 let CONVEX_HALFSPACE_GT = prove
857 (`!a b. convex {x | a dot x > b}`,
858 REWRITE_TAC[REAL_ARITH `ax > b <=> --ax < --b`] THEN
859 REWRITE_TAC[GSYM DOT_LNEG; CONVEX_HALFSPACE_LT]);;
861 let CONVEX_HALFSPACE_COMPONENT_GT = prove
862 (`!a k. convex {x:real^N | x$k > a}`,
864 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
866 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
867 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CONVEX_HALFSPACE_GT) THEN
868 ASM_SIMP_TAC[DOT_BASIS]);;
870 let CONVEX_POSITIVE_ORTHANT = prove
871 (`convex {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
873 SIMP_TAC[convex; IN_ELIM_THM; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
874 REAL_LE_MUL; REAL_LE_ADD]);;
876 let LIMPT_OF_CONVEX = prove
878 convex s /\ x IN s ==> (x limit_point_of s <=> ~(s = {x}))`,
879 REPEAT STRIP_TAC THEN
880 ASM_CASES_TAC `s = {x:real^N}` THEN ASM_REWRITE_TAC[LIMPT_SING] THEN
881 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL
882 [ASM SET_TAC[]; ALL_TAC] THEN
883 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
884 ABBREV_TAC `u = min (&1 / &2) (e / &2 / norm(y - x:real^N))` THEN
885 SUBGOAL_THEN `&0 < u /\ u < &1` STRIP_ASSUME_TAC THENL
886 [EXPAND_TAC "u" THEN REWRITE_TAC[REAL_LT_MIN; REAL_MIN_LT] THEN
887 CONV_TAC REAL_RAT_REDUCE_CONV THEN
888 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ];
890 EXISTS_TAC `(&1 - u) % x + u % y:real^N` THEN REPEAT CONJ_TAC THENL
891 [FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
892 ASM_SIMP_TAC[REAL_LT_IMP_LE];
893 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ; VECTOR_ARITH
894 `(&1 - u) % x + u % y:real^N = x <=> u % (y - x) = vec 0`] THEN
896 REWRITE_TAC[dist; NORM_MUL; VECTOR_ARITH
897 `((&1 - u) % x + u % y) - x:real^N = u % (y - x)`] THEN
898 ASM_SIMP_TAC[REAL_ARITH `&0 < u ==> abs u = u`] THEN
899 MATCH_MP_TAC(REAL_ARITH `x <= e / &2 /\ &0 < e ==> x < e`) THEN
900 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
901 ASM_REAL_ARITH_TAC]);;
903 let TRIVIAL_LIMIT_WITHIN_CONVEX = prove
905 convex s /\ x IN s ==> (trivial_limit(at x within s) <=> s = {x})`,
906 SIMP_TAC[TRIVIAL_LIMIT_WITHIN; LIMPT_OF_CONVEX]);;
908 (* ------------------------------------------------------------------------- *)
909 (* Some invariance theorems for convex sets. *)
910 (* ------------------------------------------------------------------------- *)
912 let CONVEX_TRANSLATION_EQ = prove
913 (`!a:real^N s. convex (IMAGE (\x. a + x) s) <=> convex s`,
914 REWRITE_TAC[CONVEX_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
915 REWRITE_TAC[IN_IMAGE; UNWIND_THM1; VECTOR_ARITH
916 `(&1 - u) % (a + x) + u % (a + y) = a + z <=> (&1 - u) % x + u % y = z`]);;
918 add_translation_invariants [CONVEX_TRANSLATION_EQ];;
920 let CONVEX_TRANSLATION = prove
921 (`!s a:real^N. convex s ==> convex (IMAGE (\x. a + x) s)`,
922 REWRITE_TAC[CONVEX_TRANSLATION_EQ]);;
924 let CONVEX_LINEAR_IMAGE = prove
925 (`!f s. convex s /\ linear f ==> convex(IMAGE f s)`,
926 REWRITE_TAC[convex; FORALL_IN_IMAGE; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
927 REWRITE_TAC[IN_IMAGE; linear] THEN MESON_TAC[]);;
929 let CONVEX_LINEAR_IMAGE_EQ = prove
930 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
931 ==> (convex (IMAGE f s) <=> convex s)`,
932 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONVEX_LINEAR_IMAGE));;
934 add_linear_invariants [CONVEX_LINEAR_IMAGE_EQ];;
936 (* ------------------------------------------------------------------------- *)
937 (* Explicit expressions for convexity in terms of arbitrary sums. *)
938 (* ------------------------------------------------------------------------- *)
940 let CONVEX_VSUM = prove
941 (`!s k u x:A->real^N.
942 FINITE k /\ convex s /\ sum k u = &1 /\
943 (!i. i IN k ==> &0 <= u i /\ x i IN s)
944 ==> vsum k (\i. u i % x i) IN s`,
945 GEN_TAC THEN ASM_CASES_TAC `convex(s:real^N->bool)` THEN
946 ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
947 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
948 SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FORALL_IN_INSERT] THEN
949 CONV_TAC REAL_RAT_REDUCE_CONV THEN
950 MAP_EVERY X_GEN_TAC [`i:A`; `k:A->bool`] THEN
951 GEN_REWRITE_TAC (BINOP_CONV o DEPTH_CONV) [RIGHT_IMP_FORALL_THM] THEN
952 REWRITE_TAC[IMP_IMP] THEN STRIP_TAC THEN
953 MAP_EVERY X_GEN_TAC [`u:A->real`; `x:A->real^N`] THEN
954 ASM_CASES_TAC `(u:A->real) i = &1` THENL
955 [ASM_REWRITE_TAC[REAL_ARITH `&1 + a = &1 <=> a = &0`] THEN
957 SUBGOAL_THEN `vsum k (\i:A. u i % x(i):real^N) = vec 0`
958 (fun th -> ASM_SIMP_TAC[th; VECTOR_ADD_RID; VECTOR_MUL_LID]) THEN
959 MATCH_MP_TAC VSUM_EQ_0 THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
960 REPEAT STRIP_TAC THEN DISJ1_TAC THEN
961 ASM_MESON_TAC[SUM_POS_EQ_0];
963 FIRST_X_ASSUM(MP_TAC o SPEC `\j:A. u(j) / (&1 - u(i))`) THEN
964 ASM_REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
965 ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
966 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN
967 SUBGOAL_THEN `&0 < &1 - u(i:A)` ASSUME_TAC THENL
968 [ASM_MESON_TAC[SUM_POS_LE; REAL_ADD_SYM; REAL_ARITH
969 `&0 <= a /\ &0 <= b /\ b + a = &1 /\ ~(a = &1) ==> &0 < &1 - a`];
971 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LT_IMP_LE] THEN
972 ASM_SIMP_TAC[REAL_EQ_LDIV_EQ; REAL_MUL_LID; REAL_EQ_SUB_LADD] THEN
973 DISCH_TAC THEN ONCE_REWRITE_TAC[VECTOR_ADD_SYM] THEN
974 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
975 DISCH_THEN(MP_TAC o SPECL
976 [`vsum k (\j. (u j / (&1 - u(i:A))) % x(j) :real^N)`;
977 `x(i:A):real^N`; `&1 - u(i:A)`; `u(i:A):real`]) THEN
978 REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
979 ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
980 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN
981 REWRITE_TAC[VECTOR_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN
982 ASM_SIMP_TAC[REAL_LT_IMP_LE; VSUM_LMUL] THEN
983 CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; REAL_ARITH_TAC] THEN
984 ASM_MESON_TAC[REAL_ADD_SYM]]);;
986 let CONVEX_VSUM_STRONG = prove
987 (`!s k u x:A->real^N.
990 (!i. i IN k ==> &0 <= u i /\ (u i = &0 \/ x i IN s))
991 ==> vsum k (\i. u i % x i) IN s`,
992 REPEAT STRIP_TAC THEN
994 `vsum k (\i. u i % (x:A->real^N) i) =
995 vsum {i | i IN k /\ ~(u i = &0)} (\i. u i % x i)`
997 [MATCH_MP_TAC VSUM_SUPERSET THEN REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
999 MATCH_MP_TAC CONVEX_VSUM THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
1000 [ASM_MESON_TAC[SUM_DEGENERATE; REAL_ARITH `~(&1 = &0)`];
1001 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
1002 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN ASM SET_TAC[];
1005 let CONVEX_INDEXED = prove
1008 !k u x. (!i:num. 1 <= i /\ i <= k ==> &0 <= u(i) /\ x(i) IN s) /\
1010 ==> vsum (1..k) (\i. u(i) % x(i)) IN s`,
1011 REPEAT GEN_TAC THEN EQ_TAC THENL
1012 [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN
1013 ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG];
1014 DISCH_TAC THEN REWRITE_TAC[convex] THEN
1015 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
1016 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `2`) THEN
1017 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then u else v:real`) THEN
1018 DISCH_THEN(MP_TAC o SPEC `\n. if n = 1 then x else y:real^N`) THEN
1019 REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG; VSUM_CLAUSES_NUMSEG;
1020 NUMSEG_SING; VSUM_SING; SUM_SING] THEN REWRITE_TAC[ARITH] THEN
1023 let CONVEX_EXPLICIT = prove
1026 !t u. FINITE t /\ t SUBSET s /\ (!x. x IN t ==> &0 <= u x) /\
1028 ==> vsum t (\x. u(x) % x) IN s`,
1029 REPEAT GEN_TAC THEN EQ_TAC THENL
1030 [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN
1031 ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
1032 DISCH_TAC THEN REWRITE_TAC[convex] THEN
1033 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
1034 ASM_CASES_TAC `x:real^N = y` THENL
1035 [ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID]; ALL_TAC] THEN
1036 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N,y}`) THEN
1037 DISCH_THEN(MP_TAC o SPEC `\z:real^N. if z = x then u else v:real`) THEN
1038 ASM_SIMP_TAC[FINITE_INSERT; FINITE_RULES; SUM_CLAUSES; VSUM_CLAUSES;
1040 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; REAL_ADD_RID; SUBSET] THEN
1041 REWRITE_TAC[VECTOR_ADD_RID] THEN ASM_MESON_TAC[]]);;
1046 !(s:real^N->bool) (u:real^N->real).
1047 FINITE s /\ ~(s = {}) /\ s SUBSET V /\
1048 (!x. x IN s ==> &0 <= u x) /\ sum s u = &1
1049 ==> vsum s (\x. u x % x) IN V`,
1050 GEN_TAC THEN REWRITE_TAC[CONVEX_EXPLICIT] THEN
1051 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
1052 X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[] THEN
1053 AP_TERM_TAC THEN ABS_TAC THEN
1054 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1055 ASM_REWRITE_TAC[SUM_CLAUSES] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
1057 let CONVEX_FINITE = prove
1061 !u. (!x. x IN s ==> &0 <= u x) /\
1063 ==> vsum s (\x. u(x) % x) IN s)`,
1064 REPEAT STRIP_TAC THEN REWRITE_TAC[CONVEX_EXPLICIT] THEN
1065 EQ_TAC THENL [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN
1066 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
1068 FIRST_X_ASSUM(MP_TAC o SPEC `\x:real^N. if x IN t then u x else &0`) THEN
1069 ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET] THEN
1070 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
1071 ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL; GSYM VSUM_RESTRICT_SET] THEN
1072 ASM_SIMP_TAC[COND_ID; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`]);;
1074 let AFFINE_PCROSS = prove
1075 (`!s:real^M->bool t:real^N->bool.
1076 affine s /\ affine t ==> affine(s PCROSS t)`,
1077 REWRITE_TAC[affine; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1078 SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN
1079 SIMP_TAC[PASTECART_IN_PCROSS]);;
1081 let AFFINE_PCROSS_EQ = prove
1082 (`!s:real^M->bool t:real^N->bool.
1083 affine(s PCROSS t) <=> s = {} \/ t = {} \/ affine s /\ affine t`,
1085 ASM_CASES_TAC `s:real^M->bool = {}` THEN
1086 ASM_REWRITE_TAC[PCROSS_EMPTY; AFFINE_EMPTY] THEN
1087 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1088 ASM_REWRITE_TAC[PCROSS_EMPTY; AFFINE_EMPTY] THEN
1089 EQ_TAC THEN REWRITE_TAC[AFFINE_PCROSS] THEN REPEAT STRIP_TAC THENL
1090 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
1091 `(s:real^M->bool) PCROSS (t:real^N->bool)`] AFFINE_LINEAR_IMAGE) THEN
1092 ASM_REWRITE_TAC[LINEAR_FSTCART];
1093 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
1094 `(s:real^M->bool) PCROSS (t:real^N->bool)`] AFFINE_LINEAR_IMAGE) THEN
1095 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
1096 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1097 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
1098 FSTCART_PASTECART; SNDCART_PASTECART] THEN
1101 let CONVEX_PCROSS = prove
1102 (`!s:real^M->bool t:real^N->bool.
1103 convex s /\ convex t ==> convex(s PCROSS t)`,
1104 REWRITE_TAC[convex; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1105 SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN
1106 SIMP_TAC[PASTECART_IN_PCROSS]);;
1108 let CONVEX_PCROSS_EQ = prove
1109 (`!s:real^M->bool t:real^N->bool.
1110 convex(s PCROSS t) <=> s = {} \/ t = {} \/ convex s /\ convex t`,
1112 ASM_CASES_TAC `s:real^M->bool = {}` THEN
1113 ASM_REWRITE_TAC[PCROSS_EMPTY; CONVEX_EMPTY] THEN
1114 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1115 ASM_REWRITE_TAC[PCROSS_EMPTY; CONVEX_EMPTY] THEN
1116 EQ_TAC THEN REWRITE_TAC[CONVEX_PCROSS] THEN REPEAT STRIP_TAC THENL
1117 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
1118 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_LINEAR_IMAGE) THEN
1119 ASM_REWRITE_TAC[LINEAR_FSTCART];
1120 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
1121 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_LINEAR_IMAGE) THEN
1122 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
1123 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1124 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
1125 FSTCART_PASTECART; SNDCART_PASTECART] THEN
1128 (* ------------------------------------------------------------------------- *)
1129 (* Conic sets and conic hull. *)
1130 (* ------------------------------------------------------------------------- *)
1132 let conic = new_definition
1133 `conic s <=> !x c. x IN s /\ &0 <= c ==> (c % x) IN s`;;
1135 let SUBSPACE_IMP_CONIC = prove
1136 (`!s. subspace s ==> conic s`,
1137 SIMP_TAC[subspace; conic]);;
1139 let CONIC_EMPTY = prove
1141 REWRITE_TAC[conic; NOT_IN_EMPTY]);;
1143 let CONIC_UNIV = prove
1144 (`conic (UNIV:real^N->bool)`,
1145 REWRITE_TAC[conic; IN_UNIV]);;
1147 let CONIC_INTERS = prove
1148 (`(!s. s IN f ==> conic s) ==> conic(INTERS f)`,
1149 REWRITE_TAC[conic; IN_INTERS] THEN MESON_TAC[]);;
1151 let CONIC_LINEAR_IMAGE = prove
1152 (`!f s. conic s /\ linear f ==> conic(IMAGE f s)`,
1153 REWRITE_TAC[conic; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
1154 REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[LINEAR_CMUL]);;
1156 let CONIC_LINEAR_IMAGE_EQ = prove
1157 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
1158 ==> (conic (IMAGE f s) <=> conic s)`,
1159 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONIC_LINEAR_IMAGE));;
1161 add_linear_invariants [CONIC_LINEAR_IMAGE_EQ];;
1163 let CONIC_CONIC_HULL = prove
1164 (`!s. conic(conic hull s)`,
1165 SIMP_TAC[P_HULL; CONIC_INTERS]);;
1167 let CONIC_HULL_EQ = prove
1168 (`!s. (conic hull s = s) <=> conic s`,
1169 SIMP_TAC[HULL_EQ; CONIC_INTERS]);;
1171 let CONIC_NEGATIONS = prove
1172 (`!s. conic s ==> conic (IMAGE (--) s)`,
1173 REWRITE_TAC[conic; RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_IMAGE] THEN
1174 REWRITE_TAC[IN_IMAGE; VECTOR_MUL_RNEG] THEN MESON_TAC[]);;
1176 let CONIC_SPAN = prove
1177 (`!s. conic(span s)`,
1178 SIMP_TAC[SUBSPACE_IMP_CONIC; SUBSPACE_SPAN]);;
1180 let CONIC_HULL_EXPLICIT = prove
1181 (`!s:real^N->bool. conic hull s = {c % x | &0 <= c /\ x IN s}`,
1182 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN
1183 REWRITE_TAC[conic; SUBSET; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
1184 REWRITE_TAC[FORALL_IN_GSPEC] THEN
1185 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; IN_ELIM_THM] THEN
1186 REPEAT CONJ_TAC THENL
1187 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
1188 MAP_EVERY EXISTS_TAC [`&1`; `x:real^N`] THEN
1189 ASM_SIMP_TAC[REAL_POS; VECTOR_MUL_LID];
1190 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MESON_TAC[REAL_LE_MUL];
1193 let CONIC_HULL_LINEAR_IMAGE = prove
1194 (`!f s. linear f ==> conic hull (IMAGE f s) = IMAGE f (conic hull s)`,
1195 REPEAT GEN_TAC THEN REWRITE_TAC[CONIC_HULL_EXPLICIT] THEN
1196 REWRITE_TAC[SET_RULE `IMAGE f {c % x | P c x} = {f(c % x) | P c x}`] THEN
1197 REWRITE_TAC[SET_RULE `{c % x | &0 <= c /\ x IN IMAGE f s} =
1198 {c % f(x) | &0 <= c /\ x IN s}`] THEN
1199 DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]));;
1201 add_linear_invariants [CONIC_HULL_LINEAR_IMAGE];;
1203 let CONVEX_CONIC_HULL = prove
1204 (`!s:real^N->bool. convex s ==> convex (conic hull s)`,
1205 REWRITE_TAC[CONIC_HULL_EXPLICIT] THEN
1206 REWRITE_TAC[CONVEX_ALT; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1207 REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM; IMP_IMP] THEN
1208 X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN
1209 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN
1210 MAP_EVERY X_GEN_TAC [`d:real`; `y:real^N`] THEN STRIP_TAC THEN
1211 X_GEN_TAC `u:real` THEN STRIP_TAC THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
1212 ASM_CASES_TAC `(&1 - u) * c = &0` THENL
1213 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
1214 ASM_MESON_TAC[REAL_LE_MUL];
1216 SUBGOAL_THEN `&0 < (&1 - u) * c + u * d` ASSUME_TAC THENL
1217 [MATCH_MP_TAC REAL_LTE_ADD THEN ASM_REWRITE_TAC[REAL_LT_LE] THEN
1218 CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC;
1220 EXISTS_TAC `(&1 - u) * c + u * d:real` THEN
1221 EXISTS_TAC `((&1 - u) * c) / ((&1 - u) * c + u * d) % x +
1222 (u * d) / ((&1 - u) * c + u * d) % y:real^N` THEN
1223 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN
1224 ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
1225 ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL; REAL_SUB_LE] THEN
1226 ASM_SIMP_TAC[REAL_FIELD
1227 `&0 < u + v ==> u / (u + v) = &1 - (v / (u + v))`] THEN
1228 RULE_ASSUM_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]) THEN
1229 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
1230 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ] THEN
1231 ASM_SIMP_TAC[REAL_MUL_LZERO; REAL_LE_MUL; REAL_MUL_LID; REAL_LE_ADDL;
1234 let CONIC_HALFSPACE_LE = prove
1235 (`!a. conic {x | a dot x <= &0}`,
1236 REWRITE_TAC[conic; IN_ELIM_THM; DOT_RMUL] THEN
1237 REWRITE_TAC[REAL_ARITH `a <= &0 <=> &0 <= --a`] THEN
1238 SIMP_TAC[GSYM REAL_MUL_RNEG; REAL_LE_MUL]);;
1240 let CONIC_HALFSPACE_GE = prove
1241 (`!a. conic {x | a dot x >= &0}`,
1242 SIMP_TAC[conic; IN_ELIM_THM; DOT_RMUL; real_ge; REAL_LE_MUL]);;
1244 let CONIC_HULL_EMPTY = prove
1245 (`conic hull {} = {}`,
1246 MATCH_MP_TAC HULL_UNIQUE THEN
1247 REWRITE_TAC[SUBSET_REFL; CONIC_EMPTY; EMPTY_SUBSET]);;
1249 let CONIC_CONTAINS_0 = prove
1250 (`!s:real^N->bool. conic s ==> (vec 0 IN s <=> ~(s = {}))`,
1251 REPEAT STRIP_TAC THEN EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
1252 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
1253 DISCH_THEN(X_CHOOSE_TAC `x:real^N`) THEN
1254 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [conic]) THEN
1255 DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `&0`]) THEN
1256 ASM_REWRITE_TAC[REAL_POS; VECTOR_MUL_LZERO]);;
1258 let CONIC_HULL_EQ_EMPTY = prove
1259 (`!s. (conic hull s = {}) <=> (s = {})`,
1260 GEN_TAC THEN EQ_TAC THEN
1261 MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; CONIC_HULL_EMPTY]);;
1263 let CONIC_SUMS = prove
1264 (`!s t. conic s /\ conic t ==> conic {x + y:real^N | x IN s /\ y IN t}`,
1265 REWRITE_TAC[conic; IN_ELIM_THM] THEN
1266 MESON_TAC[VECTOR_ADD_LDISTRIB]);;
1268 let CONIC_PCROSS = prove
1269 (`!s:real^M->bool t:real^N->bool.
1270 conic s /\ conic t ==> conic(s PCROSS t)`,
1271 REWRITE_TAC[conic; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1272 SIMP_TAC[FORALL_IN_PCROSS; GSYM PASTECART_CMUL; PASTECART_ADD] THEN
1273 SIMP_TAC[PASTECART_IN_PCROSS]);;
1275 let CONIC_PCROSS_EQ = prove
1276 (`!s:real^M->bool t:real^N->bool.
1277 conic(s PCROSS t) <=> s = {} \/ t = {} \/ conic s /\ conic t`,
1279 ASM_CASES_TAC `s:real^M->bool = {}` THEN
1280 ASM_REWRITE_TAC[PCROSS_EMPTY; CONIC_EMPTY] THEN
1281 ASM_CASES_TAC `t:real^N->bool = {}` THEN
1282 ASM_REWRITE_TAC[PCROSS_EMPTY; CONIC_EMPTY] THEN
1283 EQ_TAC THEN REWRITE_TAC[CONIC_PCROSS] THEN REPEAT STRIP_TAC THENL
1284 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
1285 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONIC_LINEAR_IMAGE) THEN
1286 ASM_REWRITE_TAC[LINEAR_FSTCART];
1287 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
1288 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONIC_LINEAR_IMAGE) THEN
1289 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
1290 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1291 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
1292 FSTCART_PASTECART; SNDCART_PASTECART] THEN
1295 let CONIC_POSITIVE_ORTHANT = prove
1296 (`conic {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i}`,
1297 SIMP_TAC[conic; IN_ELIM_THM; REAL_LE_MUL; VECTOR_MUL_COMPONENT]);;
1299 let SEPARATE_CLOSED_CONES = prove
1300 (`!c d:real^N->bool.
1301 conic c /\ closed c /\ conic d /\ closed d /\ c INTER d SUBSET {vec 0}
1303 !x y. x IN c /\ y IN d
1304 ==> dist(x,y) >= e * max (norm x) (norm y)`,
1307 conic c /\ closed c /\ conic d /\ closed d /\ c INTER d SUBSET {vec 0}
1309 !x y. x IN c /\ y IN d ==> dist(x,y)
1312 [REPEAT STRIP_TAC THEN REWRITE_TAC[real_ge] THEN
1313 MP_TAC(ISPECL [`c INTER sphere(vec 0:real^N,&1)`; `d:real^N->bool`]
1314 SEPARATE_COMPACT_CLOSED) THEN
1315 ASM_SIMP_TAC[CLOSED_INTER_COMPACT; COMPACT_SPHERE] THEN ANTS_TAC THENL
1316 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
1317 `c INTER d SUBSET {a} ==> ~(a IN s) ==> (c INTER s) INTER d = {}`)) THEN
1318 REWRITE_TAC[IN_SPHERE_0; NORM_0] THEN REAL_ARITH_TAC;
1319 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
1320 REWRITE_TAC[IN_INTER; IN_SPHERE_0] THEN STRIP_TAC THEN
1321 ASM_REWRITE_TAC[] THEN
1322 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1323 ASM_CASES_TAC `x:real^N = vec 0` THEN
1324 ASM_REWRITE_TAC[DIST_POS_LE; REAL_MUL_RZERO; NORM_0] THEN
1325 FIRST_X_ASSUM(MP_TAC o SPECL
1326 [`inv(norm x) % x:real^N`; `inv(norm(x:real^N)) % y:real^N`]) THEN
1327 REWRITE_TAC[dist; NORM_MUL; GSYM VECTOR_SUB_LDISTRIB] THEN
1328 REWRITE_TAC[REAL_ARITH `abs x * a = a * abs x`] THEN
1329 REWRITE_TAC[REAL_ABS_INV; GSYM real_div; REAL_ABS_NORM] THEN
1330 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
1331 DISCH_THEN MATCH_MP_TAC THEN
1332 ASM_SIMP_TAC[REAL_DIV_REFL; NORM_EQ_0] THEN
1333 RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN
1334 CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
1335 ASM_SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE]];
1336 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
1337 MP_TAC(SPECL [`c:real^N->bool`; `d:real^N->bool`] th) THEN
1338 MP_TAC(SPECL [`d:real^N->bool`; `c:real^N->bool`] th)) THEN
1339 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[INTER_COMM] THEN
1340 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; real_ge] THEN
1341 X_GEN_TAC `d:real` THEN STRIP_TAC THEN
1342 X_GEN_TAC `e:real` THEN STRIP_TAC THEN
1343 EXISTS_TAC `min d e:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
1344 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1345 REWRITE_TAC[real_max] THEN COND_CASES_TAC THEN
1346 MATCH_MP_TAC REAL_LE_TRANS THENL
1347 [EXISTS_TAC `d * norm(y:real^N)` THEN ONCE_REWRITE_TAC[DIST_SYM];
1348 EXISTS_TAC `e * norm(x:real^N)`] THEN
1349 ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_RMUL THEN NORM_ARITH_TAC]);;
1351 let CONTINUOUS_ON_COMPACT_SURFACE_PROJECTION = prove
1352 (`!s:real^N->bool v d:real^N->real.
1353 compact s /\ s SUBSET (v DELETE (vec 0)) /\ conic v /\
1354 (!x k. x IN v DELETE (vec 0) ==> (&0 < k /\ (k % x) IN s <=> d x = k))
1355 ==> (\x. d x % x) continuous_on (v DELETE (vec 0))`,
1357 (`!s:real^N->real^N p srf:real^N->bool pnc.
1358 compact srf /\ srf SUBSET pnc /\
1359 IMAGE s pnc SUBSET srf /\ (!x. x IN srf ==> s x = x) /\
1360 p continuous_on pnc /\
1361 (!x. x IN pnc ==> s(p x) = s x /\ p(s x) = p x)
1362 ==> s continuous_on pnc`,
1363 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN
1364 EXISTS_TAC `(s:real^N->real^N) o (p:real^N->real^N)` THEN
1365 CONJ_TAC THENL [ASM_SIMP_TAC[o_DEF]; ALL_TAC] THEN
1366 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN
1367 SUBGOAL_THEN `IMAGE (p:real^N->real^N) pnc = IMAGE p srf` SUBST1_TAC THENL
1369 MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN ASM_REWRITE_TAC[] THEN
1370 CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ASM SET_TAC[]]]) in
1371 REWRITE_TAC[conic; IN_DELETE; SUBSET] THEN
1372 REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN
1373 MAP_EVERY EXISTS_TAC [`\x:real^N. inv(norm x) % x`; `s:real^N->bool`] THEN
1374 ASM_REWRITE_TAC[] THEN
1375 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
1376 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
1377 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN CONJ_TAC THENL
1378 [MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; CONTINUOUS_ON_ID] THEN
1379 MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
1380 SIMP_TAC[IN_DELETE; NORM_EQ_0; SIMP_RULE[o_DEF] CONTINUOUS_ON_LIFT_NORM];
1381 REWRITE_TAC[IN_UNIV; IN_DELETE]] THEN
1383 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
1384 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `&1`]) THEN
1385 ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_LT_01; IN_DELETE] THEN
1386 ASM_MESON_TAC[VECTOR_MUL_LID; SUBSET; IN_DELETE];
1388 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN CONJ_TAC THENL
1389 [FIRST_ASSUM(MP_TAC o SPECL
1390 [`inv(norm x) % x:real^N`; `norm x * (d:real^N->real) x`]) THEN
1391 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `(d:real^N->real) x`]) THEN
1392 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0] THEN STRIP_TAC THEN
1393 ASM_SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE; REAL_LT_MUL; NORM_POS_LT] THEN
1394 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; NORM_EQ_0; REAL_FIELD
1395 `~(n = &0) ==> (n * d) * inv n = d`];
1396 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `(d:real^N->real) x`]) THEN
1397 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
1398 ASM_SIMP_TAC[NORM_MUL; VECTOR_MUL_ASSOC; REAL_INV_MUL] THEN
1399 ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN
1400 ASM_SIMP_TAC[REAL_FIELD `&0 < x ==> (inv(x) * y) * x = y`]]);;
1402 (* ------------------------------------------------------------------------- *)
1403 (* Affine dependence and consequential theorems (from Lars Schewe). *)
1404 (* ------------------------------------------------------------------------- *)
1406 let affine_dependent = new_definition
1407 `affine_dependent (s:real^N -> bool) <=>
1408 ?x. x IN s /\ x IN (affine hull (s DELETE x))`;;
1410 let AFFINE_DEPENDENT_EXPLICIT = prove
1411 (`!p. affine_dependent (p:real^N -> bool) <=>
1412 (?s u. FINITE s /\ s SUBSET p /\
1414 (?v. v IN s /\ ~(u v = &0)) /\
1415 vsum s (\v. u v % v) = (vec 0):real^N)`,
1416 X_GEN_TAC `p:real^N->bool` THEN EQ_TAC THENL
1417 [REWRITE_TAC[affine_dependent;AFFINE_HULL_EXPLICIT;
1419 REPEAT STRIP_TAC THEN
1420 EXISTS_TAC `(x:real^N) INSERT s` THEN
1421 EXISTS_TAC `\v:real^N.if v = x then -- &1 else u v` THEN
1422 ASM_SIMP_TAC[FINITE_INSERT;SUM_CLAUSES;VSUM_CLAUSES;INSERT_SUBSET] THEN
1423 REPEAT CONJ_TAC THENL
1425 COND_CASES_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN
1426 ASM_SIMP_TAC[SUM_CASES; SUM_CLAUSES; SET_RULE
1427 `~((x:real^N) IN s) ==> {v | v IN s /\ v = x} = {} /\
1428 {v | v IN s /\ ~(v = x)} = s`] THEN
1430 SET_TAC[REAL_ARITH `~(-- &1 = &0)`];
1431 MP_TAC (SET_RULE `s SUBSET p DELETE (x:real^N) ==> ~(x IN s)`) THEN
1432 ASM_REWRITE_TAC[] THEN
1434 ASM_SIMP_TAC[VECTOR_ARITH
1435 `(-- &1 % (x:real^N)) + a = vec 0 <=> a = x`] THEN
1436 MATCH_MP_TAC EQ_TRANS THEN
1437 EXISTS_TAC `vsum s (\v:real^N. u v % v)` THEN
1439 MATCH_MP_TAC VSUM_EQ THEN
1442 ASM_REWRITE_TAC[]]];
1444 REWRITE_TAC[affine_dependent;AFFINE_HULL_EXPLICIT;IN_ELIM_THM] THEN
1445 REPEAT STRIP_TAC THEN
1446 EXISTS_TAC `v:real^N` THEN
1447 CONJ_TAC THENL [ASM SET_TAC[];ALL_TAC] THEN
1448 EXISTS_TAC `s DELETE (v:real^N)` THEN
1449 EXISTS_TAC `\x:real^N. -- (&1 / (u v)) * u x` THEN
1450 ASM_SIMP_TAC[FINITE_DELETE;SUM_DELETE;VSUM_DELETE_CASES] THEN
1451 ASM_SIMP_TAC[SUM_LMUL;GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;
1452 VECTOR_MUL_RZERO;VECTOR_ARITH `vec 0 - -- a % x = a % x:real^N`;
1453 REAL_MUL_RZERO;REAL_ARITH `&0 - -- a * b = a * b`] THEN
1454 ASM_SIMP_TAC[REAL_FIELD `~(x = &0) ==> &1 / x * x = &1`;
1455 VECTOR_MUL_ASSOC;VECTOR_MUL_LID] THEN
1456 CONJ_TAC THENL [ALL_TAC;ASM SET_TAC[]] THEN
1457 ASM_SIMP_TAC[SET_RULE `v IN s ==> (s DELETE v = {} <=> s = {v})`] THEN
1458 ASM_CASES_TAC `s = {v:real^N}` THEN
1459 ASM_REWRITE_TAC[] THEN
1460 FIRST_X_ASSUM SUBST_ALL_TAC THEN
1461 FIND_ASSUM MP_TAC `sum {v:real^N} u = &0` THEN
1462 REWRITE_TAC[SUM_SING]
1463 THEN ASM_REWRITE_TAC[]);;
1465 let AFFINE_DEPENDENT_EXPLICIT_FINITE = prove
1466 (`!s. FINITE(s:real^N -> bool)
1467 ==> (affine_dependent s <=>
1469 (?v. v IN s /\ ~(u v = &0)) /\
1470 vsum s (\v. u v % v) = vec 0)`,
1471 REPEAT STRIP_TAC THEN REWRITE_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN
1472 EQ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET_REFL]] THEN
1473 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool`
1474 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)) THEN
1475 EXISTS_TAC `\x:real^N. if x IN t then u(x) else &0` THEN
1476 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
1477 ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
1478 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
1481 let AFFINE_DEPENDENT_TRANSLATION_EQ = prove
1482 (`!a s. affine_dependent (IMAGE (\x. a + x) s) <=> affine_dependent s`,
1483 REWRITE_TAC[affine_dependent] THEN GEOM_TRANSLATE_TAC[]);;
1485 add_translation_invariants [AFFINE_DEPENDENT_TRANSLATION_EQ];;
1487 let AFFINE_DEPENDENT_TRANSLATION = prove
1488 (`!s a. affine_dependent s ==> affine_dependent (IMAGE (\x. a + x) s)`,
1489 REWRITE_TAC[AFFINE_DEPENDENT_TRANSLATION_EQ]);;
1491 let AFFINE_DEPENDENT_LINEAR_IMAGE_EQ = prove
1492 (`!f:real^M->real^N s.
1493 linear f /\ (!x y. f x = f y ==> x = y)
1494 ==> (affine_dependent(IMAGE f s) <=> affine_dependent s)`,
1495 REWRITE_TAC[affine_dependent] THEN GEOM_TRANSFORM_TAC[]);;
1497 add_linear_invariants [AFFINE_DEPENDENT_LINEAR_IMAGE_EQ];;
1499 let AFFINE_DEPENDENT_LINEAR_IMAGE = prove
1500 (`!f:real^M->real^N s.
1501 linear f /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
1503 ==> affine_dependent(IMAGE f s)`,
1505 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
1506 REWRITE_TAC[affine_dependent; EXISTS_IN_IMAGE] THEN
1507 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN
1508 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1509 SUBGOAL_THEN `IMAGE (f:real^M->real^N) s DELETE f a = IMAGE f (s DELETE a)`
1510 (fun t -> ASM_SIMP_TAC[FUN_IN_IMAGE; AFFINE_HULL_LINEAR_IMAGE; t]) THEN
1513 let AFFINE_DEPENDENT_MONO = prove
1514 (`!s t:real^N->bool. affine_dependent s /\ s SUBSET t ==> affine_dependent t`,
1515 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
1516 REWRITE_TAC[affine_dependent] THEN MATCH_MP_TAC MONO_EXISTS THEN
1517 X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
1518 FIRST_ASSUM(MP_TAC o MATCH_MP HULL_MONO o SPEC `x:real^N` o MATCH_MP
1519 (SET_RULE `!x. s SUBSET t ==> (s DELETE x) SUBSET (t DELETE x)`)) THEN
1522 let AFFINE_INDEPENDENT_EMPTY = prove
1523 (`~(affine_dependent {})`,
1524 REWRITE_TAC[affine_dependent; NOT_IN_EMPTY]);;
1526 let AFFINE_INDEPENDENT_1 = prove
1527 (`!a:real^N. ~(affine_dependent {a})`,
1528 REWRITE_TAC[affine_dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
1529 REWRITE_TAC[SET_RULE `{a} DELETE a = {}`; AFFINE_HULL_EMPTY; NOT_IN_EMPTY]);;
1531 let AFFINE_INDEPENDENT_2 = prove
1532 (`!a b:real^N. ~(affine_dependent {a,b})`,
1533 REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
1534 [ASM_REWRITE_TAC[INSERT_AC; AFFINE_INDEPENDENT_1];
1535 REWRITE_TAC[affine_dependent; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
1536 ASM_SIMP_TAC[SET_RULE
1537 `~(a = b) ==> {a,b} DELETE a = {b} /\ {a,b} DELETE b = {a}`] THEN
1538 ASM_REWRITE_TAC[AFFINE_HULL_SING; IN_SING]]);;
1540 let AFFINE_INDEPENDENT_SUBSET = prove
1541 (`!s t. ~affine_dependent t /\ s SUBSET t ==> ~affine_dependent s`,
1542 REWRITE_TAC[IMP_CONJ_ALT; CONTRAPOS_THM] THEN
1543 REWRITE_TAC[GSYM IMP_CONJ_ALT; AFFINE_DEPENDENT_MONO]);;
1545 let AFFINE_INDEPENDENT_DELETE = prove
1546 (`!s a. ~affine_dependent s ==> ~affine_dependent(s DELETE a)`,
1548 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_INDEPENDENT_SUBSET) THEN
1551 (* ------------------------------------------------------------------------- *)
1552 (* Coplanarity, and collinearity in terms of affine hull. *)
1553 (* ------------------------------------------------------------------------- *)
1555 let coplanar = new_definition
1556 `coplanar s <=> ?u v w. s SUBSET affine hull {u,v,w}`;;
1558 let COLLINEAR_AFFINE_HULL = prove
1559 (`!s:real^N->bool. collinear s <=> ?u v. s SUBSET affine hull {u,v}`,
1560 GEN_TAC THEN REWRITE_TAC[collinear; AFFINE_HULL_2] THEN EQ_TAC THEN
1561 REWRITE_TAC[SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
1562 REWRITE_TAC[REAL_ARITH `u + v = &1 <=> &1 - u = v`; UNWIND_THM1] THENL
1563 [X_GEN_TAC `u:real^N` THEN DISCH_TAC THEN
1564 ASM_CASES_TAC `s:real^N->bool = {}` THEN
1565 ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
1566 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
1567 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN
1568 DISCH_TAC THEN EXISTS_TAC `x + u:real^N` THEN X_GEN_TAC `y:real^N` THEN
1570 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^N`; `y:real^N`]) THEN
1571 ASM_REWRITE_TAC[VECTOR_ARITH `x - y:real^N = z <=> x = y + z`] THEN
1572 DISCH_THEN(X_CHOOSE_THEN `c:real` SUBST1_TAC) THEN
1573 EXISTS_TAC `&1 + c` THEN VECTOR_ARITH_TAC;
1574 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN
1575 EXISTS_TAC `b - a:real^N` THEN
1576 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
1577 FIRST_X_ASSUM(fun th ->
1578 MP_TAC(SPEC `y:real^N` th) THEN MP_TAC(SPEC `x:real^N` th)) THEN
1579 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
1580 X_GEN_TAC `r:real` THEN DISCH_THEN SUBST1_TAC THEN
1581 X_GEN_TAC `s:real` THEN DISCH_THEN SUBST1_TAC THEN
1582 EXISTS_TAC `s - r:real` THEN VECTOR_ARITH_TAC]);;
1584 let COLLINEAR_IMP_COPLANAR = prove
1585 (`!s. collinear s ==> coplanar s`,
1586 REWRITE_TAC[coplanar; COLLINEAR_AFFINE_HULL] THEN MESON_TAC[INSERT_AC]);;
1588 let COPLANAR_SMALL = prove
1589 (`!s. FINITE s /\ CARD s <= 3 ==> coplanar s`,
1590 GEN_TAC THEN REWRITE_TAC[ARITH_RULE `s <= 3 <=> s <= 2 \/ s = 3`] THEN
1591 REWRITE_TAC[LEFT_OR_DISTRIB; GSYM HAS_SIZE] THEN
1592 DISCH_THEN(DISJ_CASES_THEN MP_TAC) THEN
1593 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_SMALL] THEN
1594 CONV_TAC(LAND_CONV HAS_SIZE_CONV) THEN REWRITE_TAC[coplanar] THEN
1595 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
1596 STRIP_TAC THEN ASM_REWRITE_TAC[HULL_INC; SUBSET]);;
1598 let COPLANAR_EMPTY = prove
1600 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_EMPTY]);;
1602 let COPLANAR_SING = prove
1603 (`!a. coplanar {a}`,
1604 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_SING]);;
1606 let COPLANAR_2 = prove
1607 (`!a b. coplanar {a,b}`,
1608 SIMP_TAC[COLLINEAR_IMP_COPLANAR; COLLINEAR_2]);;
1610 let COPLANAR_3 = prove
1611 (`!a b c. coplanar {a,b,c}`,
1612 REPEAT GEN_TAC THEN MATCH_MP_TAC COPLANAR_SMALL THEN
1613 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_RULES] THEN ARITH_TAC);;
1615 let COLLINEAR_AFFINE_HULL_COLLINEAR = prove
1616 (`!s. collinear(affine hull s) <=> collinear s`,
1617 REWRITE_TAC[COLLINEAR_AFFINE_HULL] THEN
1618 MESON_TAC[HULL_HULL; HULL_MONO; HULL_INC; SUBSET]);;
1620 let COPLANAR_AFFINE_HULL_COPLANAR = prove
1621 (`!s. coplanar(affine hull s) <=> coplanar s`,
1622 REWRITE_TAC[coplanar] THEN
1623 MESON_TAC[HULL_HULL; HULL_MONO; HULL_INC; SUBSET]);;
1625 let COPLANAR_TRANSLATION_EQ = prove
1626 (`!a:real^N s. coplanar(IMAGE (\x. a + x) s) <=> coplanar s`,
1627 REWRITE_TAC[coplanar] THEN GEOM_TRANSLATE_TAC[]);;
1629 let COPLANAR_TRANSLATION = prove
1630 (`!a:real^N s. coplanar s ==> coplanar(IMAGE (\x. a + x) s)`,
1631 REWRITE_TAC[COPLANAR_TRANSLATION_EQ]);;
1633 add_translation_invariants [COPLANAR_TRANSLATION_EQ];;
1635 let COPLANAR_LINEAR_IMAGE = prove
1636 (`!f:real^M->real^N s. coplanar s /\ linear f ==> coplanar(IMAGE f s)`,
1637 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
1638 REWRITE_TAC[coplanar; LEFT_IMP_EXISTS_THM] THEN
1639 MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real^M`] THEN STRIP_TAC THEN
1640 MAP_EVERY EXISTS_TAC
1641 [`(f:real^M->real^N) a`; `(f:real^M->real^N) b`; `(f:real^M->real^N) c`] THEN
1642 REWRITE_TAC[SET_RULE `{f a,f b,f c} = IMAGE f {a,b,c}`] THEN
1643 ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE; IMAGE_SUBSET]);;
1645 let COPLANAR_LINEAR_IMAGE_EQ = prove
1646 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
1647 ==> (coplanar (IMAGE f s) <=> coplanar s)`,
1648 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
1650 add_linear_invariants [COPLANAR_LINEAR_IMAGE_EQ];;
1652 let COPLANAR_SUBSET = prove
1653 (`!s t. coplanar t /\ s SUBSET t ==> coplanar s`,
1654 REWRITE_TAC[coplanar] THEN SET_TAC[]);;
1656 let AFFINE_HULL_3_IMP_COLLINEAR = prove
1657 (`!a b c. c IN affine hull {a,b} ==> collinear {a,b,c}`,
1658 ONCE_REWRITE_TAC[GSYM COLLINEAR_AFFINE_HULL_COLLINEAR] THEN
1659 SIMP_TAC[HULL_REDUNDANT_EQ; INSERT_AC] THEN
1660 REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2]);;
1662 let COLLINEAR_3_AFFINE_HULL = prove
1664 ~(a = b) ==> (collinear {a,b,c} <=> c IN affine hull {a,b})`,
1665 REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[AFFINE_HULL_3_IMP_COLLINEAR] THEN
1666 REWRITE_TAC[collinear] THEN
1667 DISCH_THEN(X_CHOOSE_THEN `u:real^N` STRIP_ASSUME_TAC) THEN
1668 FIRST_ASSUM(fun th -> MP_TAC(SPECL [`b:real^N`; `a:real^N`] th) THEN
1669 MP_TAC(SPECL [`c:real^N`; `a:real^N`] th)) THEN
1670 REWRITE_TAC[IN_INSERT; AFFINE_HULL_2; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
1671 REWRITE_TAC[VECTOR_ARITH `a - b:real^N = c <=> a = b + c`] THEN
1672 X_GEN_TAC `x:real` THEN DISCH_TAC THEN X_GEN_TAC `y:real` THEN
1673 ASM_CASES_TAC `y = &0` THEN
1674 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN DISCH_TAC THEN
1675 ASM_REWRITE_TAC[] THEN
1676 MAP_EVERY EXISTS_TAC [`&1 - x / y`; `x / y:real`] THEN
1677 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
1678 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN
1679 ASM_SIMP_TAC[REAL_DIV_RMUL] THEN VECTOR_ARITH_TAC);;
1681 let COLLINEAR_3_EQ_AFFINE_DEPENDENT = prove
1683 collinear{a,b,c} <=>
1684 a = b \/ a = c \/ b = c \/ affine_dependent {a,b,c}`,
1687 ASM_CASES_TAC t THENL [ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC])
1688 [`a:real^N = b`; `a:real^N = c`; `b:real^N = c`] THEN
1689 ASM_REWRITE_TAC[affine_dependent] THEN EQ_TAC THENL
1690 [ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL] THEN DISCH_TAC THEN
1691 EXISTS_TAC `c:real^N` THEN REWRITE_TAC[IN_INSERT];
1692 REWRITE_TAC[EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THENL
1693 [ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`];
1694 ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,a,b}`];
1696 ASM_SIMP_TAC[COLLINEAR_3_AFFINE_HULL]] THEN
1697 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
1698 `x IN s ==> s SUBSET t ==> x IN t`)) THEN
1699 MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]);;
1701 let AFFINE_DEPENDENT_IMP_COLLINEAR_3 = prove
1702 (`!a b c:real^N. affine_dependent {a,b,c} ==> collinear{a,b,c}`,
1703 REPEAT GEN_TAC THEN REWRITE_TAC[affine_dependent] THEN
1704 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; RIGHT_OR_DISTRIB] THEN
1705 REWRITE_TAC[EXISTS_OR_THM; UNWIND_THM2; COLLINEAR_AFFINE_HULL] THEN
1707 [MAP_EVERY EXISTS_TAC [`b:real^N`; `c:real^N`];
1708 MAP_EVERY EXISTS_TAC [`a:real^N`; `c:real^N`];
1709 MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`]] THEN
1710 SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET; HULL_INC; IN_INSERT] THEN
1711 POP_ASSUM MP_TAC THEN
1712 MATCH_MP_TAC(SET_RULE `s SUBSET t ==> a IN s ==> a IN t`) THEN
1713 MATCH_MP_TAC HULL_MONO THEN SET_TAC[]);;
1715 let COLLINEAR_3_IN_AFFINE_HULL = prove
1718 ==> (collinear {v0,v1,x} <=> x IN affine hull {v0,v1})`,
1719 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `v0:real^N` THEN
1720 REWRITE_TAC[COLLINEAR_LEMMA; AFFINE_HULL_2] THEN REPEAT STRIP_TAC THEN
1721 ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID; IN_ELIM_THM] THEN
1722 ASM_CASES_TAC `x:real^N = vec 0` THEN ASM_REWRITE_TAC[] THENL
1723 [MAP_EVERY EXISTS_TAC [`&1`; `&0`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
1725 MESON_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`]]);;
1727 (* ------------------------------------------------------------------------- *)
1728 (* A general lemma. *)
1729 (* ------------------------------------------------------------------------- *)
1731 let CONVEX_CONNECTED = prove
1732 (`!s:real^N->bool. convex s ==> connected s`,
1733 REWRITE_TAC[CONVEX_ALT; connected; SUBSET; EXTENSION; IN_INTER;
1734 IN_UNION; NOT_IN_EMPTY; NOT_FORALL_THM; NOT_EXISTS_THM] THEN
1735 GEN_TAC THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
1736 MAP_EVERY (K(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))) (1--4) THEN
1737 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `x1:real^N` STRIP_ASSUME_TAC)
1738 (X_CHOOSE_THEN `x2:real^N` STRIP_ASSUME_TAC)) THEN
1739 MP_TAC(ISPECL [`\u. (&1 - u) % x1 + u % (x2:real^N)`;
1740 `&0`; `&1`; `e1:real^N->bool`; `e2:real^N->bool`]
1741 (REWRITE_RULE[GSYM open_def] CONNECTED_REAL_LEMMA)) THEN
1742 ASM_REWRITE_TAC[NOT_IMP; REAL_SUB_RZERO; VECTOR_MUL_LID; VECTOR_MUL_LZERO;
1743 REAL_SUB_REFL; VECTOR_ADD_RID; VECTOR_ADD_LID; REAL_POS] THEN
1744 REPEAT(CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]]) THEN
1745 REPEAT STRIP_TAC THEN REWRITE_TAC[dist] THEN
1746 REWRITE_TAC[NORM_MUL; VECTOR_ARITH
1747 `((&1 - a) % x + a % y) - ((&1 - b) % x + b % y) = (a - b) % (y - x)`] THEN
1748 MP_TAC(ISPEC `(x2 - x1):real^N` NORM_POS_LE) THEN
1749 REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THENL
1750 [ALL_TAC; ASM_MESON_TAC[REAL_MUL_RZERO; REAL_LT_01]] THEN
1751 EXISTS_TAC `e / norm((x2 - x1):real^N)` THEN
1752 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_DIV]);;
1754 (* ------------------------------------------------------------------------- *)
1755 (* Various topological facts are queued up here, just because they rely on *)
1756 (* CONNECTED_UNIV, which is a trivial consequence of CONVEX_UNIV. It would *)
1757 (* be fairly easy to prove it earlier and move these back to the topology.ml *)
1758 (* file, which is a bit tidier intellectually. *)
1759 (* ------------------------------------------------------------------------- *)
1761 let CONNECTED_UNIV = prove
1762 (`connected (UNIV:real^N->bool)`,
1763 SIMP_TAC[CONVEX_CONNECTED; CONVEX_UNIV]);;
1765 let CONNECTED_COMPONENT_UNIV = prove
1766 (`!x. connected_component(:real^N) x = (:real^N)`,
1767 MESON_TAC[CONNECTED_CONNECTED_COMPONENT_SET; CONNECTED_UNIV; IN_UNIV]);;
1769 let CONNECTED_COMPONENT_EQ_UNIV = prove
1770 (`!s x. connected_component s x = (:real^N) <=> s = (:real^N)`,
1771 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONNECTED_COMPONENT_UNIV] THEN
1772 MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s = UNIV ==> t = UNIV`) THEN
1773 REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]);;
1775 let COMPONENTS_UNIV = prove
1776 (`components(:real^N) = {(:real^N)}`,
1777 REWRITE_TAC[COMPONENTS_EQ_SING; CONNECTED_UNIV; UNIV_NOT_EMPTY]);;
1780 (`!s. closed s /\ open s <=> s = {} \/ s = (:real^N)`,
1781 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
1782 ASM_REWRITE_TAC[CLOSED_EMPTY; OPEN_EMPTY; CLOSED_UNIV; OPEN_UNIV] THEN
1783 MATCH_MP_TAC(REWRITE_RULE[CONNECTED_CLOPEN] CONNECTED_UNIV) THEN
1784 ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; GSYM CLOSED_IN]);;
1786 let COMPACT_OPEN = prove
1787 (`!s:real^N->bool. compact s /\ open s <=> s = {}`,
1788 MESON_TAC[COMPACT_EMPTY; OPEN_EMPTY; COMPACT_IMP_CLOSED; CLOPEN;
1789 COMPACT_IMP_BOUNDED; NOT_BOUNDED_UNIV]);;
1791 let FRONTIER_NOT_EMPTY = prove
1792 (`!s. ~(s = {}) /\ ~(s = (:real^N)) ==> ~(frontier s = {})`,
1793 REPEAT STRIP_TAC THEN
1794 MP_TAC(ISPECL [`(:real^N)`; `s:real^N->bool`] CONNECTED_INTER_FRONTIER) THEN
1795 REWRITE_TAC[CONNECTED_UNIV] THEN ASM SET_TAC[]);;
1797 let FRONTIER_EQ_EMPTY = prove
1798 (`!s. frontier s = {} <=> s = {} \/ s = (:real^N)`,
1799 MESON_TAC[FRONTIER_NOT_EMPTY; FRONTIER_EMPTY; FRONTIER_UNIV]);;
1801 let EQ_INTERVAL = prove
1803 interval[a,b] = interval[c,d] <=>
1804 interval[a,b] = {} /\ interval[c,d] = {} \/ a = c /\ b = d) /\
1806 interval[a,b] = interval(c,d) <=>
1807 interval[a,b] = {} /\ interval(c,d) = {}) /\
1809 interval(a,b) = interval[c,d] <=>
1810 interval(a,b) = {} /\ interval[c,d] = {}) /\
1812 interval(a,b) = interval(c,d) <=>
1813 interval(a,b) = {} /\ interval(c,d) = {} \/ a = c /\ b = d)`,
1814 REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN
1815 (EQ_TAC THENL [ALL_TAC; STRIP_TAC THEN ASM_REWRITE_TAC[]]) THEN
1816 MATCH_MP_TAC(MESON[]
1817 `(p = {} /\ q = {} ==> r) /\ (~(p = {}) /\ ~(q = {}) ==> p = q ==> r)
1818 ==> p = q ==> r`) THEN
1820 [REWRITE_TAC[INTERVAL_NE_EMPTY; CART_EQ] THEN
1821 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
1822 SIMP_TAC[SUBSET_INTERVAL; GSYM REAL_LE_ANTISYM];
1823 STRIP_TAC THEN MATCH_MP_TAC(MESON[CLOPEN]
1824 `closed s /\ open t /\ ~(s = {}) /\ ~(s = UNIV) ==> ~(s = t)`) THEN
1825 ASM_REWRITE_TAC[CLOSED_INTERVAL; OPEN_INTERVAL; NOT_INTERVAL_UNIV];
1826 STRIP_TAC THEN MATCH_MP_TAC(MESON[CLOPEN]
1827 `closed s /\ open t /\ ~(s = {}) /\ ~(s = UNIV) ==> ~(t = s)`) THEN
1828 ASM_REWRITE_TAC[CLOSED_INTERVAL; OPEN_INTERVAL; NOT_INTERVAL_UNIV];
1829 REWRITE_TAC[INTERVAL_NE_EMPTY; CART_EQ] THEN
1830 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
1831 SIMP_TAC[SUBSET_INTERVAL; GSYM REAL_LE_ANTISYM]]);;
1833 let CLOSED_INTERVAL_EQ = prove
1834 (`(!a b:real^N. closed(interval[a,b])) /\
1835 (!a b:real^N. closed(interval(a,b)) <=> interval(a,b) = {})`,
1836 REWRITE_TAC[CLOSED_INTERVAL] THEN
1837 REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
1838 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN
1839 MP_TAC(ISPEC `interval(a:real^N,b)` CLOPEN) THEN
1840 ASM_REWRITE_TAC[OPEN_INTERVAL] THEN
1841 MESON_TAC[BOUNDED_INTERVAL; NOT_BOUNDED_UNIV]);;
1843 let OPEN_INTERVAL_EQ = prove
1844 (`(!a b:real^N. open(interval[a,b]) <=> interval[a,b] = {}) /\
1845 (!a b:real^N. open(interval(a,b)))`,
1846 REWRITE_TAC[OPEN_INTERVAL] THEN
1847 REPEAT GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
1848 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN
1849 MP_TAC(ISPEC `interval[a:real^N,b]` CLOPEN) THEN
1850 ASM_REWRITE_TAC[CLOSED_INTERVAL] THEN
1851 MESON_TAC[BOUNDED_INTERVAL; NOT_BOUNDED_UNIV]);;
1853 let COMPACT_INTERVAL_EQ = prove
1854 (`(!a b:real^N. compact(interval[a,b])) /\
1855 (!a b:real^N. compact(interval(a,b)) <=> interval(a,b) = {})`,
1856 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_INTERVAL] THEN
1857 REWRITE_TAC[CLOSED_INTERVAL_EQ]);;
1859 let CONNECTED_CHAIN = prove
1860 (`!f:(real^N->bool)->bool.
1861 (!s. s IN f ==> compact s /\ connected s) /\
1862 (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s)
1863 ==> connected(INTERS f)`,
1864 REPEAT STRIP_TAC THEN
1865 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
1866 ASM_REWRITE_TAC[INTERS_0; CONNECTED_UNIV] THEN
1867 ABBREV_TAC `c:real^N->bool = INTERS f` THEN
1868 SUBGOAL_THEN `compact(c:real^N->bool)` ASSUME_TAC THENL
1869 [EXPAND_TAC "c" THEN MATCH_MP_TAC COMPACT_INTERS THEN ASM SET_TAC[];
1871 ASM_SIMP_TAC[CONNECTED_CLOSED_SET; COMPACT_IMP_CLOSED; NOT_EXISTS_THM] THEN
1872 MAP_EVERY X_GEN_TAC [`a:real^N->bool`; `b:real^N->bool`] THEN STRIP_TAC THEN
1873 MP_TAC(ISPECL [`a:real^N->bool`; `b:real^N->bool`] SEPARATION_NORMAL) THEN
1874 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
1875 MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
1877 SUBGOAL_THEN `?k:real^N->bool. k IN f` STRIP_ASSUME_TAC THENL
1878 [ASM SET_TAC[]; ALL_TAC] THEN
1879 SUBGOAL_THEN `?n:real^N->bool. open n /\ k SUBSET n` MP_TAC THENL
1880 [ASM_MESON_TAC[BOUNDED_SUBSET_BALL; COMPACT_IMP_BOUNDED; OPEN_BALL];
1881 REWRITE_TAC[UNIONS_SUBSET] THEN STRIP_TAC] THEN
1882 MP_TAC(ISPEC `k:real^N->bool` COMPACT_IMP_HEINE_BOREL) THEN
1883 ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o SPEC
1884 `(u UNION v:real^N->bool) INSERT {n DIFF s | s IN f}`) THEN
1885 REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_INSERT; FORALL_IN_IMAGE] THEN
1886 ASM_SIMP_TAC[OPEN_UNION; OPEN_DIFF; COMPACT_IMP_CLOSED; NOT_IMP] THEN
1888 [REWRITE_TAC[UNIONS_INSERT] THEN REWRITE_TAC[SUBSET] THEN
1889 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[IN_UNION] THEN
1890 ASM_CASES_TAC `(x:real^N) IN c` THENL [ASM SET_TAC[]; DISJ2_TAC] THEN
1891 REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM] THEN
1892 UNDISCH_TAC `~((x:real^N) IN c)` THEN
1893 SUBST1_TAC(SYM(ASSUME `INTERS f:real^N->bool = c`)) THEN
1894 REWRITE_TAC[IN_INTERS; NOT_FORALL_THM] THEN
1895 MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[];
1897 DISCH_THEN(X_CHOOSE_THEN `g:(real^N->bool)->bool` MP_TAC) THEN
1898 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
1899 REWRITE_TAC[SUBSET_INSERT_DELETE] THEN
1900 SUBGOAL_THEN `FINITE(g DELETE (u UNION v:real^N->bool))` MP_TAC THENL
1901 [ASM_REWRITE_TAC[FINITE_DELETE];
1902 REWRITE_TAC[TAUT `p ==> ~q <=> ~(p /\ q)`]] THEN
1903 REWRITE_TAC[FINITE_SUBSET_IMAGE] THEN
1904 DISCH_THEN(X_CHOOSE_THEN `f':(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
1906 `?j:real^N->bool. j IN f /\
1907 UNIONS(IMAGE (\s. n DIFF s) f') SUBSET (n DIFF j)`
1908 STRIP_ASSUME_TAC THENL
1909 [ASM_CASES_TAC `f':(real^N->bool)->bool = {}` THEN
1910 ASM_REWRITE_TAC[IMAGE_CLAUSES; UNIONS_0; EMPTY_SUBSET] THENL
1911 [ASM SET_TAC[]; ALL_TAC] THEN
1913 `?j:real^N->bool. j IN f' /\
1914 UNIONS(IMAGE (\s. n DIFF s) f') SUBSET (n DIFF j)`
1915 MP_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET]] THEN
1917 `!s t:real^N->bool. s IN f' /\ t IN f' ==> s SUBSET t \/ t SUBSET s`
1918 MP_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
1919 UNDISCH_TAC `~(f':(real^N->bool)->bool = {})` THEN
1920 UNDISCH_TAC `FINITE(f':(real^N->bool)->bool)` THEN
1921 SPEC_TAC(`f':(real^N->bool)->bool`,`f':(real^N->bool)->bool`) THEN
1922 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[] THEN
1923 REWRITE_TAC[EXISTS_IN_INSERT; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1924 REWRITE_TAC[FORALL_IN_INSERT] THEN POP_ASSUM_LIST(K ALL_TAC) THEN
1925 MAP_EVERY X_GEN_TAC [`i:real^N->bool`; `f:(real^N->bool)->bool`] THEN
1926 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
1927 ASM_REWRITE_TAC[IMAGE_CLAUSES; UNIONS_INSERT; NOT_IN_EMPTY;
1928 UNIONS_0; UNION_EMPTY; SUBSET_REFL] THEN
1929 DISCH_THEN(fun th -> REPEAT DISCH_TAC THEN MP_TAC th) THEN
1930 ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
1931 DISCH_THEN(X_CHOOSE_THEN `j:real^N->bool` STRIP_ASSUME_TAC) THEN
1932 SUBGOAL_THEN `(n DIFF j) SUBSET (n DIFF i) \/
1933 (n DIFF i:real^N->bool) SUBSET (n DIFF j)`
1934 STRIP_ASSUME_TAC THENL
1935 [FIRST_X_ASSUM(MP_TAC o SPEC `j:real^N->bool` o CONJUNCT2) THEN
1937 DISJ1_TAC THEN ASM SET_TAC[];
1938 DISJ2_TAC THEN EXISTS_TAC `j:real^N->bool` THEN ASM SET_TAC[]];
1940 SUBGOAL_THEN `(j INTER k:real^N->bool) SUBSET (u UNION v)` ASSUME_TAC THENL
1941 [MATCH_MP_TAC(SET_RULE
1942 `k SUBSET (u UNION v) UNION (n DIFF j)
1943 ==> (j INTER k) SUBSET (u UNION v)`) THEN
1944 MATCH_MP_TAC SUBSET_TRANS THEN
1945 EXISTS_TAC `UNIONS g :real^N->bool` THEN ASM_REWRITE_TAC[] THEN
1946 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
1947 `UNIONS((u UNION v:real^N->bool) INSERT (g DELETE (u UNION v)))` THEN
1948 CONJ_TAC THENL [MATCH_MP_TAC SUBSET_UNIONS THEN SET_TAC[]; ALL_TAC] THEN
1949 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[UNIONS_INSERT] THEN
1952 SUBGOAL_THEN `connected(j INTER k:real^N->bool)` MP_TAC THENL
1953 [ASM_MESON_TAC[SET_RULE `s SUBSET t ==> s INTER t = s`; INTER_COMM];
1954 REWRITE_TAC[connected] THEN
1955 MAP_EVERY EXISTS_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
1956 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
1958 let CONNECTED_CHAIN_GEN = prove
1959 (`!f:(real^N->bool)->bool.
1960 (!s. s IN f ==> closed s /\ connected s) /\
1961 (?s. s IN f /\ compact s) /\
1962 (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s)
1963 ==> connected(INTERS f)`,
1964 GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
1965 FIRST_X_ASSUM(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
1967 `INTERS f = INTERS(IMAGE (\t:real^N->bool. s INTER t) f)`
1969 [REWRITE_TAC[EXTENSION; INTERS_IMAGE] THEN ASM SET_TAC[];
1970 MATCH_MP_TAC CONNECTED_CHAIN THEN
1971 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
1972 ASM_SIMP_TAC[COMPACT_INTER_CLOSED] THEN
1973 CONJ_TAC THENL [X_GEN_TAC `t:real^N->bool`; ASM SET_TAC[]] THEN
1975 SUBGOAL_THEN `s INTER t:real^N->bool = s \/ s INTER t = t`
1976 (DISJ_CASES_THEN SUBST1_TAC) THEN
1979 let CONNECTED_NEST = prove
1980 (`!s. (!n. compact(s n) /\ connected(s n)) /\
1981 (!m n. m <= n ==> s n SUBSET s m)
1982 ==> connected(INTERS {s n | n IN (:num)})`,
1983 GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CONNECTED_CHAIN THEN
1984 ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
1985 MATCH_MP_TAC WLOG_LE THEN ASM_MESON_TAC[]);;
1987 let CONNECTED_NEST_GEN = prove
1988 (`!s. (!n. closed(s n) /\ connected(s n)) /\ (?n. compact(s n)) /\
1989 (!m n. m <= n ==> s n SUBSET s m)
1990 ==> connected(INTERS {s n | n IN (:num)})`,
1992 DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN
1993 MATCH_MP_TAC CONNECTED_CHAIN_GEN THEN
1994 ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV; IMP_CONJ; RIGHT_FORALL_IMP_THM;
1995 EXISTS_IN_GSPEC] THEN
1996 MATCH_MP_TAC WLOG_LE THEN ASM_MESON_TAC[]);;
1998 let EQ_BALLS = prove
1999 (`(!a a':real^N r r'.
2000 ball(a,r) = ball(a',r') <=> a = a' /\ r = r' \/ r <= &0 /\ r' <= &0) /\
2002 ball(a,r) = cball(a',r') <=> r <= &0 /\ r' < &0) /\
2004 cball(a,r) = ball(a',r') <=> r < &0 /\ r' <= &0) /\
2006 cball(a,r) = cball(a',r') <=> a = a' /\ r = r' \/ r < &0 /\ r' < &0)`,
2007 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT STRIP_TAC THEN
2009 [ALL_TAC; REWRITE_TAC[EXTENSION; IN_BALL; IN_CBALL] THEN NORM_ARITH_TAC])
2011 [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_BALLS] THEN NORM_ARITH_TAC;
2012 ONCE_REWRITE_TAC[EQ_SYM_EQ];
2014 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_BALLS] THEN NORM_ARITH_TAC] THEN
2015 DISCH_THEN(MP_TAC o MATCH_MP (MESON[CLOPEN; BOUNDED_BALL; NOT_BOUNDED_UNIV]
2016 `s = t ==> closed s /\ open t /\ bounded t ==> s = {} /\ t = {}`)) THEN
2017 REWRITE_TAC[OPEN_BALL; CLOSED_CBALL; BOUNDED_BALL;
2018 BALL_EQ_EMPTY; CBALL_EQ_EMPTY] THEN
2021 let FINITE_CBALL = prove
2022 (`!a:real^N r. FINITE(cball(a,r)) <=> r <= &0`,
2023 REPEAT STRIP_TAC THEN ASM_CASES_TAC `r < &0` THEN
2024 ASM_SIMP_TAC[CBALL_EMPTY; REAL_LT_IMP_LE; FINITE_EMPTY] THEN
2025 ASM_CASES_TAC `r = &0` THEN
2026 ASM_REWRITE_TAC[CBALL_TRIVIAL; FINITE_SING; REAL_LE_REFL] THEN
2027 EQ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
2028 DISCH_THEN(MP_TAC o MATCH_MP EMPTY_INTERIOR_FINITE) THEN
2029 REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC);;
2031 let FINITE_BALL = prove
2032 (`!a:real^N r. FINITE(ball(a,r)) <=> r <= &0`,
2033 REPEAT STRIP_TAC THEN ASM_CASES_TAC `r <= &0` THEN
2034 ASM_SIMP_TAC[BALL_EMPTY; REAL_LT_IMP_LE; FINITE_EMPTY] THEN
2035 DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ]
2036 FINITE_IMP_NOT_OPEN)) THEN
2037 REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC);;
2039 (* ------------------------------------------------------------------------- *)
2040 (* Minimal continua. *)
2041 (* ------------------------------------------------------------------------- *)
2043 let MINIMAL_CONTINUUM = prove
2044 (`!t s:real^N->bool.
2045 t SUBSET s /\ compact s /\ connected s
2046 ==> ?u. t SUBSET u /\ u SUBSET s /\ compact u /\ connected u /\
2047 !v. v SUBSET u /\ t SUBSET v /\ compact v /\ connected v
2049 REPEAT STRIP_TAC THEN
2050 ASM_CASES_TAC `t:real^N->bool = {}` THENL
2051 [EXISTS_TAC `{}:real^N->bool` THEN
2052 ASM_MESON_TAC[COMPACT_EMPTY; CONNECTED_EMPTY; SUBSET_EMPTY; EMPTY_SUBSET];
2054 MP_TAC(ISPECL [`\u:real^N->bool. t SUBSET u /\ connected u`;
2056 BROUWER_REDUCTION_THEOREM) THEN
2057 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
2058 [CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN GEN_TAC THEN STRIP_TAC THEN
2060 [REWRITE_TAC[SUBSET_INTERS] THEN ASM SET_TAC[];
2061 MATCH_MP_TAC CONNECTED_NEST THEN ASM_REWRITE_TAC[] THEN
2062 MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN
2063 ASM_REWRITE_TAC[] THEN SET_TAC[]];
2064 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
2065 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2066 REWRITE_TAC[SET_RULE `(v SUBSET u /\ p ==> v = u) <=>
2067 (v SUBSET u /\ p ==> ~(v PSUBSET u))`] THEN
2068 GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
2069 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN ASM SET_TAC[]]);;
2071 (* ------------------------------------------------------------------------- *)
2072 (* Convex functions into the reals. *)
2073 (* ------------------------------------------------------------------------- *)
2075 parse_as_infix ("convex_on",(12,"right"));;
2077 let convex_on = new_definition
2079 !x y u v. x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
2080 ==> f(u % x + v % y) <= u * f(x) + v * f(y)`;;
2082 let CONVEX_ON_SUBSET = prove
2083 (`!f s t. f convex_on t /\ s SUBSET t ==> f convex_on s`,
2084 REWRITE_TAC[convex_on; SUBSET] THEN MESON_TAC[]);;
2086 let CONVEX_ON_EQ = prove
2087 (`!f g s. convex s /\ (!x. x IN s ==> f x = g x) /\ f convex_on s
2089 REWRITE_TAC[convex_on; convex] THEN MESON_TAC[]);;
2091 let CONVEX_ADD = prove
2092 (`!s f g. f convex_on s /\ g convex_on s ==> (\x. f(x) + g(x)) convex_on s`,
2093 REWRITE_TAC[convex_on; AND_FORALL_THM] THEN
2094 REPEAT(MATCH_MP_TAC MONO_FORALL ORELSE GEN_TAC) THEN
2096 `(b /\ c ==> d) ==> (a ==> b) /\ (a ==> c) ==> a ==> d`) THEN
2099 let CONVEX_CMUL = prove
2100 (`!s c f. &0 <= c /\ f convex_on s ==> (\x. c * f(x)) convex_on s`,
2101 SIMP_TAC[convex_on; REAL_LE_LMUL;
2102 REAL_ARITH `u * c * fx + v * c * fy = c * (u * fx + v * fy)`]);;
2104 let CONVEX_MAX = prove
2105 (`!f g s. f convex_on s /\ g convex_on s
2106 ==> (\x. max (f x) (g x)) convex_on s`,
2107 REWRITE_TAC[convex_on; REAL_MAX_LE] THEN REPEAT STRIP_TAC THEN
2108 FIRST_X_ASSUM(fun th ->
2109 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
2110 ASM_REWRITE_TAC[] THEN
2111 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
2112 MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN
2113 MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REAL_ARITH_TAC);;
2115 let CONVEX_LOWER = prove
2116 (`!f s x y. f convex_on s /\
2117 x IN s /\ y IN s /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
2118 ==> f(u % x + v % y) <= max (f(x)) (f(y))`,
2119 REWRITE_TAC[convex_on] THEN REPEAT STRIP_TAC THEN
2120 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN
2121 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [SYM th]) THEN
2122 REWRITE_TAC[REAL_ADD_RDISTRIB] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
2123 ASM_MESON_TAC[REAL_LE_ADD2; REAL_LE_LMUL; REAL_MAX_MAX]);;
2125 let CONVEX_LOWER_SEGMENT = prove
2126 (`!f s a b x:real^N.
2127 f convex_on s /\ a IN s /\ b IN s /\ x IN segment[a,b]
2128 ==> f(x) <= max (f a) (f b)`,
2129 REWRITE_TAC[IN_SEGMENT] THEN REPEAT STRIP_TAC THEN
2130 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_LOWER THEN
2131 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
2133 let CONVEX_LOCAL_GLOBAL_MINIMUM = prove
2135 f convex_on s /\ x IN t /\ open t /\ t SUBSET s /\
2136 (!y. y IN t ==> f(x) <= f(y))
2137 ==> !y. y IN s ==> f(x) <= f(y)`,
2138 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
2139 SUBGOAL_THEN `&0 < dist(x:real^N,y)` ASSUME_TAC THENL
2140 [ASM_MESON_TAC[DIST_POS_LT; REAL_LT_REFL]; ALL_TAC] THEN
2141 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
2142 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
2143 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
2144 MP_TAC(SPECL [`&1`; `e / dist(x:real^N,y)`] REAL_DOWN2) THEN
2145 ANTS_TAC THENL [ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_01]; ALL_TAC] THEN
2146 DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
2147 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
2149 SPECL [`x:real^N`; `y:real^N`; `&1 - u`; `u:real`]) THEN
2151 [ASM_SIMP_TAC[REAL_SUB_ADD; REAL_SUB_LE; REAL_LT_IMP_LE] THEN
2152 ASM_MESON_TAC[CENTRE_IN_BALL; SUBSET];
2154 REWRITE_TAC[REAL_NOT_LE] THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
2155 EXISTS_TAC `(&1 - u) * f(x) + u * f(x:real^N):real` THEN
2156 ASM_SIMP_TAC[REAL_LT_LADD; REAL_LT_LMUL] THEN
2157 REWRITE_TAC[REAL_ARITH `(&1 - x) * a + x * a = a`] THEN
2158 FIRST_X_ASSUM MATCH_MP_TAC THEN
2159 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
2160 REWRITE_TAC[IN_BALL; dist] THEN
2161 REWRITE_TAC[VECTOR_ARITH `x - ((&1 - u) % x + u % y):real^N =
2163 REWRITE_TAC[NORM_MUL; GSYM dist] THEN
2164 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ;
2165 REAL_ARITH `&0 < x /\ x < b ==> abs x < b`]);;
2167 let CONVEX_DISTANCE = prove
2168 (`!s a. (\x. dist(a,x)) convex_on s`,
2169 REWRITE_TAC[convex_on; dist] THEN REPEAT STRIP_TAC THEN
2170 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV)
2171 [GSYM VECTOR_MUL_LID] THEN
2172 FIRST_ASSUM(SUBST1_TAC o SYM) THEN
2173 REWRITE_TAC[VECTOR_ARITH
2174 `(u + v) % z - (u % x + v % y) = u % (z - x) + v % (z - y)`] THEN
2175 ASM_MESON_TAC[NORM_TRIANGLE; NORM_MUL; REAL_ABS_REFL]);;
2177 let CONVEX_NORM = prove
2178 (`!s:real^N->bool. norm convex_on s`,
2180 MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`] CONVEX_DISTANCE) THEN
2181 REWRITE_TAC[DIST_0; ETA_AX]);;
2183 let CONVEX_ON_COMPOSE_LINEAR = prove
2184 (`!f g:real^M->real^N s.
2185 f convex_on (IMAGE g s) /\ linear g ==> (f o g) convex_on s`,
2186 REWRITE_TAC[convex_on; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
2187 REWRITE_TAC[FORALL_IN_IMAGE; o_THM] THEN
2188 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM CONJ_ASSOC] THEN
2189 REPEAT STRIP_TAC THEN
2190 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_ADD th]) THEN
2191 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN
2194 let CONVEX_ON_TRANSLATION = prove
2196 f convex_on (IMAGE (\x. a + x) s) <=> (\x. f(a + x)) convex_on s`,
2197 REWRITE_TAC[convex_on; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
2198 REWRITE_TAC[FORALL_IN_IMAGE; o_THM] THEN
2199 REWRITE_TAC[VECTOR_ARITH
2200 `u % (a + x) + v % (a + y):real^N = (u + v) % a + u % x + v % y`] THEN
2201 SIMP_TAC[VECTOR_MUL_LID]);;
2203 (* ------------------------------------------------------------------------- *)
2204 (* Open and closed balls are convex and hence connected. *)
2205 (* ------------------------------------------------------------------------- *)
2207 let CONVEX_BALL = prove
2208 (`!x:real^N e. convex(ball(x,e))`,
2209 let lemma = REWRITE_RULE[convex_on; IN_UNIV]
2210 (ISPEC `(:real^N)` CONVEX_DISTANCE) in
2211 REWRITE_TAC[convex; IN_BALL] THEN REPEAT STRIP_TAC THEN
2212 W(MP_TAC o PART_MATCH (lhand o rand) lemma o lhand o snd) THEN
2213 ASM_MESON_TAC[REAL_LET_TRANS; REAL_CONVEX_BOUND_LT]);;
2215 let CONNECTED_BALL = prove
2216 (`!x:real^N e. connected(ball(x,e))`,
2217 SIMP_TAC[CONVEX_CONNECTED; CONVEX_BALL]);;
2219 let CONVEX_CBALL = prove
2220 (`!x:real^N e. convex(cball(x,e))`,
2221 REWRITE_TAC[convex; IN_CBALL; dist] THEN MAP_EVERY X_GEN_TAC
2222 [`x:real^N`; `e:real`; `y:real^N`; `z:real^N`; `u:real`; `v:real`] THEN
2223 STRIP_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH `a - b = &1 % a - b`] THEN
2224 FIRST_ASSUM(SUBST1_TAC o SYM) THEN
2225 REWRITE_TAC[VECTOR_ARITH
2226 `(a + b) % x - (a % y + b % z) = a % (x - y) + b % (x - z)`] THEN
2227 MATCH_MP_TAC REAL_LE_TRANS THEN
2228 EXISTS_TAC `norm(u % (x - y)) + norm(v % (x - z):real^N)` THEN
2229 REWRITE_TAC[NORM_TRIANGLE; NORM_MUL] THEN
2230 MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[REAL_ABS_POS] THEN
2231 ASM_SIMP_TAC[REAL_ARITH
2232 `&0 <= u /\ &0 <= v /\ (u + v = &1) ==> (abs(u) + abs(v) = &1)`]);;
2234 let CONNECTED_CBALL = prove
2235 (`!x:real^N e. connected(cball(x,e))`,
2236 SIMP_TAC[CONVEX_CONNECTED; CONVEX_CBALL]);;
2238 let CONVEX_INTERMEDIATE_BALL = prove
2239 (`!a:real^N r t. ball(a,r) SUBSET t /\ t SUBSET cball(a,r) ==> convex t`,
2240 REPEAT STRIP_TAC THEN REWRITE_TAC[CONVEX_CONTAINS_OPEN_SEGMENT] THEN
2241 REPEAT STRIP_TAC THEN
2242 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
2244 REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN GEN_TAC THEN DISCH_THEN
2245 (MP_TAC o SPEC `a:real^N` o MATCH_MP DIST_DECREASES_OPEN_SEGMENT) THEN
2246 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
2247 REWRITE_TAC[IN_CBALL] THEN ASM_MESON_TAC[REAL_LTE_TRANS]);;
2249 let FRONTIER_OF_CONNECTED_COMPONENT_SUBSET = prove
2250 (`!s c x:real^N. frontier(connected_component s x) SUBSET frontier s`,
2251 REPEAT GEN_TAC THEN REWRITE_TAC[frontier; SUBSET; IN_DIFF] THEN
2252 X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THENL
2253 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
2254 `y IN s ==> s SUBSET t ==> y IN t`)) THEN
2255 MATCH_MP_TAC SUBSET_CLOSURE THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET];
2256 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN
2257 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
2258 SUBGOAL_THEN `ball(y:real^N,e) SUBSET connected_component s y`
2260 [MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
2261 ASM_REWRITE_TAC[CONNECTED_BALL; CENTRE_IN_BALL];
2262 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSURE_APPROACHABLE]) THEN
2263 DISCH_THEN(MP_TAC o SPEC `e:real`) THEN
2264 ASM_REWRITE_TAC[ONCE_REWRITE_RULE[DIST_SYM] (GSYM IN_BALL)] THEN
2265 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
2266 REWRITE_TAC[IN_INTERIOR] THEN EXISTS_TAC `e:real` THEN
2267 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`; `y:real^N`]
2268 CONNECTED_COMPONENT_OVERLAP) THEN
2269 MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN
2270 ASM_SIMP_TAC[] THEN ASM SET_TAC[]]]);;
2272 let FRONTIER_OF_COMPONENTS_SUBSET = prove
2273 (`!s c:real^N->bool.
2274 c IN components s ==> frontier c SUBSET frontier s`,
2275 SIMP_TAC[components; FORALL_IN_GSPEC;
2276 FRONTIER_OF_CONNECTED_COMPONENT_SUBSET]);;
2278 let FRONTIER_OF_COMPONENTS_CLOSED_COMPLEMENT = prove
2279 (`!s c. closed s /\ c IN components ((:real^N) DIFF s)
2280 ==> frontier c SUBSET s`,
2281 REPEAT STRIP_TAC THEN
2282 FIRST_ASSUM(MP_TAC o MATCH_MP FRONTIER_OF_COMPONENTS_SUBSET) THEN
2283 REWRITE_TAC[FRONTIER_COMPLEMENT] THEN
2284 ASM_MESON_TAC[FRONTIER_SUBSET_EQ; SUBSET_TRANS]);;
2286 (* ------------------------------------------------------------------------- *)
2287 (* A couple of lemmas about components (see Newman IV, 3.3 and 3.4). *)
2288 (* ------------------------------------------------------------------------- *)
2290 let CONNECTED_UNION_CLOPEN_IN_COMPLEMENT = prove
2291 (`!s t u:real^N->bool.
