1 (* (c) Copyright, John Harrison 1998-2014 *)
2 (* (c) Copyright, Valentina Bruno 2010 *)
3 (* Distributed under the same license as HOL Light *)
5 (* Theorems taken directly from Multivariate/topology.ml which run after *)
6 (* loading Topology.ml. *)
8 needs "Library/card.ml";;
9 needs "Multivariate/determinants.ml";;
10 needs "RichterHilbertAxiomGeometry/Topology.ml";;
12 (* ------------------------------------------------------------------------- *)
13 (* Open and closed balls and spheres. *)
14 (* ------------------------------------------------------------------------- *)
16 let sphere = new_definition
17 `sphere(x,e) = { y | dist(x,y) = e}`;;
20 (`!x y e. y IN sphere(x,e) <=> dist(x,y) = e`,
21 REWRITE_TAC[sphere; IN_ELIM_THM]);;
24 (`!x e. x IN ball(vec 0,e) <=> norm(x) < e`,
25 REWRITE_TAC[IN_BALL; dist; VECTOR_SUB_LZERO; NORM_NEG]);;
27 let IN_CBALL_0 = prove
28 (`!x e. x IN cball(vec 0,e) <=> norm(x) <= e`,
29 REWRITE_TAC[IN_CBALL; dist; VECTOR_SUB_LZERO; NORM_NEG]);;
31 let IN_SPHERE_0 = prove
32 (`!x e. x IN sphere(vec 0,e) <=> norm(x) = e`,
33 REWRITE_TAC[IN_SPHERE; dist; VECTOR_SUB_LZERO; NORM_NEG]);;
35 let BALL_TRIVIAL = prove
36 (`!x. ball(x,&0) = {}`,
37 REWRITE_TAC[EXTENSION; IN_BALL; IN_SING; NOT_IN_EMPTY] THEN NORM_ARITH_TAC);;
39 let CBALL_TRIVIAL = prove
40 (`!x. cball(x,&0) = {x}`,
41 REWRITE_TAC[EXTENSION; IN_CBALL; IN_SING; NOT_IN_EMPTY] THEN NORM_ARITH_TAC);;
43 let CENTRE_IN_CBALL = prove
44 (`!x e. x IN cball(x,e) <=> &0 <= e`,
45 MESON_TAC[IN_CBALL; DIST_REFL]);;
47 let SPHERE_SUBSET_CBALL = prove
48 (`!x e. sphere(x,e) SUBSET cball(x,e)`,
49 REWRITE_TAC[IN_SPHERE; IN_CBALL; SUBSET] THEN REAL_ARITH_TAC);;
51 let SUBSET_BALL = prove
52 (`!x d e. d <= e ==> ball(x,d) SUBSET ball(x,e)`,
53 REWRITE_TAC[SUBSET; IN_BALL] THEN MESON_TAC[REAL_LTE_TRANS]);;
55 let SUBSET_CBALL = prove
56 (`!x d e. d <= e ==> cball(x,d) SUBSET cball(x,e)`,
57 REWRITE_TAC[SUBSET; IN_CBALL] THEN MESON_TAC[REAL_LE_TRANS]);;
59 let BALL_MAX_UNION = prove
60 (`!a r s. ball(a,max r s) = ball(a,r) UNION ball(a,s)`,
61 REWRITE_TAC[IN_BALL; IN_UNION; EXTENSION] THEN REAL_ARITH_TAC);;
63 let BALL_MIN_INTER = prove
64 (`!a r s. ball(a,min r s) = ball(a,r) INTER ball(a,s)`,
65 REWRITE_TAC[IN_BALL; IN_INTER; EXTENSION] THEN REAL_ARITH_TAC);;
67 let CBALL_MAX_UNION = prove
68 (`!a r s. cball(a,max r s) = cball(a,r) UNION cball(a,s)`,
69 REWRITE_TAC[IN_CBALL; IN_UNION; EXTENSION] THEN REAL_ARITH_TAC);;
71 let CBALL_MIN_INTER = prove
72 (`!x d e. cball(x,min d e) = cball(x,d) INTER cball(x,e)`,
73 REWRITE_TAC[EXTENSION; IN_INTER; IN_CBALL] THEN REAL_ARITH_TAC);;
75 let BALL_TRANSLATION = prove
76 (`!a x r. ball(a + x,r) = IMAGE (\y. a + y) (ball(x,r))`,
77 REWRITE_TAC[ball] THEN GEOM_TRANSLATE_TAC[]);;
79 let CBALL_TRANSLATION = prove
80 (`!a x r. cball(a + x,r) = IMAGE (\y. a + y) (cball(x,r))`,
81 REWRITE_TAC[cball] THEN GEOM_TRANSLATE_TAC[]);;
83 let SPHERE_TRANSLATION = prove
84 (`!a x r. sphere(a + x,r) = IMAGE (\y. a + y) (sphere(x,r))`,
85 REWRITE_TAC[sphere] THEN GEOM_TRANSLATE_TAC[]);;
87 add_translation_invariants
88 [BALL_TRANSLATION; CBALL_TRANSLATION; SPHERE_TRANSLATION];;
90 let BALL_LINEAR_IMAGE = prove
91 (`!f:real^M->real^N x r.
92 linear f /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x)
93 ==> ball(f x,r) = IMAGE f (ball(x,r))`,
94 REWRITE_TAC[ball] THEN GEOM_TRANSFORM_TAC[]);;
96 let CBALL_LINEAR_IMAGE = prove
97 (`!f:real^M->real^N x r.
98 linear f /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x)
99 ==> cball(f x,r) = IMAGE f (cball(x,r))`,
100 REWRITE_TAC[cball] THEN GEOM_TRANSFORM_TAC[]);;
102 let SPHERE_LINEAR_IMAGE = prove
103 (`!f:real^M->real^N x r.
104 linear f /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x)
105 ==> sphere(f x,r) = IMAGE f (sphere(x,r))`,
106 REWRITE_TAC[sphere] THEN GEOM_TRANSFORM_TAC[]);;
108 add_linear_invariants
109 [BALL_LINEAR_IMAGE; CBALL_LINEAR_IMAGE; SPHERE_LINEAR_IMAGE];;
111 let BALL_SCALING = prove
112 (`!c. &0 < c ==> !x r. ball(c % x,c * r) = IMAGE (\x. c % x) (ball(x,r))`,
113 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
114 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[] THEN CONJ_TAC THENL
115 [ASM_MESON_TAC[SURJECTIVE_SCALING; REAL_LT_IMP_NZ]; ALL_TAC] THEN
116 REWRITE_TAC[IN_BALL; DIST_MUL] THEN
117 ASM_SIMP_TAC[REAL_ARITH `&0 < c ==> abs c = c`; REAL_LT_LMUL_EQ]);;
119 let CBALL_SCALING = prove
120 (`!c. &0 < c ==> !x r. cball(c % x,c * r) = IMAGE (\x. c % x) (cball(x,r))`,
121 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
122 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN REWRITE_TAC[] THEN CONJ_TAC THENL
123 [ASM_MESON_TAC[SURJECTIVE_SCALING; REAL_LT_IMP_NZ]; ALL_TAC] THEN
124 REWRITE_TAC[IN_CBALL; DIST_MUL] THEN
125 ASM_SIMP_TAC[REAL_ARITH `&0 < c ==> abs c = c`; REAL_LE_LMUL_EQ]);;
127 add_scaling_theorems [BALL_SCALING; CBALL_SCALING];;
129 let CBALL_DIFF_BALL = prove
130 (`!a r. cball(a,r) DIFF ball(a,r) = sphere(a,r)`,
131 REWRITE_TAC[ball; cball; sphere; EXTENSION; IN_DIFF; IN_ELIM_THM] THEN
134 let BALL_UNION_SPHERE = prove
135 (`!a r. ball(a,r) UNION sphere(a,r) = cball(a,r)`,
136 REWRITE_TAC[ball; cball; sphere; EXTENSION; IN_UNION; IN_ELIM_THM] THEN
139 let SPHERE_UNION_BALL = prove
140 (`!a r. sphere(a,r) UNION ball(a,r) = cball(a,r)`,
141 REWRITE_TAC[ball; cball; sphere; EXTENSION; IN_UNION; IN_ELIM_THM] THEN
144 let CBALL_DIFF_SPHERE = prove
145 (`!a r. cball(a,r) DIFF sphere(a,r) = ball(a,r)`,
146 REWRITE_TAC[EXTENSION; IN_DIFF; IN_SPHERE; IN_BALL; IN_CBALL] THEN
149 let OPEN_CONTAINS_BALL_EQ = prove
150 (`!s. open s ==> (!x. x IN s <=> ?e. &0 < e /\ ball(x,e) SUBSET s)`,
151 MESON_TAC[OPEN_CONTAINS_BALL; SUBSET; CENTRE_IN_BALL]);;
153 let BALL_EQ_EMPTY = prove
154 (`!x e. (ball(x,e) = {}) <=> e <= &0`,
155 REWRITE_TAC[EXTENSION; IN_BALL; NOT_IN_EMPTY; REAL_NOT_LT] THEN
156 MESON_TAC[DIST_POS_LE; REAL_LE_TRANS; DIST_REFL]);;
158 let BALL_EMPTY = prove
159 (`!x e. e <= &0 ==> ball(x,e) = {}`,
160 REWRITE_TAC[BALL_EQ_EMPTY]);;
162 let OPEN_CONTAINS_CBALL = prove
163 (`!s. open s <=> !x. x IN s ==> ?e. &0 < e /\ cball(x,e) SUBSET s`,
164 GEN_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN EQ_TAC THENL
165 [ALL_TAC; ASM_MESON_TAC[SUBSET_TRANS; BALL_SUBSET_CBALL]] THEN
166 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
167 REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN
168 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
169 EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN
170 SUBGOAL_THEN `e / &2 < e` (fun th -> ASM_MESON_TAC[th; REAL_LET_TRANS]) THEN
171 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
172 UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC);;
174 let OPEN_CONTAINS_CBALL_EQ = prove
175 (`!s. open s ==> (!x. x IN s <=> ?e. &0 < e /\ cball(x,e) SUBSET s)`,
176 MESON_TAC[OPEN_CONTAINS_CBALL; SUBSET; REAL_LT_IMP_LE; CENTRE_IN_CBALL]);;
178 let SPHERE_EQ_EMPTY = prove
179 (`!a:real^N r. sphere(a,r) = {} <=> r < &0`,
180 REWRITE_TAC[sphere; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN
181 REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; CONV_TAC NORM_ARITH] THEN
182 MESON_TAC[VECTOR_CHOOSE_DIST; REAL_NOT_LE]);;
184 let SPHERE_EMPTY = prove
185 (`!a:real^N r. r < &0 ==> sphere(a,r) = {}`,
186 REWRITE_TAC[SPHERE_EQ_EMPTY]);;
188 let NEGATIONS_BALL = prove
189 (`!r. IMAGE (--) (ball(vec 0:real^N,r)) = ball(vec 0,r)`,
190 GEN_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
191 REWRITE_TAC[IN_BALL_0; NORM_NEG] THEN MESON_TAC[VECTOR_NEG_NEG]);;
193 let NEGATIONS_CBALL = prove
194 (`!r. IMAGE (--) (cball(vec 0:real^N,r)) = cball(vec 0,r)`,
195 GEN_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
196 REWRITE_TAC[IN_CBALL_0; NORM_NEG] THEN MESON_TAC[VECTOR_NEG_NEG]);;
198 let NEGATIONS_SPHERE = prove
199 (`!r. IMAGE (--) (sphere(vec 0:real^N,r)) = sphere(vec 0,r)`,
200 GEN_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
201 REWRITE_TAC[IN_SPHERE_0; NORM_NEG] THEN MESON_TAC[VECTOR_NEG_NEG]);;
203 let ORTHOGONAL_TRANSFORMATION_BALL = prove
204 (`!f:real^N->real^N r.
205 orthogonal_transformation f ==> IMAGE f (ball(vec 0,r)) = ball(vec 0,r)`,
206 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_BALL_0] THEN
207 MESON_TAC[ORTHOGONAL_TRANSFORMATION_INVERSE; ORTHOGONAL_TRANSFORMATION]);;
209 let ORTHOGONAL_TRANSFORMATION_CBALL = prove
210 (`!f:real^N->real^N r.
211 orthogonal_transformation f ==> IMAGE f (cball(vec 0,r)) = cball(vec 0,r)`,
212 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_CBALL_0] THEN
213 MESON_TAC[ORTHOGONAL_TRANSFORMATION_INVERSE; ORTHOGONAL_TRANSFORMATION]);;
215 let ORTHOGONAL_TRANSFORMATION_SPHERE = prove
216 (`!f:real^N->real^N r.