2292 connected s /\ connected u /\ s SUBSET u /\
2293 open_in (subtopology euclidean (u DIFF s)) t /\
2294 closed_in (subtopology euclidean (u DIFF s)) t
2295 ==> connected (s UNION t)`,
2297 [`c:real^N->bool`; `h:real^N->bool`; `s:real^N->bool`] THEN
2299 REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN
2300 MATCH_MP_TAC(MESON[]
2301 `!Q. (!x y. P x y <=> P y x) /\
2302 (!x y. P x y ==> Q x \/ Q y) /\
2303 (!x y. P x y /\ Q x ==> F)
2304 ==> (!x y. ~(P x y))`) THEN
2305 EXISTS_TAC `\x:real^N->bool. c SUBSET x` THEN
2306 CONJ_TAC THENL [MESON_TAC[INTER_COMM; UNION_COMM]; ALL_TAC] THEN
2307 REWRITE_TAC[] THEN CONJ_TAC THEN
2308 MAP_EVERY X_GEN_TAC [`h1:real^N->bool`; `h2:real^N->bool`] THENL
2309 [STRIP_TAC THEN UNDISCH_TAC `connected(c:real^N->bool)` THEN
2310 REWRITE_TAC[CONNECTED_CLOSED_IN; NOT_EXISTS_THM] THEN
2312 SPECL [`c INTER h1:real^N->bool`; `c INTER h2:real^N->bool`]) THEN
2314 `(p /\ q) /\ (~r ==> s) ==> ~(p /\ q /\ r) ==> s`) THEN
2315 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL
2317 `closed_in(subtopology euclidean (c UNION h)) (h1:real^N->bool)`;
2319 `closed_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)`] THEN
2320 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2323 FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN
2324 SUBGOAL_THEN `(h2:real^N->bool) SUBSET h` ASSUME_TAC THENL
2325 [ASM SET_TAC[]; ALL_TAC] THEN
2326 UNDISCH_TAC `connected(s:real^N->bool)` THEN
2327 REWRITE_TAC[CONNECTED_CLOPEN] THEN
2328 DISCH_THEN(MP_TAC o SPEC `h2:real^N->bool`) THEN REWRITE_TAC[NOT_IMP] THEN
2329 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2330 SUBGOAL_THEN `s:real^N->bool = (s DIFF c) UNION (c UNION h)`
2331 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN CONJ_TAC THENL
2332 [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_UNION THEN
2333 MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL
2334 [REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
2335 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2336 SUBGOAL_THEN `(c UNION h) DIFF h2:real^N->bool = h1`
2337 (fun th -> ASM_REWRITE_TAC[th]) THEN ASM SET_TAC[];
2338 DISCH_TAC THEN MATCH_MP_TAC OPEN_IN_TRANS THEN
2339 EXISTS_TAC `h:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2341 `open_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)` THEN
2342 REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN
2344 MATCH_MP_TAC CLOSED_IN_SUBTOPOLOGY_UNION THEN ASM_REWRITE_TAC[] THEN
2345 MATCH_MP_TAC CLOSED_IN_TRANS THEN EXISTS_TAC `h:real^N->bool` THEN
2346 ASM_REWRITE_TAC[] THEN
2348 `closed_in(subtopology euclidean (c UNION h)) (h2:real^N->bool)` THEN
2349 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2352 let COMPONENT_COMPLEMENT_CONNECTED = prove
2353 (`!s u c:real^N->bool.
2354 connected s /\ connected u /\ s SUBSET u /\ c IN components (u DIFF s)
2355 ==> connected(u DIFF c)`,
2357 [`a:real^N->bool`; `s:real^N->bool`; `c:real^N->bool`] THEN
2358 STRIP_TAC THEN UNDISCH_TAC `connected(a:real^N->bool)` THEN
2359 REWRITE_TAC[CONNECTED_CLOSED_IN_EQ; NOT_EXISTS_THM] THEN
2360 DISCH_TAC THEN MAP_EVERY X_GEN_TAC
2361 [`h3:real^N->bool`; `h4:real^N->bool`] THEN STRIP_TAC THEN
2362 FIRST_X_ASSUM(MP_TAC o SPECL
2363 [`a INTER h3:real^N->bool`; `a INTER h4:real^N->bool`]) THEN
2364 FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN
2365 FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_SUBSET) THEN
2366 EVERY_ASSUM(fun th -> try
2367 MP_TAC(CONJUNCT1(GEN_REWRITE_RULE I [closed_in] th))
2368 with Failure _ -> ALL_TAC) THEN
2369 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN REPEAT DISCH_TAC THEN
2370 REPEAT CONJ_TAC THENL
2371 [UNDISCH_TAC `closed_in (subtopology euclidean (s DIFF c))
2372 (h3:real^N->bool)` THEN
2373 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2375 UNDISCH_TAC `closed_in (subtopology euclidean (s DIFF c))
2376 (h4:real^N->bool)` THEN
2377 REWRITE_TAC[CLOSED_IN_CLOSED] THEN MATCH_MP_TAC MONO_EXISTS THEN
2382 MP_TAC(ISPECL [`s DIFF a:real^N->bool`; `c UNION h3:real^N->bool`;
2383 `c:real^N->bool`] COMPONENTS_MAXIMAL) THEN
2384 ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN
2385 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2386 MATCH_MP_TAC CONNECTED_UNION_CLOPEN_IN_COMPLEMENT THEN
2387 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2388 REPEAT CONJ_TAC THENL
2389 [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED];
2391 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
2392 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2393 SUBGOAL_THEN `s DIFF c DIFF h3:real^N->bool = h4` SUBST1_TAC THEN
2396 MP_TAC(ISPECL [`s DIFF a:real^N->bool`; `c UNION h4:real^N->bool`;
2397 `c:real^N->bool`] COMPONENTS_MAXIMAL) THEN
2398 ASM_REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC] THEN
2399 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2400 MATCH_MP_TAC CONNECTED_UNION_CLOPEN_IN_COMPLEMENT THEN
2401 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2402 REPEAT CONJ_TAC THENL
2403 [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED];
2405 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
2406 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2407 SUBGOAL_THEN `s DIFF c DIFF h4:real^N->bool = h3` SUBST1_TAC THEN
2410 (* ------------------------------------------------------------------------- *)
2411 (* Condition for an open map's image to contain a ball. *)
2412 (* ------------------------------------------------------------------------- *)
2414 let BALL_SUBSET_OPEN_MAP_IMAGE = prove
2415 (`!f:real^M->real^N s a r.
2416 bounded s /\ f continuous_on closure s /\ open(IMAGE f (interior s)) /\
2417 a IN s /\ &0 < r /\ (!z. z IN frontier s ==> r <= norm(f z - f a))
2418 ==> ball(f(a),r) SUBSET IMAGE f s`,
2419 REPEAT STRIP_TAC THEN
2420 MP_TAC(ISPECL [`ball((f:real^M->real^N) a,r)`;
2421 `(:real^N) DIFF IMAGE (f:real^M->real^N) s`]
2422 CONNECTED_INTER_FRONTIER) THEN
2423 REWRITE_TAC[CONNECTED_BALL] THEN MATCH_MP_TAC(SET_RULE
2424 `~(b INTER s = {}) /\ b INTER f = {} ==>
2425 (~(b INTER (UNIV DIFF s) = {}) /\ ~(b DIFF (UNIV DIFF s) = {})
2426 ==> ~(b INTER f = {}))
2427 ==> b SUBSET s`) THEN
2428 REWRITE_TAC[FRONTIER_COMPLEMENT] THEN CONJ_TAC THENL
2429 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
2430 EXISTS_TAC `(f:real^M->real^N) a` THEN
2431 ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ASM SET_TAC[];
2432 REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN t ==> ~(x IN s)`] THEN
2433 REWRITE_TAC[IN_BALL; REAL_NOT_LT]] THEN
2434 MP_TAC(ISPECL[`frontier(IMAGE (f:real^M->real^N) s)`; `(f:real^M->real^N) a`]
2435 DISTANCE_ATTAINS_INF) THEN
2436 REWRITE_TAC[FRONTIER_CLOSED; FRONTIER_EQ_EMPTY] THEN ANTS_TAC THENL
2437 [SIMP_TAC[DE_MORGAN_THM] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2438 MATCH_MP_TAC(MESON[NOT_BOUNDED_UNIV] `bounded s ==> ~(s = UNIV)`) THEN
2439 MATCH_MP_TAC BOUNDED_SUBSET THEN
2440 EXISTS_TAC `IMAGE (f:real^M->real^N) (closure s)` THEN
2441 SIMP_TAC[IMAGE_SUBSET; CLOSURE_SUBSET] THEN
2442 MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
2443 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
2444 ASM_REWRITE_TAC[COMPACT_CLOSURE];
2445 DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC)] THEN
2446 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [frontier]) THEN
2447 REWRITE_TAC[IN_DIFF] THEN
2448 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
2449 REWRITE_TAC[CLOSURE_SEQUENTIAL] THEN
2450 DISCH_THEN(X_CHOOSE_THEN `y:num->real^N`
2451 (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
2452 REWRITE_TAC[IN_IMAGE; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
2453 X_GEN_TAC `z:num->real^M` THEN REWRITE_TAC[FORALL_AND_THM] THEN
2454 ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN
2455 DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
2456 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM COMPACT_CLOSURE]) THEN
2457 REWRITE_TAC[compact] THEN
2458 DISCH_THEN(MP_TAC o SPEC `z:num->real^M`) THEN
2459 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; LEFT_IMP_EXISTS_THM] THEN
2460 MAP_EVERY X_GEN_TAC [`y:real^M`; `r:num->num`] THEN STRIP_TAC THEN
2462 `(((\n. (f:real^M->real^N)(z n)) o (r:num->num)) --> w) sequentially`
2464 [MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[];
2465 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[GSYM o_ASSOC]] THEN
2467 SUBGOAL_THEN `!n. ((z:num->real^M) o (r:num->num)) n IN s` MP_TAC THENL
2468 [ASM_REWRITE_TAC[o_THM];
2469 UNDISCH_THEN `((\n. (f:real^M->real^N) ((z:num->real^M) n)) --> w)
2470 sequentially` (K ALL_TAC) THEN
2471 UNDISCH_THEN `!n. (z:num->real^M) n IN s` (K ALL_TAC)] THEN
2472 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
2473 SPEC_TAC(`(z:num->real^M) o (r:num->num)`, `z:num->real^M`) THEN
2474 REPEAT STRIP_TAC THEN
2475 SUBGOAL_THEN `w = (f:real^M->real^N) y` SUBST_ALL_TAC THENL
2476 [MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
2477 EXISTS_TAC `(f:real^M->real^N) o (z:num->real^M)` THEN
2478 ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
2479 ASM_MESON_TAC[CONTINUOUS_ON_CLOSURE_SEQUENTIALLY];
2481 MATCH_MP_TAC REAL_LE_TRANS THEN
2482 EXISTS_TAC `norm(f y - (f:real^M->real^N) a)` THEN CONJ_TAC THENL
2483 [FIRST_X_ASSUM MATCH_MP_TAC; ASM_MESON_TAC[dist; NORM_SUB]] THEN
2484 ASM_REWRITE_TAC[frontier; IN_DIFF] THEN DISCH_TAC THEN
2485 FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
2486 REWRITE_TAC[interior; IN_ELIM_THM] THEN
2487 EXISTS_TAC `IMAGE (f:real^M->real^N) (interior s)` THEN
2488 ASM_SIMP_TAC[IMAGE_SUBSET; INTERIOR_SUBSET] THEN ASM SET_TAC[]);;
2490 (* ------------------------------------------------------------------------- *)
2491 (* Arithmetic operations on sets preserve convexity. *)
2492 (* ------------------------------------------------------------------------- *)
2494 let CONVEX_SCALING = prove
2495 (`!s c. convex s ==> convex (IMAGE (\x. c % x) s)`,
2496 REWRITE_TAC[convex; IN_IMAGE] THEN REPEAT STRIP_TAC THEN
2497 ASM_REWRITE_TAC[VECTOR_ARITH
2498 `u % c % x + v % c % y = c % (u % x + v % y)`] THEN
2501 let CONVEX_SCALING_EQ = prove
2502 (`!s c. ~(c = &0) ==> (convex (IMAGE (\x. c % x) s) <=> convex s)`,
2503 REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[CONVEX_SCALING] THEN
2504 DISCH_THEN(MP_TAC o SPEC `inv c` o MATCH_MP CONVEX_SCALING) THEN
2505 ASM_SIMP_TAC[GSYM IMAGE_o; o_DEF; VECTOR_MUL_ASSOC;
2506 REAL_MUL_LINV; VECTOR_MUL_LID; IMAGE_ID]);;
2508 let CONVEX_NEGATIONS = prove
2509 (`!s. convex s ==> convex (IMAGE (--) s)`,
2510 REWRITE_TAC[convex; IN_IMAGE] THEN REPEAT STRIP_TAC THEN
2511 ASM_REWRITE_TAC[VECTOR_ARITH
2512 `u % --x + v % --y = --(u % x + v % y)`] THEN
2515 let CONVEX_SUMS = prove
2516 (`!s t. convex s /\ convex t ==> convex {x + y | x IN s /\ y IN t}`,
2517 REWRITE_TAC[convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
2518 ASM_REWRITE_TAC[VECTOR_ARITH
2519 `u % (a + b) + v % (c + d) = (u % a + v % c) + (u % b + v % d)`] THEN
2522 let CONVEX_DIFFERENCES = prove
2523 (`!s t. convex s /\ convex t ==> convex {x - y | x IN s /\ y IN t}`,
2524 REWRITE_TAC[convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
2525 ASM_REWRITE_TAC[VECTOR_ARITH
2526 `u % (a - b) + v % (c - d) = (u % a + v % c) - (u % b + v % d)`] THEN
2529 let CONVEX_AFFINITY = prove
2531 convex s ==> convex (IMAGE (\x. a + c % x) s)`,
2532 REPEAT STRIP_TAC THEN
2533 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
2534 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2535 ASM_SIMP_TAC[IMAGE_o; CONVEX_TRANSLATION; CONVEX_SCALING]);;
2537 let CONVEX_LINEAR_PREIMAGE = prove
2538 (`!f:real^M->real^N.
2539 linear f /\ convex s ==> convex {x | f(x) IN s}`,
2540 REWRITE_TAC[CONVEX_ALT; IN_ELIM_THM] THEN
2541 SIMP_TAC[LINEAR_ADD; LINEAR_CMUL]);;
2543 (* ------------------------------------------------------------------------- *)
2544 (* Some interesting "cancellation" properties for sum-sets. *)
2545 (* ------------------------------------------------------------------------- *)
2547 let SUBSET_SUMS_LCANCEL = prove
2548 (`!s t u:real^N->bool.
2549 ~(s = {}) /\ bounded s /\ closed u /\ convex u /\
2550 {x + y | x IN s /\ y IN t} SUBSET {x + z | x IN s /\ z IN u}
2552 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REWRITE_TAC[IN_ELIM_THM] THEN
2553 REPEAT GEN_TAC THEN STRIP_TAC THEN
2554 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
2555 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN X_GEN_TAC `b:real^N` THEN
2558 `!n. ?w z:real^N. w IN s /\ z IN u /\ (&n + &1) % (b - z) = w - a`
2561 [FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN
2562 ASM_REWRITE_TAC[REAL_ADD_LID; VECTOR_MUL_LID] THEN
2563 REWRITE_TAC[VECTOR_ARITH `b - z:real^N = w - a <=> a + b = w + z`] THEN
2565 FIRST_X_ASSUM(X_CHOOSE_THEN `a':real^N` (X_CHOOSE_THEN `c':real^N`
2566 STRIP_ASSUME_TAC)) THEN
2567 FIRST_X_ASSUM(MP_TAC o SPECL [`a':real^N`; `b:real^N`]) THEN
2568 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
2569 MAP_EVERY X_GEN_TAC [`a'':real^N`; `c'':real^N`] THEN STRIP_TAC THEN
2570 EXISTS_TAC `a'':real^N` THEN EXISTS_TAC
2571 `(&1 - &1 / (&n + &2)) % c' + &1 / (&n + &2) % c'':real^N` THEN
2572 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
2573 [FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[CONVEX_ALT]) THEN
2574 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ;
2575 REAL_ARITH `&0 < &n + &2`] THEN
2577 FIRST_X_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I
2578 [VECTOR_ARITH `a' + b:real^N = a'' + c <=> a'' = (a' + b) - c`]) THEN
2579 REWRITE_TAC[VECTOR_ARITH
2580 `(&n + &1) % (b - c):real^N = (a' + b) - c'' - a <=>
2581 &n % b - (&n + &1) % c = (a' - c'') - a`] THEN
2582 SIMP_TAC[GSYM REAL_OF_NUM_SUC; VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB;
2583 REAL_ARITH `(n + &1) + &1 = n + &2`] THEN
2584 REWRITE_TAC[VECTOR_MUL_LID; REAL_FIELD
2585 `(&n + &2) * (&1 - (&1 / (&n + &2))) = &n + &1 /\
2586 (&n + &2) * &1 / (&n + &2) = &1`] THEN
2587 ASM_REWRITE_TAC[VECTOR_ARITH
2588 `n % b - (n % c + d):real^N = n % (b - c) - d`] THEN
2589 CONV_TAC VECTOR_ARITH]];
2590 FIRST_X_ASSUM(K ALL_TAC o check is_forall o concl) THEN
2591 MP_TAC(ISPECL [`s:real^N->bool`; `s:real^N->bool`] BOUNDED_DIFFS) THEN
2592 ASM_REWRITE_TAC[] THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_GSPEC] THEN
2593 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
2594 DISCH_TAC THEN FIRST_X_ASSUM(fun th ->
2595 ONCE_REWRITE_TAC[GSYM(MATCH_MP CLOSED_APPROACHABLE th)]) THEN
2596 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2597 MP_TAC(SPEC `e:real` REAL_ARCH) THEN ASM_REWRITE_TAC[] THEN
2598 DISCH_THEN(MP_TAC o SPEC `B:real`) THEN
2599 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
2600 MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[REAL_MUL_LZERO] THEN
2601 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; X_GEN_TAC `n:num`] THEN
2602 DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN
2603 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN
2604 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
2605 X_GEN_TAC `c:real^N` THEN
2606 DISCH_THEN(X_CHOOSE_THEN `d:real^N` STRIP_ASSUME_TAC) THEN
2607 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN
2608 EXISTS_TAC `abs(&n + &1)` THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
2609 CONJ_TAC THENL [REAL_ARITH_TAC; REWRITE_TAC[dist]] THEN
2610 ASM_REWRITE_TAC[GSYM NORM_MUL] THEN
2611 REWRITE_TAC[REAL_ARITH `abs(&n + &1) = &n + &1`] THEN
2612 ASM_MESON_TAC[REAL_LET_TRANS]]);;
2614 let SUBSET_SUMS_RCANCEL = prove
2615 (`!s t u:real^N->bool.
2616 closed t /\ convex t /\ bounded u /\ ~(u = {}) /\
2617 {x + z | x IN s /\ z IN u} SUBSET {y + z | y IN t /\ z IN u}
2619 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_SUMS_LCANCEL THEN
2620 EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
2621 ONCE_REWRITE_TAC[SUMS_SYM] THEN ASM_REWRITE_TAC[]);;
2623 let EQ_SUMS_LCANCEL = prove
2625 ~(s = {}) /\ bounded s /\
2626 closed t /\ convex t /\ closed u /\ convex u /\
2627 {x + y | x IN s /\ y IN t} = {x + z | x IN s /\ z IN u}
2629 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; EMPTY_SUBSET] THEN
2630 REWRITE_TAC[SUBSET_EMPTY] THEN MESON_TAC[SUBSET_SUMS_LCANCEL]);;
2632 let EQ_SUMS_RCANCEL = prove
2634 closed s /\ convex s /\ closed t /\ convex t /\
2635 bounded u /\ ~(u = {}) /\
2636 {x + z | x IN s /\ z IN u} = {y + z | y IN t /\ z IN u}
2638 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; EMPTY_SUBSET] THEN
2639 REWRITE_TAC[SUBSET_EMPTY] THEN MESON_TAC[SUBSET_SUMS_RCANCEL]);;
2641 (* ------------------------------------------------------------------------- *)
2643 (* ------------------------------------------------------------------------- *)
2645 let CONVEX_CONVEX_HULL = prove
2646 (`!s. convex(convex hull s)`,
2647 SIMP_TAC[P_HULL; CONVEX_INTERS]);;
2649 let CONVEX_HULL_EQ = prove
2650 (`!s. (convex hull s = s) <=> convex s`,
2651 SIMP_TAC[HULL_EQ; CONVEX_INTERS]);;
2653 let CONVEX_HULLS_EQ = prove
2654 (`!s t. s SUBSET convex hull t /\ t SUBSET convex hull s
2655 ==> convex hull s = convex hull t`,
2656 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULLS_EQ THEN
2657 ASM_SIMP_TAC[CONVEX_INTERS]);;
2659 let IS_CONVEX_HULL = prove
2660 (`!s. convex s <=> ?t. s = convex hull t`,
2661 GEN_TAC THEN MATCH_MP_TAC IS_HULL THEN SIMP_TAC[CONVEX_INTERS]);;
2663 let MIDPOINTS_IN_CONVEX_HULL = prove
2664 (`!x:real^N s. x IN convex hull s /\ y IN convex hull s
2665 ==> midpoint(x,y) IN convex hull s`,
2666 MESON_TAC[MIDPOINT_IN_CONVEX; CONVEX_CONVEX_HULL]);;
2668 let CONVEX_HULL_UNIV = prove
2669 (`convex hull (:real^N) = (:real^N)`,
2670 REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_UNIV]);;
2672 let BOUNDED_CONVEX_HULL = prove
2673 (`!s:real^N->bool. bounded s ==> bounded(convex hull s)`,
2674 GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [bounded] THEN
2675 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
2676 MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(vec 0:real^N,B)` THEN
2677 SIMP_TAC[BOUNDED_CBALL; SUBSET_HULL; CONVEX_CBALL] THEN
2678 ASM_REWRITE_TAC[IN_CBALL; SUBSET; dist; VECTOR_SUB_LZERO; NORM_NEG]);;
2680 let BOUNDED_CONVEX_HULL_EQ = prove
2681 (`!s. bounded(convex hull s) <=> bounded s`,
2682 MESON_TAC[BOUNDED_CONVEX_HULL; HULL_SUBSET; BOUNDED_SUBSET]);;
2684 let FINITE_IMP_BOUNDED_CONVEX_HULL = prove
2685 (`!s. FINITE s ==> bounded(convex hull s)`,
2686 SIMP_TAC[BOUNDED_CONVEX_HULL; FINITE_IMP_BOUNDED]);;
2688 (* ------------------------------------------------------------------------- *)
2689 (* Stepping theorems for convex hulls of finite sets. *)
2690 (* ------------------------------------------------------------------------- *)
2692 let CONVEX_HULL_EMPTY = prove
2693 (`convex hull {} = {}`,
2694 MATCH_MP_TAC HULL_UNIQUE THEN
2695 REWRITE_TAC[SUBSET_REFL; CONVEX_EMPTY; EMPTY_SUBSET]);;
2697 let CONVEX_HULL_EQ_EMPTY = prove
2698 (`!s. (convex hull s = {}) <=> (s = {})`,
2699 GEN_TAC THEN EQ_TAC THEN
2700 MESON_TAC[SUBSET_EMPTY; HULL_SUBSET; CONVEX_HULL_EMPTY]);;
2702 let CONVEX_HULL_SING = prove
2703 (`!a. convex hull {a} = {a}`,
2704 REWRITE_TAC[CONVEX_HULL_EQ; CONVEX_SING]);;
2706 let CONVEX_HULL_EQ_SING = prove
2707 (`!s a:real^N. convex hull s = {a} <=> s = {a}`,
2708 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
2709 ASM_REWRITE_TAC[CONVEX_HULL_EMPTY] THEN
2710 EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[CONVEX_HULL_SING] THEN
2711 MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET {a} ==> s = {a}`) THEN
2712 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
2713 REWRITE_TAC[HULL_SUBSET]);;
2715 let CONVEX_HULL_INSERT = prove
2717 ==> (convex hull (a INSERT s) =
2718 {x:real^N | ?u v b. &0 <= u /\ &0 <= v /\ (u + v = &1) /\
2719 b IN (convex hull s) /\
2720 (x = u % a + v % b)})`,
2721 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
2722 [MATCH_MP_TAC HULL_MINIMAL THEN CONJ_TAC THENL
2723 [REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INSERT] THEN
2724 X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL
2725 [MAP_EVERY EXISTS_TAC [`&1`; `&0`];
2726 MAP_EVERY EXISTS_TAC [`&0`; `&1`]] THEN
2727 ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO] THEN
2728 ASM_REWRITE_TAC[VECTOR_ADD_LID; VECTOR_ADD_RID] THEN
2729 CONV_TAC REAL_RAT_REDUCE_CONV THEN
2730 ASM_MESON_TAC[MEMBER_NOT_EMPTY; HULL_SUBSET; SUBSET];
2732 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
2733 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2734 MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN
2735 ASM_REWRITE_TAC[] THEN
2736 ASM_MESON_TAC[HULL_SUBSET; SUBSET; IN_INSERT; HULL_MONO]] THEN
2737 REWRITE_TAC[convex; IN_ELIM_THM] THEN
2738 REWRITE_TAC[LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
2739 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
2741 [`x:real^N`; `y:real^N`; `u:real`; `v:real`; `u1:real`; `v1:real`;
2742 `b1:real^N`; `u2:real`; `v2:real`; `b2:real^N`] THEN
2743 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2744 MAP_EVERY EXISTS_TAC [`u * u1 + v * u2`; `u * v1 + v * v2`] THEN
2745 REWRITE_TAC[VECTOR_ARITH
2746 `u % (u1 % a + v1 % b1) + v % (u2 % a + v2 % b2) =
2747 (u * u1 + v * u2) % a + (u * v1) % b1 + (v * v2) % b2`] THEN
2748 ASM_SIMP_TAC[REAL_LE_ADD; REAL_LE_MUL] THEN
2749 ASM_REWRITE_TAC[REAL_MUL_RID; REAL_ARITH
2750 `(u * u1 + v * u2) + (u * v1 + v * v2) =
2751 u * (u1 + v1) + v * (u2 + v2)`] THEN
2752 ASM_CASES_TAC `u * v1 + v * v2 = &0` THENL
2753 [FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH
2754 `(a + b = &0) ==> &0 <= a /\ &0 <= b ==> (a = &0) /\ (b = &0)`)) THEN
2755 ASM_SIMP_TAC[REAL_LE_MUL; REAL_ADD_LID; VECTOR_MUL_LZERO;
2756 VECTOR_ADD_RID] THEN
2759 EXISTS_TAC `(u * v1) / (u * v1 + v * v2) % b1 +
2760 (v * v2) / (u * v1 + v * v2) % b2 :real^N` THEN
2761 ASM_SIMP_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC; REAL_DIV_LMUL] THEN
2762 MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN
2763 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_MUL; REAL_LE_ADD] THEN
2764 ASM_SIMP_TAC[real_div; GSYM REAL_ADD_RDISTRIB; REAL_MUL_RINV]);;
2766 let CONVEX_HULL_INSERT_ALT = prove
2768 convex hull (a INSERT s) =
2770 else {(&1 - u) % a + u % x | &0 <= u /\ u <= &1 /\ x IN convex hull s}`,
2771 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
2772 ASM_REWRITE_TAC[CONVEX_HULL_SING] THEN
2773 ASM_SIMP_TAC[CONVEX_HULL_INSERT] THEN
2774 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> b /\ c /\ a /\ d`] THEN
2775 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
2776 REWRITE_TAC[RIGHT_EXISTS_AND_THM; UNWIND_THM2; REAL_SUB_LE;
2777 REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
2780 (* ------------------------------------------------------------------------- *)
2781 (* Explicit expression for convex hull. *)
2782 (* ------------------------------------------------------------------------- *)
2784 let CONVEX_HULL_INDEXED = prove
2785 (`!s. convex hull s =
2786 {y:real^N | ?k u x. (!i. 1 <= i /\ i <= k ==> &0 <= u i /\ x i IN s) /\
2787 (sum (1..k) u = &1) /\
2788 (vsum (1..k) (\i. u i % x i) = y)}`,
2789 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN REPEAT CONJ_TAC THENL
2790 [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
2791 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
2792 MAP_EVERY EXISTS_TAC [`1`; `\i:num. &1`; `\i:num. x:real^N`] THEN
2793 ASM_SIMP_TAC[FINITE_RULES; IN_SING; SUM_SING; VECTOR_MUL_LID; VSUM_SING;
2794 REAL_POS; NUMSEG_SING];
2796 REWRITE_TAC[CONVEX_INDEXED; SUBSET; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
2798 REWRITE_TAC[convex; IN_ELIM_THM] THEN
2799 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
2800 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
2801 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
2802 [`k1:num`; `u1:num->real`; `x1:num->real^N`;
2803 `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN
2804 STRIP_TAC THEN EXISTS_TAC `k1 + k2:num` THEN
2805 EXISTS_TAC `\i:num. if i <= k1 then u * u1(i) else v * u2(i - k1):real` THEN
2806 EXISTS_TAC `\i:num. if i <= k1 then x1(i) else x2(i - k1):real^N` THEN
2807 ASM_SIMP_TAC[NUMSEG_ADD_SPLIT; ARITH_RULE `1 <= x + 1 /\ x < x + 1`;
2808 IN_NUMSEG; SUM_UNION; VSUM_UNION; FINITE_NUMSEG; DISJOINT_NUMSEG;
2809 ARITH_RULE `k1 + 1 <= i ==> ~(i <= k1)`] THEN
2810 REWRITE_TAC[ONCE_REWRITE_RULE[ADD_SYM] NUMSEG_OFFSET_IMAGE] THEN
2811 ASM_SIMP_TAC[SUM_IMAGE; VSUM_IMAGE; EQ_ADD_LCANCEL; FINITE_NUMSEG] THEN
2812 ASM_SIMP_TAC[o_DEF; ADD_SUB2; SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC;
2813 FINITE_NUMSEG; REAL_MUL_RID] THEN
2814 ASM_MESON_TAC[REAL_LE_MUL; ARITH_RULE
2815 `i <= k1 + k2 /\ ~(i <= k1) ==> 1 <= i - k1 /\ i - k1 <= k2`]);;
2817 (* ------------------------------------------------------------------------- *)
2818 (* Another formulation from Lars Schewe. *)
2819 (* ------------------------------------------------------------------------- *)
2821 let CONVEX_HULL_EXPLICIT = prove
2822 (`!p. convex hull p =
2823 {y:real^N | ?s u. FINITE s /\ s SUBSET p /\
2824 (!x. x IN s ==> &0 <= u x) /\
2825 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
2826 REWRITE_TAC[CONVEX_HULL_INDEXED;EXTENSION;IN_ELIM_THM] THEN
2827 REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
2828 [MAP_EVERY EXISTS_TAC [`IMAGE (x':num->real^N) (1..k)`;
2829 `\v:real^N.sum {i | i IN (1..k) /\ x' i = v} u`]
2830 THEN ASM_SIMP_TAC[FINITE_IMAGE;FINITE_NUMSEG;IN_IMAGE] THEN
2831 REPEAT STRIP_TAC THENL
2832 [REWRITE_TAC[IMAGE;SUBSET;IN_ELIM_THM;IN_NUMSEG] THEN
2834 MATCH_MP_TAC SUM_POS_LE THEN
2835 ASM_SIMP_TAC[FINITE_NUMSEG;FINITE_RESTRICT;IN_ELIM_THM;IN_NUMSEG];
2836 ASM_SIMP_TAC[GSYM SUM_IMAGE_GEN;FINITE_IMAGE;FINITE_NUMSEG];
2837 FIRST_X_ASSUM (fun th -> REWRITE_TAC[GSYM th]) THEN
2838 ASM_SIMP_TAC[GSYM VSUM_IMAGE_GEN;FINITE_IMAGE;
2839 FINITE_NUMSEG;VSUM_VMUL;FINITE_RESTRICT] THEN
2840 MP_TAC (ISPECL [`x':num->real^N`;`\i:num.u i % (x' i):real^N`;`(1..k)`]
2841 (GSYM VSUM_IMAGE_GEN)) THEN
2842 ASM_SIMP_TAC[FINITE_NUMSEG]];ALL_TAC] THEN
2843 STRIP_ASSUME_TAC (ASM_REWRITE_RULE [ASSUME `FINITE (s:real^N->bool)`]
2844 (ISPEC `s:real^N->bool` FINITE_INDEX_NUMSEG)) THEN
2845 MAP_EVERY EXISTS_TAC [`CARD (s:real^N->bool)`;
2846 `(u:real^N->real) o (f:num->real^N)`;
2847 `(f:num->real^N)`] THEN
2848 REPEAT STRIP_TAC THENL
2849 [REWRITE_TAC[o_DEF] THEN FIRST_ASSUM MATCH_MP_TAC THEN
2850 FIRST_ASSUM SUBST1_TAC THEN
2851 REWRITE_TAC[IN_IMAGE;IN_NUMSEG] THEN
2853 MATCH_MP_TAC (REWRITE_RULE [SUBSET]
2854 (ASSUME `(s:real^N->bool) SUBSET p`)) THEN
2855 FIRST_ASSUM SUBST1_TAC THEN
2856 REWRITE_TAC[IN_IMAGE;IN_NUMSEG] THEN
2858 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sum (s:real^N->bool) u` THEN
2859 CONJ_TAC THENL [ALL_TAC;ASM_REWRITE_TAC[]] THEN
2860 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
2861 [ASSUME `(s:real^N->bool) = IMAGE f (1..CARD s)`] THEN
2862 MATCH_MP_TAC (GSYM SUM_IMAGE) THEN
2864 REWRITE_TAC[MESON [o_THM;FUN_EQ_THM]
2865 `(\i:num. (u o f) i % f i) = (\v:real^N. u v % v) o f`] THEN
2866 MATCH_MP_TAC EQ_TRANS THEN
2867 EXISTS_TAC `vsum (s:real^N->bool) (\v. u v % v)` THEN
2868 CONJ_TAC THENL [ALL_TAC;ASM_REWRITE_TAC[]] THEN
2869 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
2870 [ASSUME `(s:real^N->bool) = IMAGE f (1..CARD s)`] THEN
2871 MATCH_MP_TAC (GSYM VSUM_IMAGE) THEN
2872 ASM SET_TAC[FINITE_NUMSEG]]);;
2874 let CONVEX_HULL_FINITE = prove
2877 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
2879 vsum s (\x. u x % x) = y}`,
2880 GEN_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
2881 REWRITE_TAC[CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN
2882 X_GEN_TAC `x:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
2883 [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `f:real^N->real`] THEN
2885 EXISTS_TAC `\x:real^N. if x IN t then f x else &0` THEN
2886 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
2887 REWRITE_TAC[GSYM SUM_RESTRICT_SET; GSYM VSUM_RESTRICT_SET] THEN
2888 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
2889 REWRITE_TAC[REAL_LE_REFL; COND_ID];
2890 X_GEN_TAC `f:real^N->real` THEN
2891 ASM_CASES_TAC `s:real^N->bool = {}` THEN
2892 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN STRIP_TAC THEN
2893 EXISTS_TAC `support (+) (f:real^N->real) s` THEN
2894 EXISTS_TAC `f:real^N->real` THEN
2895 MP_TAC(ASSUME `sum s (f:real^N->real) = &1`) THEN
2896 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [sum] THEN
2897 REWRITE_TAC[iterate] THEN COND_CASES_TAC THEN
2898 ASM_REWRITE_TAC[NEUTRAL_REAL_ADD; REAL_OF_NUM_EQ; ARITH] THEN
2899 DISCH_THEN(K ALL_TAC) THEN
2900 UNDISCH_TAC `sum s (f:real^N->real) = &1` THEN
2901 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM SUM_SUPPORT] THEN
2902 ASM_CASES_TAC `support (+) (f:real^N->real) s = {}` THEN
2903 ASM_SIMP_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH] THEN
2904 DISCH_TAC THEN REWRITE_TAC[SUPPORT_SUBSET] THEN CONJ_TAC THENL
2905 [ASM_SIMP_TAC[support; IN_ELIM_THM]; ALL_TAC] THEN
2906 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
2907 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
2908 REWRITE_TAC[SUPPORT_SUBSET] THEN
2909 REWRITE_TAC[support; IN_ELIM_THM; NEUTRAL_REAL_ADD] THEN
2910 MESON_TAC[VECTOR_MUL_LZERO]]);;
2912 let CONVEX_HULL_UNION_EXPLICIT = prove
2913 (`!s t:real^N->bool.
2914 convex s /\ convex t
2915 ==> convex hull (s UNION t) =
2917 {(&1 - u) % x + u % y | x IN s /\ y IN t /\ &0 <= u /\ u <= &1}`,
2918 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
2919 [REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN GEN_REWRITE_TAC I [SUBSET] THEN
2920 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
2921 MAP_EVERY X_GEN_TAC [`y:real^N`; `u:real^N->bool`; `f:real^N->real`] THEN
2922 REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2923 SUBST1_TAC(SET_RULE `u:real^N->bool = (u INTER s) UNION (u DIFF s)`) THEN
2924 ASM_SIMP_TAC[SUM_UNION; VSUM_UNION; FINITE_INTER; FINITE_DIFF;
2925 SET_RULE `DISJOINT (u INTER s) (u DIFF s)`] THEN
2926 ASM_CASES_TAC `sum (u INTER s) (f:real^N->real) = &0` THENL
2927 [SUBGOAL_THEN `!x. x IN (u INTER s) ==> (f:real^N->real) x = &0`
2929 [ASM_MESON_TAC[SUM_POS_EQ_0; FINITE_INTER; IN_INTER];
2930 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0] THEN
2931 REWRITE_TAC[VECTOR_ADD_LID; REAL_ADD_LID] THEN
2932 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
2933 REWRITE_TAC[IN_UNION] THEN DISJ2_TAC THEN DISJ1_TAC THEN
2934 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
2935 ASM_SIMP_TAC[FINITE_DIFF; IN_DIFF] THEN ASM SET_TAC[]];
2937 ASM_CASES_TAC `sum (u DIFF s) (f:real^N->real) = &0` THENL
2938 [SUBGOAL_THEN `!x. x IN (u DIFF s) ==> (f:real^N->real) x = &0`
2940 [ASM_MESON_TAC[SUM_POS_EQ_0; FINITE_DIFF; IN_DIFF];
2941 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VSUM_0] THEN
2942 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN
2943 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
2944 REWRITE_TAC[IN_UNION] THEN DISJ1_TAC THEN
2945 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
2946 ASM_SIMP_TAC[FINITE_INTER; IN_INTER] THEN ASM SET_TAC[]];
2948 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
2949 REWRITE_TAC[IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN DISJ2_TAC THEN
2950 MAP_EVERY EXISTS_TAC
2951 [`vsum(u INTER s) (\v:real^N. (f v / sum(u INTER s) f) % v)`;
2952 `sum(u DIFF s) (f:real^N->real)`;
2953 `vsum(u DIFF s) (\v:real^N. (f v / sum(u DIFF s) f) % v)`] THEN
2954 REPEAT CONJ_TAC THENL
2955 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
2956 ASM_SIMP_TAC[INTER_SUBSET; FINITE_INTER; SUM_POS_LE; REAL_LE_DIV;
2957 IN_INTER; real_div; SUM_RMUL; REAL_MUL_RINV];
2958 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
2959 ASM_SIMP_TAC[SUBSET_DIFF; FINITE_DIFF; SUM_POS_LE; REAL_LE_DIV;
2960 IN_DIFF; real_div; SUM_RMUL; REAL_MUL_RINV] THEN
2962 ASM_SIMP_TAC[SUM_POS_LE; IN_DIFF; FINITE_DIFF];
2963 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
2964 `a + b = &1 ==> &0 <= a ==> b <= &1`)) THEN
2965 ASM_SIMP_TAC[SUM_POS_LE; IN_INTER; FINITE_INTER];
2966 ASM_SIMP_TAC[GSYM VSUM_LMUL; FINITE_INTER; FINITE_DIFF] THEN
2967 SIMP_TAC[VECTOR_MUL_ASSOC; REAL_ARITH `a * b / c:real = a / c * b`] THEN
2968 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (REAL_ARITH
2969 `a + b = &1 ==> &1 - b = a`)) THEN
2970 ASM_SIMP_TAC[REAL_DIV_REFL; REAL_MUL_LID]];
2971 REWRITE_TAC[GSYM UNION_ASSOC] THEN ONCE_REWRITE_TAC[UNION_SUBSET] THEN
2972 REWRITE_TAC[HULL_SUBSET] THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
2973 MAP_EVERY X_GEN_TAC [`x:real^N`; `u:real`; `y:real^N`] THEN STRIP_TAC THEN
2974 MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] CONVEX_CONVEX_HULL) THEN
2975 ASM_SIMP_TAC[HULL_INC; IN_UNION]]);;
2977 let CONVEX_HULL_UNION_NONEMPTY_EXPLICIT = prove
2978 (`!s t:real^N->bool.
2979 convex s /\ ~(s = {}) /\ convex t /\ ~(t = {})
2980 ==> convex hull (s UNION t) =
2981 {(&1 - u) % x + u % y | x IN s /\ y IN t /\ &0 <= u /\ u <= &1}`,
2982 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_UNION_EXPLICIT] THEN
2983 SIMP_TAC[SET_RULE `s UNION t UNION u = u <=> s SUBSET u /\ t SUBSET u`] THEN
2984 CONJ_TAC THEN REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `z:real^N` THEN
2986 [MAP_EVERY EXISTS_TAC [`z:real^N`; `&0`] THEN
2987 REWRITE_TAC[REAL_SUB_RZERO; VECTOR_MUL_LID; REAL_POS; VECTOR_MUL_LZERO;
2988 VECTOR_ADD_RID] THEN
2990 SUBGOAL_THEN `?a:real^N. a IN s` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2991 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN
2992 MAP_EVERY EXISTS_TAC [`&1`; `z:real^N`] THEN
2993 ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN VECTOR_ARITH_TAC]);;
2995 let CONVEX_HULL_UNION_UNIONS = prove
2996 (`!f s:real^N->bool.
2997 convex(UNIONS f) /\ ~(f = {})
2998 ==> convex hull (s UNION UNIONS f) =
2999 UNIONS {convex hull (s UNION t) | t IN f}`,
3000 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3002 REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN
3003 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]] THEN
3004 ASM_CASES_TAC `s:real^N->bool = {}` THENL
3005 [ASM_SIMP_TAC[UNION_EMPTY; HULL_P; UNIONS_SUBSET] THEN
3006 X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN
3007 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `convex hull u:real^N->bool` THEN
3008 REWRITE_TAC[HULL_SUBSET] THEN ASM SET_TAC[];
3010 ASM_CASES_TAC `UNIONS f :real^N->bool = {}` THENL
3011 [ASM_REWRITE_TAC[UNION_EMPTY] THEN
3012 SUBGOAL_THEN `?u:real^N->bool. u IN f` CHOOSE_TAC THENL
3013 [ASM_REWRITE_TAC[MEMBER_NOT_EMPTY]; ALL_TAC] THEN
3014 MATCH_MP_TAC SUBSET_TRANS THEN
3015 EXISTS_TAC `convex hull (s UNION u:real^N->bool)` THEN
3016 ASM_SIMP_TAC[HULL_MONO; SUBSET_UNION] THEN ASM SET_TAC[];
3018 GEN_REWRITE_TAC LAND_CONV [HULL_UNION_LEFT] THEN
3019 ASM_SIMP_TAC[CONVEX_HULL_UNION_NONEMPTY_EXPLICIT; CONVEX_HULL_EQ_EMPTY;
3020 CONVEX_CONVEX_HULL] THEN
3021 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3022 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_UNIONS] THEN
3023 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
3024 MAP_EVERY X_GEN_TAC [`a:real`; `u:real^N->bool`] THEN DISCH_TAC THEN
3025 X_GEN_TAC `y:real^N` THEN REPEAT STRIP_TAC THEN
3026 REWRITE_TAC[IN_UNIONS; EXISTS_IN_GSPEC] THEN
3027 EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
3028 MATCH_MP_TAC(REWRITE_RULE[CONVEX_ALT] CONVEX_CONVEX_HULL) THEN
3029 ASM_MESON_TAC[HULL_MONO; IN_UNION; SUBSET; HULL_INC]);;
3031 (* ------------------------------------------------------------------------- *)
3032 (* A stepping theorem for that expansion. *)
3033 (* ------------------------------------------------------------------------- *)
3035 let CONVEX_HULL_FINITE_STEP = prove
3036 (`((?u. (!x. x IN {} ==> &0 <= u x) /\
3038 vsum {} (\x. u(x) % x) = y) <=> w = &0 /\ y = vec 0) /\
3039 (FINITE(s:real^N->bool)
3040 ==> ((?u. (!x. x IN (a INSERT s) ==> &0 <= u x) /\
3041 sum (a INSERT s) u = w /\
3042 vsum (a INSERT s) (\x. u(x) % x) = y) <=>
3044 ?u. (!x. x IN s ==> &0 <= u x) /\
3046 vsum s (\x. u(x) % x) = y - v % a))`,
3047 MP_TAC(ISPEC `\x:real^N y:real. &0 <= y` AFFINE_HULL_FINITE_STEP_GEN) THEN
3048 SIMP_TAC[REAL_ARITH `&0 <= x / &2 <=> &0 <= x`; REAL_LE_ADD] THEN
3049 REWRITE_TAC[RIGHT_AND_EXISTS_THM]);;
3051 (* ------------------------------------------------------------------------- *)
3052 (* Hence some special cases. *)
3053 (* ------------------------------------------------------------------------- *)
3055 let CONVEX_HULL_2 = prove
3056 (`!a b. convex hull {a,b} =
3057 {u % a + v % b | &0 <= u /\ &0 <= v /\ u + v = &1}`,
3058 SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
3059 SIMP_TAC[CONVEX_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
3060 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
3061 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
3062 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
3064 let CONVEX_HULL_2_ALT = prove
3065 (`!a b. convex hull {a,b} = {a + u % (b - a) | &0 <= u /\ u <= &1}`,
3066 ONCE_REWRITE_TAC[SET_RULE `{a,b} = {b,a}`] THEN
3067 REWRITE_TAC[CONVEX_HULL_2; EXTENSION; IN_ELIM_THM] THEN
3068 REWRITE_TAC[REAL_ADD_ASSOC; CONJ_ASSOC] THEN
3069 REWRITE_TAC[TAUT `(a /\ x + y = &1) /\ b <=> x + y = &1 /\ a /\ b`] THEN
3070 REWRITE_TAC[REAL_ARITH `x + y = &1 <=> y = &1 - x`; UNWIND_THM2] THEN
3071 REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
3072 BINOP_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);;
3074 let CONVEX_HULL_3 = prove
3075 (`convex hull {a,b,c} =
3076 { u % a + v % b + w % c |
3077 &0 <= u /\ &0 <= v /\ &0 <= w /\ u + v + w = &1}`,
3078 SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT; FINITE_RULES] THEN
3079 SIMP_TAC[CONVEX_HULL_FINITE_STEP; FINITE_INSERT; FINITE_RULES] THEN
3080 REWRITE_TAC[REAL_ARITH `x - y = z:real <=> x = y + z`;
3081 VECTOR_ARITH `x - y = z:real^N <=> x = y + z`] THEN
3082 REWRITE_TAC[VECTOR_ADD_RID; REAL_ADD_RID] THEN SET_TAC[]);;
3084 let CONVEX_HULL_3_ALT = prove
3085 (`!a b c. convex hull {a,b,c} =
3086 {a + u % (b - a) + v % (c - a) |
3087 &0 <= u /\ &0 <= v /\ u + v <= &1}`,
3088 ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,c,a}`] THEN
3089 REWRITE_TAC[CONVEX_HULL_3; EXTENSION; IN_ELIM_THM] THEN
3090 REWRITE_TAC[REAL_ADD_ASSOC; CONJ_ASSOC] THEN
3091 REWRITE_TAC[TAUT `(a /\ x + y = &1) /\ b <=> x + y = &1 /\ a /\ b`] THEN
3092 REWRITE_TAC[REAL_ARITH `x + y = &1 <=> y = &1 - x`; UNWIND_THM2] THEN
3093 REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
3094 BINOP_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]);;
3096 let CONVEX_HULL_SUMS = prove
3097 (`!s t:real^N->bool.
3098 convex hull {x + y | x IN s /\ y IN t} =
3099 {x + y | x IN convex hull s /\ y IN convex hull t}`,
3100 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3101 [MATCH_MP_TAC HULL_MINIMAL THEN
3102 SIMP_TAC[CONVEX_SUMS; CONVEX_CONVEX_HULL] THEN
3103 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3104 REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[HULL_INC];
3105 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3106 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
3107 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [CONVEX_HULL_INDEXED] THEN
3108 REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
3109 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
3111 [`k1:num`; `u1:num->real`; `x1:num->real^N`;
3112 `k2:num`; `u2:num->real`; `x2:num->real^N`] THEN
3116 vsum(1..k1) (\i. vsum(1..k2) (\j. u1 i % u2 j % (x1 i + x2 j)))`
3118 [REWRITE_TAC[VECTOR_ADD_LDISTRIB; VSUM_ADD_NUMSEG] THEN
3119 ASM_SIMP_TAC[VSUM_LMUL; VSUM_RMUL; VECTOR_MUL_LID];
3120 REWRITE_TAC[VSUM_LMUL] THEN MATCH_MP_TAC CONVEX_VSUM THEN
3121 ASM_SIMP_TAC[FINITE_NUMSEG; CONVEX_CONVEX_HULL; IN_NUMSEG] THEN
3122 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_VSUM THEN
3123 ASM_SIMP_TAC[FINITE_NUMSEG; CONVEX_CONVEX_HULL; IN_NUMSEG] THEN
3124 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);;
3126 let AFFINE_HULL_PCROSS,CONVEX_HULL_PCROSS = (CONJ_PAIR o prove)
3127 (`(!s:real^M->bool t:real^N->bool.
3128 affine hull (s PCROSS t) =
3129 (affine hull s) PCROSS (affine hull t)) /\
3130 (!s:real^M->bool t:real^N->bool.
3131 convex hull (s PCROSS t) =
3132 (convex hull s) PCROSS (convex hull t))`,
3134 (`!u v x y:real^M z:real^N.
3136 ==> pastecart z (u % x + v % y) =
3137 u % pastecart z x + v % pastecart z y /\
3138 pastecart (u % x + v % y) z =
3139 u % pastecart x z + v % pastecart y z`,
3140 REWRITE_TAC[PASTECART_ADD; GSYM PASTECART_CMUL] THEN
3141 SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID])
3143 (`INTERS {{x | pastecart x y IN u} | y IN t} =
3144 {x | !y. y IN t ==> pastecart x y IN u}`,
3145 REWRITE_TAC[INTERS_GSPEC; EXTENSION; IN_ELIM_THM] THEN SET_TAC[]) in
3147 [REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3148 [MATCH_MP_TAC HULL_MINIMAL THEN
3149 SIMP_TAC[AFFINE_PCROSS; AFFINE_AFFINE_HULL; HULL_SUBSET; PCROSS_MONO];
3150 REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN
3151 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3152 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3153 [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
3154 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3155 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
3156 SUBGOAL_THEN `pastecart (x:real^M) (y:real^N) IN s PCROSS t` MP_TAC
3157 THENL [ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN
3158 REWRITE_TAC[HULL_INC];
3160 REWRITE_TAC[GSYM lemma2] THEN MATCH_MP_TAC AFFINE_INTERS THEN
3161 REWRITE_TAC[FORALL_IN_GSPEC]] THEN
3162 SIMP_TAC[affine; IN_ELIM_THM; lemma1;
3163 ONCE_REWRITE_RULE[affine] AFFINE_AFFINE_HULL]];
3164 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3165 [MATCH_MP_TAC HULL_MINIMAL THEN
3166 SIMP_TAC[CONVEX_PCROSS; CONVEX_CONVEX_HULL; HULL_SUBSET; PCROSS_MONO];
3167 REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN
3168 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3169 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3170 [X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
3171 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
3172 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
3173 SUBGOAL_THEN `pastecart (x:real^M) (y:real^N) IN s PCROSS t` MP_TAC
3174 THENL [ASM_REWRITE_TAC[PASTECART_IN_PCROSS]; ALL_TAC] THEN
3175 REWRITE_TAC[HULL_INC];
3177 REWRITE_TAC[GSYM lemma2] THEN MATCH_MP_TAC CONVEX_INTERS THEN
3178 REWRITE_TAC[FORALL_IN_GSPEC]] THEN
3179 SIMP_TAC[convex; IN_ELIM_THM; lemma1;
3180 ONCE_REWRITE_RULE[convex] CONVEX_CONVEX_HULL]]]);;
3182 (* ------------------------------------------------------------------------- *)
3183 (* Relations among closure notions and corresponding hulls. *)
3184 (* ------------------------------------------------------------------------- *)
3186 let SUBSPACE_IMP_AFFINE = prove
3187 (`!s. subspace s ==> affine s`,
3188 REWRITE_TAC[subspace; affine] THEN MESON_TAC[]);;
3190 let AFFINE_IMP_CONVEX = prove
3191 (`!s. affine s ==> convex s`,
3192 REWRITE_TAC[affine; convex] THEN MESON_TAC[]);;
3194 let SUBSPACE_IMP_CONVEX = prove
3195 (`!s. subspace s ==> convex s`,
3196 MESON_TAC[SUBSPACE_IMP_AFFINE; AFFINE_IMP_CONVEX]);;
3198 let AFFINE_HULL_SUBSET_SPAN = prove
3199 (`!s. (affine hull s) SUBSET (span s)`,
3200 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN
3201 REWRITE_TAC[SUBSET; IN; SUBSPACE_IMP_AFFINE]);;
3203 let CONVEX_HULL_SUBSET_SPAN = prove
3204 (`!s. (convex hull s) SUBSET (span s)`,
3205 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN
3206 REWRITE_TAC[SUBSET; IN; SUBSPACE_IMP_CONVEX]);;
3208 let CONVEX_HULL_SUBSET_AFFINE_HULL = prove
3209 (`!s. (convex hull s) SUBSET (affine hull s)`,
3210 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_ANTIMONO THEN
3211 REWRITE_TAC[SUBSET; IN; AFFINE_IMP_CONVEX]);;
3213 let COLLINEAR_CONVEX_HULL_COLLINEAR = prove
3214 (`!s:real^N->bool. collinear(convex hull s) <=> collinear s`,
3215 MESON_TAC[COLLINEAR_SUBSET; HULL_SUBSET; SUBSET_TRANS;
3216 COLLINEAR_AFFINE_HULL_COLLINEAR; CONVEX_HULL_SUBSET_AFFINE_HULL]);;
3218 let AFFINE_SPAN = prove
3219 (`!s. affine(span s)`,
3220 SIMP_TAC[SUBSPACE_IMP_AFFINE; SUBSPACE_SPAN]);;
3222 let CONVEX_SPAN = prove
3223 (`!s. convex(span s)`,
3224 SIMP_TAC[SUBSPACE_IMP_CONVEX; SUBSPACE_SPAN]);;
3226 let AFFINE_EQ_SUBSPACE = prove
3227 (`!s:real^N->bool. vec 0 IN s ==> (affine s <=> subspace s)`,
3228 REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[subspace; affine] THEN
3229 DISCH_TAC THEN MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN
3231 [MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN
3232 SUBST1_TAC(VECTOR_ARITH `c % x:real^N = c % x + (&1 - c) % vec 0`) THEN
3233 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
3234 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
3235 STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH
3236 `x + y:real^N = &2 % (&1 / &2 % x + &1 / &2 % y)`) THEN
3237 FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
3238 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);;
3240 let AFFINE_IMP_SUBSPACE = prove
3241 (`!s. affine s /\ vec 0 IN s ==> subspace s`,
3242 SIMP_TAC[GSYM AFFINE_EQ_SUBSPACE]);;
3244 let AFFINE_HULL_EQ_SPAN = prove
3245 (`!s:real^N->bool. (vec 0) IN affine hull s ==> affine hull s = span s`,
3246 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
3247 REWRITE_TAC[AFFINE_HULL_SUBSET_SPAN] THEN
3248 REWRITE_TAC[SUBSET] THEN MATCH_MP_TAC SPAN_INDUCT THEN
3249 ASM_REWRITE_TAC[SUBSET; subspace; IN_ELIM_THM; HULL_INC] THEN
3250 REPEAT STRIP_TAC THENL
3251 [SUBST1_TAC(VECTOR_ARITH
3252 `x + y:real^N = &2 % (&1 / &2 % x + &1 / &2 % y) + --(&1) % vec 0`) THEN
3253 MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN
3254 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN
3255 MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN
3256 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[];
3257 SUBST1_TAC(VECTOR_ARITH
3258 `c % x:real^N = c % x + (&1 - c) % vec 0`) THEN
3259 MATCH_MP_TAC(REWRITE_RULE[affine] AFFINE_AFFINE_HULL) THEN
3260 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);;
3262 let CLOSED_AFFINE = prove
3263 (`!s:real^N->bool. affine s ==> closed s`,
3264 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
3265 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN
3266 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3267 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
3268 SUBGOAL_THEN `affine (IMAGE (\x:real^N. --a + x) s)
3269 ==> closed (IMAGE (\x:real^N. --a + x) s)`
3271 [DISCH_THEN(fun th -> MATCH_MP_TAC CLOSED_SUBSPACE THEN MP_TAC th) THEN
3272 MATCH_MP_TAC EQ_IMP THEN MATCH_MP_TAC AFFINE_EQ_SUBSPACE THEN
3273 REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `a:real^N` THEN
3274 ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
3275 REWRITE_TAC[AFFINE_TRANSLATION_EQ; CLOSED_TRANSLATION_EQ]]);;
3277 let CLOSED_AFFINE_HULL = prove
3278 (`!s. closed(affine hull s)`,
3279 SIMP_TAC[CLOSED_AFFINE; AFFINE_AFFINE_HULL]);;
3281 let CLOSURE_SUBSET_AFFINE_HULL = prove
3282 (`!s. closure s SUBSET affine hull s`,
3283 GEN_TAC THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN
3284 REWRITE_TAC[CLOSED_AFFINE_HULL; HULL_SUBSET]);;
3286 let AFFINE_HULL_CLOSURE = prove
3287 (`!s:real^N->bool. affine hull (closure s) = affine hull s`,
3288 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN
3289 REWRITE_TAC[CLOSURE_SUBSET_AFFINE_HULL; AFFINE_AFFINE_HULL] THEN
3290 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
3291 MATCH_MP_TAC HULL_MINIMAL THEN ASM_REWRITE_TAC[] THEN
3292 ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]);;
3294 let AFFINE_HULL_EQ_SPAN_EQ = prove
3295 (`!s:real^N->bool. (affine hull s = span s) <=> (vec 0) IN affine hull s`,
3296 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[SPAN_0; AFFINE_HULL_EQ_SPAN]);;
3298 let AFFINE_DEPENDENT_IMP_DEPENDENT = prove
3299 (`!s. affine_dependent s ==> dependent s`,
3300 REWRITE_TAC[affine_dependent; dependent] THEN
3301 MESON_TAC[SUBSET; AFFINE_HULL_SUBSET_SPAN]);;
3303 let DEPENDENT_AFFINE_DEPENDENT_CASES = prove
3305 dependent s <=> affine_dependent s \/ (vec 0) IN affine hull s`,
3306 REWRITE_TAC[DEPENDENT_EXPLICIT; AFFINE_DEPENDENT_EXPLICIT;
3307 AFFINE_HULL_EXPLICIT_ALT; IN_ELIM_THM] THEN
3308 GEN_TAC THEN ONCE_REWRITE_TAC[OR_EXISTS_THM] THEN
3309 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
3310 X_GEN_TAC `t:real^N->bool` THEN
3311 ASM_CASES_TAC `FINITE(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN
3312 EQ_TAC THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN
3313 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC))
3315 [ASM_CASES_TAC `sum t (u:real^N->real) = &0` THENL
3316 [ASM_MESON_TAC[]; ALL_TAC] THEN
3317 DISJ2_TAC THEN EXISTS_TAC `\v:real^N. inv(sum t u) * u v` THEN
3318 ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
3319 ASM_SIMP_TAC[VECTOR_MUL_RZERO; REAL_MUL_LINV];
3320 EXISTS_TAC `u:real^N->real` THEN ASM_MESON_TAC[];
3321 EXISTS_TAC `u:real^N->real` THEN
3322 ASM_REWRITE_TAC[SET_RULE
3323 `(?v. v IN t /\ ~p v) <=> ~(!v. v IN t ==> p v)`] THEN
3325 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
3326 `x = &1 ==> x = &0 ==> F`)) THEN
3327 ASM_MESON_TAC[SUM_EQ_0]]);;
3329 let DEPENDENT_IMP_AFFINE_DEPENDENT = prove
3330 (`!a:real^N s. dependent {x - a | x IN s} /\ ~(a IN s)
3331 ==> affine_dependent(a INSERT s)`,
3332 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
3333 REWRITE_TAC[DEPENDENT_EXPLICIT; AFFINE_DEPENDENT_EXPLICIT] THEN
3334 REWRITE_TAC[SIMPLE_IMAGE; CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
3335 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN
3336 GEN_REWRITE_TAC LAND_CONV [SWAP_EXISTS_THM] THEN
3337 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_EXISTS_THM] THEN
3338 REWRITE_TAC[TAUT `a /\ x = IMAGE f s /\ b <=> x = IMAGE f s /\ a /\ b`] THEN
3339 REWRITE_TAC[UNWIND_THM2; EXISTS_IN_IMAGE] THEN
3340 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` (X_CHOOSE_THEN `t:real^N->bool`
3341 STRIP_ASSUME_TAC)) THEN
3342 FIRST_X_ASSUM(MP_TAC o check (is_eq o concl)) THEN
3343 ASM_SIMP_TAC[VSUM_IMAGE; VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
3344 ASM_SIMP_TAC[o_DEF; VECTOR_SUB_LDISTRIB; VSUM_SUB; VSUM_RMUL] THEN
3346 MAP_EVERY EXISTS_TAC
3347 [`(a:real^N) INSERT t`;
3348 `\x. if x = a then --sum t (\x. u (x - a))
3349 else (u:real^N->real) (x - a)`] THEN
3350 ASM_REWRITE_TAC[FINITE_INSERT; SUBSET_REFL] THEN
3351 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
3352 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3353 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM SET_TAC[]; ALL_TAC] THEN
3354 REPEAT CONJ_TAC THENL
3355 [MATCH_MP_TAC(REAL_ARITH `x = y ==> --x + y = &0`) THEN
3356 MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[];
3357 EXISTS_TAC `x:real^N` THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3358 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
3359 MATCH_MP_TAC(VECTOR_ARITH
3360 `!s. s - t % a = vec 0 /\ s = u ==> --t % a + u = vec 0`) THEN
3361 EXISTS_TAC `vsum t (\x:real^N. u(x - a) % x)` THEN
3362 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC VSUM_EQ THEN
3363 REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN
3364 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]]);;
3366 let AFFINE_DEPENDENT_BIGGERSET = prove
3368 (FINITE s ==> CARD s >= dimindex(:N) + 2) ==> affine_dependent s`,
3369 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
3370 ASM_SIMP_TAC[CARD_CLAUSES; ARITH_RULE `~(0 >= n + 2)`; FINITE_RULES] THEN
3371 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3372 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
3373 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
3374 `x IN s ==> s = x INSERT (s DELETE x)`)) THEN
3375 SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; IN_DELETE] THEN
3376 REWRITE_TAC[ARITH_RULE `SUC x >= n + 2 <=> x > n`] THEN DISCH_TAC THEN
3377 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3378 REWRITE_TAC[IN_DELETE] THEN MATCH_MP_TAC DEPENDENT_BIGGERSET THEN
3379 REWRITE_TAC[SET_RULE `{x - a:real^N | x | x IN s /\ ~(x = a)} =
3380 IMAGE (\x. x - a) (s DELETE a)`] THEN
3381 ASM_SIMP_TAC[FINITE_IMAGE_INJ_EQ;
3382 VECTOR_ARITH `x - a = y - a <=> x:real^N = y`;
3385 let AFFINE_DEPENDENT_BIGGERSET_GENERAL = prove
3386 (`!s:real^N->bool. (FINITE s ==> CARD s >= dim s + 2) ==> affine_dependent s`,
3387 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
3388 ASM_SIMP_TAC[CARD_CLAUSES; ARITH_RULE `~(0 >= n + 2)`; FINITE_RULES] THEN
3389 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3390 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
3391 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
3392 `x IN s ==> s = x INSERT (s DELETE x)`)) THEN
3393 SIMP_TAC[FINITE_INSERT; CARD_CLAUSES; IN_DELETE] THEN
3394 REWRITE_TAC[ARITH_RULE `SUC x >= n + 2 <=> x > n`] THEN DISCH_TAC THEN
3395 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3396 REWRITE_TAC[IN_DELETE] THEN
3397 MATCH_MP_TAC DEPENDENT_BIGGERSET_GENERAL THEN
3398 REWRITE_TAC[SET_RULE `{x - a:real^N | x | x IN s /\ ~(x = a)} =
3399 IMAGE (\x. x - a) (s DELETE a)`] THEN
3400 ASM_SIMP_TAC[FINITE_IMAGE_INJ_EQ; FINITE_DELETE;
3401 VECTOR_ARITH `x - a = y - a <=> x:real^N = y`;
3402 CARD_IMAGE_INJ] THEN
3403 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check(is_imp o concl)) THEN
3404 ASM_REWRITE_TAC[FINITE_DELETE] THEN
3405 MATCH_MP_TAC(ARITH_RULE `c:num <= b ==> (a > b ==> a > c)`) THEN
3406 MATCH_MP_TAC SUBSET_LE_DIM THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
3407 SIMP_TAC[SPAN_SUB; SPAN_SUPERSET; IN_INSERT]);;
3409 let AFFINE_INDEPENDENT_IMP_FINITE = prove
3410 (`!s:real^N->bool. ~(affine_dependent s) ==> FINITE s`,
3411 MESON_TAC[AFFINE_DEPENDENT_BIGGERSET]);;
3413 let AFFINE_INDEPENDENT_CARD_LE = prove
3414 (`!s:real^N->bool. ~(affine_dependent s) ==> CARD s <= dimindex(:N) + 1`,
3415 REWRITE_TAC[ARITH_RULE `s <= n + 1 <=> ~(n + 2 <= s)`; CONTRAPOS_THM] THEN
3416 REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_DEPENDENT_BIGGERSET THEN
3417 ASM_REWRITE_TAC[GE]);;
3419 let AFFINE_INDEPENDENT_CONVEX_AFFINE_HULL = prove
3420 (`!s t:real^N->bool.
3421 ~affine_dependent s /\ t SUBSET s
3422 ==> convex hull t = affine hull t INTER convex hull s`,
3423 REPEAT STRIP_TAC THEN
3424 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3425 SUBGOAL_THEN `FINITE(t:real^N->bool)` ASSUME_TAC THENL
3426 [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
3427 MATCH_MP_TAC(SET_RULE
3428 `ct SUBSET a /\ ct SUBSET cs /\ a INTER cs SUBSET ct
3429 ==> ct = a INTER cs`) THEN
3430 ASM_SIMP_TAC[HULL_MONO; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
3431 REWRITE_TAC[SUBSET; IN_INTER; CONVEX_HULL_FINITE; AFFINE_HULL_FINITE] THEN
3432 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
3433 DISCH_THEN(CONJUNCTS_THEN2
3434 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)
3435 (X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC)) THEN
3436 EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[] THEN
3437 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
3438 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
3439 [AFFINE_DEPENDENT_EXPLICIT]) THEN
3440 REWRITE_TAC[NOT_EXISTS_THM] THEN
3441 DISCH_THEN(MP_TAC o SPECL [`s:real^N->bool`;
3442 `\x:real^N. if x IN t then v x - u x:real else v x`]) THEN
3443 ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[MESON[]
3444 `(if p then a else b) % x = if p then a % x else b % x`] THEN
3445 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES; SET_RULE
3446 `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
3447 ASM_SIMP_TAC[GSYM DIFF; SUM_DIFF; VSUM_DIFF; VECTOR_SUB_RDISTRIB;
3448 SUM_SUB; VSUM_SUB] THEN
3449 REWRITE_TAC[REAL_ARITH `a - b + b - a = &0`; NOT_EXISTS_THM;
3450 VECTOR_ARITH `a - b + b - a:real^N = vec 0`] THEN
3451 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
3452 ASM_REWRITE_TAC[REAL_SUB_0] THEN ASM SET_TAC[]);;
3454 let DISJOINT_AFFINE_HULL = prove
3455 (`!s t u:real^N->bool.
3456 ~affine_dependent s /\ t SUBSET s /\ u SUBSET s /\ DISJOINT t u
3457 ==> DISJOINT (affine hull t) (affine hull u)`,
3458 REPEAT STRIP_TAC THEN
3459 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3460 SUBGOAL_THEN `FINITE(t:real^N->bool) /\ FINITE (u:real^N->bool)` ASSUME_TAC
3461 THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
3462 REWRITE_TAC[IN_DISJOINT; AFFINE_HULL_FINITE; IN_ELIM_THM] THEN
3463 DISCH_THEN(X_CHOOSE_THEN `y:real^N` MP_TAC) THEN
3464 DISCH_THEN(CONJUNCTS_THEN2
3465 (X_CHOOSE_THEN `a:real^N->real` STRIP_ASSUME_TAC)
3466 (X_CHOOSE_THEN `b:real^N->real` STRIP_ASSUME_TAC)) THEN
3467 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
3468 [AFFINE_DEPENDENT_EXPLICIT]) THEN
3469 REWRITE_TAC[NOT_EXISTS_THM] THEN
3470 MAP_EVERY EXISTS_TAC
3472 `\x:real^N. if x IN t then a x else if x IN u then --(b x) else &0`] THEN
3473 ASM_REWRITE_TAC[SUBSET_REFL] THEN REWRITE_TAC[MESON[]
3474 `(if p then a else b) % x = if p then a % x else b % x`] THEN
3475 ASM_SIMP_TAC[SUM_CASES; SUBSET_REFL; VSUM_CASES; GSYM DIFF; SUM_DIFF;
3476 VSUM_DIFF; SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
3477 ASM_SIMP_TAC[SUM_0; VSUM_0; VECTOR_MUL_LZERO; SUM_NEG; VSUM_NEG;
3478 VECTOR_MUL_LNEG; SET_RULE `DISJOINT t u ==> ~(x IN t /\ x IN u)`] THEN
3479 REWRITE_TAC[EMPTY_GSPEC; SUM_CLAUSES; VSUM_CLAUSES] THEN
3480 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
3481 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
3482 UNDISCH_TAC `sum t (a:real^N->real) = &1` THEN
3483 ASM_CASES_TAC `!x:real^N. x IN t ==> a x = &0` THEN
3484 ASM_SIMP_TAC[SUM_EQ_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN
3485 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
3486 MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);;
3488 let AFFINE_INDEPENDENT_SPAN_EQ = prove
3489 (`!s. ~(affine_dependent s) /\ CARD s = dimindex(:N) + 1
3490 ==> affine hull s = (:real^N)`,
3491 MATCH_MP_TAC SET_PROVE_CASES THEN
3492 REWRITE_TAC[CARD_CLAUSES; ARITH_RULE `~(0 = n + 1)`] THEN
3493 SIMP_TAC[IMP_CONJ; AFFINE_INDEPENDENT_IMP_FINITE; MESON[HAS_SIZE]
3494 `FINITE s ==> (CARD s = n <=> s HAS_SIZE n)`] THEN
3495 X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN
3496 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN
3497 SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; IMP_CONJ] THEN
3498 REWRITE_TAC[ARITH_RULE `SUC n = m + 1 <=> n = m`; GSYM UNIV_SUBSET] THEN
3499 REPEAT STRIP_TAC THEN MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
3500 ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV; LE_REFL; independent] THEN
3501 UNDISCH_TAC `~affine_dependent((vec 0:real^N) INSERT s)` THEN
3502 REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_TAC THEN
3503 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3504 ASM_REWRITE_TAC[VECTOR_SUB_RZERO; SET_RULE `{x | x IN s} = s`]);;
3506 let AFFINE_INDEPENDENT_SPAN_GT = prove
3508 ~(affine_dependent s) /\ dimindex(:N) < CARD s
3509 ==> affine hull s = (:real^N)`,
3510 REPEAT STRIP_TAC THEN MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_EQ THEN
3511 ASM_REWRITE_TAC[] THEN
3512 MP_TAC(SPEC `s:real^N->bool` AFFINE_DEPENDENT_BIGGERSET) THEN
3513 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE] THEN ASM_ARITH_TAC);;
3515 let EMPTY_INTERIOR_AFFINE_HULL = prove
3517 FINITE s /\ CARD(s) <= dimindex(:N)
3518 ==> interior(affine hull s) = {}`,
3519 REWRITE_TAC[IMP_CONJ] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3520 REWRITE_TAC[AFFINE_HULL_EMPTY; INTERIOR_EMPTY] THEN
3522 `!x s:real^N->bool n.
3523 ~(x IN s) /\ (x INSERT s) HAS_SIZE n /\ n <= dimindex(:N)
3524 ==> interior(affine hull(x INSERT s)) = {}`
3525 (fun th -> MESON_TAC[th; HAS_SIZE; FINITE_INSERT]) THEN
3526 X_GEN_TAC `orig:real^N` THEN GEOM_ORIGIN_TAC `orig:real^N` THEN
3527 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; SPAN_INSERT_0; HULL_INC] THEN
3528 REWRITE_TAC[HAS_SIZE; FINITE_INSERT; IMP_CONJ] THEN
3529 SIMP_TAC[CARD_CLAUSES] THEN
3530 REPEAT STRIP_TAC THEN MATCH_MP_TAC EMPTY_INTERIOR_LOWDIM THEN
3531 MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN
3532 ASM_SIMP_TAC[DIM_LE_CARD; DIM_SPAN] THEN ASM_ARITH_TAC);;
3534 let EMPTY_INTERIOR_CONVEX_HULL = prove
3536 FINITE s /\ CARD(s) <= dimindex(:N)
3537 ==> interior(convex hull s) = {}`,
3538 REPEAT STRIP_TAC THEN
3539 MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
3540 EXISTS_TAC `interior(affine hull s):real^N->bool` THEN
3541 SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
3542 ASM_SIMP_TAC[EMPTY_INTERIOR_AFFINE_HULL]);;
3544 let AFFINE_DEPENDENT_CHOOSE = prove
3546 ~(affine_dependent s)
3547 ==> (affine_dependent(a INSERT s) <=> ~(a IN s) /\ a IN affine hull s)`,
3548 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THEN
3549 ASM_SIMP_TAC[SET_RULE `a IN s ==> a INSERT s = s`] THEN
3550 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3552 [UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN
3553 ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE; AFFINE_HULL_FINITE;
3554 FINITE_INSERT; IN_ELIM_THM; SUM_CLAUSES; VSUM_CLAUSES] THEN
3555 DISCH_TAC THEN REWRITE_TAC[EXISTS_IN_INSERT] THEN
3556 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` MP_TAC) THEN
3557 ASM_CASES_TAC `(u:real^N->real) a = &0` THEN ASM_REWRITE_TAC[] THENL
3558 [REWRITE_TAC[REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
3560 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
3561 DISCH_THEN(MP_TAC o SPEC `u:real^N->real`) THEN ASM_REWRITE_TAC[];
3562 ONCE_REWRITE_TAC[REAL_ARITH `ua + sa = &0 <=> sa = --ua`;
3563 VECTOR_ARITH `va + sa:real^N = vec 0 <=> sa = --va`] THEN
3564 STRIP_TAC THEN EXISTS_TAC `(\x. --(inv(u a)) * u x):real^N->real` THEN
3565 ASM_SIMP_TAC[SUM_LMUL; GSYM VECTOR_MUL_ASSOC; VSUM_LMUL] THEN
3566 ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM VECTOR_MUL_LNEG] THEN
3567 REWRITE_TAC[REAL_ARITH `--a * --b:real = a * b`] THEN
3568 ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_LID]];
3569 DISCH_TAC THEN REWRITE_TAC[affine_dependent] THEN
3570 EXISTS_TAC `a:real^N` THEN
3571 ASM_SIMP_TAC[IN_INSERT; SET_RULE
3572 `~(a IN s) ==> (a INSERT s) DELETE a = s`]]);;
3574 let AFFINE_INDEPENDENT_INSERT = prove
3576 ~(affine_dependent s) /\ ~(a IN affine hull s)
3577 ==> ~(affine_dependent(a INSERT s))`,
3578 SIMP_TAC[AFFINE_DEPENDENT_CHOOSE]);;
3580 let AFFINE_HULL_EXPLICIT_UNIQUE = prove
3581 (`!s:real^N->bool u u'.
3582 ~(affine_dependent s) /\
3583 sum s u = &1 /\ sum s u' = &1 /\
3584 vsum s (\x. u x % x) = vsum s (\x. u' x % x)
3585 ==> !x. x IN s ==> u x = u' x`,
3586 REPEAT GEN_TAC THEN STRIP_TAC THEN
3587 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3588 FIRST_ASSUM(MP_TAC o MATCH_MP AFFINE_DEPENDENT_EXPLICIT_FINITE) THEN
3589 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
3590 DISCH_THEN(MP_TAC o SPEC `(\x. u x - u' x):real^N->real`) THEN
3591 ASM_SIMP_TAC[VSUM_SUB; SUM_SUB; REAL_SUB_REFL; VECTOR_SUB_RDISTRIB;
3592 VECTOR_SUB_REFL; VECTOR_SUB_EQ; REAL_SUB_0] THEN
3595 let INDEPENDENT_IMP_AFFINE_DEPENDENT_0 = prove
3596 (`!s. independent s ==> ~(affine_dependent(vec 0 INSERT s))`,
3597 REWRITE_TAC[independent; DEPENDENT_AFFINE_DEPENDENT_CASES] THEN
3598 SIMP_TAC[DE_MORGAN_THM; AFFINE_INDEPENDENT_INSERT]);;
3600 let AFFINE_INDEPENDENT_STDBASIS = prove
3601 (`~(affine_dependent
3602 ((vec 0:real^N) INSERT {basis i | 1 <= i /\ i <= dimindex (:N)}))`,
3603 SIMP_TAC[INDEPENDENT_IMP_AFFINE_DEPENDENT_0; INDEPENDENT_STDBASIS]);;
3605 (* ------------------------------------------------------------------------- *)
3606 (* Nonempty affine sets are translates of (unique) subspaces. *)
3607 (* ------------------------------------------------------------------------- *)
3609 let AFFINE_TRANSLATION_SUBSPACE = prove
3611 affine t /\ ~(t = {}) <=> ?a s. subspace s /\ t = IMAGE (\x. a + x) s`,
3612 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
3613 ASM_SIMP_TAC[SUBSPACE_IMP_NONEMPTY; IMAGE_EQ_EMPTY;
3614 AFFINE_TRANSLATION; SUBSPACE_IMP_AFFINE] THEN
3615 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3616 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
3617 ONCE_REWRITE_TAC[TRANSLATION_GALOIS] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
3618 REWRITE_TAC[UNWIND_THM2] THEN MATCH_MP_TAC AFFINE_IMP_SUBSPACE THEN
3619 ASM_REWRITE_TAC[AFFINE_TRANSLATION_EQ; IN_IMAGE] THEN
3620 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);;
3622 let AFFINE_TRANSLATION_UNIQUE_SUBSPACE = prove
3624 affine t /\ ~(t = {}) <=>
3625 ?!s. ?a. subspace s /\ t = IMAGE (\x. a + x) s`,
3626 GEN_TAC THEN REWRITE_TAC[AFFINE_TRANSLATION_SUBSPACE] THEN
3627 MATCH_MP_TAC(MESON[]
3628 `(!a a' s s'. P s a /\ P s' a' ==> s = s')
3629 ==> ((?a s. P s a) <=> (?!s. ?a. P s a))`) THEN
3631 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
3632 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
3633 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[TRANSLATION_GALOIS] THEN
3634 DISCH_THEN SUBST1_TAC THEN CONV_TAC SYM_CONV THEN
3635 REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_ADD_ASSOC] THEN
3636 MATCH_MP_TAC SUBSPACE_TRANSLATION_SELF THEN ASM_REWRITE_TAC[] THEN
3637 ONCE_REWRITE_TAC[VECTOR_ARITH `--a' + a:real^N = --(a' - a)`] THEN
3638 MATCH_MP_TAC SUBSPACE_NEG THEN ASM_REWRITE_TAC[] THEN
3639 UNDISCH_TAC `t = IMAGE (\x:real^N. a' + x) s'` THEN
3640 DISCH_THEN(MP_TAC o AP_TERM `\s. (a':real^N) IN s`) THEN
3641 REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a:real^N = a + x <=> x = vec 0`] THEN
3642 ASM_SIMP_TAC[UNWIND_THM2; SUBSPACE_0] THEN
3643 REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `a':real^N = a + x <=> x = a' - a`] THEN
3644 REWRITE_TAC[UNWIND_THM2]);;
3646 let AFFINE_TRANSLATION_SUBSPACE_EXPLICIT = prove
3647 (`!t:real^N->bool a.
3649 ==> subspace {x - a | x IN t} /\
3650 t = IMAGE (\x. a + x) {x - a | x IN t}`,
3651 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[AFFINE_DIFFS_SUBSPACE] THEN
3652 ASM_REWRITE_TAC[SIMPLE_IMAGE; GSYM IMAGE_o] THEN
3653 REWRITE_TAC[o_DEF; VECTOR_SUB_ADD2; IMAGE_ID]);;
3655 (* ------------------------------------------------------------------------- *)
3656 (* If we take a slice out of a set, we can do it perpendicularly, *)
3657 (* with the normal vector to the slice parallel to the affine hull. *)
3658 (* ------------------------------------------------------------------------- *)
3660 let AFFINE_PARALLEL_SLICE = prove
3663 ==> s INTER {x | a dot x <= b} = {} \/ s SUBSET {x | a dot x <= b} \/
3664 ?a' b'. ~(a' = vec 0) /\
3666 s INTER {x | a' dot x <= b'} = s INTER {x | a dot x <= b} /\
3667 s INTER {x | a' dot x = b'} = s INTER {x | a dot x = b} /\
3668 !w. w IN s ==> (w + a') IN s`,
3669 REPEAT STRIP_TAC THEN
3670 ASM_CASES_TAC `s INTER {x:real^N | a dot x = b} = {}` THENL
3671 [MATCH_MP_TAC(TAUT `~(~p /\ ~q) ==> p \/ q \/ r`) THEN
3672 REPEAT STRIP_TAC THEN SUBGOAL_THEN
3673 `?u v:real^N. u IN s /\ v IN s /\
3674 a dot u <= b /\ ~(a dot v <= b)`
3675 STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3676 SUBGOAL_THEN `(a:real^N) dot u < b` ASSUME_TAC THENL
3677 [ASM_REWRITE_TAC[REAL_LT_LE] THEN ASM SET_TAC[]; ALL_TAC] THEN
3678 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
3679 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
3680 REWRITE_TAC[NOT_IN_EMPTY; IN_INTER; NOT_FORALL_THM; IN_ELIM_THM] THEN
3682 `u + (b - a dot u) / (a dot v - a dot u) % (v - u):real^N` THEN
3683 ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF] THEN
3684 REWRITE_TAC[DOT_RADD; DOT_RMUL; DOT_RSUB] THEN
3685 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
3686 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3687 DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN
3688 REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN POP_ASSUM MP_TAC THEN
3689 GEN_GEOM_ORIGIN_TAC `z:real^N` ["a"; "a'"; "b'"; "w"] THEN
3690 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
3691 REWRITE_TAC[VECTOR_ADD_RID; FORALL_IN_IMAGE] THEN
3692 REWRITE_TAC[DOT_RADD; REAL_ARITH `a + x <= a <=> x <= &0`] THEN
3693 SUBGOAL_THEN `subspace(s:real^N->bool) /\ span s = s`
3694 STRIP_ASSUME_TAC THENL
3695 [ASM_MESON_TAC[AFFINE_IMP_SUBSPACE; SPAN_EQ_SELF]; ALL_TAC] THEN
3696 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`]
3697 ORTHOGONAL_SUBSPACE_DECOMP_EXISTS) THEN
3698 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; orthogonal] THEN
3699 MAP_EVERY X_GEN_TAC [`a':real^N`; `a'':real^N`] THEN
3700 ASM_CASES_TAC `a':real^N = vec 0` THENL
3701 [ASM_REWRITE_TAC[VECTOR_ADD_LID] THEN
3702 ASM_CASES_TAC `a'':real^N = a` THEN ASM_REWRITE_TAC[] THEN
3703 SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_LE_REFL];
3705 STRIP_TAC THEN REPEAT DISJ2_TAC THEN
3706 EXISTS_TAC `a':real^N` THEN ASM_REWRITE_TAC[] THEN
3707 EXISTS_TAC `(a':real^N) dot z` THEN
3708 REPEAT(CONJ_TAC THENL
3709 [MATCH_MP_TAC(SET_RULE
3710 `(!x. x IN s ==> (p x <=> q x))
3711 ==> s INTER {x | p x} = s INTER {x | q x}`) THEN
3712 ASM_SIMP_TAC[DOT_LADD] THEN REAL_ARITH_TAC;
3714 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
3715 EXISTS_TAC `x + a':real^N` THEN
3716 ASM_SIMP_TAC[SUBSPACE_ADD; VECTOR_ADD_ASSOC]]);;
3718 (* ------------------------------------------------------------------------- *)
3719 (* Affine dimension. *)
3720 (* ------------------------------------------------------------------------- *)
3722 let MAXIMAL_AFFINE_INDEPENDENT_SUBSET = prove
3723 (`!s b:real^N->bool.
3724 b SUBSET s /\ ~(affine_dependent b) /\
3725 (!b'. b SUBSET b' /\ b' SUBSET s /\ ~(affine_dependent b') ==> b' = b)
3726 ==> s SUBSET (affine hull b)`,
3727 REPEAT STRIP_TAC THEN
3728 MATCH_MP_TAC(SET_RULE `(!a. a IN t /\ ~(a IN s) ==> F) ==> t SUBSET s`) THEN
3729 X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN
3730 FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N) INSERT b`) THEN
3731 FIRST_ASSUM(MP_TAC o MATCH_MP
3732 (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] HULL_INC)) THEN
3733 ASM_SIMP_TAC[AFFINE_INDEPENDENT_INSERT; INSERT_SUBSET] THEN
3736 let MAXIMAL_AFFINE_INDEPENDENT_SUBSET_AFFINE = prove
3737 (`!s b:real^N->bool.
3738 affine s /\ b SUBSET s /\ ~(affine_dependent b) /\
3739 (!b'. b SUBSET b' /\ b' SUBSET s /\ ~(affine_dependent b') ==> b' = b)
3740 ==> affine hull b = s`,
3741 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3742 [ASM_MESON_TAC[HULL_MONO; HULL_P];
3743 ASM_MESON_TAC[MAXIMAL_AFFINE_INDEPENDENT_SUBSET]]);;
3745 let EXTEND_TO_AFFINE_BASIS = prove
3746 (`!s u:real^N->bool.
3747 ~(affine_dependent s) /\ s SUBSET u
3748 ==> ?t. ~(affine_dependent t) /\ s SUBSET t /\ t SUBSET u /\
3749 affine hull t = affine hull u`,
3750 REPEAT STRIP_TAC THEN
3751 MP_TAC(SPEC `\n. ?t:real^N->bool. ~(affine_dependent t) /\ s SUBSET t /\
3752 t SUBSET u /\ CARD t = n`
3754 DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN REWRITE_TAC[] THEN ANTS_TAC THENL
3755 [ASM_MESON_TAC[SUBSET_REFL; AFFINE_INDEPENDENT_CARD_LE]; ALL_TAC] THEN
3756 DISCH_THEN(X_CHOOSE_THEN `n:num` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
3757 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
3758 ASM_REWRITE_TAC[] THEN
3759 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3760 [ASM_MESON_TAC[HULL_MONO; HULL_P]; ALL_TAC] THEN
3761 MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[AFFINE_AFFINE_HULL] THEN
3762 MATCH_MP_TAC MAXIMAL_AFFINE_INDEPENDENT_SUBSET THEN ASM_REWRITE_TAC[] THEN
3763 X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN
3764 FIRST_X_ASSUM(MP_TAC o SPEC `CARD(c:real^N->bool)`) THEN
3765 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
3766 DISCH_THEN(MP_TAC o SPEC `c:real^N->bool`) THEN
3767 ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN
3768 CONV_TAC SYM_CONV THEN MATCH_MP_TAC CARD_SUBSET_LE THEN
3769 ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]);;
3771 let AFFINE_BASIS_EXISTS = prove
3773 ?b. ~(affine_dependent b) /\ b SUBSET s /\
3774 affine hull b = affine hull s`,
3776 MP_TAC(ISPECL [`{}:real^N->bool`; `s:real^N->bool`]
3777 EXTEND_TO_AFFINE_BASIS) THEN
3778 REWRITE_TAC[AFFINE_INDEPENDENT_EMPTY; EMPTY_SUBSET]);;
3780 let aff_dim = new_definition
3782 @d:int. ?b. affine hull b = affine hull s /\ ~(affine_dependent b) /\
3783 &(CARD b) = d + &1`;;
3786 (`!s. ?b. affine hull b = affine hull s /\
3787 ~(affine_dependent b) /\
3788 aff_dim s = &(CARD b) - &1`,
3790 REWRITE_TAC[aff_dim; INT_ARITH `y:int = x + &1 <=> x = y - &1`] THEN
3791 CONV_TAC SELECT_CONV THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
3792 REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN
3793 MESON_TAC[AFFINE_BASIS_EXISTS]);;
3795 let AFF_DIM_EMPTY = prove
3796 (`aff_dim {} = -- &1`,
3797 REWRITE_TAC[aff_dim; AFFINE_HULL_EMPTY; AFFINE_HULL_EQ_EMPTY] THEN
3798 REWRITE_TAC[UNWIND_THM2; AFFINE_INDEPENDENT_EMPTY; CARD_CLAUSES] THEN
3799 REWRITE_TAC[INT_ARITH `&0 = d + &1 <=> d:int = -- &1`; SELECT_REFL]);;
3801 let AFF_DIM_AFFINE_HULL = prove
3802 (`!s. aff_dim(affine hull s) = aff_dim s`,
3803 REWRITE_TAC[aff_dim; HULL_HULL]);;
3805 let AFF_DIM_TRANSLATION_EQ = prove
3806 (`!a:real^N s. aff_dim (IMAGE (\x. a + x) s) = aff_dim s`,
3807 REWRITE_TAC[aff_dim] THEN GEOM_TRANSLATE_TAC[] THEN
3808 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a /\ b ==> ~c)`] THEN
3809 SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; CARD_IMAGE_INJ;
3810 VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]);;
3812 add_translation_invariants [AFF_DIM_TRANSLATION_EQ];;
3814 let AFFINE_INDEPENDENT_CARD_DIM_DIFFS = prove
3816 ~affine_dependent s /\ a IN s
3817 ==> CARD s = dim {x - a | x IN s} + 1`,
3818 REPEAT STRIP_TAC THEN
3819 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
3820 MATCH_MP_TAC(ARITH_RULE `~(s = 0) /\ v = s - 1 ==> s = v + 1`) THEN
3821 ASM_SIMP_TAC[CARD_EQ_0] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3822 MATCH_MP_TAC DIM_UNIQUE THEN
3823 EXISTS_TAC `{b - a:real^N |b| b IN (s DELETE a)}` THEN REPEAT CONJ_TAC THENL
3825 REWRITE_TAC[SIMPLE_IMAGE; SUBSET; FORALL_IN_IMAGE] THEN
3826 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `x:real^N = a` THENL
3827 [ASM_REWRITE_TAC[VECTOR_SUB_REFL; SPAN_0];
3828 MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]];
3829 UNDISCH_TAC `~affine_dependent(s:real^N->bool)` THEN
3830 REWRITE_TAC[independent; CONTRAPOS_THM] THEN DISCH_TAC THEN
3831 SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL
3832 [ASM SET_TAC[]; ALL_TAC] THEN
3833 MATCH_MP_TAC DEPENDENT_IMP_AFFINE_DEPENDENT THEN
3834 ASM_REWRITE_TAC[IN_DELETE];
3835 REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN
3836 SIMP_TAC[VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN
3837 ASM_SIMP_TAC[HAS_SIZE; FINITE_DELETE; CARD_DELETE]]);;
3839 let AFF_DIM_DIM_AFFINE_DIFFS = prove
3840 (`!a:real^N s. affine s /\ a IN s ==> aff_dim s = &(dim {x - a | x IN s})`,
3841 REPEAT STRIP_TAC THEN
3842 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN
3843 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` MP_TAC) THEN
3844 ASM_CASES_TAC `b:real^N->bool = {}` THENL
3845 [ASM_MESON_TAC[AFFINE_HULL_EQ_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN
3847 ASM_REWRITE_TAC[INT_EQ_SUB_RADD; INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN
3848 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3849 DISCH_THEN(X_CHOOSE_TAC `c:real^N`) THEN MATCH_MP_TAC EQ_TRANS THEN
3850 EXISTS_TAC `dim {x - c:real^N | x IN b} + 1` THEN CONJ_TAC THENL
3851 [MATCH_MP_TAC AFFINE_INDEPENDENT_CARD_DIM_DIFFS THEN ASM_REWRITE_TAC[];
3853 MATCH_MP_TAC EQ_TRANS THEN
3854 EXISTS_TAC `dim {x - c:real^N | x IN affine hull b} + 1` THEN CONJ_TAC THENL
3855 [ASM_SIMP_TAC[DIFFS_AFFINE_HULL_SPAN; DIM_SPAN]; ALL_TAC] THEN
3856 ASM_REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
3857 SUBGOAL_THEN `affine hull s:real^N->bool = s` SUBST1_TAC THENL
3858 [ASM_MESON_TAC[AFFINE_HULL_EQ]; ALL_TAC] THEN
3859 SUBGOAL_THEN `(c:real^N) IN s` ASSUME_TAC THENL
3860 [ASM_MESON_TAC[AFFINE_HULL_EQ; HULL_INC]; ALL_TAC] THEN
3861 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3862 REWRITE_TAC[IN_ELIM_THM] THEN
3863 SIMP_TAC[VECTOR_ARITH `x - c:real^N = y - a <=> y = x + &1 % (a - c)`] THEN
3864 ASM_MESON_TAC[IN_AFFINE_ADD_MUL_DIFF]);;
3866 let AFF_DIM_DIM_0 = prove
3867 (`!s:real^N->bool. vec 0 IN affine hull s ==> aff_dim s = &(dim s)`,
3868 REPEAT STRIP_TAC THEN
3869 MP_TAC(ISPECL [`vec 0:real^N`; `affine hull s:real^N->bool`]
3870 AFF_DIM_DIM_AFFINE_DIFFS) THEN
3871 ASM_REWRITE_TAC[AFFINE_AFFINE_HULL; VECTOR_SUB_RZERO] THEN
3872 REWRITE_TAC[AFF_DIM_AFFINE_HULL; SET_RULE `{x | x IN s} = s`] THEN
3873 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; DIM_SPAN]);;
3875 let AFF_DIM_DIM_SUBSPACE = prove
3876 (`!s:real^N->bool. subspace s ==> aff_dim s = &(dim s)`,
3877 MESON_TAC[AFF_DIM_DIM_0; SUBSPACE_0; HULL_INC]);;
3879 let AFF_DIM_LINEAR_IMAGE_LE = prove
3880 (`!f:real^M->real^N s. linear f ==> aff_dim(IMAGE f s) <= aff_dim s`,
3881 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
3882 ASM_SIMP_TAC[AFFINE_HULL_LINEAR_IMAGE] THEN
3883 MP_TAC(ISPEC `s:real^M->bool` AFFINE_AFFINE_HULL) THEN
3884 SPEC_TAC(`affine hull s:real^M->bool`,`s:real^M->bool`) THEN
3885 GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN
3886 ASM_REWRITE_TAC[IMAGE_CLAUSES; AFF_DIM_EMPTY; INT_LE_REFL] THEN
3887 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3888 DISCH_THEN(X_CHOOSE_TAC `a:real^M`) THEN
3889 SUBGOAL_THEN `dim {x - f(a) |x| x IN IMAGE (f:real^M->real^N) s} <=
3890 dim {x - a | x IN s}`
3892 [REWRITE_TAC[SET_RULE `{f x | x IN IMAGE g s} = {f (g x) | x IN s}`] THEN
3893 ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN
3894 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN
3895 MATCH_MP_TAC DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[];
3896 MATCH_MP_TAC EQ_IMP THEN REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN
3897 BINOP_TAC THEN CONV_TAC SYM_CONV THEN
3898 MATCH_MP_TAC AFF_DIM_DIM_AFFINE_DIFFS THEN
3899 ASM_SIMP_TAC[AFFINE_LINEAR_IMAGE; FUN_IN_IMAGE]]);;
3901 let AFF_DIM_INJECTIVE_LINEAR_IMAGE = prove
3902 (`!f:real^M->real^N s.
3903 linear f /\ (!x y. f x = f y ==> x = y)
3904 ==> aff_dim(IMAGE f s) = aff_dim s`,
3905 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM INT_LE_ANTISYM] THEN
3906 CONJ_TAC THENL [ASM_MESON_TAC[AFF_DIM_LINEAR_IMAGE_LE]; ALL_TAC] THEN
3907 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
3908 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
3909 DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
3910 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC
3911 `aff_dim(IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) s))` THEN
3913 [ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; INT_LE_REFL];
3914 MATCH_MP_TAC AFF_DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[]]);;
3916 add_linear_invariants [AFF_DIM_INJECTIVE_LINEAR_IMAGE];;
3918 let AFF_DIM_AFFINE_INDEPENDENT = prove
3920 ~(affine_dependent b) ==> aff_dim b = &(CARD b) - &1`,
3921 GEN_TAC THEN ASM_CASES_TAC `b:real^N->bool = {}` THENL
3922 [ASM_REWRITE_TAC[CARD_CLAUSES; AFF_DIM_EMPTY] THEN INT_ARITH_TAC;
3924 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
3925 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN DISCH_TAC THEN
3926 MP_TAC(ISPECL [`b:real^N->bool`; `a:real^N`]
3927 AFFINE_INDEPENDENT_CARD_DIM_DIFFS) THEN
3928 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
3929 REWRITE_TAC[GSYM INT_OF_NUM_ADD; INT_ARITH `(a + b) - b:int = a`] THEN
3930 MP_TAC(ISPECL [`a:real^N`; `affine hull b:real^N->bool`]
3931 AFF_DIM_DIM_AFFINE_DIFFS) THEN
3932 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC; AFF_DIM_AFFINE_HULL] THEN
3933 DISCH_THEN(K ALL_TAC) THEN AP_TERM_TAC THEN
3934 ASM_MESON_TAC[DIFFS_AFFINE_HULL_SPAN; DIM_SPAN]);;
3936 let AFF_DIM_UNIQUE = prove
3937 (`!s b:real^N->bool.
3938 affine hull b = affine hull s /\ ~(affine_dependent b)
3939 ==> aff_dim s = &(CARD b) - &1`,
3940 MESON_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_AFFINE_INDEPENDENT]);;
3942 let AFF_DIM_SING = prove
3943 (`!a:real^N. aff_dim {a} = &0`,
3944 GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
3945 EXISTS_TAC `&(CARD {a:real^N}) - &1:int` THEN CONJ_TAC THENL
3946 [MATCH_MP_TAC AFF_DIM_AFFINE_INDEPENDENT THEN
3947 REWRITE_TAC[AFFINE_INDEPENDENT_1];
3948 SIMP_TAC[CARD_CLAUSES; FINITE_RULES; ARITH; NOT_IN_EMPTY; INT_SUB_REFL]]);;
3950 let AFF_DIM_LE_CARD = prove
3951 (`!s:real^N->bool. FINITE s ==> aff_dim s <= &(CARD s) - &1`,
3952 MATCH_MP_TAC SET_PROVE_CASES THEN
3953 SIMP_TAC[AFF_DIM_EMPTY; CARD_CLAUSES] THEN CONV_TAC INT_REDUCE_CONV THEN
3954 GEOM_ORIGIN_TAC `a:real^N` THEN
3955 SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC] THEN
3956 SIMP_TAC[CARD_IMAGE_INJ; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
3957 SIMP_TAC[DIM_INSERT_0; INT_LE_SUB_LADD; CARD_CLAUSES; FINITE_INSERT] THEN
3958 REWRITE_TAC[INT_OF_NUM_ADD; INT_OF_NUM_LE; ADD1; LE_ADD_RCANCEL] THEN
3959 SIMP_TAC[DIM_LE_CARD]);;
3961 let AFF_DIM_GE = prove
3962 (`!s:real^N->bool. -- &1 <= aff_dim s`,
3963 GEN_TAC THEN MP_TAC(ISPEC `s:real^N->bool` AFF_DIM) THEN
3964 STRIP_TAC THEN ASM_REWRITE_TAC[INT_LE_SUB_LADD; INT_ADD_LINV; INT_POS]);;
3966 let AFF_DIM_SUBSET = prove
3967 (`!s t:real^N->bool. s SUBSET t ==> aff_dim s <= aff_dim t`,
3968 MATCH_MP_TAC SET_PROVE_CASES THEN REWRITE_TAC[AFF_DIM_GE; AFF_DIM_EMPTY] THEN
3969 GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN
3970 SUBGOAL_THEN `(vec 0:real^N) IN t` ASSUME_TAC THENL
3971 [ASM SET_TAC[]; ALL_TAC] THEN
3972 ASM_SIMP_TAC[AFF_DIM_DIM_0; IN_INSERT; HULL_INC; INT_OF_NUM_LE; DIM_SUBSET]);;
3974 let AFF_DIM_LE_DIM = prove
3975 (`!s:real^N->bool. aff_dim s <= &(dim s)`,
3976 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
3977 ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE; SUBSPACE_SPAN] THEN
3978 MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[SPAN_INC]);;
3980 let AFF_DIM_CONVEX_HULL = prove
3981 (`!s:real^N->bool. aff_dim(convex hull s) = aff_dim s`,
3982 GEN_TAC THEN MATCH_MP_TAC(INT_ARITH
3983 `!c:int. c = a /\ a <= b /\ b <= c ==> b = a`) THEN
3984 EXISTS_TAC `aff_dim(affine hull s:real^N->bool)` THEN
3985 SIMP_TAC[AFF_DIM_AFFINE_HULL; AFF_DIM_SUBSET; HULL_SUBSET;
3986 CONVEX_HULL_SUBSET_AFFINE_HULL]);;
3988 let AFF_DIM_CLOSURE = prove
3989 (`!s:real^N->bool. aff_dim(closure s) = aff_dim s`,
3990 GEN_TAC THEN MATCH_MP_TAC(INT_ARITH
3991 `!h. h = s /\ s <= c /\ c <= h ==> c:int = s`) THEN
3992 EXISTS_TAC `aff_dim(affine hull s:real^N->bool)` THEN
3993 REPEAT CONJ_TAC THENL
3994 [REWRITE_TAC[AFF_DIM_AFFINE_HULL];
3995 MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET];
3996 MATCH_MP_TAC AFF_DIM_SUBSET THEN
3997 MATCH_MP_TAC CLOSURE_MINIMAL THEN
3998 REWRITE_TAC[CLOSED_AFFINE_HULL; HULL_SUBSET]]);;
4000 let AFF_DIM_2 = prove
4001 (`!a b:real^N. aff_dim {a,b} = if a = b then &0 else &1`,
4002 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
4003 [ASM_REWRITE_TAC[INSERT_AC; AFF_DIM_SING]; ALL_TAC] THEN
4004 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `&(CARD {a:real^N,b}) - &1:int` THEN
4005 ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; AFFINE_INDEPENDENT_2] THEN
4006 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES; IN_INSERT; NOT_IN_EMPTY] THEN
4007 CONV_TAC NUM_REDUCE_CONV THEN INT_ARITH_TAC);;
4009 let AFF_DIM_EQ_MINUS1 = prove
4010 (`!s:real^N->bool. aff_dim s = -- &1 <=> s = {}`,
4011 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_EMPTY] THEN
4012 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
4013 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN
4014 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
4015 MATCH_MP_TAC(INT_ARITH `&0:int <= n ==> ~(n = -- &1)`) THEN
4016 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim {a:real^N}` THEN
4017 ASM_SIMP_TAC[AFF_DIM_SUBSET; SING_SUBSET] THEN
4018 REWRITE_TAC[AFF_DIM_SING; INT_LE_REFL]);;
4020 let AFF_DIM_POS_LE = prove
4021 (`!s:real^N->bool. &0 <= aff_dim s <=> ~(s = {})`,
4022 GEN_TAC THEN REWRITE_TAC[GSYM AFF_DIM_EQ_MINUS1] THEN
4023 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_GE) THEN INT_ARITH_TAC);;
4025 let AFF_DIM_EQ_0 = prove
4026 (`!s:real^N->bool. aff_dim s = &0 <=> ?a. s = {a}`,
4027 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[AFF_DIM_SING; LEFT_IMP_EXISTS_THM] THEN
4028 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN
4029 CONV_TAC INT_REDUCE_CONV THEN DISCH_TAC THEN
4030 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
4031 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
4032 MATCH_MP_TAC(SET_RULE
4033 `(!b. ~(b = a) /\ {a,b} SUBSET s ==> F) ==> a IN s ==> s = {a}`) THEN
4034 X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN
4035 FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN
4036 MP_TAC(ISPECL [`a:real^N`; `b:real^N`] AFF_DIM_2) THEN
4037 ASM_SIMP_TAC[] THEN INT_ARITH_TAC);;
4039 let CONNECTED_IMP_PERFECT_AFF_DIM = prove
4041 connected s /\ ~(aff_dim s = &0) /\ x IN s ==> x limit_point_of s`,
4042 REWRITE_TAC[AFF_DIM_EQ_0; CONNECTED_IMP_PERFECT]);;
4044 let AFF_DIM_UNIV = prove
4045 (`aff_dim(:real^N) = &(dimindex(:N))`,
4046 SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_UNIV; DIM_UNIV]);;
4048 let AFF_DIM_EQ_AFFINE_HULL = prove
4049 (`!s t:real^N->bool.
4050 s SUBSET t /\ aff_dim t <= aff_dim s
4051 ==> affine hull s = affine hull t`,
4052 MATCH_MP_TAC SET_PROVE_CASES THEN
4053 SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; AFF_DIM_GE;
4054 INT_ARITH `a:int <= x ==> (x <= a <=> x = a)`] THEN
4055 X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN
4056 SIMP_TAC[INSERT_SUBSET; IMP_CONJ; AFF_DIM_DIM_0; IN_INSERT; DIM_EQ_SPAN;
4057 HULL_INC; AFFINE_HULL_EQ_SPAN; INT_OF_NUM_LE]);;
4059 let AFF_DIM_SUMS_INTER = prove
4060 (`!s t:real^N->bool.
4061 affine s /\ affine t /\ ~(s INTER t = {})
4062 ==> aff_dim {x + y | x IN s /\ y IN t} =
4063 (aff_dim s + aff_dim t) - aff_dim(s INTER t)`,
4064 REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN
4065 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN
4066 GEN_REWRITE_TAC BINDER_CONV [SWAP_FORALL_THM] THEN
4067 GEN_REWRITE_TAC I [SWAP_FORALL_THM] THEN X_GEN_TAC `a:real^N` THEN
4068 GEOM_ORIGIN_TAC `a:real^N` THEN
4069 REWRITE_TAC[VECTOR_ARITH `(a + x) + (a + y):real^N = &2 % a + (x + y)`] THEN
4070 ONCE_REWRITE_TAC[SET_RULE `{a + x + y:real^N | x IN s /\ y IN t} =
4071 IMAGE (\x. a + x) {x + y | x IN s /\ y IN t}`] THEN
4072 REWRITE_TAC[AFF_DIM_TRANSLATION_EQ; IN_INTER] THEN
4073 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN STRIP_TAC THEN
4075 SUBGOAL_THEN `(vec 0:real^N) IN {x + y | x IN s /\ y IN t}` ASSUME_TAC THENL
4076 [REWRITE_TAC[IN_ELIM_THM] THEN REPEAT(EXISTS_TAC `vec 0:real^N`) THEN
4077 ASM_REWRITE_TAC[VECTOR_ADD_LID];
4079 ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; IN_INTER] THEN
4080 REWRITE_TAC[INT_EQ_SUB_LADD; INT_OF_NUM_ADD; INT_OF_NUM_EQ] THEN
4081 MATCH_MP_TAC DIM_SUMS_INTER THEN ASM_SIMP_TAC[AFFINE_IMP_SUBSPACE]);;
4083 let AFF_DIM_PSUBSET = prove
4084 (`!s t. (affine hull s) PSUBSET (affine hull t) ==> aff_dim s < aff_dim t`,
4085 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4086 SIMP_TAC[PSUBSET; AFF_DIM_SUBSET; INT_LT_LE] THEN
4087 MESON_TAC[INT_EQ_IMP_LE; AFF_DIM_EQ_AFFINE_HULL; HULL_HULL]);;
4089 let AFF_DIM_EQ_FULL = prove
4090 (`!s. aff_dim s = &(dimindex(:N)) <=> affine hull s = (:real^N)`,
4091 GEN_TAC THEN EQ_TAC THENL
4092 [DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM AFFINE_HULL_UNIV] THEN
4093 MATCH_MP_TAC AFF_DIM_EQ_AFFINE_HULL THEN
4094 ASM_REWRITE_TAC[SUBSET_UNIV; AFF_DIM_UNIV; INT_LE_REFL];
4095 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4096 SIMP_TAC[AFF_DIM_UNIV]]);;
4098 let AFF_DIM_LE_UNIV = prove
4099 (`!s:real^N->bool. aff_dim s <= &(dimindex(:N))`,
4100 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN
4101 MATCH_MP_TAC AFF_DIM_SUBSET THEN REWRITE_TAC[SUBSET_UNIV]);;
4103 let AFFINE_INDEPENDENT_IFF_CARD = prove
4105 ~affine_dependent s <=> FINITE s /\ aff_dim s = &(CARD s) - &1`,
4106 GEN_TAC THEN EQ_TAC THEN
4107 SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; AFFINE_INDEPENDENT_IMP_FINITE] THEN
4108 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4109 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN DISCH_TAC THEN
4110 X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC
4111 (ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN
4112 MATCH_MP_TAC(ARITH_RULE `!b:int. a <= b - &1 /\ b < s ==> ~(a = s - &1)`) THEN
4113 EXISTS_TAC `&(CARD(b:real^N->bool)):int` THEN CONJ_TAC THENL
4114 [ASM_MESON_TAC[AFF_DIM_LE_CARD; FINITE_SUBSET; AFF_DIM_AFFINE_HULL];
4115 REWRITE_TAC[INT_OF_NUM_LT] THEN MATCH_MP_TAC CARD_PSUBSET THEN
4116 ASM_REWRITE_TAC[PSUBSET] THEN ASM_MESON_TAC[]]);;
4118 let AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR = prove
4119 (`!s t:real^N->bool.
4120 convex s /\ ~(s INTER interior t = {})
4121 ==> affine hull (s INTER t) = affine hull s`,
4122 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM;
4123 LEFT_IMP_EXISTS_THM] THEN
4124 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`; `a:real^N`] THEN
4125 GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN
4126 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
4127 SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
4128 SIMP_TAC[SUBSET_HULL; AFFINE_AFFINE_HULL] THEN
4129 FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SIMP_RULE[SUBSET] INTERIOR_SUBSET)) THEN
4130 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INTER] THEN
4131 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN
4132 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_CBALL_0] THEN
4133 X_GEN_TAC `e:real` THEN STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_UNIV] THEN
4134 X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
4135 ASM_SIMP_TAC[SPAN_SUPERSET; IN_INTER] THEN DISCH_TAC THEN
4136 ABBREV_TAC `k = min (&1 / &2) (e / norm(x:real^N))` THEN
4137 SUBGOAL_THEN `&0 < k /\ k < &1` STRIP_ASSUME_TAC THENL
4138 [EXPAND_TAC "k" THEN
4139 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; NORM_POS_LT; REAL_MIN_LT] THEN
4140 CONV_TAC REAL_RAT_REDUCE_CONV;
4142 SUBGOAL_THEN `x:real^N = inv k % k % x` SUBST1_TAC THENL
4143 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID;
4146 MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN
4147 REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL
4148 [ONCE_REWRITE_TAC[VECTOR_ARITH
4149 `k % x:real^N = (&1 - k) % vec 0 + k % x`] THEN
4150 MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE];
4151 FIRST_X_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "k" THEN
4152 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
4153 ASM_REAL_ARITH_TAC]);;
4155 let AFFINE_HULL_CONVEX_INTER_OPEN = prove
4156 (`!s t:real^N->bool.
4157 convex s /\ open t /\ ~(s INTER t = {})
4158 ==> affine hull (s INTER t) = affine hull s`,
4159 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR; INTERIOR_OPEN]);;
4161 let AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR = prove
4162 (`!s t:real^N->bool.
4163 affine s /\ ~(s INTER interior t = {})
4164 ==> affine hull (s INTER t) = s`,
4165 SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR; AFFINE_IMP_CONVEX;
4168 let AFFINE_HULL_AFFINE_INTER_OPEN = prove
4169 (`!s t:real^N->bool.
4170 affine s /\ open t /\ ~(s INTER t = {})
4171 ==> affine hull (s INTER t) = s`,
4172 SIMP_TAC[AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR; INTERIOR_OPEN]);;
4174 let CONVEX_AND_AFFINE_INTER_OPEN = prove
4175 (`!s t u:real^N->bool.
4176 convex s /\ affine t /\ open u /\
4177 s INTER u = t INTER u /\ ~(s INTER u = {})
4178 ==> affine hull s = t`,
4179 REPEAT STRIP_TAC THEN
4180 MATCH_MP_TAC(MESON[] `!u v. x = u /\ u = v /\ v = y ==> x = y`) THEN
4181 MAP_EVERY EXISTS_TAC
4182 [`affine hull (s INTER u:real^N->bool)`;
4183 `affine hull t:real^N->bool`] THEN
4184 REPEAT CONJ_TAC THENL
4185 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN
4187 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN
4188 ASM_SIMP_TAC[AFFINE_IMP_CONVEX] THEN ASM SET_TAC[];
4189 ASM_REWRITE_TAC[AFFINE_HULL_EQ]]);;
4191 let AFFINE_HULL_CONVEX_INTER_OPEN_IN = prove
4192 (`!s t:real^N->bool.
4193 convex s /\ open_in (subtopology euclidean (affine hull s)) t /\
4195 ==> affine hull (s INTER t) = affine hull s`,
4196 REPEAT STRIP_TAC THEN
4197 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4198 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4199 ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER t INTER u = s INTER u`;
4201 MATCH_MP_TAC AFFINE_HULL_CONVEX_INTER_OPEN THEN ASM SET_TAC[]);;
4203 let AFFINE_HULL_AFFINE_INTER_OPEN_IN = prove
4204 (`!s t:real^N->bool.
4205 affine s /\ open_in (subtopology euclidean s) t /\ ~(s INTER t = {})
4206 ==> affine hull (s INTER t) = s`,
4207 REPEAT STRIP_TAC THEN
4208 MP_TAC(ISPECL [`affine hull s:real^N->bool`; `t:real^N->bool`]
4209 AFFINE_HULL_CONVEX_INTER_OPEN_IN) THEN
4210 ASM_SIMP_TAC[HULL_HULL; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL; HULL_P]);;
4212 let AFFINE_HULL_OPEN_IN = prove
4213 (`!s t:real^N->bool.
4214 open_in (subtopology euclidean (affine hull t)) s /\ ~(s = {})
4215 ==> affine hull s = affine hull t`,
4216 REPEAT STRIP_TAC THEN
4217 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4218 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4219 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC AFFINE_HULL_AFFINE_INTER_OPEN THEN
4220 REWRITE_TAC[AFFINE_AFFINE_HULL] THEN ASM SET_TAC[]);;
4222 let AFFINE_HULL_OPEN = prove
4223 (`!s. open s /\ ~(s = {}) ==> affine hull s = (:real^N)`,
4224 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4225 SUBST1_TAC(SET_RULE `s = (:real^N) INTER s`) THEN
4226 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN; CONVEX_UNIV] THEN
4227 REWRITE_TAC[AFFINE_HULL_UNIV]);;
4229 let AFFINE_HULL_NONEMPTY_INTERIOR = prove
4230 (`!s. ~(interior s = {}) ==> affine hull s = (:real^N)`,
4231 REPEAT STRIP_TAC THEN
4232 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
4233 EXISTS_TAC `affine hull (interior s:real^N->bool)` THEN
4234 SIMP_TAC[HULL_MONO; INTERIOR_SUBSET] THEN
4235 ASM_SIMP_TAC[AFFINE_HULL_OPEN; OPEN_INTERIOR]);;
4237 let AFF_DIM_OPEN = prove
4238 (`!s:real^N->bool. open s /\ ~(s = {}) ==> aff_dim s = &(dimindex(:N))`,
4239 SIMP_TAC[AFF_DIM_EQ_FULL; AFFINE_HULL_OPEN]);;
4241 let AFF_DIM_NONEMPTY_INTERIOR = prove
4242 (`!s:real^N->bool. ~(interior s = {}) ==> aff_dim s = &(dimindex(:N))`,
4243 SIMP_TAC[AFF_DIM_EQ_FULL; AFFINE_HULL_NONEMPTY_INTERIOR]);;
4245 let SPAN_OPEN = prove
4246 (`!s. open s /\ ~(s = {}) ==> span s = (:real^N)`,
4247 REPEAT STRIP_TAC THEN
4248 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
4249 EXISTS_TAC `affine hull s:real^N->bool` THEN
4250 ASM_SIMP_TAC[AFFINE_HULL_OPEN; AFFINE_HULL_SUBSET_SPAN]);;
4252 let DIM_OPEN = prove
4253 (`!s:real^N->bool. open s /\ ~(s = {}) ==> dim s = dimindex(:N)`,
4254 SIMP_TAC[DIM_EQ_FULL; SPAN_OPEN]);;
4256 let AFF_DIM_INSERT = prove
4258 aff_dim (a INSERT s) =
4259 if a IN affine hull s then aff_dim s else aff_dim s + &1`,
4260 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC SET_PROVE_CASES THEN
4261 SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_SING; AFFINE_HULL_EMPTY; NOT_IN_EMPTY] THEN
4262 CONV_TAC INT_REDUCE_CONV THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN
4263 MAP_EVERY X_GEN_TAC [`b:real^N`; `s:real^N->bool`; `a:real^N`] THEN
4264 GEOM_ORIGIN_TAC `b:real^N` THEN
4265 SIMP_TAC[AFFINE_HULL_EQ_SPAN; AFF_DIM_DIM_0; HULL_INC; IN_INSERT] THEN
4266 MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`] THEN
4267 DISCH_THEN(K ALL_TAC) THEN
4268 SPEC_TAC(`(vec 0:real^N) INSERT s`,`s:real^N->bool`) THEN
4269 SIMP_TAC[DIM_INSERT; INT_OF_NUM_ADD] THEN MESON_TAC[]);;
4271 let AFFINE_BOUNDED_EQ_TRIVIAL = prove
4273 affine s ==> (bounded s <=> s = {} \/ ?a. s = {a})`,
4274 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
4275 ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN
4276 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
4277 DISCH_THEN(X_CHOOSE_THEN `b:real^N` MP_TAC) THEN
4278 GEOM_ORIGIN_TAC `b:real^N` THEN SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN
4279 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SUBSPACE_BOUNDED_EQ_TRIVIAL] THEN
4280 FIRST_ASSUM(MP_TAC o MATCH_MP SUBSPACE_0) THEN SET_TAC[]);;
4282 let AFFINE_BOUNDED_EQ_LOWDIM = prove
4284 affine s ==> (bounded s <=> aff_dim s <= &0)`,
4285 SIMP_TAC[AFF_DIM_GE; INT_ARITH
4286 `--(&1):int <= x ==> (x <= &0 <=> x = --(&1) \/ x = &0)`] THEN
4287 SIMP_TAC[AFF_DIM_EQ_0; AFF_DIM_EQ_MINUS1; AFFINE_BOUNDED_EQ_TRIVIAL]);;
4289 let COLLINEAR_AFF_DIM = prove
4290 (`!s:real^N->bool. collinear s <=> aff_dim s <= &1`,
4291 GEN_TAC THEN EQ_TAC THENL
4292 [REWRITE_TAC[COLLINEAR_AFFINE_HULL; LEFT_IMP_EXISTS_THM] THEN
4293 MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN STRIP_TAC THEN
4294 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim{u:real^N,v}` THEN
4296 [ASM_MESON_TAC[AFF_DIM_SUBSET; AFF_DIM_AFFINE_HULL];
4297 MATCH_MP_TAC INT_LE_TRANS THEN
4298 EXISTS_TAC `&(CARD{u:real^N,v}) - &1:int` THEN
4299 SIMP_TAC[AFF_DIM_LE_CARD; FINITE_INSERT; FINITE_EMPTY] THEN
4300 REWRITE_TAC[INT_ARITH `x - &1:int <= &1 <=> x <= &2`; INT_OF_NUM_LE] THEN
4301 SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN ARITH_TAC];
4302 ONCE_REWRITE_TAC[GSYM COLLINEAR_AFFINE_HULL_COLLINEAR;
4303 GSYM AFF_DIM_AFFINE_HULL] THEN
4304 MP_TAC(ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN
4305 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` (STRIP_ASSUME_TAC o GSYM)) THEN
4306 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I
4307 [AFFINE_INDEPENDENT_IFF_CARD]) THEN
4308 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4309 ASM_REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR;
4310 AFF_DIM_AFFINE_HULL] THEN
4311 REWRITE_TAC[INT_ARITH `x - &1:int <= &1 <=> x <= &2`; INT_OF_NUM_LE] THEN
4312 ASM_SIMP_TAC[COLLINEAR_SMALL]]);;
4314 let HOMEOMORPHIC_AFFINE_SETS = prove
4315 (`!s:real^M->bool t:real^N->bool.
4316 affine s /\ affine t /\ aff_dim s = aff_dim t ==> s homeomorphic t`,
4318 ASM_CASES_TAC `t:real^N->bool = {}` THEN
4319 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN
4320 POP_ASSUM MP_TAC THEN
4321 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN
4322 ASM_CASES_TAC `s:real^M->bool = {}` THEN
4323 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1; HOMEOMORPHIC_EMPTY] THEN
4324 POP_ASSUM MP_TAC THEN REWRITE_TAC
4325 [GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN
4326 MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`] THEN
4327 GEOM_ORIGIN_TAC `a:real^M` THEN GEOM_ORIGIN_TAC `b:real^N` THEN
4328 SIMP_TAC[AFFINE_EQ_SUBSPACE; AFF_DIM_DIM_0; HULL_INC; INT_OF_NUM_EQ] THEN
4329 MESON_TAC[HOMEOMORPHIC_SUBSPACES]);;
4331 let AFF_DIM_OPEN_IN = prove
4332 (`!s t:real^N->bool.
4333 ~(s = {}) /\ open_in (subtopology euclidean t) s /\ affine t
4334 ==> aff_dim s = aff_dim t`,
4336 REWRITE_TAC[IMP_CONJ; GSYM MEMBER_NOT_EMPTY; LEFT_IMP_EXISTS_THM] THEN
4337 X_GEN_TAC `a:real^N` THEN GEOM_ORIGIN_TAC `a:real^N` THEN
4338 REPEAT GEN_TAC THEN STRIP_TAC THEN DISCH_TAC THEN
4339 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
4340 SUBGOAL_THEN `(vec 0:real^N) IN t` ASSUME_TAC THENL
4341 [ASM SET_TAC[]; ALL_TAC] THEN
4342 ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; AFFINE_EQ_SUBSPACE] THEN
4343 DISCH_TAC THEN AP_TERM_TAC THEN
4344 ASM_SIMP_TAC[GSYM LE_ANTISYM; DIM_SUBSET] THEN
4345 SUBGOAL_THEN `?e. &0 < e /\ cball(vec 0:real^N,e) INTER t SUBSET s`
4347 [FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4348 FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o
4349 GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
4351 REWRITE_TAC[SUBSET; IN_INTER; IN_CBALL_0] THEN
4352 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)] THEN
4353 FIRST_ASSUM(MP_TAC o MATCH_MP ORTHONORMAL_BASIS_SUBSPACE) THEN
4354 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4355 MP_TAC(ISPECL [`s:real^N->bool`; `IMAGE (\x:real^N. e % x) b`]
4356 INDEPENDENT_CARD_LE_DIM) THEN
4357 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
4358 ASM_SIMP_TAC[CARD_IMAGE_INJ; VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ] THEN
4359 ANTS_TAC THENL [REWRITE_TAC[SUBSET]; MESON_TAC[]] THEN CONJ_TAC THENL
4360 [REWRITE_TAC[FORALL_IN_IMAGE] THEN REPEAT STRIP_TAC THEN
4361 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NORM_MUL] THEN
4362 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC SUBSPACE_MUL] THEN
4364 MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE THEN
4365 ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; LINEAR_SCALING]]);;
4367 let DIM_OPEN_IN = prove
4368 (`!s t:real^N->bool.
4369 ~(s = {}) /\ open_in (subtopology euclidean t) s /\ subspace t
4371 REPEAT STRIP_TAC THEN
4372 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
4373 ASM_SIMP_TAC[GSYM LE_ANTISYM; DIM_SUBSET] THEN
4374 REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN
4375 MATCH_MP_TAC INT_LE_TRANS THEN EXISTS_TAC `aff_dim(s:real^N->bool)` THEN
4376 REWRITE_TAC[AFF_DIM_LE_DIM] THEN ASM_SIMP_TAC[GSYM AFF_DIM_DIM_SUBSPACE] THEN
4377 MATCH_MP_TAC INT_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN
4378 MATCH_MP_TAC AFF_DIM_OPEN_IN THEN ASM_SIMP_TAC[SUBSPACE_IMP_AFFINE]);;
4380 let AFF_DIM_CONVEX_INTER_NONEMPTY_INTERIOR = prove
4381 (`!s t:real^N->bool.
4382 convex s /\ ~(s INTER interior t = {})
4383 ==> aff_dim(s INTER t) = aff_dim s`,
4384 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4385 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR] THEN
4386 REWRITE_TAC[AFF_DIM_AFFINE_HULL]);;
4388 let AFF_DIM_CONVEX_INTER_OPEN = prove
4389 (`!s t:real^N->bool.
4390 convex s /\ open t /\ ~(s INTER t = {})
4391 ==> aff_dim(s INTER t) = aff_dim s`,
4392 ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4393 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_INTER_OPEN] THEN
4394 REWRITE_TAC[AFF_DIM_AFFINE_HULL]);;
4396 let AFFINE_HULL_HALFSPACE_LT = prove
4397 (`!a b. affine hull {x | a dot x < b} =
4398 if a = vec 0 /\ b <= &0 then {} else (:real^N)`,
4399 REPEAT GEN_TAC THEN COND_CASES_TAC THEN
4400 ASM_SIMP_TAC[AFFINE_HULL_EQ_EMPTY; HALFSPACE_EQ_EMPTY_LT;
4401 AFFINE_HULL_OPEN; OPEN_HALFSPACE_LT]);;
4403 let AFFINE_HULL_HALFSPACE_LE = prove
4404 (`!a b. affine hull {x | a dot x <= b} =
4405 if a = vec 0 /\ b < &0 then {} else (:real^N)`,
4406 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL
4407 [ASM_SIMP_TAC[DOT_LZERO; SET_RULE `{x | p} = if p then UNIV else {}`] THEN
4408 COND_CASES_TAC THEN ASM_SIMP_TAC[AFFINE_HULL_EMPTY; AFFINE_HULL_UNIV] THEN
4409 COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
4410 ASM_SIMP_TAC[GSYM CLOSURE_HALFSPACE_LT; AFFINE_HULL_CLOSURE] THEN
4411 ASM_REWRITE_TAC[AFFINE_HULL_HALFSPACE_LT]]);;
4413 let AFFINE_HULL_HALFSPACE_GT = prove
4414 (`!a b. affine hull {x | a dot x > b} =
4415 if a = vec 0 /\ b >= &0 then {} else (:real^N)`,
4416 REPEAT GEN_TAC THEN COND_CASES_TAC THEN
4417 ASM_SIMP_TAC[AFFINE_HULL_EQ_EMPTY; HALFSPACE_EQ_EMPTY_GT;
4418 AFFINE_HULL_OPEN; OPEN_HALFSPACE_GT]);;
4420 let AFFINE_HULL_HALFSPACE_GE = prove
4421 (`!a b. affine hull {x | a dot x >= b} =
4422 if a = vec 0 /\ b > &0 then {} else (:real^N)`,
4424 MP_TAC(ISPECL [`--a:real^N`; `--b:real`] AFFINE_HULL_HALFSPACE_LE) THEN
4425 SIMP_TAC[real_ge; DOT_LNEG; REAL_LE_NEG2; VECTOR_NEG_EQ_0] THEN
4426 REWRITE_TAC[REAL_ARITH `--b < &0 <=> b > &0`]);;
4428 let AFF_DIM_HALFSPACE_LT = prove
4430 aff_dim {x | a dot x < b} =
4431 if a = vec 0 /\ b <= &0 then --(&1) else &(dimindex(:N))`,
4432 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4433 SIMP_TAC[AFFINE_HULL_HALFSPACE_LT] THEN COND_CASES_TAC THEN
4434 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4436 let AFF_DIM_HALFSPACE_LE = prove
4438 aff_dim {x | a dot x <= b} =
4439 if a = vec 0 /\ b < &0 then --(&1) else &(dimindex(:N))`,
4440 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4441 SIMP_TAC[AFFINE_HULL_HALFSPACE_LE] THEN COND_CASES_TAC THEN
4442 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4444 let AFF_DIM_HALFSPACE_GT = prove
4446 aff_dim {x | a dot x > b} =
4447 if a = vec 0 /\ b >= &0 then --(&1) else &(dimindex(:N))`,
4448 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4449 SIMP_TAC[AFFINE_HULL_HALFSPACE_GT] THEN COND_CASES_TAC THEN
4450 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4452 let AFF_DIM_HALFSPACE_GE = prove
4454 aff_dim {x | a dot x >= b} =
4455 if a = vec 0 /\ b > &0 then --(&1) else &(dimindex(:N))`,
4456 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
4457 SIMP_TAC[AFFINE_HULL_HALFSPACE_GE] THEN COND_CASES_TAC THEN
4458 ASM_REWRITE_TAC[AFF_DIM_EMPTY; AFF_DIM_UNIV]);;
4460 let CHOOSE_AFFINE_SUBSET = prove
4461 (`!s:real^N->bool d.
4462 affine s /\ --(&1) <= d /\ d <= aff_dim s
4463 ==> ?t. affine t /\ t SUBSET s /\ aff_dim t = d`,
4464 REPEAT GEN_TAC THEN ASM_CASES_TAC `d:int = --(&1)` THENL
4465 [STRIP_TAC THEN EXISTS_TAC `{}:real^N->bool` THEN
4466 ASM_REWRITE_TAC[EMPTY_SUBSET; AFFINE_EMPTY; AFF_DIM_EMPTY];
4467 ASM_SIMP_TAC[INT_ARITH
4468 `~(d:int = --(&1)) ==> (--(&1) <= d <=> &0 <= d)`] THEN
4469 POP_ASSUM(K ALL_TAC)] THEN
4470 ASM_CASES_TAC `s:real^N->bool = {}` THENL
4471 [ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN INT_ARITH_TAC;
4472 POP_ASSUM MP_TAC] THEN
4473 GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN
4474 DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN
4475 GEOM_ORIGIN_TAC `a:real^N` THEN
4476 SIMP_TAC[IMP_CONJ; AFF_DIM_DIM_SUBSPACE; AFFINE_EQ_SUBSPACE] THEN
4477 REPEAT GEN_TAC THEN DISCH_TAC THEN DISCH_TAC THEN
4478 REWRITE_TAC[GSYM INT_OF_NUM_EXISTS] THEN
4479 DISCH_THEN(X_CHOOSE_THEN `n:num` SUBST1_TAC) THEN
4480 REWRITE_TAC[INT_OF_NUM_LE] THEN DISCH_TAC THEN
4481 MP_TAC(ISPECL [`s:real^N->bool`; `n:num`]
4482 CHOOSE_SUBSPACE_OF_SUBSPACE) THEN
4483 ASM_SIMP_TAC[SPAN_OF_SUBSPACE] THEN
4484 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real^N->bool` THEN
4485 ASM_SIMP_TAC[AFF_DIM_DIM_SUBSPACE; SUBSPACE_IMP_AFFINE]);;
4487 (* ------------------------------------------------------------------------- *)
4488 (* Existence of a rigid transform between congruent sets. *)
4489 (* ------------------------------------------------------------------------- *)
4491 let RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS = prove
4492 (`!x:A->real^N y:A->real^N s.
4493 (!i j. i IN s /\ j IN s ==> dist(x i,x j) = dist(y i,y j))
4494 ==> ?a f. orthogonal_transformation f /\
4495 !i. i IN s ==> y i = a + f(x i)`,
4497 (`!x:(real^N)^M y:(real^N)^M.
4498 (!i j. 1 <= i /\ i <= dimindex(:M) /\
4499 1 <= j /\ j <= dimindex(:M)
4500 ==> dist(x$i,x$j) = dist(y$i,y$j))
4501 ==> ?a f. orthogonal_transformation f /\
4502 !i. 1 <= i /\ i <= dimindex(:M)
4503 ==> y$i = a + f(x$i)`,
4504 REPEAT STRIP_TAC THEN
4505 ABBREV_TAC `(X:real^M^N) = lambda i j. (x:real^N^M)$j$i - x$1$i` THEN
4506 ABBREV_TAC `(Y:real^M^N) = lambda i j. (y:real^N^M)$j$i - y$1$i` THEN
4507 SUBGOAL_THEN `transp(X:real^M^N) ** X = transp(Y:real^M^N) ** Y`
4509 [REWRITE_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN] THEN
4510 MAP_EVERY EXPAND_TAC ["X"; "Y"] THEN
4511 SIMP_TAC[CART_EQ; column; LAMBDA_BETA; dot] THEN
4512 REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT; GSYM dot] THEN
4513 REWRITE_TAC[DOT_NORM_SUB; VECTOR_ARITH
4514 `(x - a) - (y - a):real^N = x - y`] THEN
4515 ASM_SIMP_TAC[GSYM dist; DIMINDEX_GE_1; LE_REFL];
4518 `?M:real^N^N. orthogonal_matrix M /\ (Y:real^M^N) = M ** (X:real^M^N)`
4519 (CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL
4521 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [CART_EQ] THEN
4522 MAP_EVERY EXPAND_TAC ["X"; "Y"] THEN
4523 SIMP_TAC[LAMBDA_BETA; matrix_mul] THEN
4524 REWRITE_TAC[REAL_ARITH `x - y:real = z <=> x = y + z`] THEN STRIP_TAC THEN
4525 EXISTS_TAC `(y:real^N^M)$1 - (M:real^N^N) ** (x:real^N^M)$1` THEN
4526 EXISTS_TAC `\x:real^N. (M:real^N^N) ** x` THEN
4527 ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX;
4528 MATRIX_OF_MATRIX_VECTOR_MUL; MATRIX_VECTOR_MUL_LINEAR] THEN
4529 SIMP_TAC[CART_EQ; matrix_vector_mul; LAMBDA_BETA;
4530 VECTOR_ADD_COMPONENT] THEN
4531 ASM_SIMP_TAC[REAL_SUB_LDISTRIB; SUM_SUB_NUMSEG] THEN
4532 REWRITE_TAC[VECTOR_SUB_COMPONENT; REAL_ARITH
4533 `a + y - b:real = a - z + y <=> z = b`] THEN
4534 SIMP_TAC[LAMBDA_BETA]] THEN
4535 MP_TAC(ISPEC `transp(X:real^M^N) ** X`
4536 SYMMETRIC_MATRIX_DIAGONALIZABLE_EXPLICIT) THEN
4537 REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_TRANSP; LEFT_IMP_EXISTS_THM] THEN
4538 MAP_EVERY X_GEN_TAC [`P:real^M^M`; `d:num->real`] THEN
4539 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4540 DISCH_THEN(fun th -> MP_TAC th THEN ASM_REWRITE_TAC[] THEN MP_TAC th) THEN
4541 REWRITE_TAC[MATRIX_MUL_ASSOC; GSYM MATRIX_TRANSP_MUL] THEN
4542 REWRITE_TAC[GSYM MATRIX_MUL_ASSOC; LEFT_IMP_EXISTS_THM] THEN
4543 REWRITE_TAC[IMP_IMP] THEN
4544 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [CART_EQ] THEN
4545 SIMP_TAC[MATRIX_MUL_LTRANSP_DOT_COLUMN; LAMBDA_BETA] THEN STRIP_TAC THEN
4546 MP_TAC(ISPECL [`\i. column i ((X:real^M^N) ** (P:real^M^M))`;
4547 `\i. column i ((Y:real^M^N) ** (P:real^M^M))`;
4549 ORTHOGONAL_TRANSFORMATION_BETWEEN_ORTHOGONAL_SETS) THEN
4550 REWRITE_TAC[IN_NUMSEG] THEN ANTS_TAC THENL
4551 [ASM_SIMP_TAC[pairwise; IN_NUMSEG; NORM_EQ; orthogonal]; ALL_TAC] THEN
4552 DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
4553 EXISTS_TAC `matrix(f:real^N->real^N)` THEN CONJ_TAC THENL
4554 [ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX]; ALL_TAC] THEN
4556 `!M:real^M^N. M = M ** (P:real^M^M) ** transp P`
4557 (fun th -> GEN_REWRITE_TAC BINOP_CONV [th])
4559 [ASM_MESON_TAC[orthogonal_matrix; MATRIX_MUL_RID];
4560 REWRITE_TAC[MATRIX_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC] THEN
4561 REWRITE_TAC[GSYM MATRIX_MUL_ASSOC] THEN
4562 ASM_SIMP_TAC[MATRIX_EQUAL_COLUMNS] THEN
4563 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
4564 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [orthogonal_transformation]) THEN
4565 DISCH_THEN(ASSUME_TAC o GSYM o MATCH_MP MATRIX_WORKS o CONJUNCT1) THEN
4566 ASM_REWRITE_TAC[] THEN
4567 SIMP_TAC[CART_EQ; matrix_vector_mul; column; LAMBDA_BETA] THEN
4568 X_GEN_TAC `j:num` THEN STRIP_TAC THEN
4569 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [matrix_mul] THEN
4570 ASM_SIMP_TAC[LAMBDA_BETA]) in
4571 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:A->bool = {}` THENL
4572 [REPEAT STRIP_TAC THEN
4573 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `\x:real^N. x`] THEN
4574 ASM_REWRITE_TAC[NOT_IN_EMPTY; ORTHOGONAL_TRANSFORMATION_ID];
4575 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
4576 DISCH_THEN(X_CHOOSE_TAC `m:A`) THEN DISCH_TAC] THEN
4578 `?r. IMAGE r (1..dimindex(:(N,1)finite_sum)) SUBSET s /\
4579 affine hull (IMAGE (y o r) (1..dimindex(:(N,1)finite_sum))) =
4580 affine hull (IMAGE (y:A->real^N) s)`
4582 [REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
4583 SIMP_TAC[IMAGE_o; TAUT `p /\ q <=> ~(p ==> ~q)`;
4584 HULL_MONO; IMAGE_SUBSET] THEN REWRITE_TAC[NOT_IMP] THEN
4585 MP_TAC(ISPEC `IMAGE (y:A->real^N) s` AFFINE_BASIS_EXISTS) THEN
4586 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4587 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [AFFINE_INDEPENDENT_IFF_CARD]) THEN
4589 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN
4590 DISCH_THEN(X_CHOOSE_THEN `f:num->real^N` STRIP_ASSUME_TAC) THEN
4591 SUBGOAL_THEN `CARD(b:real^N->bool) <= dimindex(:(N,1)finite_sum)`
4593 [REWRITE_TAC[GSYM INT_OF_NUM_LE] THEN
4594 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH
4595 `a:int = c - &1 ==> a + &1 <= n ==> c <= n`)) THEN
4596 REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1; GSYM INT_OF_NUM_ADD] THEN
4597 REWRITE_TAC[INT_LE_RADD; AFF_DIM_LE_UNIV];
4599 UNDISCH_TAC `b SUBSET IMAGE (y:A->real^N) s` THEN
4600 ONCE_ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
4601 REWRITE_TAC[IN_IMAGE] THEN
4602 GEN_REWRITE_TAC (LAND_CONV o DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
4603 REWRITE_TAC[SKOLEM_THM; IN_NUMSEG] THEN
4604 DISCH_THEN(X_CHOOSE_THEN `r:num->A` STRIP_ASSUME_TAC) THEN
4605 EXISTS_TAC `\i. if i <= CARD(b:real^N->bool) then r i else (m:A)` THEN
4606 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
4607 UNDISCH_THEN `affine hull b:real^N->bool = affine hull IMAGE y (s:A->bool)`
4608 (SUBST1_TAC o SYM) THEN
4609 REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC HULL_MONO THEN
4610 ONCE_ASM_REWRITE_TAC[] THEN
4611 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG] THEN
4612 X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM IMAGE_o] THEN
4613 REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN EXISTS_TAC `i:num` THEN
4614 ASM_REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[LE_TRANS];
4615 REWRITE_TAC[SUBSET; IN_NUMSEG; FORALL_IN_IMAGE] THEN
4616 STRIP_TAC THEN MP_TAC(ISPECL
4617 [`(lambda i. x(r i:A)):real^N^(N,1)finite_sum`;
4618 `(lambda i. y(r i:A)):real^N^(N,1)finite_sum`] lemma) THEN
4619 ASM_SIMP_TAC[LAMBDA_BETA] THEN
4620 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
4621 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN
4622 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4623 X_GEN_TAC `k:A` THEN STRIP_TAC THEN
4626 affine hull IMAGE (y o (r:num->A)) (1..dimindex(:(N,1)finite_sum))
4627 ==> dist(z,y k) = dist(z,a + (f:real^N->real^N)(x k))`
4629 [MATCH_MP_TAC SAME_DISTANCES_TO_AFFINE_HULL THEN
4630 REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_NUMSEG] THEN
4631 X_GEN_TAC `j:num` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
4632 EXISTS_TAC `dist(x(r(j:num)),(x:A->real^N) k)` THEN CONJ_TAC THENL
4633 [CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[];
4634 REWRITE_TAC[dist] THEN ASM_SIMP_TAC[NORM_ARITH
4635 `(a + x) - (a + y):real^N = x - y`] THEN
4636 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION; LINEAR_SUB]];
4637 ASM_SIMP_TAC[NORM_ARITH
4638 `a:real^N = b <=> dist(a:real^N,a) = dist(a,b)`] THEN
4639 DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HULL_INC THEN
4640 REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN ASM_MESON_TAC[]]]);;
4642 let RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS_STRONG = prove
4643 (`!x:A->real^N y:A->real^N s t.
4644 t SUBSET s /\ affine hull (IMAGE y t) = affine hull (IMAGE y s) /\
4645 (!i j. i IN s /\ j IN t ==> dist(x i,x j) = dist(y i,y j))
4646 ==> ?a f. orthogonal_transformation f /\
4647 !i. i IN s ==> y i = a + f(x i)`,
4648 REPEAT STRIP_TAC THEN
4649 MP_TAC(ISPECL [`x:A->real^N`; `y:A->real^N`; `t:A->bool`]
4650 RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS) THEN
4651 ANTS_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
4652 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
4653 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^N->real^N` THEN
4654 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:A` THEN DISCH_TAC THEN
4656 `!z. z IN affine hull (IMAGE (y:A->real^N) t)
4657 ==> dist(z,y i) = dist(z,a + (f:real^N->real^N)(x i))`
4659 [MATCH_MP_TAC SAME_DISTANCES_TO_AFFINE_HULL THEN
4660 REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_NUMSEG] THEN
4661 X_GEN_TAC `j:A` THEN STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
4662 EXISTS_TAC `dist(a + f(x(j:A):real^N):real^N,a + f(x i))` THEN
4663 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
4664 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = dist(x,y)`] THEN
4665 ASM_MESON_TAC[ORTHOGONAL_TRANSFORMATION_ISOMETRY; DIST_SYM];
4666 ASM_SIMP_TAC[NORM_ARITH
4667 `a:real^N = b <=> dist(a:real^N,a) = dist(a,b)`] THEN
4668 DISCH_THEN MATCH_MP_TAC THEN MATCH_MP_TAC HULL_INC THEN
4669 REWRITE_TAC[IN_IMAGE] THEN ASM_MESON_TAC[]]);;
4671 let RIGID_TRANSFORMATION_BETWEEN_3 = prove
4672 (`!a b c a' b' c':real^N.
4673 dist(a,b) = dist(a',b') /\
4674 dist(b,c) = dist(b',c') /\
4675 dist(c,a) = dist(c',a')
4676 ==> ?k f. orthogonal_transformation f /\
4677 a' = k + f a /\ b' = k + f b /\ c' = k + f c`,
4678 REPEAT STRIP_TAC THEN
4680 [`FST:real^N#real^N->real^N`; `SND:real^N#real^N->real^N`;
4681 `{(a:real^N,a':real^N), (b,b'), (c,c')}`]
4682 RIGID_TRANSFORMATION_BETWEEN_CONGRUENT_SETS) THEN
4683 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_INSERT] THEN
4684 REWRITE_TAC[NOT_IN_EMPTY; IMP_IMP] THEN DISCH_THEN MATCH_MP_TAC THEN
4685 ASM_MESON_TAC[DIST_REFL; DIST_SYM]);;
4687 let RIGID_TRANSFORMATION_BETWEEN_2 = prove
4688 (`!a b a' b':real^N.
4689 dist(a,b) = dist(a',b')
4690 ==> ?k f. orthogonal_transformation f /\
4691 a' = k + f a /\ b' = k + f b`,
4692 REPEAT STRIP_TAC THEN
4693 MP_TAC(ISPECL [`a:real^N`; `b:real^N`; `a:real^N`;
4694 `a':real^N`; `b':real^N`; `a':real^N`]
4695 RIGID_TRANSFORMATION_BETWEEN_3) THEN
4696 ASM_MESON_TAC[DIST_EQ_0; DIST_SYM]);;
4698 (* ------------------------------------------------------------------------- *)
4699 (* Caratheodory's theorem. *)
4700 (* ------------------------------------------------------------------------- *)
4702 let CONVEX_HULL_CARATHEODORY_AFF_DIM = prove
4703 (`!p. convex hull p =
4704 {y:real^N | ?s u. FINITE s /\ s SUBSET p /\
4705 &(CARD s) <= aff_dim p + &1 /\
4706 (!x. x IN s ==> &0 <= u x) /\
4707 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
4708 GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_EXPLICIT] THEN
4709 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN
4710 EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
4711 MATCH_MP_TAC(TAUT `!q. (p ==> q) /\ (q ==> r) ==> (p ==> r)`) THEN
4712 EXISTS_TAC `?n s u. CARD s = n /\
4713 FINITE s /\ s SUBSET p /\
4714 (!x. x IN s ==> &0 <= u x) /\
4715 sum s u = &1 /\ vsum s (\v. u v % v) = (y:real^N)` THEN
4716 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
4717 GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
4718 DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN
4719 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4720 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
4721 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4722 GEN_REWRITE_TAC I [GSYM INT_NOT_LT] THEN DISCH_TAC THEN
4723 FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN REWRITE_TAC[NOT_IMP] THEN
4725 [MATCH_MP_TAC(ARITH_RULE `~(n = 0) ==> n - 1 < n`) THEN
4726 DISCH_THEN SUBST_ALL_TAC THEN
4727 UNDISCH_TAC `aff_dim(p:real^N->bool) + &1 < &0` THEN
4728 REWRITE_TAC[INT_ARITH `p + &1:int < &0 <=> ~(-- &1 <= p)`] THEN
4729 REWRITE_TAC[AFF_DIM_GE];
4731 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_AFFINE_INDEPENDENT) THEN
4732 ASM_REWRITE_TAC[] THEN
4733 SUBGOAL_THEN `~(aff_dim(s:real^N->bool) = &n - &1)` ASSUME_TAC THENL
4734 [FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN
4736 UNDISCH_TAC `aff_dim(p:real^N->bool) + &1 < &n` THEN
4739 ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE] THEN
4740 DISCH_THEN(X_CHOOSE_THEN `w:real^N->real` STRIP_ASSUME_TAC) THEN
4742 `?t. (!v:real^N. v IN s ==> u(v) + t * w(v) >= &0) /\
4743 ?a. a IN s /\ u(a) + t * w(a) = &0`
4744 STRIP_ASSUME_TAC THENL
4746 `i = IMAGE (\v. u(v) / --w(v)) {v:real^N | v IN s /\ w v < &0}` THEN
4747 EXISTS_TAC `inf i` THEN MP_TAC(SPEC `i:real->bool` INF_FINITE) THEN
4749 [EXPAND_TAC "i" THEN
4750 ASM_SIMP_TAC[FINITE_IMAGE; FINITE_RESTRICT; IMAGE_EQ_EMPTY] THEN
4751 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
4752 MP_TAC(ISPECL [`w:real^N->real`; `s:real^N->bool`] SUM_ZERO_EXISTS) THEN
4753 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
4755 ABBREV_TAC `t = inf i` THEN
4756 EXPAND_TAC "i" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
4757 REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN
4758 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `a:real^N`
4759 STRIP_ASSUME_TAC) MP_TAC) THEN
4760 SIMP_TAC[REAL_LE_RDIV_EQ; REAL_ARITH `x < &0 ==> &0 < --x`; real_ge] THEN
4761 REWRITE_TAC[REAL_ARITH `t * --w <= u <=> &0 <= u + t * w`] THEN
4762 STRIP_TAC THEN CONJ_TAC THENL
4763 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
4764 DISJ_CASES_TAC(REAL_ARITH `(w:real^N->real) x < &0 \/ &0 <= w x`) THEN
4765 ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_ADD THEN ASM_SIMP_TAC[] THEN
4766 MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[] THEN
4767 MATCH_MP_TAC REAL_LE_DIV THEN ASM_SIMP_TAC[] THEN
4768 MATCH_MP_TAC(REAL_ARITH `w < &0 ==> &0 <= --w`) THEN ASM_REWRITE_TAC[];
4769 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN
4770 UNDISCH_TAC `w(a:real^N) < &0` THEN CONV_TAC REAL_FIELD];
4772 MAP_EVERY EXISTS_TAC
4773 [`s DELETE (a:real^N)`; `(\v. u(v) + t * w(v)):real^N->real`] THEN
4774 ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; CARD_DELETE; FINITE_DELETE] THEN
4775 ASM_SIMP_TAC[SUM_ADD; VECTOR_ADD_RDISTRIB; VSUM_ADD] THEN
4776 ASM_SIMP_TAC[GSYM VECTOR_MUL_ASSOC; SUM_LMUL; VSUM_LMUL] THEN
4777 REPEAT CONJ_TAC THENL
4778 [ASM SET_TAC[]; ASM SET_TAC[real_ge]; REAL_ARITH_TAC; VECTOR_ARITH_TAC]);;
4780 let CARATHEODORY_AFF_DIM = prove
4781 (`!p. convex hull p =
4782 {x:real^N | ?s. FINITE s /\ s SUBSET p /\
4783 &(CARD s) <= aff_dim p + &1 /\
4784 x IN convex hull s}`,
4785 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL
4786 [GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
4787 [CONVEX_HULL_CARATHEODORY_AFF_DIM] THEN
4788 REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
4789 ASM_MESON_TAC[HULL_SUBSET; CONVEX_EXPLICIT; CONVEX_CONVEX_HULL];
4790 MESON_TAC[SUBSET; HULL_MONO]]);;
4792 let CONVEX_HULL_CARATHEODORY = prove
4793 (`!p. convex hull p =
4794 {y:real^N | ?s u. FINITE s /\ s SUBSET p /\
4795 CARD(s) <= dimindex(:N) + 1 /\
4796 (!x. x IN s ==> &0 <= u x) /\
4797 sum s u = &1 /\ vsum s (\v. u v % v) = y}`,
4799 GEN_TAC THEN REWRITE_TAC[EXTENSION] THEN X_GEN_TAC `y:real^N` THEN
4801 [REWRITE_TAC[CONVEX_HULL_CARATHEODORY_AFF_DIM; IN_ELIM_THM] THEN
4802 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
4804 ASM_REWRITE_TAC[GSYM INT_OF_NUM_LE; GSYM INT_OF_NUM_ADD] THEN
4805 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (INT_ARITH
4806 `a:int <= x + &1 ==> x <= y ==> a <= y + &1`)) THEN
4807 REWRITE_TAC[AFF_DIM_LE_UNIV];
4808 REWRITE_TAC[CONVEX_HULL_EXPLICIT; IN_ELIM_THM] THEN MESON_TAC[]]);;
4810 let CARATHEODORY = prove
4811 (`!p. convex hull p =
4812 {x:real^N | ?s. FINITE s /\ s SUBSET p /\
4813 CARD(s) <= dimindex(:N) + 1 /\
4814 x IN convex hull s}`,
4815 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL
4816 [GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
4817 [CONVEX_HULL_CARATHEODORY] THEN
4818 REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
4819 ASM_MESON_TAC[HULL_SUBSET; CONVEX_EXPLICIT; CONVEX_CONVEX_HULL];
4820 MESON_TAC[SUBSET; HULL_MONO]]);;
4822 (* ------------------------------------------------------------------------- *)
4823 (* Some results on decomposing convex hulls, e.g. simplicial subdivision. *)
4824 (* ------------------------------------------------------------------------- *)
4826 let AFFINE_HULL_INTER,CONVEX_HULL_INTER = (CONJ_PAIR o prove)
4827 (`(!s t:real^N->bool.
4828 ~(affine_dependent(s UNION t))
4829 ==> affine hull s INTER affine hull t = affine hull (s INTER t)) /\
4831 ~(affine_dependent (s UNION t))
4832 ==> convex hull s INTER convex hull t = convex hull (s INTER t))`,
4834 (REPEAT STRIP_TAC THEN
4835 FIRST_ASSUM(MP_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
4836 REWRITE_TAC[FINITE_UNION] THEN STRIP_TAC THEN
4837 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET_INTER] THEN
4838 SIMP_TAC[HULL_MONO; INTER_SUBSET] THEN
4839 REWRITE_TAC[SUBSET; AFFINE_HULL_FINITE; CONVEX_HULL_FINITE;
4840 IN_ELIM_THM; IN_INTER] THEN
4841 X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2
4842 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)
4843 (X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC)) THEN
4844 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
4845 [AFFINE_DEPENDENT_EXPLICIT]) THEN
4846 REWRITE_TAC[NOT_EXISTS_THM] THEN
4847 DISCH_THEN(MP_TAC o SPEC `(s UNION t):real^N->bool`) THEN
4848 ASM_REWRITE_TAC[SUBSET_REFL] THEN
4849 DISCH_THEN(MP_TAC o SPEC
4850 `\x:real^N. (if x IN s then u x else &0) -
4851 (if x IN t then v x else &0)`) THEN
4852 ASM_SIMP_TAC[SUM_SUB; FINITE_UNION; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN
4854 `(if p then a else b) % x = (if p then a % x else b % x)`] THEN
4855 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO; FINITE_UNION] THEN
4856 ASM_REWRITE_TAC[SUM_0; VSUM_0;
4857 SET_RULE `{x | x IN (s UNION t) /\ x IN s} = s`;
4858 SET_RULE `{x | x IN (s UNION t) /\ x IN t} = t`] THEN
4859 MATCH_MP_TAC(TAUT `a /\ c /\ (~b ==> d) ==> ~(a /\ b /\ c) ==> d`) THEN
4860 REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC; ALL_TAC] THEN
4861 DISCH_TAC THEN EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[] THEN
4862 CONJ_TAC THEN MATCH_MP_TAC EQ_TRANS THENL
4863 [EXISTS_TAC `sum s (u:real^N->real)`;
4864 EXISTS_TAC `vsum s (\x. (u:real^N->real) x % x)`] THEN
4865 (CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM ACCEPT_TAC]) THEN
4866 CONV_TAC SYM_CONV THENL
4867 [MATCH_MP_TAC SUM_EQ_SUPERSET; MATCH_MP_TAC VSUM_EQ_SUPERSET] THEN
4868 ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET; IN_INTER] THEN
4869 X_GEN_TAC `x:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4870 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISCH_TAC THEN
4871 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
4872 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
4873 ASM_REWRITE_TAC[REAL_SUB_RZERO] THEN ASM SET_TAC[]));;
4875 let AFFINE_HULL_INTERS = prove
4876 (`!s:(real^N->bool)->bool.
4877 ~(affine_dependent(UNIONS s))
4878 ==> affine hull (INTERS s) = INTERS {affine hull t | t IN s}`,
4879 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
4880 MP_TAC th THEN MP_TAC(MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE th)) THEN
4881 SPEC_TAC(`s:(real^N->bool)->bool`,`s:(real^N->bool)->bool`) THEN
4882 REWRITE_TAC[FINITE_UNIONS; IMP_CONJ] THEN
4883 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
4884 REWRITE_TAC[UNIONS_0; INTERS_0; UNIONS_INSERT; INTERS_INSERT;
4885 SET_RULE `{f x | x IN {}} = {}`; AFFINE_HULL_UNIV] THEN
4886 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN
4887 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
4888 REWRITE_TAC[FORALL_IN_INSERT] THEN
4889 DISCH_THEN(fun th -> STRIP_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
4890 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
4891 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
4892 REWRITE_TAC[CONTRAPOS_THM] THEN
4893 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
4896 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL
4897 [ASM_REWRITE_TAC[INTERS_0; INTER_UNIV; IN_SING] THEN
4898 REWRITE_TAC[SET_RULE `{f x | x = a} = {f a}`; INTERS_1];
4900 W(MP_TAC o PART_MATCH (rhs o rand) AFFINE_HULL_INTER o lhand o snd) THEN
4902 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
4903 REWRITE_TAC[CONTRAPOS_THM] THEN
4904 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
4905 UNDISCH_TAC `~(f:(real^N->bool)->bool = {})` THEN SET_TAC[];
4906 DISCH_THEN(SUBST1_TAC o SYM)] THEN
4907 REWRITE_TAC[SET_RULE
4908 `{f x | x IN (a INSERT s)} = (f a) INSERT {f x | x IN s}`] THEN
4909 ASM_REWRITE_TAC[INTERS_INSERT]);;
4911 let CONVEX_HULL_INTERS = prove
4912 (`!s:(real^N->bool)->bool.
4913 ~(affine_dependent(UNIONS s))
4914 ==> convex hull (INTERS s) = INTERS {convex hull t | t IN s}`,
4915 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
4916 MP_TAC th THEN MP_TAC(MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE th)) THEN
4917 SPEC_TAC(`s:(real^N->bool)->bool`,`s:(real^N->bool)->bool`) THEN
4918 REWRITE_TAC[FINITE_UNIONS; IMP_CONJ] THEN
4919 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
4920 REWRITE_TAC[UNIONS_0; INTERS_0; UNIONS_INSERT; INTERS_INSERT;
4921 SET_RULE `{f x | x IN {}} = {}`; CONVEX_HULL_UNIV] THEN
4922 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `f:(real^N->bool)->bool`] THEN
4923 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
4924 REWRITE_TAC[FORALL_IN_INSERT] THEN
4925 DISCH_THEN(fun th -> STRIP_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
4926 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
4927 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
4928 REWRITE_TAC[CONTRAPOS_THM] THEN
4929 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
4932 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL
4933 [ASM_REWRITE_TAC[INTERS_0; INTER_UNIV; IN_SING] THEN
4934 REWRITE_TAC[SET_RULE `{f x | x = a} = {f a}`; INTERS_1];
4936 W(MP_TAC o PART_MATCH (rhs o rand) CONVEX_HULL_INTER o lhand o snd) THEN
4938 [UNDISCH_TAC `~affine_dependent((s UNION UNIONS f):real^N->bool)` THEN
4939 REWRITE_TAC[CONTRAPOS_THM] THEN
4940 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO) THEN
4941 UNDISCH_TAC `~(f:(real^N->bool)->bool = {})` THEN SET_TAC[];
4942 DISCH_THEN(SUBST1_TAC o SYM)] THEN
4943 REWRITE_TAC[SET_RULE
4944 `{f x | x IN (a INSERT s)} = (f a) INSERT {f x | x IN s}`] THEN
4945 ASM_REWRITE_TAC[INTERS_INSERT]);;
4947 let IN_CONVEX_HULL_EXCHANGE = prove
4949 a IN convex hull s /\ x IN convex hull s
4950 ==> ?b. b IN s /\ x IN convex hull (a INSERT (s DELETE b))`,
4951 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL
4952 [EXISTS_TAC `a:real^N` THEN ASM_SIMP_TAC[INSERT_DELETE]; ALL_TAC] THEN
4953 ASM_CASES_TAC `FINITE(s:real^N->bool) /\ CARD s <= dimindex(:N) + 1` THENL
4955 UNDISCH_TAC `(x:real^N) IN convex hull s` THEN
4956 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [CARATHEODORY] THEN
4957 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
4958 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
4959 ASM_CASES_TAC `t:real^N->bool = s` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
4960 SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b IN t)` MP_TAC THENL
4961 [ASM SET_TAC[]; ALL_TAC] THEN
4962 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN
4963 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4964 UNDISCH_TAC `(x:real^N) IN convex hull t` THEN
4965 SPEC_TAC(`x:real^N`,`x:real^N`) THEN REWRITE_TAC[GSYM SUBSET] THEN
4966 MATCH_MP_TAC HULL_MONO THEN ASM SET_TAC[]] THEN
4967 MP_TAC(ASSUME `(a:real^N) IN convex hull s`) THEN
4968 MP_TAC(ASSUME `(x:real^N) IN convex hull s`) THEN
4969 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
4970 X_GEN_TAC `v:real^N->real` THEN STRIP_TAC THEN
4971 X_GEN_TAC `u:real^N->real` THEN STRIP_TAC THEN
4972 ASM_CASES_TAC `?b. b IN s /\ (v:real^N->real) b = &0` THENL
4973 [FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_EXISTS) THEN
4974 X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
4975 EXISTS_TAC `\x:real^N. if x = a then &0 else v x` THEN
4976 ASM_SIMP_TAC[FORALL_IN_INSERT; REAL_LE_REFL] THEN
4977 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_DELETE] THEN
4978 ASM_REWRITE_TAC[IN_DELETE] THEN
4979 ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; COND_ID] THEN
4980 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
4981 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; REAL_LE_REFL; COND_ID] THEN
4982 REWRITE_TAC[VECTOR_MUL_LZERO; SUM_0; VSUM_0] THEN
4983 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ ~(x = a)} = s`] THEN
4984 CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC];
4986 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
4987 REWRITE_TAC[TAUT `~(a /\ b) <=> a ==> ~b`] THEN DISCH_TAC THEN
4988 MP_TAC(ISPEC `IMAGE (\b. (u:real^N->real) b / v b) s` SUP_FINITE) THEN
4989 ASM_CASES_TAC `s:real^N->bool = {}` THENL
4990 [ASM_MESON_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY]; ALL_TAC] THEN
4991 ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
4992 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4993 REWRITE_TAC[IN_IMAGE] THEN MATCH_MP_TAC MONO_EXISTS THEN
4994 X_GEN_TAC `b:real^N` THEN
4995 DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
4996 ASM_REWRITE_TAC[] THEN
4997 SUBGOAL_THEN `!b. b IN s ==> &0 < (v:real^N->real) b` ASSUME_TAC THENL
4998 [ASM_SIMP_TAC[REAL_LT_LE]; ALL_TAC] THEN
4999 SUBGOAL_THEN `&0 < (u:real^N->real) b /\ &0 < v b` STRIP_ASSUME_TAC THENL
5000 [ASM_SIMP_TAC[REAL_LT_LE] THEN
5001 UNDISCH_TAC `!x. x IN s ==> (u:real^N->real) x / v x <= u b / v b` THEN
5002 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN
5003 DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN
5004 REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN
5005 ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (x <= &0 <=> x = &0)`] THEN
5006 DISCH_TAC THEN UNDISCH_TAC `sum s (u:real^N->real) = &1` THEN
5007 MATCH_MP_TAC(REAL_ARITH `x = &0 ==> x = &1 ==> F`) THEN
5008 ASM_SIMP_TAC[SUM_EQ_0];
5010 EXISTS_TAC `(\x. if x = a then v b / u b else v x - (v b / u b) * u x):
5012 ASM_SIMP_TAC[FORALL_IN_INSERT; REAL_LE_DIV; REAL_LT_IMP_LE] THEN
5013 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
5014 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_DELETE] THEN
5015 ASM_SIMP_TAC[SUM_DELETE; VSUM_DELETE; IN_DELETE] THEN
5016 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; FINITE_DELETE] THEN
5017 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> {x | x IN s /\ ~(x = a)} = s`;
5018 SET_RULE `~(a IN s) ==> {x | x IN s /\ x = a} = {}`] THEN
5019 REWRITE_TAC[VSUM_CLAUSES; SUM_CLAUSES] THEN
5020 ASM_CASES_TAC `b:real^N = a` THENL [ASM_MESON_TAC[]; ASM_REWRITE_TAC[]] THEN
5021 ASM_SIMP_TAC[VECTOR_SUB_RDISTRIB; VSUM_SUB; SUM_SUB] THEN
5022 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC; VECTOR_ADD_LID; REAL_ADD_LID] THEN
5023 ASM_SIMP_TAC[SUM_LMUL; VSUM_LMUL] THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
5024 ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN REPEAT CONJ_TAC THENL
5025 [ALL_TAC; REAL_ARITH_TAC; VECTOR_ARITH_TAC] THEN
5026 X_GEN_TAC `c:real^N` THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
5027 STRIP_TAC THEN ASM_CASES_TAC `(u:real^N->real) c = &0` THENL
5028 [ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO]; ALL_TAC] THEN
5029 REWRITE_TAC[REAL_SUB_LE] THEN
5030 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_LT_LE] THEN
5031 ONCE_REWRITE_TAC[GSYM REAL_INV_DIV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
5032 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_LE]);;
5034 let IN_CONVEX_HULL_EXCHANGE_UNIQUE = prove
5035 (`!s t t' a x:real^N.