217 orthogonal_transformation f
218 ==> IMAGE f (sphere(vec 0,r)) = sphere(vec 0,r)`,
219 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_SPHERE_0] THEN
220 MESON_TAC[ORTHOGONAL_TRANSFORMATION_INVERSE; ORTHOGONAL_TRANSFORMATION]);;
222 (* ------------------------------------------------------------------------- *)
223 (* Also some invariance theorems for relative topology. *)
224 (* ------------------------------------------------------------------------- *)
226 let OPEN_IN_TRANSLATION_EQ = prove
227 (`!a s t. open_in (subtopology euclidean (IMAGE (\x. a + x) t))
228 (IMAGE (\x. a + x) s) <=>
229 open_in (subtopology euclidean t) s`,
230 REWRITE_TAC[open_in] THEN GEOM_TRANSLATE_TAC[]);;
232 add_translation_invariants [OPEN_IN_TRANSLATION_EQ];;
234 let CLOSED_IN_TRANSLATION_EQ = prove
235 (`!a s t. closed_in (subtopology euclidean (IMAGE (\x. a + x) t))
236 (IMAGE (\x. a + x) s) <=>
237 closed_in (subtopology euclidean t) s`,
238 REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
239 GEOM_TRANSLATE_TAC[]);;
241 add_translation_invariants [CLOSED_IN_TRANSLATION_EQ];;
243 (* ------------------------------------------------------------------------- *)
245 (* ------------------------------------------------------------------------- *)
247 let LIMPT_APPROACHABLE = prove
248 (`!x s. x limit_point_of s <=>
249 !e. &0 < e ==> ?x'. x' IN s /\ ~(x' = x) /\ dist(x',x) < e`,
250 REPEAT GEN_TAC THEN REWRITE_TAC[limit_point_of] THEN
251 MESON_TAC[open_def; DIST_SYM; OPEN_BALL; CENTRE_IN_BALL; IN_BALL]);;
253 let LIMPT_APPROACHABLE_LE = prove
254 (`!x s. x limit_point_of s <=>
255 !e. &0 < e ==> ?x'. x' IN s /\ ~(x' = x) /\ dist(x',x) <= e`,
256 REPEAT GEN_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN
257 MATCH_MP_TAC(TAUT `(~a <=> ~b) ==> (a <=> b)`) THEN
258 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN
259 REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> c ==> ~(a /\ b)`; APPROACHABLE_LT_LE]);;
261 let LIMPT_UNIV = prove
262 (`!x:real^N. x limit_point_of UNIV`,
263 GEN_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE; IN_UNIV] THEN
264 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
265 SUBGOAL_THEN `?c:real^N. norm(c) = e / &2` CHOOSE_TAC THENL
266 [ASM_SIMP_TAC[VECTOR_CHOOSE_SIZE; REAL_HALF; REAL_LT_IMP_LE];
268 EXISTS_TAC `x + c:real^N` THEN
269 REWRITE_TAC[dist; VECTOR_EQ_ADDR] THEN ASM_REWRITE_TAC[VECTOR_ADD_SUB] THEN
270 SUBGOAL_THEN `&0 < e / &2 /\ e / &2 < e`
271 (fun th -> ASM_MESON_TAC[th; NORM_0; REAL_LT_REFL]) THEN
272 SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
273 UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC);;
275 let CLOSED_POSITIVE_ORTHANT = prove
276 (`closed {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
278 REWRITE_TAC[CLOSED_LIMPT; LIMPT_APPROACHABLE] THEN
279 REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
280 X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN
281 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `--(x:real^N $ i)`) THEN
282 ASM_REWRITE_TAC[REAL_LT_RNEG; REAL_ADD_LID; NOT_EXISTS_THM] THEN
283 X_GEN_TAC `y:real^N` THEN
284 MATCH_MP_TAC(TAUT `(a ==> ~c) ==> ~(a /\ b /\ c)`) THEN DISCH_TAC THEN
285 MATCH_MP_TAC(REAL_ARITH `!b. abs x <= b /\ b <= a ==> ~(a + x < &0)`) THEN
286 EXISTS_TAC `abs((y - x :real^N)$i)` THEN
287 ASM_SIMP_TAC[dist; COMPONENT_LE_NORM] THEN
288 ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; REAL_ARITH
289 `x < &0 /\ &0 <= y ==> abs(x) <= abs(y - x)`]);;
291 let FINITE_SET_AVOID = prove
292 (`!a:real^N s. FINITE s
293 ==> ?d. &0 < d /\ !x. x IN s /\ ~(x = a) ==> d <= dist(a,x)`,
294 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
295 REWRITE_TAC[NOT_IN_EMPTY] THEN
296 CONJ_TAC THENL [MESON_TAC[REAL_LT_01]; ALL_TAC] THEN
297 MAP_EVERY X_GEN_TAC [`x:real^N`; `s:real^N->bool`] THEN
298 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
299 FIRST_X_ASSUM(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
300 ASM_CASES_TAC `x:real^N = a` THEN REWRITE_TAC[IN_INSERT] THENL
301 [ASM_MESON_TAC[]; ALL_TAC] THEN
302 EXISTS_TAC `min d (dist(a:real^N,x))` THEN
303 ASM_REWRITE_TAC[REAL_LT_MIN; GSYM DIST_NZ; REAL_MIN_LE] THEN
304 ASM_MESON_TAC[REAL_LE_REFL]);;
306 let LIMIT_POINT_FINITE = prove
307 (`!s a. FINITE s ==> ~(a limit_point_of s)`,
308 REWRITE_TAC[LIMPT_APPROACHABLE; GSYM REAL_NOT_LE] THEN
309 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM; REAL_NOT_LE;
310 REAL_NOT_LT; TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`] THEN
311 MESON_TAC[FINITE_SET_AVOID; DIST_SYM]);;
313 let LIMPT_SING = prove
314 (`!x y:real^N. ~(x limit_point_of {y})`,
315 SIMP_TAC[LIMIT_POINT_FINITE; FINITE_SING]);;
317 let LIMPT_INSERT = prove
318 (`!s x y:real^N. x limit_point_of (y INSERT s) <=> x limit_point_of s`,
319 ONCE_REWRITE_TAC[SET_RULE `y INSERT s = {y} UNION s`] THEN
320 REWRITE_TAC[LIMIT_POINT_UNION] THEN
321 SIMP_TAC[FINITE_SING; LIMIT_POINT_FINITE]);;
323 let LIMPT_OF_LIMPTS = prove
325 x limit_point_of {y | y limit_point_of s} ==> x limit_point_of s`,
326 REWRITE_TAC[LIMPT_APPROACHABLE; IN_ELIM_THM] THEN REPEAT GEN_TAC THEN
327 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
328 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
329 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
330 FIRST_X_ASSUM(MP_TAC o SPEC `dist(y:real^N,x)`) THEN
331 ASM_SIMP_TAC[DIST_POS_LT] THEN MATCH_MP_TAC MONO_EXISTS THEN
332 GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
333 REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);;
335 let CLOSED_LIMPTS = prove
336 (`!s. closed {x:real^N | x limit_point_of s}`,
337 REWRITE_TAC[CLOSED_LIMPT; IN_ELIM_THM; LIMPT_OF_LIMPTS]);;
339 let DISCRETE_IMP_CLOSED = prove
342 (!x y. x IN s /\ y IN s /\ norm(y - x) < e ==> y = x)
344 REPEAT STRIP_TAC THEN
345 SUBGOAL_THEN `!x:real^N. ~(x limit_point_of s)`
346 (fun th -> MESON_TAC[th; CLOSED_LIMPT]) THEN
347 GEN_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN DISCH_TAC THEN
348 FIRST_ASSUM(MP_TAC o SPEC `e / &2`) THEN
349 REWRITE_TAC[REAL_HALF; ASSUME `&0 < e`] THEN
350 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
351 FIRST_X_ASSUM(MP_TAC o SPEC `min (e / &2) (dist(x:real^N,y))`) THEN
352 ASM_SIMP_TAC[REAL_LT_MIN; DIST_POS_LT; REAL_HALF] THEN
353 DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
354 FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^N`; `z:real^N`]) THEN
355 ASM_REWRITE_TAC[] THEN ASM_NORM_ARITH_TAC);;
357 let LIMPT_OF_UNIV = prove
358 (`!x. x limit_point_of (:real^N)`,
359 GEN_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE; IN_UNIV] THEN
360 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
361 MP_TAC(ISPECL [`x:real^N`; `e / &2`] VECTOR_CHOOSE_DIST) THEN
362 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN
363 POP_ASSUM MP_TAC THEN CONV_TAC NORM_ARITH);;
365 let LIMPT_OF_OPEN_IN = prove
367 open_in (subtopology euclidean s) t /\ x limit_point_of s /\ x IN t
368 ==> x limit_point_of t`,
369 REWRITE_TAC[open_in; SUBSET; LIMPT_APPROACHABLE] THEN
370 REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
371 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
372 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
373 FIRST_X_ASSUM(MP_TAC o SPEC `min d e / &2`) THEN
374 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN
375 GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN
376 TRY(FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN
377 ASM_REAL_ARITH_TAC);;
379 let LIMPT_OF_OPEN = prove
380 (`!s x:real^N. open s /\ x IN s ==> x limit_point_of s`,
381 REWRITE_TAC[OPEN_IN] THEN ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN
382 MESON_TAC[LIMPT_OF_OPEN_IN; LIMPT_OF_UNIV]);;
384 let OPEN_IN_SING = prove
385 (`!s a. open_in (subtopology euclidean s) {a} <=>
386 a IN s /\ ~(a limit_point_of s)`,
387 REWRITE_TAC[open_in; LIMPT_APPROACHABLE; SING_SUBSET; IN_SING] THEN
388 REWRITE_TAC[FORALL_UNWIND_THM2] THEN MESON_TAC[]);;
390 (* ------------------------------------------------------------------------- *)
391 (* Interior of a set. *)
392 (* ------------------------------------------------------------------------- *)
394 let INTERIOR_LIMIT_POINT = prove
395 (`!s x:real^N. x IN interior s ==> x limit_point_of s`,
397 REWRITE_TAC[IN_INTERIOR; IN_ELIM_THM; SUBSET; IN_BALL] THEN
398 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
399 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `d:real` THEN
401 MP_TAC(ISPECL [`x:real^N`; `min d e / &2`] VECTOR_CHOOSE_DIST) THEN
402 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
403 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
404 REPEAT CONJ_TAC THENL
405 [FIRST_X_ASSUM MATCH_MP_TAC;
406 CONV_TAC (RAND_CONV SYM_CONV) THEN REWRITE_TAC[GSYM DIST_EQ_0];
407 ONCE_REWRITE_TAC[DIST_SYM]] THEN
408 ASM_REAL_ARITH_TAC);;
410 let INTERIOR_SING = prove
411 (`!a:real^N. interior {a} = {}`,
412 REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN
413 MESON_TAC[INTERIOR_LIMIT_POINT; LIMPT_SING]);;
415 (* ------------------------------------------------------------------------- *)
416 (* Closure of a set. *)
417 (* ------------------------------------------------------------------------- *)
419 let LIMPT_OF_CLOSURE = prove
420 (`!x:real^N s. x limit_point_of closure s <=> x limit_point_of s`,
421 REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM; LIMIT_POINT_UNION] THEN
422 REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT `(q ==> p) ==> (p \/ q <=> p)`) THEN
423 REWRITE_TAC[LIMPT_OF_LIMPTS]);;
425 (* ------------------------------------------------------------------------- *)
426 (* A variant of nets (slightly non-standard but good for our purposes). *)
427 (* ------------------------------------------------------------------------- *)
429 let net_tybij = new_type_definition "net" ("mk_net","netord")
431 (`?g:A->A->bool. !x y. (!z. g z x ==> g z y) \/ (!z. g z y ==> g z x)`,
432 EXISTS_TAC `\x:A y:A. F` THEN REWRITE_TAC[]));;
435 (`!n x y. (!z. netord n z x ==> netord n z y) \/
436 (!z. netord n z y ==> netord n z x)`,
437 REWRITE_TAC[net_tybij; ETA_AX]);;
440 (`!n x y. netord n x x /\ netord n y y
441 ==> ?z. netord n z z /\
442 !w. netord n w z ==> netord n w x /\ netord n w y`,
445 let NET_DILEMMA = prove
446 (`!net. (?a. (?x. netord net x a) /\ (!x. netord net x a ==> P x)) /\
447 (?b. (?x. netord net x b) /\ (!x. netord net x b ==> Q x))
448 ==> ?c. (?x. netord net x c) /\ (!x. netord net x c ==> P x /\ Q x)`,
451 (* ------------------------------------------------------------------------- *)
452 (* Common nets and the "within" modifier for nets. *)
453 (* ------------------------------------------------------------------------- *)
455 parse_as_infix("within",(14,"right"));;
456 parse_as_infix("in_direction",(14,"right"));;
458 let at = new_definition
459 `at a = mk_net(\x y. &0 < dist(x,a) /\ dist(x,a) <= dist(y,a))`;;
461 let at_infinity = new_definition
462 `at_infinity = mk_net(\x y. norm(x) >= norm(y))`;;
464 let sequentially = new_definition
465 `sequentially = mk_net(\m:num n. m >= n)`;;
467 let within = new_definition
468 `net within s = mk_net(\x y. netord net x y /\ x IN s)`;;
470 let in_direction = new_definition
471 `a in_direction v = (at a) within {b | ?c. &0 <= c /\ (b - a = c % v)}`;;
473 (* ------------------------------------------------------------------------- *)
474 (* Prove that they are all nets. *)
475 (* ------------------------------------------------------------------------- *)
477 let NET_PROVE_TAC[def] =
478 REWRITE_TAC[GSYM FUN_EQ_THM; def] THEN
479 REWRITE_TAC[ETA_AX] THEN
480 ASM_SIMP_TAC[GSYM(CONJUNCT2 net_tybij)];;
484 netord(at a) x y <=> &0 < dist(x,a) /\ dist(x,a) <= dist(y,a)`,
485 GEN_TAC THEN NET_PROVE_TAC[at] THEN
486 MESON_TAC[REAL_LE_TOTAL; REAL_LE_REFL; REAL_LE_TRANS; REAL_LET_TRANS]);;
488 let AT_INFINITY = prove
489 (`!x y. netord at_infinity x y <=> norm(x) >= norm(y)`,
490 NET_PROVE_TAC[at_infinity] THEN
491 REWRITE_TAC[real_ge; REAL_LE_REFL] THEN
492 MESON_TAC[REAL_LE_TOTAL; REAL_LE_REFL; REAL_LE_TRANS]);;
494 let SEQUENTIALLY = prove
495 (`!m n. netord sequentially m n <=> m >= n`,
496 NET_PROVE_TAC[sequentially] THEN REWRITE_TAC[GE; LE_REFL] THEN
497 MESON_TAC[LE_CASES; LE_REFL; LE_TRANS]);;
500 (`!n s x y. netord(n within s) x y <=> netord n x y /\ x IN s`,
501 GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[within; GSYM FUN_EQ_THM] THEN
502 REWRITE_TAC[GSYM(CONJUNCT2 net_tybij); ETA_AX] THEN
505 let IN_DIRECTION = prove
506 (`!a v x y. netord(a in_direction v) x y <=>
507 &0 < dist(x,a) /\ dist(x,a) <= dist(y,a) /\
508 ?c. &0 <= c /\ (x - a = c % v)`,
509 REWRITE_TAC[WITHIN; AT; in_direction; IN_ELIM_THM; CONJ_ACI]);;
511 let WITHIN_UNIV = prove
512 (`!x:real^N. at x within UNIV = at x`,
513 REWRITE_TAC[within; at; IN_UNIV] THEN REWRITE_TAC[ETA_AX; net_tybij]);;
515 let WITHIN_WITHIN = prove
516 (`!net s t. (net within s) within t = net within (s INTER t)`,
517 ONCE_REWRITE_TAC[within] THEN
518 REWRITE_TAC[WITHIN; IN_INTER; GSYM CONJ_ASSOC]);;
520 (* ------------------------------------------------------------------------- *)
521 (* Identify trivial limits, where we can't approach arbitrarily closely. *)
522 (* ------------------------------------------------------------------------- *)
524 let trivial_limit = new_definition
525 `trivial_limit net <=>
527 ?a:A b. ~(a = b) /\ !x. ~(netord(net) x a) /\ ~(netord(net) x b)`;;
529 let TRIVIAL_LIMIT_WITHIN = prove
530 (`!a:real^N. trivial_limit (at a within s) <=> ~(a limit_point_of s)`,
531 REWRITE_TAC[trivial_limit; LIMPT_APPROACHABLE_LE; WITHIN; AT; DIST_NZ] THEN
532 REPEAT GEN_TAC THEN EQ_TAC THENL
533 [DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL
534 [MESON_TAC[REAL_LT_01; REAL_LT_REFL; VECTOR_CHOOSE_DIST;
535 DIST_REFL; REAL_LT_IMP_LE];
536 DISCH_THEN(X_CHOOSE_THEN `b:real^N` (X_CHOOSE_THEN `c:real^N`
537 STRIP_ASSUME_TAC)) THEN
538 SUBGOAL_THEN `&0 < dist(a,b:real^N) \/ &0 < dist(a,c:real^N)` MP_TAC THEN
539 ASM_MESON_TAC[DIST_TRIANGLE; DIST_SYM; GSYM DIST_NZ; GSYM DIST_EQ_0;
540 REAL_ARITH `x <= &0 + &0 ==> ~(&0 < x)`]];
541 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
542 X_GEN_TAC `e:real` THEN DISCH_TAC THEN DISJ2_TAC THEN
543 EXISTS_TAC `a:real^N` THEN
544 SUBGOAL_THEN `?b:real^N. dist(a,b) = e` MP_TAC THENL
545 [ASM_SIMP_TAC[VECTOR_CHOOSE_DIST; REAL_LT_IMP_LE]; ALL_TAC] THEN
546 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN
547 DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
548 ASM_MESON_TAC[REAL_NOT_LE; DIST_REFL; DIST_NZ; DIST_SYM]]);;
550 let TRIVIAL_LIMIT_AT = prove
551 (`!a. ~(trivial_limit (at a))`,
552 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
553 REWRITE_TAC[TRIVIAL_LIMIT_WITHIN; LIMPT_UNIV]);;
555 let TRIVIAL_LIMIT_AT_INFINITY = prove
556 (`~(trivial_limit at_infinity)`,
557 REWRITE_TAC[trivial_limit; AT_INFINITY; real_ge] THEN
558 MESON_TAC[REAL_LE_REFL; VECTOR_CHOOSE_SIZE; REAL_LT_01; REAL_LT_LE]);;
560 let TRIVIAL_LIMIT_SEQUENTIALLY = prove
561 (`~(trivial_limit sequentially)`,
562 REWRITE_TAC[trivial_limit; SEQUENTIALLY] THEN
563 MESON_TAC[GE_REFL; NOT_SUC]);;
565 let LIM_WITHIN_CLOSED_TRIVIAL = prove
566 (`!a s. closed s /\ ~(a IN s) ==> trivial_limit (at a within s)`,
567 REWRITE_TAC[TRIVIAL_LIMIT_WITHIN] THEN MESON_TAC[CLOSED_LIMPT]);;
569 let NONTRIVIAL_LIMIT_WITHIN = prove
570 (`!net s. trivial_limit net ==> trivial_limit(net within s)`,
571 REWRITE_TAC[trivial_limit; WITHIN] THEN MESON_TAC[]);;
573 (* ------------------------------------------------------------------------- *)
574 (* Some property holds "sufficiently close" to the limit point. *)
575 (* ------------------------------------------------------------------------- *)
577 let eventually = new_definition
578 `eventually p net <=>
580 ?y. (?x. netord net x y) /\ (!x. netord net x y ==> p x)`;;
582 let EVENTUALLY_HAPPENS = prove
583 (`!net p. eventually p net ==> trivial_limit net \/ ?x. p x`,
584 REWRITE_TAC[eventually] THEN MESON_TAC[]);;
586 let EVENTUALLY_WITHIN_LE = prove
588 eventually p (at a within s) <=>
589 ?d. &0 < d /\ !x. x IN s /\ &0 < dist(x,a) /\ dist(x,a) <= d ==> p(x)`,
590 REWRITE_TAC[eventually; AT; WITHIN; TRIVIAL_LIMIT_WITHIN] THEN
591 REWRITE_TAC[LIMPT_APPROACHABLE_LE; DIST_NZ] THEN
592 REPEAT GEN_TAC THEN EQ_TAC THENL [MESON_TAC[REAL_LTE_TRANS]; ALL_TAC] THEN
593 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
594 MATCH_MP_TAC(TAUT `(a ==> b) ==> ~a \/ b`) THEN DISCH_TAC THEN
595 SUBGOAL_THEN `?b:real^M. dist(a,b) = d` MP_TAC THENL
596 [ASM_SIMP_TAC[VECTOR_CHOOSE_DIST; REAL_LT_IMP_LE]; ALL_TAC] THEN
597 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^M` THEN
598 DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
599 ASM_MESON_TAC[REAL_NOT_LE; DIST_REFL; DIST_NZ; DIST_SYM]);;
601 let EVENTUALLY_WITHIN = prove
603 eventually p (at a within s) <=>
604 ?d. &0 < d /\ !x. x IN s /\ &0 < dist(x,a) /\ dist(x,a) < d ==> p(x)`,
605 REWRITE_TAC[EVENTUALLY_WITHIN_LE] THEN
606 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> c ==> a /\ b ==> d`] THEN
607 REWRITE_TAC[APPROACHABLE_LT_LE]);;
609 let EVENTUALLY_AT = prove
610 (`!a p. eventually p (at a) <=>
611 ?d. &0 < d /\ !x. &0 < dist(x,a) /\ dist(x,a) < d ==> p(x)`,
612 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
613 REWRITE_TAC[EVENTUALLY_WITHIN; IN_UNIV]);;
615 let EVENTUALLY_SEQUENTIALLY = prove
616 (`!p. eventually p sequentially <=> ?N. !n. N <= n ==> p n`,
617 REWRITE_TAC[eventually; SEQUENTIALLY; GE; LE_REFL;
618 TRIVIAL_LIMIT_SEQUENTIALLY] THEN MESON_TAC[LE_REFL]);;
620 let EVENTUALLY_AT_INFINITY = prove
621 (`!p. eventually p at_infinity <=> ?b. !x. norm(x) >= b ==> p x`,
622 REWRITE_TAC[eventually; AT_INFINITY; TRIVIAL_LIMIT_AT_INFINITY] THEN
623 REPEAT GEN_TAC THEN EQ_TAC THENL [MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN
624 MESON_TAC[real_ge; REAL_LE_REFL; VECTOR_CHOOSE_SIZE;
625 REAL_ARITH `&0 <= b \/ (!x. x >= &0 ==> x >= b)`]);;
627 let EVENTUALLY_AT_INFINITY_POS = prove
629 eventually p at_infinity <=> ?b. &0 < b /\ !x. norm x >= b ==> p x`,
630 GEN_TAC THEN REWRITE_TAC[EVENTUALLY_AT_INFINITY; real_ge] THEN
631 MESON_TAC[REAL_ARITH `&0 < abs b + &1 /\ (abs b + &1 <= x ==> b <= x)`]);;
633 let ALWAYS_EVENTUALLY = prove
634 (`(!x. p x) ==> eventually p net`,
635 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[eventually; trivial_limit] THEN
638 (* ------------------------------------------------------------------------- *)
639 (* Combining theorems for "eventually". *)
640 (* ------------------------------------------------------------------------- *)
642 let EVENTUALLY_AND = prove
644 eventually (\x. p x /\ q x) net <=>
645 eventually p net /\ eventually q net`,
646 REPEAT GEN_TAC THEN REWRITE_TAC[eventually] THEN
647 ASM_CASES_TAC `trivial_limit(net:(A net))` THEN ASM_REWRITE_TAC[] THEN
648 EQ_TAC THEN SIMP_TAC[NET_DILEMMA] THEN MESON_TAC[]);;
650 let EVENTUALLY_MONO = prove
652 (!x. p x ==> q x) /\ eventually p net
653 ==> eventually q net`,
654 REWRITE_TAC[eventually] THEN MESON_TAC[]);;
656 let EVENTUALLY_MP = prove
658 eventually (\x. p x ==> q x) net /\ eventually p net
659 ==> eventually q net`,
660 REWRITE_TAC[GSYM EVENTUALLY_AND] THEN
661 REWRITE_TAC[eventually] THEN MESON_TAC[]);;
663 let EVENTUALLY_FALSE = prove
664 (`!net. eventually (\x. F) net <=> trivial_limit net`,
665 REWRITE_TAC[eventually] THEN MESON_TAC[]);;
667 let EVENTUALLY_TRUE = prove
668 (`!net. eventually (\x. T) net <=> T`,
669 REWRITE_TAC[eventually; trivial_limit] THEN MESON_TAC[]);;
671 let NOT_EVENTUALLY = prove
672 (`!net p. (!x. ~(p x)) /\ ~(trivial_limit net) ==> ~(eventually p net)`,
673 REWRITE_TAC[eventually] THEN MESON_TAC[]);;
675 let EVENTUALLY_FORALL = prove
676 (`!net:(A net) p s:B->bool.
677 FINITE s /\ ~(s = {})
678 ==> (eventually (\x. !a. a IN s ==> p a x) net <=>
679 !a. a IN s ==> eventually (p a) net)`,
680 GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
681 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
682 REWRITE_TAC[FORALL_IN_INSERT; EVENTUALLY_AND; ETA_AX] THEN
683 MAP_EVERY X_GEN_TAC [`b:B`; `t:B->bool`] THEN
684 ASM_CASES_TAC `t:B->bool = {}` THEN
685 ASM_SIMP_TAC[NOT_IN_EMPTY; EVENTUALLY_TRUE]);;
687 let FORALL_EVENTUALLY = prove
688 (`!net:(A net) p s:B->bool.
689 FINITE s /\ ~(s = {})
690 ==> ((!a. a IN s ==> eventually (p a) net) <=>
691 eventually (\x. !a. a IN s ==> p a x) net)`,
692 SIMP_TAC[EVENTUALLY_FORALL]);;
694 (* ------------------------------------------------------------------------- *)
695 (* Limits, defined as vacuously true when the limit is trivial. *)
696 (* ------------------------------------------------------------------------- *)
698 parse_as_infix("-->",(12,"right"));;
700 let tendsto = new_definition
701 `(f --> l) net <=> !e. &0 < e ==> eventually (\x. dist(f(x),l) < e) net`;;
703 let lim = new_definition
704 `lim net f = @l. (f --> l) net`;;
709 !e. &0 < e ==> ?y. (?x. netord(net) x y) /\
710 !x. netord(net) x y ==> dist(f(x),l) < e`,
711 REWRITE_TAC[tendsto; eventually] THEN MESON_TAC[]);;
713 (* ------------------------------------------------------------------------- *)
714 (* Show that they yield usual definitions in the various cases. *)
715 (* ------------------------------------------------------------------------- *)
717 let LIM_WITHIN_LE = prove
718 (`!f:real^M->real^N l a s.
719 (f --> l)(at a within s) <=>
720 !e. &0 < e ==> ?d. &0 < d /\
721 !x. x IN s /\ &0 < dist(x,a) /\ dist(x,a) <= d
722 ==> dist(f(x),l) < e`,
723 REWRITE_TAC[tendsto; EVENTUALLY_WITHIN_LE]);;
725 let LIM_WITHIN = prove
726 (`!f:real^M->real^N l a s.
727 (f --> l) (at a within s) <=>
730 !x. x IN s /\ &0 < dist(x,a) /\ dist(x,a) < d
731 ==> dist(f(x),l) < e`,
732 REWRITE_TAC[tendsto; EVENTUALLY_WITHIN] THEN MESON_TAC[]);;
734 let LIM_AT_LE = prove
735 (`!f l a. (f --> l) (at a) <=>
738 !x. &0 < dist(x,a) /\ dist(x,a) <= d
739 ==> dist (f x,l) < e`,
740 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
741 REWRITE_TAC[LIM_WITHIN_LE; IN_UNIV]);;
744 (`!f l:real^N a:real^M.
747 ==> ?d. &0 < d /\ !x. &0 < dist(x,a) /\ dist(x,a) < d
748 ==> dist(f(x),l) < e`,
749 REWRITE_TAC[tendsto; EVENTUALLY_AT] THEN MESON_TAC[]);;
751 let LIM_AT_INFINITY = prove
752 (`!f l. (f --> l) at_infinity <=>
753 !e. &0 < e ==> ?b. !x. norm(x) >= b ==> dist(f(x),l) < e`,
754 REWRITE_TAC[tendsto; EVENTUALLY_AT_INFINITY] THEN MESON_TAC[]);;
756 let LIM_AT_INFINITY_POS = prove
757 (`!f l. (f --> l) at_infinity <=>
758 !e. &0 < e ==> ?b. &0 < b /\ !x. norm x >= b ==> dist(f x,l) < e`,
759 REPEAT GEN_TAC THEN REWRITE_TAC[LIM_AT_INFINITY] THEN
760 MESON_TAC[REAL_ARITH `&0 < abs b + &1 /\ (x >= abs b + &1 ==> x >= b)`]);;
762 let LIM_SEQUENTIALLY = prove
763 (`!s l. (s --> l) sequentially <=>
764 !e. &0 < e ==> ?N. !n. N <= n ==> dist(s(n),l) < e`,
765 REWRITE_TAC[tendsto; EVENTUALLY_SEQUENTIALLY] THEN MESON_TAC[]);;
767 let LIM_EVENTUALLY = prove
768 (`!net f l. eventually (\x. f x = l) net ==> (f --> l) net`,
769 REWRITE_TAC[eventually; LIM] THEN MESON_TAC[DIST_REFL]);;
771 (* ------------------------------------------------------------------------- *)
772 (* The expected monotonicity property. *)
773 (* ------------------------------------------------------------------------- *)
775 let LIM_WITHIN_EMPTY = prove
776 (`!f l x. (f --> l) (at x within {})`,
777 REWRITE_TAC[LIM_WITHIN; NOT_IN_EMPTY] THEN MESON_TAC[REAL_LT_01]);;
779 let LIM_WITHIN_SUBSET = prove
781 (f --> l) (at a within s) /\ t SUBSET s ==> (f --> l) (at a within t)`,
782 REWRITE_TAC[LIM_WITHIN; SUBSET] THEN MESON_TAC[]);;
784 let LIM_UNION = prove
786 (f --> l) (at x within s) /\ (f --> l) (at x within t)
787 ==> (f --> l) (at x within (s UNION t))`,
788 REPEAT GEN_TAC THEN REWRITE_TAC[LIM_WITHIN; IN_UNION] THEN
789 REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN
790 X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_SIMP_TAC[] THEN
791 DISCH_THEN(CONJUNCTS_THEN2
792 (X_CHOOSE_TAC `d1:real`) (X_CHOOSE_TAC `d2:real`)) THEN
793 EXISTS_TAC `min d1 d2` THEN ASM_MESON_TAC[REAL_LT_MIN]);;
795 let LIM_UNION_UNIV = prove
797 (f --> l) (at x within s) /\ (f --> l) (at x within t) /\
798 s UNION t = (:real^N)
799 ==> (f --> l) (at x)`,
800 MESON_TAC[LIM_UNION; WITHIN_UNIV]);;
802 (* ------------------------------------------------------------------------- *)
803 (* Composition of limits. *)
804 (* ------------------------------------------------------------------------- *)
806 let LIM_COMPOSE_WITHIN = prove
807 (`!net f:real^M->real^N g:real^N->real^P s y z.
809 eventually (\w. f w IN s /\ (f w = y ==> g y = z)) net /\
810 (g --> z) (at y within s)
811 ==> ((g o f) --> z) net`,
812 REPEAT GEN_TAC THEN REWRITE_TAC[tendsto; CONJ_ASSOC] THEN
813 ONCE_REWRITE_TAC[LEFT_AND_FORALL_THM] THEN
814 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
815 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
816 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
817 REWRITE_TAC[EVENTUALLY_WITHIN; GSYM DIST_NZ; o_DEF] THEN
818 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
819 FIRST_X_ASSUM(MP_TAC o SPEC `d:real`) THEN
820 ASM_REWRITE_TAC[GSYM EVENTUALLY_AND] THEN
821 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
822 ASM_MESON_TAC[DIST_REFL]);;
824 let LIM_COMPOSE_AT = prove
825 (`!net f:real^M->real^N g:real^N->real^P y z.
827 eventually (\w. f w = y ==> g y = z) net /\
829 ==> ((g o f) --> z) net`,
830 REPEAT STRIP_TAC THEN
831 MP_TAC(ISPECL [`net:(real^M)net`; `f:real^M->real^N`; `g:real^N->real^P`;
832 `(:real^N)`; `y:real^N`; `z:real^P`]
833 LIM_COMPOSE_WITHIN) THEN
834 ASM_REWRITE_TAC[IN_UNIV; WITHIN_UNIV]);;
836 (* ------------------------------------------------------------------------- *)
837 (* Interrelations between restricted and unrestricted limits. *)
838 (* ------------------------------------------------------------------------- *)
840 let LIM_AT_WITHIN = prove
841 (`!f l a s. (f --> l)(at a) ==> (f --> l)(at a within s)`,
842 REWRITE_TAC[LIM_AT; LIM_WITHIN] THEN MESON_TAC[]);;
844 let LIM_WITHIN_OPEN = prove
846 a IN s /\ open s ==> ((f --> l)(at a within s) <=> (f --> l)(at a))`,
847 REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[LIM_AT_WITHIN] THEN
848 REWRITE_TAC[LIM_AT; LIM_WITHIN] THEN
849 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
850 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
851 DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
852 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M` o GEN_REWRITE_RULE I [open_def]) THEN
853 ASM_REWRITE_TAC[] THEN
854 DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
855 MP_TAC(SPECL [`d1:real`; `d2:real`] REAL_DOWN2) THEN ASM_REWRITE_TAC[] THEN
856 ASM_MESON_TAC[REAL_LT_TRANS]);;
858 (* ------------------------------------------------------------------------- *)
859 (* More limit point characterizations. *)
860 (* ------------------------------------------------------------------------- *)
862 let LIMPT_SEQUENTIAL_INJ = prove
864 x limit_point_of s <=>
865 ?f. (!n. f(n) IN (s DELETE x)) /\
866 (!m n. f m = f n <=> m = n) /\
867 (f --> x) sequentially`,
869 REWRITE_TAC[LIMPT_APPROACHABLE; LIM_SEQUENTIALLY; IN_DELETE] THEN
870 EQ_TAC THENL [ALL_TAC; MESON_TAC[GE; LE_REFL]] THEN
871 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
872 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
873 X_GEN_TAC `y:real->real^N` THEN DISCH_TAC THEN
874 (STRIP_ASSUME_TAC o prove_recursive_functions_exist num_RECURSION)
876 (!n. z (SUC n):real^N = y(min (inv(&2 pow (SUC n))) (dist(z n,x))))` THEN
877 EXISTS_TAC `z:num->real^N` THEN
879 `!n. z(n) IN s /\ ~(z n:real^N = x) /\ dist(z n,x) < inv(&2 pow n)`
881 [INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
882 ASM_SIMP_TAC[REAL_LT_01] THEN FIRST_X_ASSUM(MP_TAC o SPEC
883 `min (inv(&2 pow (SUC n))) (dist(z n:real^N,x))`) THEN
884 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_INV_EQ; REAL_LT_POW2; DIST_POS_LT];
885 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
886 [MATCH_MP_TAC WLOG_LT THEN REWRITE_TAC[EQ_SYM_EQ] THEN
887 SUBGOAL_THEN `!m n:num. m < n ==> dist(z n:real^N,x) < dist(z m,x)`
888 (fun th -> MESON_TAC[th; REAL_LT_REFL; LT_REFL]) THEN
889 MATCH_MP_TAC TRANSITIVE_STEPWISE_LT THEN
890 CONJ_TAC THENL [REAL_ARITH_TAC; GEN_TAC THEN ASM_REWRITE_TAC[]] THEN
891 FIRST_X_ASSUM(MP_TAC o SPEC
892 `min (inv(&2 pow (SUC n))) (dist(z n:real^N,x))`) THEN
893 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_INV_EQ; REAL_LT_POW2; DIST_POS_LT];
894 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
895 MP_TAC(ISPECL [`inv(&2)`; `e:real`] REAL_ARCH_POW_INV) THEN
896 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN
897 X_GEN_TAC `N:num` THEN REWRITE_TAC[REAL_POW_INV] THEN DISCH_TAC THEN
898 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
899 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
901 MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `inv(&2 pow n)` THEN
902 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
903 ASM_REWRITE_TAC[REAL_LT_POW2] THEN MATCH_MP_TAC REAL_POW_MONO THEN
904 REWRITE_TAC[REAL_OF_NUM_LE] THEN ASM_ARITH_TAC]]);;
906 let LIMPT_SEQUENTIAL = prove
908 x limit_point_of s <=>
909 ?f. (!n. f(n) IN (s DELETE x)) /\ (f --> x) sequentially`,
910 REPEAT GEN_TAC THEN EQ_TAC THENL
911 [REWRITE_TAC[LIMPT_SEQUENTIAL_INJ] THEN MESON_TAC[];
912 REWRITE_TAC[LIMPT_APPROACHABLE; LIM_SEQUENTIALLY; IN_DELETE] THEN
913 MESON_TAC[GE; LE_REFL]]);;
915 let [LIMPT_INFINITE_OPEN; LIMPT_INFINITE_BALL; LIMPT_INFINITE_CBALL] =
918 x limit_point_of s <=> !t. x IN t /\ open t ==> INFINITE(s INTER t)) /\
920 x limit_point_of s <=> !e. &0 < e ==> INFINITE(s INTER ball(x,e))) /\
922 x limit_point_of s <=> !e. &0 < e ==> INFINITE(s INTER cball(x,e)))`,
923 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT
924 `(q ==> p) /\ (r ==> s) /\ (s ==> q) /\ (p ==> r)
925 ==> (p <=> q) /\ (p <=> r) /\ (p <=> s)`) THEN
926 REPEAT CONJ_TAC THENL
927 [REWRITE_TAC[limit_point_of; INFINITE; SET_RULE
928 `(?y. ~(y = x) /\ y IN s /\ y IN t) <=> ~(s INTER t SUBSET {x})`] THEN
929 MESON_TAC[FINITE_SUBSET; FINITE_SING];
930 MESON_TAC[INFINITE_SUPERSET; BALL_SUBSET_CBALL;
931 SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`];
932 MESON_TAC[INFINITE_SUPERSET; OPEN_CONTAINS_CBALL;
933 SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`];
934 REWRITE_TAC[LIMPT_SEQUENTIAL_INJ; IN_DELETE; FORALL_AND_THM] THEN
935 DISCH_THEN(X_CHOOSE_THEN `f:num->real^N` STRIP_ASSUME_TAC) THEN
936 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
937 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
938 DISCH_THEN(MP_TAC o SPEC `e:real`) THEN
939 ASM_REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[DIST_SYM] IN_BALL)] THEN
940 DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
941 MATCH_MP_TAC INFINITE_SUPERSET THEN
942 EXISTS_TAC `IMAGE (f:num->real^N) (from N)` THEN
943 ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_FROM; IN_INTER] THEN
944 ASM_MESON_TAC[INFINITE_IMAGE_INJ; INFINITE_FROM]]);;
946 let INFINITE_OPEN_IN = prove
948 open_in (subtopology euclidean u) s /\ (?x. x IN s /\ x limit_point_of u)
950 REPEAT STRIP_TAC THEN
951 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
952 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
953 FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool` o
954 GEN_REWRITE_RULE I [LIMPT_INFINITE_OPEN]) THEN
955 FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM SET_TAC[]);;
957 (* ------------------------------------------------------------------------- *)
958 (* Condensation points. *)
959 (* ------------------------------------------------------------------------- *)
961 parse_as_infix ("condensation_point_of",(12,"right"));;
963 let condensation_point_of = new_definition
964 `x condensation_point_of s <=>
965 !t. x IN t /\ open t ==> ~COUNTABLE(s INTER t)`;;
967 let CONDENSATION_POINT_OF_SUBSET = prove
969 x condensation_point_of s /\ s SUBSET t ==> x condensation_point_of t`,
971 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
972 REWRITE_TAC[condensation_point_of] THEN
973 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
974 REWRITE_TAC[CONTRAPOS_THM] THEN
975 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COUNTABLE_SUBSET) THEN
978 let CONDENSATION_POINT_IMP_LIMPT = prove
979 (`!x s. x condensation_point_of s ==> x limit_point_of s`,
980 REWRITE_TAC[condensation_point_of; LIMPT_INFINITE_OPEN; INFINITE] THEN
981 MESON_TAC[FINITE_IMP_COUNTABLE]);;
983 let CONDENSATION_POINT_INFINITE_BALL,CONDENSATION_POINT_INFINITE_CBALL =
986 x condensation_point_of s <=>
987 !e. &0 < e ==> ~COUNTABLE(s INTER ball(x,e))) /\
989 x condensation_point_of s <=>
990 !e. &0 < e ==> ~COUNTABLE(s INTER cball(x,e)))`,
991 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT
992 `(p ==> q) /\ (q ==> r) /\ (r ==> p)
993 ==> (p <=> q) /\ (p <=> r)`) THEN
994 REWRITE_TAC[condensation_point_of] THEN REPEAT CONJ_TAC THENL
995 [MESON_TAC[OPEN_BALL; CENTRE_IN_BALL];
996 MESON_TAC[BALL_SUBSET_CBALL; COUNTABLE_SUBSET;
997 SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`];
998 MESON_TAC[COUNTABLE_SUBSET; OPEN_CONTAINS_CBALL;
999 SET_RULE `t SUBSET u ==> s INTER t SUBSET s INTER u`]]);;
1001 (* ------------------------------------------------------------------------- *)
1002 (* Basic arithmetical combining theorems for limits. *)
1003 (* ------------------------------------------------------------------------- *)
1005 let LIM_LINEAR = prove
1006 (`!net:(A)net h f l.