5036 ~(affine_dependent s) /\
5037 a IN convex hull s /\
5038 t SUBSET s /\ t' SUBSET s /\
5039 x IN convex hull (a INSERT t) /\
5040 x IN convex hull (a INSERT t')
5041 ==> x IN convex hull (a INSERT (t INTER t'))`,
5042 REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL
5043 [REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE
5044 `a INSERT (s INTER t) = (a INSERT s) INTER (a INSERT t)`] THEN
5045 W(MP_TAC o PART_MATCH (rand o rand) CONVEX_HULL_INTER o rand o snd) THEN
5047 [UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN
5048 REWRITE_TAC[CONTRAPOS_THM] THEN
5049 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] AFFINE_DEPENDENT_MONO);
5050 DISCH_THEN(SUBST1_TAC o SYM)] THEN
5053 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5054 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
5055 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN
5056 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `b:real^N->real` STRIP_ASSUME_TAC)
5058 REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5059 SUBGOAL_THEN `~((a:real^N) IN t) /\ ~(a IN t')` STRIP_ASSUME_TAC THENL
5060 [ASM SET_TAC[]; ALL_TAC] THEN
5061 SUBGOAL_THEN `FINITE(t:real^N->bool) /\ FINITE(t':real^N->bool)`
5062 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
5063 ASM_SIMP_TAC[AFFINE_HULL_FINITE_STEP_GEN; REAL_LE_ADD;
5064 REAL_ARITH `&0 <= a / &2 <=> &0 <= a`] THEN
5065 REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
5066 MAP_EVERY X_GEN_TAC [`u':real`; `u:real^N->real`] THEN REPEAT DISCH_TAC THEN
5067 MAP_EVERY X_GEN_TAC [`v':real`; `v:real^N->real`] THEN REPEAT DISCH_TAC THEN
5068 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
5069 [AFFINE_DEPENDENT_EXPLICIT]) THEN
5070 REWRITE_TAC[NOT_EXISTS_THM] THEN
5071 DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN
5072 ASM_REWRITE_TAC[SUBSET_REFL] THEN
5073 DISCH_THEN(MP_TAC o SPEC
5074 `\x:real^N. (if x IN t then u x else &0) - (if x IN t' then v x else &0) +
5075 (u' - v') * b x`) THEN
5076 ASM_SIMP_TAC[SUM_ADD; VSUM_ADD; SUM_LMUL; VSUM_LMUL; VECTOR_ADD_RDISTRIB] THEN
5077 ASM_SIMP_TAC[SUM_SUB; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN
5079 `(if p then a else b) % x = (if p then a % x else b % x)`] THEN
5080 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO; SUM_0; VSUM_0] THEN
5081 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`] THEN
5082 ASM_SIMP_TAC[SUM_ADD; SUM_LMUL; VSUM_ADD; VSUM_LMUL; VECTOR_ADD_RDISTRIB;
5083 GSYM VECTOR_MUL_ASSOC] THEN
5084 MATCH_MP_TAC(TAUT `a /\ c /\ (~b ==> d) ==> ~(a /\ b /\ c) ==> d`) THEN
5085 REPEAT CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC; ALL_TAC] THEN
5088 `(!x. x IN s ==> (if x IN t then u x else &0) <=
5089 (if x IN t' then v x else &0)) \/
5090 (!x:real^N. x IN s ==> (if x IN t' then v x else &0) <=
5091 (if x IN t then u x else &0))`
5092 (DISJ_CASES_THEN(LABEL_TAC "*")) THENL
5093 [MP_TAC(REAL_ARITH `&0 <= (u' - v') \/ &0 <= (v' - u')`) THEN
5094 MATCH_MP_TAC MONO_OR THEN CONJ_TAC THEN
5095 DISCH_TAC THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
5096 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
5097 DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THENL
5098 [MATCH_MP_TAC(REAL_ARITH `&0 <= c ==> a - b + c = &0 ==> a <= b`);
5099 MATCH_MP_TAC(REAL_ARITH `&0 <= --c ==> a - b + c = &0 ==> b <= a`)] THEN
5100 ASM_SIMP_TAC[REAL_LE_MUL; GSYM REAL_MUL_LNEG; REAL_NEG_SUB];
5101 EXISTS_TAC `(\x. if x = a then u' else u x):real^N->real`;
5102 EXISTS_TAC `(\x. if x = a then v' else v x):real^N->real`] THEN
5103 ASM_SIMP_TAC[FORALL_IN_INSERT] THEN
5104 (CONJ_TAC THENL [ASM_MESON_TAC[IN_INTER]; ALL_TAC]) THEN
5105 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; FINITE_INTER] THEN
5106 ASM_REWRITE_TAC[IN_INTER] THEN
5107 REWRITE_TAC[REAL_ARITH `u' + u = &1 <=> u = &1 - u'`;
5108 VECTOR_ARITH `u' + u:real^N = y <=> u = y - u'`] THEN
5110 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
5111 CONV_TAC SYM_CONV THENL
5112 [MATCH_MP_TAC SUM_EQ_SUPERSET; MATCH_MP_TAC VSUM_EQ_SUPERSET]) THEN
5113 ASM_SIMP_TAC[FINITE_INTER; INTER_SUBSET; IN_INTER] THEN
5114 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
5115 X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5116 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISCH_TAC THEN
5117 REMOVE_THEN "*" (MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN
5118 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN ASM SET_TAC[]);;
5120 let CONVEX_HULL_EXCHANGE_UNION = prove
5124 UNIONS {convex hull (a INSERT (s DELETE b)) |b| b IN s}`,
5125 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
5126 [ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN REWRITE_TAC[UNIONS_IMAGE] THEN
5127 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
5128 ASM_MESON_TAC[IN_CONVEX_HULL_EXCHANGE];
5129 REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_GSPEC;
5130 IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5131 GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[GSYM SUBSET] THEN
5132 ASM_SIMP_TAC[SUBSET_HULL; CONVEX_CONVEX_HULL] THEN
5133 ASM_REWRITE_TAC[INSERT_SUBSET] THEN
5134 MESON_TAC[HULL_INC; SUBSET; IN_DELETE]]);;
5136 let CONVEX_HULL_EXCHANGE_INTER = prove
5138 ~affine_dependent s /\
5139 a IN convex hull s /\
5142 ==> (convex hull (a INSERT t)) INTER (convex hull (a INSERT t')) =
5143 convex hull (a INSERT (t INTER t'))`,
5144 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
5145 [REWRITE_TAC[SUBSET; IN_INTER] THEN REPEAT STRIP_TAC THEN
5146 MATCH_MP_TAC IN_CONVEX_HULL_EXCHANGE_UNIQUE THEN
5147 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[];
5148 REWRITE_TAC[SUBSET_INTER] THEN CONJ_TAC THEN
5149 MATCH_MP_TAC HULL_MONO THEN SET_TAC[]]);;
5151 (* ------------------------------------------------------------------------- *)
5152 (* Representing affine hull as hyperplane or finite intersection of them. *)
5153 (* ------------------------------------------------------------------------- *)
5155 let AFF_DIM_EQ_HYPERPLANE = prove
5156 (`!s. aff_dim s = &(dimindex(:N)) - &1 <=>
5157 ?a b. ~(a = vec 0) /\ affine hull s = {x:real^N | a dot x = b}`,
5158 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
5159 [ASM_REWRITE_TAC[AFF_DIM_EMPTY; INT_ARITH `--a:int = b - a <=> b = &0`] THEN
5160 SIMP_TAC[INT_OF_NUM_EQ; LE_1; DIMINDEX_GE_1; AFFINE_HULL_EMPTY] THEN
5161 REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; NOT_EXISTS_THM] THEN
5162 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN
5163 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5164 DISCH_THEN(MP_TAC o SPEC `(b / (a dot a)) % a:real^N`) THEN
5165 ASM_SIMP_TAC[DOT_RMUL; REAL_DIV_RMUL; DOT_EQ_0];
5166 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
5167 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `c:real^N` THEN
5168 GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN
5169 SIMP_TAC[AFF_DIM_DIM_0; HULL_INC] THEN
5170 SIMP_TAC[INT_OF_NUM_SUB; DIMINDEX_GE_1; INT_OF_NUM_EQ] THEN
5171 SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; DIM_EQ_HYPERPLANE] THEN
5172 REPEAT STRIP_TAC THEN AP_TERM_TAC THEN
5173 GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `a:real^N` THEN
5174 REWRITE_TAC[] THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
5175 ASM_REWRITE_TAC[DOT_RADD; REAL_ARITH `a + b:real = c <=> b = c - a`] THEN
5176 EQ_TAC THEN STRIP_TAC THENL
5177 [EXISTS_TAC `(a:real^N) dot c` THEN ASM_REWRITE_TAC[REAL_SUB_REFL];
5178 ASM_REWRITE_TAC[] THEN
5179 FIRST_X_ASSUM(MP_TAC o AP_TERM `\s. (vec 0:real^N) IN s`) THEN
5180 ASM_SIMP_TAC[SPAN_SUPERSET; IN_ELIM_THM; DOT_RZERO]]]);;
5182 let AFF_DIM_HYPERPLANE = prove
5183 (`!a b. ~(a = vec 0)
5184 ==> aff_dim {x:real^N | a dot x = b} = &(dimindex(:N)) - &1`,
5185 REPEAT STRIP_TAC THEN REWRITE_TAC[AFF_DIM_EQ_HYPERPLANE] THEN
5186 MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real`] THEN
5187 ASM_REWRITE_TAC[AFFINE_HULL_EQ; AFFINE_HYPERPLANE]);;
5189 let BOUNDED_HYPERPLANE_EQ_TRIVIAL = prove
5190 (`!a b. bounded {x:real^N | a dot x = b} <=>
5191 if a = vec 0 then ~(b = &0) else dimindex(:N) = 1`,
5192 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
5193 ASM_REWRITE_TAC[DOT_LZERO] THENL
5194 [ASM_CASES_TAC `b = &0` THEN
5195 ASM_REWRITE_TAC[EMPTY_GSPEC; BOUNDED_EMPTY] THEN
5196 REWRITE_TAC[NOT_BOUNDED_UNIV; SET_RULE `{x | T} = UNIV`];
5197 ASM_SIMP_TAC[AFFINE_BOUNDED_EQ_LOWDIM; AFF_DIM_HYPERPLANE;
5198 AFFINE_HYPERPLANE] THEN
5199 REWRITE_TAC[INT_ARITH `a - &1:int <= &0 <=> a <= &1`; INT_OF_NUM_LE] THEN
5200 MATCH_MP_TAC(ARITH_RULE `1 <= n ==> (n <= 1 <=> n = 1)`) THEN
5201 REWRITE_TAC[DIMINDEX_GE_1]]);;
5203 let AFFINE_HULL_FINITE_INTERSECTION_HYPERPLANES = prove
5206 &(CARD f) + aff_dim s = &(dimindex(:N)) /\
5207 affine hull s = INTERS f /\
5208 (!h. h IN f ==> ?a b. ~(a = vec 0) /\ h = {x | a dot x = b})`,
5209 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM AFF_DIM_AFFINE_HULL] THEN
5210 MP_TAC(ISPEC `s:real^N->bool` AFFINE_BASIS_EXISTS) THEN
5211 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
5212 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
5213 MP_TAC(ISPECL [`b:real^N->bool`; `(:real^N)`] EXTEND_TO_AFFINE_BASIS) THEN
5214 ASM_REWRITE_TAC[SUBSET_UNIV; AFFINE_HULL_UNIV] THEN
5215 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
5216 SUBGOAL_THEN `FINITE(c:real^N->bool)` ASSUME_TAC THENL
5217 [ASM_MESON_TAC[AFFINE_INDEPENDENT_IMP_FINITE]; ALL_TAC] THEN
5218 REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
5219 REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN
5220 ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; CARD_DIFF] THEN
5221 REWRITE_TAC[INT_ARITH `f + b - &1:int = c - &1 <=> f = c - b`] THEN
5222 ASM_SIMP_TAC[INT_OF_NUM_SUB; CARD_SUBSET; GSYM CARD_DIFF; INT_OF_NUM_EQ] THEN
5223 ASM_CASES_TAC `c:real^N->bool = b` THENL
5224 [EXISTS_TAC `{}:(real^N->bool)->bool` THEN
5225 ASM_REWRITE_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY; INTERS_0;
5229 EXISTS_TAC `{affine hull (c DELETE a) |a| (a:real^N) IN (c DIFF b)}` THEN
5230 REWRITE_TAC[FORALL_IN_GSPEC] THEN REWRITE_TAC[SIMPLE_IMAGE] THEN
5231 ASM_SIMP_TAC[FINITE_IMAGE; FINITE_DIFF] THEN REPEAT CONJ_TAC THENL
5232 [MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_SIMP_TAC[FINITE_DIFF] THEN
5233 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN REWRITE_TAC[IN_DIFF] THEN
5234 STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN ASM_REWRITE_TAC[] THEN
5235 UNDISCH_TAC `~affine_dependent(c:real^N->bool)` THEN
5236 REWRITE_TAC[affine_dependent] THEN EXISTS_TAC `x:real^N` THEN
5237 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[];
5238 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[IMAGE_o] THEN
5239 ONCE_REWRITE_TAC[GSYM SIMPLE_IMAGE] THEN
5240 W(MP_TAC o PART_MATCH (rhs o rand) AFFINE_HULL_INTERS o rand o snd) THEN
5242 [MATCH_MP_TAC AFFINE_INDEPENDENT_SUBSET THEN
5243 EXISTS_TAC `c:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
5244 REWRITE_TAC[SUBSET; FORALL_IN_UNIONS; FORALL_IN_IMAGE;
5245 IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SET_TAC[];
5246 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN
5247 GEN_REWRITE_TAC I [EXTENSION] THEN
5248 REWRITE_TAC[IN_INTERS; FORALL_IN_IMAGE] THEN ASM SET_TAC[]];
5249 REWRITE_TAC[GSYM AFF_DIM_EQ_HYPERPLANE] THEN
5250 ASM_SIMP_TAC[IN_DIFF; AFFINE_INDEPENDENT_DELETE;
5251 AFF_DIM_AFFINE_INDEPENDENT; CARD_DELETE] THEN
5252 REWRITE_TAC[GSYM AFF_DIM_UNIV] THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
5253 REWRITE_TAC[AFF_DIM_AFFINE_HULL] THEN
5254 ASM_SIMP_TAC[AFF_DIM_AFFINE_INDEPENDENT; CARD_DIFF] THEN
5255 REPEAT STRIP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
5256 MATCH_MP_TAC(GSYM INT_OF_NUM_SUB) THEN
5257 MATCH_MP_TAC(ARITH_RULE `~(c = 0) ==> 1 <= c`) THEN
5258 ASM_SIMP_TAC[CARD_EQ_0] THEN ASM SET_TAC[]]);;
5260 let AFFINE_HYPERPLANE_SUMS_EQ_UNIV = prove
5263 ~(s INTER {v | a dot v = b} = {}) /\
5264 ~(s DIFF {v | a dot v = b} = {})
5265 ==> {x + y | x IN s /\ y IN {v | a dot v = b}} = (:real^N)`,
5266 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL
5267 [ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[]; ALL_TAC] THEN
5268 REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> b ==> a /\ c ==> d`] THEN
5269 GEN_REWRITE_TAC LAND_CONV [GSYM MEMBER_NOT_EMPTY] THEN
5270 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN X_GEN_TAC `c:real^N` THEN
5271 ONCE_REWRITE_TAC[SET_RULE
5272 `{x + y:real^N | x IN s /\ P y} =
5273 {z | ?x y. x IN s /\ P y /\ z = x + y}`] THEN
5274 GEOM_ORIGIN_TAC `c:real^N` THEN REPEAT GEN_TAC THEN
5275 REWRITE_TAC[DOT_RADD; REAL_ARITH `b dot c + a = d <=> a = d - b dot c`] THEN
5276 REWRITE_TAC[IN_INTER; IN_ELIM_THM; DOT_RZERO] THEN
5277 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
5278 ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE; HULL_INC] THEN STRIP_TAC THEN
5279 REWRITE_TAC[VECTOR_ARITH `c + z:real^N = (c + x) + (c + y) <=>
5280 z = c + x + y`] THEN
5281 REWRITE_TAC[SET_RULE
5282 `{z | ?x y. x IN s /\ P y /\ z = c + x + y} =
5283 IMAGE (\x. c + x) {x + y:real^N | x IN s /\ y IN {v | P v}}`] THEN
5284 MATCH_MP_TAC(SET_RULE
5285 `!f. (!x. g(f x) = x) /\ s = UNIV ==> IMAGE g s = UNIV`) THEN
5286 EXISTS_TAC `\x:real^N. x - c` THEN
5287 REWRITE_TAC[VECTOR_ARITH `c + x - c:real^N = x`] THEN
5288 MATCH_MP_TAC(MESON[SPAN_EQ_SELF] `subspace s /\ span s = t ==> s = t`) THEN
5290 [ASM_SIMP_TAC[SUBSPACE_SUMS; SUBSPACE_HYPERPLANE];
5292 REWRITE_TAC[GSYM DIM_EQ_FULL] THEN
5293 REWRITE_TAC[GSYM LE_ANTISYM; DIM_SUBSET_UNIV] THEN
5294 MATCH_MP_TAC(ARITH_RULE `m - 1 < n ==> m <= n`) THEN
5295 MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `dim {x:real^N | a dot x = &0}` THEN
5296 CONJ_TAC THENL [ASM_SIMP_TAC[DIM_HYPERPLANE; LE_REFL]; ALL_TAC] THEN
5297 MATCH_MP_TAC DIM_PSUBSET THEN
5298 ASM_SIMP_TAC[snd(EQ_IMP_RULE(SPEC_ALL SPAN_EQ_SELF));
5299 SUBSPACE_SUMS; SUBSPACE_HYPERPLANE] THEN
5300 REWRITE_TAC[PSUBSET; SUBSET; FORALL_IN_GSPEC] THEN
5301 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN CONJ_TAC THENL
5302 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
5303 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN
5304 ASM_SIMP_TAC[SUBSPACE_0; VECTOR_ADD_LID];
5305 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
5306 REWRITE_TAC[NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
5307 X_GEN_TAC `x:real^N` THEN SIMP_TAC[IN_DIFF; IN_ELIM_THM] THEN
5308 DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
5309 ASM_REWRITE_TAC[DOT_RZERO; VECTOR_ADD_RID]]);;
5311 let AFF_DIM_AFFINE_INTER_HYPERPLANE = prove
5312 (`!a b s:real^N->bool.
5314 ==> aff_dim(s INTER {x | a dot x = b}) =
5315 if s INTER {v | a dot v = b} = {} then -- &1
5316 else if s SUBSET {v | a dot v = b} then aff_dim s
5317 else aff_dim s - &1`,
5318 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THENL
5319 [ASM_REWRITE_TAC[DOT_LZERO] THEN ASM_CASES_TAC `b = &0` THEN
5320 ASM_REWRITE_TAC[EMPTY_GSPEC; INTER_EMPTY; AFF_DIM_EMPTY] THEN
5321 SIMP_TAC[SET_RULE `{x | T} = UNIV`; IN_UNIV; INTER_UNIV; SUBSET_UNIV] THEN
5322 COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY];
5323 STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[AFF_DIM_EMPTY] THEN
5324 COND_CASES_TAC THENL [AP_TERM_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN
5325 MP_TAC(ISPECL [`s:real^N->bool`; `{x:real^N | a dot x = b}`]
5326 AFF_DIM_SUMS_INTER) THEN
5327 ASM_SIMP_TAC[AFFINE_HYPERPLANE; AFF_DIM_HYPERPLANE] THEN
5328 ASM_SIMP_TAC[AFFINE_HYPERPLANE_SUMS_EQ_UNIV; AFF_DIM_UNIV;
5329 SET_RULE `~(s SUBSET t) ==> ~(s DIFF t = {})`] THEN
5330 SPEC_TAC(`aff_dim (s INTER {x:real^N | a dot x = b})`,`i:int`) THEN
5333 let AFF_DIM_LT_FULL = prove
5334 (`!s. aff_dim s < &(dimindex(:N)) <=> ~(affine hull s = (:real^N))`,
5335 GEN_TAC THEN REWRITE_TAC[GSYM AFF_DIM_EQ_FULL] THEN
5336 MP_TAC(ISPEC `s:real^N->bool` AFF_DIM_LE_UNIV) THEN ARITH_TAC);;
5338 let AFF_LOWDIM_SUBSET_HYPERPLANE = prove
5340 aff_dim s < &(dimindex(:N))
5341 ==> ?a b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`,
5342 MATCH_MP_TAC SET_PROVE_CASES THEN CONJ_TAC THENL
5343 [DISCH_TAC THEN EXISTS_TAC `basis 1:real^N` THEN
5344 SIMP_TAC[EMPTY_SUBSET; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1];
5345 MAP_EVERY X_GEN_TAC [`c:real^N`; `s:real^N->bool`] THEN
5346 CONV_TAC(ONCE_DEPTH_CONV(GEN_ALPHA_CONV `a:real^N`)) THEN
5347 GEN_GEOM_ORIGIN_TAC `c:real^N` ["a"] THEN
5348 SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; IN_INSERT; INT_OF_NUM_LT] THEN
5349 REPEAT GEN_TAC THEN DISCH_TAC THEN
5350 DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN
5351 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N` THEN
5352 STRIP_TAC THEN EXISTS_TAC `(u:real^N) dot c` THEN
5353 ASM_REWRITE_TAC[DOT_RADD; REAL_EQ_ADD_LCANCEL_0] THEN
5354 ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS]]);;
5356 (* ------------------------------------------------------------------------- *)
5357 (* An additional lemma about hyperplanes. *)
5358 (* ------------------------------------------------------------------------- *)
5360 let SUBSET_HYPERPLANES = prove
5362 {x | a dot x = b} SUBSET {x | a' dot x = b'} <=>
5363 {x | a dot x = b} = {} \/ {x | a' dot x = b'} = (:real^N) \/
5364 {x | a dot x = b} = {x | a' dot x = b'}`,
5366 ASM_CASES_TAC `{x:real^N | a dot x = b} = {}` THEN
5367 ASM_REWRITE_TAC[EMPTY_SUBSET] THEN
5368 ASM_CASES_TAC `{x | a' dot x = b'} = (:real^N)` THEN
5369 ASM_REWRITE_TAC[SUBSET_UNIV] THEN
5370 RULE_ASSUM_TAC(REWRITE_RULE
5371 [HYPERPLANE_EQ_EMPTY; HYPERPLANE_EQ_UNIV]) THEN
5372 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
5373 ASM_CASES_TAC `{x:real^N | a dot x = b} SUBSET {x | a' dot x = b'}` THEN
5374 ASM_REWRITE_TAC[] THEN
5375 MP_TAC(ISPECL [`{x:real^N | a dot x = b}`; `{x:real^N | a' dot x = b'}`]
5376 AFF_DIM_PSUBSET) THEN
5377 ASM_SIMP_TAC[PSUBSET;
5378 REWRITE_RULE[GSYM AFFINE_HULL_EQ] AFFINE_HYPERPLANE] THEN
5379 ASM_CASES_TAC `{x:real^N | a dot x = b} = {x | a' dot x = b'}` THEN
5380 ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_CASES_TAC `a':real^N = vec 0` THENL
5381 [ASM_CASES_TAC `b' = &0` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
5382 ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[];
5384 ASM_CASES_TAC `a:real^N = vec 0` THENL
5385 [ASM_CASES_TAC `b = &0` THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
5386 ASM_REWRITE_TAC[DOT_LZERO] THEN SET_TAC[];
5388 ASM_SIMP_TAC[AFF_DIM_HYPERPLANE; INT_LT_REFL]);;
5390 (* ------------------------------------------------------------------------- *)
5391 (* Openness and compactness are preserved by convex hull operation. *)
5392 (* ------------------------------------------------------------------------- *)
5394 let OPEN_CONVEX_HULL = prove
5395 (`!s:real^N->bool. open s ==> open(convex hull s)`,
5397 REWRITE_TAC[CONVEX_HULL_EXPLICIT; OPEN_CONTAINS_CBALL] THEN
5398 REWRITE_TAC[IN_ELIM_THM; SUBSET; LEFT_IMP_EXISTS_THM] THEN DISCH_TAC THEN
5399 MAP_EVERY X_GEN_TAC [`a:real^N`; `t:real^N->bool`; `u:real^N->real`] THEN
5401 SUBGOAL_THEN `?b. !x:real^N. x IN t ==> &0 < b(x) /\ cball(x,b(x)) SUBSET s`
5402 STRIP_ASSUME_TAC THENL
5403 [REWRITE_TAC[GSYM SKOLEM_THM] THEN ASM_MESON_TAC[SUBSET]; ALL_TAC] THEN
5404 ABBREV_TAC `i = IMAGE (b:real^N->real) t` THEN
5405 EXISTS_TAC `inf i` THEN MP_TAC(SPEC `i:real->bool` INF_FINITE) THEN
5406 EXPAND_TAC "i" THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_IMAGE] THEN
5408 [EXPAND_TAC "i" THEN CONJ_TAC THENL
5409 [ASM_SIMP_TAC[FINITE_IMAGE]; ALL_TAC] THEN
5410 REWRITE_TAC[IMAGE_EQ_EMPTY] THEN
5411 ASM_MESON_TAC[SUM_CLAUSES; REAL_ARITH `~(&1 = &0)`];
5413 STRIP_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
5414 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_CBALL; dist] THEN
5415 DISCH_TAC THEN EXISTS_TAC `IMAGE (\v:real^N. v + (y - a)) t` THEN
5416 EXISTS_TAC `\v. (u:real^N->real)(v - (y - a))` THEN
5417 ASM_SIMP_TAC[FINITE_IMAGE; FORALL_IN_IMAGE; SUM_IMAGE; VSUM_IMAGE;
5418 VECTOR_ARITH `v + a:real^N = w + a <=> v = w`] THEN
5419 ASM_REWRITE_TAC[o_DEF; VECTOR_ARITH `(v + a) - a:real^N = v`] THEN
5420 ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB; ETA_AX] THEN
5421 ASM_SIMP_TAC[VSUM_ADD; VSUM_RMUL] THEN
5422 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
5423 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
5424 SUBGOAL_THEN `z + (y - a) IN cball(z:real^N,b z)`
5425 (fun th -> ASM_MESON_TAC[th; SUBSET]) THEN
5426 REWRITE_TAC[IN_CBALL; dist; NORM_ARITH
5427 `norm(z - (z + a - y)) = norm(y - a)`] THEN
5428 ASM_MESON_TAC[REAL_LE_TRANS]);;
5430 let COMPACT_CONVEX_COMBINATIONS = prove
5431 (`!s t. compact s /\ compact t
5432 ==> compact { (&1 - u) % x + u % y :real^N |
5433 &0 <= u /\ u <= &1 /\ x IN s /\ y IN t}`,
5434 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5435 `{ (&1 - u) % x + u % y :real^N | &0 <= u /\ u <= &1 /\ x IN s /\ y IN t} =
5436 IMAGE (\z. (&1 - drop(fstcart z)) % fstcart(sndcart z) +
5437 drop(fstcart z) % sndcart(sndcart z))
5438 { pastecart u w | u IN interval[vec 0,vec 1] /\
5439 w IN { pastecart x y | x IN s /\ y IN t} }`
5441 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN
5442 X_GEN_TAC `x:real^N` THEN
5443 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN
5444 CONV_TAC(ONCE_DEPTH_CONV UNWIND_CONV) THEN
5445 REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN
5446 REWRITE_TAC[IN_INTERVAL_1; GSYM EXISTS_DROP; DROP_VEC] THEN MESON_TAC[];
5448 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
5449 ASM_SIMP_TAC[COMPACT_PCROSS; GSYM PCROSS; COMPACT_INTERVAL] THEN
5450 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
5451 X_GEN_TAC `z:real^(1,(N,N)finite_sum)finite_sum` THEN
5452 DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[PCROSS] THEN
5453 MATCH_MP_TAC CONTINUOUS_ADD THEN CONJ_TAC THEN
5454 MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_DROP] THEN
5455 CONJ_TAC THEN TRY(MATCH_MP_TAC CONTINUOUS_SUB) THEN
5456 REWRITE_TAC[CONTINUOUS_CONST] THEN
5457 MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
5458 REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART; ETA_AX] THEN
5459 GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN
5460 MATCH_MP_TAC LINEAR_COMPOSE THEN
5461 REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);;
5463 let COMPACT_CONVEX_HULL = prove
5464 (`!s:real^N->bool. compact s ==> compact(convex hull s)`,
5465 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CARATHEODORY] THEN
5466 SPEC_TAC(`dimindex(:N) + 1`,`n:num`) THEN
5467 ASM_CASES_TAC `s:real^N->bool = {}` THENL
5468 [ASM_REWRITE_TAC[SUBSET_EMPTY] THEN
5469 CONV_TAC(ONCE_DEPTH_CONV UNWIND_CONV) THEN
5470 REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
5471 REWRITE_TAC[SET_RULE `{x | F} = {}`; COMPACT_EMPTY];
5473 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
5474 DISCH_THEN(X_CHOOSE_TAC `w:real^N`) THEN INDUCT_TAC THENL
5476 `{x:real^N | ?t. FINITE t /\ t SUBSET s /\ CARD t <= 0 /\
5477 x IN convex hull t} = {}`
5478 (fun th -> REWRITE_TAC[th; COMPACT_EMPTY]) THEN
5479 REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; LE; IN_ELIM_THM] THEN
5480 MESON_TAC[CARD_EQ_0; CONVEX_HULL_EMPTY; NOT_IN_EMPTY];
5482 ASM_CASES_TAC `n = 0` THENL
5483 [ASM_REWRITE_TAC[ARITH_RULE `s <= SUC 0 <=> s = 0 \/ s = 1`] THEN
5484 UNDISCH_TAC `compact(s:real^N->bool)` THEN
5485 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
5486 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
5487 REWRITE_TAC[TAUT `a /\ b /\ (c \/ d) /\ e <=>
5488 (a /\ c) /\ (b /\ e) \/ (a /\ d) /\ (b /\ e)`] THEN
5489 REWRITE_TAC[GSYM HAS_SIZE; num_CONV `1`; HAS_SIZE_CLAUSES] THEN
5490 REWRITE_TAC[EXISTS_OR_THM; LEFT_AND_EXISTS_THM; RIGHT_AND_EXISTS_THM] THEN
5491 CONV_TAC(TOP_DEPTH_CONV UNWIND_CONV) THEN
5492 REWRITE_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY] THEN
5493 REWRITE_TAC[CONVEX_HULL_SING] THEN SET_TAC[];
5496 `{x:real^N | ?t. FINITE t /\ t SUBSET s /\ CARD t <= SUC n /\
5497 x IN convex hull t} =
5498 { (&1 - u) % x + u % y :real^N |
5499 &0 <= u /\ u <= &1 /\ x IN s /\
5500 y IN {x | ?t. FINITE t /\ t SUBSET s /\
5501 CARD t <= n /\ x IN convex hull t}}`
5502 (fun th -> ASM_SIMP_TAC[th; COMPACT_CONVEX_COMBINATIONS]) THEN
5503 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
5504 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
5506 REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_AND_EXISTS_THM;
5507 LEFT_AND_EXISTS_THM] THEN
5508 MAP_EVERY X_GEN_TAC [`u:real^N`; `c:real`; `v:real^N`;
5509 `t:real^N->bool`] THEN
5510 STRIP_TAC THEN EXISTS_TAC `(u:real^N) INSERT t` THEN
5511 ASM_REWRITE_TAC[FINITE_INSERT; INSERT_SUBSET] THEN
5512 ASM_SIMP_TAC[CARD_CLAUSES] THEN CONJ_TAC THENL
5513 [ASM_ARITH_TAC; ALL_TAC] THEN
5514 MATCH_MP_TAC IN_CONVEX_SET THEN
5515 ASM_REWRITE_TAC[CONVEX_CONVEX_HULL] THEN CONJ_TAC THEN
5516 ASM_MESON_TAC[HULL_SUBSET; SUBSET; IN_INSERT; HULL_MONO]] THEN
5517 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
5518 ASM_CASES_TAC `CARD(t:real^N->bool) <= n` THENL
5519 [MAP_EVERY EXISTS_TAC [`w:real^N`; `&1`; `x:real^N`] THEN
5520 ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN
5521 CONJ_TAC THENL [ASM_MESON_TAC[]; VECTOR_ARITH_TAC];
5523 SUBGOAL_THEN `(t:real^N->bool) HAS_SIZE (SUC n)` MP_TAC THENL
5524 [ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_ARITH_TAC;
5526 REWRITE_TAC[HAS_SIZE_CLAUSES] THEN
5527 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `u:real^N->bool`
5528 STRIP_ASSUME_TAC)) THEN
5529 FIRST_X_ASSUM SUBST_ALL_TAC THEN
5530 UNDISCH_TAC `(x:real^N) IN convex hull (a INSERT u)` THEN
5531 RULE_ASSUM_TAC(REWRITE_RULE[FINITE_INSERT]) THEN
5532 ASM_CASES_TAC `(u:real^N->bool) = {}` THENL
5533 [ASM_REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN
5534 DISCH_THEN SUBST_ALL_TAC THEN
5535 MAP_EVERY EXISTS_TAC [`a:real^N`; `&1`; `a:real^N`] THEN
5536 ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL] THEN
5537 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
5538 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5539 EXISTS_TAC `{a:real^N}` THEN SIMP_TAC[FINITE_RULES] THEN
5540 REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN
5541 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5542 SIMP_TAC[CARD_CLAUSES; FINITE_RULES; NOT_IN_EMPTY] THEN
5543 UNDISCH_TAC `~(n = 0)` THEN ARITH_TAC;
5545 ASM_SIMP_TAC[CONVEX_HULL_INSERT; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
5546 MAP_EVERY X_GEN_TAC [`c:real`; `d:real`; `z:real^N`] THEN STRIP_TAC THEN
5547 MAP_EVERY EXISTS_TAC [`a:real^N`; `d:real`; `z:real^N`] THEN
5548 FIRST_X_ASSUM(SUBST_ALL_TAC o MATCH_MP (REAL_ARITH
5549 `c + d = &1 ==> c = (&1 - d)`)) THEN
5550 ASM_REWRITE_TAC[REAL_ARITH `d <= &1 <=> &0 <= &1 - d`] THEN
5551 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5552 EXISTS_TAC `u:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
5553 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5554 UNDISCH_TAC `CARD ((a:real^N) INSERT u) <= SUC n` THEN
5555 ASM_SIMP_TAC[CARD_CLAUSES; LE_SUC]);;
5557 let FINITE_IMP_COMPACT_CONVEX_HULL = prove
5558 (`!s:real^N->bool. FINITE s ==> compact(convex hull s)`,
5559 SIMP_TAC[FINITE_IMP_COMPACT; COMPACT_CONVEX_HULL]);;
5561 let CONVEX_HULL_INTERIOR_SUBSET = prove
5562 (`!s:real^N->bool. convex hull (interior s) SUBSET interior (convex hull s)`,
5563 GEN_TAC THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN
5564 SIMP_TAC[OPEN_CONVEX_HULL; OPEN_INTERIOR; HULL_MONO; INTERIOR_SUBSET]);;
5566 (* ------------------------------------------------------------------------- *)
5567 (* Extremal points of a simplex are some vertices. *)
5568 (* ------------------------------------------------------------------------- *)
5570 let DIST_INCREASES_ONLINE = prove
5571 (`!a b d. ~(d = vec 0)
5572 ==> dist(a,b + d) > dist(a,b) \/ dist(a,b - d) > dist(a,b)`,
5573 REWRITE_TAC[dist; vector_norm; real_gt; GSYM NORM_POS_LT] THEN
5574 SIMP_TAC[SQRT_MONO_LT_EQ; DOT_POS_LE; SQRT_LT_0] THEN
5575 REWRITE_TAC[DOT_RSUB; DOT_RADD; DOT_LSUB; DOT_LADD] THEN REAL_ARITH_TAC);;
5577 let NORM_INCREASES_ONLINE = prove
5578 (`!a:real^N d. ~(d = vec 0)
5579 ==> norm(a + d) > norm(a) \/ norm(a - d) > norm(a)`,
5580 MP_TAC(ISPEC `vec 0 :real^N` DIST_INCREASES_ONLINE) THEN
5581 REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG]);;
5583 let SIMPLEX_FURTHEST_LT = prove
5586 ==> !x. x IN (convex hull s) /\ ~(x IN s)
5587 ==> ?y. y IN (convex hull s) /\ norm(x - a) < norm(y - a)`,
5588 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
5589 REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
5590 MAP_EVERY X_GEN_TAC [`x:real^N`; `s:real^N->bool`] THEN
5591 ASM_CASES_TAC `s:real^N->bool = {}` THENL
5592 [ASM_REWRITE_TAC[CONVEX_HULL_SING; IN_SING] THEN MESON_TAC[];
5594 ASM_SIMP_TAC[CONVEX_HULL_INSERT] THEN
5595 STRIP_TAC THEN X_GEN_TAC `y:real^N` THEN
5596 REWRITE_TAC[IN_ELIM_THM; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
5597 MAP_EVERY X_GEN_TAC [`u:real`; `v:real`; `b:real^N`] THEN
5598 ASM_CASES_TAC `y:real^N IN (convex hull s)` THENL
5599 [REWRITE_TAC[IN_INSERT; DE_MORGAN_THM] THEN STRIP_TAC THEN
5600 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN
5601 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
5602 X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN
5603 MAP_EVERY EXISTS_TAC [`&0`; `&1`; `c:real^N`] THEN
5604 ASM_REWRITE_TAC[REAL_ADD_LID; REAL_POS] THEN VECTOR_ARITH_TAC;
5606 ASM_CASES_TAC `u = &0` THENL
5607 [ASM_SIMP_TAC[REAL_ADD_LID; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
5608 ASM_MESON_TAC[VECTOR_MUL_LID];
5610 ASM_CASES_TAC `v = &0` THENL
5611 [ASM_SIMP_TAC[REAL_ADD_RID; VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN
5612 ASM_CASES_TAC `u = &1` THEN ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN
5613 ASM_CASES_TAC `y = a:real^N` THEN ASM_REWRITE_TAC[IN_INSERT] THEN
5616 REWRITE_TAC[IN_INSERT; DE_MORGAN_THM] THEN STRIP_TAC THEN
5617 MP_TAC(SPECL [`u:real`; `v:real`] REAL_DOWN2) THEN ANTS_TAC THENL
5618 [ASM_REWRITE_TAC[REAL_LT_LE]; ALL_TAC] THEN
5619 DISCH_THEN(X_CHOOSE_THEN `w:real` STRIP_ASSUME_TAC) THEN
5620 MP_TAC(ISPECL [`a:real^N`; `y:real^N`; `w % (x - b):real^N`]
5621 DIST_INCREASES_ONLINE) THEN
5623 [ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ] THEN
5624 REWRITE_TAC[VECTOR_ARITH `(x - y = vec 0) <=> (x = y)`] THEN
5625 DISCH_THEN SUBST_ALL_TAC THEN
5626 UNDISCH_TAC `~(y:real^N IN convex hull s)` THEN
5627 ASM_REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID];
5629 ASM_REWRITE_TAC[dist; real_gt] THEN
5630 REWRITE_TAC[VECTOR_ARITH
5631 `((u % x + v % b) + w % (x - b) = (u + w) % x + (v - w) % b) /\
5632 ((u % x + v % b) - w % (x - b) = (u - w) % x + (v + w) % b)`] THEN
5634 [MAP_EVERY EXISTS_TAC
5635 [`(u + w) % x + (v - w) % b:real^N`; `u + w`; `v - w`; `b:real^N`];
5636 MAP_EVERY EXISTS_TAC
5637 [`(u - w) % x + (v + w) % b:real^N`; `u - w`; `v + w`; `b:real^N`]] THEN
5638 ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_REWRITE_TAC[] THEN
5639 ASM_SIMP_TAC[REAL_LE_ADD; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
5640 UNDISCH_TAC `u + v = &1` THEN REAL_ARITH_TAC);;
5642 let SIMPLEX_FURTHEST_LE = prove
5644 FINITE s /\ ~(s = {})
5646 !x. x IN (convex hull s) ==> norm(x - a) <= norm(y - a)`,
5647 REPEAT GEN_TAC THEN DISCH_TAC THEN
5648 MP_TAC(ISPEC `convex hull (s:real^N->bool)` DISTANCE_ATTAINS_SUP) THEN
5649 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ANTS_TAC THENL
5650 [ASM_SIMP_TAC[FINITE_IMP_COMPACT_CONVEX_HULL] THEN
5651 ASM_MESON_TAC[SUBSET_EMPTY; HULL_SUBSET];
5653 ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[dist] THEN
5654 ASM_MESON_TAC[SIMPLEX_FURTHEST_LT; REAL_NOT_LE]);;
5656 let SIMPLEX_FURTHEST_LE_EXISTS = prove
5659 ==> !x. x IN (convex hull s)
5660 ==> ?y. y IN s /\ norm(x - a) <= norm(y - a)`,
5661 MESON_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY; SIMPLEX_FURTHEST_LE]);;
5663 let SIMPLEX_EXTREMAL_LE = prove
5665 FINITE s /\ ~(s = {})
5666 ==> ?u v. u IN s /\ v IN s /\
5667 !x y. x IN convex hull s /\ y IN convex hull s
5668 ==> norm(x - y) <= norm(u - v)`,
5669 REPEAT STRIP_TAC THEN
5670 MP_TAC(ISPEC `convex hull (s:real^N->bool)` COMPACT_SUP_MAXDISTANCE) THEN
5672 [ASM_SIMP_TAC[FINITE_IMP_COMPACT_CONVEX_HULL] THEN
5673 ASM_MESON_TAC[SUBSET_EMPTY; HULL_SUBSET];
5675 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
5676 SIMP_TAC[] THEN ASM_MESON_TAC[SIMPLEX_FURTHEST_LT; REAL_NOT_LE; NORM_SUB]);;
5678 let SIMPLEX_EXTREMAL_LE_EXISTS = prove
5679 (`!s:real^N->bool x y. FINITE s /\ x IN convex hull s /\ y IN convex hull s
5680 ==> ?u v. u IN s /\ v IN s /\
5681 norm(x - y) <= norm(u - v)`,
5682 MESON_TAC[NOT_IN_EMPTY; CONVEX_HULL_EMPTY; SIMPLEX_EXTREMAL_LE]);;
5684 let DIAMETER_CONVEX_HULL = prove
5685 (`!s:real^N->bool. diameter(convex hull s) = diameter s`,
5687 (`!a b s. (!x. x IN s ==> dist(a,x) <= b)
5688 ==> (!x. x IN convex hull s ==> dist(a,x) <= b)`,
5689 REPEAT GEN_TAC THEN DISCH_TAC THEN
5690 MATCH_MP_TAC HULL_INDUCT THEN ASM_REWRITE_TAC[GSYM cball; CONVEX_CBALL]) in
5691 GEN_TAC THEN REWRITE_TAC[diameter; CONVEX_HULL_EQ_EMPTY] THEN
5692 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUP_EQ THEN
5693 REWRITE_TAC[FORALL_IN_GSPEC] THEN X_GEN_TAC `b:real` THEN
5694 EQ_TAC THENL [MESON_TAC[SUBSET; HULL_SUBSET]; ALL_TAC] THEN
5695 MATCH_MP_TAC(TAUT `!b. (a ==> b) /\ (b ==> c) ==> a ==> c`) THEN
5696 EXISTS_TAC `!x:real^N y. x IN s /\ y IN convex hull s ==> norm(x - y) <= b`
5698 [MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^N` THEN
5699 ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
5700 REWRITE_TAC[GSYM dist; lemma];
5701 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
5702 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^N` THEN
5703 ASM_CASES_TAC `(y:real^N) IN convex hull s` THEN ASM_REWRITE_TAC[] THEN
5704 REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[DIST_SYM] dist); lemma]]);;
5706 let DIAMETER_SIMPLEX = prove
5709 ==> diameter(convex hull s) = sup { dist(x,y) | x IN s /\ y IN s}`,
5710 REWRITE_TAC[DIAMETER_CONVEX_HULL] THEN SIMP_TAC[diameter; dist]);;
5712 (* ------------------------------------------------------------------------- *)
5713 (* Closest point of a convex set is unique, with a continuous projection. *)
5714 (* ------------------------------------------------------------------------- *)
5716 let CLOSER_POINTS_LEMMA = prove
5720 !v. &0 < v /\ v <= u ==> norm(v % z - y) < norm y`,
5721 REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB; DOT_LMUL; DOT_RMUL;
5722 REAL_SUB_LDISTRIB; real_gt] THEN REPEAT GEN_TAC THEN
5723 REWRITE_TAC[REAL_ARITH `(a - b) - (c - d) < d <=> a < b + c`] THEN
5724 STRIP_TAC THEN SUBST1_TAC(VECTOR_ARITH `(z:real^N) dot y = y dot z`) THEN
5725 SIMP_TAC[GSYM REAL_ADD_LDISTRIB; REAL_LT_LMUL_EQ] THEN
5726 EXISTS_TAC `(y dot (z:real^N)) / (z dot z)` THEN
5727 SUBGOAL_THEN `&0 < z dot (z:real^N)` ASSUME_TAC THENL
5728 [ASM_MESON_TAC[DOT_POS_LT; DOT_RZERO; REAL_LT_REFL]; ALL_TAC] THEN
5729 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LE_RDIV_EQ] THEN
5730 ASM_SIMP_TAC[REAL_ARITH `&0 < y /\ x <= y ==> x < y + y`; REAL_LT_MUL]);;
5732 let CLOSER_POINT_LEMMA = prove
5733 (`!x y z. (y - x) dot (z - x) > &0
5734 ==> ?u. &0 < u /\ u <= &1 /\ dist(x + u % (z - x),y) < dist(x,y)`,
5735 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CLOSER_POINTS_LEMMA) THEN
5736 ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[dist; NORM_LT] THEN
5737 REWRITE_TAC[VECTOR_ARITH
5738 `(y - (x + z)) dot (y - (x + z)) = (z - (y - x)) dot (z - (y - x))`] THEN
5739 DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
5740 EXISTS_TAC `min u (&1)` THEN
5741 ASM_SIMP_TAC[REAL_LT_MIN; REAL_MIN_LE; REAL_LT_01; REAL_LE_REFL]);;
5743 let ANY_CLOSEST_POINT_DOT = prove
5745 convex s /\ closed s /\ x IN s /\ y IN s /\
5746 (!z. z IN s ==> dist(a,x) <= dist(a,z))
5747 ==> (a - x) dot (y - x) <= &0`,
5748 REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN
5749 DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN
5750 DISCH_THEN(X_CHOOSE_THEN `u:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5751 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5752 REWRITE_TAC[REAL_NOT_LT] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
5753 FIRST_X_ASSUM MATCH_MP_TAC THEN
5754 REWRITE_TAC[VECTOR_ARITH `x + u % (y - x) = (&1 - u) % x + u % y`] THEN
5755 MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]);;
5757 let ANY_CLOSEST_POINT_UNIQUE = prove
5759 convex s /\ closed s /\ x IN s /\ y IN s /\
5760 (!z. z IN s ==> dist(a,x) <= dist(a,z)) /\
5761 (!z. z IN s ==> dist(a,y) <= dist(a,z))
5763 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
5764 REWRITE_TAC[GSYM NORM_LE_0; NORM_LE_SQUARE] THEN
5765 SUBGOAL_THEN `(a - x:real^N) dot (y - x) <= &0 /\ (a - y) dot (x - y) <= &0`
5766 MP_TAC THENL [ASM_MESON_TAC[ANY_CLOSEST_POINT_DOT]; ALL_TAC] THEN
5767 REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB] THEN REAL_ARITH_TAC);;
5769 let CLOSEST_POINT_UNIQUE = prove
5771 convex s /\ closed s /\ x IN s /\
5772 (!z. z IN s ==> dist(a,x) <= dist(a,z))
5773 ==> x = closest_point s a`,
5774 REPEAT STRIP_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_UNIQUE THEN
5775 MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `a:real^N`] THEN
5776 ASM_MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);;
5778 let CLOSEST_POINT_DOT = prove
5780 convex s /\ closed s /\ x IN s
5781 ==> (a - closest_point s a) dot (x - closest_point s a) <= &0`,
5782 REPEAT STRIP_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_DOT THEN
5783 EXISTS_TAC `s:real^N->bool` THEN
5784 ASM_MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);;
5786 let CLOSEST_POINT_LT = prove
5787 (`!s a x. convex s /\ closed s /\ x IN s /\ ~(x = closest_point s a)
5788 ==> dist(a,closest_point s a) < dist(a,x)`,
5790 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5791 REWRITE_TAC[GSYM REAL_NOT_LE; CONTRAPOS_THM] THEN
5792 DISCH_TAC THEN MATCH_MP_TAC CLOSEST_POINT_UNIQUE THEN
5793 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[CLOSEST_POINT_LE; REAL_LE_TRANS]);;
5795 let CLOSEST_POINT_LIPSCHITZ = prove
5797 convex s /\ closed s /\ ~(s = {})
5798 ==> dist(closest_point s x,closest_point s y) <= dist(x,y)`,
5799 REPEAT STRIP_TAC THEN REWRITE_TAC[dist; NORM_LE] THEN
5801 `(x - closest_point s x :real^N) dot
5802 (closest_point s y - closest_point s x) <= &0 /\
5803 (y - closest_point s y) dot
5804 (closest_point s x - closest_point s y) <= &0`
5806 [CONJ_TAC THEN MATCH_MP_TAC ANY_CLOSEST_POINT_DOT THEN
5807 EXISTS_TAC `s:real^N->bool` THEN ASM_MESON_TAC[CLOSEST_POINT_EXISTS];
5808 MP_TAC(ISPEC `(x - closest_point s x :real^N) - (y - closest_point s y)`
5810 REWRITE_TAC[NORM_LT; DOT_LSUB; DOT_RSUB; DOT_SYM] THEN REAL_ARITH_TAC]);;
5812 let CONTINUOUS_AT_CLOSEST_POINT = prove
5813 (`!s x. convex s /\ closed s /\ ~(s = {})
5814 ==> (closest_point s) continuous (at x)`,
5815 REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at] THEN
5816 ASM_MESON_TAC[CLOSEST_POINT_LIPSCHITZ; REAL_LET_TRANS]);;
5818 let CONTINUOUS_ON_CLOSEST_POINT = prove
5819 (`!s t. convex s /\ closed s /\ ~(s = {})
5820 ==> (closest_point s) continuous_on t`,
5821 MESON_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON; CONTINUOUS_AT_CLOSEST_POINT]);;
5823 (* ------------------------------------------------------------------------- *)
5824 (* Relating closest points and orthogonality. *)
5825 (* ------------------------------------------------------------------------- *)
5827 let ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL = prove
5829 affine s /\ b IN s /\ (!x. x IN s ==> dist(a,b) <= dist(a,x))
5830 ==> (!x. x IN s ==> orthogonal (x - b) (a - b))`,
5831 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `b:real^N` THEN
5832 REWRITE_TAC[DIST_0; VECTOR_SUB_RZERO; orthogonal; dist; NORM_LE] THEN
5833 REWRITE_TAC[DOT_LSUB] THEN REWRITE_TAC[DOT_RSUB] THEN
5834 REWRITE_TAC[DOT_SYM; REAL_ARITH `a <= a - y - (y - x) <=> &2 * y <= x`] THEN
5835 REPEAT STRIP_TAC THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
5836 ASM_REWRITE_TAC[DOT_RZERO] THEN FIRST_X_ASSUM(fun th ->
5837 MP_TAC(SPEC `vec 0 + --((a dot x) / (x dot x)) % (x - vec 0:real^N)` th) THEN
5838 MP_TAC(SPEC `vec 0 + (a dot x) / (x dot x) % (x - vec 0:real^N)` th)) THEN
5839 ASM_SIMP_TAC[IN_AFFINE_ADD_MUL_DIFF] THEN
5840 REWRITE_TAC[VECTOR_SUB_RZERO; VECTOR_ADD_LID; DOT_RMUL] THEN
5841 REWRITE_TAC[DOT_LMUL; IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
5842 `&2 * x * a <= b * c * z /\ &2 * --x * a <= --b * --c * z
5843 ==> &2 * abs(x * a) <= b * c * z`)) THEN
5844 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
5845 ASM_SIMP_TAC[REAL_NOT_LE; REAL_DIV_RMUL; DOT_EQ_0] THEN
5846 MATCH_MP_TAC(REAL_ARITH `~(x = &0) ==> x < &2 * abs x`) THEN
5847 RULE_ASSUM_TAC(REWRITE_RULE[GSYM DOT_EQ_0]) THEN
5848 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD);;
5850 let ORTHOGONAL_ANY_CLOSEST_POINT = prove
5852 b IN s /\ (!x. x IN s ==> orthogonal (x - b) (a - b))
5853 ==> (!x. x IN s ==> dist(a,b) <= dist(a,x))`,
5854 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `b:real^N` THEN
5855 REWRITE_TAC[dist; NORM_LE; orthogonal; VECTOR_SUB_RZERO] THEN
5856 SIMP_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN
5857 REWRITE_TAC[DOT_POS_LE; REAL_ARITH `a <= a - &0 - (&0 - x) <=> &0 <= x`]);;
5859 let CLOSEST_POINT_AFFINE_ORTHOGONAL = prove
5861 affine s /\ ~(s = {}) /\ x IN s
5862 ==> orthogonal (x - closest_point s a) (a - closest_point s a)`,
5863 GEN_TAC THEN REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5864 DISCH_TAC THEN DISCH_TAC THEN GEN_TAC THEN
5865 MATCH_MP_TAC ANY_CLOSEST_POINT_AFFINE_ORTHOGONAL THEN
5866 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSEST_POINT_EXISTS THEN
5867 ASM_SIMP_TAC[CLOSED_AFFINE]);;
5869 let CLOSEST_POINT_AFFINE_ORTHOGONAL_EQ = prove
5872 ==> (closest_point s a = b <=>
5873 !x. x IN s ==> orthogonal (x - b) (a - b))`,
5874 REPEAT STRIP_TAC THEN EQ_TAC THENL
5875 [ASM_MESON_TAC[CLOSEST_POINT_AFFINE_ORTHOGONAL; MEMBER_NOT_EMPTY];
5876 DISCH_TAC THEN CONV_TAC SYM_CONV THEN
5877 MATCH_MP_TAC CLOSEST_POINT_UNIQUE THEN
5878 ASM_SIMP_TAC[CLOSED_AFFINE; AFFINE_IMP_CONVEX] THEN
5879 MATCH_MP_TAC ORTHOGONAL_ANY_CLOSEST_POINT THEN ASM_REWRITE_TAC[]]);;
5881 (* ------------------------------------------------------------------------- *)
5882 (* Various point-to-set separating/supporting hyperplane theorems. *)
5883 (* ------------------------------------------------------------------------- *)
5885 let SUPPORTING_HYPERPLANE_COMPACT_POINT_SUP = prove
5886 (`!a c s:real^N->bool.
5887 compact s /\ ~(s = {})
5888 ==> ?b y. y IN s /\ a dot (y - c) = b /\
5889 (!x. x IN s ==> a dot (x - c) <= b)`,
5890 REPEAT STRIP_TAC THEN
5891 MP_TAC(ISPECL [`\x:real^N. a dot (x - c)`; `s:real^N->bool`]
5892 CONTINUOUS_ATTAINS_SUP) THEN
5893 ASM_REWRITE_TAC[] THEN
5894 ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
5895 SUBGOAL_THEN `(\x:real^N. a dot (x - c)) = (\x. a dot x) o (\x. x - c)`
5896 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
5897 REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
5898 SIMP_TAC[CONTINUOUS_ON_LIFT_DOT; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST;
5899 CONTINUOUS_ON_ID]);;
5901 let SUPPORTING_HYPERPLANE_COMPACT_POINT_INF = prove
5902 (`!a c s:real^N->bool.
5903 compact s /\ ~(s = {})
5904 ==> ?b y. y IN s /\ a dot (y - c) = b /\
5905 (!x. x IN s ==> a dot (x - c) >= b)`,
5906 REPEAT STRIP_TAC THEN
5907 MP_TAC(ISPECL [`--a:real^N`; `c:real^N`; `s:real^N->bool`]
5908 SUPPORTING_HYPERPLANE_COMPACT_POINT_SUP) THEN
5909 ASM_REWRITE_TAC[] THEN
5910 DISCH_THEN(X_CHOOSE_THEN `b:real`
5911 (fun th -> EXISTS_TAC `--b:real` THEN MP_TAC th)) THEN
5912 REWRITE_TAC[DOT_LNEG; REAL_ARITH `x >= -- b <=> --x <= b`] THEN
5913 REWRITE_TAC[REAL_NEG_EQ]);;
5915 let SUPPORTING_HYPERPLANE_CLOSED_POINT = prove
5916 (`!s z:real^N. convex s /\ closed s /\ ~(s = {}) /\ ~(z IN s)
5917 ==> ?a b y. a dot z < b /\ y IN s /\ (a dot y = b) /\
5918 (!x. x IN s ==> a dot x >= b)`,
5919 REPEAT STRIP_TAC THEN
5920 MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN
5921 ASM_REWRITE_TAC[] THEN
5922 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
5923 EXISTS_TAC `y - z:real^N` THEN EXISTS_TAC `(y - z:real^N) dot y` THEN
5924 EXISTS_TAC `y:real^N` THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
5925 ASM_REWRITE_TAC[GSYM DOT_RSUB; DOT_POS_LT; VECTOR_SUB_EQ] THEN
5926 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN X_GEN_TAC `x:real^N` THEN
5927 DISCH_TAC THEN SUBGOAL_THEN
5928 `!u. &0 <= u /\ u <= &1 ==> dist(z:real^N,y) <= dist(z,(&1 - u) % y + u % x)`
5929 MP_TAC THENL [ASM_MESON_TAC[CONVEX_ALT]; ALL_TAC] THEN
5930 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
5931 REWRITE_TAC[real_ge; REAL_NOT_LE; NOT_FORALL_THM; NOT_IMP] THEN
5932 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `x < y <=> y - x > &0`] THEN
5933 REWRITE_TAC[VECTOR_ARITH
5934 `(a - b) dot x - (a - b) dot y = (b - a) dot (y - x)`] THEN
5935 DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN
5936 REWRITE_TAC[VECTOR_ARITH `y + u % (x - y) = (&1 - u) % y + u % x`] THEN
5937 MESON_TAC[REAL_LT_IMP_LE]);;
5939 let SEPARATING_HYPERPLANE_CLOSED_POINT_INSET = prove
5940 (`!s z:real^N. convex s /\ closed s /\ ~(s = {}) /\ ~(z IN s)
5942 (a - z) dot z < b /\
5943 (!x. x IN s ==> (a - z) dot x > b)`,
5944 REPEAT STRIP_TAC THEN
5945 MP_TAC(ISPECL [`s:real^N->bool`; `z:real^N`] DISTANCE_ATTAINS_INF) THEN
5946 ASM_REWRITE_TAC[] THEN
5947 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
5948 EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN
5949 EXISTS_TAC `(y - z:real^N) dot z + norm(y - z) pow 2 / &2` THEN
5950 SUBGOAL_THEN `&0 < norm(y - z:real^N)` ASSUME_TAC THENL
5951 [ASM_MESON_TAC[NORM_POS_LT; VECTOR_SUB_EQ]; ALL_TAC] THEN
5952 ASM_SIMP_TAC[REAL_LT_ADDR; REAL_LT_DIV; REAL_POW_LT;
5953 REAL_OF_NUM_LT; ARITH] THEN
5954 REWRITE_TAC[NORM_POW_2; REAL_ARITH `a > b + c <=> c < a - b`] THEN
5955 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
5956 SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
5957 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
5958 REWRITE_TAC[VECTOR_ARITH
5959 `((y - z) dot x - (y - z) dot z) * &2 - (y - z) dot (y - z) =
5960 &2 * ((y - z) dot (x - y)) + (y - z) dot (y - z)`] THEN
5961 MATCH_MP_TAC(REAL_ARITH `~(--x > &0) /\ &0 < y ==> &0 < &2 * x + y`) THEN
5962 ASM_SIMP_TAC[GSYM NORM_POW_2; REAL_POW_LT] THEN
5963 REWRITE_TAC[GSYM DOT_LNEG; VECTOR_NEG_SUB] THEN
5964 DISCH_THEN(MP_TAC o MATCH_MP CLOSER_POINT_LEMMA) THEN
5965 REWRITE_TAC[NOT_EXISTS_THM] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
5966 GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5967 REWRITE_TAC[REAL_NOT_LT] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
5968 REWRITE_TAC[VECTOR_ARITH `y + u % (x - y) = (&1 - u) % y + u % x`] THEN
5969 ASM_MESON_TAC[CONVEX_ALT; REAL_LT_IMP_LE]);;
5971 let SEPARATING_HYPERPLANE_CLOSED_0_INSET = prove
5973 convex s /\ closed s /\ ~(s = {}) /\ ~(vec 0 IN s)
5974 ==> ?a b. a IN s /\ ~(a = vec 0) /\ &0 < b /\
5975 (!x. x IN s ==> a dot x > b)`,
5976 REPEAT GEN_TAC THEN DISCH_TAC THEN
5977 FIRST_ASSUM(MP_TAC o MATCH_MP SEPARATING_HYPERPLANE_CLOSED_POINT_INSET) THEN
5978 REWRITE_TAC[DOT_RZERO; real_gt] THEN
5979 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
5980 SIMP_TAC[VECTOR_SUB_RZERO] THEN ASM_MESON_TAC[]);;
5982 let SEPARATING_HYPERPLANE_CLOSED_POINT = prove
5983 (`!s z:real^N. convex s /\ closed s /\ ~(z IN s)
5984 ==> ?a b. a dot z < b /\ (!x. x IN s ==> a dot x > b)`,
5985 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
5986 [MAP_EVERY EXISTS_TAC [`--z:real^N`; `&1`] THEN
5987 SIMP_TAC[DOT_LNEG; REAL_ARITH `&0 <= x ==> --x < &1`; DOT_POS_LE] THEN
5988 ASM_MESON_TAC[NOT_IN_EMPTY];
5990 ASM_MESON_TAC[SEPARATING_HYPERPLANE_CLOSED_POINT_INSET]);;
5992 let SEPARATING_HYPERPLANE_CLOSED_0 = prove
5994 convex s /\ closed s /\ ~(vec 0 IN s)
5995 ==> ?a b. ~(a = vec 0) /\ &0 < b /\ (!x. x IN s ==> a dot x > b)`,
5996 REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
5997 [EXISTS_TAC `basis 1:real^N` THEN EXISTS_TAC `&1` THEN
5998 ASM_REWRITE_TAC[NOT_IN_EMPTY; REAL_LT_01; GSYM NORM_POS_LT] THEN
5999 ASM_SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_LT_01];
6000 FIRST_X_ASSUM(MP_TAC o MATCH_MP SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6001 REWRITE_TAC[DOT_RZERO; real_gt] THEN
6002 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
6003 ASM_MESON_TAC[MEMBER_NOT_EMPTY; DOT_LZERO; REAL_LT_ANTISYM]]);;
6005 (* ------------------------------------------------------------------------- *)
6006 (* Now set-to-set for closed/compact sets. *)
6007 (* ------------------------------------------------------------------------- *)
6009 let SEPARATING_HYPERPLANE_CLOSED_COMPACT = prove
6010 (`!s t. convex s /\ closed s /\
6011 convex t /\ compact t /\ ~(t = {}) /\ DISJOINT s t
6012 ==> ?a:real^N b. (!x. x IN s ==> a dot x < b) /\
6013 (!x. x IN t ==> a dot x > b)`,
6014 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6015 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
6016 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
6017 REWRITE_TAC[BOUNDED_POS] THEN
6018 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN
6019 SUBGOAL_THEN `?z:real^N. norm(z) = b + &1` CHOOSE_TAC THENL
6020 [ASM_SIMP_TAC[VECTOR_CHOOSE_SIZE; REAL_ARITH `&0 < b ==> &0 <= b + &1`];
6022 MP_TAC(SPECL [`t:real^N->bool`; `z:real^N`]
6023 SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6024 ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
6025 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN
6026 ASM_MESON_TAC[REAL_ARITH `~(b + &1 <= b)`];
6028 MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0 :real^N`]
6029 SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6030 ASM_SIMP_TAC[CLOSED_COMPACT_DIFFERENCES; CONVEX_DIFFERENCES] THEN
6032 [REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
6033 REWRITE_TAC[VECTOR_SUB_EQ] THEN
6034 ASM_MESON_TAC[DISJOINT; NOT_IN_EMPTY; IN_INTER; EXTENSION];
6036 SIMP_TAC[DOT_RZERO; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
6037 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN
6038 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6039 GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN
6040 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
6041 ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
6042 REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL; DOT_RSUB] THEN
6043 REWRITE_TAC[real_gt; REAL_LT_SUB_LADD] THEN DISCH_TAC THEN
6044 EXISTS_TAC `--a:real^N` THEN
6045 MP_TAC(SPEC `IMAGE (\x:real^N. a dot x) t` SUP) THEN
6046 ABBREV_TAC `k = sup (IMAGE (\x:real^N. a dot x) t)` THEN
6047 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN ANTS_TAC THENL
6048 [ASM_MESON_TAC[REAL_ARITH `b + x < y ==> x <= y - b`; MEMBER_NOT_EMPTY];
6050 STRIP_TAC THEN EXISTS_TAC `--(k + b / &2)` THEN
6051 REWRITE_TAC[DOT_LNEG; REAL_LT_NEG2] THEN REPEAT STRIP_TAC THEN
6052 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH;
6053 REAL_ARITH `&0 < b /\ x <= k ==> x < k + b`] THEN
6054 FIRST_X_ASSUM(MP_TAC o SPEC `k - b / &2`) THEN
6055 ASM_SIMP_TAC[REAL_ARITH `k <= k - b2 <=> ~(&0 < b2)`; REAL_LT_DIV;
6056 REAL_OF_NUM_LT; ARITH; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM; NOT_IMP] THEN
6057 X_GEN_TAC `y:real^N` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6058 MATCH_MP_TAC(REAL_ARITH
6059 `!b. (b2 + b2 = b) /\ b + ay < ax ==> ~(ay <= k - b2) ==> k + b2 < ax`) THEN
6060 ASM_MESON_TAC[REAL_HALF]);;
6062 let SEPARATING_HYPERPLANE_COMPACT_CLOSED = prove
6063 (`!s t. convex s /\ compact s /\ ~(s = {}) /\
6064 convex t /\ closed t /\ DISJOINT s t
6065 ==> ?a:real^N b. (!x. x IN s ==> a dot x < b) /\
6066 (!x. x IN t ==> a dot x > b)`,
6067 REPEAT STRIP_TAC THEN
6068 MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`]
6069 SEPARATING_HYPERPLANE_CLOSED_COMPACT) THEN
6070 ANTS_TAC THENL [ASM_MESON_TAC[DISJOINT_SYM]; ALL_TAC] THEN
6071 REWRITE_TAC[real_gt] THEN
6072 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real`
6073 STRIP_ASSUME_TAC)) THEN
6074 MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`] THEN
6075 ASM_REWRITE_TAC[REAL_LT_NEG2; DOT_LNEG]);;
6077 let SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO = prove
6078 (`!s t:real^N->bool.
6079 convex s /\ compact s /\ ~(s = {}) /\
6080 convex t /\ closed t /\ DISJOINT s t
6081 ==> ?a b. ~(a = vec 0) /\
6082 (!x. x IN s ==> a dot x < b) /\
6083 (!x. x IN t ==> a dot x > b)`,
6084 REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL
6085 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN
6086 EXISTS_TAC `basis 1:real^N` THEN
6088 `bounded(IMAGE (\x:real^N. lift(basis 1 dot x)) s)`
6090 [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
6091 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
6092 ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DOT];
6093 REWRITE_TAC[BOUNDED_POS_LT; FORALL_IN_IMAGE; NORM_LIFT] THEN
6094 SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN
6095 MESON_TAC[REAL_ARITH `abs x < b ==> x < b`]];
6097 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`]
6098 SEPARATING_HYPERPLANE_COMPACT_CLOSED) THEN
6099 ASM_REWRITE_TAC[] THEN
6100 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
6101 ASM_CASES_TAC `a:real^N = vec 0` THEN ASM_REWRITE_TAC[] THEN
6102 REWRITE_TAC[DOT_LZERO; real_gt] THEN
6103 ASM_MESON_TAC[REAL_LT_ANTISYM; MEMBER_NOT_EMPTY]]);;
6105 let SEPARATING_HYPERPLANE_COMPACT_COMPACT = prove
6106 (`!s t:real^N->bool.
6107 convex s /\ compact s /\ convex t /\ compact t /\ DISJOINT s t
6108 ==> ?a b. ~(a = vec 0) /\
6109 (!x. x IN s ==> a dot x < b) /\
6110 (!x. x IN t ==> a dot x > b)`,
6111 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6112 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN
6113 EXISTS_TAC `--basis 1:real^N` THEN
6115 `bounded(IMAGE (\x:real^N. lift(basis 1 dot x)) t)`
6117 [MATCH_MP_TAC COMPACT_IMP_BOUNDED THEN
6118 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
6119 ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DOT];
6120 REWRITE_TAC[BOUNDED_POS_LT; FORALL_IN_IMAGE; NORM_LIFT] THEN
6121 SIMP_TAC[VECTOR_NEG_EQ_0; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN
6122 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN
6123 EXISTS_TAC `--b:real` THEN REWRITE_TAC[DOT_LNEG] THEN
6124 REWRITE_TAC[REAL_ARITH `--x > --y <=> x < y`] THEN
6125 ASM_MESON_TAC[REAL_ARITH `abs x < b ==> x < b`]];
6127 MATCH_MP_TAC SEPARATING_HYPERPLANE_COMPACT_CLOSED_NONZERO THEN
6128 ASM_SIMP_TAC[COMPACT_IMP_CLOSED]]);;
6130 (* ------------------------------------------------------------------------- *)
6131 (* General case without assuming closure and getting non-strict separation. *)
6132 (* ------------------------------------------------------------------------- *)
6134 let SEPARATING_HYPERPLANE_SET_0_INSPAN = prove
6136 convex s /\ ~(s = {}) /\ ~(vec 0 IN s)
6137 ==> ?a b. a IN span s /\ ~(a = vec 0) /\
6138 !x. x IN s ==> &0 <= a dot x`,
6139 REPEAT STRIP_TAC THEN
6140 ABBREV_TAC `k = \c:real^N. {x | &0 <= c dot x}` THEN
6142 `~((span s INTER frontier(cball(vec 0:real^N,&1))) INTER
6143 (INTERS (IMAGE k (s:real^N->bool))) = {})`
6146 SIMP_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_INTERS; NOT_FORALL_THM;
6147 FORALL_IN_IMAGE; FRONTIER_CBALL; REAL_LT_01] THEN
6148 EXPAND_TAC "k" THEN REWRITE_TAC[IN_SPHERE_0; IN_ELIM_THM; NORM_NEG] THEN
6149 MESON_TAC[NORM_EQ_0; REAL_ARITH `~(&1 = &0)`; DOT_SYM]] THEN
6150 MATCH_MP_TAC COMPACT_IMP_FIP THEN
6151 SIMP_TAC[COMPACT_CBALL; COMPACT_FRONTIER; FORALL_IN_IMAGE;
6152 CLOSED_INTER_COMPACT; CLOSED_SPAN] THEN
6154 [EXPAND_TAC "k" THEN REWRITE_TAC[GSYM real_ge; CLOSED_HALFSPACE_GE];
6156 REWRITE_TAC[FINITE_SUBSET_IMAGE] THEN GEN_TAC THEN
6157 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` MP_TAC) THEN
6158 ASM_CASES_TAC `c:real^N->bool = {}` THENL
6159 [ASM_SIMP_TAC[INTERS_0; INTER_UNIV; IMAGE_CLAUSES] THEN
6160 DISCH_THEN(K ALL_TAC) THEN
6161 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
6162 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
6163 SUBGOAL_THEN `~(a:real^N = vec 0)` ASSUME_TAC THENL
6164 [ASM_MESON_TAC[]; ALL_TAC] THEN
6165 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
6166 EXISTS_TAC `inv(norm a) % a:real^N` THEN
6167 ASM_SIMP_TAC[IN_INTER; FRONTIER_CBALL; SPAN_CLAUSES; IN_SPHERE_0] THEN
6168 REWRITE_TAC[DIST_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN
6169 ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0];
6171 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
6172 MP_TAC(ISPEC `convex hull (c:real^N->bool)`
6173 SEPARATING_HYPERPLANE_CLOSED_0_INSET) THEN
6175 [ASM_REWRITE_TAC[CONVEX_HULL_EQ_EMPTY] THEN
6176 ASM_MESON_TAC[CONVEX_CONVEX_HULL; SUBSET; SUBSET_HULL; HULL_SUBSET;
6177 FINITE_IMP_COMPACT_CONVEX_HULL; COMPACT_IMP_CLOSED];
6179 REWRITE_TAC[DOT_RZERO; real_gt] THEN
6180 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real`
6181 STRIP_ASSUME_TAC)) THEN
6182 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_INTERS; FORALL_IN_IMAGE] THEN
6183 EXPAND_TAC "k" THEN SIMP_TAC[IN_ELIM_THM; FRONTIER_CBALL; REAL_LT_01] THEN
6184 REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG] THEN
6185 EXISTS_TAC `inv(norm(a)) % a:real^N` THEN REWRITE_TAC[DOT_RMUL] THEN
6186 SUBGOAL_THEN `(a:real^N) IN s` ASSUME_TAC THENL
6187 [ASM_MESON_TAC[SUBSET; HULL_MINIMAL]; ASM_SIMP_TAC[SPAN_CLAUSES]] THEN
6188 REWRITE_TAC[IN_SPHERE_0; VECTOR_SUB_LZERO; NORM_NEG; NORM_MUL] THEN
6189 REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NORM] THEN
6190 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN
6191 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_EQ_LDIV_EQ; NORM_POS_LT] THEN
6192 REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID] THEN
6193 ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TRANS; HULL_SUBSET; SUBSET; DOT_SYM]);;
6195 let SEPARATING_HYPERPLANE_SET_POINT_INAFF = prove
6197 convex s /\ ~(s = {}) /\ ~(z IN s)
6198 ==> ?a b. (z + a) IN affine hull (z INSERT s) /\ ~(a = vec 0) /\
6199 a dot z <= b /\ (!x. x IN s ==> a dot x >= b)`,
6200 REPEAT STRIP_TAC THEN
6201 MP_TAC(ISPEC `IMAGE (\x:real^N. --z + x) s`
6202 SEPARATING_HYPERPLANE_SET_0_INSPAN) THEN
6203 ASM_SIMP_TAC[FORALL_IN_IMAGE; CONVEX_TRANSLATION; IMAGE_EQ_EMPTY] THEN
6204 REWRITE_TAC[IN_IMAGE; VECTOR_ARITH `vec 0:real^N = --z + x <=> x = z`] THEN
6205 ASM_SIMP_TAC[UNWIND_THM2; AFFINE_HULL_INSERT_SPAN; IN_ELIM_THM] THEN
6206 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
6207 REWRITE_TAC[GSYM SIMPLE_IMAGE; VECTOR_ARITH `--x + y:real^N = y - x`] THEN
6208 STRIP_TAC THEN ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
6209 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
6210 EXISTS_TAC `(a:real^N) dot z` THEN REWRITE_TAC[REAL_LE_REFL] THEN
6211 ASM_REWRITE_TAC[REAL_ARITH `x >= y <=> &0 <= x - y`; GSYM DOT_RSUB]);;
6213 let SEPARATING_HYPERPLANE_SET_0 = prove
6215 convex s /\ ~(vec 0 IN s)
6216 ==> ?a b. ~(a = vec 0) /\ !x. x IN s ==> &0 <= a dot x`,
6217 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
6218 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
6219 MESON_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1];
6220 ASM_MESON_TAC[SEPARATING_HYPERPLANE_SET_0_INSPAN]]);;
6222 let SEPARATING_HYPERPLANE_SETS = prove
6223 (`!s t. convex s /\ convex t /\ ~(s = {}) /\ ~(t = {}) /\ DISJOINT s t
6224 ==> ?a:real^N b. ~(a = vec 0) /\
6225 (!x. x IN s ==> a dot x <= b) /\
6226 (!x. x IN t ==> a dot x >= b)`,
6227 REPEAT STRIP_TAC THEN
6228 MP_TAC(ISPEC `{y - x:real^N | y IN t /\ x IN s}`
6229 SEPARATING_HYPERPLANE_SET_0) THEN
6230 ASM_SIMP_TAC[CONVEX_DIFFERENCES] THEN ANTS_TAC THENL
6231 [REWRITE_TAC[IN_ELIM_THM] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
6232 REWRITE_TAC[VECTOR_SUB_EQ] THEN
6233 ASM_MESON_TAC[DISJOINT; NOT_IN_EMPTY; IN_INTER; EXTENSION];
6235 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
6236 SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
6237 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6238 GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN
6239 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
6240 ONCE_REWRITE_TAC[IMP_CONJ] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
6241 REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL; DOT_RSUB; REAL_SUB_LE] THEN
6243 MP_TAC(SPEC `IMAGE (\x:real^N. a dot x) s` SUP) THEN
6244 ABBREV_TAC `k = sup (IMAGE (\x:real^N. a dot x) s)` THEN
6245 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IMAGE_EQ_EMPTY; real_ge] THEN ANTS_TAC THENL
6246 [ASM_MESON_TAC[MEMBER_NOT_EMPTY]; ASM_MESON_TAC[]]);;
6248 (* ------------------------------------------------------------------------- *)
6249 (* More convexity generalities. *)
6250 (* ------------------------------------------------------------------------- *)
6252 let CONVEX_CLOSURE = prove
6253 (`!s:real^N->bool. convex s ==> convex(closure s)`,
6254 REWRITE_TAC[convex; CLOSURE_SEQUENTIAL] THEN
6255 GEN_TAC THEN DISCH_TAC THEN
6256 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
6257 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `a:num->real^N`) MP_TAC) THEN
6258 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `b:num->real^N`) MP_TAC) THEN
6259 STRIP_TAC THEN EXISTS_TAC `\n:num. u % a(n) + v % b(n) :real^N` THEN
6260 ASM_SIMP_TAC[LIM_ADD; LIM_CMUL]);;
6262 let CONVEX_INTERIOR = prove
6263 (`!s:real^N->bool. convex s ==> convex(interior s)`,
6264 REWRITE_TAC[CONVEX_ALT; IN_INTERIOR; SUBSET; IN_BALL; dist] THEN
6265 REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT GEN_TAC THEN
6266 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `d:real`) MP_TAC) THEN
6267 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `e:real`) STRIP_ASSUME_TAC) THEN
6268 EXISTS_TAC `min d e` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
6269 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
6270 SUBST1_TAC(VECTOR_ARITH `z:real^N =
6271 (&1 - u) % (z - u % (y - x)) + u % (z + (&1 - u) % (y - x))`) THEN
6272 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
6273 ASM_MESON_TAC[VECTOR_ARITH `x - (z - u % (y - x)) =
6274 ((&1 - u) % x + u % y) - z:real^N`;
6275 VECTOR_ARITH `y - (z + (&1 - u) % (y - x)) =
6276 ((&1 - u) % x + u % y) - z:real^N`]);;
6278 let CONVEX_ON_SETDIST = prove
6279 (`!s t:real^N->bool. convex t ==> (\x. setdist ({x},t)) convex_on s`,
6281 `!s t:real^N->bool. convex t /\ closed t
6282 ==> (\x. setdist ({x},t)) convex_on s`
6285 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o
6286 SPECL [`s:real^N->bool`; `closure t:real^N->bool`]) THEN
6287 ASM_SIMP_TAC[CLOSED_CLOSURE; SETDIST_CLOSURE; CONVEX_CLOSURE]] THEN
6288 REPEAT STRIP_TAC THEN REWRITE_TAC[convex_on] THEN
6289 ASM_CASES_TAC `t:real^N->bool = {}` THEN
6290 ASM_SIMP_TAC[SETDIST_EMPTY; REAL_MUL_RZERO; REAL_ADD_RID; REAL_LE_REFL] THEN
6291 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`; `u:real`; `v:real`] THEN
6293 MP_TAC(ISPECL [`{x:real^N}`; `t:real^N->bool`] SETDIST_COMPACT_CLOSED) THEN
6294 MP_TAC(ISPECL [`{y:real^N}`; `t:real^N->bool`] SETDIST_COMPACT_CLOSED) THEN
6295 ASM_REWRITE_TAC[NOT_INSERT_EMPTY; COMPACT_SING; UNWIND_THM2; SETDIST_CLOSURE;
6296 CLOSURE_EQ_EMPTY; RIGHT_EXISTS_AND_THM; IN_SING] THEN
6297 DISCH_THEN(X_CHOOSE_THEN `y':real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
6298 DISCH_THEN(X_CHOOSE_THEN `x':real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
6299 MATCH_MP_TAC REAL_LE_TRANS THEN
6300 EXISTS_TAC `dist(u % x + v % y:real^N,u % x' + v % y')` THEN CONJ_TAC THENL
6301 [MATCH_MP_TAC SETDIST_LE_DIST THEN REWRITE_TAC[IN_SING] THEN
6302 ASM_MESON_TAC[convex];
6303 REWRITE_TAC[dist] THEN MATCH_MP_TAC(NORM_ARITH
6304 `norm(a - a':real^N) + norm(b - b') <= r
6305 ==> norm((a + b) - (a' + b')) <= r`) THEN
6306 ASM_REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL; dist] THEN
6307 ASM_REWRITE_TAC[real_abs; REAL_LE_REFL]]);;
6309 (* ------------------------------------------------------------------------- *)
6310 (* Moving and scaling convex hulls. *)
6311 (* ------------------------------------------------------------------------- *)
6313 let CONVEX_HULL_TRANSLATION = prove
6315 convex hull (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (convex hull s)`,
6316 REPEAT GEN_TAC THEN MATCH_MP_TAC HULL_IMAGE THEN
6317 REWRITE_TAC[CONVEX_TRANSLATION_EQ; CONVEX_CONVEX_HULL] THEN
6318 REWRITE_TAC[VECTOR_ARITH `a + x:real^N = y <=> x = y - a`; EXISTS_REFL] THEN
6321 add_translation_invariants [CONVEX_HULL_TRANSLATION];;
6323 let CONVEX_HULL_SCALING = prove
6324 (`!s:real^N->bool c.