1007 (f --> l) net /\ linear h ==> ((\x. h(f x)) --> h l) net`,
1008 REPEAT GEN_TAC THEN REWRITE_TAC[LIM] THEN
1009 ASM_CASES_TAC `trivial_limit (net:(A)net)` THEN ASM_REWRITE_TAC[] THEN
1010 STRIP_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o
1011 MATCH_MP LINEAR_BOUNDED_POS) THEN
1012 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1013 FIRST_X_ASSUM(MP_TAC o SPEC `e / B`) THEN
1014 ASM_SIMP_TAC[REAL_LT_DIV; dist; GSYM LINEAR_SUB; REAL_LT_RDIV_EQ] THEN
1015 ASM_MESON_TAC[REAL_LET_TRANS; REAL_MUL_SYM]);;
1017 let LIM_CONST = prove
1018 (`!net a:real^N. ((\x. a) --> a) net`,
1019 SIMP_TAC[LIM; DIST_REFL; trivial_limit] THEN MESON_TAC[]);;
1021 let LIM_CMUL = prove
1022 (`!f l c. (f --> l) net ==> ((\x. c % f x) --> c % l) net`,
1023 REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_LINEAR THEN
1024 ASM_REWRITE_TAC[REWRITE_RULE[ETA_AX]
1025 (MATCH_MP LINEAR_COMPOSE_CMUL LINEAR_ID)]);;
1027 let LIM_CMUL_EQ = prove
1029 ~(c = &0) ==> (((\x. c % f x) --> c % l) net <=> (f --> l) net)`,
1030 REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[LIM_CMUL] THEN
1031 DISCH_THEN(MP_TAC o SPEC `inv c:real` o MATCH_MP LIM_CMUL) THEN
1032 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID; ETA_AX]);;
1035 (`!net f l:real^N. (f --> l) net ==> ((\x. --(f x)) --> --l) net`,
1036 REPEAT GEN_TAC THEN REWRITE_TAC[LIM; dist] THEN
1037 REWRITE_TAC[VECTOR_ARITH `--x - --y = --(x - y:real^N)`; NORM_NEG]);;
1039 let LIM_NEG_EQ = prove
1040 (`!net f l:real^N. ((\x. --(f x)) --> --l) net <=> (f --> l) net`,
1041 REPEAT GEN_TAC THEN EQ_TAC THEN
1042 DISCH_THEN(MP_TAC o MATCH_MP LIM_NEG) THEN
1043 REWRITE_TAC[VECTOR_NEG_NEG; ETA_AX]);;
1046 (`!net:(A)net f g l m.
1047 (f --> l) net /\ (g --> m) net ==> ((\x. f(x) + g(x)) --> l + m) net`,
1048 REPEAT GEN_TAC THEN REWRITE_TAC[LIM] THEN
1049 ASM_CASES_TAC `trivial_limit (net:(A)net)` THEN
1050 ASM_REWRITE_TAC[AND_FORALL_THM] THEN
1051 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1052 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
1053 DISCH_THEN(MP_TAC o MATCH_MP NET_DILEMMA) THEN MATCH_MP_TAC MONO_EXISTS THEN
1054 MESON_TAC[REAL_HALF; DIST_TRIANGLE_ADD; REAL_LT_ADD2; REAL_LET_TRANS]);;
1057 (`!net:(A)net f:A->real^N l.
1059 ==> ((\x. lambda i. (abs(f(x)$i))) --> (lambda i. abs(l$i)):real^N) net`,
1060 REPEAT GEN_TAC THEN REWRITE_TAC[LIM] THEN
1061 ASM_CASES_TAC `trivial_limit (net:(A)net)` THEN ASM_REWRITE_TAC[] THEN
1062 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
1063 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
1064 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
1065 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
1066 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
1067 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
1068 MATCH_MP_TAC(NORM_ARITH
1069 `norm(x - y) <= norm(a - b) ==> dist(a,b) < e ==> dist(x,y) < e`) THEN
1070 MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN
1071 SIMP_TAC[LAMBDA_BETA; VECTOR_SUB_COMPONENT] THEN
1075 (`!net:(A)net f g l m.
1076 (f --> l) net /\ (g --> m) net ==> ((\x. f(x) - g(x)) --> l - m) net`,
1077 REWRITE_TAC[real_sub; VECTOR_SUB] THEN ASM_SIMP_TAC[LIM_ADD; LIM_NEG]);;
1080 (`!net:(A)net f g l:real^N m:real^N.
1081 (f --> l) net /\ (g --> m) net
1082 ==> ((\x. lambda i. max (f(x)$i) (g(x)$i))
1083 --> (lambda i. max (l$i) (m$i)):real^N) net`,
1084 REPEAT GEN_TAC THEN DISCH_TAC THEN
1085 FIRST_ASSUM(MP_TAC o MATCH_MP LIM_ADD) THEN
1086 FIRST_ASSUM(MP_TAC o MATCH_MP LIM_SUB) THEN
1087 DISCH_THEN(MP_TAC o MATCH_MP LIM_ABS) THEN
1088 REWRITE_TAC[IMP_IMP] THEN
1089 DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD) THEN
1090 DISCH_THEN(MP_TAC o SPEC `inv(&2)` o MATCH_MP LIM_CMUL) THEN
1091 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THEN
1092 SIMP_TAC[FUN_EQ_THM; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
1093 VECTOR_SUB_COMPONENT; LAMBDA_BETA] THEN
1097 (`!net:(A)net f g l:real^N m:real^N.
1098 (f --> l) net /\ (g --> m) net
1099 ==> ((\x. lambda i. min (f(x)$i) (g(x)$i))
1100 --> (lambda i. min (l$i) (m$i)):real^N) net`,
1102 DISCH_THEN(CONJUNCTS_THEN(MP_TAC o MATCH_MP LIM_NEG)) THEN
1103 REWRITE_TAC[IMP_IMP] THEN
1104 DISCH_THEN(MP_TAC o MATCH_MP LIM_NEG o MATCH_MP LIM_MAX) THEN
1105 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THEN
1106 SIMP_TAC[FUN_EQ_THM; CART_EQ; LAMBDA_BETA; VECTOR_NEG_COMPONENT] THEN
1109 let LIM_NORM = prove
1110 (`!net f:A->real^N l.
1111 (f --> l) net ==> ((\x. lift(norm(f x))) --> lift(norm l)) net`,
1112 REPEAT GEN_TAC THEN REWRITE_TAC[tendsto; DIST_LIFT] THEN
1113 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
1115 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
1116 REWRITE_TAC[] THEN NORM_ARITH_TAC);;
1118 let LIM_NULL = prove
1119 (`!net f l. (f --> l) net <=> ((\x. f(x) - l) --> vec 0) net`,
1120 REWRITE_TAC[LIM; dist; VECTOR_SUB_RZERO]);;
1122 let LIM_NULL_NORM = prove
1123 (`!net f. (f --> vec 0) net <=> ((\x. lift(norm(f x))) --> vec 0) net`,
1124 REWRITE_TAC[LIM; dist; VECTOR_SUB_RZERO; REAL_ABS_NORM; NORM_LIFT]);;
1126 let LIM_NULL_CMUL_EQ = prove
1128 ~(c = &0) ==> (((\x. c % f x) --> vec 0) net <=> (f --> vec 0) net)`,
1129 MESON_TAC[LIM_CMUL_EQ; VECTOR_MUL_RZERO]);;
1131 let LIM_NULL_CMUL = prove
1132 (`!net f c. (f --> vec 0) net ==> ((\x. c % f x) --> vec 0) net`,
1133 REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THEN
1134 ASM_SIMP_TAC[LIM_NULL_CMUL_EQ; VECTOR_MUL_LZERO; LIM_CONST]);;
1136 let LIM_NULL_COMPARISON = prove
1137 (`!net f g. eventually (\x. norm(f x) <= g x) net /\
1138 ((\x. lift(g x)) --> vec 0) net
1139 ==> (f --> vec 0) net`,
1140 REPEAT GEN_TAC THEN REWRITE_TAC[tendsto; RIGHT_AND_FORALL_THM] THEN
1141 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
1142 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[GSYM EVENTUALLY_AND] THEN
1143 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
1144 REWRITE_TAC[dist; VECTOR_SUB_RZERO; NORM_LIFT] THEN REAL_ARITH_TAC);;
1146 let LIM_COMPONENT = prove
1147 (`!net f i l:real^N. (f --> l) net /\ 1 <= i /\ i <= dimindex(:N)
1148 ==> ((\a. lift(f(a)$i)) --> lift(l$i)) net`,
1149 REWRITE_TAC[LIM; dist; GSYM LIFT_SUB; NORM_LIFT] THEN
1150 SIMP_TAC[GSYM VECTOR_SUB_COMPONENT] THEN
1151 MESON_TAC[COMPONENT_LE_NORM; REAL_LET_TRANS]);;
1153 let LIM_TRANSFORM_BOUND = prove
1154 (`!f g. eventually (\n. norm(f n) <= norm(g n)) net /\ (g --> vec 0) net
1155 ==> (f --> vec 0) net`,
1157 REWRITE_TAC[tendsto; RIGHT_AND_FORALL_THM] THEN
1158 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
1159 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[GSYM EVENTUALLY_AND] THEN
1160 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
1161 REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN REAL_ARITH_TAC);;
1163 let LIM_NULL_CMUL_BOUNDED = prove
1165 eventually (\a. g a = vec 0 \/ abs(f a) <= B) net /\
1167 ==> ((\n. f n % g n) --> vec 0) net`,
1168 REPEAT GEN_TAC THEN REWRITE_TAC[tendsto] THEN STRIP_TAC THEN
1169 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1170 FIRST_X_ASSUM(MP_TAC o SPEC `e / (abs B + &1)`) THEN
1171 ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < abs x + &1`] THEN
1172 UNDISCH_TAC `eventually (\a. g a:real^N = vec 0 \/ abs(f a) <= B)
1174 REWRITE_TAC[IMP_IMP; GSYM EVENTUALLY_AND] THEN
1175 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MP) THEN
1176 REWRITE_TAC[dist; VECTOR_SUB_RZERO; o_THM; NORM_LIFT; NORM_MUL] THEN
1177 MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `x:A` THEN REWRITE_TAC[] THEN
1178 ASM_CASES_TAC `(g:A->real^N) x = vec 0` THEN
1179 ASM_REWRITE_TAC[NORM_0; REAL_MUL_RZERO] THEN
1180 STRIP_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN
1181 EXISTS_TAC `B * e / (abs B + &1)` THEN
1182 ASM_SIMP_TAC[REAL_LE_MUL2; REAL_ABS_POS; NORM_POS_LE; REAL_LT_IMP_LE] THEN
1183 REWRITE_TAC[REAL_ARITH `c * (a / b) = (c * a) / b`] THEN
1184 SIMP_TAC[REAL_LT_LDIV_EQ; REAL_ARITH `&0 < abs x + &1`] THEN
1185 MATCH_MP_TAC(REAL_ARITH
1186 `e * B <= e * abs B /\ &0 < e ==> B * e < e * (abs B + &1)`) THEN
1187 ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN REAL_ARITH_TAC);;
1189 let LIM_NULL_VMUL_BOUNDED = prove
1191 ((lift o f) --> vec 0) net /\
1192 eventually (\a. f a = &0 \/ norm(g a) <= B) net
1193 ==> ((\n. f n % g n) --> vec 0) net`,
1194 REPEAT GEN_TAC THEN REWRITE_TAC[tendsto] THEN STRIP_TAC THEN
1195 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1196 FIRST_X_ASSUM(MP_TAC o SPEC `e / (abs B + &1)`) THEN
1197 ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < abs x + &1`] THEN
1198 UNDISCH_TAC `eventually(\a. f a = &0 \/ norm((g:A->real^N) a) <= B) net` THEN
1199 REWRITE_TAC[IMP_IMP; GSYM EVENTUALLY_AND] THEN
1200 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MP) THEN
1201 REWRITE_TAC[dist; VECTOR_SUB_RZERO; o_THM; NORM_LIFT; NORM_MUL] THEN
1202 MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `x:A` THEN REWRITE_TAC[] THEN
1203 ASM_CASES_TAC `(f:A->real) x = &0` THEN
1204 ASM_REWRITE_TAC[REAL_ABS_NUM; REAL_MUL_LZERO] THEN
1205 STRIP_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN
1206 EXISTS_TAC `e / (abs B + &1) * B` THEN
1207 ASM_SIMP_TAC[REAL_LE_MUL2; REAL_ABS_POS; NORM_POS_LE; REAL_LT_IMP_LE] THEN
1208 REWRITE_TAC[REAL_ARITH `(a / b) * c = (a * c) / b`] THEN
1209 SIMP_TAC[REAL_LT_LDIV_EQ; REAL_ARITH `&0 < abs x + &1`] THEN
1210 MATCH_MP_TAC(REAL_ARITH
1211 `e * B <= e * abs B /\ &0 < e ==> e * B < e * (abs B + &1)`) THEN
1212 ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN REAL_ARITH_TAC);;
1214 let LIM_VSUM = prove
1215 (`!f:A->B->real^N s.