6325 convex hull (IMAGE (\x. c % x) s) = IMAGE (\x. c % x) (convex hull s)`,
6326 REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THENL
6327 [ASM_SIMP_TAC[IMAGE_CONST; VECTOR_MUL_LZERO; CONVEX_HULL_EQ_EMPTY] THEN
6328 COND_CASES_TAC THEN REWRITE_TAC[CONVEX_HULL_EMPTY; CONVEX_HULL_SING];
6330 MATCH_MP_TAC HULL_IMAGE THEN
6331 ASM_SIMP_TAC[CONVEX_SCALING_EQ; CONVEX_CONVEX_HULL] THEN
6332 REWRITE_TAC[VECTOR_ARITH `c % x = c % y <=> c % (x - y) = vec 0`] THEN
6333 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
6334 X_GEN_TAC `x:real^N` THEN EXISTS_TAC `inv c % x:real^N` THEN
6335 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]);;
6337 let CONVEX_HULL_AFFINITY = prove
6339 convex hull (IMAGE (\x. a + c % x) s) =
6340 IMAGE (\x. a + c % x) (convex hull s)`,
6342 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
6343 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
6344 ASM_SIMP_TAC[IMAGE_o; CONVEX_HULL_TRANSLATION; CONVEX_HULL_SCALING]);;
6346 (* ------------------------------------------------------------------------- *)
6347 (* Convex set as intersection of halfspaces. *)
6348 (* ------------------------------------------------------------------------- *)
6350 let CONVEX_HALFSPACE_INTERSECTION = prove
6351 (`!s. closed(s:real^N->bool) /\ convex s
6352 ==> s = INTERS {h | s SUBSET h /\ ?a b. h = {x | a dot x <= b}}`,
6353 REPEAT STRIP_TAC THEN
6354 GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_INTERS] THEN
6355 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
6356 REWRITE_TAC[MESON[] `(!t. (P t /\ ?a b. t = x a b) ==> Q t) <=>
6357 (!a b. P(x a b) ==> Q(x a b))`] THEN
6358 EQ_TAC THENL [SET_TAC[]; ALL_TAC] THEN STRIP_TAC THEN
6359 MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
6360 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`]
6361 SEPARATING_HYPERPLANE_CLOSED_POINT) THEN
6362 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
6363 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN
6364 FIRST_X_ASSUM(MP_TAC o SPECL [`--a:real^N`; `--b:real`]) THEN
6365 ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; DOT_LNEG; NOT_IMP] THEN
6366 ASM_SIMP_TAC[REAL_LE_NEG2; REAL_LT_NEG2; REAL_NOT_LE;
6367 REAL_ARITH `a > b ==> b <= a`]);;
6369 (* ------------------------------------------------------------------------- *)
6370 (* Radon's theorem (from Lars Schewe). *)
6371 (* ------------------------------------------------------------------------- *)
6373 let RADON_EX_LEMMA = prove
6374 (`!(c:real^N->bool).
6375 FINITE c /\ affine_dependent c
6376 ==> (?u. sum c u = &0 /\ (?v. v IN c /\ ~(u v = &0)) /\
6377 vsum c (\v. u v % v) = (vec 0):real^N)`,
6378 REWRITE_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN
6379 REPEAT STRIP_TAC THEN
6380 EXISTS_TAC `\v:real^N. if v IN s then u v else &0` THEN
6381 ASM_SIMP_TAC[GSYM SUM_RESTRICT_SET] THEN
6382 ASM_SIMP_TAC[COND_RAND;COND_RATOR;
6383 VECTOR_MUL_LZERO;GSYM VSUM_RESTRICT_SET] THEN
6384 ASM_SIMP_TAC[SET_RULE `s SUBSET c ==> {x | x IN c /\ x IN s} = s`] THEN
6385 EXISTS_TAC `v:real^N` THEN
6386 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
6388 let RADON_S_LEMMA = prove
6390 FINITE s /\ sum s f = &0
6391 ==> sum {x | x IN s /\ &0 < f x} f =
6392 -- sum {x | x IN s /\ f x < &0} f`,
6393 REWRITE_TAC[REAL_ARITH `a = --b <=> a + b = &0`] THEN
6394 REPEAT STRIP_TAC THEN
6395 ASM_SIMP_TAC[FINITE_RESTRICT;GSYM SUM_UNION;
6396 REWRITE_RULE [REAL_ARITH `&0 < f x ==> ~(f x < &0)`]
6397 (SET_RULE `(!x:A. &0 < f x ==> ~(f x < &0))
6398 ==> DISJOINT {x | x IN s /\ &0 < f x}
6399 {x | x IN s /\ f x < &0}`)] THEN
6400 MATCH_MP_TAC (REAL_ARITH `!a b.a = &0 /\ a + b = &0 ==> b = &0`) THEN
6401 EXISTS_TAC `sum {x:A | x IN s /\ f x = &0} f` THEN
6403 [ASM_SIMP_TAC[SUM_RESTRICT_SET] THEN REWRITE_TAC[COND_ID;SUM_0];
6405 SUBGOAL_THEN `DISJOINT {x:A | x IN s /\ f x = &0}
6406 ({x | x IN s /\ &0 < f x} UNION
6407 {x | x IN s /\ f x < &0})` ASSUME_TAC THENL
6408 [REWRITE_TAC[DISJOINT;UNION;INTER;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN
6411 ASM_SIMP_TAC[FINITE_UNION;FINITE_RESTRICT;GSYM SUM_UNION] THEN
6412 FIRST_X_ASSUM (SUBST1_TAC o GSYM) THEN
6413 MATCH_MP_TAC (MESON[] `a = b ==> sum a f = sum b f`) THEN
6414 REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION] THEN
6415 MESON_TAC[REAL_LT_TOTAL]);;
6417 let RADON_V_LEMMA = prove
6419 FINITE s /\ vsum s f = vec 0 /\ (!x. g x = &0 ==> f x = vec 0)
6420 ==> (vsum {x | x IN s /\ &0 < g x} f) :real^N =
6421 -- vsum {x | x IN s /\ g x < &0} f`,
6422 REWRITE_TAC[VECTOR_ARITH `a:real^N = --b <=> a + b = vec 0`] THEN
6423 REPEAT STRIP_TAC THEN
6424 ASM_SIMP_TAC[FINITE_RESTRICT;GSYM VSUM_UNION;
6425 REWRITE_RULE [REAL_ARITH `&0 < f x ==> ~(f x < &0)`]
6426 (SET_RULE `(!x:A. &0 < f x ==> ~(f x < &0))
6427 ==> DISJOINT {x | x IN s /\ &0 < f x}
6428 {x | x IN s /\ f x < &0}`)] THEN
6429 MATCH_MP_TAC (VECTOR_ARITH
6430 `!a b. (a:real^N) = vec 0 /\ a + b = vec 0 ==> b = vec 0`) THEN
6431 EXISTS_TAC `(vsum {x:A | x IN s /\ g x = &0} f):real^N` THEN
6433 [ASM_SIMP_TAC[VSUM_RESTRICT_SET;COND_ID;VSUM_0];ALL_TAC] THEN
6434 SUBGOAL_THEN `DISJOINT {x:A | x IN s /\ g x = &0}
6435 ({x | x IN s /\ &0 < g x} UNION
6436 {x | x IN s /\ g x < &0})` ASSUME_TAC THENL
6437 [REWRITE_TAC[DISJOINT;UNION;INTER;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN
6440 ASM_SIMP_TAC[FINITE_UNION;FINITE_RESTRICT;GSYM VSUM_UNION] THEN
6441 FIRST_X_ASSUM (SUBST1_TAC o GSYM) THEN
6442 MATCH_MP_TAC (MESON[] `a = b ==> vsum a f = vsum b f`) THEN
6443 REWRITE_TAC[EXTENSION;IN_ELIM_THM;UNION] THEN
6444 MESON_TAC[REAL_LT_TOTAL]);;
6446 let RADON_PARTITION = prove
6447 (`!(c:real^N->bool).
6448 FINITE c /\ affine_dependent c
6449 ==> ?(m:real^N->bool) (p:real^N->bool).
6452 ~(DISJOINT (convex hull m) (convex hull p))`,
6453 REPEAT STRIP_TAC THEN
6454 MP_TAC (ISPEC `c:real^N->bool` RADON_EX_LEMMA) THEN
6455 ASM_REWRITE_TAC[] THEN
6456 REPEAT STRIP_TAC THEN
6457 MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v <= &0}`;
6458 `{v:real^N | v IN c /\ u v > &0}`] THEN
6459 REPEAT CONJ_TAC THENL
6460 [REWRITE_TAC[DISJOINT;INTER;
6461 IN_ELIM_THM;REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN
6463 REWRITE_TAC[UNION;IN_ELIM_THM;REAL_ARITH `x <= &0 <=> ~(x > &0)`] THEN
6466 SUBGOAL_THEN `~(sum {x:real^N | x IN c /\ u x > &0} u = &0)` ASSUME_TAC THENL
6467 [MATCH_MP_TAC (REAL_ARITH `a > &0 ==> ~(a = &0)`) THEN
6468 REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN
6469 MATCH_MP_TAC (REWRITE_RULE[SUM_0] (ISPEC `\x. &0` SUM_LT_ALL)) THEN
6470 ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM;EXTENSION;NOT_IN_EMPTY] THEN
6471 REWRITE_TAC[MESON[]`~(!x. ~(P x /\ Q x)) = ?x. P x /\ Q x`] THEN
6472 ASM_CASES_TAC `&0 < u (v:real^N)` THENL
6473 [ASM SET_TAC[];ALL_TAC] THEN
6474 POP_ASSUM MP_TAC THEN POP_ASSUM (K ALL_TAC) THEN POP_ASSUM MP_TAC THEN
6475 REWRITE_TAC[IMP_IMP;REAL_ARITH `~(a = &0) /\ ~(&0 < a) <=> a < &0`] THEN
6477 REWRITE_TAC[MESON[REAL_NOT_LT]
6478 `(?x:real^N. P x /\ &0 < u x) <=> (!x. P x ==> u x <= &0) ==> F`] THEN
6480 MP_TAC (ISPECL [`u:real^N->real`;`\x:real^N. &0`;`c:real^N->bool`]
6482 ASM_REWRITE_TAC[SUM_0;REAL_ARITH `~(&0 < &0)`] THEN
6483 ASM_MESON_TAC[];ALL_TAC] THEN
6484 REWRITE_TAC[SET_RULE `~DISJOINT a b <=> ?y. y IN a /\ y IN b`] THEN
6485 EXISTS_TAC `&1 / (sum {x:real^N | x IN c /\ u x > &0} u) %
6486 vsum {x:real^N | x IN c /\ u x > &0} (\x. u x % x)` THEN
6487 REWRITE_TAC[CONVEX_HULL_EXPLICIT;IN_ELIM_THM] THEN
6489 [MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v < &0}`;
6491 &1 / (sum {x:real^N | x IN c /\ u x > &0} u) *
6493 ASM_SIMP_TAC[FINITE_RESTRICT;SUBSET;IN_ELIM_THM] THEN
6494 REPEAT CONJ_TAC THENL
6496 REPEAT STRIP_TAC THEN
6497 MATCH_MP_TAC REAL_LE_MUL THEN
6498 CONJ_TAC THENL [ALL_TAC;
6499 ASM_REWRITE_TAC[REAL_NEG_GE0;REAL_LE_LT]] THEN
6500 MATCH_MP_TAC REAL_LE_DIV THEN
6501 REWRITE_TAC[REAL_LE_01] THEN
6502 MATCH_MP_TAC SUM_POS_LE THEN
6503 ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM] THEN
6505 ASM_SIMP_TAC[FINITE_RESTRICT;SUM_LMUL] THEN
6506 MATCH_MP_TAC (REAL_FIELD `!a. ~(a = &0) /\ a * b = a * c ==> b = c`) THEN
6507 EXISTS_TAC `sum {x:real^N | x IN c /\ u x > &0} u` THEN
6508 REWRITE_TAC[SUM_LMUL] THEN
6509 ASM_SIMP_TAC[REAL_FIELD `~(a = &0) ==> a * &1 / a * b = b`] THEN
6510 REWRITE_TAC[SUM_NEG;REAL_MUL_RID] THEN
6511 REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN
6512 MATCH_MP_TAC (GSYM RADON_S_LEMMA) THEN
6515 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;VECTOR_MUL_LCANCEL] THEN
6516 REWRITE_TAC[VECTOR_MUL_LNEG;VSUM_NEG] THEN
6518 MATCH_MP_TAC (REWRITE_RULE[REAL_ARITH `&0 < a <=> a > &0`]
6519 (GSYM RADON_V_LEMMA)) THEN
6520 ASM_REWRITE_TAC[] THEN
6521 MESON_TAC[VECTOR_MUL_LZERO];ALL_TAC] THEN
6522 MAP_EVERY EXISTS_TAC [`{v:real^N | v IN c /\ u v > &0}`;
6524 &1 / (sum {x:real^N | x IN c /\ u x > &0} u) *
6526 ASM_SIMP_TAC[FINITE_RESTRICT;SUBSET;IN_ELIM_THM] THEN
6527 REPEAT CONJ_TAC THENL
6528 [REPEAT STRIP_TAC THEN
6529 MATCH_MP_TAC REAL_LE_MUL THEN
6530 CONJ_TAC THENL [ALL_TAC;
6531 ASM_SIMP_TAC[REAL_ARITH `a > &0 ==> &0 <= a`]] THEN
6532 MATCH_MP_TAC REAL_LE_DIV THEN
6533 REWRITE_TAC[REAL_LE_01] THEN
6534 MATCH_MP_TAC SUM_POS_LE THEN
6535 ASM_SIMP_TAC[FINITE_RESTRICT;IN_ELIM_THM] THEN
6537 ASM_SIMP_TAC[FINITE_RESTRICT;SUM_LMUL] THEN
6538 MATCH_MP_TAC (REAL_FIELD `!a. ~(a = &0) /\ a * b = a * c ==> b = c`) THEN
6539 EXISTS_TAC `sum {x:real^N | x IN c /\ u x > &0} u` THEN
6540 REWRITE_TAC[SUM_LMUL] THEN
6541 ASM_SIMP_TAC[REAL_FIELD `~(a = &0) ==> a * &1 / a * b = b`] THEN
6542 REWRITE_TAC[SUM_NEG;REAL_MUL_RID] THEN
6543 REWRITE_TAC[REAL_ARITH `a > &0 <=> &0 < a`] THEN
6544 MATCH_MP_TAC (GSYM RADON_S_LEMMA) THEN
6547 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC;VSUM_LMUL;VECTOR_MUL_LCANCEL] THEN
6548 REWRITE_TAC[VECTOR_MUL_LNEG;VSUM_NEG] THEN
6550 MATCH_MP_TAC (REWRITE_RULE[REAL_ARITH `&0 < a <=> a > &0`]
6551 (GSYM RADON_V_LEMMA)) THEN
6552 ASM_REWRITE_TAC[] THEN
6553 MESON_TAC[VECTOR_MUL_LZERO]);;
6556 (`!(c:real^N->bool).
6558 ==> ?(m:real^N->bool) (p:real^N->bool).
6562 ~(DISJOINT (convex hull m) (convex hull p))`,
6563 REPEAT STRIP_TAC THEN MP_TAC
6564 (ISPEC `c:real^N->bool` AFFINE_DEPENDENT_EXPLICIT) THEN
6565 ASM_SIMP_TAC[] THEN REPEAT STRIP_TAC THEN MP_TAC
6566 (ISPEC `s:real^N->bool` RADON_PARTITION) THEN
6568 [ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT] THEN
6569 MAP_EVERY EXISTS_TAC [`s:real^N->bool`;`u:real^N->real`] THEN
6570 ASM SET_TAC[];ALL_TAC] THEN
6571 DISCH_THEN STRIP_ASSUME_TAC THEN
6572 MAP_EVERY EXISTS_TAC [`m:real^N->bool`;`p:real^N->bool`] THEN
6575 (* ------------------------------------------------------------------------- *)
6576 (* Helly's theorem. *)
6577 (* ------------------------------------------------------------------------- *)
6579 let HELLY_INDUCT = prove
6580 (`!n f. f HAS_SIZE n /\ n >= dimindex(:N) + 1 /\
6581 (!s:real^N->bool. s IN f ==> convex s) /\
6582 (!t. t SUBSET f /\ CARD(t) = dimindex(:N) + 1
6583 ==> ~(INTERS t = {}))
6584 ==> ~(INTERS f = {})`,
6585 INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `~(0 >= n + 1)`] THEN GEN_TAC THEN
6586 POP_ASSUM(LABEL_TAC "*") THEN STRIP_TAC THEN
6587 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_SIZE_SUC]) THEN
6588 STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
6589 FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE
6590 `SUC n >= m + 1 ==> m = n \/ n >= m + 1`))
6592 [FIRST_X_ASSUM MATCH_MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
6593 ASM_SIMP_TAC[CARD_CLAUSES; SUBSET_REFL] THEN ARITH_TAC;
6596 `?X. !s:real^N->bool. s IN f ==> X(s) IN INTERS (f DELETE s)`
6597 STRIP_ASSUME_TAC THENL
6598 [REWRITE_TAC[GSYM SKOLEM_THM; MEMBER_NOT_EMPTY; RIGHT_EXISTS_IMP_THM] THEN
6599 GEN_TAC THEN STRIP_TAC THEN REMOVE_THEN "*" MATCH_MP_TAC THEN
6600 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM SET_TAC[];
6603 `?s t:real^N->bool. s IN f /\ t IN f /\ ~(s = t) /\ X s:real^N = X t`
6605 [FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC) THEN
6606 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
6607 EXISTS_TAC `(X:(real^N->bool)->real^N) t` THEN
6608 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC ONCE_DEPTH_CONV
6611 ==> INTERS f = INTERS(f DELETE s) INTER INTERS(f DELETE t)`)
6613 REWRITE_TAC[IN_INTER] THEN ASM_MESON_TAC[];
6615 MP_TAC(ISPEC `IMAGE (X:(real^N->bool)->real^N) f` RADON_PARTITION) THEN
6617 [ASM_SIMP_TAC[FINITE_IMAGE] THEN
6618 MATCH_MP_TAC AFFINE_DEPENDENT_BIGGERSET THEN
6619 ASM_SIMP_TAC[FINITE_IMAGE] THEN
6620 MATCH_MP_TAC(ARITH_RULE
6621 `!f n. n >= d + 1 /\ f = SUC n /\ c = f ==> c >= d + 2`) THEN
6622 MAP_EVERY EXISTS_TAC [`CARD(f:(real^N->bool)->bool)`; `n:num`] THEN
6623 REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN
6624 MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_MESON_TAC[];
6626 ONCE_REWRITE_TAC[SET_RULE
6627 `P /\ m UNION p = s /\ Q <=>
6628 m SUBSET s /\ p SUBSET s /\ m UNION p = s /\ P /\ Q`] THEN
6629 REWRITE_TAC[SUBSET_IMAGE; DISJOINT] THEN
6631 `(?m p. (?u. P u /\ m = t u) /\ (?u. P u /\ p = t u) /\ Q m p) ==> r <=>
6632 (!u v. P u /\ P v /\ Q (t u) (t v) ==> r)`] THEN
6633 MAP_EVERY X_GEN_TAC [`g:(real^N->bool)->bool`; `h:(real^N->bool)->bool`] THEN
6634 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
6635 SUBGOAL_THEN `(f:(real^N->bool)->bool) = h UNION g` SUBST1_TAC THENL
6636 [MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[UNION_SUBSET] THEN
6637 REWRITE_TAC[SUBSET; IN_UNION] THEN X_GEN_TAC `s:real^N->bool` THEN
6638 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
6639 DISCH_THEN(MP_TAC o ISPEC `X:(real^N->bool)->real^N` o
6640 MATCH_MP FUN_IN_IMAGE) THEN
6641 FIRST_X_ASSUM(fun th ->
6642 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN
6643 ONCE_REWRITE_TAC[DISJ_SYM] THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN
6644 MATCH_MP_TAC MONO_OR THEN ASM_MESON_TAC[SUBSET];
6646 MATCH_MP_TAC(SET_RULE
6647 `g SUBSET INTERS g' /\ h SUBSET INTERS h'
6648 ==> ~(g INTER h = {}) ==> ~(INTERS(g' UNION h') = {})`) THEN
6649 FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE
6650 `IMAGE X s INTER IMAGE X t = {} ==> s INTER t = {}`)) THEN
6651 CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN
6652 (CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[SUBSET; CONVEX_INTERS]]) THEN
6653 REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE] THEN ASM SET_TAC[]);;
6656 (`!f:(real^N->bool)->bool.
6657 FINITE f /\ CARD(f) >= dimindex(:N) + 1 /\
6658 (!s. s IN f ==> convex s) /\
6659 (!t. t SUBSET f /\ CARD(t) = dimindex(:N) + 1 ==> ~(INTERS t = {}))
6660 ==> ~(INTERS f = {})`,
6661 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC HELLY_INDUCT THEN
6662 ASM_REWRITE_TAC[HAS_SIZE] THEN ASM_MESON_TAC[]);;
6664 let HELLY_ALT = prove
6665 (`!f:(real^N->bool)->bool.
6667 (!s. s IN f ==> convex s) /\
6668 (!t. t SUBSET f /\ CARD(t) <= dimindex(:N) + 1 ==> ~(INTERS t = {}))
6669 ==> ~(INTERS f = {})`,
6670 GEN_TAC THEN STRIP_TAC THEN
6671 ASM_CASES_TAC `CARD(f:(real^N->bool)->bool) < dimindex(:N) + 1` THEN
6672 ASM_SIMP_TAC[SUBSET_REFL; LT_IMP_LE] THEN MATCH_MP_TAC HELLY THEN
6673 ASM_SIMP_TAC[GE; GSYM NOT_LT] THEN ASM_MESON_TAC[LE_REFL]);;
6675 let HELLY_CLOSED_ALT = prove
6676 (`!f:(real^N->bool)->bool.
6677 (!s. s IN f ==> convex s /\ closed s) /\ (?s. s IN f /\ bounded s) /\
6678 (!t. t SUBSET f /\ FINITE t /\ CARD(t) <= dimindex(:N) + 1
6679 ==> ~(INTERS t = {}))
6680 ==> ~(INTERS f = {})`,
6682 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
6683 MATCH_MP_TAC CLOSED_FIP THEN ASM_SIMP_TAC[] THEN
6684 X_GEN_TAC `g:(real^N->bool)->bool` THEN STRIP_TAC THEN
6685 MATCH_MP_TAC HELLY_ALT THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
6687 ASM_MESON_TAC[SUBSET_TRANS; FINITE_SUBSET]]);;
6689 let HELLY_COMPACT_ALT = prove
6690 (`!f:(real^N->bool)->bool.
6691 (!s. s IN f ==> convex s /\ compact s) /\
6692 (!t. t SUBSET f /\ FINITE t /\ CARD(t) <= dimindex(:N) + 1
6693 ==> ~(INTERS t = {}))
6694 ==> ~(INTERS f = {})`,
6695 GEN_TAC THEN STRIP_TAC THEN
6696 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
6697 ASM_REWRITE_TAC[INTERS_0; UNIV_NOT_EMPTY] THEN
6698 MATCH_MP_TAC HELLY_CLOSED_ALT THEN
6699 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN
6700 ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPACT_IMP_BOUNDED]);;
6702 let HELLY_CLOSED = prove
6703 (`!f:(real^N->bool)->bool.
6704 (FINITE f ==> CARD f >= dimindex (:N) + 1) /\
6705 (!s. s IN f ==> convex s /\ closed s) /\ (?s. s IN f /\ bounded s) /\
6706 (!t. t SUBSET f /\ FINITE t /\ CARD(t) = dimindex(:N) + 1
6707 ==> ~(INTERS t = {}))
6708 ==> ~(INTERS f = {})`,
6709 GEN_TAC THEN REWRITE_TAC[GE] THEN
6710 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
6711 MATCH_MP_TAC HELLY_CLOSED_ALT THEN ASM_REWRITE_TAC[] THEN
6712 X_GEN_TAC `g:(real^N->bool)->bool` THEN STRIP_TAC THEN
6713 MP_TAC(ISPECL [`dimindex(:N) + 1`; `g:(real^N->bool)->bool`;
6714 `f:(real^N->bool)->bool`] CHOOSE_SUBSET_BETWEEN) THEN
6715 ASM_REWRITE_TAC[] THEN
6716 DISCH_THEN(X_CHOOSE_THEN `h:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
6717 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ ~(s = {}) ==> ~(t = {})`) THEN
6718 EXISTS_TAC `INTERS h: real^N->bool` THEN
6719 CONJ_TAC THENL [ASM SET_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC] THEN
6720 ASM_MESON_TAC[HAS_SIZE]);;
6722 let HELLY_COMPACT = prove
6723 (`!f:(real^N->bool)->bool.
6724 (FINITE f ==> CARD f >= dimindex (:N) + 1) /\
6725 (!s. s IN f ==> convex s /\ compact s) /\
6726 (!t. t SUBSET f /\ FINITE t /\ CARD(t) = dimindex(:N) + 1
6727 ==> ~(INTERS t = {}))
6728 ==> ~(INTERS f = {})`,
6729 GEN_TAC THEN STRIP_TAC THEN
6730 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
6731 ASM_REWRITE_TAC[INTERS_0; UNIV_NOT_EMPTY] THEN
6732 MATCH_MP_TAC HELLY_CLOSED THEN
6733 ASM_SIMP_TAC[COMPACT_IMP_CLOSED] THEN
6734 ASM_MESON_TAC[MEMBER_NOT_EMPTY; COMPACT_IMP_BOUNDED]);;
6736 (* ------------------------------------------------------------------------- *)
6737 (* Kirchberger's theorem *)
6738 (* ------------------------------------------------------------------------- *)
6740 let KIRCHBERGER = prove
6741 (`!s t:real^N->bool.
6742 compact s /\ compact t /\
6743 (!s' t'. s' SUBSET s /\ t' SUBSET t /\ FINITE s' /\ FINITE t' /\
6744 CARD(s') + CARD(t') <= dimindex(:N) + 2
6745 ==> ?a b. (!x. x IN s' ==> a dot x < b) /\
6746 (!x. x IN t' ==> a dot x > b))
6747 ==> ?a b. ~(a = vec 0) /\
6748 (!x. x IN s ==> a dot x < b) /\
6749 (!x. x IN t ==> a dot x > b)`,
6751 (`(!x. x IN convex hull s ==> a dot x < b) /\
6752 (!x. x IN convex hull t ==> a dot x > b) <=>
6753 (!x. x IN s ==> a dot x < b) /\ (!x. x IN t ==> a dot x > b)`,
6754 REWRITE_TAC[SET_RULE `(!x. x IN s ==> P x) <=> s SUBSET {x | P x}`] THEN
6755 SIMP_TAC[SUBSET_HULL; CONVEX_HALFSPACE_LT; CONVEX_HALFSPACE_GT])
6756 and KIRCH_LEMMA = prove
6757 (`!s t:real^N->bool.
6758 FINITE s /\ FINITE t /\
6759 (!s' t'. s' SUBSET s /\ t' SUBSET t /\
6760 CARD(s') + CARD(t') <= dimindex(:N) + 2
6761 ==> ?a b. (!x. x IN s' ==> a dot x < b) /\
6762 (!x. x IN t' ==> a dot x > b))
6763 ==> ?a b. (!x. x IN s ==> a dot x < b) /\
6764 (!x. x IN t ==> a dot x > b)`,
6765 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
6766 [`IMAGE (\r. {z:real^(N,1)finite_sum |
6767 fstcart z dot r < drop(sndcart z)}) s UNION
6768 IMAGE (\r. {z:real^(N,1)finite_sum |
6769 fstcart z dot r > drop(sndcart z)}) t`]
6771 REWRITE_TAC[FORALL_SUBSET_UNION; IN_UNION; IMP_CONJ] THEN
6772 REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_SUBSET_IMAGE] THEN
6773 ASM_SIMP_TAC[FINITE_UNION; FINITE_IMAGE; INTERS_UNION] THEN
6774 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE; IN_INTER;
6775 EXISTS_PASTECART; IN_ELIM_PASTECART_THM;
6776 FSTCART_PASTECART; SNDCART_PASTECART] THEN
6777 REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
6778 REWRITE_TAC[FORALL_AND_THM; FORALL_IN_IMAGE; RIGHT_IMP_FORALL_THM] THEN
6779 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; GSYM EXISTS_DROP] THEN
6780 DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
6781 [REWRITE_TAC[REAL_ARITH `a > b <=> --a < --b`; GSYM DOT_RNEG] THEN
6782 REWRITE_TAC[convex; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN
6783 SIMP_TAC[PASTECART_ADD; GSYM PASTECART_CMUL; IN_ELIM_PASTECART_THM] THEN
6784 SIMP_TAC[DOT_LADD; DOT_LMUL; DROP_ADD; DROP_CMUL; GSYM FORALL_DROP] THEN
6785 REWRITE_TAC[REAL_ARITH `--(a * x + b * y):real = a * --x + b * --y`] THEN
6786 REPEAT STRIP_TAC THEN
6787 FIRST_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH
6789 ==> &0 <= u /\ &0 <= v
6790 ==> u = &0 /\ v = &1 \/ u = &1 /\ v = &0 \/ &0 < u /\ &0 < v`)) THEN
6791 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
6792 ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID;
6793 REAL_ADD_LID; REAL_ADD_RID] THEN
6794 MATCH_MP_TAC REAL_LT_ADD2 THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ];
6795 REWRITE_TAC[DIMINDEX_FINITE_SUM; DIMINDEX_1;
6796 ARITH_RULE `(n + 1) + 1 = n + 2`] THEN
6797 MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `v:real^N->bool`] THEN
6798 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
6799 SUBGOAL_THEN `FINITE(u:real^N->bool) /\ FINITE(v:real^N->bool)`
6800 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
6801 W(MP_TAC o PART_MATCH (lhs o rand) CARD_UNION o lhand o lhand o snd) THEN
6802 ASM_SIMP_TAC[FINITE_IMAGE] THEN ANTS_TAC THENL
6803 [REWRITE_TAC[SET_RULE `IMAGE f s INTER IMAGE g t = {} <=>
6804 !x y. x IN s /\ y IN t ==> ~(f x = g y)`] THEN
6805 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
6806 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
6807 DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN
6808 REWRITE_TAC[GSYM FORALL_DROP; DOT_LZERO] THEN
6809 DISCH_THEN(MP_TAC o SPEC `&1`) THEN REAL_ARITH_TAC;
6810 DISCH_THEN SUBST1_TAC] THEN
6811 DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN
6812 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(ARITH_RULE
6813 `a = a' /\ b = b' ==> a + b <= n + 2 ==> a' + b' <= n + 2`) THEN
6814 CONJ_TAC THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN
6815 ASM_REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
6816 SIMP_TAC[GSYM FORALL_DROP; real_gt; VECTOR_EQ_LDOT;
6817 MESON[REAL_LT_TOTAL; REAL_LT_REFL]
6818 `((!y:real. a < y <=> b < y) <=> a = b) /\
6819 ((!y:real. y < a <=> y < b) <=> a = b)`]]) in
6820 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM lemma] THEN
6821 MATCH_MP_TAC SEPARATING_HYPERPLANE_COMPACT_COMPACT THEN
6822 ASM_SIMP_TAC[CONVEX_CONVEX_HULL; COMPACT_CONVEX_HULL;
6823 CONVEX_HULL_EQ_EMPTY] THEN
6825 `!s' t'. (s':real^N->bool) SUBSET s /\ t' SUBSET t /\
6826 FINITE s' /\ CARD(s') <= dimindex(:N) + 1 /\
6827 FINITE t' /\ CARD(t') <= dimindex(:N) + 1
6828 ==> DISJOINT (convex hull s') (convex hull t')`
6830 [REPEAT STRIP_TAC THEN
6831 MP_TAC(ISPECL [`s':real^N->bool`; `t':real^N->bool`] KIRCH_LEMMA) THEN
6833 [ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
6834 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; FINITE_SUBSET];
6835 ONCE_REWRITE_TAC[GSYM lemma] THEN SET_TAC[REAL_LT_ANTISYM; real_gt]];
6836 POP_ASSUM_LIST(K ALL_TAC) THEN STRIP_TAC THEN
6837 REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s /\ x IN t ==> F`] THEN
6838 X_GEN_TAC `x:real^N` THEN ONCE_REWRITE_TAC[CARATHEODORY] THEN
6839 REWRITE_TAC[IN_ELIM_THM] THEN
6840 DISCH_THEN(CONJUNCTS_THEN2
6841 (X_CHOOSE_THEN `s':real^N->bool` STRIP_ASSUME_TAC)
6842 (X_CHOOSE_THEN `t':real^N->bool` STRIP_ASSUME_TAC)) THEN
6843 FIRST_X_ASSUM(MP_TAC o SPECL [`s':real^N->bool`; `t':real^N->bool`]) THEN
6844 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
6846 (* ------------------------------------------------------------------------- *)
6847 (* Convex hull is "preserved" by a linear function. *)
6848 (* ------------------------------------------------------------------------- *)
6850 let CONVEX_HULL_LINEAR_IMAGE = prove
6851 (`!f s. linear f ==> convex hull (IMAGE f s) = IMAGE f (convex hull s)`,
6852 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
6853 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
6854 CONJ_TAC THEN MATCH_MP_TAC HULL_INDUCT THEN
6855 REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC[FUN_IN_IMAGE; HULL_INC] THEN
6856 REWRITE_TAC[convex; IN_ELIM_THM] THEN
6857 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THENL
6858 [FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN
6859 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN
6860 REWRITE_TAC[IN_IMAGE] THEN
6861 MESON_TAC[REWRITE_RULE[convex] CONVEX_CONVEX_HULL];
6862 ASM_SIMP_TAC[LINEAR_ADD; LINEAR_CMUL] THEN
6863 MESON_TAC[REWRITE_RULE[convex] CONVEX_CONVEX_HULL]]);;
6865 add_linear_invariants [CONVEX_HULL_LINEAR_IMAGE];;
6867 let IN_CONVEX_HULL_LINEAR_IMAGE = prove
6868 (`!f:real^M->real^N s x.
6869 linear f /\ x IN convex hull s ==> (f x) IN convex hull (IMAGE f s)`,
6870 SIMP_TAC[CONVEX_HULL_LINEAR_IMAGE] THEN SET_TAC[]);;
6872 (* ------------------------------------------------------------------------- *)
6873 (* Convexity of general and special intervals. *)
6874 (* ------------------------------------------------------------------------- *)
6876 let IS_INTERVAL_CONVEX = prove
6877 (`!s:real^N->bool. is_interval s ==> convex s`,
6878 REWRITE_TAC[is_interval; convex] THEN
6879 REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
6880 MAP_EVERY EXISTS_TAC [`x:real^N`; `y:real^N`] THEN
6881 ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
6882 GEN_TAC THEN STRIP_TAC THEN
6883 DISJ_CASES_TAC(SPECL [`(x:real^N)$i`; `(y:real^N)$i`] REAL_LE_TOTAL) THENL
6884 [DISJ1_TAC; DISJ2_TAC] THEN
6885 MATCH_MP_TAC(REAL_ARITH
6886 `&1 * a <= b /\ b <= &1 * c ==> a <= b /\ b <= c`) THEN
6887 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
6888 ASM_SIMP_TAC[GSYM VECTOR_MUL_COMPONENT;
6889 VECTOR_ADD_RDISTRIB; VECTOR_ADD_COMPONENT] THEN
6890 ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_LE_LMUL;
6891 REAL_LE_LADD; REAL_LE_RADD]);;
6893 let IS_INTERVAL_CONNECTED = prove
6894 (`!s:real^N->bool. is_interval s ==> connected s`,
6895 MESON_TAC[IS_INTERVAL_CONVEX; CONVEX_CONNECTED]);;
6897 let IS_INTERVAL_CONNECTED_1 = prove
6898 (`!s:real^1->bool. is_interval s <=> connected s`,
6899 GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[IS_INTERVAL_CONNECTED] THEN
6900 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
6901 REWRITE_TAC[IS_INTERVAL_1; connected; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM;
6902 NOT_IMP; FORALL_LIFT; LIFT_DROP] THEN
6903 MAP_EVERY X_GEN_TAC [`a:real`; `b:real`; `x:real`] THEN STRIP_TAC THEN
6904 MAP_EVERY EXISTS_TAC
6905 [`{z:real^1 | basis 1 dot z < x}`; `{z:real^1 | basis 1 dot z > x}`] THEN
6906 REWRITE_TAC[OPEN_HALFSPACE_LT; OPEN_HALFSPACE_GT] THEN
6907 SIMP_TAC[SUBSET; EXTENSION; IN_UNION; IN_INTER; GSYM drop; NOT_FORALL_THM;
6908 real_gt; NOT_IN_EMPTY; IN_ELIM_THM; DOT_BASIS; DIMINDEX_1; ARITH] THEN
6909 REPEAT CONJ_TAC THENL
6910 [ASM_MESON_TAC[REAL_LT_TOTAL; LIFT_DROP];
6912 EXISTS_TAC `lift a`;
6913 EXISTS_TAC `lift b`] THEN
6914 ASM_REWRITE_TAC[REAL_LT_LE; LIFT_DROP] THEN ASM_MESON_TAC[]);;
6916 let CONVEX_INTERVAL = prove
6917 (`!a b:real^N. convex(interval [a,b]) /\ convex(interval (a,b))`,
6918 SIMP_TAC[IS_INTERVAL_CONVEX; IS_INTERVAL_INTERVAL]);;
6920 let CONNECTED_INTERVAL = prove
6921 (`(!a b:real^N. connected(interval[a,b])) /\
6922 (!a b:real^N. connected(interval(a,b)))`,
6923 SIMP_TAC[CONVEX_CONNECTED; CONVEX_INTERVAL]);;
6925 let CONVEX_CONNECTED_COLLINEAR = prove
6926 (`!s:real^N->bool. collinear s ==> (convex s <=> connected s)`,
6927 REPEAT STRIP_TAC THEN EQ_TAC THEN REWRITE_TAC[CONVEX_CONNECTED] THEN
6928 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COLLINEAR_AFFINE_HULL]) THEN
6929 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
6930 MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN
6931 GEOM_ORIGIN_TAC `u:real^N` THEN
6932 SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; IN_INSERT; SPAN_INSERT_0] THEN
6933 GEOM_BASIS_MULTIPLE_TAC 1 `v:real^N` THEN
6934 GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
6935 REWRITE_TAC[SPAN_SPECIAL_SCALE] THEN COND_CASES_TAC THENL
6936 [REWRITE_TAC[SPAN_EMPTY; SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN
6937 STRIP_TAC THEN ASM_REWRITE_TAC[CONVEX_EMPTY; CONVEX_SING];
6939 REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; connected; NOT_EXISTS_THM] THEN
6941 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN
6942 ASM_CASES_TAC `a:real^N = b` THEN
6943 ASM_REWRITE_TAC[SEGMENT_REFL; SING_SUBSET] THEN
6944 REWRITE_TAC[SUBSET; IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN
6945 MAP_EVERY X_GEN_TAC [`x:real^N`; `u:real`] THEN
6946 MAP_EVERY ASM_CASES_TAC [`u = &0`; `u = &1`] THEN
6947 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_LID; REAL_SUB_REFL;
6948 REAL_SUB_RZERO; VECTOR_ADD_LID; VECTOR_ADD_RID] THEN
6949 ASM_REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THEN
6950 FIRST_X_ASSUM(MP_TAC o SPECL
6951 [`{y:real^N | basis 1 dot y < basis 1 dot (x:real^N)}`;
6952 `{y:real^N | basis 1 dot y > basis 1 dot (x:real^N)}`]) THEN
6953 REWRITE_TAC[OPEN_HALFSPACE_LT; OPEN_HALFSPACE_GT] THEN
6954 MATCH_MP_TAC(TAUT `q /\ r /\ (~p ==> s) ==> ~(p /\ q /\ r) ==> s`) THEN
6956 [REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; NOT_IN_EMPTY] THEN
6957 REWRITE_TAC[CONJ_ASSOC; REAL_ARITH `~(x:real < a /\ x > a)`];
6959 REWRITE_TAC[real_gt] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
6960 REWRITE_TAC[GSYM DOT_RSUB; SET_RULE
6961 `~(s SUBSET {x | P x} UNION {x | Q x}) <=>
6962 ?x. x IN s /\ ~(P x \/ Q x)`] THEN
6964 `!p q:real^N. p IN span {basis 1} /\ q IN span {basis 1} /\
6965 basis 1 dot p = basis 1 dot q
6968 [SIMP_TAC[SPAN_SING; IMP_CONJ; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
6969 SIMP_TAC[DOT_RMUL; BASIS_NONZERO; DOT_BASIS_BASIS; DIMINDEX_GE_1;
6970 LE_REFL; REAL_MUL_RID];
6972 SUBGOAL_THEN `(x:real^N) IN span {basis 1}` ASSUME_TAC THENL
6973 [ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_ADD THEN CONJ_TAC THEN
6974 MATCH_MP_TAC SPAN_MUL THEN ASM SET_TAC[];
6977 [CONJ_TAC THEN MATCH_MP_TAC(SET_RULE
6978 `(a:real^N) IN s \/ b IN s ==> ~(s = {})`) THEN
6979 ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM; DOT_RADD; DOT_RMUL;
6980 VECTOR_ARITH `((&1 - u) % a + u % b) - b:real^N = (u - &1) % (b - a)`;
6981 VECTOR_ARITH `((&1 - u) % a + u % b) - a:real^N = u % (b - a)`;
6982 VECTOR_ARITH `b - ((&1 - u) % a + u % b):real^N = (u - &1) % (a - b)`;
6983 VECTOR_ARITH `a - ((&1 - u) % a + u % b):real^N = u % (a - b)`] THEN
6984 MATCH_MP_TAC(REAL_ARITH
6985 `(&0 < x ==> &0 < u * x) /\ (&0 < --x ==> &0 < (&1 - u) * --x) /\
6987 ==> &0 < u * x \/ &0 < (u - &1) * x`) THEN
6988 ASM_SIMP_TAC[REAL_LT_MUL; REAL_SUB_LT] THEN
6989 REWRITE_TAC[DOT_RSUB; REAL_SUB_0];
6990 REWRITE_TAC[DOT_RSUB; REAL_ARITH
6991 `~(&0 < x - y \/ &0 < y - x) <=> y = x`]] THEN
6994 let CONVEX_EQ_CONVEX_LINE_INTERSECTION = prove
6995 (`!s:real^N->bool. convex s <=> !a b. convex(s INTER affine hull {a,b})`,
6996 GEN_TAC THEN EQ_TAC THEN
6997 SIMP_TAC[CONVEX_INTER; AFFINE_IMP_CONVEX; AFFINE_AFFINE_HULL] THEN
6998 REWRITE_TAC[CONVEX_CONTAINS_SEGMENT] THEN DISCH_TAC THEN
6999 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN
7000 FIRST_X_ASSUM(MP_TAC o SPECL
7001 [`a:real^N`; `b:real^N`; `a:real^N`; `b:real^N`]) THEN
7002 ASM_SIMP_TAC[IN_INTER; HULL_INC; IN_INSERT] THEN SET_TAC[]);;
7004 let CONVEX_EQ_CONNECTED_LINE_INTERSECTION = prove
7005 (`!s:real^N->bool. convex s <=> !a b. connected(s INTER affine hull {a,b})`,
7007 GEN_REWRITE_TAC LAND_CONV [CONVEX_EQ_CONVEX_LINE_INTERSECTION] THEN
7008 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
7009 MATCH_MP_TAC CONVEX_CONNECTED_COLLINEAR THEN
7010 MATCH_MP_TAC COLLINEAR_SUBSET THEN
7011 EXISTS_TAC `affine hull {a:real^N,b}` THEN
7012 REWRITE_TAC[COLLINEAR_AFFINE_HULL_COLLINEAR; COLLINEAR_2] THEN
7015 (* ------------------------------------------------------------------------- *)
7016 (* On real^1, is_interval, convex and connected are all equivalent. *)
7017 (* ------------------------------------------------------------------------- *)
7019 let IS_INTERVAL_CONVEX_1 = prove
7020 (`!s:real^1->bool. is_interval s <=> convex s`,
7021 MESON_TAC[IS_INTERVAL_CONVEX; CONVEX_CONNECTED; IS_INTERVAL_CONNECTED_1]);;
7023 let CONVEX_CONNECTED_1 = prove
7024 (`!s:real^1->bool. convex s <=> connected s`,
7025 REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; GSYM IS_INTERVAL_CONNECTED_1]);;
7027 let CONNECTED_CONVEX_1 = prove
7028 (`!s:real^1->bool. connected s <=> convex s`,
7029 REWRITE_TAC[GSYM IS_INTERVAL_CONVEX_1; GSYM IS_INTERVAL_CONNECTED_1]);;
7031 let CONNECTED_COMPACT_INTERVAL_1 = prove
7032 (`!s:real^1->bool. connected s /\ compact s <=> ?a b. s = interval[a,b]`,
7033 REWRITE_TAC[GSYM IS_INTERVAL_CONNECTED_1; IS_INTERVAL_COMPACT]);;
7035 let CONVEX_CONNECTED_1_GEN = prove
7037 dimindex(:N) = 1 ==> (convex s <=> connected s)`,
7038 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
7039 REWRITE_TAC[GSYM DIMINDEX_1] THEN
7040 DISCH_THEN(ACCEPT_TAC o C GEOM_EQUAL_DIMENSION_RULE CONVEX_CONNECTED_1));;
7042 let CONNECTED_CONVEX_1_GEN = prove
7044 dimindex(:N) = 1 ==> (convex s <=> connected s)`,
7045 SIMP_TAC[CONVEX_CONNECTED_1_GEN]);;
7047 (* ------------------------------------------------------------------------- *)
7048 (* Jung's theorem. *)
7049 (* Proof taken from http://cstheory.wordpress.com/2010/08/07/jungs-theorem/ *)
7050 (* ------------------------------------------------------------------------- *)
7053 (`!s:real^N->bool r.
7055 sqrt(&(dimindex(:N)) / &(2 * dimindex(:N) + 2)) * diameter s <= r
7056 ==> ?a. s SUBSET cball(a,r)`,
7058 (`&0 < x /\ x <= y ==> (x - &1) / x <= (y - &1) / y`,
7059 SIMP_TAC[REAL_LE_LDIV_EQ] THEN REPEAT STRIP_TAC THEN
7060 ONCE_REWRITE_TAC[REAL_ARITH `x / y * z:real = (x * z) / y`] THEN
7061 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL
7062 [ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_LE_RDIV_EQ]] THEN
7063 ASM_REAL_ARITH_TAC) in
7064 REPEAT STRIP_TAC THEN
7065 SUBGOAL_THEN `&0 <= r` ASSUME_TAC THENL
7066 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
7067 REAL_LE_TRANS)) THEN
7068 MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[DIAMETER_POS_LE] THEN
7069 SIMP_TAC[SQRT_POS_LE; REAL_LE_DIV; REAL_POS];
7071 MP_TAC(ISPEC `IMAGE (\x:real^N. cball(x,r)) s` HELLY_COMPACT_ALT) THEN
7072 REWRITE_TAC[FORALL_IN_IMAGE; COMPACT_CBALL; CONVEX_CBALL] THEN
7073 REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> q /\ p ==> r ==> s`] THEN
7074 REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN
7075 REWRITE_TAC[INTERS_IMAGE; GSYM MEMBER_NOT_EMPTY] THEN
7076 REWRITE_TAC[SUBSET; IN_CBALL; IN_ELIM_THM] THEN
7077 ANTS_TAC THENL [ALL_TAC; MESON_TAC[DIST_SYM]] THEN
7078 X_GEN_TAC `t:real^N->bool` THEN REWRITE_TAC[GSYM SUBSET] THEN
7080 ASM_SIMP_TAC[CARD_IMAGE_INJ; EQ_BALLS; GSYM REAL_NOT_LE] THEN
7081 UNDISCH_TAC `FINITE(t:real^N->bool)` THEN
7082 SUBGOAL_THEN `bounded(t:real^N->bool)` MP_TAC THENL
7083 [ASM_MESON_TAC[BOUNDED_SUBSET]; ALL_TAC] THEN
7084 UNDISCH_TAC `&0 <= r` THEN
7086 `sqrt(&(dimindex(:N)) / &(2 * dimindex(:N) + 2)) *
7087 diameter(t:real^N->bool) <= r`
7089 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
7090 REAL_LE_TRANS)) THEN
7091 MATCH_MP_TAC REAL_LE_LMUL THEN
7092 ASM_SIMP_TAC[DIAMETER_SUBSET; SQRT_POS_LE; REAL_POS; REAL_LE_DIV];
7093 POP_ASSUM_LIST(K ALL_TAC) THEN
7094 SPEC_TAC(`t:real^N->bool`,`s:real^N->bool`) THEN
7095 REPEAT STRIP_TAC] THEN
7096 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
7097 MP_TAC(ISPEC `{d | &0 <= d /\ ?a:real^N. s SUBSET cball(a,d)}` INF) THEN
7098 ABBREV_TAC `d = inf {d | &0 <= d /\ ?a:real^N. s SUBSET cball(a,d)}` THEN
7099 REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL
7100 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
7101 ASM_MESON_TAC[BOUNDED_SUBSET_CBALL; REAL_LT_IMP_LE];
7102 DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "P") (LABEL_TAC "M"))] THEN
7103 SUBGOAL_THEN `&0 <= d` ASSUME_TAC THENL
7104 [ASM_MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN
7105 SUBGOAL_THEN `?a:real^N. s SUBSET cball(a,d)` MP_TAC THENL
7107 `!n. ?a:real^N. s SUBSET cball(a,d + inv(&n + &1))`
7109 [X_GEN_TAC `n:num` THEN
7110 REMOVE_THEN "M" (MP_TAC o SPEC `d + inv(&n + &1)`) THEN
7111 REWRITE_TAC[REAL_ARITH `d + i <= d <=> ~(&0 < i)`] THEN
7112 REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
7113 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_NOT_LE] THEN
7114 MESON_TAC[SUBSET_CBALL; REAL_LT_IMP_LE; SUBSET_TRANS];
7116 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SKOLEM_THM] THEN
7117 X_GEN_TAC `aa:num->real^N` THEN DISCH_TAC THEN
7118 SUBGOAL_THEN `?t. compact t /\ !n. (aa:num->real^N) n IN t` MP_TAC THENL
7119 [FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N` o
7120 MATCH_MP BOUNDED_SUBSET_CBALL) THEN
7121 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_CBALL_0] THEN
7122 X_GEN_TAC `B:real` THEN STRIP_TAC THEN
7123 EXISTS_TAC `cball(vec 0:real^N,B + d + &1)` THEN
7124 REWRITE_TAC[COMPACT_CBALL; IN_CBALL_0] THEN X_GEN_TAC `n:num` THEN
7125 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_CBALL]) THEN
7126 MATCH_MP_TAC(NORM_ARITH
7127 `(?x:real^N. norm(x) <= B /\ dist(a,x) <= d) ==> norm(a) <= B + d`) THEN
7128 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7129 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN
7130 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `d + inv(&n + &1)` THEN
7131 ASM_SIMP_TAC[REAL_LE_LADD] THEN
7132 MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC;
7134 REWRITE_TAC[compact; LEFT_IMP_EXISTS_THM] THEN
7135 X_GEN_TAC `t:real^N->bool` THEN
7136 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
7137 DISCH_THEN(MP_TAC o SPEC `aa:num->real^N`) THEN ASM_REWRITE_TAC[] THEN
7138 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
7139 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
7140 REWRITE_TAC[SUBSET; IN_CBALL] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7141 REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
7142 MP_TAC(SPEC `(dist(a:real^N,x) - d) / &2` REAL_ARCH_INV) THEN
7143 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
7144 DISCH_THEN(MP_TAC o SPEC `(dist(a:real^N,x) - d) / &2`) THEN
7145 ASM_SIMP_TAC[REAL_SUB_LT; REAL_HALF; o_THM] THEN
7146 DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
7147 DISCH_THEN(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC) THEN
7148 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [SUBSET]) THEN
7149 DISCH_THEN(MP_TAC o SPECL [`(r:num->num)(N1 + N2)`; `x:real^N`]) THEN
7150 ASM_REWRITE_TAC[IN_CBALL; REAL_NOT_LE] THEN
7151 FIRST_X_ASSUM(MP_TAC o SPEC `N1 + N2:num`) THEN
7152 ASM_REWRITE_TAC[LE_ADD] THEN
7153 SUBGOAL_THEN `inv(&(r (N1 + N2:num)) + &1) < (dist(a:real^N,x) - d) / &2`
7154 MP_TAC THENL [ALL_TAC; NORM_ARITH_TAC] THEN
7155 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N2)` THEN
7156 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
7157 CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_INV_EQ]; ALL_TAC] THEN
7158 REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_ADD] THEN
7159 MATCH_MP_TAC(ARITH_RULE
7160 `N1 + N2 <= r(N1 + N2) ==> N2 <= r(N1 + N2) + 1`) THEN
7161 ASM_MESON_TAC[MONOTONE_BIGGER];
7163 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
7164 ONCE_REWRITE_TAC[DIST_SYM] THEN
7165 REWRITE_TAC[GSYM IN_CBALL; GSYM SUBSET] THEN
7166 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
7167 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS) THEN
7168 MATCH_MP_TAC SUBSET_CBALL THEN
7169 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
7170 `a * s <= r ==> d <= a * s ==> d <= r`)) THEN
7171 UNDISCH_THEN `&0 <= r` (K ALL_TAC) THEN REMOVE_THEN "M" (K ALL_TAC) THEN
7172 FIRST_X_ASSUM(K ALL_TAC o SYM) THEN REMOVE_THEN "P" MP_TAC THEN
7173 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
7174 ABBREV_TAC `n = CARD(s:real^N->bool)` THEN
7175 SUBGOAL_THEN `(s:real^N->bool) HAS_SIZE n` MP_TAC THENL
7176 [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN
7177 UNDISCH_THEN `CARD(s:real^N->bool) = n` (K ALL_TAC) THEN
7178 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
7179 SPEC_TAC(`d:real`,`r:real`) THEN GEN_TAC THEN
7180 GEOM_ORIGIN_TAC `a:real^N` THEN SIMP_TAC[HAS_SIZE] THEN
7181 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
7182 ABBREV_TAC `t = {x:real^N | x IN s /\ norm(x) = r}` THEN
7183 SUBGOAL_THEN `FINITE(t:real^N->bool)` ASSUME_TAC THENL
7184 [EXPAND_TAC "t" THEN ASM_SIMP_TAC[FINITE_RESTRICT]; ALL_TAC] THEN
7185 SUBGOAL_THEN `(vec 0:real^N) IN convex hull t` MP_TAC THENL
7186 [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
7187 MP_TAC(ISPEC `convex hull t:real^N->bool`
7188 SEPARATING_HYPERPLANE_CLOSED_0) THEN
7189 ASM_SIMP_TAC[CONVEX_CONVEX_HULL; NOT_IMP; COMPACT_CONVEX_HULL;
7190 FINITE_IMP_COMPACT; COMPACT_IMP_CLOSED] THEN
7191 REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN
7192 X_GEN_TAC `v:real^N` THEN
7193 ABBREV_TAC `k = CARD(s:real^N->bool)` THEN
7194 SUBGOAL_THEN `(s:real^N->bool) HAS_SIZE k` MP_TAC THENL
7195 [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN
7196 UNDISCH_THEN `CARD(s:real^N->bool) = k` (K ALL_TAC) THEN
7197 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
7198 GEOM_BASIS_MULTIPLE_TAC 1 `v:real^N` THEN X_GEN_TAC `m:real` THEN
7199 GEN_REWRITE_TAC LAND_CONV [REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN
7200 STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_EQ_0] THEN
7201 ASM_SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL; REAL_LT_IMP_NZ] THEN
7202 REPEAT GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[HAS_SIZE] THEN
7203 DISCH_THEN(SUBST_ALL_TAC o SYM) THEN X_GEN_TAC `b:real` THEN DISCH_TAC THEN
7204 ASM_SIMP_TAC[DOT_LMUL; DOT_BASIS; DIMINDEX_GE_1; LE_REFL] THEN
7205 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
7206 ASM_SIMP_TAC[real_gt; GSYM REAL_LT_LDIV_EQ] THEN
7207 SUBGOAL_THEN `&0 < b / m` MP_TAC THENL
7208 [ASM_SIMP_TAC[REAL_LT_DIV];
7209 UNDISCH_THEN `&0 < b` (K ALL_TAC) THEN
7210 SPEC_TAC(`b / m:real`,`b:real`)] THEN
7211 X_GEN_TAC `b:real` THEN DISCH_TAC THEN DISCH_TAC THEN
7213 `!x:real^N e. &0 < e /\ e < b /\ x IN t ==> norm(x - e % basis 1) < r`
7215 [MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real`] THEN STRIP_TAC THEN
7216 SUBGOAL_THEN `r = norm(x:real^N)` SUBST1_TAC THENL
7217 [ASM SET_TAC[]; REWRITE_TAC[NORM_LT; dot]] THEN
7218 SIMP_TAC[SUM_CLAUSES_LEFT; DIMINDEX_GE_1] THEN
7219 SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT;
7220 BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL;
7221 ARITH_RULE `2 <= n ==> 1 <= n /\ ~(n = 1)`; ARITH] THEN
7222 REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO; REAL_LT_RADD] THEN
7223 REWRITE_TAC[GSYM REAL_POW_2; GSYM REAL_LT_SQUARE_ABS] THEN
7224 MATCH_MP_TAC(REAL_ARITH
7225 `!b. &0 < e /\ e < b /\ b < x ==> abs(x - e * &1) < abs x`) THEN
7226 EXISTS_TAC `b:real` THEN ASM_REWRITE_TAC[] THEN
7227 ASM_MESON_TAC[HULL_INC];
7231 !x:real^N a. x IN (s DIFF t) /\ norm(a) < d ==> norm(x - a) < r`
7232 STRIP_ASSUME_TAC THENL
7233 [ASM_CASES_TAC `s DIFF t:real^N->bool = {}` THENL
7234 [ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN MESON_TAC[REAL_LT_01]; ALL_TAC] THEN
7235 EXISTS_TAC `inf (IMAGE (\x:real^N. r - norm x) (s DIFF t))` THEN
7236 SUBGOAL_THEN `FINITE(s DIFF t:real^N->bool)` ASSUME_TAC THENL
7237 [ASM_MESON_TAC[FINITE_DIFF]; ALL_TAC] THEN
7238 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
7239 REWRITE_TAC[FORALL_IN_IMAGE] THEN SIMP_TAC
7240 [NORM_ARITH `norm a < r - norm x ==> norm(x - a:real^N) < r`] THEN
7241 EXPAND_TAC "t" THEN REWRITE_TAC[IN_DIFF; IN_ELIM_THM; REAL_SUB_LT] THEN
7242 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_CBALL_0]) THEN
7243 ASM_MESON_TAC[REAL_LT_LE];
7246 `?a. !x. x IN s ==> norm(x - a:real^N) < r`
7247 STRIP_ASSUME_TAC THENL
7248 [EXISTS_TAC `min (b / &2) (d / &2) % basis 1:real^N` THEN
7249 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7250 ASM_CASES_TAC `(x:real^N) IN t` THENL
7251 [MATCH_MP_TAC(ASSUME
7252 `!x:real^N e. &0 < e /\ e < b /\ x IN t
7253 ==> norm (x - e % basis 1) < r`) THEN
7254 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
7256 `!x:real^N a. x IN s DIFF t /\ norm a < d ==> norm (x - a) < r`) THEN
7257 ASM_SIMP_TAC[IN_DIFF; NORM_MUL; LE_REFL; NORM_BASIS;
7259 ASM_REAL_ARITH_TAC];
7260 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL
7261 [ASM_MESON_TAC[MEMBER_NOT_EMPTY; NORM_ARITH
7262 `norm(x:real^N) < r ==> &0 < r`];
7265 `!x a:real^N. &0 <= x /\ s SUBSET cball (a,x) ==> r <= x` (MP_TAC o
7266 SPECL [`max (&0) (r - inf (IMAGE (\x:real^N. r - norm(x - a)) s))`;
7268 ASM_SIMP_TAC[REAL_ARITH `&0 < r ==> (r <= max (&0) a <=> r <= a)`] THEN
7269 REWRITE_TAC[SUBSET; IN_CBALL; REAL_ARITH `a <= max a b`] THEN
7270 REWRITE_TAC[NOT_IMP; REAL_ARITH `~(r <= r - x) <=> &0 < x`] THEN
7271 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
7272 ASM_REWRITE_TAC[FORALL_IN_IMAGE; REAL_SUB_LT] THEN
7273 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7274 MATCH_MP_TAC(REAL_ARITH `d <= b ==> d <= max a b`) THEN
7275 ONCE_REWRITE_TAC[REAL_ARITH `a <= b - c <=> c <= b - a`] THEN
7276 ASM_SIMP_TAC[REAL_INF_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
7277 REWRITE_TAC[EXISTS_IN_IMAGE; ONCE_REWRITE_RULE[NORM_SUB] dist] THEN
7278 ASM_MESON_TAC[REAL_LE_REFL]];
7280 ASM_CASES_TAC `t:real^N->bool = {}` THEN
7281 ASM_REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
7282 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN
7283 DISCH_THEN(X_CHOOSE_THEN `l:real^N->real` STRIP_ASSUME_TAC) THEN
7284 MATCH_MP_TAC REAL_LE_TRANS THEN
7285 EXISTS_TAC `sqrt((&(dimindex (:N)) / &(2 * dimindex (:N) + 2)) *
7286 diameter(s:real^N->bool) pow 2)` THEN
7288 [MATCH_MP_TAC REAL_LE_RSQRT;
7289 ASM_SIMP_TAC[SQRT_MUL; DIAMETER_POS_LE; REAL_POW_LE; REAL_LE_DIV;
7290 REAL_POS; POW_2_SQRT; REAL_LE_REFL]] THEN
7293 `sum t (\y:real^N. &2 * r pow 2) <=
7294 sum t (\y. (&1 - l y) * diameter(s:real^N->bool) pow 2)`
7296 [MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN
7297 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
7298 MATCH_MP_TAC REAL_LE_TRANS THEN
7299 EXISTS_TAC `sum (t DELETE x) (\x:real^N. l(x)) *
7300 diameter(s:real^N->bool) pow 2` THEN CONJ_TAC THENL
7301 [ALL_TAC; ASM_SIMP_TAC[SUM_DELETE; ETA_AX; REAL_LE_REFL]] THEN
7302 REWRITE_TAC[GSYM SUM_RMUL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
7303 EXISTS_TAC `sum (t DELETE x) (\y:real^N. l y * norm(y - x) pow 2)` THEN
7306 MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FINITE_DELETE; IN_DELETE] THEN
7307 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_LMUL THEN
7308 ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_POW_LE2 THEN
7309 REWRITE_TAC[NORM_POS_LE] THEN
7310 MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN ASM SET_TAC[]] THEN
7311 MATCH_MP_TAC REAL_LE_TRANS THEN
7312 EXISTS_TAC `sum t (\y:real^N. l y * norm (y - x) pow 2)` THEN
7315 MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ_SUPERSET THEN
7316 ASM_REWRITE_TAC[FINITE_DELETE] THEN
7317 CONJ_TAC THENL [SET_TAC[]; REWRITE_TAC[IN_DELETE]] THEN
7318 SIMP_TAC[TAUT `p /\ ~(p /\ ~q) <=> p /\ q`] THEN
7319 REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN REAL_ARITH_TAC] THEN
7320 REWRITE_TAC[NORM_POW_2; VECTOR_ARITH
7321 `(y - x:real^N) dot (y - x) = (x dot x + y dot y) - &2 * x dot y`] THEN
7322 MATCH_MP_TAC REAL_LE_TRANS THEN
7323 EXISTS_TAC `sum t (\y:real^N. l y * (&2 * r pow 2 - &2 * (x dot y)))` THEN
7326 MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC SUM_EQ THEN
7327 UNDISCH_TAC `(x:real^N) IN t` THEN EXPAND_TAC "t" THEN
7328 REWRITE_TAC[IN_DELETE; IN_ELIM_THM] THEN
7329 SIMP_TAC[NORM_EQ_SQUARE; NORM_POW_2] THEN REAL_ARITH_TAC] THEN
7330 REWRITE_TAC[REAL_ARITH `x * (&2 * y - &2 * z) = &2 * (x * y - x * z)`] THEN
7331 REWRITE_TAC[SUM_LMUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN
7332 REWRITE_TAC[REAL_POS] THEN
7333 ASM_SIMP_TAC[SUM_SUB; FINITE_DELETE; SUM_RMUL] THEN
7334 REWRITE_TAC[GSYM DOT_RMUL] THEN
7335 ASM_SIMP_TAC[GSYM DOT_RSUM; DOT_RZERO] THEN REAL_ARITH_TAC;
7336 ASM_SIMP_TAC[SUM_CONST; SUM_RMUL; SUM_SUB] THEN
7337 REWRITE_TAC[REAL_OF_NUM_MUL; MULT_CLAUSES] THEN
7338 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [REAL_MUL_SYM] THEN
7339 SUBGOAL_THEN `&0 < &(CARD(t:real^N->bool) * 2)` ASSUME_TAC THENL
7340 [REWRITE_TAC[REAL_OF_NUM_LT; ARITH_RULE `0 < n * 2 <=> ~(n = 0)`] THEN
7341 ASM_SIMP_TAC[CARD_EQ_0];
7342 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
7343 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
7344 REWRITE_TAC[REAL_ARITH `(a * b) / c:real = a / c * b`] THEN
7345 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[REAL_LE_POW_2] THEN
7346 REWRITE_TAC[ARITH_RULE `2 * n + 2 = (n + 1) * 2`; GSYM REAL_OF_NUM_MUL;
7347 real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN
7348 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[GSYM real_div] THEN
7349 CONV_TAC REAL_RAT_REDUCE_CONV THEN
7350 SUBGOAL_THEN `&(dimindex(:N)) = &(dimindex(:N) + 1) - &1`
7352 [REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC;
7353 MATCH_MP_TAC lemma THEN
7354 ASM_SIMP_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; CARD_EQ_0;
7355 ARITH_RULE `0 < n <=> ~(n = 0)`] THEN
7356 MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `CARD(s:real^N->bool)` THEN
7357 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CARD_SUBSET THEN
7360 (* ------------------------------------------------------------------------- *)
7361 (* The Dugundji extension theorem, and Tietze variants as corollaries. *)
7362 (* ------------------------------------------------------------------------- *)
7364 let DUGUNDJI = prove
7365 (`!f:real^M->real^N c u s.
7366 convex c /\ ~(c = {}) /\
7367 closed_in (subtopology euclidean u) s /\
7368 f continuous_on s /\ IMAGE f s SUBSET c
7369 ==> ?g. g continuous_on u /\ IMAGE g u SUBSET c /\
7370 !x. x IN s ==> g x = f x`,
7371 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THENL
7372 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7373 DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN
7374 EXISTS_TAC `(\x. y):real^M->real^N` THEN
7375 REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[];
7377 MP_TAC(ISPECL [`u DIFF s:real^M->bool`;
7378 `{ ball(x:real^M,setdist({x},s) / &2) |x| x IN u DIFF s}`]
7380 REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC; OPEN_BALL] THEN ANTS_TAC THENL
7381 [REWRITE_TAC[SUBSET; IN_DIFF; IN_ELIM_THM; UNIONS_GSPEC] THEN
7382 X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN EXISTS_TAC `x:real^M` THEN
7383 ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN
7384 MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ ~(x = &0) ==> &0 < x / &2`) THEN
7385 ASM_MESON_TAC[SETDIST_POS_LE; SETDIST_EQ_0_CLOSED_IN];
7386 DISCH_THEN(X_CHOOSE_THEN `c:(real^M->bool)->bool` STRIP_ASSUME_TAC)] THEN
7389 ==> ?v a:real^M. v IN u /\ ~(v IN s) /\ a IN s /\
7390 t SUBSET ball(v,setdist({v},s) / &2) /\
7391 dist(v,a) <= &2 * setdist({v},s)`
7393 [X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN
7394 FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN
7395 ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
7396 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^M` THEN
7397 REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
7398 MP_TAC(ISPECL[`{v:real^M}`; `s:real^M->bool`; `&2 * setdist({v:real^M},s)`]
7399 REAL_SETDIST_LT_EXISTS) THEN
7400 ASM_SIMP_TAC[NOT_INSERT_EMPTY; SETDIST_POS_LE; REAL_ARITH
7401 `&0 <= x ==> (x < &2 * x <=> ~(x = &0))`] THEN
7402 ASM_MESON_TAC[REAL_LT_IMP_LE; IN_SING; SETDIST_EQ_0_CLOSED_IN];
7403 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
7404 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
7405 [`vv:(real^M->bool)->real^M`; `aa:(real^M->bool)->real^M`] THEN
7408 `!t v:real^M. t IN c /\ v IN t ==> setdist({vv t},s) <= &2 * setdist({v},s)`
7410 [REPEAT STRIP_TAC THEN
7411 FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN
7412 ASM_REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o el 3 o CONJUNCTS) THEN
7413 DISCH_THEN(MP_TAC o SPEC `v:real^M`) THEN ASM_REWRITE_TAC[IN_BALL] THEN
7414 MP_TAC(ISPECL [`s:real^M->bool`; `(vv:(real^M->bool)->real^M) t`;
7415 `v:real^M`] SETDIST_SING_TRIANGLE) THEN
7419 `!t v a:real^M. t IN c /\ v IN t /\ a IN s
7420 ==> dist(a,aa t) <= &6 * dist(a,v)`
7422 [REPEAT STRIP_TAC THEN
7423 FIRST_X_ASSUM(MP_TAC o SPECL [`t:real^M->bool`; `v:real^M`]) THEN
7424 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN
7425 ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o funpow 3 CONJUNCT2) THEN
7426 REWRITE_TAC[IMP_CONJ; SUBSET; IN_BALL] THEN
7427 DISCH_THEN(MP_TAC o SPEC `v:real^M`) THEN ASM_REWRITE_TAC[] THEN
7428 MP_TAC(ISPECL [`{v:real^M}`; `s:real^M->bool`; `v:real^M`; `a:real^M`]
7429 SETDIST_LE_DIST) THEN
7430 ASM_REWRITE_TAC[IN_SING] THEN CONV_TAC NORM_ARITH;
7432 MP_TAC(ISPECL [`c:(real^M->bool)->bool`; `u DIFF s:real^M->bool`]
7433 SUBORDINATE_PARTITION_OF_UNITY) THEN
7434 ASM_SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN
7435 X_GEN_TAC `h:(real^M->bool)->real^M->real` THEN STRIP_TAC THEN
7437 `\x. if x IN s then (f:real^M->real^N) x
7438 else vsum c (\t:real^M->bool. h t x % f(aa t))` THEN
7439 SIMP_TAC[] THEN CONJ_TAC THENL
7441 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
7442 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
7443 COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
7444 MATCH_MP_TAC CONVEX_VSUM_STRONG THEN ASM SET_TAC[]] THEN
7445 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
7446 X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN
7447 ASM_CASES_TAC `(a:real^M) IN s` THENL
7449 MATCH_MP_TAC CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN THEN
7450 MAP_EVERY EXISTS_TAC
7452 vsum c (\t:real^M->bool. h t x % (f:real^M->real^N) (aa t))`;
7453 `u DIFF s:real^M->bool`] THEN
7454 ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; IN_DIFF] THEN
7455 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M`) THEN
7456 ASM_REWRITE_TAC[IN_DIFF] THEN
7457 DISCH_THEN(X_CHOOSE_THEN `n:real^M->bool` STRIP_ASSUME_TAC) THEN
7458 MATCH_MP_TAC CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN THEN MAP_EVERY EXISTS_TAC
7459 [`\x. vsum {u | u IN c /\ ~(!x:real^M. x IN n ==> h u x = &0)}
7460 (\t:real^M->bool. h t x % (f:real^M->real^N) (aa t))`;
7461 `(u DIFF s) INTER n:real^M->bool`] THEN
7462 ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL; OPEN_IN_INTER_OPEN;
7463 IN_INTER; IN_DIFF] THEN
7465 [REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
7466 MATCH_MP_TAC VSUM_SUPERSET THEN
7467 REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM SET_TAC[];
7468 MATCH_MP_TAC CONTINUOUS_VSUM THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN
7469 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_VMUL THEN
7470 FIRST_X_ASSUM(MP_TAC o SPEC `t:real^M->bool`) THEN
7471 ASM_REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
7472 DISCH_THEN(MP_TAC o SPEC `a:real^M` o CONJUNCT1) THEN
7473 ASM_REWRITE_TAC[IN_DIFF; ETA_AX] THEN
7474 REWRITE_TAC[CONTINUOUS_WITHIN] THEN MATCH_MP_TAC EQ_IMP THEN
7475 MATCH_MP_TAC LIM_TRANSFORM_WITHIN_SET THEN
7476 SUBGOAL_THEN `open_in (subtopology euclidean u) (u DIFF s:real^M->bool)`
7477 MP_TAC THENL [ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL]; ALL_TAC] THEN
7478 REWRITE_TAC[EVENTUALLY_AT; OPEN_IN_CONTAINS_BALL] THEN
7479 DISCH_THEN(MP_TAC o SPEC `a:real^M` o CONJUNCT2) THEN
7480 ASM_REWRITE_TAC[IN_DIFF] THEN MATCH_MP_TAC MONO_EXISTS THEN
7481 REWRITE_TAC[SUBSET; IN_BALL; IN_INTER; IN_DIFF] THEN
7482 MESON_TAC[DIST_SYM]]] THEN
7483 ASM_REWRITE_TAC[CONTINUOUS_WITHIN_OPEN] THEN
7484 X_GEN_TAC `w:real^N->bool` THEN STRIP_TAC THEN
7485 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
7486 DISCH_THEN(MP_TAC o SPEC `(f:real^M->real^N) a`) THEN ASM_REWRITE_TAC[] THEN
7487 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
7488 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
7489 [CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN]) THEN
7490 DISCH_THEN(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[] THEN
7491 REWRITE_TAC[continuous_within] THEN
7492 DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
7493 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
7494 EXISTS_TAC `ball(a:real^M,d / &6)` THEN
7495 ASM_REWRITE_TAC[CENTRE_IN_BALL; OPEN_BALL] THEN
7496 ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &6 <=> &0 < e`] THEN
7497 X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_BALL] THEN STRIP_TAC THEN
7498 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
7499 COND_CASES_TAC THENL
7500 [REWRITE_TAC[IN_BALL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
7501 FIRST_X_ASSUM MATCH_MP_TAC THEN
7502 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC;
7504 MATCH_MP_TAC CONVEX_VSUM_STRONG THEN
7505 ASM_SIMP_TAC[CONVEX_BALL; IN_DIFF] THEN
7506 X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN
7507 ASM_CASES_TAC `(x:real^M) IN t` THENL [DISJ2_TAC; ASM SET_TAC[]] THEN
7508 REWRITE_TAC[IN_BALL] THEN
7509 ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
7510 ASM_SIMP_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
7511 `dist(a:real^M,v) < d / &6
7512 ==> dist(a,a') <= &6 * dist(a,v) ==> dist(a',a) < d`)) THEN
7513 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
7516 (`!f:real^M->real^N u s B.
7518 closed_in (subtopology euclidean u) s /\
7519 f continuous_on s /\
7520 (!x. x IN s ==> norm(f x) <= B)
7521 ==> ?g. g continuous_on u /\
7522 (!x. x IN s ==> g x = f x) /\
7523 (!x. x IN u ==> norm(g x) <= B)`,
7524 REPEAT STRIP_TAC THEN
7525 MP_TAC(ISPECL [`f:real^M->real^N`; `cball(vec 0:real^N,B)`; `u:real^M->bool`;
7526 `s:real^M->bool`] DUGUNDJI) THEN
7527 ASM_REWRITE_TAC[CONVEX_CBALL; CBALL_EQ_EMPTY; REAL_NOT_LT] THEN
7528 ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_CBALL_0] THEN
7531 let TIETZE_CLOSED_INTERVAL = prove
7532 (`!f:real^M->real^N u s a b.
7533 ~(interval[a,b] = {}) /\
7534 closed_in (subtopology euclidean u) s /\
7535 f continuous_on s /\
7536 (!x. x IN s ==> f x IN interval[a,b])
7537 ==> ?g. g continuous_on u /\
7538 (!x. x IN s ==> g x = f x) /\
7539 (!x. x IN u ==> g(x) IN interval[a,b])`,
7540 REPEAT STRIP_TAC THEN
7541 MP_TAC(ISPECL [`f:real^M->real^N`; `interval[a:real^N,b]`; `u:real^M->bool`;
7542 `s:real^M->bool`] DUGUNDJI) THEN
7543 ASM_REWRITE_TAC[CONVEX_INTERVAL; SUBSET; FORALL_IN_IMAGE] THEN
7546 let TIETZE_CLOSED_INTERVAL_1 = prove
7547 (`!f:real^N->real^1 u s a b.
7549 closed_in (subtopology euclidean u) s /\
7550 f continuous_on s /\
7551 (!x. x IN s ==> f x IN interval[a,b])
7552 ==> ?g. g continuous_on u /\
7553 (!x. x IN s ==> g x = f x) /\
7554 (!x. x IN u ==> g(x) IN interval[a,b])`,
7555 REPEAT STRIP_TAC THEN MATCH_MP_TAC TIETZE_CLOSED_INTERVAL THEN
7556 ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1]);;
7558 let TIETZE_OPEN_INTERVAL = prove
7559 (`!f:real^M->real^N u s a b.
7560 ~(interval(a,b) = {}) /\
7561 closed_in (subtopology euclidean u) s /\
7562 f continuous_on s /\
7563 (!x. x IN s ==> f x IN interval(a,b))
7564 ==> ?g. g continuous_on u /\
7565 (!x. x IN s ==> g x = f x) /\
7566 (!x. x IN u ==> g(x) IN interval(a,b))`,
7567 REPEAT STRIP_TAC THEN
7568 MP_TAC(ISPECL [`f:real^M->real^N`; `interval(a:real^N,b)`; `u:real^M->bool`;
7569 `s:real^M->bool`] DUGUNDJI) THEN
7570 ASM_REWRITE_TAC[CONVEX_INTERVAL; SUBSET; FORALL_IN_IMAGE] THEN
7573 let TIETZE_OPEN_INTERVAL_1 = prove
7574 (`!f:real^N->real^1 u s a b.
7576 closed_in (subtopology euclidean u) s /\
7577 f continuous_on s /\
7578 (!x. x IN s ==> f x IN interval(a,b))
7579 ==> ?g. g continuous_on u /\
7580 (!x. x IN s ==> g x = f x) /\
7581 (!x. x IN u ==> g(x) IN interval(a,b))`,
7582 REPEAT STRIP_TAC THEN MATCH_MP_TAC TIETZE_OPEN_INTERVAL THEN
7583 ASM_REWRITE_TAC[INTERVAL_NE_EMPTY_1]);;
7585 let TIETZE_UNBOUNDED = prove
7586 (`!f:real^M->real^N u s.
7587 closed_in (subtopology euclidean u) s /\ f continuous_on s
7588 ==> ?g. g continuous_on u /\
7589 (!x. x IN s ==> g x = f x)`,
7590 REPEAT STRIP_TAC THEN
7591 MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^N)`; `u:real^M->bool`;
7592 `s:real^M->bool`] DUGUNDJI) THEN
7593 ASM_REWRITE_TAC[CONVEX_UNIV; UNIV_NOT_EMPTY; SUBSET_UNIV]);;
7595 (* ------------------------------------------------------------------------- *)
7596 (* Convex cones and corresponding hulls. *)
7597 (* ------------------------------------------------------------------------- *)
7599 let convex_cone = new_definition
7600 `convex_cone s <=> ~(s = {}) /\ convex s /\ conic s`;;
7602 let CONVEX_CONE = prove
7606 (!x y. x IN s /\ y IN s ==> (x + y) IN s) /\
7607 (!x c. x IN s /\ &0 <= c ==> (c % x) IN s)`,
7608 GEN_TAC THEN REWRITE_TAC[convex_cone; GSYM conic] THEN
7609 ASM_CASES_TAC `conic(s:real^N->bool)` THEN
7610 ASM_SIMP_TAC[CONIC_CONTAINS_0] THEN AP_TERM_TAC THEN
7611 RULE_ASSUM_TAC(REWRITE_RULE[conic]) THEN
7612 REWRITE_TAC[convex] THEN EQ_TAC THEN
7613 ASM_SIMP_TAC[REAL_SUB_LE] THEN DISCH_TAC THEN
7614 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
7615 FIRST_X_ASSUM(MP_TAC o SPECL
7616 [`&2 % (x:real^N)`; `&2 % (y:real^N)`; `&1 / &2`; `&1 / &2`]) THEN
7617 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
7618 ASM_SIMP_TAC[VECTOR_MUL_LID; REAL_POS]);;
7620 let CONVEX_CONE_ADD = prove
7621 (`!s x y. convex_cone s /\ x IN s /\ y IN s ==> (x + y) IN s`,
7622 MESON_TAC[CONVEX_CONE]);;
7624 let CONVEX_CONE_MUL = prove
7625 (`!s c x. convex_cone s /\ &0 <= c /\ x IN s ==> (c % x) IN s`,
7626 MESON_TAC[CONVEX_CONE]);;
7628 let CONVEX_CONE_NONEMPTY = prove
7629 (`!s. convex_cone s ==> ~(s = {})`,
7630 MESON_TAC[CONVEX_CONE; MEMBER_NOT_EMPTY]);;
7632 let CONVEX_CONE_LINEAR_IMAGE = prove
7633 (`!f:real^M->real^N s.
7634 convex_cone s /\ linear f ==> convex_cone(IMAGE f s)`,
7635 SIMP_TAC[convex_cone; CONVEX_LINEAR_IMAGE; IMAGE_EQ_EMPTY;
7636 CONIC_LINEAR_IMAGE]);;
7638 let CONVEX_CONE_LINEAR_IMAGE_EQ = prove
7639 (`!f:real^M->real^N s.