1216 FINITE s /\ (!i. i IN s ==> ((f i) --> (l i)) net)
1217 ==> ((\x. vsum s (\i. f i x)) --> vsum s l) net`,
1218 GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
1219 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1220 SIMP_TAC[VSUM_CLAUSES; LIM_CONST; LIM_ADD; IN_INSERT; ETA_AX]);;
1222 (* ------------------------------------------------------------------------- *)
1223 (* Deducing things about the limit from the elements. *)
1224 (* ------------------------------------------------------------------------- *)
1226 let LIM_IN_CLOSED_SET = prove
1227 (`!net f:A->real^N s l.
1228 closed s /\ eventually (\x. f(x) IN s) net /\
1229 ~(trivial_limit net) /\ (f --> l) net
1231 REWRITE_TAC[closed] THEN REPEAT STRIP_TAC THEN
1232 MATCH_MP_TAC(SET_RULE `~(x IN (UNIV DIFF s)) ==> x IN s`) THEN
1234 FIRST_ASSUM(MP_TAC o SPEC `l:real^N` o GEN_REWRITE_RULE I
1235 [OPEN_CONTAINS_BALL]) THEN
1236 ASM_REWRITE_TAC[SUBSET; IN_BALL; IN_DIFF; IN_UNION] THEN
1237 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
1238 FIRST_X_ASSUM(MP_TAC o SPEC `e:real` o GEN_REWRITE_RULE I [tendsto]) THEN
1239 UNDISCH_TAC `eventually (\x. (f:A->real^N) x IN s) net` THEN
1240 ASM_REWRITE_TAC[GSYM EVENTUALLY_AND; TAUT `a ==> ~b <=> ~(a /\ b)`] THEN
1241 MATCH_MP_TAC NOT_EVENTUALLY THEN ASM_MESON_TAC[DIST_SYM]);;
1243 (* ------------------------------------------------------------------------- *)
1244 (* Need to prove closed(cball(x,e)) before deducing this as a corollary. *)
1245 (* ------------------------------------------------------------------------- *)
1247 let LIM_NORM_UBOUND = prove
1248 (`!net:(A)net f (l:real^N) b.
1249 ~(trivial_limit net) /\
1251 eventually (\x. norm(f x) <= b) net
1253 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1254 ASM_REWRITE_TAC[LIM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1255 ASM_REWRITE_TAC[eventually] THEN
1256 STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN
1257 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN DISCH_TAC THEN
1259 `?x:A. dist(f(x):real^N,l) < norm(l:real^N) - b /\ norm(f x) <= b`
1260 (CHOOSE_THEN MP_TAC) THENL [ASM_MESON_TAC[NET]; ALL_TAC] THEN
1261 REWRITE_TAC[REAL_NOT_LT; REAL_LE_SUB_RADD; DE_MORGAN_THM; dist] THEN
1264 let LIM_NORM_LBOUND = prove
1265 (`!net:(A)net f (l:real^N) b.
1266 ~(trivial_limit net) /\ (f --> l) net /\
1267 eventually (\x. b <= norm(f x)) net
1269 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1270 ASM_REWRITE_TAC[LIM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1271 ASM_REWRITE_TAC[eventually] THEN
1272 STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN
1273 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN DISCH_TAC THEN
1275 `?x:A. dist(f(x):real^N,l) < b - norm(l:real^N) /\ b <= norm(f x)`
1276 (CHOOSE_THEN MP_TAC) THENL [ASM_MESON_TAC[NET]; ALL_TAC] THEN
1277 REWRITE_TAC[REAL_NOT_LT; REAL_LE_SUB_RADD; DE_MORGAN_THM; dist] THEN
1280 (* ------------------------------------------------------------------------- *)
1281 (* Uniqueness of the limit, when nontrivial. *)
1282 (* ------------------------------------------------------------------------- *)
1284 let LIM_UNIQUE = prove
1285 (`!net:(A)net f l:real^N l'.
1286 ~(trivial_limit net) /\ (f --> l) net /\ (f --> l') net ==> (l = l')`,
1287 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1288 DISCH_THEN(ASSUME_TAC o REWRITE_RULE[VECTOR_SUB_REFL] o MATCH_MP LIM_SUB) THEN
1289 SUBGOAL_THEN `!e. &0 < e ==> norm(l:real^N - l') <= e` MP_TAC THENL
1290 [GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC LIM_NORM_UBOUND THEN
1291 MAP_EVERY EXISTS_TAC [`net:(A)net`; `\x:A. vec 0 : real^N`] THEN
1292 ASM_SIMP_TAC[NORM_0; REAL_LT_IMP_LE; eventually] THEN
1293 ASM_MESON_TAC[trivial_limit];
1294 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DIST_NZ; dist] THEN
1295 DISCH_TAC THEN DISCH_THEN(MP_TAC o SPEC `norm(l - l':real^N) / &2`) THEN
1296 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
1297 UNDISCH_TAC `&0 < norm(l - l':real^N)` THEN REAL_ARITH_TAC]);;
1299 let TENDSTO_LIM = prove
1300 (`!net f l. ~(trivial_limit net) /\ (f --> l) net ==> lim net f = l`,
1301 REWRITE_TAC[lim] THEN MESON_TAC[LIM_UNIQUE]);;
1303 let LIM_CONST_EQ = prove
1304 (`!net:(A net) c d:real^N.
1305 ((\x. c) --> d) net <=> trivial_limit net \/ c = d`,
1307 ASM_CASES_TAC `trivial_limit (net:A net)` THEN ASM_REWRITE_TAC[] THENL
1308 [ASM_REWRITE_TAC[LIM]; ALL_TAC] THEN
1309 EQ_TAC THEN SIMP_TAC[LIM_CONST] THEN DISCH_TAC THEN
1310 MATCH_MP_TAC(SPEC `net:A net` LIM_UNIQUE) THEN
1311 EXISTS_TAC `(\x. c):A->real^N` THEN ASM_REWRITE_TAC[LIM_CONST]);;
1313 (* ------------------------------------------------------------------------- *)
1314 (* Some unwieldy but occasionally useful theorems about uniform limits. *)
1315 (* ------------------------------------------------------------------------- *)
1317 let UNIFORM_LIM_ADD = prove
1318 (`!net:(A)net P f g l m.
1320 ==> eventually (\x. !n:B. P n ==> norm(f n x - l n) < e) net) /\
1322 ==> eventually (\x. !n. P n ==> norm(g n x - m n) < e) net)
1326 ==> norm((f n x + g n x) - (l n + m n)) < e)
1328 REPEAT GEN_TAC THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_TAC THEN
1329 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1330 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
1331 ASM_REWRITE_TAC[REAL_HALF; GSYM EVENTUALLY_AND] THEN
1332 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
1333 GEN_TAC THEN REWRITE_TAC[AND_FORALL_THM] THEN
1334 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:B` THEN
1335 ASM_CASES_TAC `(P:B->bool) n` THEN ASM_REWRITE_TAC[] THEN
1336 CONV_TAC NORM_ARITH);;
1338 let UNIFORM_LIM_SUB = prove
1339 (`!net:(A)net P f g l m.
1341 ==> eventually (\x. !n:B. P n ==> norm(f n x - l n) < e) net) /\
1343 ==> eventually (\x. !n. P n ==> norm(g n x - m n) < e) net)
1347 ==> norm((f n x - g n x) - (l n - m n)) < e)
1349 REPEAT GEN_TAC THEN REWRITE_TAC[AND_FORALL_THM] THEN DISCH_TAC THEN
1350 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1351 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
1352 ASM_REWRITE_TAC[REAL_HALF; GSYM EVENTUALLY_AND] THEN
1353 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
1354 GEN_TAC THEN REWRITE_TAC[AND_FORALL_THM] THEN
1355 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:B` THEN
1356 ASM_CASES_TAC `(P:B->bool) n` THEN ASM_REWRITE_TAC[] THEN
1357 CONV_TAC NORM_ARITH);;
1359 (* ------------------------------------------------------------------------- *)
1360 (* Limit under bilinear function, uniform version first. *)
1361 (* ------------------------------------------------------------------------- *)
1363 let UNIFORM_LIM_BILINEAR = prove
1364 (`!net:(A)net P (h:real^M->real^N->real^P) f g l m b1 b2.
1366 eventually (\x. !n. P n ==> norm(l n) <= b1) net /\
1367 eventually (\x. !n. P n ==> norm(m n) <= b2) net /\
1369 ==> eventually (\x. !n:B. P n ==> norm(f n x - l n) < e) net) /\
1371 ==> eventually (\x. !n. P n ==> norm(g n x - m n) < e) net)
1375 ==> norm(h (f n x) (g n x) - h (l n) (m n)) < e)
1378 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1379 FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o MATCH_MP
1380 BILINEAR_BOUNDED_POS) THEN
1381 REWRITE_TAC[AND_FORALL_THM; RIGHT_AND_FORALL_THM] THEN DISCH_TAC THEN
1382 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1383 FIRST_X_ASSUM(MP_TAC o SPEC
1384 `min (abs b2 + &1) (e / &2 / (B * (abs b1 + abs b2 + &2)))`) THEN
1385 ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; REAL_LT_MUL; REAL_LT_MIN;
1386 REAL_ARITH `&0 < abs x + &1`;
1387 REAL_ARITH `&0 < abs x + abs y + &2`] THEN
1388 REWRITE_TAC[GSYM EVENTUALLY_AND] THEN
1389 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
1390 X_GEN_TAC `x:A` THEN REWRITE_TAC[AND_FORALL_THM] THEN
1391 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:B` THEN
1392 ASM_CASES_TAC `(P:B->bool) n` THEN ASM_REWRITE_TAC[] THEN
1394 ONCE_REWRITE_TAC[VECTOR_ARITH
1395 `h a b - h c d :real^N = (h a b - h a d) + (h a d - h c d)`] THEN
1396 ASM_SIMP_TAC[GSYM BILINEAR_LSUB; GSYM BILINEAR_RSUB] THEN
1397 MATCH_MP_TAC NORM_TRIANGLE_LT THEN
1398 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
1399 (MESON[REAL_LE_ADD2; REAL_LET_TRANS]
1400 `(!x y. norm(h x y:real^P) <= B * norm x * norm y)
1401 ==> B * norm a * norm b + B * norm c * norm d < e
1402 ==> norm(h a b) + norm(h c d) < e`)) THEN
1403 MATCH_MP_TAC(REAL_ARITH
1404 `x * B < e / &2 /\ y * B < e / &2 ==> B * x + B * y < e`) THEN
1405 CONJ_TAC THEN ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THENL
1406 [ONCE_REWRITE_TAC[REAL_MUL_SYM]; ALL_TAC] THEN
1407 MATCH_MP_TAC REAL_LET_TRANS THEN
1408 EXISTS_TAC `e / &2 / (B * (abs b1 + abs b2 + &2)) *
1409 (abs b1 + abs b2 + &1)` THEN
1411 [MATCH_MP_TAC REAL_LE_MUL2 THEN
1412 ASM_SIMP_TAC[NORM_POS_LE; REAL_LT_IMP_LE] THEN
1413 ASM_SIMP_TAC[REAL_ARITH `a <= b2 ==> a <= abs b1 + abs b2 + &1`] THEN
1414 ASM_MESON_TAC[NORM_ARITH
1415 `norm(f - l:real^P) < abs b2 + &1 /\ norm(l) <= b1
1416 ==> norm(f) <= abs b1 + abs b2 + &1`];
1417 ONCE_REWRITE_TAC[real_div] THEN
1418 ASM_SIMP_TAC[REAL_LT_LMUL_EQ; REAL_HALF; GSYM REAL_MUL_ASSOC;
1420 REWRITE_TAC[REAL_ARITH `B * inv x * y < B <=> B * y / x < B * &1`] THEN
1421 ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_LT_LMUL_EQ; REAL_LT_LDIV_EQ;
1422 REAL_ARITH `&0 < abs x + abs y + &2`] THEN
1425 let LIM_BILINEAR = prove
1426 (`!net:(A)net (h:real^M->real^N->real^P) f g l m.
1427 (f --> l) net /\ (g --> m) net /\ bilinear h
1428 ==> ((\x. h (f x) (g x)) --> (h l m)) net`,
1429 REPEAT STRIP_TAC THEN
1431 [`net:(A)net`; `\x:one. T`; `h:real^M->real^N->real^P`;
1432 `\n:one. (f:A->real^M)`; `\n:one. (g:A->real^N)`;
1433 `\n:one. (l:real^M)`; `\n:one. (m:real^N)`;
1434 `norm(l:real^M)`; `norm(m:real^N)`]
1435 UNIFORM_LIM_BILINEAR) THEN
1436 ASM_REWRITE_TAC[REAL_LE_REFL; EVENTUALLY_TRUE] THEN
1437 ASM_REWRITE_TAC[GSYM dist; GSYM tendsto]);;
1439 (* ------------------------------------------------------------------------- *)
1440 (* These are special for limits out of the same vector space. *)
1441 (* ------------------------------------------------------------------------- *)
1443 let LIM_WITHIN_ID = prove
1444 (`!a s. ((\x. x) --> a) (at a within s)`,
1445 REWRITE_TAC[LIM_WITHIN] THEN MESON_TAC[]);;
1447 let LIM_AT_ID = prove
1448 (`!a. ((\x. x) --> a) (at a)`,
1449 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN REWRITE_TAC[LIM_WITHIN_ID]);;
1451 let LIM_AT_ZERO = prove
1452 (`!f:real^M->real^N l a.
1453 (f --> l) (at a) <=> ((\x. f(a + x)) --> l) (at(vec 0))`,
1454 REPEAT GEN_TAC THEN REWRITE_TAC[LIM_AT] THEN
1455 AP_TERM_TAC THEN ABS_TAC THEN
1456 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
1457 AP_TERM_TAC THEN ABS_TAC THEN
1458 ASM_CASES_TAC `&0 < d` THEN ASM_REWRITE_TAC[] THEN
1459 EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `x:real^M` THENL
1460 [FIRST_X_ASSUM(MP_TAC o SPEC `a + x:real^M`) THEN
1461 REWRITE_TAC[dist; VECTOR_ADD_SUB; VECTOR_SUB_RZERO];
1462 FIRST_X_ASSUM(MP_TAC o SPEC `x - a:real^M`) THEN
1463 REWRITE_TAC[dist; VECTOR_SUB_RZERO; VECTOR_SUB_ADD2]]);;
1465 (* ------------------------------------------------------------------------- *)
1466 (* It's also sometimes useful to extract the limit point from the net. *)
1467 (* ------------------------------------------------------------------------- *)
1469 let netlimit = new_definition
1470 `netlimit net = @a. !x. ~(netord net x a)`;;
1472 let NETLIMIT_WITHIN = prove
1473 (`!a:real^N s. ~(trivial_limit (at a within s))
1474 ==> (netlimit (at a within s) = a)`,
1475 REWRITE_TAC[trivial_limit; netlimit; AT; WITHIN; DE_MORGAN_THM] THEN
1476 REPEAT STRIP_TAC THEN MATCH_MP_TAC SELECT_UNIQUE THEN REWRITE_TAC[] THEN
1478 `!x:real^N. ~(&0 < dist(x,a) /\ dist(x,a) <= dist(a,a) /\ x IN s)`
1480 [ASM_MESON_TAC[DIST_REFL; REAL_NOT_LT]; ASM_MESON_TAC[]]);;
1482 let NETLIMIT_AT = prove
1483 (`!a. netlimit(at a) = a`,
1484 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
1485 MATCH_MP_TAC NETLIMIT_WITHIN THEN
1486 SIMP_TAC[TRIVIAL_LIMIT_AT; WITHIN_UNIV]);;
1488 (* ------------------------------------------------------------------------- *)
1489 (* Transformation of limit. *)
1490 (* ------------------------------------------------------------------------- *)
1492 let LIM_TRANSFORM = prove
1494 ((\x. f x - g x) --> vec 0) net /\ (f --> l) net ==> (g --> l) net`,
1495 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_SUB) THEN
1496 DISCH_THEN(MP_TAC o MATCH_MP LIM_NEG) THEN MATCH_MP_TAC EQ_IMP THEN
1497 AP_THM_TAC THEN BINOP_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
1500 let LIM_TRANSFORM_EVENTUALLY = prove
1502 eventually (\x. f x = g x) net /\ (f --> l) net ==> (g --> l) net`,
1503 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
1504 DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o MATCH_MP LIM_EVENTUALLY) MP_TAC) THEN
1505 MESON_TAC[LIM_TRANSFORM]);;
1507 let LIM_TRANSFORM_WITHIN = prove
1510 (!x'. x' IN s /\ &0 < dist(x',x) /\ dist(x',x) < d ==> f(x') = g(x')) /\
1511 (f --> l) (at x within s)
1512 ==> (g --> l) (at x within s)`,
1513 REPEAT GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
1514 DISCH_TAC THEN DISCH_TAC THEN
1515 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM) THEN
1516 REWRITE_TAC[LIM_WITHIN] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `d:real` THEN
1517 ASM_SIMP_TAC[VECTOR_SUB_REFL; DIST_REFL]);;
1519 let LIM_TRANSFORM_AT = prove
1522 (!x'. &0 < dist(x',x) /\ dist(x',x) < d ==> f(x') = g(x')) /\
1524 ==> (g --> l) (at x)`,
1525 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN MESON_TAC[LIM_TRANSFORM_WITHIN]);;
1527 let LIM_TRANSFORM_EQ = prove
1528 (`!net f:A->real^N g l.
1529 ((\x. f x - g x) --> vec 0) net ==> ((f --> l) net <=> (g --> l) net)`,
1530 REPEAT STRIP_TAC THEN EQ_TAC THEN
1531 DISCH_TAC THEN MATCH_MP_TAC LIM_TRANSFORM THENL
1532 [EXISTS_TAC `f:A->real^N` THEN ASM_REWRITE_TAC[];
1533 EXISTS_TAC `g:A->real^N` THEN ASM_REWRITE_TAC[] THEN
1534 ONCE_REWRITE_TAC[GSYM LIM_NEG_EQ] THEN
1535 ASM_REWRITE_TAC[VECTOR_NEG_SUB; VECTOR_NEG_0]]);;
1537 let LIM_TRANSFORM_WITHIN_SET = prove
1539 eventually (\x. x IN s <=> x IN t) (at a)
1540 ==> ((f --> l) (at a within s) <=> (f --> l) (at a within t))`,
1541 REPEAT GEN_TAC THEN REWRITE_TAC[EVENTUALLY_AT; LIM_WITHIN] THEN
1542 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
1543 EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1544 FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
1545 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
1546 EXISTS_TAC `min d k:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
1549 (* ------------------------------------------------------------------------- *)
1550 (* Common case assuming being away from some crucial point like 0. *)
1551 (* ------------------------------------------------------------------------- *)
1553 let LIM_TRANSFORM_AWAY_WITHIN = prove
1554 (`!f:real^M->real^N g a b s.
1556 (!x. x IN s /\ ~(x = a) /\ ~(x = b) ==> f(x) = g(x)) /\
1557 (f --> l) (at a within s)
1558 ==> (g --> l) (at a within s)`,
1559 REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_TRANSFORM_WITHIN THEN
1560 MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `dist(a:real^M,b)`] THEN
1561 ASM_REWRITE_TAC[GSYM DIST_NZ] THEN X_GEN_TAC `y:real^M` THEN
1562 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
1563 ASM_MESON_TAC[DIST_SYM; REAL_LT_REFL]);;
1565 let LIM_TRANSFORM_AWAY_AT = prove
1566 (`!f:real^M->real^N g a b.
1568 (!x. ~(x = a) /\ ~(x = b) ==> f(x) = g(x)) /\
1570 ==> (g --> l) (at a)`,
1571 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
1572 MESON_TAC[LIM_TRANSFORM_AWAY_WITHIN]);;
1574 (* ------------------------------------------------------------------------- *)
1575 (* Alternatively, within an open set. *)
1576 (* ------------------------------------------------------------------------- *)
1578 let LIM_TRANSFORM_WITHIN_OPEN = prove
1579 (`!f g:real^M->real^N s a l.
1581 (!x. x IN s /\ ~(x = a) ==> f x = g x) /\
1583 ==> (g --> l) (at a)`,
1584 REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_TRANSFORM_AT THEN
1585 EXISTS_TAC `f:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN
1586 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
1587 DISCH_THEN(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[] THEN
1588 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[SUBSET; IN_BALL] THEN
1589 ASM_MESON_TAC[DIST_NZ; DIST_SYM]);;
1591 let LIM_TRANSFORM_WITHIN_OPEN_IN = prove
1592 (`!f g:real^M->real^N s t a l.
1593 open_in (subtopology euclidean t) s /\ a IN s /\
1594 (!x. x IN s /\ ~(x = a) ==> f x = g x) /\
1595 (f --> l) (at a within t)
1596 ==> (g --> l) (at a within t)`,
1597 REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_TRANSFORM_WITHIN THEN
1598 EXISTS_TAC `f:real^M->real^N` THEN ASM_REWRITE_TAC[] THEN
1599 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_CONTAINS_BALL]) THEN
1600 DISCH_THEN(MP_TAC o SPEC `a:real^M` o CONJUNCT2) THEN ASM_REWRITE_TAC[] THEN
1601 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[SUBSET; IN_INTER; IN_BALL] THEN
1602 ASM_MESON_TAC[DIST_NZ; DIST_SYM]);;
1604 (* ------------------------------------------------------------------------- *)
1605 (* Another quite common idiom of an explicit conditional in a sequence. *)
1606 (* ------------------------------------------------------------------------- *)
1608 let LIM_CASES_FINITE_SEQUENTIALLY = prove
1609 (`!f g l. FINITE {n | P n}
1610 ==> (((\n. if P n then f n else g n) --> l) sequentially <=>
1611 (g --> l) sequentially)`,
1612 REPEAT STRIP_TAC THEN EQ_TAC THEN
1613 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
1614 FIRST_ASSUM(MP_TAC o SPEC `\n:num. n` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN
1615 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
1616 X_GEN_TAC `N:num` THEN DISCH_TAC THEN SIMP_TAC[EVENTUALLY_SEQUENTIALLY] THEN
1617 EXISTS_TAC `N + 1` THEN
1618 ASM_MESON_TAC[ARITH_RULE `~(x <= n /\ n + 1 <= x)`]);;
1620 let LIM_CASES_COFINITE_SEQUENTIALLY = prove
1621 (`!f g l. FINITE {n | ~P n}
1622 ==> (((\n. if P n then f n else g n) --> l) sequentially <=>
1623 (f --> l) sequentially)`,
1624 ONCE_REWRITE_TAC[TAUT `(if p then x else y) = (if ~p then y else x)`] THEN
1625 REWRITE_TAC[LIM_CASES_FINITE_SEQUENTIALLY]);;
1627 let LIM_CASES_SEQUENTIALLY = prove
1628 (`!f g l m. (((\n. if m <= n then f n else g n) --> l) sequentially <=>
1629 (f --> l) sequentially) /\
1630 (((\n. if m < n then f n else g n) --> l) sequentially <=>
1631 (f --> l) sequentially) /\
1632 (((\n. if n <= m then f n else g n) --> l) sequentially <=>
1633 (g --> l) sequentially) /\
1634 (((\n. if n < m then f n else g n) --> l) sequentially <=>
1635 (g --> l) sequentially)`,
1636 SIMP_TAC[LIM_CASES_FINITE_SEQUENTIALLY; LIM_CASES_COFINITE_SEQUENTIALLY;
1637 NOT_LE; NOT_LT; FINITE_NUMSEG_LT; FINITE_NUMSEG_LE]);;
1639 (* ------------------------------------------------------------------------- *)
1640 (* A congruence rule allowing us to transform limits assuming not at point. *)
1641 (* ------------------------------------------------------------------------- *)
1643 let LIM_CONG_WITHIN = prove
1644 (`(!x. ~(x = a) ==> f x = g x)
1645 ==> (((\x. f x) --> l) (at a within s) <=> ((g --> l) (at a within s)))`,
1646 REWRITE_TAC[LIM_WITHIN; GSYM DIST_NZ] THEN SIMP_TAC[]);;
1648 let LIM_CONG_AT = prove
1649 (`(!x. ~(x = a) ==> f x = g x)
1650 ==> (((\x. f x) --> l) (at a) <=> ((g --> l) (at a)))`,
1651 REWRITE_TAC[LIM_AT; GSYM DIST_NZ] THEN SIMP_TAC[]);;
1653 extend_basic_congs [LIM_CONG_WITHIN; LIM_CONG_AT];;
1655 (* ------------------------------------------------------------------------- *)
1656 (* Useful lemmas on closure and set of possible sequential limits. *)
1657 (* ------------------------------------------------------------------------- *)
1659 let CLOSURE_SEQUENTIAL = prove
1661 l IN closure(s) <=> ?x. (!n. x(n) IN s) /\ (x --> l) sequentially`,
1662 REWRITE_TAC[closure; IN_UNION; LIMPT_SEQUENTIAL; IN_ELIM_THM; IN_DELETE] THEN
1663 REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT
1664 `((b ==> c) /\ (~a /\ c ==> b)) /\ (a ==> c) ==> (a \/ b <=> c)`) THEN
1665 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_TAC THEN
1666 EXISTS_TAC `\n:num. l:real^N` THEN
1667 ASM_REWRITE_TAC[LIM_CONST]);;
1669 let CLOSED_CONTAINS_SEQUENTIAL_LIMIT = prove
1671 closed s /\ (!n. x n IN s) /\ (x --> l) sequentially ==> l IN s`,
1672 MESON_TAC[CLOSURE_SEQUENTIAL; CLOSURE_CLOSED]);;
1674 let CLOSED_SEQUENTIAL_LIMITS = prove
1676 !x l. (!n. x(n) IN s) /\ (x --> l) sequentially ==> l IN s`,
1677 MESON_TAC[CLOSURE_SEQUENTIAL; CLOSURE_CLOSED;
1678 CLOSED_LIMPT; LIMPT_SEQUENTIAL; IN_DELETE]);;
1680 let CLOSURE_APPROACHABLE = prove
1681 (`!x s. x IN closure(s) <=> !e. &0 < e ==> ?y. y IN s /\ dist(y,x) < e`,
1682 REWRITE_TAC[closure; LIMPT_APPROACHABLE; IN_UNION; IN_ELIM_THM] THEN
1683 MESON_TAC[DIST_REFL]);;
1685 let CLOSED_APPROACHABLE = prove
1687 ==> ((!e. &0 < e ==> ?y. y IN s /\ dist(y,x) < e) <=> x IN s)`,
1688 MESON_TAC[CLOSURE_CLOSED; CLOSURE_APPROACHABLE]);;
1690 let IN_CLOSURE_DELETE = prove
1691 (`!s x:real^N. x IN closure(s DELETE x) <=> x limit_point_of s`,
1692 SIMP_TAC[CLOSURE_APPROACHABLE; LIMPT_APPROACHABLE; IN_DELETE; CONJ_ASSOC]);;
1694 (* ------------------------------------------------------------------------- *)
1695 (* Some other lemmas about sequences. *)
1696 (* ------------------------------------------------------------------------- *)
1698 let SEQ_OFFSET = prove
1699 (`!f l k. (f --> l) sequentially ==> ((\i. f(i + k)) --> l) sequentially`,
1700 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
1701 MESON_TAC[ARITH_RULE `N <= n ==> N <= n + k:num`]);;
1703 let SEQ_OFFSET_NEG = prove
1704 (`!f l k. (f --> l) sequentially ==> ((\i. f(i - k)) --> l) sequentially`,
1705 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
1706 MESON_TAC[ARITH_RULE `N + k <= n ==> N <= n - k:num`]);;
1708 let SEQ_OFFSET_REV = prove
1709 (`!f l k. ((\i. f(i + k)) --> l) sequentially ==> (f --> l) sequentially`,
1710 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
1711 MESON_TAC[ARITH_RULE `N + k <= n ==> N <= n - k /\ (n - k) + k = n:num`]);;
1713 let SEQ_HARMONIC = prove
1714 (`((\n. lift(inv(&n))) --> vec 0) sequentially`,
1715 REWRITE_TAC[LIM_SEQUENTIALLY] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
1716 FIRST_ASSUM(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC o
1717 GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN
1718 EXISTS_TAC `N:num` THEN REPEAT STRIP_TAC THEN
1719 REWRITE_TAC[dist; VECTOR_SUB_RZERO; NORM_LIFT] THEN
1720 ASM_REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NUM] THEN
1721 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN
1722 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
1723 ASM_REWRITE_TAC[REAL_OF_NUM_LT; REAL_OF_NUM_LE; LT_NZ]);;
1725 (* ------------------------------------------------------------------------- *)
1726 (* More properties of closed balls. *)
1727 (* ------------------------------------------------------------------------- *)
1729 let CLOSED_CBALL = prove
1730 (`!x:real^N e. closed(cball(x,e))`,
1731 REWRITE_TAC[CLOSED_SEQUENTIAL_LIMITS; IN_CBALL; dist] THEN
1732 GEN_TAC THEN GEN_TAC THEN X_GEN_TAC `s:num->real^N` THEN
1733 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
1734 MATCH_MP_TAC(ISPEC `sequentially` LIM_NORM_UBOUND) THEN
1735 EXISTS_TAC `\n. x - (s:num->real^N) n` THEN
1736 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN
1737 ASM_SIMP_TAC[LIM_SUB; LIM_CONST; SEQUENTIALLY] THEN MESON_TAC[GE_REFL]);;
1739 let IN_INTERIOR_CBALL = prove
1740 (`!x s. x IN interior s <=> ?e. &0 < e /\ cball(x,e) SUBSET s`,
1741 REWRITE_TAC[interior; IN_ELIM_THM] THEN
1742 MESON_TAC[OPEN_CONTAINS_CBALL; SUBSET_TRANS;
1743 BALL_SUBSET_CBALL; CENTRE_IN_BALL; OPEN_BALL]);;
1745 let LIMPT_BALL = prove
1746 (`!x:real^N y e. y limit_point_of ball(x,e) <=> &0 < e /\ y IN cball(x,e)`,
1747 REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 < e` THENL
1748 [ALL_TAC; ASM_MESON_TAC[LIMPT_EMPTY; REAL_NOT_LT; BALL_EQ_EMPTY]] THEN
1749 ASM_REWRITE_TAC[] THEN EQ_TAC THENL
1750 [MESON_TAC[CLOSED_CBALL; CLOSED_LIMPT; LIMPT_SUBSET; BALL_SUBSET_CBALL];
1751 REWRITE_TAC[IN_CBALL; LIMPT_APPROACHABLE; IN_BALL]] THEN
1752 DISCH_TAC THEN X_GEN_TAC `d:real` THEN DISCH_TAC THEN
1753 ASM_CASES_TAC `y:real^N = x` THEN ASM_REWRITE_TAC[DIST_NZ] THENL
1754 [MP_TAC(SPECL [`d:real`; `e:real`] REAL_DOWN2) THEN
1755 ASM_REWRITE_TAC[] THEN
1756 GEN_MESON_TAC 0 40 1 [VECTOR_CHOOSE_DIST; DIST_SYM; REAL_LT_IMP_LE];
1758 MP_TAC(SPECL [`norm(y:real^N - x)`; `d:real`] REAL_DOWN2) THEN
1759 RULE_ASSUM_TAC(REWRITE_RULE[DIST_NZ; dist]) THEN ASM_REWRITE_TAC[] THEN
1760 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
1761 EXISTS_TAC `(y:real^N) - (k / dist(y,x)) % (y - x)` THEN
1762 REWRITE_TAC[dist; VECTOR_ARITH `(y - c % z) - y = --c % z`] THEN
1763 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_ABS_NEG] THEN
1764 ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ] THEN
1765 REWRITE_TAC[VECTOR_ARITH `x - (y - k % (y - x)) = (&1 - k) % (x - y)`] THEN
1766 ASM_SIMP_TAC[REAL_ARITH `&0 < k ==> &0 < abs k`; NORM_MUL] THEN
1767 ASM_SIMP_TAC[REAL_ARITH `&0 < k /\ k < d ==> abs k < d`] THEN
1768 MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `norm(x:real^N - y)` THEN
1769 ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN
1770 MATCH_MP_TAC REAL_LT_RMUL THEN CONJ_TAC THENL
1771 [ALL_TAC; ASM_MESON_TAC[NORM_SUB]] THEN
1772 MATCH_MP_TAC(REAL_ARITH `&0 < k /\ k < &1 ==> abs(&1 - k) < &1`) THEN
1773 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_MUL_LZERO;
1776 let CLOSURE_BALL = prove
1777 (`!x:real^N e. &0 < e ==> (closure(ball(x,e)) = cball(x,e))`,
1778 SIMP_TAC[EXTENSION; closure; IN_ELIM_THM; IN_UNION; LIMPT_BALL] THEN
1779 REWRITE_TAC[IN_BALL; IN_CBALL] THEN REAL_ARITH_TAC);;
1781 let INTERIOR_BALL = prove
1782 (`!a r. interior(ball(a,r)) = ball(a,r)`,
1783 SIMP_TAC[INTERIOR_OPEN; OPEN_BALL]);;
1785 let INTERIOR_CBALL = prove
1786 (`!x:real^N e. interior(cball(x,e)) = ball(x,e)`,
1787 REPEAT GEN_TAC THEN ASM_CASES_TAC `&0 <= e` THENL
1789 SUBGOAL_THEN `cball(x:real^N,e) = {} /\ ball(x:real^N,e) = {}`
1790 (fun th -> REWRITE_TAC[th; INTERIOR_EMPTY]) THEN
1791 REWRITE_TAC[IN_BALL; IN_CBALL; EXTENSION; NOT_IN_EMPTY] THEN
1792 CONJ_TAC THEN X_GEN_TAC `y:real^N` THEN
1793 MP_TAC(ISPECL [`x:real^N`; `y:real^N`] DIST_POS_LE) THEN
1794 POP_ASSUM MP_TAC THEN REAL_ARITH_TAC] THEN
1795 MATCH_MP_TAC INTERIOR_UNIQUE THEN
1796 REWRITE_TAC[BALL_SUBSET_CBALL; OPEN_BALL] THEN
1797 X_GEN_TAC `t:real^N->bool` THEN
1798 SIMP_TAC[SUBSET; IN_CBALL; IN_BALL; REAL_LT_LE] THEN STRIP_TAC THEN
1799 X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