7640 linear f /\ (!x y. f x = f y ==> x = y)
7641 ==> (convex_cone(IMAGE f s) <=> convex_cone s)`,
7642 REWRITE_TAC[convex_cone] THEN
7643 MESON_TAC[IMAGE_EQ_EMPTY; CONVEX_LINEAR_IMAGE_EQ; CONIC_LINEAR_IMAGE_EQ]);;
7645 add_linear_invariants [CONVEX_CONE_LINEAR_IMAGE_EQ];;
7647 let CONVEX_CONE_HALFSPACE_GE = prove
7648 (`!a. convex_cone {x | a dot x >= &0}`,
7649 SIMP_TAC[CONVEX_CONE; real_ge; IN_ELIM_THM; DOT_RZERO; DOT_RADD; DOT_RMUL;
7650 REAL_LE_ADD; REAL_LE_MUL; REAL_LE_REFL]);;
7652 let CONVEX_CONE_HALFSPACE_LE = prove
7653 (`!a. convex_cone {x | a dot x <= &0}`,
7654 REWRITE_TAC[REAL_ARITH `x <= &0 <=> &0 <= --x`; GSYM DOT_LNEG] THEN
7655 REWRITE_TAC[GSYM real_ge; CONVEX_CONE_HALFSPACE_GE]);;
7657 let CONVEX_CONE_CONTAINS_0 = prove
7658 (`!s:real^N->bool. convex_cone s ==> vec 0 IN s`,
7659 SIMP_TAC[CONVEX_CONE]);;
7661 let CONVEX_CONE_INTERS = prove
7662 (`!f. (!s:real^N->bool. s IN f ==> convex_cone s) ==> convex_cone(INTERS f)`,
7663 SIMP_TAC[convex_cone; CONIC_INTERS; CONVEX_INTERS] THEN
7664 REWRITE_TAC[GSYM convex_cone] THEN GEN_TAC THEN DISCH_TAC THEN
7665 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN
7666 ASM_SIMP_TAC[IN_INTERS; CONVEX_CONE_CONTAINS_0]);;
7668 let CONVEX_CONE_CONVEX_CONE_HULL = prove
7669 (`!s. convex_cone(convex_cone hull s)`,
7670 SIMP_TAC[P_HULL; CONVEX_CONE_INTERS]);;
7672 let CONVEX_CONVEX_CONE_HULL = prove
7673 (`!s. convex(convex_cone hull s)`,
7674 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);;
7676 let CONIC_CONVEX_CONE_HULL = prove
7677 (`!s. conic(convex_cone hull s)`,
7678 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);;
7680 let CONVEX_CONE_HULL_NONEMPTY = prove
7681 (`!s. ~(convex_cone hull s = {})`,
7682 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; convex_cone]);;
7684 let CONVEX_CONE_HULL_CONTAINS_0 = prove
7685 (`!s. vec 0 IN convex_cone hull s`,
7686 MESON_TAC[CONVEX_CONE_CONVEX_CONE_HULL; CONVEX_CONE]);;
7688 let CONVEX_CONE_HULL_ADD = prove
7690 x IN convex_cone hull s /\ y IN convex_cone hull s
7691 ==> x + y IN convex_cone hull s`,
7692 MESON_TAC[CONVEX_CONE; CONVEX_CONE_CONVEX_CONE_HULL]);;
7694 let CONVEX_CONE_HULL_MUL = prove
7696 &0 <= c /\ x IN convex_cone hull s
7697 ==> (c % x) IN convex_cone hull s`,
7698 MESON_TAC[CONVEX_CONE; CONVEX_CONE_CONVEX_CONE_HULL]);;
7700 let CONVEX_CONE_SUMS = prove
7701 (`!s t. convex_cone s /\ convex_cone t
7702 ==> convex_cone {x + y:real^N | x IN s /\ y IN t}`,
7703 SIMP_TAC[convex_cone; CONIC_SUMS; CONVEX_SUMS] THEN SET_TAC[]);;
7705 let CONVEX_CONE_PCROSS = prove
7706 (`!s:real^M->bool t:real^N->bool.
7707 convex_cone s /\ convex_cone t ==> convex_cone(s PCROSS t)`,
7708 SIMP_TAC[convex_cone; CONVEX_PCROSS; CONIC_PCROSS; PCROSS_EQ_EMPTY]);;
7710 let CONVEX_CONE_PCROSS_EQ = prove
7711 (`!s:real^M->bool t:real^N->bool.
7712 convex_cone(s PCROSS t) <=> convex_cone s /\ convex_cone t`,
7714 ASM_CASES_TAC `s:real^M->bool = {}` THENL
7715 [ASM_REWRITE_TAC[PCROSS_EMPTY; convex_cone]; ALL_TAC] THEN
7716 ASM_CASES_TAC `t:real^N->bool = {}` THENL
7717 [ASM_REWRITE_TAC[PCROSS_EMPTY; convex_cone]; ALL_TAC] THEN
7718 EQ_TAC THEN REWRITE_TAC[CONVEX_CONE_PCROSS] THEN REPEAT STRIP_TAC THENL
7719 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
7720 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_CONE_LINEAR_IMAGE) THEN
7721 ASM_REWRITE_TAC[LINEAR_FSTCART];
7722 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
7723 `(s:real^M->bool) PCROSS (t:real^N->bool)`] CONVEX_CONE_LINEAR_IMAGE) THEN
7724 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
7725 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
7726 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
7727 FSTCART_PASTECART; SNDCART_PASTECART] THEN
7730 let CONVEX_CONE_HULL_UNION = prove
7731 (`!s t. convex_cone hull(s UNION t) =
7732 {x + y:real^N | x IN convex_cone hull s /\ y IN convex_cone hull t}`,
7733 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
7734 [MATCH_MP_TAC HULL_MINIMAL THEN
7735 SIMP_TAC[CONVEX_CONE_SUMS; CONVEX_CONE_CONVEX_CONE_HULL] THEN
7736 REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN
7737 X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL
7738 [MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
7739 ASM_SIMP_TAC[HULL_INC; CONVEX_CONE_HULL_CONTAINS_0; VECTOR_ADD_RID];
7740 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN
7741 ASM_SIMP_TAC[HULL_INC; CONVEX_CONE_HULL_CONTAINS_0; VECTOR_ADD_LID]];
7742 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
7743 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_CONE_HULL_ADD THEN
7744 ASM_MESON_TAC[HULL_MONO; SUBSET_UNION; SUBSET]]);;
7746 let CONVEX_CONE_SING = prove
7747 (`convex_cone {vec 0}`,
7748 SIMP_TAC[CONVEX_CONE; IN_SING; VECTOR_ADD_LID; VECTOR_MUL_RZERO]);;
7750 let CONVEX_HULL_SUBSET_CONVEX_CONE_HULL = prove
7751 (`!s. convex hull s SUBSET convex_cone hull s`,
7752 GEN_TAC THEN MATCH_MP_TAC HULL_ANTIMONO THEN
7753 SIMP_TAC[convex_cone; SUBSET; IN]);;
7755 let CONIC_HULL_SUBSET_CONVEX_CONE_HULL = prove
7756 (`!s. conic hull s SUBSET convex_cone hull s`,
7757 GEN_TAC THEN MATCH_MP_TAC HULL_ANTIMONO THEN
7758 SIMP_TAC[convex_cone; SUBSET; IN]);;
7760 let CONVEX_CONE_HULL_SEPARATE_NONEMPTY = prove
7763 ==> convex_cone hull s = conic hull (convex hull s)`,
7764 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THEN
7765 MATCH_MP_TAC HULL_MINIMAL THEN
7766 REWRITE_TAC[CONIC_CONVEX_CONE_HULL; CONVEX_HULL_SUBSET_CONVEX_CONE_HULL] THEN
7767 ASM_SIMP_TAC[CONVEX_CONIC_HULL; CONVEX_CONVEX_HULL; CONIC_CONIC_HULL;
7768 convex_cone; CONIC_HULL_EQ_EMPTY; CONVEX_HULL_EQ_EMPTY] THEN
7769 ASM_MESON_TAC[HULL_SUBSET; SUBSET_REFL; SUBSET_TRANS]);;
7771 let CONVEX_CONE_HULL_EMPTY = prove
7772 (`convex_cone hull {} = {vec 0}`,
7773 MATCH_MP_TAC HULL_UNIQUE THEN
7774 REWRITE_TAC[CONVEX_CONE_CONTAINS_0; EMPTY_SUBSET; CONVEX_CONE_SING;
7775 SET_RULE `{a} SUBSET s <=> a IN s`; CONVEX_CONE_CONTAINS_0]);;
7777 let CONVEX_CONE_HULL_SEPARATE = prove
7779 convex_cone hull s = vec 0 INSERT conic hull (convex hull s)`,
7780 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
7781 ASM_SIMP_TAC[CONVEX_CONE_HULL_EMPTY; CONVEX_HULL_EMPTY; CONIC_HULL_EMPTY] THEN
7782 ASM_SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY] THEN
7783 MATCH_MP_TAC(SET_RULE `a IN s ==> s = a INSERT s`) THEN
7784 ASM_SIMP_TAC[CONIC_CONTAINS_0; CONIC_CONIC_HULL] THEN
7785 ASM_REWRITE_TAC[CONIC_HULL_EQ_EMPTY; CONVEX_HULL_EQ_EMPTY]);;
7787 let CONVEX_CONE_HULL_CONVEX_HULL_NONEMPTY = prove
7790 ==> convex_cone hull s = {c % x | &0 <= c /\ x IN convex hull s}`,
7791 SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY; CONIC_HULL_EXPLICIT]);;
7793 let CONVEX_CONE_HULL_CONVEX_HULL = prove
7795 convex_cone hull s =
7796 vec 0 INSERT {c % x | &0 <= c /\ x IN convex hull s}`,
7797 REWRITE_TAC[CONVEX_CONE_HULL_SEPARATE; CONIC_HULL_EXPLICIT]);;
7799 let CONVEX_CONE_HULL_LINEAR_IMAGE = prove
7800 (`!f:real^M->real^N s.
7802 ==> convex_cone hull (IMAGE f s) = IMAGE f (convex_cone hull s)`,
7803 REPEAT STRIP_TAC THEN
7804 ASM_CASES_TAC `s:real^M-> bool = {}` THEN
7805 ASM_SIMP_TAC[CONVEX_CONE_HULL_SEPARATE_NONEMPTY; IMAGE_EQ_EMPTY;
7806 CONVEX_HULL_LINEAR_IMAGE; CONIC_HULL_LINEAR_IMAGE] THEN
7807 REWRITE_TAC[IMAGE_CLAUSES; CONVEX_CONE_HULL_EMPTY] THEN
7808 MATCH_MP_TAC(SET_RULE `f x = y ==> {y} = {f x}`) THEN
7809 ASM_MESON_TAC[LINEAR_0]);;
7811 add_linear_invariants [CONVEX_CONE_HULL_LINEAR_IMAGE];;
7813 let SUBSPACE_IMP_CONVEX_CONE = prove
7814 (`!s. subspace s ==> convex_cone s`,
7815 SIMP_TAC[subspace; CONVEX_CONE]);;
7817 let CONVEX_CONE_SPAN = prove
7818 (`!s. convex_cone(span s)`,
7819 SIMP_TAC[convex_cone; CONVEX_SPAN; CONIC_SPAN; GSYM MEMBER_NOT_EMPTY] THEN
7820 MESON_TAC[SPAN_0]);;
7822 let CONVEX_CONE_NEGATIONS = prove
7823 (`!s. convex_cone s ==> convex_cone (IMAGE (--) s)`,
7824 SIMP_TAC[convex_cone; IMAGE_EQ_EMPTY; CONIC_NEGATIONS; CONVEX_NEGATIONS]);;
7826 let SUBSPACE_CONVEX_CONE_SYMMETRIC = prove
7828 subspace s <=> convex_cone s /\ (!x. x IN s ==> --x IN s)`,
7829 GEN_TAC THEN REWRITE_TAC[subspace; CONVEX_CONE] THEN
7830 EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THENL
7831 [ASM_MESON_TAC[VECTOR_ARITH `--x:real^N = -- &1 % x`];
7832 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN DISCH_TAC THEN
7833 DISJ_CASES_TAC(SPEC `c:real` REAL_LE_NEGTOTAL) THEN ASM_SIMP_TAC[] THEN
7834 ASM_MESON_TAC[VECTOR_ARITH `c % x:real^N = --(--c % x)`]]);;
7836 let SPAN_CONVEX_CONE_ALLSIGNS = prove
7837 (`!s:real^N->bool. span s = convex_cone hull (s UNION IMAGE (--) s)`,
7838 GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
7839 [MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN CONJ_TAC THENL
7840 [MESON_TAC[HULL_SUBSET; SUBSET_UNION; SUBSET_TRANS]; ALL_TAC] THEN
7841 REWRITE_TAC[SUBSPACE_CONVEX_CONE_SYMMETRIC;
7842 CONVEX_CONE_CONVEX_CONE_HULL] THEN
7843 MATCH_MP_TAC HULL_INDUCT THEN CONJ_TAC THENL
7844 [X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_UNION; IN_IMAGE] THEN
7845 DISCH_TAC THEN MATCH_MP_TAC HULL_INC THEN
7846 REWRITE_TAC[IN_UNION; IN_IMAGE] THEN ASM_MESON_TAC[VECTOR_NEG_NEG];
7847 SUBGOAL_THEN `!s. {x:real^N | (--x) IN s} = IMAGE (--) s`
7848 (fun th -> SIMP_TAC[th; CONVEX_CONE_NEGATIONS;
7849 CONVEX_CONE_CONVEX_CONE_HULL]) THEN
7850 GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
7851 REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[VECTOR_NEG_NEG]];
7852 MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_CONE_SPAN] THEN
7853 REWRITE_TAC[UNION_SUBSET; SPAN_INC] THEN
7854 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
7855 MESON_TAC[SPAN_SUPERSET; SPAN_NEG]]);;
7857 (* ------------------------------------------------------------------------- *)
7858 (* Epigraphs of convex functions. *)
7859 (* ------------------------------------------------------------------------- *)
7861 let epigraph = new_definition
7862 `epigraph s (f:real^N->real) =
7863 {xy:real^((N,1)finite_sum) |
7864 fstcart xy IN s /\ f(fstcart xy) <= drop(sndcart xy)}`;;
7866 let IN_EPIGRAPH = prove
7867 (`!x y. (pastecart x (lift y)) IN epigraph s f <=> x IN s /\ f(x) <= y`,
7868 REWRITE_TAC[epigraph; IN_ELIM_THM; FSTCART_PASTECART; SNDCART_PASTECART;
7871 let CONVEX_EPIGRAPH = prove
7872 (`!f s. f convex_on s /\ convex s <=> convex(epigraph s f)`,
7873 REWRITE_TAC[convex; convex_on; IN_ELIM_THM; SNDCART_ADD; SNDCART_CMUL;
7874 epigraph; FSTCART_ADD; FSTCART_CMUL; FORALL_PASTECART; FSTCART_PASTECART;
7875 SNDCART_PASTECART] THEN
7876 REWRITE_TAC[GSYM FORALL_DROP; DROP_ADD; DROP_CMUL] THEN
7877 MESON_TAC[REAL_LE_REFL; REAL_LE_ADD2; REAL_LE_LMUL; REAL_LE_TRANS]);;
7879 let CONVEX_EPIGRAPH_CONVEX = prove
7880 (`!f s. convex s ==> (f convex_on s <=> convex(epigraph s f))`,
7881 REWRITE_TAC[GSYM CONVEX_EPIGRAPH] THEN CONV_TAC TAUT);;
7883 let CONVEX_ON_EPIGRAPH_SLICE_LE = prove
7884 (`!f:real^N->real s a.
7885 f convex_on s /\ convex s ==> convex {x | x IN s /\ f(x) <= a}`,
7886 SIMP_TAC[convex_on; convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
7887 FIRST_X_ASSUM(fun th ->
7888 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
7889 ASM_REWRITE_TAC[] THEN
7890 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
7891 MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REWRITE_TAC[]);;
7893 let CONVEX_ON_EPIGRAPH_SLICE_LT = prove
7894 (`!f:real^N->real s a.
7895 f convex_on s /\ convex s ==> convex {x | x IN s /\ f(x) < a}`,
7896 SIMP_TAC[convex_on; convex; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
7897 FIRST_X_ASSUM(fun th ->
7898 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
7899 ASM_REWRITE_TAC[] THEN
7900 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN
7901 MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REWRITE_TAC[]);;
7903 (* ------------------------------------------------------------------------- *)
7904 (* Use this to derive general bound property of convex function. *)
7905 (* ------------------------------------------------------------------------- *)
7907 let FORALL_OF_PASTECART = prove
7908 (`(!p. P (fstcart o p) (sndcart o p)) <=> (!x:A->B^M y:A->B^N. P x y)`,
7909 EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THEN
7910 FIRST_ASSUM(MP_TAC o SPEC `\a:A. pastecart (x a :B^M) (y a :B^N)`) THEN
7911 REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART; ETA_AX]);;
7913 let FORALL_OF_DROP = prove
7914 (`(!v. P (drop o v)) <=> (!x:A->real. P x)`,
7915 EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN REPEAT STRIP_TAC THEN
7916 FIRST_ASSUM(MP_TAC o SPEC `\a:A. lift(x a)`) THEN
7917 REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]);;
7919 let CONVEX_ON_JENSEN = prove
7920 (`!f:real^N->real s.
7922 ==> (f convex_on s <=>
7924 (!i:num. 1 <= i /\ i <= k ==> &0 <= u(i) /\ x(i) IN s) /\
7926 ==> f(vsum (1..k) (\i. u(i) % x(i)))
7927 <= sum (1..k) (\i. u(i) * f(x(i))))`,
7929 (`(!x. P x ==> (Q x = R x)) ==> (!x. P x) ==> ((!x. Q x) <=> (!x. R x))`,
7931 REPEAT STRIP_TAC THEN FIRST_ASSUM
7932 (fun th -> REWRITE_TAC[MATCH_MP CONVEX_EPIGRAPH_CONVEX th]) THEN
7933 REWRITE_TAC[CONVEX_INDEXED; epigraph] THEN
7934 SIMP_TAC[IN_ELIM_THM; SNDCART_ADD; SNDCART_CMUL; FINITE_NUMSEG;
7935 FSTCART_ADD; FSTCART_CMUL; FORALL_PASTECART; DROP_CMUL;
7936 FSTCART_PASTECART; SNDCART_PASTECART;
7937 FSTCART_VSUM; SNDCART_VSUM; DROP_VSUM; o_DEF] THEN
7938 REWRITE_TAC[GSYM(ISPEC `fstcart` o_THM); GSYM(ISPEC `sndcart` o_THM)] THEN
7939 REWRITE_TAC[GSYM(ISPEC `drop` o_THM)] THEN
7940 REWRITE_TAC[FORALL_OF_PASTECART; FORALL_OF_DROP] THEN
7941 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_INDEXED]) THEN
7942 REPEAT(MATCH_MP_TAC lemma THEN GEN_TAC) THEN SIMP_TAC[] THEN
7943 REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN
7944 REWRITE_TAC[FORALL_AND_THM] THEN DISCH_THEN(K ALL_TAC) THEN
7945 EQ_TAC THEN SIMP_TAC[REAL_LE_REFL] THEN
7946 DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN
7947 ASM_REWRITE_TAC[] THEN
7948 MATCH_MP_TAC(REAL_ARITH `a <= b ==> x <= a ==> x <= b`) THEN
7949 ASM_SIMP_TAC[SUM_LE_NUMSEG; REAL_LE_LMUL]);;
7951 let CONVEX_ON_IMP_JENSEN = prove
7952 (`!f:real^N->real s k:A->bool u x.
7953 f convex_on s /\ convex s /\ FINITE k /\
7954 (!i. i IN k ==> &0 <= u i /\ x i IN s) /\ sum k u = &1
7955 ==> f(vsum k (\i. u i % x i)) <= sum k (\i. u i * f(x i))`,
7957 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
7958 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN
7959 ABBREV_TAC `n = CARD(k:A->bool)` THEN
7960 REWRITE_TAC[INJECTIVE_ON_ALT] THEN
7961 DISCH_THEN(X_CHOOSE_THEN `g:num->A`
7962 (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
7963 ASM_SIMP_TAC[VSUM_IMAGE; SUM_IMAGE; FINITE_NUMSEG; IMP_CONJ; o_DEF] THEN
7964 DISCH_TAC THEN MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`]
7965 CONVEX_ON_JENSEN) THEN
7966 ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
7967 ASM_REWRITE_TAC[GSYM IN_NUMSEG] THEN ASM SET_TAC[]);;
7969 (* ------------------------------------------------------------------------- *)
7970 (* Another intermediate value theorem formulation. *)
7971 (* ------------------------------------------------------------------------- *)
7973 let IVT_INCREASING_COMPONENT_ON_1 = prove
7974 (`!f:real^1->real^N a b y k.
7975 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
7976 f continuous_on interval[a,b] /\
7977 f(a)$k <= y /\ y <= f(b)$k
7978 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
7979 REPEAT STRIP_TAC THEN
7980 MP_TAC(ISPECL [`IMAGE (f:real^1->real^N) (interval[a,b])`]
7981 CONNECTED_IVT_COMPONENT) THEN
7982 REWRITE_TAC[EXISTS_IN_IMAGE] THEN DISCH_THEN MATCH_MP_TAC THEN
7983 REWRITE_TAC[RIGHT_EXISTS_AND_THM; EXISTS_IN_IMAGE] THEN
7984 ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; CONVEX_CONNECTED;
7985 CONVEX_INTERVAL] THEN
7986 EXISTS_TAC `a:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL] THEN
7987 EXISTS_TAC `b:real^1` THEN ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL]);;
7989 let IVT_INCREASING_COMPONENT_1 = prove
7990 (`!f:real^1->real^N a b y k.
7991 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
7992 (!x. x IN interval[a,b] ==> f continuous at x) /\
7993 f(a)$k <= y /\ y <= f(b)$k
7994 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
7995 REPEAT STRIP_TAC THEN MATCH_MP_TAC IVT_INCREASING_COMPONENT_ON_1 THEN
7996 ASM_SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
7998 let IVT_DECREASING_COMPONENT_ON_1 = prove
7999 (`!f:real^1->real^N a b y k.
8000 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
8001 f continuous_on interval[a,b] /\
8002 f(b)$k <= y /\ y <= f(a)$k
8003 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
8004 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EQ_NEG2] THEN
8005 ASM_SIMP_TAC[GSYM VECTOR_NEG_COMPONENT] THEN
8006 MATCH_MP_TAC IVT_INCREASING_COMPONENT_ON_1 THEN
8007 ASM_SIMP_TAC[VECTOR_NEG_COMPONENT; CONTINUOUS_ON_NEG; REAL_LE_NEG2]);;
8009 let IVT_DECREASING_COMPONENT_1 = prove
8010 (`!f:real^1->real^N a b y k.
8011 drop a <= drop b /\ 1 <= k /\ k <= dimindex(:N) /\
8012 (!x. x IN interval[a,b] ==> f continuous at x) /\
8013 f(b)$k <= y /\ y <= f(a)$k
8014 ==> ?x. x IN interval[a,b] /\ f(x)$k = y`,
8015 REPEAT STRIP_TAC THEN MATCH_MP_TAC IVT_DECREASING_COMPONENT_ON_1 THEN
8016 ASM_SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
8018 (* ------------------------------------------------------------------------- *)
8019 (* A bound within a convex hull, and so an interval. *)
8020 (* ------------------------------------------------------------------------- *)
8022 let CONVEX_ON_CONVEX_HULL_BOUND = prove
8023 (`!f s b. f convex_on (convex hull s) /\
8024 (!x:real^N. x IN s ==> f(x) <= b)
8025 ==> !x. x IN convex hull s ==> f(x) <= b`,
8026 REPEAT GEN_TAC THEN SIMP_TAC[CONVEX_ON_JENSEN; CONVEX_CONVEX_HULL] THEN
8027 STRIP_TAC THEN GEN_TAC THEN REWRITE_TAC[CONVEX_HULL_INDEXED] THEN
8028 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
8029 MAP_EVERY X_GEN_TAC [`k:num`; `u:num->real`; `v:num->real^N`] THEN
8030 DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN ASM_REWRITE_TAC[] THEN
8031 MATCH_MP_TAC REAL_LE_TRANS THEN
8032 EXISTS_TAC `sum(1..k) (\i. u i * f(v i :real^N))` THEN CONJ_TAC THENL
8033 [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[SUBSET; HULL_SUBSET];
8035 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum(1..k) (\i. u i * b)` THEN
8037 [MATCH_MP_TAC SUM_LE_NUMSEG THEN ASM_SIMP_TAC[REAL_LE_LMUL];
8039 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[SUM_LMUL] THEN
8040 ASM_MESON_TAC[REAL_LE_REFL; REAL_MUL_RID]);;
8042 let UNIT_INTERVAL_CONVEX_HULL = prove
8043 (`interval [vec 0,vec 1:real^N] =
8045 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
8046 ==> ((x$i = &0) \/ (x$i = &1))}`,
8048 (`FINITE {i | 1 <= i /\ i <= n /\ P(i)} /\
8049 CARD {i | 1 <= i /\ i <= n /\ P(i)} <= n`,
8051 [MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `1..n`;
8052 GEN_REWRITE_TAC RAND_CONV [ARITH_RULE `x = (x + 1) - 1`] THEN
8053 REWRITE_TAC[GSYM CARD_NUMSEG] THEN MATCH_MP_TAC CARD_SUBSET] THEN
8054 SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; SUBSET; IN_ELIM_THM]) in
8055 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
8057 MATCH_MP_TAC HULL_MINIMAL THEN
8058 REWRITE_TAC[CONVEX_INTERVAL; SUBSET; IN_INTERVAL; IN_ELIM_THM] THEN
8059 SIMP_TAC[VEC_COMPONENT] THEN MESON_TAC[REAL_LE_REFL; REAL_POS]] THEN
8062 x IN interval[vec 0,vec 1] /\
8063 n <= dimindex(:N) /\
8064 CARD {i | 1 <= i /\ i <= dimindex(:N) /\ ~(x$i = &0)} <= n
8065 ==> x IN convex hull
8066 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
8067 ==> ((x$i = &0) \/ (x$i = &1))}`
8070 REWRITE_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN
8071 FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `dimindex(:N)` THEN
8072 ASM_REWRITE_TAC[LE_REFL; lemma]] THEN
8073 INDUCT_TAC THEN X_GEN_TAC `x:real^N` THENL
8074 [SIMP_TAC[LE; lemma; CARD_EQ_0] THEN
8075 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
8076 [EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; BETA_THM] THEN
8077 REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN STRIP_TAC THEN
8078 SUBGOAL_THEN `x = vec 0:real^N` SUBST1_TAC THENL
8079 [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT]; ALL_TAC] THEN
8080 MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
8081 SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT];
8084 `{i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)} = {}`
8086 [DISCH_THEN(K ALL_TAC) THEN
8087 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
8088 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
8089 [EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY; BETA_THM] THEN
8090 REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN STRIP_TAC THEN
8091 SUBGOAL_THEN `x = vec 0:real^N` SUBST1_TAC THENL
8092 [ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT]; ALL_TAC] THEN
8093 MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
8094 SIMP_TAC[IN_ELIM_THM; VEC_COMPONENT];
8098 {i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)}`
8100 ABBREV_TAC `xi = inf
8102 {i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)})` THEN
8103 ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; lemma] THEN
8104 REWRITE_TAC[FORALL_IN_IMAGE] THEN
8105 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [IN_IMAGE; IN_ELIM_THM] THEN
8106 REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
8107 FIRST_X_ASSUM(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC) THEN
8108 FIRST_X_ASSUM SUBST_ALL_TAC THEN REPEAT STRIP_TAC THEN
8109 SUBGOAL_THEN `&0 <= (x:real^N)$i /\ x$i <= &1` STRIP_ASSUME_TAC THENL
8110 [UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN
8111 ASM_SIMP_TAC[IN_INTERVAL; VEC_COMPONENT];
8113 FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
8114 `x <= &1 ==> (x = &1) \/ x < &1`))
8117 `x = lambda i. if (x:real^N)$i = &0 then &0 else &1`
8119 [UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN
8120 ASM_SIMP_TAC[CART_EQ; IN_INTERVAL; VEC_COMPONENT; LAMBDA_BETA] THEN
8121 ASM_MESON_TAC[REAL_LE_ANTISYM];
8123 MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
8124 SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA] THEN MESON_TAC[];
8128 x$i % (lambda j. if x$j = &0 then &0 else &1) +
8130 (lambda j. if x$j = &0 then &0 else (x$j - x$i) / (&1 - x$i))`
8132 [SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
8133 LAMBDA_BETA; VEC_COMPONENT] THEN
8134 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
8135 ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_ADD_LID] THEN
8136 ASM_SIMP_TAC[REAL_DIV_LMUL; ARITH_RULE `x < &1 ==> ~(&1 - x = &0)`] THEN
8137 REPEAT STRIP_TAC THEN REAL_ARITH_TAC;
8139 MATCH_MP_TAC(REWRITE_RULE[convex] CONVEX_CONVEX_HULL) THEN
8140 ASM_SIMP_TAC[REAL_ARITH `x < &1 ==> &0 <= &1 - x`;
8141 REAL_ARITH `x + &1 - x = &1`] THEN
8143 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] HULL_SUBSET) THEN
8144 SIMP_TAC[LAMBDA_BETA; IN_ELIM_THM] THEN MESON_TAC[];
8146 FIRST_X_ASSUM MATCH_MP_TAC THEN
8147 ASM_SIMP_TAC[ARITH_RULE `SUC k <= n ==> k <= n`] THEN CONJ_TAC THENL
8148 [SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; VEC_COMPONENT] THEN
8149 GEN_TAC THEN STRIP_TAC THEN
8150 COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL; REAL_POS] THEN
8151 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ;
8152 REAL_ARITH `x < &1 ==> &0 < &1 - x`] THEN
8153 ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_LE; REAL_MUL_LID] THEN
8154 ASM_SIMP_TAC[REAL_ARITH `a - b <= &1 - b <=> a <= &1`] THEN
8155 UNDISCH_TAC `x:real^N IN interval [vec 0,vec 1]` THEN
8156 ASM_SIMP_TAC[CART_EQ; IN_INTERVAL; VEC_COMPONENT; LAMBDA_BETA];
8158 MATCH_MP_TAC LE_TRANS THEN
8160 `CARD({i | 1 <= i /\ i <= dimindex(:N) /\ ~((x:real^N)$i = &0)}
8163 [MATCH_MP_TAC CARD_SUBSET THEN REWRITE_TAC[lemma; FINITE_DELETE] THEN
8164 REWRITE_TAC[SUBSET; IN_DELETE; IN_ELIM_THM] THEN
8165 GEN_TAC THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
8166 ASM_SIMP_TAC[LAMBDA_BETA] THEN
8167 COND_CASES_TAC THEN ASM_REWRITE_TAC[CONTRAPOS_THM] THEN
8168 SIMP_TAC[real_div; REAL_SUB_REFL; REAL_MUL_LZERO];
8169 SIMP_TAC[lemma; CARD_DELETE] THEN COND_CASES_TAC THEN
8170 ASM_SIMP_TAC[ARITH_RULE `x <= SUC n ==> x - 1 <= n`] THEN
8171 RULE_ASSUM_TAC(REWRITE_RULE[IN_ELIM_THM]) THEN
8174 (* ------------------------------------------------------------------------- *)
8175 (* Some convexity-related properties of Hausdorff distance *)
8176 (* ------------------------------------------------------------------------- *)
8178 let HAUSDIST_CONVEX_HULLS = prove
8179 (`!s t:real^N->bool.
8180 bounded s /\ bounded t
8181 ==> hausdist(convex hull s,convex hull t) <= hausdist(s,t)`,
8182 REPEAT STRIP_TAC THEN
8183 ASM_CASES_TAC `s:real^N->bool = {}` THENL
8184 [ASM_MESON_TAC[HAUSDIST_EMPTY; CONVEX_HULL_EMPTY; REAL_LE_REFL];
8186 ASM_CASES_TAC `t:real^N->bool = {}` THENL
8187 [ASM_MESON_TAC[HAUSDIST_EMPTY; CONVEX_HULL_EMPTY; REAL_LE_REFL];
8189 MATCH_MP_TAC REAL_HAUSDIST_LE THEN ASM_REWRITE_TAC[CONVEX_HULL_EQ_EMPTY] THEN
8190 CONJ_TAC THEN MATCH_MP_TAC CONVEX_ON_CONVEX_HULL_BOUND THEN
8191 CONJ_TAC THEN SIMP_TAC[CONVEX_ON_SETDIST; CONVEX_CONVEX_HULL] THEN
8192 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_HAUSDIST THEN
8193 ASM_REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM; CONJ_ASSOC] THEN
8195 [CONJ_TAC; ASM_MESON_TAC[SETDIST_SUBSET_RIGHT; HULL_SUBSET]]) THEN
8196 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
8197 ASM_REWRITE_TAC[bounded; FORALL_IN_GSPEC; GSYM dist] THEN
8198 MATCH_MP_TAC MONO_EXISTS THEN
8199 ASM_MESON_TAC[SETDIST_LE_DIST; dist; DIST_SYM; REAL_LE_TRANS;
8200 MEMBER_NOT_EMPTY; IN_SING]);;
8202 let HAUSDIST_SUMS = prove
8203 (`!s t:real^N->bool u.
8204 bounded s /\ bounded t /\ convex s /\ convex t /\ bounded u /\
8205 ~(s = {}) /\ ~(t = {}) /\ ~(u = {})
8206 ==> hausdist({x + e | x IN s /\ e IN u},
8207 {y + e | y IN t /\ e IN u}) =
8210 ONCE_REWRITE_TAC[MESON[HAUSDIST_CLOSURE]
8211 `hausdist(s:real^N->bool,t) = hausdist(closure s,closure t)`] THEN
8212 SIMP_TAC[CLOSURE_SUMS] THEN
8213 SIMP_TAC[CLOSURE_CLOSED; CLOSED_CBALL; GSYM COMPACT_CLOSURE] THEN
8214 ONCE_REWRITE_TAC[GSYM CLOSURE_EQ_EMPTY] THEN
8216 `convex(closure s:real^N->bool) /\ convex(closure t:real^N->bool)`
8217 THENL [POP_ASSUM MP_TAC; ASM_MESON_TAC[CONVEX_CLOSURE]] THEN
8218 ASM_CASES_TAC `convex(s:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN
8219 ASM_CASES_TAC `convex(t:real^N->bool)` THEN ASM_REWRITE_TAC[] THEN
8220 POP_ASSUM_LIST(K ALL_TAC) THEN
8221 SPEC_TAC(`closure u:real^N->bool`,`u:real^N->bool`) THEN
8222 SPEC_TAC(`closure t:real^N->bool`,`t:real^N->bool`) THEN
8223 SPEC_TAC(`closure s:real^N->bool`,`s:real^N->bool`) THEN
8224 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN REPEAT STRIP_TAC THEN
8225 MATCH_MP_TAC REAL_HAUSDIST_LE_SUMS THEN
8226 MAP_EVERY ABBREV_TAC
8227 [`a = hausdist(s:real^N->bool,t)`;
8228 `b = hausdist({x + e:real^N | x IN s /\ e IN u},
8229 {y + e | y IN t /\ e IN u})`] THEN
8230 ASM_REWRITE_TAC[CBALL_EQ_EMPTY; REAL_NOT_LT; SET_RULE
8231 `{f x y | x IN s /\ y IN t} = {} <=> s = {} \/ t = {}`]
8233 [REWRITE_TAC[SUMS_ASSOC] THEN
8234 GEN_REWRITE_TAC (BINOP_CONV o RAND_CONV o RAND_CONV o ONCE_DEPTH_CONV)
8236 REWRITE_TAC[GSYM SUMS_ASSOC] THEN CONJ_TAC THEN
8237 MATCH_MP_TAC(SET_RULE
8239 ==> {f x y | x IN s /\ y IN t} SUBSET {f x y | x IN s' /\ y IN t}`) THEN
8240 EXPAND_TAC "a" THENL
8241 [ALL_TAC; ONCE_REWRITE_TAC[HAUSDIST_SYM]] THEN
8242 MATCH_MP_TAC HAUSDIST_COMPACT_SUMS THEN ASM_SIMP_TAC[COMPACT_IMP_BOUNDED];
8243 CONJ_TAC THEN MATCH_MP_TAC SUBSET_SUMS_RCANCEL THEN
8244 EXISTS_TAC `u:real^N->bool` THEN
8245 ASM_SIMP_TAC[CLOSED_COMPACT_SUMS; COMPACT_CBALL; COMPACT_IMP_CLOSED;
8246 CONVEX_CBALL; COMPACT_IMP_BOUNDED; CONVEX_SUMS; REAL_NOT_LT] THEN
8247 REWRITE_TAC[SUMS_ASSOC] THEN
8248 GEN_REWRITE_TAC (RAND_CONV o RAND_CONV o ONCE_DEPTH_CONV) [SUMS_SYM] THEN
8249 REWRITE_TAC[GSYM SUMS_ASSOC] THEN
8250 EXPAND_TAC "b" THENL
8251 [ALL_TAC; ONCE_REWRITE_TAC[HAUSDIST_SYM]] THEN
8252 MATCH_MP_TAC HAUSDIST_COMPACT_SUMS THEN
8253 ASM_SIMP_TAC[BOUNDED_SUMS; COMPACT_SUMS; COMPACT_CBALL;
8254 COMPACT_IMP_BOUNDED; CBALL_EQ_EMPTY; REAL_NOT_LT; SET_RULE
8255 `{f x y | x IN s /\ y IN t} = {} <=> s = {} \/ t = {}`]]);;
8257 (* ------------------------------------------------------------------------- *)
8258 (* Representation of any interval as a finite convex hull. *)
8259 (* ------------------------------------------------------------------------- *)
8261 let CLOSED_INTERVAL_AS_CONVEX_HULL = prove
8262 (`!a b:real^N. ?s. FINITE s /\ interval[a,b] = convex hull s`,
8263 REPEAT GEN_TAC THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL
8264 [ASM_MESON_TAC[CONVEX_HULL_EMPTY; FINITE_EMPTY]; ALL_TAC] THEN
8265 ASM_SIMP_TAC[CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL] THEN
8267 `?s:real^N->bool. FINITE s /\ interval[vec 0,vec 1] = convex hull s`
8268 STRIP_ASSUME_TAC THENL
8270 `{x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
8271 ==> ((x$i = &0) \/ (x$i = &1))}` THEN
8272 REWRITE_TAC[UNIT_INTERVAL_CONVEX_HULL] THEN
8273 MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC
8274 `IMAGE (\s. (lambda i. if i IN s then &1 else &0):real^N)
8275 {t | t SUBSET (1..dimindex(:N))}` THEN
8276 ASM_SIMP_TAC[FINITE_POWERSET; FINITE_IMAGE; FINITE_NUMSEG] THEN
8277 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_IMAGE] THEN
8278 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC
8279 `{i | 1 <= i /\ i <= dimindex(:N) /\ ((x:real^N)$i = &1)}` THEN
8280 SIMP_TAC[CART_EQ; IN_ELIM_THM; IN_NUMSEG; LAMBDA_BETA] THEN
8282 EXISTS_TAC `IMAGE (\x:real^N. a + x)
8283 (IMAGE (\x. (lambda i. ((b:real^N)$i - a$i) * x$i))
8284 (s:real^N->bool))` THEN
8285 ASM_SIMP_TAC[FINITE_IMAGE; CONVEX_HULL_TRANSLATION] THEN
8286 AP_TERM_TAC THEN MATCH_MP_TAC(GSYM CONVEX_HULL_LINEAR_IMAGE) THEN
8287 SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
8288 VECTOR_MUL_COMPONENT] THEN
8289 REPEAT STRIP_TAC THEN REAL_ARITH_TAC]);;
8291 (* ------------------------------------------------------------------------- *)
8292 (* Bounded convex function on open set is continuous. *)
8293 (* ------------------------------------------------------------------------- *)
8295 let CONVEX_ON_BOUNDED_CONTINUOUS = prove
8296 (`!f:real^N->real s b.
8297 open s /\ f convex_on s /\ (!x. x IN s ==> abs(f x) <= b)
8298 ==> (lift o f) continuous_on s`,
8299 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
8300 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
8301 REWRITE_TAC[CONTINUOUS_AT_LIFT_RANGE] THEN
8302 ABBREV_TAC `B = abs(b) + &1` THEN
8303 SUBGOAL_THEN `&0 < B /\ !x:real^N. x IN s ==> abs(f x) <= B`
8304 STRIP_ASSUME_TAC THENL
8305 [EXPAND_TAC "B" THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
8306 ASM_MESON_TAC[REAL_ARITH `x <= b ==> x <= abs b + &1`];
8308 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
8309 REWRITE_TAC[REAL_ARITH `abs(x - y) < e <=> x - y < e /\ y - x < e`] THEN
8310 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
8311 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[SUBSET; IN_CBALL] THEN
8312 ASM_REWRITE_TAC[] THEN
8313 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
8314 EXISTS_TAC `min (k / &2) (e / (&2 * B) * k)` THEN
8315 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; REAL_LT_MUL;
8316 REAL_OF_NUM_LT; ARITH] THEN
8317 X_GEN_TAC `y:real^N` THEN
8318 ASM_CASES_TAC `y:real^N = x` THEN ASM_REWRITE_TAC[REAL_SUB_REFL] THEN
8320 ABBREV_TAC `t = k / norm(y - x:real^N)` THEN
8321 SUBGOAL_THEN `&2 < t` ASSUME_TAC THENL
8322 [EXPAND_TAC "t" THEN
8323 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
8324 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8325 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH];
8327 FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP (REAL_ARITH
8328 `&2 < t ==> &0 < t /\ ~(t = &0) /\ &0 < t - &1 /\
8329 &0 < &1 + t /\ ~(&1 + t = &0)`)) THEN
8330 SUBGOAL_THEN `y:real^N IN s` ASSUME_TAC THENL
8331 [FIRST_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[dist] THEN
8332 ONCE_REWRITE_TAC[NORM_SUB] THEN
8333 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
8334 `x < k / &2 ==> k / &2 <= k ==> x <= k`)) THEN
8335 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
8336 UNDISCH_TAC `&0 < k` THEN REAL_ARITH_TAC;
8339 [ABBREV_TAC `w:real^N = x + t % (y - x)` THEN
8340 SUBGOAL_THEN `w:real^N IN s` STRIP_ASSUME_TAC THENL
8341 [FIRST_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "w" THEN
8342 REWRITE_TAC[dist; VECTOR_ARITH `x - (x + t) = --t:real^N`] THEN
8343 EXPAND_TAC "t" THEN REWRITE_TAC[NORM_NEG; NORM_MUL; REAL_ABS_DIV] THEN
8344 REWRITE_TAC[REAL_ABS_NORM; real_div; GSYM REAL_MUL_ASSOC] THEN
8345 ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; NORM_POS_LT; VECTOR_SUB_EQ;
8346 REAL_MUL_RID; REAL_ARITH `&0 < x ==> abs(x) <= x`];
8348 SUBGOAL_THEN `(&1 / t) % w + (t - &1) / t % x = y:real^N` ASSUME_TAC THENL
8349 [EXPAND_TAC "w" THEN
8350 REWRITE_TAC[VECTOR_ARITH
8351 `b % (x + c % (y - x)) + a % x =
8352 (a + b - b * c) % x + (b * c) % y`] THEN
8353 ASM_SIMP_TAC[REAL_DIV_RMUL; VECTOR_MUL_LID] THEN
8354 ASM_SIMP_TAC[real_div; REAL_MUL_RINV; REAL_SUB_REFL;
8355 VECTOR_MUL_LZERO; VECTOR_ADD_LID;
8356 REAL_ARITH `(a - &1) * b + &1 * b - &1 = a * b - &1`];
8358 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
8359 DISCH_THEN(MP_TAC o SPECL
8360 [`w:real^N`; `x:real^N`; `&1 / t`; `(t - &1) / t`]) THEN
8361 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_DIV; REAL_LT_01] THEN
8362 REWRITE_TAC[real_div; GSYM REAL_ADD_RDISTRIB] THEN
8363 ASM_SIMP_TAC[REAL_SUB_ADD2; REAL_MUL_RINV] THEN
8364 MATCH_MP_TAC(REAL_ARITH
8365 `a * fw + (b - &1) * fx < e
8366 ==> fy <= a * fw + b * fx ==> fy - fx < e`) THEN
8367 ASM_SIMP_TAC[real_div; REAL_SUB_RDISTRIB; REAL_MUL_RINV; REAL_MUL_LID;
8368 REAL_ARITH `a * x + y - a * y - y = a * (x - y)`] THEN
8369 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8370 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
8371 MATCH_MP_TAC(REAL_ARITH
8372 `!b. abs(x) <= b /\ abs(y) <= b /\ &2 * b < z ==> x - y < z`) THEN
8373 EXISTS_TAC `B:real` THEN ASM_SIMP_TAC[] THEN EXPAND_TAC "t" THEN
8374 REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN
8375 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
8376 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8377 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN
8378 REWRITE_TAC[real_div; REAL_ARITH `(a * b) * inv c = (b * inv c) * a`] THEN
8379 ASM_REWRITE_TAC[GSYM real_div];
8381 ABBREV_TAC `w:real^N = x - t % (y - x)` THEN
8382 SUBGOAL_THEN `w:real^N IN s` STRIP_ASSUME_TAC THENL
8383 [FIRST_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "w" THEN
8384 REWRITE_TAC[dist; VECTOR_ARITH `x - (x - t) = t:real^N`] THEN
8385 EXPAND_TAC "t" THEN REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN
8386 REWRITE_TAC[REAL_ABS_NORM; real_div; GSYM REAL_MUL_ASSOC] THEN
8387 ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; NORM_POS_LT; VECTOR_SUB_EQ;
8388 REAL_MUL_RID; REAL_ARITH `&0 < x ==> abs(x) <= x`];
8390 SUBGOAL_THEN `(&1 / (&1 + t)) % w + t / (&1 + t) % y = x:real^N`
8392 [EXPAND_TAC "w" THEN
8393 REWRITE_TAC[VECTOR_ARITH
8394 `b % (x - c % (y - x)) + a % y =
8395 (b * (&1 + c)) % x + (a - b * c) % y`] THEN
8396 ASM_SIMP_TAC[REAL_DIV_RMUL; VECTOR_MUL_LID] THEN
8397 REWRITE_TAC[real_div; REAL_MUL_AC; REAL_MUL_LID; REAL_MUL_RID] THEN
8398 REWRITE_TAC[REAL_SUB_REFL; VECTOR_MUL_LZERO; VECTOR_ADD_RID];
8400 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
8401 DISCH_THEN(MP_TAC o SPECL
8402 [`w:real^N`; `y:real^N`; `&1 / (&1 + t)`; `t / (&1 + t)`]) THEN
8403 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LT_DIV; REAL_LT_01] THEN
8404 REWRITE_TAC[real_div; GSYM REAL_ADD_RDISTRIB] THEN
8405 ASM_SIMP_TAC[REAL_SUB_ADD2; REAL_MUL_RINV] THEN
8406 MATCH_MP_TAC(REAL_ARITH
8407 `a * fw + (b - &1) * fx < e
8408 ==> fy <= a * fw + b * fx ==> fy - fx < e`) THEN
8409 SUBGOAL_THEN `t * inv(&1 + t) - &1 = --(inv(&1 + t))` SUBST1_TAC THENL
8410 [REWRITE_TAC[REAL_ARITH `(a * b - &1 = --b) <=> ((&1 + a) * b = &1)`] THEN
8411 ASM_SIMP_TAC[REAL_MUL_RINV];
8413 REWRITE_TAC[REAL_ARITH `(&1 * a) * x + --a * y = a * (x - y)`] THEN
8414 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8415 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
8416 MATCH_MP_TAC(REAL_ARITH
8417 `!b. abs(x) <= b /\ abs(y) <= b /\ &2 * b < z ==> x - y < z`) THEN
8418 EXISTS_TAC `B:real` THEN ASM_SIMP_TAC[] THEN
8419 MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x < e * k ==> x < e * (&1 + k)`) THEN
8420 EXPAND_TAC "t" THEN REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN
8421 REWRITE_TAC[GSYM real_div] THEN
8422 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
8423 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8424 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN
8425 REWRITE_TAC[real_div; REAL_ARITH `(a * b) * inv c = (b * inv c) * a`] THEN
8426 ASM_REWRITE_TAC[GSYM real_div]]);;
8428 (* ------------------------------------------------------------------------- *)
8429 (* Upper bound on a ball implies upper and lower bounds. *)
8430 (* ------------------------------------------------------------------------- *)
8432 let CONVEX_BOUNDS_LEMMA = prove
8434 f convex_on cball(x,e) /\
8435 (!y. y IN cball(x,e) ==> f(y) <= b)
8436 ==> !y. y IN cball(x,e) ==> abs(f(y)) <= b + &2 * abs(f(x))`,
8437 REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 <= e` THENL
8439 REWRITE_TAC[IN_CBALL] THEN ASM_MESON_TAC[DIST_POS_LE; REAL_LE_TRANS]] THEN
8440 REPEAT STRIP_TAC THEN
8441 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [convex_on]) THEN
8442 DISCH_THEN(MP_TAC o SPECL
8443 [`y:real^N`; `&2 % x - y:real^N`; `&1 / &2`; `&1 / &2`]) THEN
8444 REWRITE_TAC[GSYM VECTOR_ADD_LDISTRIB; GSYM REAL_ADD_LDISTRIB] THEN
8445 REWRITE_TAC[VECTOR_ARITH `y + x - y = x:real^N`] THEN
8446 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
8447 ABBREV_TAC `z = &2 % x - y:real^N` THEN
8448 SUBGOAL_THEN `z:real^N IN cball(x,e)` ASSUME_TAC THENL
8449 [UNDISCH_TAC `y:real^N IN cball(x,e)` THEN
8450 EXPAND_TAC "z" THEN REWRITE_TAC[dist; IN_CBALL] THEN
8451 REWRITE_TAC[VECTOR_ARITH `x - (&2 % x - y) = y - x`] THEN
8452 REWRITE_TAC[NORM_SUB];
8454 ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8455 REWRITE_TAC[real_div; REAL_MUL_LID] THEN REWRITE_TAC[GSYM real_div] THEN
8456 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
8457 FIRST_X_ASSUM(fun th ->
8458 MAP_EVERY (MP_TAC o C SPEC th) [`y:real^N`; `z:real^N`]) THEN
8459 ASM_REWRITE_TAC[CENTRE_IN_CBALL] THEN REAL_ARITH_TAC);;
8461 (* ------------------------------------------------------------------------- *)
8462 (* Hence a convex function on an open set is continuous. *)
8463 (* ------------------------------------------------------------------------- *)
8465 let CONVEX_ON_CONTINUOUS = prove
8466 (`!f s:real^N->bool. open s /\ f convex_on s ==> lift o f continuous_on s`,
8467 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT] THEN
8468 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
8469 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
8470 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
8471 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
8472 ABBREV_TAC `d = e / &(dimindex(:N))` THEN
8473 SUBGOAL_THEN `&0 < d` ASSUME_TAC THENL
8474 [EXPAND_TAC "d" THEN MATCH_MP_TAC REAL_LT_DIV THEN
8475 ASM_REWRITE_TAC[REAL_OF_NUM_LT; DIMINDEX_GE_1;
8476 ARITH_RULE `0 < d <=> 1 <= d`];
8479 `?b. !y:real^N. y IN interval[(x - lambda i. d),(x + lambda i. d)]
8481 STRIP_ASSUME_TAC THENL
8482 [MP_TAC(ISPECL [`x - (lambda i. d):real^N`; `x + (lambda i. d):real^N`]
8483 CLOSED_INTERVAL_AS_CONVEX_HULL) THEN
8484 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
8485 ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `c = {}:real^N->bool` THEN
8486 ASM_REWRITE_TAC[CONVEX_HULL_EMPTY; NOT_IN_EMPTY] THEN
8487 MP_TAC(ISPEC `IMAGE (f:real^N->real) c` SUP_FINITE) THEN
8488 ASM_SIMP_TAC[FINITE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
8489 ABBREV_TAC `k = sup(IMAGE (f:real^N->real) c)` THEN
8490 STRIP_TAC THEN EXISTS_TAC `k:real` THEN
8491 MATCH_MP_TAC CONVEX_ON_CONVEX_HULL_BOUND THEN
8492 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_ON_SUBSET THEN
8493 EXISTS_TAC `s:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
8494 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `cball (x:real^N,e)` THEN
8495 ASM_REWRITE_TAC[] THEN
8496 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
8497 REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL] THEN
8498 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; LAMBDA_BETA] THEN
8499 X_GEN_TAC `z:real^N` THEN
8500 REWRITE_TAC[REAL_ARITH `x - d <= z /\ z <= x + d <=> abs(x - z) <= d`] THEN
8501 DISCH_TAC THEN REWRITE_TAC[dist] THEN
8502 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
8503 `sum(1..dimindex(:N)) (\i. abs((x - z:real^N)$i))` THEN
8504 REWRITE_TAC[NORM_LE_L1] THEN
8505 MATCH_MP_TAC SUM_BOUND_GEN THEN
8506 REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; CARD_NUMSEG] THEN
8507 ASM_SIMP_TAC[IN_NUMSEG; NOT_LT; DIMINDEX_GE_1; ADD_SUB;
8508 VECTOR_SUB_COMPONENT];
8510 SUBGOAL_THEN `cball(x:real^N,d) SUBSET cball(x,e)` ASSUME_TAC THENL
8511 [REWRITE_TAC[SUBSET; IN_CBALL] THEN GEN_TAC THEN
8512 MATCH_MP_TAC(REAL_ARITH `d <= e ==> x <= d ==> x <= e`) THEN
8514 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; DIMINDEX_GE_1;
8515 ARITH_RULE `0 < x <=> 1 <= x`] THEN
8516 GEN_REWRITE_TAC LAND_CONV [GSYM REAL_MUL_RID] THEN
8517 ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_OF_NUM_LE; DIMINDEX_GE_1];
8520 `!y:real^N. y IN cball(x,d) ==> abs(f(y)) <= b + &2 * abs(f(x))`
8522 [MATCH_MP_TAC CONVEX_BOUNDS_LEMMA THEN CONJ_TAC THENL
8523 [ASM_MESON_TAC[CONVEX_ON_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN
8524 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
8525 UNDISCH_TAC `y:real^N IN cball(x,d)` THEN REWRITE_TAC[IN_CBALL] THEN
8526 REWRITE_TAC[IN_INTERVAL; IN_CBALL; dist] THEN DISCH_TAC THEN
8527 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; LAMBDA_BETA] THEN
8528 REWRITE_TAC[REAL_ARITH `x - d <= z /\ z <= x + d <=> abs(x - z) <= d`] THEN
8529 SIMP_TAC[GSYM VECTOR_SUB_COMPONENT] THEN REPEAT STRIP_TAC THEN
8530 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(x - y:real^N)` THEN
8531 ASM_SIMP_TAC[COMPONENT_LE_NORM];
8533 SUBGOAL_THEN `(lift o f) continuous_on (ball(x:real^N,d))` MP_TAC THENL
8534 [MATCH_MP_TAC CONVEX_ON_BOUNDED_CONTINUOUS THEN REWRITE_TAC[OPEN_BALL] THEN
8535 EXISTS_TAC `b + &2 * abs(f(x:real^N))` THEN
8536 ASM_MESON_TAC[SUBSET; CONVEX_ON_SUBSET; SUBSET_TRANS; BALL_SUBSET_CBALL];
8538 ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_BALL; CENTRE_IN_BALL]);;
8540 (* ------------------------------------------------------------------------- *)
8541 (* Characterizations of convex functions in terms of sequents. *)
8542 (* ------------------------------------------------------------------------- *)
8544 let CONVEX_ON_LEFT_SECANT_MUL,CONVEX_ON_RIGHT_SECANT_MUL = (CONJ_PAIR o prove)
8545 (`(!f s:real^N->bool.
8547 !a b x. a IN s /\ b IN s /\ x IN segment[a,b]
8548 ==> (f x - f a) * norm(b - a) <= (f b - f a) * norm(x - a)) /\
8551 !a b x. a IN s /\ b IN s /\ x IN segment[a,b]
8552 ==> (f b - f a) * norm(b - x) <= (f b - f x) * norm(b - a))`,
8554 REPEAT GEN_TAC THEN REWRITE_TAC[convex_on] THEN
8555 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8556 X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN
8557 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8558 X_GEN_TAC `b:real^N` THEN REWRITE_TAC[] THEN
8559 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8560 ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8561 REWRITE_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN
8562 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
8563 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8564 X_GEN_TAC `u:real` THEN REWRITE_TAC[] THEN
8565 REWRITE_TAC[TAUT `a /\ x = y <=> x = y /\ a`;
8566 TAUT `a /\ x = y /\ b <=> x = y /\ a /\ b`] THEN
8567 REWRITE_TAC[REAL_ARITH `v + u = &1 <=> v = &1 - u`] THEN
8568 REWRITE_TAC[FORALL_UNWIND_THM2; IMP_CONJ] THEN
8569 REWRITE_TAC[REAL_SUB_LE] THEN
8570 ASM_CASES_TAC `&0 <= u` THEN ASM_REWRITE_TAC[] THEN
8571 ASM_CASES_TAC `u <= &1` THEN ASM_REWRITE_TAC[] THEN
8572 REWRITE_TAC[VECTOR_ARITH `((&1 - u) % a + u % b) - a:real^N = u % (b - a)`;
8573 VECTOR_ARITH `b - ((&1 - u) % a + u % b):real^N = (&1 - u) % (b - a)`] THEN
8574 REWRITE_TAC[NORM_MUL; REAL_MUL_ASSOC] THEN
8575 (ASM_CASES_TAC `b:real^N = a` THENL
8576 [ASM_REWRITE_TAC[VECTOR_SUB_REFL; REAL_SUB_REFL;
8577 VECTOR_ARITH `(&1 - u) % a + u % a:real^N = a`] THEN
8579 ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
8580 ASM_SIMP_TAC[REAL_ARITH
8581 `&0 <= u /\ u <= &1 ==> abs u = u /\ abs(&1 - u) = &1 - u`] THEN
8584 let CONVEX_ON_LEFT_SECANT,CONVEX_ON_RIGHT_SECANT = (CONJ_PAIR o prove)
8585 (`(!f s:real^N->bool.
8587 !a b x. a IN s /\ b IN s /\ x IN segment(a,b)
8588 ==> (f x - f a) / norm(x - a) <= (f b - f a) / norm(b - a)) /\
8591 !a b x. a IN s /\ b IN s /\ x IN segment(a,b)
8592 ==> (f b - f a) / norm(b - a) <= (f b - f x) / norm(b - x))`,
8593 CONJ_TAC THEN REPEAT GEN_TAC THENL
8594 [REWRITE_TAC[CONVEX_ON_LEFT_SECANT_MUL];
8595 REWRITE_TAC[CONVEX_ON_RIGHT_SECANT_MUL]] THEN
8596 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8597 X_GEN_TAC `a:real^N` THEN REWRITE_TAC[] THEN
8598 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8599 X_GEN_TAC `b:real^N` THEN REWRITE_TAC[] THEN
8600 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8601 ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
8602 ASM_CASES_TAC `a:real^N = b` THEN
8603 ASM_REWRITE_TAC[SEGMENT_REFL; NOT_IN_EMPTY; REAL_SUB_REFL; VECTOR_SUB_REFL;
8604 NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LE_REFL] THEN
8605 AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
8606 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[] THEN
8607 REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN
8608 MAP_EVERY ASM_CASES_TAC [`x:real^N = a`; `x:real^N = b`] THEN
8609 ASM_REWRITE_TAC[REAL_LE_REFL; REAL_SUB_REFL; VECTOR_SUB_REFL; NORM_0;
8610 REAL_MUL_LZERO; REAL_MUL_RZERO] THEN
8611 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; GSYM REAL_LE_LDIV_EQ; NORM_POS_LT;
8613 AP_TERM_TAC THEN REAL_ARITH_TAC);;
8615 (* ------------------------------------------------------------------------- *)
8616 (* Starlike sets and more stuff about line segments. *)
8617 (* ------------------------------------------------------------------------- *)
8619 let starlike = new_definition
8620 `starlike s <=> ?a. a IN s /\ !x. x IN s ==> segment[a,x] SUBSET s`;;
8622 let CONVEX_IMP_STARLIKE = prove
8623 (`!s. convex s /\ ~(s = {}) ==> starlike s`,
8624 REWRITE_TAC[CONVEX_CONTAINS_SEGMENT; starlike; GSYM MEMBER_NOT_EMPTY] THEN
8627 let SEGMENT_CONVEX_HULL = prove
8628 (`!a b. segment[a,b] = convex hull {a,b}`,
8630 SIMP_TAC[CONVEX_HULL_INSERT; CONVEX_HULL_SING; NOT_INSERT_EMPTY] THEN
8631 REWRITE_TAC[IN_SING; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
8632 REWRITE_TAC[segment; EXTENSION; IN_ELIM_THM] THEN
8633 REWRITE_TAC[REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
8634 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
8635 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> c /\ a /\ b /\ d`] THEN
8636 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN
8637 REWRITE_TAC[REAL_LE_SUB_LADD; REAL_ADD_LID] THEN MESON_TAC[]);;
8639 let SEGMENT_FURTHEST_LE = prove
8641 x IN segment[a,b] ==> norm(y - x) <= norm(y - a) \/
8642 norm(y - x) <= norm(y - b)`,
8643 REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN REPEAT STRIP_TAC THEN
8644 MP_TAC(ISPECL [`y:real^N`; `{a:real^N,b}`] SIMPLEX_FURTHEST_LE) THEN
8645 ASM_REWRITE_TAC[FINITE_INSERT; FINITE_RULES; NOT_INSERT_EMPTY] THEN
8646 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN STRIP_TAC THEN
8647 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
8648 ASM_MESON_TAC[NORM_SUB]);;
8650 let SEGMENT_BOUND = prove
8652 x IN segment[a,b] ==> norm(x - a) <= norm(b - a) /\
8653 norm(x - b) <= norm(b - a)`,
8654 REPEAT STRIP_TAC THEN
8655 MP_TAC(SPECL [`a:real^N`; `b:real^N`; `x:real^N`] SEGMENT_FURTHEST_LE) THENL
8656 [DISCH_THEN(MP_TAC o SPEC `a:real^N`);
8657 DISCH_THEN(MP_TAC o SPEC `b:real^N`)] THEN
8658 REWRITE_TAC[VECTOR_SUB_REFL; NORM_0] THEN
8659 ASM_MESON_TAC[NORM_POS_LE; REAL_LE_TRANS; NORM_SUB]);;
8661 let BETWEEN_IN_CONVEX_HULL = prove
8662 (`!x a b:real^N. between x (a,b) <=> x IN convex hull {a,b}`,
8663 REWRITE_TAC[BETWEEN_IN_SEGMENT; SEGMENT_CONVEX_HULL]);;
8665 let STARLIKE_LINEAR_IMAGE = prove
8666 (`!f s. starlike s /\ linear f ==> starlike(IMAGE f s)`,
8667 REWRITE_TAC[starlike; FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN
8668 SIMP_TAC[CLOSED_SEGMENT_LINEAR_IMAGE] THEN SET_TAC[]);;
8670 let STARLIKE_LINEAR_IMAGE_EQ = prove
8671 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
8672 ==> (starlike (IMAGE f s) <=> starlike s)`,
8673 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE STARLIKE_LINEAR_IMAGE));;
8675 add_linear_invariants [STARLIKE_LINEAR_IMAGE_EQ];;
8677 let STARLIKE_TRANSLATION_EQ = prove
8678 (`!a s. starlike (IMAGE (\x. a + x) s) <=> starlike s`,
8679 REWRITE_TAC[starlike] THEN GEOM_TRANSLATE_TAC[]);;
8681 add_translation_invariants [STARLIKE_TRANSLATION_EQ];;
8683 let BETWEEN_LINEAR_IMAGE_EQ = prove
8684 (`!f x y z. linear f /\ (!x y. f x = f y ==> x = y)
8685 ==> (between (f x) (f y,f z) <=> between x (y,z))`,
8686 SIMP_TAC[BETWEEN_IN_SEGMENT; CLOSED_SEGMENT_LINEAR_IMAGE] THEN SET_TAC[]);;
8688 add_linear_invariants [BETWEEN_LINEAR_IMAGE_EQ];;
8690 let BETWEEN_TRANSLATION = prove
8691 (`!a x y. between (a + x) (a + y,a + z) <=> between x (y,z)`,
8692 REWRITE_TAC[between] THEN NORM_ARITH_TAC);;
8694 add_translation_invariants [STARLIKE_TRANSLATION_EQ];;
8696 let STARLIKE_CLOSURE = prove
8697 (`!s:real^N->bool. starlike s ==> starlike(closure s)`,
8698 GEN_TAC THEN REWRITE_TAC[starlike; SUBSET; segment; FORALL_IN_GSPEC] THEN
8699 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
8700 STRIP_TAC THEN ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN
8701 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE] THEN
8702 DISCH_TAC THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN
8703 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN
8704 ASM_REWRITE_TAC[] THEN
8705 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
8706 EXISTS_TAC `(&1 - u) % a + u % y:real^N` THEN
8707 ASM_SIMP_TAC[dist; NORM_MUL; VECTOR_ARITH
8708 `(v % a + u % y) - (v % a + u % z):real^N = u % (y - z)`] THEN
8709 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
8710 REAL_LET_TRANS)) THEN
8711 REWRITE_TAC[dist; REAL_ARITH `u * n <= n <=> &0 <= n * (&1 - u)`] THEN
8712 MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
8713 ASM_REAL_ARITH_TAC);;
8715 let STARLIKE_UNIV = prove
8716 (`starlike(:real^N)`,
8717 MESON_TAC[CONVEX_IMP_STARLIKE; CONVEX_UNIV;
8718 BOUNDED_EMPTY; NOT_BOUNDED_UNIV]);;
8720 let STARLIKE_PCROSS = prove
8721 (`!s:real^M->bool t:real^N->bool.
8722 starlike s /\ starlike t ==> starlike(s PCROSS t)`,
8723 SIMP_TAC[starlike; EXISTS_IN_PCROSS; SUBSET; IN_SEGMENT] THEN
8724 REPEAT GEN_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
8725 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
8726 REWRITE_TAC[FORALL_IN_PCROSS; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
8727 REWRITE_TAC[FORALL_UNWIND_THM2; IMP_IMP] THEN
8728 REWRITE_TAC[GSYM PASTECART_CMUL; PASTECART_ADD] THEN
8729 REWRITE_TAC[PASTECART_IN_PCROSS] THEN MESON_TAC[]);;
8731 let STARLIKE_PCROSS_EQ = prove
8732 (`!s:real^M->bool t:real^N->bool.
8733 starlike(s PCROSS t) <=> starlike s /\ starlike t`,
8735 ASM_CASES_TAC `s:real^M->bool = {}` THENL
8736 [ASM_REWRITE_TAC[PCROSS_EMPTY] THEN MESON_TAC[starlike; NOT_IN_EMPTY];
8738 ASM_CASES_TAC `t:real^N->bool = {}` THENL
8739 [ASM_REWRITE_TAC[PCROSS_EMPTY] THEN MESON_TAC[starlike; NOT_IN_EMPTY];
8741 EQ_TAC THEN REWRITE_TAC[STARLIKE_PCROSS] THEN REPEAT STRIP_TAC THENL
8742 [MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
8743 `(s:real^M->bool) PCROSS (t:real^N->bool)`] STARLIKE_LINEAR_IMAGE) THEN
8744 ASM_REWRITE_TAC[LINEAR_FSTCART];
8745 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
8746 `(s:real^M->bool) PCROSS (t:real^N->bool)`] STARLIKE_LINEAR_IMAGE) THEN
8747 ASM_REWRITE_TAC[LINEAR_SNDCART]] THEN
8748 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
8749 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; PASTECART_IN_PCROSS;
8750 FSTCART_PASTECART; SNDCART_PASTECART] THEN
8753 let BETWEEN_DIST_LT = prove
8755 dist(c,a) < r /\ dist(c,b) < r /\ between x (a,b) ==> dist(c,x) < r`,
8756 REPEAT STRIP_TAC THEN
8757 SUBGOAL_THEN `convex hull {a,b} SUBSET ball(c:real^N,r)` MP_TAC THENL
8758 [MATCH_MP_TAC HULL_MINIMAL THEN
8759 ASM_REWRITE_TAC[CONVEX_BALL; INSERT_SUBSET; EMPTY_SUBSET; IN_BALL];
8760 ASM_SIMP_TAC[SUBSET; GSYM BETWEEN_IN_CONVEX_HULL; IN_BALL]]);;
8762 let BETWEEN_DIST_LE = prove
8764 dist(c,a) <= r /\ dist(c,b) <= r /\ between x (a,b) ==> dist(c,x) <= r`,
8766 REPEAT STRIP_TAC THEN
8767 SUBGOAL_THEN `convex hull {a,b} SUBSET cball(c:real^N,r)` MP_TAC THENL
8768 [MATCH_MP_TAC HULL_MINIMAL THEN
8769 ASM_REWRITE_TAC[CONVEX_CBALL; INSERT_SUBSET; EMPTY_SUBSET; IN_CBALL];
8770 ASM_SIMP_TAC[SUBSET; GSYM BETWEEN_IN_CONVEX_HULL; IN_CBALL]]);;
8772 let BETWEEN_NORM_LT = prove
8774 norm a < r /\ norm b < r /\ between x (a,b) ==> norm x < r`,
8775 REWRITE_TAC[GSYM(CONJUNCT2(SPEC_ALL DIST_0)); BETWEEN_DIST_LT]);;
8777 let BETWEEN_NORM_LE = prove
8779 norm a <= r /\ norm b <= r /\ between x (a,b) ==> norm x <= r`,
8780 REWRITE_TAC[GSYM(CONJUNCT2(SPEC_ALL DIST_0)); BETWEEN_DIST_LE]);;
8782 let UNION_SEGMENT = prove
8785 ==> segment[a,b] UNION segment[b,c] = segment[a,c]`,
8786 REPEAT GEN_TAC THEN ASM_CASES_TAC `c:real^N = a` THENL
8787 [ASM_SIMP_TAC[SEGMENT_REFL; IN_SING; UNION_IDEMPOT];
8788 ONCE_REWRITE_TAC[UNION_COMM] THEN REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN
8789 DISCH_THEN(SUBST1_TAC o MATCH_MP CONVEX_HULL_EXCHANGE_UNION) THEN
8790 ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
8791 REWRITE_TAC[IMAGE_CLAUSES; UNIONS_2] THEN
8792 BINOP_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[]]);;
8794 (* ------------------------------------------------------------------------- *)
8795 (* Shrinking towards the interior of a convex set. *)
8796 (* ------------------------------------------------------------------------- *)
8798 let IN_INTERIOR_CONVEX_SHRINK = prove
8800 convex s /\ c IN interior s /\
8801 x IN s /\ &0 < e /\ e <= &1
8802 ==> x - e % (x - c) IN interior s`,
8803 REPEAT STRIP_TAC THEN
8804 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN
8805 REWRITE_TAC[IN_INTERIOR; SUBSET; IN_BALL; dist] THEN
8806 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
8807 EXISTS_TAC `e * d:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
8808 X_GEN_TAC `y':real^N` THEN DISCH_TAC THEN
8809 FIRST_X_ASSUM(MP_TAC o SPEC `(&1 / e) % y' - ((&1 - e) / e) % x:real^N`) THEN
8811 [UNDISCH_TAC `norm (x - e % (x - c) - y':real^N) < e * d` THEN
8812 SUBGOAL_THEN `x - e % (x - c) - y':real^N =
8813 e % (c - (&1 / e % y' - (&1 - e) / e % x))`
8815 [ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC;
8816 REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC;
8817 ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL_EQ; real_abs; REAL_LT_IMP_LE]];
8819 SUBGOAL_THEN `y' = (&1 - (&1 - e)) % (&1 / e % y' - (&1 - e) / e % x) +
8820 (&1 - e) % x:real^N`
8822 [ASM_SIMP_TAC[REAL_ARITH `&1 - (&1 - e) = e`; VECTOR_SUB_LDISTRIB;
8823 VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
8825 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
8826 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);;
8828 let IN_INTERIOR_CLOSURE_CONVEX_SHRINK = prove
8830 convex s /\ c IN interior s /\
8831 x IN closure s /\ &0 < e /\ e <= &1
8832 ==> x - e % (x - c) IN interior s`,
8833 REPEAT STRIP_TAC THEN
8834 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR]) THEN
8835 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
8836 SUBGOAL_THEN `?y:real^N. y IN s /\ norm(y - x) * (&1 - e) < e * d`
8837 STRIP_ASSUME_TAC THENL
8838 [ASM_CASES_TAC `(x:real^N) IN s` THENL
8839 [EXISTS_TAC `x:real^N` THEN
8840 ASM_SIMP_TAC[REAL_LT_MUL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO];
8842 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [closure]) THEN
8843 ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; LIMPT_APPROACHABLE; dist] THEN
8844 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
8845 `e <= &1 ==> e = &1 \/ e < &1`)) THEN
8846 ASM_SIMP_TAC[REAL_SUB_REFL; GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THENL
8847 [DISCH_THEN(MP_TAC o SPEC `&1`) THEN
8848 REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_01];
8849 DISCH_THEN(MP_TAC o SPEC `(e * d) / (&1 - e)`)] THEN
8850 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_SUB_LT; REAL_MUL_LZERO; REAL_LT_MUL;
8852 MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[];
8854 ABBREV_TAC `z:real^N = c + ((&1 - e) / e) % (x - y)` THEN
8855 SUBGOAL_THEN `x - e % (x - c):real^N = y - e % (y - z)` SUBST1_TAC THENL
8856 [EXPAND_TAC "z" THEN
8857 REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB] THEN
8858 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
8861 MATCH_MP_TAC IN_INTERIOR_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN
8862 FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET] o
8863 MATCH_MP SUBSET_INTERIOR) THEN
8864 SIMP_TAC[INTERIOR_OPEN; OPEN_BALL] THEN
8865 REWRITE_TAC[IN_BALL; dist] THEN EXPAND_TAC "z" THEN
8866 REWRITE_TAC[NORM_ARITH `norm(c - (c + x)) = norm(x)`] THEN
8867 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN
8868 ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
8869 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8870 REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN
8871 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
8872 ASM_MESON_TAC[REAL_MUL_SYM; NORM_SUB]);;
8874 let IN_INTERIOR_CLOSURE_CONVEX_SEGMENT = prove
8876 convex s /\ a IN interior s /\ b IN closure s
8877 ==> segment(a,b) SUBSET interior s`,
8878 REWRITE_TAC[SUBSET; IN_SEGMENT] THEN REPEAT STRIP_TAC THEN
8879 ASM_REWRITE_TAC[VECTOR_ARITH
8880 `(&1 - u) % a + u % b:real^N = b - (&1 - u) % (b - a)`] THEN
8881 MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
8882 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
8884 (* ------------------------------------------------------------------------- *)
8885 (* Relative interior of a set. *)
8886 (* ------------------------------------------------------------------------- *)
8888 let relative_interior = new_definition
8889 `relative_interior s =
8890 {x | ?t. open_in (subtopology euclidean (affine hull s)) t /\
8891 x IN t /\ t SUBSET s}`;;
8893 let relative_frontier = new_definition
8894 `relative_frontier s = closure s DIFF relative_interior s`;;
8896 let RELATIVE_INTERIOR = prove
8897 (`!s. relative_interior s =
8899 ?t. open t /\ x IN t /\ t INTER (affine hull s) SUBSET s}`,
8900 REWRITE_TAC[EXTENSION; relative_interior; IN_ELIM_THM] THEN
8901 REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN
8902 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
8903 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> b /\ a /\ c /\ d`] THEN
8904 REWRITE_TAC[UNWIND_THM2; SUBSET; IN_INTER; RIGHT_AND_EXISTS_THM] THEN
8905 AP_TERM_TAC THEN ABS_TAC THEN MESON_TAC[HULL_INC]);;
8907 let RELATIVE_INTERIOR_EQ = prove
8908 (`!s. relative_interior s = s <=>
8909 open_in(subtopology euclidean (affine hull s)) s`,
8910 GEN_TAC THEN REWRITE_TAC[EXTENSION; relative_interior; IN_ELIM_THM] THEN
8911 GEN_REWRITE_TAC RAND_CONV [OPEN_IN_SUBOPEN] THEN MESON_TAC[SUBSET]);;
8913 let RELATIVE_INTERIOR_OPEN_IN = prove
8914 (`!s. open_in(subtopology euclidean (affine hull s)) s
8915 ==> relative_interior s = s`,
8916 REWRITE_TAC[RELATIVE_INTERIOR_EQ]);;
8918 let RELATIVE_INTERIOR_EMPTY = prove
8919 (`relative_interior {} = {}`,
8920 SIMP_TAC[RELATIVE_INTERIOR_OPEN_IN; OPEN_IN_EMPTY]);;
8922 let RELATIVE_FRONTIER_EMPTY = prove
8923 (`relative_frontier {} = {}`,
8924 REWRITE_TAC[relative_frontier; CLOSURE_EMPTY; EMPTY_DIFF]);;
8926 let RELATIVE_INTERIOR_AFFINE = prove
8927 (`!s:real^N->bool. affine s ==> relative_interior s = s`,
8928 SIMP_TAC[RELATIVE_INTERIOR_EQ; OPEN_IN_SUBTOPOLOGY_REFL; HULL_P] THEN
8929 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);;
8931 let RELATIVE_INTERIOR_UNIV = prove
8932 (`!s. relative_interior(affine hull s) = affine hull s`,
8933 REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_OPEN_IN THEN
8934 REWRITE_TAC[HULL_HULL; OPEN_IN_SUBTOPOLOGY_REFL] THEN
8935 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV]);;
8937 let OPEN_IN_RELATIVE_INTERIOR = prove
8938 (`!s. open_in (subtopology euclidean (affine hull s))
8939 (relative_interior s)`,
8940 GEN_TAC THEN REWRITE_TAC[relative_interior] THEN
8941 GEN_REWRITE_TAC I [OPEN_IN_SUBOPEN] THEN
8942 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8944 let RELATIVE_INTERIOR_SUBSET = prove
8945 (`!s. (relative_interior s) SUBSET s`,
8946 REWRITE_TAC[SUBSET; relative_interior; IN_ELIM_THM] THEN MESON_TAC[]);;
8948 let OPEN_IN_SET_RELATIVE_INTERIOR = prove
8949 (`!s:real^N->bool. open_in (subtopology euclidean s) (relative_interior s)`,
8950 GEN_TAC THEN MATCH_MP_TAC OPEN_IN_SUBSET_TRANS THEN
8951 EXISTS_TAC `affine hull s:real^N->bool` THEN
8952 REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; RELATIVE_INTERIOR_SUBSET;
8955 let SUBSET_RELATIVE_INTERIOR = prove
8956 (`!s t. s SUBSET t /\ affine hull s = affine hull t
8957 ==> (relative_interior s) SUBSET (relative_interior t)`,
8958 REWRITE_TAC[relative_interior; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8960 let RELATIVE_INTERIOR_MAXIMAL = prove
8961 (`!s t. t SUBSET s /\
8962 open_in(subtopology euclidean (affine hull s)) t
8963 ==> t SUBSET (relative_interior s)`,
8964 REWRITE_TAC[relative_interior; SUBSET; IN_ELIM_THM] THEN MESON_TAC[]);;
8966 let RELATIVE_INTERIOR_UNIQUE = prove
8967 (`!s t. t SUBSET s /\
8968 open_in(subtopology euclidean (affine hull s)) t /\
8969 (!t'. t' SUBSET s /\
8970 open_in(subtopology euclidean (affine hull s)) t'
8972 ==> (relative_interior s = t)`,
8973 MESON_TAC[SUBSET_ANTISYM; RELATIVE_INTERIOR_MAXIMAL; RELATIVE_INTERIOR_SUBSET;
8974 OPEN_IN_RELATIVE_INTERIOR]);;
8976 let IN_RELATIVE_INTERIOR = prove
8978 x IN relative_interior s <=>
8979 x IN s /\ ?e. &0 < e /\ (ball(x,e) INTER (affine hull s)) SUBSET s`,
8980 REPEAT GEN_TAC THEN REWRITE_TAC[relative_interior; IN_ELIM_THM] THEN
8981 REWRITE_TAC[OPEN_IN_OPEN; LEFT_AND_EXISTS_THM] THEN
8982 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
8983 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> b /\ a /\ c /\ d`] THEN
8984 REWRITE_TAC[UNWIND_THM2; SUBSET; IN_INTER] THEN EQ_TAC THENL
8985 [ASM_MESON_TAC[SUBSET; OPEN_CONTAINS_BALL];
8986 STRIP_TAC THEN EXISTS_TAC `ball(x:real^N,e)` THEN
8987 ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL; HULL_INC]]);;
8989 let IN_RELATIVE_INTERIOR_CBALL = prove
8991 x IN relative_interior s <=>
8992 x IN s /\ ?e. &0 < e /\ (cball(x,e) INTER affine hull s) SUBSET s`,
8993 REPEAT GEN_TAC THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN
8994 AP_TERM_TAC THEN EQ_TAC THEN
8995 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THENL
8996 [EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN
8997 MATCH_MP_TAC SUBSET_TRANS THEN
8998 EXISTS_TAC `ball(x:real^N,e) INTER affine hull s` THEN
8999 ASM_REWRITE_TAC[] THEN
9000 REWRITE_TAC[SUBSET; IN_INTER; IN_BALL; IN_CBALL] THEN
9001 ASM_SIMP_TAC[REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`];
9002 EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN
9003 MATCH_MP_TAC SUBSET_TRANS THEN
9004 EXISTS_TAC `cball(x:real^N,e) INTER affine hull s` THEN
9005 ASM_REWRITE_TAC[] THEN
9006 SIMP_TAC[SUBSET; IN_INTER; IN_BALL; IN_CBALL; REAL_LT_IMP_LE]]);;
9008 let OPEN_IN_SUBSET_RELATIVE_INTERIOR = prove
9009 (`!s t. open_in(subtopology euclidean (affine hull t)) s
9010 ==> (s SUBSET relative_interior t <=> s SUBSET t)`,
9011 MESON_TAC[RELATIVE_INTERIOR_MAXIMAL; RELATIVE_INTERIOR_SUBSET;
9014 let RELATIVE_INTERIOR_TRANSLATION = prove
9016 relative_interior (IMAGE (\x. a + x) s) =
9017 IMAGE (\x. a + x) (relative_interior s)`,
9018 REWRITE_TAC[relative_interior; OPEN_IN_OPEN] THEN GEOM_TRANSLATE_TAC[]);;
9020 add_translation_invariants [RELATIVE_INTERIOR_TRANSLATION];;
9022 let RELATIVE_FRONTIER_TRANSLATION = prove
9024 relative_frontier (IMAGE (\x. a + x) s) =
9025 IMAGE (\x. a + x) (relative_frontier s)`,
9026 REWRITE_TAC[relative_frontier] THEN GEOM_TRANSLATE_TAC[]);;
9028 add_translation_invariants [RELATIVE_FRONTIER_TRANSLATION];;
9030 let RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE = prove
9031 (`!f:real^M->real^N s.
9032 linear f /\ (!x y. f x = f y ==> x = y)
9033 ==> relative_interior(IMAGE f s) = IMAGE f (relative_interior s)`,
9034 REPEAT GEN_TAC THEN DISCH_TAC THEN
9035 ASM_SIMP_TAC[relative_interior; AFFINE_HULL_LINEAR_IMAGE] THEN
9036 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> c /\ a /\ b`] THEN
9037 REWRITE_TAC[EXISTS_SUBSET_IMAGE] THEN
9038 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_INJECTIVE_LINEAR_IMAGE) THEN
9039 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
9041 add_linear_invariants [RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE];;
9043 let RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE = prove
9044 (`!f:real^M->real^N s.