1800 FIRST_X_ASSUM(MP_TAC o SPEC `z:real^N` o GEN_REWRITE_RULE I [open_def]) THEN
1801 ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` MP_TAC) THEN
1802 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
1803 ASM_CASES_TAC `z:real^N = x` THENL
1804 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
1805 FIRST_X_ASSUM(X_CHOOSE_TAC `k:real` o MATCH_MP REAL_DOWN) THEN
1806 SUBGOAL_THEN `?w:real^N. dist(w,x) = k` STRIP_ASSUME_TAC THENL
1807 [ASM_MESON_TAC[VECTOR_CHOOSE_DIST; DIST_SYM; REAL_LT_IMP_LE];
1808 ASM_MESON_TAC[REAL_NOT_LE; DIST_REFL; DIST_SYM]];
1809 RULE_ASSUM_TAC(REWRITE_RULE[DIST_NZ]) THEN
1810 DISCH_THEN(MP_TAC o SPEC `z + ((d / &2) / dist(z,x)) % (z - x:real^N)`) THEN
1811 REWRITE_TAC[dist; VECTOR_ADD_SUB; NORM_MUL; REAL_ABS_DIV;
1812 REAL_ABS_NORM; REAL_ABS_NUM] THEN
1813 ASM_SIMP_TAC[REAL_DIV_RMUL; GSYM dist; REAL_LT_IMP_NZ] THEN
1814 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
1815 ASM_REWRITE_TAC[REAL_ARITH `abs d < d * &2 <=> &0 < d`] THEN
1816 DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[dist] THEN
1817 REWRITE_TAC[VECTOR_ARITH `x - (z + k % (z - x)) = (&1 + k) % (x - z)`] THEN
1818 REWRITE_TAC[REAL_NOT_LE; NORM_MUL] THEN
1819 GEN_REWRITE_TAC LAND_CONV [GSYM REAL_MUL_LID] THEN
1820 ONCE_REWRITE_TAC[NORM_SUB] THEN
1821 ASM_SIMP_TAC[REAL_LT_RMUL_EQ; GSYM dist] THEN
1822 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> &1 < abs(&1 + x)`) THEN
1823 ONCE_REWRITE_TAC[DIST_SYM] THEN
1824 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH]]);;
1826 let FRONTIER_BALL = prove
1827 (`!a e. &0 < e ==> frontier(ball(a,e)) = sphere(a,e)`,
1828 SIMP_TAC[frontier; sphere; CLOSURE_BALL; INTERIOR_OPEN; OPEN_BALL;
1829 REAL_LT_IMP_LE] THEN
1830 REWRITE_TAC[EXTENSION; IN_DIFF; IN_ELIM_THM; IN_BALL; IN_CBALL] THEN
1833 let FRONTIER_CBALL = prove
1834 (`!a e. frontier(cball(a,e)) = sphere(a,e)`,
1835 SIMP_TAC[frontier; sphere; INTERIOR_CBALL; CLOSED_CBALL; CLOSURE_CLOSED;
1836 REAL_LT_IMP_LE] THEN
1837 REWRITE_TAC[EXTENSION; IN_DIFF; IN_ELIM_THM; IN_BALL; IN_CBALL] THEN
1840 let CBALL_EQ_EMPTY = prove
1841 (`!x e. (cball(x,e) = {}) <=> e < &0`,
1842 REWRITE_TAC[EXTENSION; IN_CBALL; NOT_IN_EMPTY; REAL_NOT_LE] THEN
1843 MESON_TAC[DIST_POS_LE; DIST_REFL; REAL_LTE_TRANS]);;
1845 let CBALL_EMPTY = prove
1846 (`!x e. e < &0 ==> cball(x,e) = {}`,
1847 REWRITE_TAC[CBALL_EQ_EMPTY]);;
1849 let CBALL_EQ_SING = prove
1850 (`!x:real^N e. (cball(x,e) = {x}) <=> e = &0`,
1851 REPEAT GEN_TAC THEN REWRITE_TAC[EXTENSION; IN_CBALL; IN_SING] THEN
1852 EQ_TAC THENL [ALL_TAC; MESON_TAC[DIST_LE_0]] THEN
1853 DISCH_THEN(fun th -> MP_TAC(SPEC `x + (e / &2) % basis 1:real^N` th) THEN
1854 MP_TAC(SPEC `x:real^N` th)) THEN
1855 REWRITE_TAC[dist; VECTOR_ARITH `x - (x + e):real^N = --e`;
1856 VECTOR_ARITH `x + e = x <=> e:real^N = vec 0`] THEN
1857 REWRITE_TAC[NORM_NEG; NORM_MUL; VECTOR_MUL_EQ_0; NORM_0; VECTOR_SUB_REFL] THEN
1858 SIMP_TAC[NORM_BASIS; BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1] THEN
1861 let CBALL_SING = prove
1862 (`!x e. e = &0 ==> cball(x,e) = {x}`,
1863 REWRITE_TAC[CBALL_EQ_SING]);;
1865 let SPHERE_SING = prove
1866 (`!x e. e = &0 ==> sphere(x,e) = {x}`,
1867 SIMP_TAC[sphere; DIST_EQ_0; SING_GSPEC]);;
1869 let SPHERE_EQ_SING = prove
1870 (`!a:real^N r x. sphere(a,r) = {x} <=> x = a /\ r = &0`,
1871 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[SPHERE_SING] THEN
1872 ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[SPHERE_EMPTY; NOT_INSERT_EMPTY] THEN
1873 ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[SPHERE_SING] THENL
1874 [ASM SET_TAC[]; ALL_TAC] THEN
1875 MATCH_MP_TAC(SET_RULE
1876 `!y. (x IN s ==> y IN s /\ ~(y = x)) ==> ~(s = {x})`) THEN
1877 EXISTS_TAC `a - (x - a):real^N` THEN REWRITE_TAC[IN_SPHERE] THEN
1878 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC NORM_ARITH);;
1880 (* ------------------------------------------------------------------------- *)
1881 (* For points in the interior, localization of limits makes no difference. *)
1882 (* ------------------------------------------------------------------------- *)
1884 let EVENTUALLY_WITHIN_INTERIOR = prove
1887 ==> (eventually p (at x within s) <=> eventually p (at x))`,
1888 REWRITE_TAC[EVENTUALLY_WITHIN; EVENTUALLY_AT; IN_INTERIOR] THEN
1889 REPEAT GEN_TAC THEN SIMP_TAC[SUBSET; IN_BALL; LEFT_IMP_FORALL_THM] THEN
1890 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
1891 EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
1892 EXISTS_TAC `min (d:real) e` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
1893 ASM_MESON_TAC[DIST_SYM]);;
1895 let LIM_WITHIN_INTERIOR = prove
1898 ==> ((f --> l) (at x within s) <=> (f --> l) (at x))`,
1899 SIMP_TAC[tendsto; EVENTUALLY_WITHIN_INTERIOR]);;
1901 let NETLIMIT_WITHIN_INTERIOR = prove
1902 (`!s x:real^N. x IN interior s ==> netlimit(at x within s) = x`,
1903 REPEAT STRIP_TAC THEN MATCH_MP_TAC NETLIMIT_WITHIN THEN
1904 REWRITE_TAC[TRIVIAL_LIMIT_WITHIN] THEN
1905 FIRST_ASSUM(MP_TAC o MATCH_MP(REWRITE_RULE[OPEN_CONTAINS_BALL]
1906 (SPEC_ALL OPEN_INTERIOR))) THEN
1907 ASM_MESON_TAC[LIMPT_SUBSET; LIMPT_BALL; CENTRE_IN_CBALL; REAL_LT_IMP_LE;
1908 SUBSET_TRANS; INTERIOR_SUBSET]);;
1910 (* ------------------------------------------------------------------------- *)
1911 (* A non-singleton connected set is perfect (i.e. has no isolated points). *)
1912 (* ------------------------------------------------------------------------- *)
1914 let CONNECTED_IMP_PERFECT = prove
1916 connected s /\ ~(?a. s = {a}) /\ x IN s ==> x limit_point_of s`,
1917 REPEAT STRIP_TAC THEN REWRITE_TAC[limit_point_of] THEN
1918 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
1919 MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
1920 FIRST_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I
1921 [OPEN_CONTAINS_CBALL]) THEN
1922 ASM_REWRITE_TAC[] THEN
1923 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
1924 FIRST_X_ASSUM(MP_TAC o SPEC `{x:real^N}` o
1925 GEN_REWRITE_RULE I [CONNECTED_CLOPEN]) THEN
1926 REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL
1927 [REWRITE_TAC[OPEN_IN_OPEN] THEN EXISTS_TAC `t:real^N->bool` THEN
1929 REWRITE_TAC[CLOSED_IN_CLOSED] THEN
1930 EXISTS_TAC `cball(x:real^N,e)` THEN REWRITE_TAC[CLOSED_CBALL] THEN
1931 REWRITE_TAC[EXTENSION; IN_INTER; IN_SING] THEN
1932 ASM_MESON_TAC[CENTRE_IN_CBALL; SUBSET; REAL_LT_IMP_LE];
1935 (* ------------------------------------------------------------------------- *)
1937 (* ------------------------------------------------------------------------- *)
1939 let bounded = new_definition
1940 `bounded s <=> ?a. !x:real^N. x IN s ==> norm(x) <= a`;;
1942 let BOUNDED_EMPTY = prove
1944 REWRITE_TAC[bounded; NOT_IN_EMPTY]);;
1946 let BOUNDED_SUBSET = prove
1947 (`!s t. bounded t /\ s SUBSET t ==> bounded s`,
1948 MESON_TAC[bounded; SUBSET]);;
1950 let BOUNDED_INTERIOR = prove
1951 (`!s:real^N->bool. bounded s ==> bounded(interior s)`,
1952 MESON_TAC[BOUNDED_SUBSET; INTERIOR_SUBSET]);;
1954 let BOUNDED_CLOSURE = prove
1955 (`!s:real^N->bool. bounded s ==> bounded(closure s)`,
1956 REWRITE_TAC[bounded; CLOSURE_SEQUENTIAL] THEN
1957 GEN_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
1958 MESON_TAC[REWRITE_RULE[eventually] LIM_NORM_UBOUND;
1959 TRIVIAL_LIMIT_SEQUENTIALLY; trivial_limit]);;
1961 let BOUNDED_CLOSURE_EQ = prove
1962 (`!s:real^N->bool. bounded(closure s) <=> bounded s`,
1963 GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[BOUNDED_CLOSURE] THEN
1964 MESON_TAC[BOUNDED_SUBSET; CLOSURE_SUBSET]);;
1966 let BOUNDED_CBALL = prove
1967 (`!x:real^N e. bounded(cball(x,e))`,
1968 REPEAT GEN_TAC THEN REWRITE_TAC[bounded] THEN
1969 EXISTS_TAC `norm(x:real^N) + e` THEN REWRITE_TAC[IN_CBALL; dist] THEN
1972 let BOUNDED_BALL = prove
1973 (`!x e. bounded(ball(x,e))`,
1974 MESON_TAC[BALL_SUBSET_CBALL; BOUNDED_CBALL; BOUNDED_SUBSET]);;
1976 let FINITE_IMP_BOUNDED = prove
1977 (`!s:real^N->bool. FINITE s ==> bounded s`,
1978 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN REWRITE_TAC[BOUNDED_EMPTY] THEN
1979 REWRITE_TAC[bounded; IN_INSERT] THEN X_GEN_TAC `x:real^N` THEN GEN_TAC THEN
1980 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `B:real`) STRIP_ASSUME_TAC) THEN
1981 EXISTS_TAC `norm(x:real^N) + abs B` THEN REPEAT STRIP_TAC THEN
1982 ASM_MESON_TAC[NORM_POS_LE; REAL_ARITH
1983 `(y <= b /\ &0 <= x ==> y <= x + abs b) /\ x <= x + abs b`]);;
1985 let BOUNDED_UNION = prove
1986 (`!s t. bounded (s UNION t) <=> bounded s /\ bounded t`,
1987 REWRITE_TAC[bounded; IN_UNION] THEN MESON_TAC[REAL_LE_MAX]);;
1989 let BOUNDED_UNIONS = prove
1990 (`!f. FINITE f /\ (!s. s IN f ==> bounded s) ==> bounded(UNIONS f)`,
1991 REWRITE_TAC[IMP_CONJ] THEN
1992 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1993 REWRITE_TAC[UNIONS_0; BOUNDED_EMPTY; IN_INSERT; UNIONS_INSERT] THEN
1994 MESON_TAC[BOUNDED_UNION]);;
1996 let BOUNDED_POS = prove
1997 (`!s. bounded s <=> ?b. &0 < b /\ !x. x IN s ==> norm(x) <= b`,
1998 REWRITE_TAC[bounded] THEN
1999 MESON_TAC[REAL_ARITH `&0 < &1 + abs(y) /\ (x <= y ==> x <= &1 + abs(y))`]);;
2001 let BOUNDED_POS_LT = prove
2002 (`!s. bounded s <=> ?b. &0 < b /\ !x. x IN s ==> norm(x) < b`,
2003 REWRITE_TAC[bounded] THEN
2004 MESON_TAC[REAL_LT_IMP_LE;
2005 REAL_ARITH `&0 < &1 + abs(y) /\ (x <= y ==> x < &1 + abs(y))`]);;
2007 let BOUNDED_INTER = prove
2008 (`!s t. bounded s \/ bounded t ==> bounded (s INTER t)`,
2009 MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET]);;
2011 let BOUNDED_DIFF = prove
2012 (`!s t. bounded s ==> bounded (s DIFF t)`,
2013 MESON_TAC[BOUNDED_SUBSET; SUBSET_DIFF]);;
2015 let BOUNDED_INSERT = prove
2016 (`!x s. bounded(x INSERT s) <=> bounded s`,
2017 ONCE_REWRITE_TAC[SET_RULE `x INSERT s = {x} UNION s`] THEN
2018 SIMP_TAC[BOUNDED_UNION; FINITE_IMP_BOUNDED; FINITE_RULES]);;
2020 let BOUNDED_SING = prove
2022 REWRITE_TAC[BOUNDED_INSERT; BOUNDED_EMPTY]);;
2024 let BOUNDED_INTERS = prove
2025 (`!f:(real^N->bool)->bool.
2026 (?s:real^N->bool. s IN f /\ bounded s) ==> bounded(INTERS f)`,
2027 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN REPEAT GEN_TAC THEN
2028 DISCH_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET) THEN
2031 let NOT_BOUNDED_UNIV = prove
2032 (`~(bounded (:real^N))`,
2033 REWRITE_TAC[BOUNDED_POS; NOT_FORALL_THM; NOT_EXISTS_THM; IN_UNIV;
2034 DE_MORGAN_THM; REAL_NOT_LE] THEN
2035 X_GEN_TAC `B:real` THEN ASM_CASES_TAC `&0 < B` THEN ASM_REWRITE_TAC[] THEN
2036 MP_TAC(SPEC `B + &1` VECTOR_CHOOSE_SIZE) THEN
2037 ASM_SIMP_TAC[REAL_ARITH `&0 < B ==> &0 <= B + &1`] THEN
2038 MATCH_MP_TAC MONO_EXISTS THEN REAL_ARITH_TAC);;
2040 let COBOUNDED_IMP_UNBOUNDED = prove
2041 (`!s. bounded((:real^N) DIFF s) ==> ~bounded s`,
2042 GEN_TAC THEN REWRITE_TAC[TAUT `a ==> ~b <=> ~(a /\ b)`] THEN
2043 REWRITE_TAC[GSYM BOUNDED_UNION; SET_RULE `UNIV DIFF s UNION s = UNIV`] THEN
2044 REWRITE_TAC[NOT_BOUNDED_UNIV]);;
2046 let BOUNDED_LINEAR_IMAGE = prove
2047 (`!f:real^M->real^N s. bounded s /\ linear f ==> bounded(IMAGE f s)`,
2048 REPEAT GEN_TAC THEN REWRITE_TAC[BOUNDED_POS] THEN
2049 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `B1:real`) MP_TAC) THEN
2050 DISCH_THEN(X_CHOOSE_TAC `B2:real` o MATCH_MP LINEAR_BOUNDED_POS) THEN
2051 EXISTS_TAC `B2 * B1` THEN ASM_SIMP_TAC[REAL_LT_MUL; FORALL_IN_IMAGE] THEN
2052 X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
2053 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `B2 * norm(x:real^M)` THEN
2054 ASM_SIMP_TAC[REAL_LE_LMUL_EQ]);;
2056 let BOUNDED_LINEAR_IMAGE_EQ = prove
2057 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
2058 ==> (bounded (IMAGE f s) <=> bounded s)`,
2059 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE BOUNDED_LINEAR_IMAGE));;
2061 add_linear_invariants [BOUNDED_LINEAR_IMAGE_EQ];;
2063 let BOUNDED_SCALING = prove
2064 (`!c s. bounded s ==> bounded (IMAGE (\x. c % x) s)`,
2065 REPEAT STRIP_TAC THEN MATCH_MP_TAC BOUNDED_LINEAR_IMAGE THEN
2066 ASM_SIMP_TAC[LINEAR_COMPOSE_CMUL; LINEAR_ID]);;
2068 let BOUNDED_NEGATIONS = prove
2069 (`!s. bounded s ==> bounded (IMAGE (--) s)`,
2071 DISCH_THEN(MP_TAC o SPEC `-- &1` o MATCH_MP BOUNDED_SCALING) THEN
2072 REWRITE_TAC[bounded; IN_IMAGE; VECTOR_MUL_LNEG; VECTOR_MUL_LID]);;
2074 let BOUNDED_TRANSLATION = prove
2075 (`!a:real^N s. bounded s ==> bounded (IMAGE (\x. a + x) s)`,
2076 REPEAT GEN_TAC THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE] THEN
2077 DISCH_THEN(X_CHOOSE_TAC `B:real`) THEN
2078 EXISTS_TAC `B + norm(a:real^N)` THEN POP_ASSUM MP_TAC THEN
2079 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [NORM_ARITH_TAC; ALL_TAC] THEN
2080 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
2081 REWRITE_TAC[] THEN NORM_ARITH_TAC);;
2083 let BOUNDED_TRANSLATION_EQ = prove
2084 (`!a s. bounded (IMAGE (\x:real^N. a + x) s) <=> bounded s`,
2085 REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[BOUNDED_TRANSLATION] THEN
2086 DISCH_THEN(MP_TAC o SPEC `--a:real^N` o MATCH_MP BOUNDED_TRANSLATION) THEN
2087 REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
2088 VECTOR_ARITH `--a + a + x:real^N = x`]);;
2090 add_translation_invariants [BOUNDED_TRANSLATION_EQ];;
2092 let BOUNDED_DIFFS = prove
2093 (`!s t:real^N->bool.