9045 linear f /\ (!x y. f x = f y ==> x = y)
9046 ==> relative_frontier(IMAGE f s) = IMAGE f (relative_frontier s)`,
9047 REWRITE_TAC[relative_frontier] THEN GEOM_TRANSFORM_TAC[]);;
9049 add_linear_invariants [RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE];;
9051 let RELATIVE_INTERIOR_EQ_EMPTY = prove
9053 convex s ==> (relative_interior s = {} <=> s = {})`,
9056 vec 0 IN s /\ convex s ==> ~(relative_interior s = {})`
9059 GEN_TAC THEN DISCH_TAC THEN
9060 ASM_CASES_TAC `s:real^N->bool = {}` THEN
9061 ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY] THEN
9062 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9063 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
9064 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\x:real^N. --a + x) s`) THEN
9065 REWRITE_TAC[CONVEX_TRANSLATION_EQ; RELATIVE_INTERIOR_TRANSLATION] THEN
9066 ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; IN_IMAGE] THEN
9067 DISCH_THEN MATCH_MP_TAC THEN EXISTS_TAC `a:real^N` THEN
9068 ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC] THEN
9069 GEN_TAC THEN STRIP_TAC THEN
9070 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_RELATIVE_INTERIOR] THEN
9071 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
9072 X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC
9073 (ISPEC `s:real^N->bool` BASIS_EXISTS) THEN
9074 SUBGOAL_THEN `span(s:real^N->bool) = span b` SUBST_ALL_TAC THENL
9075 [ASM_SIMP_TAC[SPAN_EQ] THEN ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS];
9077 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
9078 ABBREV_TAC `n = dim(s:real^N->bool)` THEN
9080 `!c. (!v. v IN b ==> &0 <= c(v)) /\ sum b c <= &1
9081 ==> vsum b (\v:real^N. c(v) % v) IN s`
9083 [REPEAT STRIP_TAC THEN SUBGOAL_THEN
9084 `vsum (vec 0 INSERT b :real^N->bool)
9085 (\v. (if v = vec 0 then &1 - sum b c else c v) % v) IN s`
9087 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_EXPLICIT]) THEN
9088 ASM_SIMP_TAC[INSERT_SUBSET; FINITE_INSERT; SUM_CLAUSES;
9089 INDEPENDENT_NONZERO; IN_INSERT] THEN
9090 CONJ_TAC THENL [ASM_MESON_TAC[REAL_SUB_LE]; ALL_TAC] THEN
9091 REWRITE_TAC[REAL_ARITH `&1 - x + y = &1 <=> x = y`] THEN
9092 MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[INDEPENDENT_NONZERO];
9093 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
9094 ASM_SIMP_TAC[VSUM_CLAUSES; INDEPENDENT_NONZERO] THEN
9095 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
9096 MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[INDEPENDENT_NONZERO]];
9098 ABBREV_TAC `a:real^N = vsum b (\v. inv(&2 * &n + &1) % v)` THEN
9099 EXISTS_TAC `a:real^N` THEN CONJ_TAC THENL
9100 [EXPAND_TAC "a" THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
9101 ASM_SIMP_TAC[SUM_CONST; REAL_LE_INV_EQ; REAL_ARITH `&0 < &2 * &n + &1`;
9102 GSYM real_div; REAL_LT_IMP_LE; REAL_LE_LDIV_EQ] THEN
9105 MP_TAC(ISPECL [`b:real^N->bool`; `inv(&2 * &n + &1)`]
9106 BASIS_COORDINATES_CONTINUOUS) THEN
9107 ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
9108 ANTS_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
9109 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
9110 ASM_SIMP_TAC[SUBSET; IN_INTER; IMP_CONJ_ALT] THEN
9111 ASM_SIMP_TAC[SPAN_FINITE; LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
9112 GEN_TAC THEN X_GEN_TAC `u:real^N->real` THEN
9113 DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[IN_BALL; dist] THEN
9114 EXPAND_TAC "a" THEN ASM_SIMP_TAC[GSYM VSUM_SUB] THEN
9115 DISCH_THEN(fun th -> FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC th) THEN
9116 REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB] THEN
9117 DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o C MATCH_MP th)) THEN
9118 REWRITE_TAC[REAL_ARITH `abs(x - y) < x <=> &0 < y /\ abs(y) < &2 * x`] THEN
9119 SIMP_TAC[REAL_LT_IMP_LE] THEN DISCH_TAC THEN
9120 MATCH_MP_TAC REAL_LE_TRANS THEN
9121 EXISTS_TAC `&(CARD(b:real^N->bool)) * &2 * inv(&2 * &n + &1)` THEN
9123 [MATCH_MP_TAC SUM_BOUND THEN
9124 ASM_SIMP_TAC[REAL_ARITH `abs x < a ==> x <= a`];
9125 ASM_REWRITE_TAC[REAL_MUL_ASSOC] THEN REWRITE_TAC[GSYM real_div] THEN
9126 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &2 * &n + &1`] THEN
9129 let RELATIVE_INTERIOR_INTERIOR = prove
9130 (`!s. affine hull s = (:real^N)
9131 ==> relative_interior s = interior s`,
9132 SIMP_TAC[relative_interior; interior; SUBTOPOLOGY_UNIV; OPEN_IN]);;
9134 let RELATIVE_INTERIOR_OPEN = prove
9135 (`!s:real^N->bool. open s ==> relative_interior s = s`,
9136 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9137 ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY] THEN
9138 ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_OPEN; INTERIOR_EQ]);;
9140 let RELATIVE_INTERIOR_NONEMPTY_INTERIOR = prove
9141 (`!s. ~(interior s = {}) ==> relative_interior s = interior s`,
9142 MESON_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_NONEMPTY_INTERIOR]);;
9144 let RELATIVE_FRONTIER_NONEMPTY_INTERIOR = prove
9145 (`!s. ~(interior s = {}) ==> relative_frontier s = frontier s`,
9146 SIMP_TAC[relative_frontier; frontier; RELATIVE_INTERIOR_NONEMPTY_INTERIOR]);;
9148 let RELATIVE_FRONTIER_FRONTIER = prove
9149 (`!s. affine hull s = (:real^N) ==> relative_frontier s = frontier s`,
9150 SIMP_TAC[relative_frontier; frontier; RELATIVE_INTERIOR_INTERIOR]);;
9152 let AFFINE_HULL_CONVEX_HULL = prove
9153 (`!s. affine hull (convex hull s) = affine hull s`,
9154 GEN_TAC THEN MATCH_MP_TAC HULL_UNIQUE THEN
9155 REWRITE_TAC[AFFINE_AFFINE_HULL; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
9156 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN
9157 ASM_MESON_TAC[SUBSET_TRANS; HULL_SUBSET]);;
9159 let INTERIOR_SIMPLEX_NONEMPTY = prove
9161 independent s /\ s HAS_SIZE (dimindex(:N))
9162 ==> ?a. a IN interior(convex hull (vec 0 INSERT s))`,
9163 REPEAT STRIP_TAC THEN
9164 MP_TAC(ISPEC `convex hull (vec 0 INSERT s):real^N->bool`
9165 RELATIVE_INTERIOR_EQ_EMPTY) THEN
9166 ASM_SIMP_TAC[AFFINE_HULL_CONVEX_HULL] THEN
9167 REWRITE_TAC[CONVEX_HULL_EQ_EMPTY; CONVEX_CONVEX_HULL; NOT_INSERT_EMPTY] THEN
9168 REWRITE_TAC[MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
9169 AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
9170 SIMP_TAC[AFFINE_HULL_EQ_SPAN; IN_INSERT; HULL_INC] THEN
9171 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
9172 EXISTS_TAC `span s:real^N->bool` THEN CONJ_TAC THENL
9173 [MATCH_MP_TAC SPAN_MONO THEN MATCH_MP_TAC(SET_RULE
9174 `(a INSERT s) SUBSET P hull (a INSERT s)
9175 ==> s SUBSET P hull (a INSERT s)`) THEN REWRITE_TAC[HULL_SUBSET];
9176 MATCH_MP_TAC(SET_RULE `UNIV SUBSET s ==> s = UNIV`) THEN
9177 MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
9178 ASM_REWRITE_TAC[DIM_UNIV; SUBSET_UNIV] THEN
9179 ASM_MESON_TAC[LE_REFL;HAS_SIZE]]);;
9181 let INTERIOR_SUBSET_RELATIVE_INTERIOR = prove
9182 (`!s. interior s SUBSET relative_interior s`,
9183 REWRITE_TAC[SUBSET; IN_INTERIOR; IN_RELATIVE_INTERIOR; IN_INTER] THEN
9184 MESON_TAC[CENTRE_IN_BALL]);;
9186 let CONVEX_RELATIVE_INTERIOR = prove
9187 (`!s:real^N->bool. convex s ==> convex(relative_interior s)`,
9188 REWRITE_TAC[CONVEX_ALT; IN_RELATIVE_INTERIOR; IN_INTER;
9189 SUBSET; IN_BALL; dist] THEN
9190 GEN_TAC THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
9191 REWRITE_TAC[TAUT `(a /\ b) /\ (c /\ d) /\ e ==> f <=>
9192 a /\ c /\ e ==> b /\ d ==> f`] THEN
9193 STRIP_TAC THEN ASM_SIMP_TAC[] THEN
9194 MATCH_MP_TAC(MESON[] `(!d e. P d /\ Q e ==> R(min d e))
9195 ==> (?e. P e) /\ (?e. Q e) ==> (?e. R e)`) THEN
9196 REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
9197 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
9198 SUBST1_TAC(VECTOR_ARITH `z:real^N =
9199 (&1 - u) % (z - u % (y - x)) + u % (z + (&1 - u) % (y - x))`) THEN
9200 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
9201 FIRST_X_ASSUM(CONJUNCTS_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
9202 REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN MATCH_MP_TAC MONO_AND THEN
9203 CONJ_TAC THEN DISCH_THEN MATCH_MP_TAC THEN
9205 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
9206 `norm x < e ==> norm x = y ==> y < e`)) THEN
9207 AP_TERM_TAC THEN VECTOR_ARITH_TAC;
9208 REWRITE_TAC[VECTOR_ARITH `a - b % c:real^N = a + --b % c`] THEN
9209 MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
9210 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]]));;
9212 let IN_RELATIVE_INTERIOR_CONVEX_SHRINK = prove
9214 convex s /\ c IN relative_interior s /\
9215 x IN s /\ &0 < e /\ e <= &1
9216 ==> x - e % (x - c) IN relative_interior s`,
9217 REPEAT STRIP_TAC THEN
9218 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN
9219 REWRITE_TAC[IN_RELATIVE_INTERIOR; SUBSET; IN_INTER; IN_BALL; dist] THEN
9220 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9221 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL
9222 [REWRITE_TAC[VECTOR_ARITH
9223 `x - e % (x - c):real^N = (&1 - e) % x + e % c`] THEN
9224 FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
9225 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
9227 EXISTS_TAC `e * d:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
9228 X_GEN_TAC `y':real^N` THEN STRIP_TAC THEN
9229 FIRST_X_ASSUM(MP_TAC o SPEC `(&1 / e) % y' - ((&1 - e) / e) % x:real^N`) THEN
9232 [UNDISCH_TAC `norm (x - e % (x - c) - y':real^N) < e * d` THEN
9233 SUBGOAL_THEN `x - e % (x - c) - y':real^N =
9234 e % (c - (&1 / e % y' - (&1 - e) / e % x))`
9236 [ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC;
9237 REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN VECTOR_ARITH_TAC;
9238 ASM_SIMP_TAC[NORM_MUL; REAL_LT_LMUL_EQ; real_abs; REAL_LT_IMP_LE]];
9239 REWRITE_TAC[real_div; REAL_SUB_RDISTRIB] THEN
9240 ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN
9241 REWRITE_TAC[VECTOR_ARITH `a % y - (b - c) % x:real^N =
9242 (c - b) % x + a % y`] THEN
9243 MATCH_MP_TAC(REWRITE_RULE[AFFINE_ALT] AFFINE_AFFINE_HULL) THEN
9244 ASM_SIMP_TAC[HULL_INC]];
9246 SUBGOAL_THEN `y' = (&1 - (&1 - e)) % (&1 / e % y' - (&1 - e) / e % x) +
9247 (&1 - e) % x:real^N`
9249 [ASM_SIMP_TAC[REAL_ARITH `&1 - (&1 - e) = e`; VECTOR_SUB_LDISTRIB;
9250 VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
9252 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
9253 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]]);;
9255 let IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK = prove
9257 convex s /\ c IN relative_interior s /\
9258 x IN closure s /\ &0 < e /\ e <= &1
9259 ==> x - e % (x - c) IN relative_interior s`,
9260 REPEAT STRIP_TAC THEN
9261 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR]) THEN
9262 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9263 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
9264 SUBGOAL_THEN `?y:real^N. y IN s /\ norm(y - x) * (&1 - e) < e * d`
9265 STRIP_ASSUME_TAC THENL
9266 [ASM_CASES_TAC `(x:real^N) IN s` THENL
9267 [EXISTS_TAC `x:real^N` THEN
9268 ASM_SIMP_TAC[REAL_LT_MUL; VECTOR_SUB_REFL; NORM_0; REAL_MUL_LZERO];
9270 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [closure]) THEN
9271 ASM_REWRITE_TAC[IN_UNION; IN_ELIM_THM; LIMPT_APPROACHABLE; dist] THEN
9272 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
9273 `e <= &1 ==> e = &1 \/ e < &1`)) THEN
9274 ASM_SIMP_TAC[REAL_SUB_REFL; GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THENL
9275 [DISCH_THEN(MP_TAC o SPEC `&1`) THEN
9276 REWRITE_TAC[REAL_MUL_RZERO; REAL_LT_01];
9277 DISCH_THEN(MP_TAC o SPEC `(e * d) / (&1 - e)`)] THEN
9278 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_SUB_LT; REAL_MUL_LZERO; REAL_LT_MUL;
9280 MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[];
9282 ABBREV_TAC `z:real^N = c + ((&1 - e) / e) % (x - y)` THEN
9283 SUBGOAL_THEN `x - e % (x - c):real^N = y - e % (y - z)` SUBST1_TAC THENL
9284 [EXPAND_TAC "z" THEN
9285 REWRITE_TAC[VECTOR_SUB_LDISTRIB; VECTOR_ADD_LDISTRIB] THEN
9286 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
9289 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CONVEX_SHRINK THEN ASM_REWRITE_TAC[] THEN
9290 SUBGOAL_THEN `dist(c:real^N,z) < d` ASSUME_TAC THENL
9291 [EXPAND_TAC "z" THEN
9292 REWRITE_TAC[NORM_ARITH `dist(c:real^N,c + x) = norm x`] THEN
9293 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN
9294 REWRITE_TAC[REAL_ARITH `a / b * c:real = (c * a) / b`] THEN
9295 ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_LT_LDIV_EQ] THEN
9298 SUBGOAL_THEN `(z:real^N) IN affine hull s` ASSUME_TAC THENL
9299 [EXPAND_TAC "z" THEN MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
9300 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC] THEN
9301 MATCH_MP_TAC(SET_RULE `!t. x IN t /\ t = s ==> x IN s`) THEN
9302 EXISTS_TAC `closure(affine hull s):real^N->bool` THEN
9303 SIMP_TAC[CLOSURE_EQ; CLOSED_AFFINE_HULL] THEN
9304 ASM_MESON_TAC[SUBSET_CLOSURE; HULL_INC; SUBSET];
9306 ASM_REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN CONJ_TAC THENL
9307 [ASM_MESON_TAC[IN_BALL; IN_INTER; SUBSET]; ALL_TAC] THEN
9308 EXISTS_TAC `d - dist(c:real^N,z)` THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN
9309 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
9310 (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN
9311 REWRITE_TAC[SUBSET; IN_INTER] THEN GEN_TAC THEN
9312 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
9313 UNDISCH_TAC `dist(c:real^N,z) < d` THEN REWRITE_TAC[IN_BALL] THEN
9316 let IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT = prove
9318 convex s /\ a IN relative_interior s /\ b IN closure s
9319 ==> segment(a,b) SUBSET relative_interior s`,
9320 REWRITE_TAC[SUBSET; IN_SEGMENT] THEN REPEAT STRIP_TAC THEN
9321 ASM_REWRITE_TAC[VECTOR_ARITH
9322 `(&1 - u) % a + u % b:real^N = b - (&1 - u) % (b - a)`] THEN
9323 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
9324 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
9326 let CONVEX_OPEN_SEGMENT_CASES = prove
9328 convex s /\ a IN closure s /\ b IN closure s
9329 ==> segment(a,b) SUBSET relative_frontier s \/
9330 segment(a,b) SUBSET relative_interior s`,
9331 REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier] THEN
9332 MATCH_MP_TAC(SET_RULE
9333 `s SUBSET c /\ (!a. a IN i /\ a IN s ==> s SUBSET i)
9334 ==> s SUBSET c DIFF i \/ s SUBSET i`) THEN
9336 [ASM_MESON_TAC[CONVEX_CONTAINS_OPEN_SEGMENT; CONVEX_CLOSURE];
9337 X_GEN_TAC `c:real^N` THEN ONCE_REWRITE_TAC[segment]] THEN
9338 REWRITE_TAC[IN_DIFF; IN_INSERT; DE_MORGAN_THM; NOT_IN_EMPTY] THEN
9339 STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP UNION_SEGMENT) THEN
9340 MP_TAC(ISPECL [`s:real^N->bool`; `c:real^N`]
9341 IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN
9342 DISCH_THEN(fun th ->
9343 MP_TAC(SPEC `b:real^N` th) THEN MP_TAC(SPEC `a:real^N` th)) THEN
9344 ASM_REWRITE_TAC[SEGMENT_SYM; CONJUNCT2 segment] THEN ASM SET_TAC[]);;
9346 let RELATIVE_INTERIOR_SING = prove
9347 (`!a. relative_interior {a} = {a}`,
9348 GEN_TAC THEN MATCH_MP_TAC(SET_RULE
9349 `s SUBSET {a} /\ ~(s = {}) ==> s = {a}`) THEN
9350 SIMP_TAC[RELATIVE_INTERIOR_SUBSET; RELATIVE_INTERIOR_EQ_EMPTY;
9354 let RELATIVE_FRONTIER_SING = prove
9355 (`!a:real^N. relative_frontier {a} = {}`,
9356 REWRITE_TAC[relative_frontier; RELATIVE_INTERIOR_SING; CLOSURE_SING] THEN
9359 let RELATIVE_INTERIOR_CBALL = prove
9360 (`!a r. relative_interior(cball(a,r)) = if r = &0 then {a} else ball(a,r)`,
9361 REPEAT GEN_TAC THEN ASM_CASES_TAC `r < &0` THEN
9362 ASM_SIMP_TAC[REAL_LT_IMP_NE; CBALL_EMPTY; BALL_EMPTY;
9363 RELATIVE_INTERIOR_EMPTY; REAL_LT_IMP_LE] THEN
9364 COND_CASES_TAC THEN ASM_SIMP_TAC[CBALL_SING; RELATIVE_INTERIOR_SING] THEN
9365 REWRITE_TAC[GSYM INTERIOR_CBALL] THEN
9366 MATCH_MP_TAC RELATIVE_INTERIOR_NONEMPTY_INTERIOR THEN
9367 ASM_REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC);;
9369 let RELATIVE_INTERIOR_BALL = prove
9370 (`!a r. relative_interior(ball(a,r)) = ball(a,r)`,
9371 SIMP_TAC[RELATIVE_INTERIOR_OPEN; OPEN_BALL]);;
9373 let RELATIVE_FRONTIER_CBALL = prove
9374 (`!a:real^N r. relative_frontier(cball(a,r)) =
9375 if r = &0 then {} else sphere(a,r)`,
9376 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
9377 ASM_SIMP_TAC[CBALL_SING; RELATIVE_FRONTIER_SING] THEN
9378 ASM_CASES_TAC `r < &0` THEN
9379 ASM_SIMP_TAC[CBALL_EMPTY; SPHERE_EMPTY; RELATIVE_FRONTIER_EMPTY] THEN
9380 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
9381 ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; INTERIOR_CBALL;
9382 BALL_EQ_EMPTY; GSYM REAL_NOT_LT; FRONTIER_CBALL]);;
9384 let RELATIVE_FRONTIER_BALL = prove
9385 (`!a:real^N r. relative_frontier(ball(a,r)) =
9386 if r = &0 then {} else sphere(a,r)`,
9387 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
9388 ASM_SIMP_TAC[BALL_EMPTY; REAL_LE_REFL; RELATIVE_FRONTIER_EMPTY] THEN
9389 ASM_CASES_TAC `r < &0` THEN
9390 ASM_SIMP_TAC[BALL_EMPTY; REAL_LT_IMP_LE; SPHERE_EMPTY;
9391 RELATIVE_FRONTIER_EMPTY] THEN
9392 SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
9393 ASM_SIMP_TAC[RELATIVE_FRONTIER_NONEMPTY_INTERIOR; INTERIOR_OPEN; OPEN_BALL;
9394 BALL_EQ_EMPTY; GSYM REAL_NOT_LT; FRONTIER_BALL]);;
9396 let STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS = prove
9397 (`!s t:real^N->bool.
9398 convex s /\ ~(s = {}) /\
9399 relative_interior s SUBSET t /\ t SUBSET closure s
9401 REPEAT STRIP_TAC THEN
9402 SUBGOAL_THEN `~(relative_interior s:real^N->bool = {})` MP_TAC THENL
9403 [ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY]; REWRITE_TAC[starlike]] THEN
9404 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN
9405 X_GEN_TAC `a:real^N` THEN
9406 REPEAT STRIP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
9407 MATCH_MP_TAC(SET_RULE
9408 `a IN s /\ b IN s /\ segment[a,b] DIFF {a,b} SUBSET s
9409 ==> segment[a:real^N,b] SUBSET s`) THEN
9410 ASM_REWRITE_TAC[GSYM open_segment] THEN
9411 ASM_MESON_TAC[IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT; SUBSET]);;
9413 let RELATIVE_INTERIOR_PROLONG = prove
9415 x IN relative_interior s /\ y IN s
9416 ==> ?t. &1 < t /\ (y + t % (x - y)) IN s`,
9418 REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL; IN_ELIM_THM] THEN
9419 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
9420 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real`
9421 STRIP_ASSUME_TAC)) THEN
9422 ASM_CASES_TAC `y:real^N = x` THENL
9423 [ASM_REWRITE_TAC[VECTOR_ARITH `y + t % (x - x):real^N = y`] THEN
9424 EXISTS_TAC `&2` THEN CONV_TAC REAL_RAT_REDUCE_CONV;
9425 EXISTS_TAC `&1 + e / norm(x - y:real^N)` THEN
9426 ASM_SIMP_TAC[REAL_LT_ADDR; REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ] THEN
9427 REWRITE_TAC[VECTOR_ARITH
9428 `y + (&1 + e) % (x - y):real^N = x + e % (x - y)`] THEN
9429 FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
9430 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; IN_INTER; IN_AFFINE_ADD_MUL_DIFF;
9431 HULL_INC; IN_CBALL] THEN
9432 REWRITE_TAC[NORM_ARITH `dist(x:real^N,x + y) = norm y`] THEN
9433 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
9434 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
9435 ASM_REAL_ARITH_TAC]);;
9437 let RELATIVE_INTERIOR_CONVEX_PROLONG = prove
9439 ==> relative_interior s =
9440 {x:real^N | x IN s /\
9441 !y. y IN s ==> ?t. &1 < t /\ (y + t % (x - y)) IN s}`,
9442 REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
9443 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
9444 [SIMP_TAC[RELATIVE_INTERIOR_PROLONG] THEN
9445 MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET];
9447 SUBGOAL_THEN `?y:real^N. y IN relative_interior s` STRIP_ASSUME_TAC THENL
9448 [ASM_SIMP_TAC[MEMBER_NOT_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY] THEN
9451 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ANTS_TAC THENL
9452 [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]; ALL_TAC] THEN
9453 ASM_CASES_TAC `y:real^N = x` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
9454 DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN
9455 MP_TAC(ISPECL [`s:real^N->bool`; `y:real^N`; `y + t % (x - y):real^N`]
9456 IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN
9457 ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; ALL_TAC] THEN
9458 REWRITE_TAC[SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN
9459 REWRITE_TAC[IN_SEGMENT; IN_ELIM_THM] THEN
9460 ASM_REWRITE_TAC[VECTOR_ARITH `y:real^N = y + x <=> x = vec 0`;
9461 VECTOR_ARITH `(&1 - u) % y + u % (y + t % (x - y)):real^N =
9462 y + t % u % (x - y)`] THEN
9463 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
9464 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
9465 EXISTS_TAC `inv t:real` THEN
9466 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_INV_EQ;
9467 REAL_INV_LT_1; REAL_LT_IMP_NZ; REAL_ARITH `&1 < x ==> &0 < x`] THEN
9468 VECTOR_ARITH_TAC]);;
9470 let RELATIVE_INTERIOR_EQ_CLOSURE = prove
9472 relative_interior s = closure s <=> affine s`,
9473 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9474 ASM_REWRITE_TAC[RELATIVE_INTERIOR_EMPTY; CLOSURE_EMPTY; AFFINE_EMPTY] THEN
9476 SIMP_TAC[RELATIVE_INTERIOR_AFFINE; CLOSURE_CLOSED; CLOSED_AFFINE] THEN
9477 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
9478 `relative_interior s = closure s
9479 ==> relative_interior s SUBSET s /\ s SUBSET closure s
9480 ==> relative_interior s = s /\ closure s = s`)) THEN
9481 REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET] THEN
9482 REWRITE_TAC[RELATIVE_INTERIOR_EQ; CLOSURE_EQ; GSYM AFFINE_HULL_EQ] THEN
9483 STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
9484 `~(s = {}) ==> s = {} \/ s = a ==> a = s`)) THEN
9485 MP_TAC(ISPEC `affine hull s:real^N->bool` CONNECTED_CLOPEN) THEN
9486 SIMP_TAC[AFFINE_IMP_CONVEX; CONVEX_CONNECTED; AFFINE_AFFINE_HULL] THEN
9487 DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
9488 MATCH_MP_TAC CLOSED_SUBSET THEN ASM_REWRITE_TAC[HULL_SUBSET]);;
9490 let RAY_TO_RELATIVE_FRONTIER = prove
9492 bounded s /\ a IN relative_interior s /\
9493 (a + l) IN affine hull s /\ ~(l = vec 0)
9495 (a + d % l) IN relative_frontier s /\
9496 !e. &0 <= e /\ e < d ==> (a + e % l) IN relative_interior s`,
9497 REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier] THEN
9499 `{d | &0 < d /\ ~((a + d % l:real^N) IN relative_interior(s))}` INF) THEN
9501 `d = inf {d | &0 < d /\ ~((a + d % l:real^N) IN relative_interior(s))}` THEN
9503 `?e. &0 < e /\ !d. &0 <= d /\ d < e
9504 ==> (a + d % l:real^N) IN relative_interior s`
9505 (X_CHOOSE_THEN `k:real` (LABEL_TAC "0"))
9507 [MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN
9508 REWRITE_TAC[open_in; GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
9509 DISCH_THEN(MP_TAC o SPEC `a:real^N` o CONJUNCT2) THEN
9510 ASM_REWRITE_TAC[] THEN
9511 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9512 EXISTS_TAC `e / norm(l:real^N)` THEN
9513 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT] THEN X_GEN_TAC `x:real` THEN
9514 STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
9515 [MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
9516 ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN
9517 ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET];
9518 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a) = norm x`] THEN
9519 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN
9520 ASM_REAL_ARITH_TAC];
9523 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
9524 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE]] THEN
9525 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N` o
9526 MATCH_MP BOUNDED_SUBSET_BALL) THEN
9527 REWRITE_TAC[SUBSET; IN_BALL] THEN
9528 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
9529 EXISTS_TAC `B / norm(l:real^N)` THEN
9530 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT] THEN
9531 DISCH_THEN(MP_TAC o MATCH_MP
9532 (REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET)) THEN
9533 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV
9534 [GSYM CONTRAPOS_THM]) THEN
9535 REWRITE_TAC[REAL_NOT_LT] THEN DISCH_THEN MATCH_MP_TAC THEN
9536 REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN
9537 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM;
9538 REAL_DIV_RMUL; NORM_EQ_0] THEN
9540 REWRITE_TAC[IN_ELIM_THM] THEN
9541 DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "1") (LABEL_TAC "2")) THEN
9542 EXISTS_TAC `d:real` THEN
9543 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
9544 [MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `k:real` THEN
9545 ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE];
9547 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
9548 [REWRITE_TAC[REAL_LE_LT] THEN
9549 ASM_MESON_TAC[VECTOR_ARITH `a + &0 % l:real^N = a`;
9550 REAL_NOT_LT; REAL_LT_IMP_LE];
9552 REWRITE_TAC[IN_DIFF] THEN CONJ_TAC THENL
9553 [REWRITE_TAC[CLOSURE_APPROACHABLE] THEN
9554 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
9555 EXISTS_TAC `a + (d - min d (x / &2 / norm(l:real^N))) % l` THEN
9557 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET) THEN
9558 FIRST_X_ASSUM MATCH_MP_TAC THEN
9559 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
9560 MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < d ==> d - min d x < d`) THEN
9561 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT];
9562 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = norm(x - y)`] THEN
9563 REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN
9564 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN
9565 MATCH_MP_TAC(REAL_ARITH
9566 `&0 < x /\ x < y /\ &0 < d ==> abs((d - min d x) - d) < y`) THEN
9567 REWRITE_TAC[REAL_ARITH `x / &2 / y < x / y <=> &0 < x / y`] THEN
9568 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; NORM_POS_LT]];
9570 MP_TAC(ISPEC `s:real^N->bool` OPEN_IN_RELATIVE_INTERIOR) THEN
9571 REWRITE_TAC[open_in; GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
9572 DISCH_THEN(MP_TAC o SPEC `a + d % l:real^N` o CONJUNCT2) THEN
9573 ASM_REWRITE_TAC[] THEN
9574 DISCH_THEN(X_CHOOSE_THEN `e:real`
9575 (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "3"))) THEN
9576 REMOVE_THEN "2" (MP_TAC o SPEC `d + e / norm(l:real^N)`) THEN
9577 ASM_SIMP_TAC[NOT_IMP; REAL_ARITH `~(d + l <= d) <=> &0 < l`;
9578 REAL_LT_DIV; NORM_POS_LT] THEN
9579 X_GEN_TAC `x:real` THEN
9580 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9581 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
9582 REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN
9583 ASM_CASES_TAC `x < d` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
9584 REMOVE_THEN "3" MATCH_MP_TAC THEN CONJ_TAC THENL
9585 [MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
9586 ASM_REWRITE_TAC[AFFINE_AFFINE_HULL] THEN
9587 ASM_MESON_TAC[SUBSET; HULL_SUBSET; RELATIVE_INTERIOR_SUBSET];
9588 REWRITE_TAC[NORM_ARITH `dist(a + x:real^N,a + y) = norm(x - y)`] THEN
9589 REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; NORM_MUL] THEN
9590 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT] THEN
9591 ASM_REAL_ARITH_TAC]]]);;
9593 let RAY_TO_FRONTIER = prove
9595 bounded s /\ a IN interior s /\ ~(l = vec 0)
9596 ==> ?d. &0 < d /\ (a + d % l) IN frontier s /\
9597 !e. &0 <= e /\ e < d ==> (a + e % l) IN interior s`,
9598 REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN
9599 SUBGOAL_THEN `interior s:real^N->bool = relative_interior s` SUBST1_TAC THENL
9601 REWRITE_TAC[GSYM relative_frontier] THEN
9602 MATCH_MP_TAC RAY_TO_RELATIVE_FRONTIER THEN ASM_REWRITE_TAC[]] THEN
9603 ASM_MESON_TAC[NOT_IN_EMPTY; RELATIVE_INTERIOR_NONEMPTY_INTERIOR; IN_UNIV;
9604 AFFINE_HULL_NONEMPTY_INTERIOR]);;
9606 let RELATIVE_FRONTIER_NOT_SING = prove
9607 (`!s a:real^N. bounded s ==> ~(relative_frontier s = {a})`,
9608 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9609 ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY; NOT_INSERT_EMPTY] THEN
9610 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9611 DISCH_THEN(X_CHOOSE_TAC `z:real^N`) THEN
9612 ASM_CASES_TAC `s = {z:real^N}` THEN
9613 ASM_REWRITE_TAC[RELATIVE_FRONTIER_SING; NOT_INSERT_EMPTY] THEN
9614 SUBGOAL_THEN `?w:real^N. w IN s /\ ~(w = z)` STRIP_ASSUME_TAC THENL
9615 [ASM SET_TAC[]; REPEAT STRIP_TAC] THEN
9617 `~((w:real^N) IN relative_frontier s /\ z IN relative_frontier s)`
9618 MP_TAC THENL [ASM SET_TAC[]; DISCH_TAC] THEN
9619 MAP_EVERY UNDISCH_TAC
9620 [`relative_frontier s = {a:real^N}`; `bounded(s:real^N->bool)`;
9621 `~(w:real^N = z)`; `(z:real^N) IN s`; `(w:real^N) IN s`;
9622 `~((w:real^N) IN relative_frontier s /\ z IN relative_frontier s)`] THEN
9623 POP_ASSUM_LIST(K ALL_TAC) THEN REWRITE_TAC[DE_MORGAN_THM] THEN
9624 MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`z:real^N`; `w:real^N`] THEN
9625 MATCH_MP_TAC(MESON[]
9626 `(!w z. Q w z <=> Q z w) /\ (!w z. P z ==> Q w z)
9627 ==> !w z. P w \/ P z ==> Q w z`) THEN
9628 CONJ_TAC THENL [MESON_TAC[]; REPEAT GEN_TAC] THEN
9629 DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN
9630 REWRITE_TAC[relative_frontier; IN_DIFF] THEN
9631 CONJ_TAC THENL [ASM_MESON_TAC[SUBSET; CLOSURE_SUBSET]; DISCH_TAC] THEN
9633 (ISPECL [`s:real^N->bool`; `z:real^N`; `d % (w - z):real^N`]
9634 RAY_TO_RELATIVE_FRONTIER)) THEN
9635 ASM_SIMP_TAC[VECTOR_SUB_EQ; IN_AFFINE_ADD_MUL_DIFF; AFFINE_AFFINE_HULL;
9636 HULL_INC; VECTOR_MUL_EQ_0] THEN
9637 DISCH_THEN(fun th -> MP_TAC(SPEC `&1` th) THEN MP_TAC(SPEC `--(&1)` th)) THEN
9638 CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[IN_SING] THEN
9639 DISCH_THEN(X_CHOOSE_THEN `e:real` (STRIP_ASSUME_TAC o GSYM)) THEN
9640 ASM_REWRITE_TAC[VECTOR_MUL_RCANCEL; VECTOR_MUL_ASSOC; VECTOR_SUB_EQ;
9641 VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
9642 ASM_REAL_ARITH_TAC);;
9644 let RELATIVE_INTERIOR_PCROSS = prove
9645 (`!s:real^M->bool t:real^N->bool.
9646 relative_interior(s PCROSS t) =
9647 relative_interior s PCROSS relative_interior t`,
9648 REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC
9649 [`s:real^M->bool = {}`; `t:real^N->bool = {}`] THEN
9650 ASM_REWRITE_TAC[PCROSS_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN
9651 REWRITE_TAC[relative_interior; AFFINE_HULL_PCROSS] THEN
9652 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM;
9653 PASTECART_IN_PCROSS] THEN
9654 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN EQ_TAC THENL
9655 [ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ q /\ p`] THEN
9656 DISCH_THEN(X_CHOOSE_THEN `u:real^(M,N)finite_sum->bool`
9657 (CONJUNCTS_THEN ASSUME_TAC)) THEN
9658 FIRST_ASSUM(MP_TAC o MATCH_MP PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN
9659 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
9660 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
9661 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
9662 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
9663 W(MP_TAC o PART_MATCH (funpow 3 rand) SUBSET_PCROSS o snd) THEN
9665 DISCH_THEN(CONJUNCTS_THEN2
9666 (X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC)
9667 (X_CHOOSE_THEN `w:real^N->bool` STRIP_ASSUME_TAC)) THEN
9668 EXISTS_TAC `(v:real^M->bool) PCROSS (w:real^N->bool)` THEN
9669 ASM_SIMP_TAC[PASTECART_IN_PCROSS; SUBSET_PCROSS; OPEN_IN_PCROSS]]);;
9671 let RELATIVE_FRONTIER_EQ_EMPTY = prove
9672 (`!s:real^N->bool. relative_frontier s = {} <=> affine s`,
9673 GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN
9674 REWRITE_TAC[GSYM RELATIVE_INTERIOR_EQ_CLOSURE] THEN
9675 MP_TAC(ISPEC `s:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN
9676 MP_TAC(ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN SET_TAC[]);;
9678 let DIAMETER_BOUNDED_BOUND_LT = prove
9680 bounded s /\ x IN relative_interior s /\ y IN closure s /\
9682 ==> norm(x - y) < diameter s`,
9685 bounded s /\ x IN relative_interior s /\ y IN s /\
9687 ==> norm(x - y) < diameter s`,
9688 REPEAT STRIP_TAC THEN FIRST_X_ASSUM
9689 (MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN
9690 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_THEN `e:real`
9691 STRIP_ASSUME_TAC)) THEN
9692 ASM_SIMP_TAC[REAL_LT_LE; DIAMETER_BOUNDED_BOUND] THEN
9693 ASM_CASES_TAC `y:real^N = x` THEN
9694 ASM_SIMP_TAC[VECTOR_SUB_REFL; NORM_0] THEN
9696 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
9697 DISCH_THEN(MP_TAC o SPEC `x + e / norm(x - y) % (x - y):real^N`) THEN
9698 REWRITE_TAC[NOT_IMP; IN_INTER] THEN REPEAT CONJ_TAC THENL
9699 [REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(x:real^M,x + y) = norm y`] THEN
9700 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL;
9701 NORM_EQ_0; VECTOR_SUB_EQ] THEN ASM_REAL_ARITH_TAC;
9702 MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
9703 ASM_SIMP_TAC[HULL_INC; AFFINE_AFFINE_HULL];
9704 DISCH_TAC THEN MP_TAC(ISPECL
9705 [`s:real^N->bool`; `x + e / norm(x - y) % (x - y):real^N`; `y:real^N`]
9706 DIAMETER_BOUNDED_BOUND) THEN
9707 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
9708 REWRITE_TAC[VECTOR_ARITH
9709 `(x + e % (x - y)) - y:real^N = (&1 + e) % (x - y)`] THEN
9710 SIMP_TAC[NORM_MUL; REAL_ARITH `~(a * n <= n) <=> &0 < n * (a - &1)`] THEN
9711 MATCH_MP_TAC REAL_LT_MUL THEN
9712 ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN
9713 MATCH_MP_TAC(REAL_ARITH `&0 < e ==> &0 < abs(&1 + e) - &1`) THEN
9714 MATCH_MP_TAC REAL_LT_DIV THEN
9715 ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ]]) in
9716 REPEAT STRIP_TAC THEN
9717 MP_TAC(ISPECL [`closure s:real^N->bool`; `x:real^N`; `y:real^N`]
9719 ASM_SIMP_TAC[DIAMETER_CLOSURE; BOUNDED_CLOSURE] THEN
9720 DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
9721 (SET_RULE `x IN s ==> s SUBSET t ==> x IN t`)) THEN
9722 MATCH_MP_TAC SUBSET_RELATIVE_INTERIOR THEN
9723 REWRITE_TAC[CLOSURE_SUBSET; AFFINE_HULL_CLOSURE]);;
9725 let DIAMETER_ATTAINED_RELATIVE_FRONTIER = prove
9727 bounded s /\ ~(diameter s = &0)
9728 ==> ?x y. x IN relative_frontier s /\
9729 y IN relative_frontier s /\
9730 norm(x - y) = diameter s`,
9731 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9732 ASM_REWRITE_TAC[DIAMETER_EMPTY; relative_frontier] THEN REPEAT STRIP_TAC THEN
9733 MP_TAC(ISPEC `closure s:real^N->bool` DIAMETER_COMPACT_ATTAINED) THEN
9734 ASM_SIMP_TAC[COMPACT_CLOSURE; CLOSURE_EQ_EMPTY; DIAMETER_CLOSURE] THEN
9735 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
9736 STRIP_TAC THEN ASM_REWRITE_TAC[IN_DIFF] THEN REPEAT STRIP_TAC THEN
9737 MP_TAC(ISPEC `s:real^N->bool` DIAMETER_BOUNDED_BOUND_LT) THENL
9738 [DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `y:real^N`]);
9739 DISCH_THEN(MP_TAC o SPECL [`y:real^N`; `x:real^N`])] THEN
9740 ASM_MESON_TAC[REAL_LT_REFL; NORM_SUB]);;
9742 let DIAMETER_RELATIVE_FRONTIER = prove
9744 bounded s /\ ~(?a. s = {a})
9745 ==> diameter(relative_frontier s) = diameter s`,
9746 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9747 ASM_REWRITE_TAC[RELATIVE_FRONTIER_EMPTY] THEN
9748 REWRITE_TAC[relative_frontier] THEN
9749 ASM_SIMP_TAC[GSYM DIAMETER_CLOSURE; GSYM REAL_LE_ANTISYM] THEN
9750 ASM_SIMP_TAC[SUBSET_DIFF; DIAMETER_SUBSET; BOUNDED_CLOSURE] THEN
9751 ASM_SIMP_TAC[DIAMETER_CLOSURE] THEN
9752 MP_TAC(ISPEC `s:real^N->bool` DIAMETER_ATTAINED_RELATIVE_FRONTIER) THEN
9753 ASM_SIMP_TAC[DIAMETER_EQ_0; relative_frontier] THEN STRIP_TAC THEN
9754 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN
9755 ASM_SIMP_TAC[BOUNDED_CLOSURE; BOUNDED_DIFF]);;
9757 let DIAMETER_ATTAINED_FRONTIER = prove
9759 bounded s /\ ~(diameter s = &0)
9760 ==> ?x y. x IN frontier s /\ y IN frontier s /\
9761 norm(x - y) = diameter s`,
9763 DISCH_THEN(MP_TAC o MATCH_MP DIAMETER_ATTAINED_RELATIVE_FRONTIER) THEN
9764 REWRITE_TAC[frontier; relative_frontier; IN_DIFF] THEN
9765 MESON_TAC[REWRITE_RULE[SUBSET] INTERIOR_SUBSET_RELATIVE_INTERIOR]);;
9767 let DIAMETER_FRONTIER = prove
9768 (`!s:real^N->bool. bounded s ==> diameter(frontier s) = diameter s`,
9769 REPEAT STRIP_TAC THEN ASM_CASES_TAC `?a:real^N. s = {a}` THENL
9770 [ASM_MESON_TAC[FRONTIER_SING]; ALL_TAC] THEN
9771 MATCH_MP_TAC(REAL_ARITH
9772 `!r. r <= f /\ f <= s /\ r = s ==> f = s`) THEN
9773 EXISTS_TAC `diameter(closure s DIFF relative_interior s:real^N->bool)` THEN
9774 REPEAT CONJ_TAC THENL
9775 [ASM_SIMP_TAC[GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_SUBSET THEN
9776 ASM_SIMP_TAC[BOUNDED_FRONTIER] THEN REWRITE_TAC[frontier] THEN
9777 MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET_RELATIVE_INTERIOR) THEN
9779 ASM_SIMP_TAC[GSYM DIAMETER_CLOSURE] THEN MATCH_MP_TAC DIAMETER_SUBSET THEN
9780 ASM_SIMP_TAC[BOUNDED_CLOSURE; frontier; SUBSET_DIFF];
9781 ASM_SIMP_TAC[DIAMETER_RELATIVE_FRONTIER; GSYM relative_frontier]]);;
9783 let DIAMETER_SPHERE = prove
9784 (`!a:real^N r. diameter(sphere(a,r)) = if r < &0 then &0 else &2 * r`,
9785 REWRITE_TAC[GSYM FRONTIER_CBALL] THEN
9786 ASM_SIMP_TAC[DIAMETER_FRONTIER; BOUNDED_CBALL; DIAMETER_CBALL]);;
9788 let CLOSEST_POINT_IN_RELATIVE_INTERIOR = prove
9790 closed s /\ ~(s = {}) /\ x IN affine hull s
9791 ==> ((closest_point s x) IN relative_interior s <=>
9792 x IN relative_interior s)`,
9793 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN
9794 ASM_SIMP_TAC[CLOSEST_POINT_SELF] THEN
9795 MATCH_MP_TAC(TAUT `~q /\ ~p ==> (p <=> q)`) THEN CONJ_TAC THENL
9796 [ASM_MESON_TAC[RELATIVE_INTERIOR_SUBSET; SUBSET]; STRIP_TAC] THEN
9797 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_RELATIVE_INTERIOR_CBALL]) THEN
9798 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9799 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9800 SUBGOAL_THEN `~(closest_point s (x:real^N) = x)` ASSUME_TAC THENL
9801 [ASM_MESON_TAC[]; ALL_TAC] THEN
9802 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`;
9803 `closest_point s x -
9804 (min (&1) (e / norm(closest_point s x - x))) %
9805 (closest_point s x - x):real^N`]
9806 CLOSEST_POINT_LE) THEN
9807 ASM_REWRITE_TAC[dist; NOT_IMP; VECTOR_ARITH
9808 `x - (y - e % (y - x)):real^N = (&1 - e) % (x - y)`] THEN
9810 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
9811 REWRITE_TAC[IN_CBALL; IN_INTER] THEN CONJ_TAC THENL
9812 [REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - x) = norm x`] THEN
9813 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
9814 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
9815 MATCH_MP_TAC(REAL_ARITH `&0 <= a ==> abs(min (&1) a) <= a`) THEN
9816 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_DIV; NORM_POS_LE];
9817 MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN
9818 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC]];
9819 REWRITE_TAC[NORM_MUL; REAL_ARITH
9820 `~(n <= a * n) <=> &0 < (&1 - a) * n`] THEN
9821 MATCH_MP_TAC REAL_LT_MUL THEN
9822 ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN
9823 MATCH_MP_TAC(REAL_ARITH
9824 `&0 < e /\ e <= &1 ==> &0 < &1 - abs(&1 - e)`) THEN
9825 REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LT_01; REAL_LE_REFL] THEN
9826 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]);;
9828 let CLOSEST_POINT_IN_RELATIVE_FRONTIER = prove
9830 closed s /\ ~(s = {}) /\ x IN affine hull s DIFF relative_interior s
9831 ==> closest_point s x IN relative_frontier s`,
9832 SIMP_TAC[relative_frontier; IN_DIFF; CLOSEST_POINT_IN_RELATIVE_INTERIOR] THEN
9833 MESON_TAC[CLOSURE_SUBSET; CLOSEST_POINT_IN_SET; SUBSET]);;
9835 (* ------------------------------------------------------------------------- *)
9836 (* Interior, relative interior and closure interrelations. *)
9837 (* ------------------------------------------------------------------------- *)
9839 let CONVEX_CLOSURE_INTERIOR = prove
9841 convex s /\ ~(interior s = {})
9842 ==> closure(interior s) = closure s`,
9843 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
9844 SIMP_TAC[SUBSET_CLOSURE; INTERIOR_SUBSET] THEN
9845 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9846 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN REWRITE_TAC[SUBSET] THEN
9847 X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
9848 [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; ALL_TAC] THEN
9849 REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN
9850 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
9851 EXISTS_TAC `b - min (e / &2 / norm(b - a)) (&1) % (b - a):real^N` THEN
9852 REPEAT CONJ_TAC THENL
9853 [MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
9854 ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LE_REFL; REAL_LT_01];
9855 REWRITE_TAC[VECTOR_ARITH `b - x:real^N = b <=> x = vec 0`] THEN
9856 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
9857 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min x (&1) = &0)`);
9858 REWRITE_TAC[NORM_ARITH `dist(b - x:real^N,b) = norm x`] THEN
9859 REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LET_TRANS THEN
9860 EXISTS_TAC `e / &2 / norm(b - a:real^N) * norm(b - a)` THEN CONJ_TAC THENL
9861 [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
9862 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> abs(min x (&1)) <= x`);
9863 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_POS_LT; REAL_LT_IMP_NZ;
9865 ASM_REAL_ARITH_TAC]] THEN
9866 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_OF_NUM_LT;
9867 VECTOR_SUB_EQ; ARITH]);;
9869 let EMPTY_INTERIOR_SUBSET_HYPERPLANE = prove
9870 (`!s. convex s /\ interior s = {}
9871 ==> ?a:real^N b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`,
9873 (`!s. convex s /\ (vec 0) IN s /\ interior s = {}
9874 ==> ?a:real^N b. ~(a = vec 0) /\ s SUBSET {x | a dot x = b}`,
9876 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
9877 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
9878 SUBGOAL_THEN `~(relative_interior(s:real^N->bool) = {})` MP_TAC THENL
9879 [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; MEMBER_NOT_EMPTY]; ALL_TAC] THEN
9880 ASM_REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN
9881 AP_THM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
9882 MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
9883 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC] THEN
9884 ONCE_REWRITE_TAC[GSYM SPAN_UNIV] THEN MATCH_MP_TAC DIM_EQ_SPAN THEN
9885 REWRITE_TAC[SUBSET_UNIV; DIM_UNIV; GSYM NOT_LT] THEN
9886 DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN
9887 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
9888 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
9889 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN
9890 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN EXISTS_TAC `&0` THEN
9891 ASM_MESON_TAC[SUBSET_TRANS; SPAN_INC]) in
9892 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
9893 [ASM_MESON_TAC[EMPTY_SUBSET; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1];
9896 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9897 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
9898 MP_TAC(ISPEC `IMAGE (\x:real^N. --a + x) s` lemma) THEN
9899 ASM_REWRITE_TAC[CONVEX_TRANSLATION_EQ; INTERIOR_TRANSLATION;
9900 IMAGE_EQ_EMPTY; IN_IMAGE; UNWIND_THM2;
9901 VECTOR_ARITH `vec 0:real^N = --a + x <=> x = a`] THEN
9902 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN
9903 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; DOT_RADD] THEN
9904 MESON_TAC[REAL_ARITH `a + x:real = b <=> x = b - a`]);;
9906 let CONVEX_INTERIOR_CLOSURE = prove
9907 (`!s:real^N->bool. convex s ==> interior(closure s) = interior s`,
9908 REPEAT STRIP_TAC THEN
9909 ASM_CASES_TAC `interior(s:real^N->bool) = {}` THENL
9910 [MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN
9911 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
9912 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN
9913 MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
9914 EXISTS_TAC `interior {x:real^N | a dot x = b}` THEN CONJ_TAC THENL
9915 [ALL_TAC; ASM_SIMP_TAC[INTERIOR_HYPERPLANE]] THEN
9916 MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN
9917 ASM_REWRITE_TAC[CLOSED_HYPERPLANE];
9919 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9920 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
9921 MATCH_MP_TAC SUBSET_ANTISYM THEN
9922 SIMP_TAC[SUBSET_INTERIOR; CLOSURE_SUBSET] THEN
9923 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN
9924 MP_TAC(ASSUME `(b:real^N) IN interior(closure s)`) THEN
9925 GEN_REWRITE_TAC LAND_CONV [IN_INTERIOR_CBALL] THEN
9926 REWRITE_TAC[SUBSET; IN_CBALL; LEFT_IMP_EXISTS_THM] THEN
9927 X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
9928 ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[] THEN
9929 DISCH_THEN(MP_TAC o SPEC `b + e / norm(b - a) % (b - a):real^N`) THEN
9930 ASM_SIMP_TAC[NORM_ARITH `dist(b:real^N,b + e) = norm e`; NORM_MUL;
9931 REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ;
9932 REAL_ARITH `&0 < e ==> abs e <= e`] THEN
9935 `b = (b + e / norm(b - a) % (b - a)) -
9936 e / norm(b - a) / (&1 + e / norm(b - a)) %
9937 ((b + e / norm(b - a) % (b - a)) - a):real^N`
9939 [REWRITE_TAC[VECTOR_ARITH
9940 `b = (b + e % (b - a)) - d % ((b + e % (b - a)) - a) <=>
9941 (e - d * (&1 + e)) % (b - a) = vec 0`] THEN
9942 ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0];
9943 MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SHRINK] THEN
9944 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_DIV; NORM_POS_LT;
9945 VECTOR_SUB_EQ; REAL_ARITH `&0 < x ==> &0 < &1 + x`;
9946 REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`;
9947 REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_DIV_RMUL;
9948 REAL_LT_IMP_NZ; REAL_LE_ADDL; NORM_POS_LE; REAL_SUB_REFL]);;
9950 let FRONTIER_CLOSURE_CONVEX = prove
9951 (`!s:real^N->bool. convex s ==> frontier(closure s) = frontier s`,
9952 SIMP_TAC[frontier; CLOSURE_CLOSURE; CONVEX_INTERIOR_CLOSURE]);;
9954 let CONVEX_CLOSURE_RELATIVE_INTERIOR = prove
9956 convex s ==> closure(relative_interior s) = closure s`,
9957 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
9958 SIMP_TAC[SUBSET_CLOSURE; RELATIVE_INTERIOR_SUBSET] THEN
9959 ASM_CASES_TAC `relative_interior(s:real^N->bool) = {}` THENL
9960 [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; SUBSET_REFL]; ALL_TAC] THEN
9961 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
9962 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN REWRITE_TAC[SUBSET] THEN
9963 X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN ASM_CASES_TAC `b:real^N = a` THENL
9964 [ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]; ALL_TAC] THEN
9965 REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN
9966 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
9967 EXISTS_TAC `b - min (e / &2 / norm(b - a)) (&1) % (b - a):real^N` THEN
9968 REPEAT CONJ_TAC THENL
9969 [MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
9970 ASM_REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LE_REFL; REAL_LT_01];
9971 REWRITE_TAC[VECTOR_ARITH `b - x:real^N = b <=> x = vec 0`] THEN
9972 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
9973 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(min x (&1) = &0)`);
9974 REWRITE_TAC[NORM_ARITH `dist(b - x:real^N,b) = norm x`] THEN
9975 REWRITE_TAC[NORM_MUL] THEN MATCH_MP_TAC REAL_LET_TRANS THEN
9976 EXISTS_TAC `e / &2 / norm(b - a:real^N) * norm(b - a)` THEN CONJ_TAC THENL
9977 [MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
9978 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> abs(min x (&1)) <= x`);
9979 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_POS_LT; REAL_LT_IMP_NZ;
9981 ASM_REAL_ARITH_TAC]] THEN
9982 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_OF_NUM_LT;
9983 VECTOR_SUB_EQ; ARITH]);;
9985 let AFFINE_HULL_RELATIVE_INTERIOR = prove
9987 ==> affine hull (relative_interior s) = affine hull s`,
9988 MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR; AFFINE_HULL_CLOSURE]);;
9990 let CONVEX_RELATIVE_INTERIOR_CLOSURE = prove
9992 convex s ==> relative_interior(closure s) = relative_interior s`,
9993 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
9994 ASM_REWRITE_TAC[CLOSURE_EMPTY; RELATIVE_INTERIOR_EMPTY] THEN
9995 SUBGOAL_THEN `?a:real^N. a IN relative_interior s` STRIP_ASSUME_TAC THENL
9996 [ASM_SIMP_TAC[MEMBER_NOT_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY];
9998 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[SUBSET] THEN CONJ_TAC THENL
10000 REWRITE_TAC[IN_RELATIVE_INTERIOR; AFFINE_HULL_CLOSURE; SUBSET] THEN
10001 MESON_TAC[CLOSURE_SUBSET; SUBSET]] THEN
10002 X_GEN_TAC `b:real^N` THEN DISCH_TAC THEN
10003 MP_TAC(ASSUME `(b:real^N) IN relative_interior(closure s)`) THEN
10004 GEN_REWRITE_TAC LAND_CONV [IN_RELATIVE_INTERIOR_CBALL] THEN
10005 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
10006 REWRITE_TAC[SUBSET; IN_CBALL; IN_INTER; LEFT_IMP_EXISTS_THM;
10007 AFFINE_HULL_CLOSURE] THEN
10008 X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
10009 ASM_CASES_TAC `b:real^N = a` THEN ASM_REWRITE_TAC[] THEN
10010 DISCH_THEN(MP_TAC o SPEC `b + e / norm(b - a) % (b - a):real^N`) THEN
10011 ASM_SIMP_TAC[NORM_ARITH `dist(b:real^N,b + e) = norm e`; NORM_MUL;
10012 REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ;
10013 REAL_ARITH `&0 < e ==> abs e <= e`] THEN
10015 [MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
10016 ASM_MESON_TAC[SUBSET; AFFINE_AFFINE_HULL; RELATIVE_INTERIOR_SUBSET;
10017 CLOSURE_SUBSET_AFFINE_HULL; HULL_INC];
10021 `b = (b + e / norm(b - a) % (b - a)) -
10022 e / norm(b - a) / (&1 + e / norm(b - a)) %
10023 ((b + e / norm(b - a) % (b - a)) - a):real^N`
10025 [REWRITE_TAC[VECTOR_ARITH
10026 `b = (b + e % (b - a)) - d % ((b + e % (b - a)) - a) <=>
10027 (e - d * (&1 + e)) % (b - a) = vec 0`] THEN
10028 ASM_REWRITE_TAC[VECTOR_SUB_EQ; VECTOR_MUL_EQ_0];
10029 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK] THEN
10030 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_DIV; NORM_POS_LT;
10031 VECTOR_SUB_EQ; REAL_ARITH `&0 < x ==> &0 < &1 + x`;
10032 REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`;
10033 REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_DIV_RMUL;
10034 REAL_LT_IMP_NZ; REAL_LE_ADDL; NORM_POS_LE; REAL_SUB_REFL]);;
10036 let RELATIVE_FRONTIER_CLOSURE = prove
10037 (`!s. convex s ==> relative_frontier(closure s) = relative_frontier s`,
10038 SIMP_TAC[relative_frontier; CLOSURE_CLOSURE;
10039 CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
10041 let RELATIVE_FRONTIER_RELATIVE_INTERIOR = prove
10044 ==> relative_frontier(relative_interior s) = relative_frontier s`,
10045 ASM_MESON_TAC[RELATIVE_FRONTIER_CLOSURE; CONVEX_CLOSURE_RELATIVE_INTERIOR;
10046 CONVEX_RELATIVE_INTERIOR]);;
10048 let CONNECTED_INTER_RELATIVE_FRONTIER = prove
10049 (`!s t:real^N->bool.
10050 connected s /\ s SUBSET affine hull t /\
10051 ~(s INTER t = {}) /\ ~(s DIFF t = {})
10052 ==> ~(s INTER relative_frontier t = {})`,
10053 REWRITE_TAC[relative_frontier] THEN REPEAT STRIP_TAC THEN
10054 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_OPEN_IN]) THEN
10055 REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC
10056 [`s INTER relative_interior t:real^N->bool`;
10057 `s DIFF closure t:real^N->bool`] THEN
10058 REPEAT CONJ_TAC THENL
10059 [MATCH_MP_TAC OPEN_IN_SUBTOPOLOGY_INTER_SUBSET THEN
10060 EXISTS_TAC `affine hull t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
10061 MATCH_MP_TAC OPEN_IN_INTER THEN
10062 REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR; OPEN_IN_SUBTOPOLOGY_REFL] THEN
10063 REWRITE_TAC[TOPSPACE_EUCLIDEAN; SUBSET_UNIV];
10064 ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN
10065 MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN
10066 REWRITE_TAC[GSYM closed; CLOSED_CLOSURE];
10068 MATCH_MP_TAC(SET_RULE
10069 `i SUBSET t /\ t SUBSET c ==> (s INTER i) INTER (s DIFF c) = {}`) THEN
10070 REWRITE_TAC[RELATIVE_INTERIOR_SUBSET; CLOSURE_SUBSET];
10071 MP_TAC(ISPEC `t:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[];
10072 MP_TAC(ISPEC `t:real^N->bool` RELATIVE_INTERIOR_SUBSET) THEN
10075 let CLOSED_RELATIVE_FRONTIER = prove
10076 (`!s:real^N->bool. closed(relative_frontier s)`,
10077 REPEAT GEN_TAC THEN REWRITE_TAC[relative_frontier] THEN
10078 MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN
10079 EXISTS_TAC `affine hull s:real^N->bool` THEN
10080 REWRITE_TAC[CLOSED_AFFINE_HULL] THEN MATCH_MP_TAC CLOSED_IN_DIFF THEN
10081 REWRITE_TAC[OPEN_IN_RELATIVE_INTERIOR] THEN
10082 MATCH_MP_TAC CLOSED_SUBSET THEN REWRITE_TAC[CLOSED_CLOSURE] THEN
10083 MATCH_MP_TAC(SET_RULE
10084 `s SUBSET closure t /\ closure t = t ==> s SUBSET t`) THEN
10085 SIMP_TAC[SUBSET_CLOSURE; HULL_SUBSET; CLOSURE_EQ; CLOSED_AFFINE_HULL]);;
10087 let CLOSED_RELATIVE_BOUNDARY = prove
10088 (`!s. closed s ==> closed(s DIFF relative_interior s)`,
10089 MESON_TAC[CLOSED_RELATIVE_FRONTIER; relative_frontier; CLOSURE_CLOSED]);;
10091 let COMPACT_RELATIVE_BOUNDARY = prove
10092 (`!s. compact s ==> compact(s DIFF relative_interior s)`,
10093 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_RELATIVE_BOUNDARY;
10096 let BOUNDED_RELATIVE_FRONTIER = prove
10097 (`!s:real^N->bool. bounded s ==> bounded(relative_frontier s)`,
10098 REWRITE_TAC[relative_frontier] THEN
10099 MESON_TAC[BOUNDED_CLOSURE; BOUNDED_SUBSET; SUBSET_DIFF]);;
10101 let COMPACT_RELATIVE_FRONTIER_BOUNDED = prove
10102 (`!s:real^N->bool. bounded s ==> compact(relative_frontier s)`,
10103 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_RELATIVE_FRONTIER;
10104 BOUNDED_RELATIVE_FRONTIER]);;
10106 let COMPACT_RELATIVE_FRONTIER = prove
10107 (`!s:real^N->bool. compact s ==> compact(relative_frontier s)`,
10108 SIMP_TAC[COMPACT_RELATIVE_FRONTIER_BOUNDED; COMPACT_IMP_BOUNDED]);;
10110 let CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE = prove
10111 (`!s t. convex s /\ convex t
10112 ==> (relative_interior s = relative_interior t <=>
10113 closure s = closure t)`,
10114 MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR;
10115 CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
10117 let CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE_STRADDLE = prove
10118 (`!s t. convex s /\ convex t
10119 ==> (relative_interior s = relative_interior t <=>
10120 relative_interior s SUBSET t /\ t SUBSET closure s)`,
10121 MESON_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR;
10122 CONVEX_RELATIVE_INTERIOR_CLOSURE; SUBSET_CLOSURE;
10123 SUBSET_ANTISYM; RELATIVE_INTERIOR_SUBSET;
10124 CLOSURE_SUBSET; CLOSURE_CLOSURE]);;
10126 let RELATIVE_INTERIOR_LINEAR_IMAGE_CONVEX = prove
10127 (`!f:real^M->real^N s.
10128 linear f /\ convex s
10129 ==> relative_interior(IMAGE f s) = IMAGE f (relative_interior s)`,
10130 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
10132 `relative_interior (IMAGE f (relative_interior s)) =
10133 relative_interior (IMAGE (f:real^M->real^N) s)`
10134 (fun th -> REWRITE_TAC[SYM th; RELATIVE_INTERIOR_SUBSET]) THEN
10135 ASM_SIMP_TAC[CONVEX_SAME_RELATIVE_INTERIOR_CLOSURE_STRADDLE;
10136 CONVEX_RELATIVE_INTERIOR; CONVEX_LINEAR_IMAGE] THEN
10138 [MATCH_MP_TAC SUBSET_TRANS THEN
10139 EXISTS_TAC `IMAGE (f:real^M->real^N) (relative_interior s)` THEN
10140 SIMP_TAC[RELATIVE_INTERIOR_SUBSET; IMAGE_SUBSET];
10141 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC
10142 `IMAGE (f:real^M->real^N) (closure(relative_interior s))` THEN
10143 ASM_SIMP_TAC[CLOSURE_LINEAR_IMAGE_SUBSET] THEN
10144 ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN
10145 MATCH_MP_TAC IMAGE_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET]];
10146 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `z:real^M` THEN
10148 ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_PROLONG; CONVEX_LINEAR_IMAGE] THEN
10149 REWRITE_TAC[IN_ELIM_THM; FORALL_IN_IMAGE] THEN CONJ_TAC THENL
10150 [MATCH_MP_TAC FUN_IN_IMAGE THEN
10151 ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET];
10153 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
10154 MP_TAC(ISPECL [`s:real^M->bool`; `z:real^M`; `x:real^M`]
10155 RELATIVE_INTERIOR_PROLONG) THEN ASM_REWRITE_TAC[] THEN
10156 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN
10157 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
10158 DISCH_THEN(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP FUN_IN_IMAGE) THEN
10159 ASM_MESON_TAC[LINEAR_ADD; LINEAR_SUB; LINEAR_CMUL]]);;
10161 let CLOSURE_INTERS_CONVEX = prove
10162 (`!f:(real^N->bool)->bool.
10163 (!s. s IN f ==> convex s) /\
10164 ~(INTERS(IMAGE relative_interior f) = {})
10165 ==> closure(INTERS f) = INTERS(IMAGE closure f)`,
10166 REPEAT STRIP_TAC THEN
10167 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[CLOSURE_INTERS_SUBSET] THEN
10168 REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE] THEN
10169 X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN
10170 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
10171 REWRITE_TAC[INTERS_IMAGE; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
10172 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
10173 REWRITE_TAC[CLOSURE_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
10174 ASM_CASES_TAC `b:real^N = a` THENL
10175 [EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[DIST_REFL; IN_INTERS] THEN
10176 ASM_MESON_TAC[SUBSET; RELATIVE_INTERIOR_SUBSET];
10178 EXISTS_TAC `b - min (&1 / &2) (e / &2 / norm(b - a)) % (b - a):real^N` THEN
10181 REWRITE_TAC[NORM_ARITH `dist(b - a:real^N,b) = norm a`; NORM_MUL] THEN
10182 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
10183 MATCH_MP_TAC(REAL_ARITH
10184 `&0 < a /\ &0 < x /\ x < y ==> abs(min a x) < y`) THEN
10185 ASM_SIMP_TAC[REAL_LT_DIV2_EQ; REAL_HALF; REAL_LT_DIV; NORM_POS_LT;
10186 VECTOR_SUB_EQ] THEN
10187 ASM_REAL_ARITH_TAC] THEN
10188 REWRITE_TAC[IN_INTERS] THEN X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN
10190 (MESON[RELATIVE_INTERIOR_SUBSET; SUBSET]
10191 `!x. x IN relative_interior s ==> x IN s`) THEN
10192 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
10193 ASM_SIMP_TAC[REAL_LT_MIN; REAL_HALF; REAL_LT_DIV; NORM_POS_LT;
10194 VECTOR_SUB_EQ] THEN
10197 let CLOSURE_INTERS_CONVEX_OPEN = prove
10198 (`!f:(real^N->bool)->bool.
10199 (!s. s IN f ==> convex s /\ open s)
10200 ==> closure(INTERS f) =
10201 if INTERS f = {} then {}
10202 else INTERS(IMAGE closure f)`,
10203 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CLOSURE_EMPTY] THEN
10204 MATCH_MP_TAC CLOSURE_INTERS_CONVEX THEN ASM_SIMP_TAC[] THEN
10205 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
10206 `~(s = {}) ==> s = t ==> ~(t = {})`)) THEN
10207 AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE
10208 `(!x. x IN s ==> f x = x) ==> s = IMAGE f s`) THEN
10209 ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN; INTERIOR_EQ]);;
10211 let CLOSURE_INTER_CONVEX = prove
10212 (`!s t:real^N->bool.
10213 convex s /\ convex t /\
10214 ~(relative_interior s INTER relative_interior t = {})
10215 ==> closure(s INTER t) = closure(s) INTER closure(t)`,
10216 REPEAT STRIP_TAC THEN
10217 MP_TAC(ISPEC `{s:real^N->bool,t}` CLOSURE_INTERS_CONVEX) THEN
10218 ASM_SIMP_TAC[IMAGE_CLAUSES; INTERS_2] THEN
10219 ASM_REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY]);;
10221 let CLOSURE_INTER_CONVEX_OPEN = prove
10222 (`!s t. convex s /\ open s /\ convex t /\ open t
10223 ==> closure(s INTER t) =
10224 if s INTER t = {} then {} else closure(s) INTER closure(t)`,
10225 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CLOSURE_EMPTY] THEN
10226 MATCH_MP_TAC CLOSURE_INTER_CONVEX THEN
10227 ASM_SIMP_TAC[RELATIVE_INTERIOR_OPEN]);;
10229 let CLOSURE_CONVEX_INTER_SUPERSET = prove
10230 (`!s t:real^N->bool.
10231 convex s /\ ~(interior s = {}) /\ interior s SUBSET closure t
10232 ==> closure(s INTER t) = closure s`,
10233 REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
10234 SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET; SUBSET_INTER] THEN
10235 MATCH_MP_TAC SUBSET_TRANS THEN
10236 EXISTS_TAC `closure(interior s):real^N->bool` THEN CONJ_TAC THENL
10237 [ASM_SIMP_TAC[CONVEX_CLOSURE_INTERIOR; SUBSET_REFL];
10238 ASM_SIMP_TAC[GSYM CLOSURE_OPEN_INTER_SUPERSET; OPEN_INTERIOR] THEN
10239 MATCH_MP_TAC SUBSET_CLOSURE THEN
10240 MP_TAC(ISPEC `s:real^N->bool` INTERIOR_SUBSET) THEN SET_TAC[]]);;
10242 let CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET = prove
10244 convex s /\ ~(interior s = {})
10245 ==> closure(s INTER
10246 { inv(&2 pow n) % x | n,x |
10247 !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }) =
10249 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_CONVEX_INTER_SUPERSET THEN
10250 ASM_REWRITE_TAC[CLOSURE_DYADIC_RATIONALS; SUBSET_UNIV]);;
10252 let CLOSURE_RATIONALS_IN_CONVEX_SET = prove
10254 convex s /\ ~(interior s = {})
10255 ==> closure(s INTER
10256 { x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) }) =
10258 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_CONVEX_INTER_SUPERSET THEN
10259 ASM_REWRITE_TAC[CLOSURE_RATIONAL_COORDINATES; SUBSET_UNIV]);;
10261 let RELATIVE_INTERIOR_CONVEX_INTER_AFFINE = prove
10262 (`!s t:real^N->bool.
10263 convex s /\ affine t /\ ~(interior s INTER t = {})
10264 ==> relative_interior(s INTER t) = interior s INTER t`,
10265 REPEAT GEN_TAC THEN
10266 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; RIGHT_AND_EXISTS_THM] THEN
10267 DISCH_THEN(X_CHOOSE_THEN `a:real^N` MP_TAC) THEN
10268 GEOM_ORIGIN_TAC `a:real^N` THEN REWRITE_TAC[IN_INTER] THEN
10269 REPEAT GEN_TAC THEN ASM_CASES_TAC `(vec 0:real^N) IN t` THEN
10270 ASM_SIMP_TAC[AFFINE_EQ_SUBSPACE] THEN STRIP_TAC THEN
10271 GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `x:real^N` THEN
10272 MP_TAC(ISPECL [`t:real^N->bool`; `s:real^N->bool`]
10273 (ONCE_REWRITE_RULE[INTER_COMM]
10274 AFFINE_HULL_AFFINE_INTER_NONEMPTY_INTERIOR)) THEN
10275 ASM_SIMP_TAC[SUBSPACE_IMP_AFFINE; IN_RELATIVE_INTERIOR_CBALL] THEN
10276 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_INTER; IN_INTERIOR_CBALL]] THEN
10277 DISCH_THEN SUBST1_TAC THEN
10278 ASM_CASES_TAC `(x:real^N) IN t` THEN ASM_REWRITE_TAC[] THEN
10279 SIMP_TAC[SUBSET; IN_INTER] THEN
10280 ASM_CASES_TAC `(x:real^N) IN s` THENL
10281 [ASM_REWRITE_TAC[]; ASM_MESON_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE]] THEN
10282 EQ_TAC THENL [REWRITE_TAC[IN_CBALL]; MESON_TAC[]] THEN
10283 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
10284 ASM_CASES_TAC `x:real^N = vec 0` THENL
10285 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN
10286 ASM_REWRITE_TAC[SUBSET; IN_CBALL];
10289 [`s:real^N->bool`; `vec 0:real^N`; `(&1 + e / norm x) % x:real^N`]
10290 IN_INTERIOR_CLOSURE_CONVEX_SEGMENT) THEN
10291 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
10292 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN
10293 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[SUBSPACE_MUL] THEN
10294 REWRITE_TAC[VECTOR_ADD_RDISTRIB; VECTOR_MUL_LID;
10295 NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN
10296 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM;
10297 REAL_DIV_RMUL; NORM_EQ_0] THEN
10298 ASM_REAL_ARITH_TAC;
10299 REWRITE_TAC[SUBSET; IN_INTERIOR_CBALL; IN_CBALL] THEN
10300 DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_SEGMENT] THEN
10301 CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
10302 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
10303 REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
10304 EXISTS_TAC `inv(&1 + e / norm(x:real^N))` THEN
10305 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_LT_DIV; NORM_POS_LT; VECTOR_MUL_LID;
10306 REAL_LT_INV_EQ; REAL_MUL_LINV; REAL_INV_LT_1; REAL_ARITH
10307 `&0 < x ==> &1 < &1 + x /\ &0 < &1 + x /\ ~(&1 + x = &0)`]]);;
10309 (* ------------------------------------------------------------------------- *)
10310 (* Homeomorphism of all convex compact sets with same affine dimension, and *)
10311 (* in particular all those with nonempty interior. *)
10312 (* ------------------------------------------------------------------------- *)
10314 let COMPACT_FRONTIER_LINE_LEMMA = prove
10315 (`!s x. compact s /\ (vec 0 IN s) /\ ~(x = vec 0 :real^N)
10316 ==> ?u. &0 <= u /\ (u % x) IN frontier s /\
10317 !v. u < v ==> ~((v % x) IN s)`,
10318 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
10319 REWRITE_TAC[BOUNDED_POS] THEN
10320 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN
10322 [`{y:real^N | ?u. &0 <= u /\ u <= b / norm(x) /\ (y = u % x)} INTER s`;
10324 DISTANCE_ATTAINS_SUP) THEN
10328 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `vec 0:real^N` THEN
10329 ASM_REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN
10330 EXISTS_TAC `&0` THEN
10331 ASM_SIMP_TAC[VECTOR_MUL_LZERO; REAL_LE_REFL; REAL_LT_IMP_LE;
10332 REAL_LT_DIV; NORM_POS_LT]] THEN
10333 MATCH_MP_TAC COMPACT_INTER THEN ASM_REWRITE_TAC[] THEN
10335 `{y:real^N | ?u. &0 <= u /\ u <= b / norm(x) /\ (y = u % x)} =
10336 IMAGE (\u. drop u % x) (interval [vec 0,lambda i. b / norm(x:real^N)])`
10338 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; IN_INTERVAL] THEN
10339 SIMP_TAC[LAMBDA_BETA] THEN
10340 SIMP_TAC[DIMINDEX_1; ARITH_RULE `1 <= i /\ i <= 1 <=> (i = 1)`] THEN
10341 REWRITE_TAC[GSYM drop; LEFT_FORALL_IMP_THM; EXISTS_REFL; DROP_VEC] THEN
10342 REWRITE_TAC[EXISTS_LIFT; LIFT_DROP] THEN MESON_TAC[];
10344 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
10345 REWRITE_TAC[COMPACT_INTERVAL] THEN
10346 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REPEAT STRIP_TAC THEN
10347 MATCH_MP_TAC CONTINUOUS_VMUL THEN
10348 REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_AT_ID];
10350 REWRITE_TAC[IN_INTER; IN_ELIM_THM; LEFT_AND_EXISTS_THM] THEN
10351 ONCE_REWRITE_TAC[TAUT `(a /\ b /\ c) /\ d <=> c /\ a /\ b /\ d`] THEN
10352 SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
10353 GEN_REWRITE_TAC (BINDER_CONV o ONCE_DEPTH_CONV) [SWAP_FORALL_THM] THEN
10354 SIMP_TAC[IMP_CONJ] THEN
10355 REWRITE_TAC[LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
10356 REWRITE_TAC[IMP_IMP] THEN REWRITE_TAC[LEFT_FORALL_IMP_THM] THEN
10357 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real` THEN
10358 REWRITE_TAC[dist; VECTOR_SUB_LZERO; NORM_NEG; NORM_MUL] THEN
10359 ASM_SIMP_TAC[REAL_LE_RMUL_EQ; NORM_POS_LT] THEN
10360 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
10361 ASM_SIMP_TAC[real_abs] THEN REPEAT STRIP_TAC THENL
10362 [REWRITE_TAC[FRONTIER_STRADDLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
10364 [EXISTS_TAC `u % x :real^N` THEN ASM_REWRITE_TAC[DIST_REFL];
10366 EXISTS_TAC `(u + (e / &2) / norm(x)) % x :real^N` THEN
10367 REWRITE_TAC[dist; VECTOR_ARITH `u % x - (u + a) % x = --(a % x)`] THEN
10368 ASM_SIMP_TAC[NORM_NEG; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; NORM_EQ_0;
10369 REAL_DIV_RMUL; REAL_ABS_NUM; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT;
10370 ARITH; REAL_ARITH `abs e < e * &2 <=> &0 < e`] THEN
10372 FIRST_X_ASSUM(MP_TAC o SPEC `u + (e / &2) / norm(x:real^N)`) THEN
10373 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
10374 `&0 < e /\ &0 <= u /\ u + e <= b
10375 ==> ~(&0 <= u + e /\ u + e <= b ==> u + e <= u)`) THEN
10376 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NORM_POS_LT] THEN
10377 FIRST_X_ASSUM(MP_TAC o SPEC `(u + (e / &2) / norm(x:real^N)) % x`) THEN
10378 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
10381 FIRST_X_ASSUM(MP_TAC o SPEC `v:real`) THEN
10382 ASM_REWRITE_TAC[GSYM REAL_NOT_LT] THEN ASM_REWRITE_TAC[REAL_NOT_LT] THEN
10383 CONJ_TAC THENL [ASM_MESON_TAC[REAL_LET_TRANS; REAL_LT_IMP_LE]; ALL_TAC] THEN
10384 FIRST_X_ASSUM(MP_TAC o SPEC `v % x:real^N`) THEN
10385 ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LE_RDIV_EQ; NORM_POS_LT] THEN
10388 let STARLIKE_COMPACT_PROJECTIVE = prove
10389 (`!s:real^N->bool a.
10390 compact s /\ a IN relative_interior s /\
10391 (!x. x IN s ==> segment(a,x) SUBSET relative_interior s)
10392 ==> s DIFF relative_interior s homeomorphic
10393 sphere(a,&1) INTER affine hull s /\
10394 s homeomorphic cball(a,&1) INTER affine hull s`,
10395 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN
10396 REWRITE_TAC[SUBSET; IMP_IMP; RIGHT_IMP_FORALL_THM] THEN
10397 GEN_TAC THEN STRIP_TAC THEN
10399 `!x:real^N u. x IN s /\ &0 <= u /\ u < &1
10400 ==> (u % x) IN relative_interior s`
10402 [REWRITE_TAC[REAL_ARITH `&0 <= u <=> u = &0 \/ &0 < u`] THEN
10403 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
10404 ASM_CASES_TAC `x:real^N = vec 0` THEN
10405 ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
10406 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_SEGMENT] THEN
10407 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN ASM_MESON_TAC[];
10408 FIRST_X_ASSUM(K ALL_TAC o SPECL [`x:real^N`; `x:real^N`])] THEN
10409 FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET]
10410 RELATIVE_INTERIOR_SUBSET)) THEN
10411 ABBREV_TAC `proj = \x:real^N. inv(norm(x)) % x` THEN
10413 `!x:real^N y. (proj(x) = proj(y):real^N) /\ (norm x = norm y) <=> (x = y)`
10415 [REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
10416 ASM_CASES_TAC `y:real^N = vec 0` THEN
10417 ASM_SIMP_TAC[NORM_EQ_0; NORM_0] THEN
10418 ASM_CASES_TAC `x:real^N = vec 0` THENL
10419 [ASM_MESON_TAC[NORM_EQ_0]; ALL_TAC] THEN
10420 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
10421 EXPAND_TAC "proj" THEN REWRITE_TAC[] THEN
10422 ASM_REWRITE_TAC[VECTOR_ARITH
10423 `a % x = a % y <=> a % (x - y):real^N = vec 0`] THEN
10424 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0; VECTOR_SUB_EQ];
10427 `(!x. x IN affine hull s ==> proj x IN affine hull s) /\
10428 (!x. ~(x = vec 0) ==> norm(proj x) = &1) /\
10429 (!x:real^N. proj x = vec 0 <=> x = vec 0)`
10430 STRIP_ASSUME_TAC THENL
10431 [EXPAND_TAC "proj" THEN REWRITE_TAC[NORM_MUL; VECTOR_MUL_EQ_0] THEN
10432 REWRITE_TAC[REAL_INV_EQ_0; NORM_EQ_0; REAL_ABS_INV; REAL_ABS_NORM] THEN
10433 SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0] THEN REPEAT STRIP_TAC THEN
10434 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN
10435 MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
10436 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID; HULL_INC];
10438 SUBGOAL_THEN `(proj:real^N->real^N) continuous_on (UNIV DELETE vec 0)`
10440 [MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
10441 REWRITE_TAC[IN_DELETE; IN_UNIV] THEN EXPAND_TAC "proj" THEN
10442 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
10443 ASM_SIMP_TAC[CONTINUOUS_AT_ID] THEN
10444 REWRITE_TAC[GSYM(ISPEC `lift` o_DEF);
10445 GSYM(ISPEC `inv:real->real` o_DEF)] THEN
10446 MATCH_MP_TAC CONTINUOUS_AT_INV THEN
10447 ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ; CONTINUOUS_AT_LIFT_NORM];
10449 ABBREV_TAC `usph = {x:real^N | x IN affine hull s /\ norm x = &1}` THEN
10450 SUBGOAL_THEN ` sphere(vec 0:real^N,&1) INTER affine hull s = usph`
10452 [EXPAND_TAC "usph" THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_SPHERE_0] THEN
10456 `!x. x IN affine hull s /\ ~(x = vec 0)
10457 ==> (proj:real^N->real^N) x IN usph`
10458 ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
10459 SUBGOAL_THEN `?surf. homeomorphism (s DIFF relative_interior s,usph)
10460 (proj:real^N->real^N,surf)`
10462 [MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN
10463 ASM_SIMP_TAC[COMPACT_RELATIVE_BOUNDARY] THEN REPEAT CONJ_TAC THENL
10464 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10465 CONTINUOUS_ON_SUBSET)) THEN
10467 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
10468 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_DIFF] THEN
10469 EXPAND_TAC "usph" THEN REWRITE_TAC[IN_ELIM_THM] THEN
10470 ASM_MESON_TAC[HULL_INC];
10471 MAP_EVERY EXPAND_TAC ["proj"; "usph"] THEN
10472 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN
10473 ASM_CASES_TAC `x:real^N = vec 0` THEN
10474 ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN STRIP_TAC THEN
10475 MP_TAC(ISPECL [`s:real^N->bool`; `vec 0:real^N`; `x:real^N`]
10476 RAY_TO_RELATIVE_FRONTIER) THEN
10477 REWRITE_TAC[relative_frontier] THEN
10478 ASM_SIMP_TAC[COMPACT_IMP_BOUNDED; CLOSURE_CLOSED; COMPACT_IMP_CLOSED;
10479 VECTOR_ADD_LID] THEN
10480 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
10481 EXPAND_TAC "proj" THEN REWRITE_TAC[IN_IMAGE] THEN
10482 EXISTS_TAC `d % x:real^N` THEN ASM_REWRITE_TAC[NORM_MUL] THEN
10483 ASM_SIMP_TAC[REAL_MUL_RID; real_abs; REAL_LT_IMP_LE] THEN
10484 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ;
10486 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
10487 REWRITE_TAC[IN_DIFF] THEN STRIP_TAC THEN
10488 ASM_CASES_TAC `x:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN
10489 ASM_CASES_TAC `y:real^N = vec 0` THENL [ASM SET_TAC[]; ALL_TAC] THEN
10490 UNDISCH_TAC `(proj:real^N->real^N) x = proj y` THEN
10491 EXPAND_TAC "proj" THEN
10492 REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH
10493 `norm(x:real^N) = norm(y:real^N) \/
10494 norm x < norm y \/ norm y < norm x`)
10496 [ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL; REAL_INV_EQ_0; NORM_EQ_0];
10497 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
10498 [`y:real^N`; `norm(x:real^N) / norm(y:real^N)`]);
10499 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
10500 [`x:real^N`; `norm(y:real^N) / norm(x:real^N)`])] THEN
10501 ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_LDIV_EQ; NORM_POS_LT;
10503 ASM_REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THENL
10504 [FIRST_X_ASSUM(SUBST1_TAC o SYM); ALL_TAC] THEN
10505 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; NORM_EQ_0] THEN
10506 ASM_REWRITE_TAC[VECTOR_MUL_LID]];
10507 DISCH_THEN(fun th ->
10509 [MESON_TAC[homeomorphic; th];
10510 ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
10511 MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN
10512 SIMP_TAC[COMPACT_INTER_CLOSED; CLOSED_AFFINE_HULL; COMPACT_CBALL] THEN
10514 REWRITE_TAC[HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN
10515 X_GEN_TAC `surf:real^N->real^N` THEN STRIP_TAC THEN
10516 EXISTS_TAC `\x:real^N. norm(x) % (surf:real^N->real^N)(proj(x))` THEN
10517 REWRITE_TAC[]] THEN
10519 `(proj:real^N->real^N) continuous_on s DIFF relative_interior s`
10521 REWRITE_TAC[] THEN CONJ_TAC THENL
10522 [REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; IN_INTER] THEN
10523 X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
10524 ASM_CASES_TAC `x = vec 0:real^N` THENL
10525 [ASM_REWRITE_TAC[CONTINUOUS_WITHIN; VECTOR_MUL_LZERO; NORM_0] THEN
10526 MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN
10527 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
10528 REWRITE_TAC[BOUNDED_POS] THEN MATCH_MP_TAC MONO_EXISTS THEN
10529 REPEAT STRIP_TAC THENL
10530 [REWRITE_TAC[LIM_WITHIN; o_THM; DIST_0; NORM_LIFT; REAL_ABS_NORM] THEN
10532 REWRITE_TAC[EVENTUALLY_WITHIN] THEN EXISTS_TAC `&1` THEN
10533 REWRITE_TAC[REAL_LT_01; IN_INTER; DIST_0; NORM_POS_LT] THEN
10535 MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
10536 EXISTS_TAC `affine hull s:real^N->bool` THEN
10537 REWRITE_TAC[INTER_SUBSET] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
10538 SIMP_TAC[CONTINUOUS_LIFT_NORM_COMPOSE; CONTINUOUS_WITHIN_ID; o_DEF] THEN
10540 `((surf:real^N->real^N) o (proj:real^N->real^N)) continuous_on
10541 (affine hull s DELETE vec 0)`
10543 [MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
10544 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN
10545 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10546 CONTINUOUS_ON_SUBSET)) THEN
10547 SIMP_TAC[SUBSET; IN_DELETE; IN_UNIV; FORALL_IN_IMAGE] THEN
10548 EXPAND_TAC "usph" THEN ASM_SIMP_TAC[IN_ELIM_THM];
10549 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
10550 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_SIMP_TAC[IN_DELETE] THEN
10551 REWRITE_TAC[CONTINUOUS_WITHIN; o_DEF] THEN MATCH_MP_TAC EQ_IMP THEN
10552 MATCH_MP_TAC LIM_TRANSFORM_WITHIN_SET THEN
10553 REWRITE_TAC[EVENTUALLY_AT] THEN EXISTS_TAC `norm(x:real^N)` THEN
10554 ASM_REWRITE_TAC[IN_DELETE; IN_INTER; IN_CBALL; NORM_POS_LT] THEN
10555 X_GEN_TAC `y:real^N` THEN
10556 ASM_CASES_TAC `(y:real^N) IN affine hull s` THEN ASM_REWRITE_TAC[] THEN
10557 CONV_TAC NORM_ARITH]];
10560 `!a x. &0 < a ==> (proj:real^N->real^N)(a % x) = proj x`
10562 [REPEAT GEN_TAC THEN EXPAND_TAC "proj" THEN
10563 REWRITE_TAC[NORM_MUL; REAL_INV_MUL; VECTOR_MUL_ASSOC] THEN
10564 SIMP_TAC[REAL_FIELD `&0 < a ==> (inv(a) * x) * a = x`; real_abs;
10569 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
10570 ASM_CASES_TAC `y:real^N = vec 0` THENL
10571 [ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_EQ_0; NORM_0; NORM_EQ_0] THEN
10574 ASM_CASES_TAC `x:real^N = vec 0` THENL
10575 [CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
10576 ASM_SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_MUL_EQ_0; NORM_0; NORM_EQ_0] THEN
10579 REWRITE_TAC[IN_INTER; IN_CBALL_0] THEN
10580 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
10581 DISCH_THEN(fun th -> MP_TAC th THEN
10582 MP_TAC(AP_TERM `proj:real^N->real^N` th)) THEN
10583 ASM_SIMP_TAC[NORM_POS_LT; VECTOR_MUL_RCANCEL] THEN ASM SET_TAC[]] THEN
10584 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
10585 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_CBALL_0] THEN
10586 X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
10587 ASM_REWRITE_TAC[NORM_0; VECTOR_MUL_LZERO; IN_INTER] THEN
10588 REWRITE_TAC[IN_CBALL_0; REAL_LE_LT] THEN STRIP_TAC THENL
10589 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] RELATIVE_INTERIOR_SUBSET) THEN
10590 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[NORM_POS_LE] THEN
10592 ASM_REWRITE_TAC[VECTOR_MUL_LID] THEN ASM SET_TAC[]];
10594 REWRITE_TAC[SUBSET; IN_IMAGE; IN_CBALL_0; IN_INTER] THEN
10595 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
10596 ASM_CASES_TAC `x:real^N = vec 0` THENL
10597 [EXISTS_TAC `vec 0:real^N` THEN
10598 ASM_SIMP_TAC[NORM_0; VECTOR_MUL_LZERO; HULL_INC; REAL_POS];
10601 `!x. x IN usph ==> ~((surf:real^N->real^N) x = vec 0)`
10602 ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
10603 EXISTS_TAC `inv(norm(surf(proj x:real^N):real^N)) % x:real^N` THEN
10604 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [GSYM th]) THEN
10605 REWRITE_TAC[GSYM CONJ_ASSOC] THEN
10606 ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
10607 [NORM_POS_LT; REAL_LT_INV_EQ; HULL_INC; REAL_LT_MUL; NORM_MUL;
10608 REAL_ABS_INV; REAL_ABS_NORM] THEN
10609 REPEAT CONJ_TAC THENL
10610 [MATCH_MP_TAC(REAL_FIELD `~(y = &0) ==> x = (inv y * x) * y`) THEN
10611 ASM_SIMP_TAC[NORM_EQ_0; HULL_INC];
10612 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
10613 ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
10614 [GSYM real_div; REAL_LE_LDIV_EQ; NORM_POS_LT; HULL_INC; REAL_MUL_LID] THEN
10615 FIRST_X_ASSUM(MP_TAC o SPECL
10616 [`x:real^N`; `norm(surf(proj x:real^N):real^N) / norm(x:real^N)`]) THEN
10617 ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_LDIV_EQ; NORM_POS_LT] THEN
10618 GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN
10619 REWRITE_TAC[REAL_NOT_LT; REAL_MUL_LID] THEN DISCH_THEN MATCH_MP_TAC THEN
10621 `norm(surf(proj x)) / norm x % x:real^N = surf(proj x:real^N)`
10623 [FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC I [GSYM th]) THEN
10624 ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
10625 [NORM_POS_LT; REAL_LT_INV_EQ; HULL_INC; REAL_LT_MUL; NORM_MUL;
10626 REAL_ABS_INV; REAL_ABS_NORM; REAL_ABS_DIV; REAL_LT_DIV;
10627 REAL_DIV_RMUL; NORM_EQ_0];
10628 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
10629 `IMAGE f s SUBSET t DIFF u ==> x IN s ==> ~(f x IN u)`)) THEN
10630 ASM_SIMP_TAC[HULL_INC]];
10631 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN
10632 MATCH_MP_TAC IN_AFFINE_ADD_MUL THEN
10633 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; VECTOR_ADD_LID; HULL_INC]]);;
10635 let HOMEOMORPHIC_CONVEX_COMPACT_SETS,
10636 HOMEOMORPHIC_RELATIVE_FRONTIERS_CONVEX_BOUNDED_SETS = (CONJ_PAIR o prove)
10637 (`(!s:real^M->bool t:real^N->bool.
10638 convex s /\ compact s /\ convex t /\ compact t /\ aff_dim s = aff_dim t
10639 ==> s homeomorphic t) /\
10640 (!s:real^M->bool t:real^N->bool.
10641 convex s /\ bounded s /\ convex t /\ bounded t /\ aff_dim s = aff_dim t
10642 ==> relative_frontier s homeomorphic relative_frontier t)`,
10644 (`!s:real^M->bool t:real^N->bool.
10645 convex s /\ compact s /\ convex t /\ compact t /\
10646 aff_dim s = aff_dim t
10647 ==> (s DIFF relative_interior s) homeomorphic
10648 (t DIFF relative_interior t) /\
10650 REPEAT GEN_TAC THEN
10651 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
10652 ASM_CASES_TAC `relative_interior t:real^N->bool = {}` THENL
10653 [UNDISCH_TAC `relative_interior t:real^N->bool = {}` THEN
10654 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1;
10655 EMPTY_DIFF; HOMEOMORPHIC_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY];
10656 FIRST_X_ASSUM(X_CHOOSE_THEN `b:real^N` MP_TAC o
10657 GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN
10658 CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
10659 ASM_CASES_TAC `relative_interior s:real^M->bool = {}` THENL
10660 [UNDISCH_TAC `relative_interior s:real^M->bool = {}` THEN
10661 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_EQ_MINUS1;
10662 EMPTY_DIFF; HOMEOMORPHIC_EMPTY; RELATIVE_INTERIOR_EQ_EMPTY];
10663 FIRST_X_ASSUM(X_CHOOSE_THEN `a:real^M` MP_TAC o
10664 GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY])] THEN
10665 REPEAT(POP_ASSUM MP_TAC) THEN
10666 GEOM_ORIGIN_TAC `b:real^N` THEN REPEAT GEN_TAC THEN
10667 GEOM_ORIGIN_TAC `a:real^M` THEN REPEAT GEN_TAC THEN
10668 REPEAT DISCH_TAC THEN
10669 MP_TAC(ISPECL [`s:real^M->bool`; `vec 0:real^M`]
10670 STARLIKE_COMPACT_PROJECTIVE) THEN
10671 MP_TAC(ISPECL [`t:real^N->bool`; `vec 0:real^N`]
10672 STARLIKE_COMPACT_PROJECTIVE) THEN
10673 ASM_SIMP_TAC[IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT;
10674 REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN
10675 DISCH_THEN(fun th -> MATCH_MP_TAC MONO_AND THEN MP_TAC th) THEN
10676 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
10677 DISCH_THEN(fun th ->
10678 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHIC_TRANS) THEN
10679 MP_TAC(ONCE_REWRITE_RULE[HOMEOMORPHIC_SYM] th)) THEN
10680 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] HOMEOMORPHIC_TRANS) THEN
10681 REPEAT(FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[SUBSET]
10682 RELATIVE_INTERIOR_SUBSET))) THEN
10683 FIRST_X_ASSUM(MP_TAC o SYM) THEN
10684 ASM_SIMP_TAC[AFFINE_HULL_EQ_SPAN; HULL_INC; AFF_DIM_DIM_0] THEN
10685 REWRITE_TAC[INT_OF_NUM_EQ] THEN DISCH_TAC THEN
10686 MP_TAC(ISPECL [`span s:real^M->bool`; `span t:real^N->bool`]
10687 ISOMETRIES_SUBSPACES) THEN
10688 ASM_REWRITE_TAC[SUBSPACE_SPAN; DIM_SPAN; homeomorphic; HOMEOMORPHISM] THEN
10689 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN
10690 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN
10691 SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_CBALL_0; IN_SPHERE_0] THEN
10692 SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN ASM SET_TAC[]) in
10693 SIMP_TAC[lemma; relative_frontier] THEN REPEAT STRIP_TAC THEN
10694 MP_TAC(SPECL [`closure s:real^M->bool`; `closure t:real^N->bool`] lemma) THEN
10695 ASM_SIMP_TAC[CONVEX_CLOSURE; COMPACT_CLOSURE; AFF_DIM_CLOSURE] THEN
10696 ASM_SIMP_TAC[CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
10698 let HOMEOMORPHIC_CONVEX_COMPACT = prove
10699 (`!s:real^N->bool t:real^N->bool.
10700 convex s /\ compact s /\ ~(interior s = {}) /\
10701 convex t /\ compact t /\ ~(interior t = {})
10702 ==> s homeomorphic t`,
10703 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT_SETS THEN
10704 ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR]);;
10706 let HOMEOMORPHIC_CONVEX_COMPACT_CBALL = prove
10707 (`!s:real^N->bool b:real^N e.
10708 convex s /\ compact s /\ ~(interior s = {}) /\ &0 < e
10709 ==> s homeomorphic cball(b,e)`,
10710 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN
10711 ASM_REWRITE_TAC[COMPACT_CBALL; INTERIOR_CBALL; CONVEX_CBALL] THEN
10712 ASM_REWRITE_TAC[BALL_EQ_EMPTY; REAL_NOT_LE]);;
10714 let HOMEOMORPHIC_CLOSED_INTERVALS = prove
10715 (`!a b:real^N c d:real^N.
10716 ~(interval(a,b) = {}) /\ ~(interval(c,d) = {})
10717 ==> interval[a,b] homeomorphic interval[c,d]`,
10718 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_CONVEX_COMPACT THEN
10719 REWRITE_TAC[CONVEX_INTERVAL; COMPACT_INTERVAL] THEN
10720 ASM_REWRITE_TAC[INTERIOR_CLOSED_INTERVAL]);;
10722 (* ------------------------------------------------------------------------- *)
10723 (* More about affine dimension of special sets. *)
10724 (* ------------------------------------------------------------------------- *)
10726 let AFF_DIM_NONEMPTY_INTERIOR_EQ = prove
10728 convex s ==> (aff_dim s = &(dimindex (:N)) <=> ~(interior s = {}))`,
10729 REPEAT STRIP_TAC THEN EQ_TAC THEN
10730 ASM_SIMP_TAC[AFF_DIM_NONEMPTY_INTERIOR] THEN
10731 REPEAT STRIP_TAC THEN
10732 MP_TAC(ISPEC `s:real^N->bool` EMPTY_INTERIOR_SUBSET_HYPERPLANE) THEN
10733 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
10734 FIRST_ASSUM(MP_TAC o MATCH_MP AFF_DIM_SUBSET) THEN
10735 ASM_SIMP_TAC[AFF_DIM_HYPERPLANE] THEN INT_ARITH_TAC);;
10737 let AFF_DIM_BALL = prove
10739 aff_dim(ball(a,r)) = if &0 < r then &(dimindex(:N)) else --(&1)`,
10740 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
10741 [MATCH_MP_TAC AFF_DIM_OPEN THEN
10742 ASM_REWRITE_TAC[OPEN_BALL; BALL_EQ_EMPTY; REAL_NOT_LE];
10743 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT; GSYM BALL_EQ_EMPTY]) THEN
10744 ASM_REWRITE_TAC[AFF_DIM_EMPTY]]);;
10746 let AFF_DIM_CBALL = prove
10748 aff_dim(cball(a,r)) =
10749 if &0 < r then &(dimindex(:N))
10750 else if r = &0 then &0 else --(&1)`,
10751 REPEAT GEN_TAC THEN REPEAT COND_CASES_TAC THENL
10752 [MATCH_MP_TAC AFF_DIM_NONEMPTY_INTERIOR THEN
10753 ASM_REWRITE_TAC[INTERIOR_CBALL; BALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC;
10754 ASM_SIMP_TAC[CBALL_SING; AFF_DIM_SING];
10755 MATCH_MP_TAC(MESON[AFF_DIM_EMPTY] `s = {} ==> aff_dim s = --(&1)`) THEN
10756 REWRITE_TAC[CBALL_EQ_EMPTY] THEN ASM_REAL_ARITH_TAC]);;
10758 let AFF_DIM_INTERVAL = prove
10760 aff_dim(interval[a,b]) =
10761 if interval[a,b] = {} then --(&1)
10762 else &(CARD {i | 1 <= i /\ i <= dimindex(:N) /\ a$i < b$i})) /\
10764 aff_dim(interval(a,b)) =
10765 if interval(a,b) = {} then --(&1)
10766 else &(dimindex(:N)))`,
10767 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
10768 ASM_SIMP_TAC[AFF_DIM_EMPTY; AFF_DIM_OPEN; OPEN_INTERVAL] THEN
10769 POP_ASSUM MP_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN
10770 REPEAT STRIP_TAC THEN
10771 REWRITE_TAC[VECTOR_ADD_COMPONENT; VEC_COMPONENT; REAL_LT_LADD] THEN
10772 ASM_SIMP_TAC[AFF_DIM_DIM_0; HULL_INC; ENDS_IN_INTERVAL] THEN AP_TERM_TAC THEN
10773 ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN MATCH_MP_TAC DIM_UNIQUE THEN EXISTS_TAC
10774 `{basis i:real^N | 1 <= i /\ i <= dimindex(:N) /\ &0 < (b:real^N)$i}` THEN
10775 RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY; VEC_COMPONENT]) THEN
10776 REPEAT CONJ_TAC THENL
10777 [REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
10778 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
10779 SUBGOAL_THEN `basis i:real^N = inv(b$i) % (b:real^N)$i % basis i`
10781 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN
10782 REWRITE_TAC[VECTOR_MUL_LID];
10783 MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN
10784 SIMP_TAC[IN_INTERVAL; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN
10785 X_GEN_TAC `j:num` THEN REWRITE_TAC[VEC_COMPONENT] THEN
10786 COND_CASES_TAC THEN
10787 ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_MUL_RID; REAL_LE_REFL]];
10788 MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN
10789 REWRITE_TAC[SUBSPACE_SPAN; SUBSET; IN_INTERVAL; VEC_COMPONENT] THEN
10790 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
10791 GEN_REWRITE_TAC LAND_CONV [GSYM BASIS_EXPANSION] THEN
10792 MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG] THEN
10793 X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN
10794 ASM_CASES_TAC `&0 < (b:real^N)$i` THENL
10795 [MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[];
10796 SUBGOAL_THEN `(x:real^N)$i = &0`
10797 (fun th -> REWRITE_TAC[th; VECTOR_MUL_LZERO; SPAN_0]) THEN
10798 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN
10799 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC];
10800 MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN
10801 REWRITE_TAC[SET_RULE `~(a IN {f x | P x}) <=> !x. P x ==> ~(f x = a)`] THEN
10802 SIMP_TAC[BASIS_NONZERO; pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
10803 SIMP_TAC[FORALL_IN_GSPEC; BASIS_INJ_EQ; ORTHOGONAL_BASIS_BASIS];
10804 GEN_REWRITE_TAC LAND_CONV [SIMPLE_IMAGE_GEN] THEN
10805 MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN
10806 SIMP_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC; BASIS_INJ_EQ;
10808 SIMP_TAC[CONJ_ASSOC; GSYM IN_NUMSEG; FINITE_RESTRICT; FINITE_NUMSEG]]);;
10810 (* ------------------------------------------------------------------------- *)
10811 (* Deducing convexity from midpoint convexity in common cases. *)
10812 (* ------------------------------------------------------------------------- *)
10814 let MIDPOINT_CONVEX_DYADIC_RATIONALS = prove
10815 (`!f:real^N->real s.
10816 (!x y. x IN s /\ y IN s
10817 ==> midpoint(x,y) IN s /\
10818 f(midpoint(x,y)) <= (f(x) + f(y)) / &2)
10820 x IN s /\ y IN s /\ m + p = 2 EXP n
10821 ==> (&m / &2 pow n % x + &p / &2 pow n % y) IN s /\
10822 f(&m / &2 pow n % x + &p / &2 pow n % y)
10823 <= &m / &2 pow n * f x + &p / &2 pow n * f y`,
10824 REPEAT GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THENL
10825 [REWRITE_TAC[ARITH_RULE
10826 `m + p = 2 EXP 0 <=> m = 0 /\ p = 1 \/ m = 1 /\ p = 0`] THEN
10827 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
10828 CONV_TAC REAL_RAT_REDUCE_CONV THEN
10829 ASM_REWRITE_TAC[VECTOR_MUL_LID; VECTOR_MUL_LZERO;
10830 VECTOR_ADD_LID; VECTOR_ADD_RID] THEN
10832 MATCH_MP_TAC WLOG_LE THEN CONJ_TAC THENL
10833 [REWRITE_TAC[VECTOR_ADD_SYM; REAL_ADD_SYM; ADD_SYM] THEN MESON_TAC[];
10835 MAP_EVERY X_GEN_TAC [`m:num`; `p:num`] THEN DISCH_TAC THEN
10836 REPEAT GEN_TAC THEN REWRITE_TAC[EXP; real_pow] THEN STRIP_TAC THEN
10837 REWRITE_TAC[real_div; REAL_INV_MUL] THEN
10838 ONCE_REWRITE_TAC[REAL_ARITH `x * inv(&2) * y = inv(&2) * x * y`] THEN
10839 ONCE_REWRITE_TAC[GSYM REAL_MUL_ASSOC; GSYM VECTOR_MUL_ASSOC] THEN
10840 REWRITE_TAC[GSYM REAL_ADD_LDISTRIB; GSYM VECTOR_ADD_LDISTRIB] THEN
10841 SUBGOAL_THEN `2 EXP n <= p` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
10842 SUBGOAL_THEN `&p * inv(&2 pow n) = &(p - 2 EXP n) * inv(&2 pow n) + &1`
10844 [ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW] THEN
10845 ASM_SIMP_TAC[REAL_SUB_RDISTRIB; REAL_MUL_RINV; REAL_LT_IMP_NZ;
10846 REAL_LT_POW2] THEN REAL_ARITH_TAC;
10847 REWRITE_TAC[VECTOR_ADD_RDISTRIB; REAL_ADD_RDISTRIB] THEN
10848 REWRITE_TAC[VECTOR_MUL_LID; REAL_MUL_LID] THEN
10849 REWRITE_TAC[VECTOR_ADD_ASSOC; REAL_ADD_ASSOC] THEN
10850 REWRITE_TAC[GSYM midpoint; GSYM real_div] THEN FIRST_X_ASSUM(fun th ->
10851 W(MP_TAC o PART_MATCH (lhand o rand) th o lhand o snd)) THEN
10852 FIRST_X_ASSUM(fun th ->
10853 W(MP_TAC o PART_MATCH (lhand o rand) th o funpow 3 lhand o snd)) THEN
10854 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
10855 [ASM_ARITH_TAC; SIMP_TAC[] THEN REAL_ARITH_TAC]]]);;
10857 let CONTINUOUS_MIDPOINT_CONVEX = prove
10858 (`!f:real^N->real s.
10859 (lift o f) continuous_on s /\ convex s /\
10860 (!x y. x IN s /\ y IN s ==> f(midpoint(x,y)) <= (f(x) + f(y)) / &2)
10861 ==> f convex_on s`,
10862 REWRITE_TAC[midpoint] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[convex_on] THEN
10863 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
10864 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
10865 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
10866 REWRITE_TAC[REAL_ARITH `u + v = &1 <=> v = &1 - u`; IMP_CONJ] THEN
10867 REWRITE_TAC[FORALL_UNWIND_THM2; REAL_SUB_LE] THEN
10868 REWRITE_TAC[FORALL_DROP; GSYM DROP_VEC; IMP_IMP; GSYM IN_INTERVAL_1] THEN
10869 MP_TAC(ISPEC `interval[vec 0:real^1,vec 1]`
10870 CLOSURE_DYADIC_RATIONALS_IN_CONVEX_SET) THEN
10871 SIMP_TAC[CONVEX_INTERVAL; INTERIOR_CLOSED_INTERVAL;
10872 CLOSURE_CLOSED; CLOSED_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN
10873 REWRITE_TAC[DIMINDEX_1; FORALL_1; GSYM drop] THEN
10874 DISCH_THEN(fun th -> SUBST1_TAC(SYM th) THEN ASSUME_TAC th) THEN
10875 ONCE_REWRITE_TAC[REAL_ARITH `a <= b <=> a - b <= &0`] THEN
10876 MATCH_MP_TAC CONTINUOUS_LE_ON_CLOSURE THEN
10877 REWRITE_TAC[IN_INTER; IMP_CONJ_ALT; FORALL_IN_GSPEC] THEN
10878 FIRST_X_ASSUM SUBST1_TAC THEN
10879 REWRITE_TAC[IN_INTERVAL_1; DROP_CMUL; GSYM FORALL_DROP; DROP_VEC] THEN
10881 [REWRITE_TAC[o_DEF; LIFT_SUB; LIFT_ADD; LIFT_CMUL] THEN
10882 MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THENL
10883 [REPLICATE_TAC 2 (ONCE_REWRITE_TAC[GSYM o_DEF]) THEN
10884 REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
10887 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10888 CONTINUOUS_ON_SUBSET)) THEN
10889 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; GSYM FORALL_DROP;
10890 DROP_VEC] THEN REPEAT STRIP_TAC THEN
10891 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
10892 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC];
10894 MATCH_MP_TAC CONTINUOUS_ON_ADD THEN CONJ_TAC THEN
10895 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
10896 SIMP_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST;
10897 LIFT_SUB; CONTINUOUS_ON_SUB];
10898 MAP_EVERY X_GEN_TAC [`n:num`; `i:real`] THEN
10899 ASM_SIMP_TAC[REAL_LE_MUL_EQ; REAL_LT_INV_EQ; REAL_LT_POW2] THEN
10900 ASM_CASES_TAC `&0 <= i` THEN ASM_SIMP_TAC[INTEGER_POS] THEN
10901 DISCH_THEN(X_CHOOSE_THEN `m:num` SUBST_ALL_TAC) THEN
10902 REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] (GSYM real_div)] THEN
10903 SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_POW2; REAL_MUL_LID] THEN
10904 GEN_REWRITE_TAC (LAND_CONV o DEPTH_CONV)
10905 [REAL_OF_NUM_POW; REAL_OF_NUM_LE] THEN DISCH_TAC THEN
10906 MP_TAC(ISPECL [`f:real^N->real`; `s:real^N->bool`]
10907 MIDPOINT_CONVEX_DYADIC_RATIONALS) THEN
10909 [ASM_SIMP_TAC[midpoint] THEN REWRITE_TAC[VECTOR_ADD_LDISTRIB] THEN
10910 REPEAT STRIP_TAC THEN
10911 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [convex]) THEN
10912 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
10913 DISCH_THEN(MP_TAC o SPECL
10914 [`n:num`; `m:num`; `2 EXP n - m`; `x:real^N`; `y:real^N`]) THEN
10915 ASM_REWRITE_TAC[] THEN
10916 ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT2)] THEN
10917 ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_POW] THEN
10918 ASM_SIMP_TAC[REAL_LT_POW2; REAL_FIELD
10919 `&0 < y ==> (y - x) / y = &1 - x / y`] THEN
10920 REAL_ARITH_TAC]]);;
10922 (* ------------------------------------------------------------------------- *)
10923 (* Slightly shaper separating/supporting hyperplane results. *)
10924 (* ------------------------------------------------------------------------- *)
10926 let SEPARATING_HYPERPLANE_RELATIVE_INTERIORS = prove
10927 (`!s t. convex s /\ convex t /\
10928 ~(s = {} /\ t = (:real^N) \/ s = (:real^N) /\ t = {}) /\
10929 DISJOINT (relative_interior s) (relative_interior t)
10930 ==> ?a b. ~(a = vec 0) /\
10931 (!x. x IN s ==> a dot x <= b) /\
10932 (!x. x IN t ==> a dot x >= b)`,
10933 REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC
10934 [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
10935 ASM_REWRITE_TAC[NOT_IN_EMPTY; UNIV_NOT_EMPTY; CONVEX_EMPTY;
10936 RELATIVE_INTERIOR_EMPTY] THEN
10938 [EXISTS_TAC `basis 1:real^N` THEN
10939 SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL];
10940 FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^N` o MATCH_MP (SET_RULE
10941 `~(s = UNIV) ==> ?a. ~(a IN s)`)) THEN
10942 MP_TAC(ISPECL [`t:real^N->bool`; `x:real^N`]
10943 SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN
10945 FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^N` o MATCH_MP (SET_RULE
10946 `~(s = UNIV) ==> ?a. ~(a IN s)`)) THEN
10947 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`]
10948 SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN
10949 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; real_ge] THEN
10950 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real`] THEN STRIP_TAC THEN
10951 MAP_EVERY EXISTS_TAC [`--a:real^N`; `--b:real`] THEN
10952 ASM_REWRITE_TAC[VECTOR_NEG_EQ_0; DOT_LNEG; REAL_LE_NEG2];
10953 MP_TAC(ISPECL [`relative_interior s:real^N->bool`;
10954 `relative_interior t:real^N->bool`]
10955 SEPARATING_HYPERPLANE_SETS) THEN
10956 ASM_SIMP_TAC[RELATIVE_INTERIOR_EQ_EMPTY; CONVEX_RELATIVE_INTERIOR] THEN
10957 SIMP_TAC[real_ge] THEN MATCH_MP_TAC MONO_EXISTS THEN
10958 X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC MONO_EXISTS THEN
10959 X_GEN_TAC `b:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
10960 CONJ_TAC THEN MATCH_MP_TAC
10961 (MESON[CONVEX_CLOSURE_RELATIVE_INTERIOR; CLOSURE_SUBSET; SUBSET]
10962 `convex s /\ (!x. x IN closure(relative_interior s) ==> P x)
10963 ==> !x. x IN s ==> P x`) THEN
10964 ASM_REWRITE_TAC[] THENL
10965 [MATCH_MP_TAC CONTINUOUS_LE_ON_CLOSURE;
10966 MATCH_MP_TAC CONTINUOUS_GE_ON_CLOSURE] THEN
10967 ASM_REWRITE_TAC[CONTINUOUS_ON_LIFT_DOT]]);;
10969 let SUPPORTING_HYPERPLANE_RELATIVE_BOUNDARY = prove
10971 convex s /\ x IN s /\ ~(x IN relative_interior s)
10972 ==> ?a. ~(a = vec 0) /\
10973 (!y. y IN s ==> a dot x <= a dot y) /\
10974 (!y. y IN relative_interior s ==> a dot x < a dot y)`,
10975 REPEAT STRIP_TAC THEN
10976 MP_TAC(ISPECL [`relative_interior s:real^N->bool`; `x:real^N`]
10977 SEPARATING_HYPERPLANE_SET_POINT_INAFF) THEN
10978 ASM_SIMP_TAC[CONVEX_SING; CONVEX_RELATIVE_INTERIOR;
10979 RELATIVE_INTERIOR_EQ_EMPTY; real_ge] THEN
10980 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
10981 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
10982 REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
10983 DISCH_THEN(X_CHOOSE_THEN `b:real` STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[] THEN
10984 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
10985 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
10986 MP_TAC(ISPECL [`lift o (\x:real^N. a dot x)`;
10987 `relative_interior s:real^N->bool`;
10988 `y:real^N`; `(a:real^N) dot x`; `1`]
10989 CONTINUOUS_ON_CLOSURE_COMPONENT_GE) THEN
10990 REWRITE_TAC[CONTINUOUS_ON_LIFT_DOT; GSYM drop; o_THM; LIFT_DROP] THEN
10991 ASM_SIMP_TAC[CONVEX_CLOSURE_RELATIVE_INTERIOR] THEN
10992 ASM_MESON_TAC[CLOSURE_SUBSET; REAL_LE_TRANS; SUBSET];
10994 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
10995 REWRITE_TAC[REAL_LT_LE] THEN CONJ_TAC THENL
10996 [ASM_MESON_TAC[REAL_LE_TRANS]; ALL_TAC] THEN
10997 DISCH_TAC THEN UNDISCH_TAC `(y:real^N) IN relative_interior s` THEN
10998 REWRITE_TAC[IN_RELATIVE_INTERIOR_CBALL] THEN
10999 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
11000 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_INTER; IN_CBALL] THEN
11001 X_GEN_TAC `e:real` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
11002 DISCH_THEN(MP_TAC o SPEC `y + --(e / norm(a)) % ((x + a) - x):real^N`) THEN
11003 REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL
11004 [SIMP_TAC[NORM_ARITH `dist(y:real^N,y + e) = norm e`; VECTOR_ADD_SUB] THEN
11005 REWRITE_TAC[NORM_MUL; REAL_ABS_NEG; REAL_ABS_DIV; REAL_ABS_NORM] THEN
11006 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_REAL_ARITH_TAC;
11007 MATCH_MP_TAC IN_AFFINE_ADD_MUL_DIFF THEN
11008 ASM_SIMP_TAC[AFFINE_AFFINE_HULL; HULL_INC] THEN
11009 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
11010 `x IN s ==> s SUBSET t ==> x IN t`)) THEN
11011 MATCH_MP_TAC HULL_MONO THEN
11012 ASM_REWRITE_TAC[INSERT_SUBSET; RELATIVE_INTERIOR_SUBSET];
11013 REWRITE_TAC[VECTOR_ADD_SUB] THEN DISCH_TAC THEN
11014 UNDISCH_TAC `!y:real^N. y IN s ==> a dot x <= a dot y` THEN
11015 DISCH_THEN(MP_TAC o SPEC `y + --(e / norm(a)) % a:real^N`) THEN
11016 ASM_REWRITE_TAC[DOT_RMUL; DOT_RNEG; DOT_RADD] THEN
11017 MATCH_MP_TAC(REAL_ARITH `&0 < x * y ==> ~(a <= a + --x * y)`) THEN
11018 ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; NORM_POS_LT; DOT_POS_LT]]);;
11020 let SUPPORTING_HYPERPLANE_RELATIVE_FRONTIER = prove
11022 convex s /\ x IN closure s /\ ~(x IN relative_interior s)
11023 ==> ?a. ~(a = vec 0) /\
11024 (!y. y IN closure s ==> a dot x <= a dot y) /\
11025 (!y. y IN relative_interior s ==> a dot x < a dot y)`,
11026 REPEAT STRIP_TAC THEN
11027 MP_TAC(ISPECL [`closure s:real^N->bool`; `x:real^N`]
11028 SUPPORTING_HYPERPLANE_RELATIVE_BOUNDARY) THEN
11029 ASM_SIMP_TAC[CONVEX_CLOSURE; CONVEX_RELATIVE_INTERIOR_CLOSURE]);;
11031 (* ------------------------------------------------------------------------- *)
11032 (* Containment of rays in unbounded convex sets. *)
11033 (* ------------------------------------------------------------------------- *)
11035 let UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY = prove
11037 convex s /\ ~bounded s /\ closed s /\ a IN s
11038 ==> ?l. ~(l = vec 0) /\ !t. &0 <= t ==> (a + t % l) IN s`,
11039 GEN_GEOM_ORIGIN_TAC `a:real^N` ["l"] THEN
11040 REWRITE_TAC[VECTOR_ADD_LID] THEN REPEAT STRIP_TAC THEN
11041 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [BOUNDED_POS]) THEN
11042 REWRITE_TAC[NOT_EXISTS_THM; TAUT `~(p /\ q) <=> p ==> ~q`] THEN
11043 DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&n + &1:real`) THEN
11044 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; REAL_ARITH `&0 < &n + &1`] THEN
11045 REWRITE_TAC[REAL_NOT_LE; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
11046 X_GEN_TAC `x:num->real^N` THEN REWRITE_TAC[FORALL_AND_THM] THEN
11048 SUBGOAL_THEN `!n. ~((x:num->real^N) n = vec 0)` ASSUME_TAC THENL
11049 [ASM_MESON_TAC[NORM_ARITH `~(&n + &1 < norm(vec 0:real^N))`]; ALL_TAC] THEN
11050 MP_TAC(ISPEC `sphere(vec 0:real^N,&1)` compact) THEN
11051 REWRITE_TAC[COMPACT_SPHERE] THEN
11052 DISCH_THEN(MP_TAC o SPEC `\n. inv(norm(x n)) % (x:num->real^N) n`) THEN
11053 ASM_SIMP_TAC[IN_SPHERE_0; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM;
11054 REAL_MUL_LINV; NORM_EQ_0; o_DEF] THEN
11055 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN
11056 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN CONJ_TAC THENL
11057 [ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`]; ALL_TAC] THEN
11058 X_GEN_TAC `t:real` THEN DISCH_TAC THEN
11059 MATCH_MP_TAC CLOSED_CONTAINS_SEQUENTIAL_LIMIT THEN
11061 `?N:num. !n. N <= n ==> t / norm(x n:real^N) <= &1`
11062 STRIP_ASSUME_TAC THENL
11063 [ASM_SIMP_TAC[REAL_LE_LDIV_EQ; NORM_POS_LT] THEN
11064 MP_TAC(SPEC `t:real` REAL_ARCH_SIMPLE) THEN
11065 MATCH_MP_TAC MONO_EXISTS THEN
11066 REWRITE_TAC[GSYM REAL_OF_NUM_LE; REAL_MUL_LID] THEN
11067 ASM_MESON_TAC[REAL_ARITH `t <= m /\ m <= n /\ n + &1 < x ==> t <= x`];
11068 EXISTS_TAC `\n:num. t / norm((x:num->real^N)(r(N + n))) % x(r(N + n))` THEN
11069 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
11070 [X_GEN_TAC `n:num` THEN
11071 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
11072 DISCH_THEN(MP_TAC o SPEC `vec 0:real^N`) THEN
11073 ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
11074 DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE] THEN
11075 FIRST_X_ASSUM MATCH_MP_TAC THEN
11076 FIRST_ASSUM(MP_TAC o SPEC `N + n:num` o MATCH_MP MONOTONE_BIGGER) THEN
11078 REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THEN
11079 MATCH_MP_TAC LIM_CMUL THEN ONCE_REWRITE_TAC[ADD_SYM] THEN
11080 FIRST_ASSUM(MP_TAC o SPEC `N:num` o MATCH_MP SEQ_OFFSET) THEN
11081 ASM_REWRITE_TAC[]]]);;
11083 let CONVEX_CLOSED_CONTAINS_SAME_RAY = prove
11085 convex s /\ closed s /\ b IN s /\ (!t. &0 <= t ==> (a + t % l) IN s)
11086 ==> !t. &0 <= t ==> (b + t % l) IN s`,
11087 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `&0`) THEN
11088 REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID] THEN DISCH_TAC THEN
11089 MATCH_MP_TAC(ISPEC `sequentially` LIM_IN_CLOSED_SET) THEN
11090 EXISTS_TAC `\n. (&1 - t / (&n + &1)) % b +
11091 t / (&n + &1) % (a + (&n + &1) % l):real^N` THEN
11092 ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL
11093 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
11094 MP_TAC(SPEC `t:real` REAL_ARCH_SIMPLE) THEN
11095 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
11096 DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
11097 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
11098 ASM_SIMP_TAC[REAL_LE_DIV; REAL_ARITH `&0 <= &n + &1`] THEN
11099 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
11100 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN ASM_REAL_ARITH_TAC;
11101 REWRITE_TAC[VECTOR_ARITH
11102 `(&1 - u) % b + u % c:real^N = b + u % (c - b)`] THEN
11103 MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
11104 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_SUB_LDISTRIB] THEN
11105 SIMP_TAC[VECTOR_MUL_ASSOC; REAL_FIELD `t / (&n + &1) * (&n + &1) = t`] THEN
11106 SIMP_TAC[VECTOR_ARITH `(v % a + b) - v % c:real^N = b + v % (a - c)`] THEN
11107 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN
11108 MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
11109 REWRITE_TAC[real_div; VECTOR_ARITH `(x * y) % a:real^N = y % x % a`] THEN
11110 MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN
11111 EXISTS_TAC `norm(t % (a - b):real^N)` THEN
11112 REWRITE_TAC[REAL_LE_REFL; EVENTUALLY_TRUE; o_DEF] THEN
11113 MP_TAC(MATCH_MP SEQ_OFFSET SEQ_HARMONIC) THEN
11114 SIMP_TAC[REAL_OF_NUM_ADD]]);;
11116 let UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAYS = prove
11118 convex s /\ ~bounded s /\ closed s
11119 ==> ?l. ~(l = vec 0) /\ !a t. a IN s /\ &0 <= t ==> (a + t % l) IN s`,
11120 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
11121 ASM_REWRITE_TAC[BOUNDED_EMPTY] THEN
11122 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
11123 ASM_MESON_TAC[UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY;
11124 CONVEX_CLOSED_CONTAINS_SAME_RAY]);;
11126 let RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAY = prove
11128 convex s /\ ~bounded s /\ a IN relative_interior s
11129 ==> ?l. ~(l = vec 0) /\
11130 !t. &0 <= t ==> (a + t % l) IN relative_interior s`,
11131 REPEAT STRIP_TAC THEN
11132 MP_TAC(ISPECL [`closure s:real^N->bool`; `a:real^N`]
11133 UNBOUNDED_CONVEX_CLOSED_CONTAINS_RAY) THEN
11134 ASM_SIMP_TAC[CONVEX_CLOSURE; CLOSED_CLOSURE] THEN ANTS_TAC THENL
11135 [ASM_MESON_TAC[BOUNDED_SUBSET; SUBSET; CLOSURE_SUBSET;
11136 RELATIVE_INTERIOR_SUBSET];
11137 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN STRIP_TAC THEN
11138 ASM_REWRITE_TAC[] THEN
11139 ONCE_REWRITE_TAC[VECTOR_ARITH
11140 `a + t % l:real^N =
11141 (a + (&2 * t) % l) - inv(&2) % ((a + (&2 * t) % l) - a)`] THEN
11142 REPEAT STRIP_TAC THEN
11143 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
11144 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN
11145 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]);;
11147 let RELATIVE_INTERIOR_CONVEX_CONTAINS_SAME_RAY = prove
11149 convex s /\ b IN relative_interior s /\
11150 (!t. &0 <= t ==> (a + t % l) IN relative_interior s)
11151 ==> !t. &0 <= t ==> (b + t % l) IN relative_interior s`,
11152 REPEAT STRIP_TAC THEN
11153 MP_TAC(ISPECL [`closure s:real^N->bool`; `a:real^N`; `b:real^N`; `l:real^N`]
11154 CONVEX_CLOSED_CONTAINS_SAME_RAY) THEN
11155 ASM_SIMP_TAC[CONVEX_CLOSURE; CLOSED_CLOSURE] THEN ANTS_TAC THENL
11156 [ASM_MESON_TAC[BOUNDED_SUBSET; SUBSET; CLOSURE_SUBSET;
11157 RELATIVE_INTERIOR_SUBSET];
11159 ONCE_REWRITE_TAC[VECTOR_ARITH
11160 `a + t % l:real^N =
11161 (a + (&2 * t) % l) - inv(&2) % ((a + (&2 * t) % l) - a)`] THEN
11162 REPEAT STRIP_TAC THEN
11163 MATCH_MP_TAC IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SHRINK THEN
11164 CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN
11165 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC]);;
11167 let RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAYS = prove
11169 convex s /\ ~bounded s
11170 ==> ?l. ~(l = vec 0) /\
11171 !a t. a IN relative_interior s /\ &0 <= t
11172 ==> (a + t % l) IN relative_interior s`,
11173 REPEAT GEN_TAC THEN
11174 ASM_CASES_TAC `relative_interior s:real^N->bool = {}` THENL
11175 [ASM_MESON_TAC[RELATIVE_INTERIOR_EQ_EMPTY; BOUNDED_EMPTY]; ALL_TAC] THEN
11176 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
11177 ASM_MESON_TAC[RELATIVE_INTERIOR_UNBOUNDED_CONVEX_CONTAINS_RAY;
11178 RELATIVE_INTERIOR_CONVEX_CONTAINS_SAME_RAY]);;
11180 (* ------------------------------------------------------------------------- *)
11181 (* Explicit formulas for interior and relative interior of convex hull. *)
11182 (* ------------------------------------------------------------------------- *)
11184 let EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL = prove
11186 ==> {y:real^N | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\
11188 vsum s (\x. u x % x) = y}
11189 SUBSET relative_interior(convex hull s)`,
11190 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
11191 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ] THEN
11192 REWRITE_TAC[EMPTY_GSPEC; EMPTY_SUBSET] THEN
11193 REPEAT STRIP_TAC THEN MATCH_MP_TAC RELATIVE_INTERIOR_MAXIMAL THEN
11194 REWRITE_TAC[AFFINE_HULL_CONVEX_HULL] THEN CONJ_TAC THENL
11195 [REWRITE_TAC[CONVEX_HULL_FINITE; SUBSET; IN_ELIM_THM] THEN
11196 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_LT_IMP_LE];
11198 REWRITE_TAC[open_in; IN_ELIM_THM] THEN CONJ_TAC THENL
11199 [REWRITE_TAC[AFFINE_HULL_FINITE; SUBSET; IN_ELIM_THM] THEN
11200 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[REAL_LT_IMP_LE];
11202 X_GEN_TAC `y:real^N` THEN
11203 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
11204 ABBREV_TAC `e = inf (IMAGE (\x:real^N. min (&1 - u x) (u x)) s)` THEN
11205 SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL
11206 [EXPAND_TAC "e" THEN
11207 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
11208 ASM_SIMP_TAC[REAL_LT_MIN; REAL_SUB_LT; FORALL_IN_IMAGE];
11210 MP_TAC(ISPEC `IMAGE (\z:real^N. z - y) (affine hull s)` BASIS_EXISTS) THEN
11211 REWRITE_TAC[SUBSET_IMAGE] THEN
11212 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool`
11213 (CONJUNCTS_THEN2 (X_CHOOSE_THEN `c:real^N->bool` (STRIP_ASSUME_TAC o GSYM))
11215 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; HAS_SIZE] THEN
11216 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
11217 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
11218 ASM_SIMP_TAC[SPAN_FINITE; IN_ELIM_THM] THEN
11219 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
11220 REWRITE_TAC[SKOLEM_THM] THEN
11221 DISCH_THEN(X_CHOOSE_TAC `compo:real^N->real^N->real`) THEN
11222 FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o
11223 MATCH_MP BASIS_COORDINATES_LIPSCHITZ) THEN
11225 `!i. i IN b ==> ?u. sum s u = &0 /\ vsum s (\x:real^N. u x % x) = i`
11227 [EXPAND_TAC "b" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
11228 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
11229 SUBGOAL_THEN `(x:real^N) IN affine hull s` MP_TAC THENL
11230 [ASM SET_TAC[]; REWRITE_TAC[AFFINE_HULL_FINITE; IN_ELIM_THM]] THEN
11231 DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN
11232 EXISTS_TAC `(\x. v x - u x):real^N->real` THEN
11233 ASM_SIMP_TAC[SUM_SUB; VSUM_SUB; VECTOR_SUB_RDISTRIB] THEN
11234 REWRITE_TAC[REAL_SUB_REFL; VECTOR_SUB_RZERO];
11235 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
11236 [RIGHT_IMP_EXISTS_THM; SKOLEM_THM; FORALL_AND_THM;
11237 TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN
11238 DISCH_THEN(X_CHOOSE_THEN `w:real^N->real^N->real` STRIP_ASSUME_TAC)] THEN
11239 EXISTS_TAC `e / B /
11240 (&1 + sum (b:real^N->bool)
11241 (\i. abs(sup(IMAGE (abs o w i) (s:real^N->bool)))))` THEN
11242 ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 <= x ==> &0 < &1 + x`;
11243 SUM_POS_LE; REAL_ABS_POS] THEN
11244 X_GEN_TAC `z:real^N` THEN STRIP_TAC THEN
11246 `\x:real^N. u x + sum (b:real^N->bool)
11247 (\i. compo (z:real^N) i * w i x)` THEN
11248 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
11250 ASM_SIMP_TAC[SUM_ADD; REAL_ARITH `&1 + x = &1 <=> x = &0`] THEN
11251 W(MP_TAC o PART_MATCH (lhs o rand) SUM_SWAP o lhand o snd) THEN
11252 ASM_REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN
11253 MATCH_MP_TAC SUM_EQ_0 THEN
11254 ASM_SIMP_TAC[SUM_LMUL; ETA_AX; REAL_MUL_RZERO; SUM_0];
11255 ASM_SIMP_TAC[VSUM_ADD; VECTOR_ADD_RDISTRIB] THEN
11256 ONCE_REWRITE_TAC[VECTOR_ARITH `y + w:real^N = z <=> w = z - y`] THEN
11257 ASM_SIMP_TAC[GSYM VSUM_LMUL; GSYM VSUM_RMUL; GSYM VECTOR_MUL_ASSOC] THEN
11258 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_SWAP o lhand o snd) THEN
11259 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
11260 ASM_SIMP_TAC[VSUM_LMUL] THEN MATCH_MP_TAC EQ_TRANS THEN
11261 EXISTS_TAC `vsum b (\v:real^N. compo (z:real^N) v % v)` THEN
11262 CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[]] THEN
11263 MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[]] THEN
11264 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
11265 MATCH_MP_TAC(REAL_ARITH
11266 `abs(x) < min u (&1 - u) ==> &0 < u + x /\ u + x < &1`) THEN
11267 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC
11268 `B * norm(z - y:real^N) * sum (b:real^N->bool)
11269 (\i. abs(sup(IMAGE (abs o w i) (s:real^N->bool))))` THEN
11271 [REWRITE_TAC[GSYM SUM_LMUL] THEN MATCH_MP_TAC SUM_ABS_LE THEN
11272 ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_MUL_ASSOC] THEN
11273 X_GEN_TAC `i:real^N` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
11274 REWRITE_TAC[REAL_ABS_POS] THEN CONJ_TAC THENL
11275 [FIRST_X_ASSUM(MP_TAC o SPECL [`(compo:real^N->real^N->real) z`;
11278 MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs a`) THEN
11279 ASM_SIMP_TAC[REAL_LE_SUP_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
11280 REWRITE_TAC[EXISTS_IN_IMAGE; o_THM] THEN ASM_MESON_TAC[REAL_LE_REFL]];
11282 GEN_REWRITE_TAC LAND_CONV [REAL_MUL_SYM] THEN
11283 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
11284 MATCH_MP_TAC(REAL_ARITH
11285 `&0 <= x /\ x * (&1 + e) < d ==> x * e < d`) THEN
11286 REWRITE_TAC[NORM_POS_LE] THEN
11287 ASM_SIMP_TAC[NORM_POS_LE; GSYM REAL_LT_RDIV_EQ;
11288 REAL_ARITH `&0 <= x ==> &0 < &1 + x`;
11289 SUM_POS_LE; REAL_ABS_POS] THEN
11290 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
11291 `dist(z:real^N,y) < k ==> k <= d ==> norm(z - y) < d`)) THEN
11292 ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_ARITH `&0 <= x ==> &0 < &1 + x`;
11293 SUM_POS_LE; REAL_ABS_POS] THEN
11294 EXPAND_TAC "e" THEN
11295 ASM_SIMP_TAC[REAL_INF_LE_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
11296 REWRITE_TAC[EXISTS_IN_IMAGE] THEN EXISTS_TAC `x:real^N` THEN
11297 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
11299 let EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL = prove
11301 ==> {y:real^N | ?u. (!x. x IN s ==> &0 < u x) /\
11303 vsum s (\x. u x % x) = y}
11304 SUBSET relative_interior(convex hull s)`,
11305 REPEAT STRIP_TAC THEN
11306 ASM_CASES_TAC `s:real^N->bool = {}` THEN
11307 ASM_REWRITE_TAC[SUM_CLAUSES; REAL_OF_NUM_EQ; ARITH_EQ] THEN
11308 REWRITE_TAC[EMPTY_GSPEC; EMPTY_SUBSET] THEN
11309 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
11310 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
11311 ASM_CASES_TAC `s = {a:real^N}` THENL
11312 [ASM_REWRITE_TAC[SUM_SING; VSUM_SING; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
11313 REWRITE_TAC[RELATIVE_INTERIOR_SING; CONVEX_HULL_SING] THEN
11314 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_SING] THEN
11315 MESON_TAC[VECTOR_MUL_LID];
11316 FIRST_ASSUM(MP_TAC o MATCH_MP
11317 EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL) THEN
11318 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUBSET_TRANS) THEN
11319 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `w:real^N` THEN
11320 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->real` THEN
11321 STRIP_TAC THEN ASM_SIMP_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
11322 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL
11323 [ASM SET_TAC[]; ALL_TAC] THEN
11324 SUBGOAL_THEN `sum {x,y} u <= sum s (u:real^N->real)` MP_TAC THENL
11325 [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
11326 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; REAL_LT_IMP_LE; IN_DIFF] THEN
11328 ASM_SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11329 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
11330 MATCH_MP_TAC(REAL_ARITH `&0 < y ==> x + y + &0 <= &1 ==> x < &1`) THEN
11331 ASM_SIMP_TAC[]]]);;
11333 let RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT = prove
11334 (`!s. ~(affine_dependent s)
11335 ==> relative_interior(convex hull s) =
11336 {y:real^N | ?u. (!x. x IN s ==> &0 < u x) /\
11338 vsum s (\x. u x % x) = y}`,
11339 REPEAT STRIP_TAC THEN
11340 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11341 MATCH_MP_TAC SUBSET_ANTISYM THEN
11342 ASM_SIMP_TAC[EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL] THEN
11343 ASM_CASES_TAC `?a:real^N. s = {a}` THENL
11344 [FIRST_X_ASSUM(CHOOSE_THEN SUBST1_TAC) THEN
11345 ASM_REWRITE_TAC[SUM_SING; VSUM_SING; CONVEX_HULL_SING;
11346 RELATIVE_INTERIOR_SING] THEN
11347 REWRITE_TAC[IN_ELIM_THM; SUBSET; IN_SING] THEN
11348 REPEAT STRIP_TAC THEN EXISTS_TAC `\x:real^N. &1` THEN
11349 ASM_REWRITE_TAC[VECTOR_MUL_LID; REAL_LT_01];
11351 MATCH_MP_TAC(SET_RULE
11352 `relative_interior s SUBSET s /\
11353 (!x. x IN s /\ ~(x IN t) ==> ~(x IN relative_interior s))
11354 ==> relative_interior s SUBSET t`) THEN
11355 REWRITE_TAC[RELATIVE_INTERIOR_SUBSET] THEN
11356 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[IN_RELATIVE_INTERIOR] THEN
11357 REWRITE_TAC[AFFINE_HULL_CONVEX_HULL; IN_ELIM_THM; NOT_EXISTS_THM] THEN
11358 REWRITE_TAC[CONVEX_HULL_FINITE; IN_ELIM_THM] THEN
11359 DISCH_THEN(CONJUNCTS_THEN2
11360 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC)
11361 (MP_TAC o SPEC `u:real^N->real`)) THEN
11362 ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; IN_RELATIVE_INTERIOR; DE_MORGAN_THM;
11363 SUBSET; IN_ELIM_THM; IN_BALL; IN_INTER] THEN
11364 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN DISJ2_TAC THEN
11365 DISCH_THEN(X_CHOOSE_THEN `e:real`
11366 (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))) THEN
11367 SUBGOAL_THEN `(u:real^N->real) a = &0` ASSUME_TAC THENL
11368 [ASM_SIMP_TAC[REAL_ARITH `&0 <= x /\ ~(&0 < x) ==> x = &0`]; ALL_TAC] THEN
11369 SUBGOAL_THEN `?b:real^N. b IN s /\ ~(b = a)` STRIP_ASSUME_TAC THENL
11370 [ASM SET_TAC[];ALL_TAC] THEN
11371 SUBGOAL_THEN `?d. &0 < d /\ norm(d % (a - b):real^N) < e`
11372 STRIP_ASSUME_TAC THENL
11373 [EXISTS_TAC `e / &2 / norm(a - b:real^N)` THEN
11374 ASM_SIMP_TAC[NORM_MUL; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; NORM_POS_LT;
11375 REAL_ABS_DIV; REAL_ABS_NORM; REAL_ABS_NUM;
11376 REAL_DIV_RMUL; REAL_LT_IMP_NZ; VECTOR_SUB_EQ] THEN
11377 ASM_REAL_ARITH_TAC;
11379 REMOVE_THEN "*" (MP_TAC o SPEC `y - d % (a - b):real^N`) THEN
11380 ASM_REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - b) = norm b`] THEN
11381 REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL
11382 [MATCH_MP_TAC IN_AFFINE_SUB_MUL_DIFF THEN
11383 ASM_SIMP_TAC[HULL_INC; AFFINE_AFFINE_HULL] THEN
11384 REWRITE_TAC[AFFINE_HULL_FINITE; IN_ELIM_THM] THEN
11385 EXISTS_TAC `u:real^N->real` THEN ASM_REWRITE_TAC[];
11387 DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN
11388 UNDISCH_TAC `~(affine_dependent(s:real^N->bool))` THEN
11389 ASM_SIMP_TAC[AFFINE_DEPENDENT_EXPLICIT_FINITE] THEN
11390 EXISTS_TAC `\x:real^N. (v x - u x) -
11391 (if x = a then --d else if x = b then d else &0)` THEN
11392 REWRITE_TAC[VECTOR_SUB_RDISTRIB; MESON[]
11393 `(if p then a else b) % x = (if p then a % x else b % x)`] THEN
11394 ASM_SIMP_TAC[SUM_SUB; VSUM_SUB] THEN
11395 ASM_SIMP_TAC[VSUM_CASES; SUM_CASES; FINITE_RESTRICT; IN_ELIM_THM] THEN
11396 ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`;
11397 SET_RULE `b IN s /\ ~(b = a)
11398 ==> {x | (x IN s /\ ~(x = a)) /\ x = b} = {b}`] THEN
11399 ASM_SIMP_TAC[VECTOR_MUL_LZERO; SUM_0; VSUM_0; SUM_SING; VSUM_SING] THEN
11400 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
11401 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
11402 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN
11403 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN
11404 ASM_REAL_ARITH_TAC);;
11406 let EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL = prove
11407 (`!s. FINITE s /\ affine hull s = (:real^N)
11408 ==> {y | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\
11410 vsum s (\x. u x % x) = y}
11411 SUBSET interior(convex hull s)`,
11412 REPEAT STRIP_TAC THEN
11413 FIRST_ASSUM(MP_TAC o
11414 MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL) THEN
11415 ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_CONVEX_HULL]);;
11417 let EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL_MINIMAL = prove
11418 (`!s. FINITE s /\ affine hull s = (:real^N)
11419 ==> {y | ?u. (!x. x IN s ==> &0 < u x) /\
11421 vsum s (\x. u x % x) = y}
11422 SUBSET interior(convex hull s)`,
11423 REPEAT STRIP_TAC THEN
11424 FIRST_ASSUM(MP_TAC o
11425 MATCH_MP EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL) THEN
11426 ASM_SIMP_TAC[RELATIVE_INTERIOR_INTERIOR; AFFINE_HULL_CONVEX_HULL]);;
11428 let INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL = prove
11430 ~(affine_dependent s)
11431 ==> interior(convex hull s) =
11432 if CARD(s) <= dimindex(:N) then {}
11433 else {y | ?u. (!x. x IN s ==> &0 < u x) /\
11435 vsum s (\x. u x % x) = y}`,
11436 REPEAT STRIP_TAC THEN
11437 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11438 COND_CASES_TAC THEN ASM_SIMP_TAC[EMPTY_INTERIOR_CONVEX_HULL] THEN
11439 MATCH_MP_TAC EQ_TRANS THEN
11440 EXISTS_TAC `relative_interior(convex hull s):real^N->bool` THEN
11442 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
11443 REWRITE_TAC[AFFINE_HULL_CONVEX_HULL] THEN
11444 MATCH_MP_TAC AFFINE_INDEPENDENT_SPAN_GT THEN
11445 ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC;
11446 ASM_SIMP_TAC[RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT]]);;
11448 let INTERIOR_CONVEX_HULL_EXPLICIT = prove
11450 ~(affine_dependent s)
11451 ==> interior(convex hull s) =
11452 if CARD(s) <= dimindex(:N) then {}
11453 else {y | ?u. (!x. x IN s ==> &0 < u x /\ u x < &1) /\
11455 vsum s (\x. u x % x) = y}`,
11456 REPEAT STRIP_TAC THEN
11457 ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN
11458 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
11459 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `v:real^N` THEN
11460 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `u:real^N->real` THEN
11461 EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN
11462 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
11463 MP_TAC(ISPEC `s:real^N->bool` CHOOSE_SUBSET) THEN
11464 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE] THEN
11465 DISCH_THEN(MP_TAC o SPEC `2`) THEN ANTS_TAC THENL
11466 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ARITH_RULE
11467 `~(c <= n) ==> 1 <= n ==> 2 <= c`)) THEN
11468 REWRITE_TAC[DIMINDEX_GE_1];
11470 CONV_TAC(ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN
11471 REWRITE_TAC[SUBSET] THEN
11472 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` (CONJUNCTS_THEN2 ASSUME_TAC
11474 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (X_CHOOSE_THEN `b:real^N`
11475 STRIP_ASSUME_TAC)) THEN
11476 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` STRIP_ASSUME_TAC THENL
11477 [ASM SET_TAC[]; ALL_TAC] THEN
11478 SUBGOAL_THEN `sum {x,y} u <= sum s (u:real^N->real)` MP_TAC THENL
11479 [MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
11480 ASM_SIMP_TAC[AFFINE_INDEPENDENT_IMP_FINITE; REAL_LT_IMP_LE; IN_DIFF] THEN
11483 ASM_SIMP_TAC[SUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11484 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
11485 MATCH_MP_TAC(REAL_ARITH `&0 < y ==> x + y + &0 <= &1 ==> x < &1`) THEN
11488 let INTERIOR_CONVEX_HULL_3_MINIMAL = prove
11491 ==> interior(convex hull {a,b,c}) =
11492 {v | ?x y z. &0 < x /\
11496 x % a + y % b + z % c = v}`,
11497 REWRITE_TAC[COLLINEAR_3_EQ_AFFINE_DEPENDENT; DE_MORGAN_THM] THEN
11498 REPEAT STRIP_TAC THEN
11499 ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN
11500 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11501 CONV_TAC(LAND_CONV(RATOR_CONV(LAND_CONV(ONCE_DEPTH_CONV(REWRITE_CONV
11502 [IN_INSERT; NOT_IN_EMPTY]))))) THEN
11503 ASM_REWRITE_TAC[DIMINDEX_2; ARITH] THEN
11504 SIMP_TAC[FINITE_INSERT; FINITE_UNION; FINITE_EMPTY; RIGHT_EXISTS_AND_THM;
11505 AFFINE_HULL_FINITE_STEP_GEN; REAL_LT_ADD; REAL_HALF] THEN
11506 REWRITE_TAC[REAL_ARITH `&1 - a - b - c = &0 <=> a + b + c = &1`;
11507 VECTOR_ARITH `y - a - b - c:real^N = vec 0 <=> a + b + c = y`]);;
11509 let INTERIOR_CONVEX_HULL_3 = prove
11512 ==> interior(convex hull {a,b,c}) =
11513 {v | ?x y z. &0 < x /\ x < &1 /\
11514 &0 < y /\ y < &1 /\
11515 &0 < z /\ z < &1 /\
11517 x % a + y % b + z % c = v}`,
11518 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_3_MINIMAL] THEN
11519 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN GEN_TAC THEN
11520 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN EQ_TAC THEN STRIP_TAC THEN
11521 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
11523 (* ------------------------------------------------------------------------- *)
11524 (* Similar results for closure and (relative or absolute) frontier. *)
11525 (* ------------------------------------------------------------------------- *)
11527 let CLOSURE_CONVEX_HULL = prove
11528 (`!s. compact s ==> closure(convex hull s) = convex hull s`,
11529 SIMP_TAC[CLOSURE_CLOSED; COMPACT_IMP_CLOSED; COMPACT_CONVEX_HULL]);;
11531 let RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT = prove
11533 ~(affine_dependent s)
11534 ==> relative_frontier(convex hull s) =
11535 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
11536 (?x. x IN s /\ u x = &0) /\
11538 vsum s (\x. u x % x) = y}`,
11539 REPEAT STRIP_TAC THEN REWRITE_TAC[relative_frontier; UNIONS_GSPEC] THEN
11540 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11541 ASM_SIMP_TAC[CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11542 ASM_SIMP_TAC[CONVEX_HULL_FINITE; RELATIVE_INTERIOR_CONVEX_HULL_EXPLICIT] THEN
11543 GEN_REWRITE_TAC I [EXTENSION] THEN X_GEN_TAC `y:real^N` THEN
11544 REWRITE_TAC[IN_DIFF; IN_ELIM_THM] THEN EQ_TAC THENL
11545 [DISCH_THEN(CONJUNCTS_THEN2
11546 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) ASSUME_TAC) THEN
11547 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
11548 DISCH_THEN(MP_TAC o SPEC `u:real^N->real`) THEN
11549 ASM_REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
11550 DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
11551 ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> (~(&0 < x) <=> x = &0)`] THEN
11552 DISCH_TAC THEN EXISTS_TAC `u:real^N->real` THEN
11553 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
11554 DISCH_THEN(X_CHOOSE_THEN `u:real^N->real`
11555 (REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC)) THEN
11557 [EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[]; ALL_TAC] THEN
11558 DISCH_THEN(X_CHOOSE_THEN `v:real^N->real` STRIP_ASSUME_TAC) THEN
11559 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV
11560 [AFFINE_DEPENDENT_EXPLICIT]) THEN
11561 REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC
11562 [`s:real^N->bool`; `(\x. u x - v x):real^N->real`] THEN
11563 ASM_SIMP_TAC[SUBSET_REFL; VECTOR_SUB_RDISTRIB; SUM_SUB; VSUM_SUB] THEN
11564 REWRITE_TAC[REAL_SUB_0; VECTOR_SUB_EQ] THEN ASM_MESON_TAC[REAL_LT_REFL]]);;
11566 let FRONTIER_CONVEX_HULL_EXPLICIT = prove
11568 ~(affine_dependent s)
11569 ==> frontier(convex hull s) =
11570 {y | ?u. (!x. x IN s ==> &0 <= u x) /\
11571 (dimindex(:N) < CARD s ==> ?x. x IN s /\ u x = &0) /\
11573 vsum s (\x. u x % x) = y}`,
11574 REPEAT STRIP_TAC THEN REWRITE_TAC[frontier] THEN
11575 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11577 (ARITH_RULE `CARD(s:real^N->bool) <= dimindex(:N) \/
11578 dimindex(:N) < CARD(s:real^N->bool)`)
11580 [ASM_SIMP_TAC[GSYM NOT_LE; INTERIOR_CONVEX_HULL_EXPLICIT] THEN
11581 ASM_SIMP_TAC[CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT; DIFF_EMPTY] THEN
11582 REWRITE_TAC[CONVEX_HULL_FINITE];
11583 ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT] THEN
11584 REWRITE_TAC[relative_frontier] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
11585 MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
11586 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
11587 EXISTS_TAC `affine hull s:real^N->bool` THEN
11588 ASM_SIMP_TAC[AFFINE_INDEPENDENT_SPAN_GT; HULL_MONO; HULL_SUBSET]]);;
11590 let RELATIVE_FRONTIER_CONVEX_HULL_CASES = prove
11592 ~(affine_dependent s)
11593 ==> relative_frontier(convex hull s) =
11594 UNIONS { convex hull (s DELETE a) |a| a IN s }`,
11595 REPEAT STRIP_TAC THEN REWRITE_TAC[UNIONS_GSPEC] THEN
11596 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11597 ASM_SIMP_TAC[RELATIVE_FRONTIER_CONVEX_HULL_EXPLICIT] THEN
11598 REWRITE_TAC[EXTENSION; IN_ELIM_THM; CONVEX_HULL_FINITE] THEN
11599 X_GEN_TAC `y:real^N` THEN EQ_TAC THENL
11600 [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` MP_TAC) THEN
11601 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
11602 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
11603 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
11604 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXISTS_TAC `u:real^N->real` THEN
11605 ASM_SIMP_TAC[IN_DELETE; SUM_DELETE; VSUM_DELETE; REAL_SUB_RZERO] THEN
11607 REWRITE_TAC[IN_DELETE] THEN
11608 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (CONJUNCTS_THEN2 ASSUME_TAC
11609 (X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC))) THEN
11610 EXISTS_TAC `(\x. if x = a then &0 else u x):real^N->real` THEN
11611 ASM_SIMP_TAC[COND_RAND; COND_RATOR; REAL_LE_REFL; COND_ID] THEN
11612 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
11613 ASM_SIMP_TAC[SUM_CASES; VSUM_CASES; VECTOR_MUL_LZERO] THEN
11614 ASM_SIMP_TAC[GSYM DELETE; SUM_0; VSUM_0; REAL_ADD_LID; VECTOR_ADD_LID]]);;
11616 let FRONTIER_CONVEX_HULL_CASES = prove
11618 ~(affine_dependent s)
11619 ==> frontier(convex hull s) =
11620 if CARD(s) <= dimindex(:N) then convex hull s
11621 else UNIONS { convex hull (s DELETE a) |a| a IN s }`,
11622 REPEAT STRIP_TAC THEN
11623 FIRST_ASSUM(ASSUME_TAC o MATCH_MP AFFINE_INDEPENDENT_IMP_FINITE) THEN
11624 ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11625 COND_CASES_TAC THENL
11626 [ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT; DIFF_EMPTY]; ALL_TAC] THEN
11627 ASM_SIMP_TAC[GSYM RELATIVE_FRONTIER_CONVEX_HULL_CASES] THEN
11628 ASM_SIMP_TAC[relative_frontier; frontier;
11629 CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11630 AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
11631 RULE_ASSUM_TAC(REWRITE_RULE[NOT_LE]) THEN
11632 MATCH_MP_TAC RELATIVE_INTERIOR_INTERIOR THEN
11633 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
11634 EXISTS_TAC `affine hull s:real^N->bool` THEN
11635 ASM_SIMP_TAC[AFFINE_INDEPENDENT_SPAN_GT; HULL_MONO; HULL_SUBSET]);;
11637 let IN_FRONTIER_CONVEX_HULL = prove
11639 FINITE s /\ CARD s <= dimindex(:N) + 1 /\ x IN s
11640 ==> x IN frontier(convex hull s)`,
11641 REPEAT STRIP_TAC THEN ASM_CASES_TAC `affine_dependent(s:real^N->bool)` THENL
11642 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [affine_dependent]) THEN
11643 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
11644 ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11645 ASM_SIMP_TAC[HULL_INC; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE
11646 `!t. s SUBSET t /\ t = {} ==> ~(x IN s)`) THEN
11647 EXISTS_TAC `interior(affine hull s):real^N->bool` THEN
11648 SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
11649 SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL
11650 [ASM SET_TAC[]; ALL_TAC] THEN
11651 ASM_SIMP_TAC[HULL_REDUNDANT] THEN
11652 MATCH_MP_TAC EMPTY_INTERIOR_AFFINE_HULL THEN
11653 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC;
11654 ASM_SIMP_TAC[FRONTIER_CONVEX_HULL_CASES] THEN
11655 COND_CASES_TAC THEN ASM_SIMP_TAC[HULL_INC] THEN
11656 REWRITE_TAC[UNIONS_GSPEC; IN_ELIM_THM] THEN
11657 SUBGOAL_THEN `?y:real^N. y IN s /\ ~(y = x)` MP_TAC THENL
11658 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
11659 `x IN s ==> ~(s = {x}) ==> ?y. y IN s /\ ~(y = x)`)) THEN DISCH_TAC THEN
11660 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_LE]) THEN
11661 ASM_SIMP_TAC[CARD_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
11662 REWRITE_TAC[NOT_LT; NOT_IN_EMPTY; ARITH_SUC; DIMINDEX_GE_1];
11663 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN
11664 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_INC THEN ASM SET_TAC[]]]);;
11666 let NOT_IN_INTERIOR_CONVEX_HULL = prove
11668 FINITE s /\ CARD s <= dimindex(:N) + 1 /\ x IN s
11669 ==> ~(x IN interior(convex hull s))`,
11670 REPEAT GEN_TAC THEN
11671 DISCH_THEN(MP_TAC o MATCH_MP IN_FRONTIER_CONVEX_HULL) THEN
11672 SIMP_TAC[frontier; IN_DIFF]);;
11674 let INTERIOR_CONVEX_HULL_EQ_EMPTY = prove
11676 s HAS_SIZE (dimindex(:N) + 1)
11677 ==> (interior(convex hull s) = {} <=> affine_dependent s)`,
11678 REPEAT GEN_TAC THEN REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC THEN
11679 ASM_CASES_TAC `affine_dependent(s:real^N->bool)` THENL
11680 [ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
11681 [affine_dependent]) THEN
11682 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
11683 ASM_SIMP_TAC[frontier; CLOSURE_CONVEX_HULL; FINITE_IMP_COMPACT] THEN
11684 ASM_SIMP_TAC[HULL_INC; IN_DIFF] THEN MATCH_MP_TAC(SET_RULE
11685 `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
11686 EXISTS_TAC `interior(affine hull s):real^N->bool` THEN
11687 SIMP_TAC[SUBSET_INTERIOR; CONVEX_HULL_SUBSET_AFFINE_HULL] THEN
11688 SUBGOAL_THEN `s = (a:real^N) INSERT (s DELETE a)` SUBST1_TAC THENL
11689 [ASM SET_TAC[]; ALL_TAC] THEN
11690 ASM_SIMP_TAC[HULL_REDUNDANT] THEN
11691 MATCH_MP_TAC EMPTY_INTERIOR_AFFINE_HULL THEN
11692 ASM_SIMP_TAC[FINITE_DELETE; CARD_DELETE] THEN ASM_ARITH_TAC;
11693 ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_EXPLICIT_MINIMAL] THEN
11694 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; ARITH_RULE `~(n + 1 <= n)`] THEN
11695 EXISTS_TAC `vsum s (\x:real^N. inv(&(dimindex(:N)) + &1) % x)` THEN
11696 REWRITE_TAC[IN_ELIM_THM] THEN
11697 EXISTS_TAC `\x:real^N. inv(&(dimindex(:N)) + &1)` THEN
11698 ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
11699 ASM_SIMP_TAC[SUM_CONST; GSYM REAL_OF_NUM_ADD] THEN
11700 CONV_TAC REAL_FIELD]);;
11702 (* ------------------------------------------------------------------------- *)
11703 (* Similar things in special case (could use above as lemmas here instead). *)
11704 (* ------------------------------------------------------------------------- *)
11706 let SIMPLEX_EXPLICIT = prove
11708 FINITE s /\ ~(vec 0 IN s)
11709 ==> convex hull (vec 0 INSERT s) =
11710 { y | ?u. (!x. x IN s ==> &0 <= u x) /\
11712 vsum s (\x. u x % x) = y}`,
11713 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CONVEX_HULL_FINITE; FINITE_INSERT] THEN
11714 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN X_GEN_TAC `y:real^N` THEN
11715 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; IN_INSERT] THEN
11716 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
11717 EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THENL
11718 [EXISTS_TAC `u:real^N->real` THEN ASM_SIMP_TAC[REAL_LE_REFL] THEN
11719 FIRST_X_ASSUM(MP_TAC o SPEC `vec 0:real^N`) THEN REWRITE_TAC[] THEN
11720 ASM_REAL_ARITH_TAC;
11721 EXISTS_TAC `\x:real^N. if x = vec 0 then &1 - sum (s:real^N->bool) u
11723 ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
11724 [X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `x:real^N = vec 0` THEN
11725 ASM_REWRITE_TAC[REAL_SUB_LE];
11726 MATCH_MP_TAC(REAL_ARITH `s = t ==> &1 - s + t = &1`) THEN
11727 MATCH_MP_TAC SUM_EQ THEN ASM_MESON_TAC[];
11728 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
11729 MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[]]]);;
11731 let STD_SIMPLEX = prove
11732 (`convex hull (vec 0 INSERT { basis i | 1 <= i /\ i <= dimindex(:N)}) =
11733 {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 <= x$i) /\
11734 sum (1..dimindex(:N)) (\i. x$i) <= &1 }`,
11735 W(MP_TAC o PART_MATCH (lhs o rand) SIMPLEX_EXPLICIT o lhs o snd) THEN ANTS_TAC THENL
11736 [REWRITE_TAC[SIMPLE_IMAGE; GSYM IN_NUMSEG] THEN
11737 SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG; IN_IMAGE] THEN
11738 REWRITE_TAC[IN_NUMSEG] THEN MESON_TAC[BASIS_NONZERO];
11740 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION] THEN
11741 ONCE_REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN
11742 REWRITE_TAC[SIMPLE_IMAGE; GSYM IN_NUMSEG] THEN
11743 SUBGOAL_THEN `!u. sum (IMAGE (basis:num->real^N) (1..dimindex(:N))) u =
11744 sum (1..dimindex(:N)) (u o basis)`
11745 (fun th -> REWRITE_TAC[th])
11747 [GEN_TAC THEN MATCH_MP_TAC SUM_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN
11748 REWRITE_TAC[GSYM CONJ_ASSOC; BASIS_INJ];
11750 SUBGOAL_THEN `!u. vsum (IMAGE (basis:num->real^N) (1..dimindex(:N))) u =
11751 vsum (1..dimindex(:N)) ((u:real^N->real^N) o basis)`
11752 (fun th -> REWRITE_TAC[th])
11754 [GEN_TAC THEN MATCH_MP_TAC VSUM_IMAGE THEN REWRITE_TAC[IN_NUMSEG] THEN
11755 REWRITE_TAC[GSYM CONJ_ASSOC; BASIS_INJ; FINITE_NUMSEG];
11757 REWRITE_TAC[o_DEF; BASIS_EXPANSION_UNIQUE; FORALL_IN_IMAGE] THEN
11758 REWRITE_TAC[IN_NUMSEG] THEN EQ_TAC THENL
11759 [DISCH_THEN(X_CHOOSE_THEN `u:real^N->real` STRIP_ASSUME_TAC) THEN
11760 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
11761 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
11762 `x <= &1 ==> x = y ==> y <= &1`)) THEN
11763 MATCH_MP_TAC SUM_EQ THEN ASM_SIMP_TAC[IN_NUMSEG];
11764 STRIP_TAC THEN EXISTS_TAC `\y:real^N. y dot x` THEN
11765 ASM_SIMP_TAC[DOT_BASIS]]);;
11767 let INTERIOR_STD_SIMPLEX = prove
11769 (convex hull (vec 0 INSERT { basis i | 1 <= i /\ i <= dimindex(:N)})) =
11770 {x:real^N | (!i. 1 <= i /\ i <= dimindex(:N) ==> &0 < x$i) /\
11771 sum (1..dimindex(:N)) (\i. x$i) < &1 }`,
11772 REWRITE_TAC[EXTENSION; IN_INTERIOR; IN_ELIM_THM; STD_SIMPLEX] THEN
11773 REWRITE_TAC[SUBSET; IN_BALL; IN_ELIM_THM] THEN
11774 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
11775 [DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
11776 FIRST_ASSUM(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[DIST_REFL] THEN
11777 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_SIMP_TAC[REAL_LT_LE] THEN
11779 [X_GEN_TAC `k:num` THEN STRIP_TAC THEN
11780 FIRST_X_ASSUM(MP_TAC o SPEC `x - (e / &2) % basis k:real^N`) THEN
11781 REWRITE_TAC[NORM_ARITH `dist(x,x - e) = norm(e)`; NORM_MUL] THEN
11782 ASM_SIMP_TAC[NORM_BASIS; REAL_ARITH `&0 < e ==> abs(e / &2) * &1 < e`;
11783 VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN
11784 DISCH_THEN(MP_TAC o SPEC `k:num` o CONJUNCT1) THEN ASM_REWRITE_TAC[] THEN
11785 ASM_SIMP_TAC[BASIS_COMPONENT] THEN UNDISCH_TAC `&0 < e` THEN
11787 FIRST_X_ASSUM(MP_TAC o SPEC `x + (e / &2) % basis 1:real^N`) THEN
11788 REWRITE_TAC[NORM_ARITH `dist(x,x + e) = norm(e)`; NORM_MUL] THEN
11789 ASM_SIMP_TAC[NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN
11790 ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> abs(e / &2) * &1 < e`] THEN
11791 DISCH_THEN(MP_TAC o CONJUNCT2) THEN
11792 MATCH_MP_TAC(REAL_ARITH `x < y ==> y <= &1 ==> ~(x = &1)`) THEN
11793 MATCH_MP_TAC SUM_LT THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
11794 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c <=> ~(a /\ b ==> ~c)`] THEN
11795 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
11796 BASIS_COMPONENT] THEN
11798 [GEN_TAC THEN COND_CASES_TAC;
11799 EXISTS_TAC `1` THEN REWRITE_TAC[LE_REFL; DIMINDEX_GE_1]] THEN
11800 ASM_REAL_ARITH_TAC];
11803 `min (inf(IMAGE (\i. (x:real^N)$i) (1..dimindex(:N))))
11804 ((&1 - sum (1..dimindex(:N)) (\i. x$i)) / &(dimindex(:N)))` THEN
11805 ASM_SIMP_TAC[REAL_LT_MIN] THEN
11806 SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; FINITE_NUMSEG;
11807 IMAGE_EQ_EMPTY; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN
11808 REWRITE_TAC[FORALL_IN_IMAGE] THEN
11809 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT;
11810 ARITH_RULE `0 < x <=> 1 <= x`; DIMINDEX_GE_1] THEN
11811 ASM_REWRITE_TAC[IN_NUMSEG; REAL_MUL_LZERO; REAL_SUB_LT] THEN
11812 REPEAT(POP_ASSUM(K ALL_TAC)) THEN X_GEN_TAC `y:real^N` THEN
11813 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
11814 [MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `k:num` THEN
11815 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
11816 ASM_REWRITE_TAC[] THEN
11817 MATCH_MP_TAC(REAL_ARITH `abs(xk - yk) <= d ==> d < xk ==> &0 <= yk`);
11818 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV o RAND_CONV)
11819 [GSYM CARD_NUMSEG_1] THEN
11820 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
11821 SIMP_TAC[GSYM SUM_CONST; FINITE_NUMSEG] THEN
11822 MATCH_MP_TAC(REAL_ARITH
11823 `s2 <= s0 + s1 ==> s0 < &1 - s1 ==> s2 <= &1`) THEN
11824 REWRITE_TAC[GSYM SUM_ADD_NUMSEG] THEN
11825 MATCH_MP_TAC SUM_LE_NUMSEG THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
11826 MATCH_MP_TAC(REAL_ARITH `abs(y - x) <= z ==> x <= z + y`)] THEN
11827 ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; dist; COMPONENT_LE_NORM]]);;
git://colo12-c703.uibk.ac.at / hl193./.git / blob