2094 bounded s /\ bounded t ==> bounded {x - y | x IN s /\ y IN t}`,
2095 REPEAT GEN_TAC THEN REWRITE_TAC[BOUNDED_POS] THEN
2096 DISCH_THEN(CONJUNCTS_THEN2
2097 (X_CHOOSE_TAC `B:real`) (X_CHOOSE_TAC `C:real`)) THEN
2098 EXISTS_TAC `B + C:real` THEN REWRITE_TAC[IN_ELIM_THM] THEN
2099 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REPEAT STRIP_TAC] THEN
2100 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH
2101 `norm x <= a /\ norm y <= b ==> norm(x - y) <= a + b`) THEN
2104 let BOUNDED_SUMS = prove
2105 (`!s t:real^N->bool.
2106 bounded s /\ bounded t ==> bounded {x + y | x IN s /\ y IN t}`,
2107 REPEAT GEN_TAC THEN REWRITE_TAC[BOUNDED_POS] THEN
2108 DISCH_THEN(CONJUNCTS_THEN2
2109 (X_CHOOSE_TAC `B:real`) (X_CHOOSE_TAC `C:real`)) THEN
2110 EXISTS_TAC `B + C:real` THEN REWRITE_TAC[IN_ELIM_THM] THEN
2111 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REPEAT STRIP_TAC] THEN
2112 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH
2113 `norm x <= a /\ norm y <= b ==> norm(x + y) <= a + b`) THEN
2116 let BOUNDED_SUMS_IMAGE = prove
2117 (`!f g t. bounded {f x | x IN t} /\ bounded {g x | x IN t}
2118 ==> bounded {f x + g x | x IN t}`,
2119 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_SUMS) THEN
2120 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] BOUNDED_SUBSET) THEN
2123 let BOUNDED_SUMS_IMAGES = prove
2124 (`!f:A->B->real^N t s.
2126 (!a. a IN s ==> bounded {f x a | x IN t})
2127 ==> bounded { vsum s (f x) | x IN t}`,
2128 GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
2129 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
2130 SIMP_TAC[VSUM_CLAUSES] THEN CONJ_TAC THENL
2131 [DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC BOUNDED_SUBSET THEN
2132 EXISTS_TAC `{vec 0:real^N}` THEN
2133 SIMP_TAC[FINITE_IMP_BOUNDED; FINITE_RULES] THEN SET_TAC[];
2135 REPEAT STRIP_TAC THEN MATCH_MP_TAC BOUNDED_SUMS_IMAGE THEN
2136 ASM_SIMP_TAC[IN_INSERT]);;
2138 let BOUNDED_SUBSET_BALL = prove
2139 (`!s x:real^N. bounded(s) ==> ?r. &0 < r /\ s SUBSET ball(x,r)`,
2140 REPEAT GEN_TAC THEN REWRITE_TAC[BOUNDED_POS] THEN
2141 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
2142 EXISTS_TAC `&2 * B + norm(x:real^N)` THEN
2143 ASM_SIMP_TAC[NORM_POS_LE; REAL_ARITH
2144 `&0 < B /\ &0 <= x ==> &0 < &2 * B + x`] THEN
2145 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
2146 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[IN_BALL] THEN
2147 UNDISCH_TAC `&0 < B` THEN NORM_ARITH_TAC);;
2149 let BOUNDED_SUBSET_CBALL = prove
2150 (`!s x:real^N. bounded(s) ==> ?r. &0 < r /\ s SUBSET cball(x,r)`,
2151 MESON_TAC[BOUNDED_SUBSET_BALL; SUBSET_TRANS; BALL_SUBSET_CBALL]);;
2153 let UNBOUNDED_INTER_COBOUNDED = prove
2154 (`!s t. ~bounded s /\ bounded((:real^N) DIFF t) ==> ~(s INTER t = {})`,
2155 REWRITE_TAC[SET_RULE `s INTER t = {} <=> s SUBSET (:real^N) DIFF t`] THEN
2156 MESON_TAC[BOUNDED_SUBSET]);;
2158 let COBOUNDED_INTER_UNBOUNDED = prove
2159 (`!s t. bounded((:real^N) DIFF s) /\ ~bounded t ==> ~(s INTER t = {})`,
2160 REWRITE_TAC[SET_RULE `s INTER t = {} <=> t SUBSET (:real^N) DIFF s`] THEN
2161 MESON_TAC[BOUNDED_SUBSET]);;
2163 let SUBSPACE_BOUNDED_EQ_TRIVIAL = prove
2164 (`!s:real^N->bool. subspace s ==> (bounded s <=> s = {vec 0})`,
2165 REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[BOUNDED_SING] THEN
2166 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
2167 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
2168 `~(s = {a}) ==> a IN s ==> ?b. b IN s /\ ~(b = a)`)) THEN
2169 ASM_SIMP_TAC[SUBSPACE_0] THEN
2170 DISCH_THEN(X_CHOOSE_THEN `v:real^N` STRIP_ASSUME_TAC) THEN
2171 REWRITE_TAC[bounded; NOT_EXISTS_THM] THEN X_GEN_TAC `B:real` THEN
2172 DISCH_THEN(MP_TAC o SPEC `(B + &1) / norm v % v:real^N`) THEN
2173 ASM_SIMP_TAC[SUBSPACE_MUL; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
2174 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN REAL_ARITH_TAC);;
2176 let BOUNDED_COMPONENTWISE = prove
2178 bounded s <=> !i. 1 <= i /\ i <= dimindex(:N)
2179 ==> bounded (IMAGE (\x. lift(x$i)) s)`,
2180 GEN_TAC THEN REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE; NORM_LIFT] THEN
2181 EQ_TAC THENL [ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]; ALL_TAC] THEN
2182 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
2183 SIMP_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:num->real` THEN
2184 DISCH_TAC THEN EXISTS_TAC `sum(1..dimindex(:N)) b` THEN CONJ_TAC THENL
2185 [MATCH_MP_TAC REAL_LET_TRANS THEN
2186 EXISTS_TAC `sum(1..dimindex(:N)) (\i. &0)` THEN
2187 SIMP_TAC[SUM_POS_LE_NUMSEG; REAL_POS] THEN
2188 MATCH_MP_TAC SUM_LT_ALL THEN
2189 ASM_SIMP_TAC[IN_NUMSEG; FINITE_NUMSEG; NUMSEG_EMPTY] THEN
2190 REWRITE_TAC[NOT_LT; DIMINDEX_GE_1];
2191 REPEAT STRIP_TAC THEN
2192 W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN
2193 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
2194 MATCH_MP_TAC SUM_LE THEN ASM_SIMP_TAC[IN_NUMSEG; FINITE_NUMSEG]]);;
2196 (* ------------------------------------------------------------------------- *)
2197 (* Some theorems on sups and infs using the notion "bounded". *)
2198 (* ------------------------------------------------------------------------- *)
2200 let BOUNDED_LIFT = prove
2201 (`!s. bounded(IMAGE lift s) <=> ?a. !x. x IN s ==> abs(x) <= a`,
2202 REWRITE_TAC[bounded; FORALL_LIFT; NORM_LIFT; LIFT_IN_IMAGE_LIFT]);;
2204 let BOUNDED_HAS_SUP = prove
2205 (`!s. bounded(IMAGE lift s) /\ ~(s = {})
2206 ==> (!x. x IN s ==> x <= sup s) /\
2207 (!b. (!x. x IN s ==> x <= b) ==> sup s <= b)`,
2208 REWRITE_TAC[BOUNDED_LIFT; IMAGE_EQ_EMPTY] THEN
2209 MESON_TAC[SUP; REAL_ARITH `abs(x) <= a ==> x <= a`]);;
2211 let SUP_INSERT = prove
2212 (`!x s. bounded (IMAGE lift s)
2213 ==> sup(x INSERT s) = if s = {} then x else max x (sup s)`,
2214 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_SUP_UNIQUE THEN
2215 COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_SING] THENL
2216 [MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN
2217 REWRITE_TAC[REAL_LE_MAX; REAL_LT_MAX; IN_INSERT] THEN
2218 MP_TAC(ISPEC `s:real->bool` BOUNDED_HAS_SUP) THEN ASM_REWRITE_TAC[] THEN
2219 REPEAT STRIP_TAC THEN ASM_MESON_TAC[REAL_LE_REFL; REAL_NOT_LT]);;
2221 let BOUNDED_HAS_INF = prove
2222 (`!s. bounded(IMAGE lift s) /\ ~(s = {})
2223 ==> (!x. x IN s ==> inf s <= x) /\
2224 (!b. (!x. x IN s ==> b <= x) ==> b <= inf s)`,
2225 REWRITE_TAC[BOUNDED_LIFT; IMAGE_EQ_EMPTY] THEN
2226 MESON_TAC[INF; REAL_ARITH `abs(x) <= a ==> --a <= x`]);;
2228 let INF_INSERT = prove
2229 (`!x s. bounded (IMAGE lift s)
2230 ==> inf(x INSERT s) = if s = {} then x else min x (inf s)`,
2231 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_INF_UNIQUE THEN
2232 COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_SING] THENL
2233 [MESON_TAC[REAL_LE_REFL]; ALL_TAC] THEN
2234 REWRITE_TAC[REAL_MIN_LE; REAL_MIN_LT; IN_INSERT] THEN
2235 MP_TAC(ISPEC `s:real->bool` BOUNDED_HAS_INF) THEN ASM_REWRITE_TAC[] THEN
2236 REPEAT STRIP_TAC THEN ASM_MESON_TAC[REAL_LE_REFL; REAL_NOT_LT]);;
2238 (* ------------------------------------------------------------------------- *)
2239 (* Subset and overlapping relations on balls. *)
2240 (* ------------------------------------------------------------------------- *)
2242 let SUBSET_BALLS = prove
2243 (`(!a a':real^N r r'.
2244 ball(a,r) SUBSET ball(a',r') <=> dist(a,a') + r <= r' \/ r <= &0) /\
2246 ball(a,r) SUBSET cball(a',r') <=> dist(a,a') + r <= r' \/ r <= &0) /\
2248 cball(a,r) SUBSET ball(a',r') <=> dist(a,a') + r < r' \/ r < &0) /\
2250 cball(a,r) SUBSET cball(a',r') <=> dist(a,a') + r <= r' \/ r < &0)`,
2253 cball(a,r) SUBSET cball(a',r') <=> dist(a,a') + r <= r' \/ r < &0) /\
2255 cball(a,r) SUBSET ball(a',r') <=> dist(a,a') + r < r' \/ r < &0)`,
2257 (GEOM_ORIGIN_TAC `a':real^N` THEN
2258 REPEAT GEN_TAC THEN REWRITE_TAC[SUBSET; IN_CBALL; IN_BALL] THEN
2259 EQ_TAC THENL [REWRITE_TAC[DIST_0]; NORM_ARITH_TAC] THEN
2260 DISJ_CASES_TAC(REAL_ARITH `r < &0 \/ &0 <= r`) THEN
2261 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN DISJ1_TAC THEN
2262 ASM_CASES_TAC `a:real^N = vec 0` THENL
2263 [FIRST_X_ASSUM(MP_TAC o SPEC `r % basis 1:real^N`) THEN
2264 ASM_SIMP_TAC[DIST_0; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL];
2265 FIRST_X_ASSUM(MP_TAC o SPEC `(&1 + r / norm(a)) % a:real^N`) THEN
2266 SIMP_TAC[dist; VECTOR_ARITH `a - (&1 + x) % a:real^N = --(x % a)`] THEN
2267 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; NORM_NEG; REAL_POS;
2268 REAL_LE_DIV; NORM_POS_LE; REAL_ADD_RDISTRIB; REAL_DIV_RMUL;
2269 NORM_EQ_0; REAL_ARITH `&0 <= x ==> abs(&1 + x) = &1 + x`]] THEN
2270 UNDISCH_TAC `&0 <= r` THEN NORM_ARITH_TAC))
2271 and tac = DISCH_THEN(MP_TAC o MATCH_MP SUBSET_CLOSURE) THEN
2272 ASM_SIMP_TAC[CLOSED_CBALL; CLOSURE_CLOSED; CLOSURE_BALL] in
2273 REWRITE_TAC[AND_FORALL_THM] THEN GEOM_ORIGIN_TAC `a':real^N` THEN
2274 REPEAT STRIP_TAC THEN
2276 [ALL_TAC; REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL] THEN NORM_ARITH_TAC]) THEN
2277 MATCH_MP_TAC(SET_RULE
2278 `(s = {} <=> q) /\ (s SUBSET t /\ ~(s = {}) /\ ~(t = {}) ==> p)
2279 ==> s SUBSET t ==> p \/ q`) THEN
2280 REWRITE_TAC[BALL_EQ_EMPTY; CBALL_EQ_EMPTY; REAL_NOT_LE; REAL_NOT_LT] THEN
2281 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THENL
2282 [tac; tac; ALL_TAC; ALL_TAC] THEN REWRITE_TAC[lemma] THEN
2283 REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);;
2285 let INTER_BALLS_EQ_EMPTY = prove
2286 (`(!a b:real^N r s. ball(a,r) INTER ball(b,s) = {} <=>
2287 r <= &0 \/ s <= &0 \/ r + s <= dist(a,b)) /\
2288 (!a b:real^N r s. ball(a,r) INTER cball(b,s) = {} <=>
2289 r <= &0 \/ s < &0 \/ r + s <= dist(a,b)) /\
2290 (!a b:real^N r s. cball(a,r) INTER ball(b,s) = {} <=>
2291 r < &0 \/ s <= &0 \/ r + s <= dist(a,b)) /\
2292 (!a b:real^N r s. cball(a,r) INTER cball(b,s) = {} <=>
2293 r < &0 \/ s < &0 \/ r + s < dist(a,b))`,
2294 REPEAT STRIP_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN
2295 GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN REPEAT STRIP_TAC THEN
2296 REWRITE_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_CBALL; IN_BALL] THEN
2299 SPEC_TAC(`b % basis 1:real^N`,`v:real^N`) THEN CONV_TAC NORM_ARITH]) THEN
2300 DISCH_THEN(MP_TAC o GEN `c:real` o SPEC `c % basis 1:real^N`) THEN
2301 SIMP_TAC[NORM_MUL; NORM_BASIS; LE_REFL; DIMINDEX_GE_1; dist; NORM_NEG;
2302 VECTOR_SUB_LZERO; GSYM VECTOR_SUB_RDISTRIB; REAL_MUL_RID] THEN
2303 ASM_REWRITE_TAC[real_abs] THEN REWRITE_TAC[GSYM real_abs] THEN
2304 DISCH_THEN(fun th ->
2305 MP_TAC(SPEC `min b r:real` th) THEN
2306 MP_TAC(SPEC `max (&0) (b - s:real)` th) THEN
2307 MP_TAC(SPEC `(r + (b - s)) / &2` th)) THEN
2308 ASM_REAL_ARITH_TAC);;
2310 (* ------------------------------------------------------------------------- *)
2311 (* Every closed set is a G_Delta. *)
2312 (* ------------------------------------------------------------------------- *)
2314 let CLOSED_AS_GDELTA = prove
2317 ==> ?g. COUNTABLE g /\
2318 (!u. u IN g ==> open u) /\
2320 REPEAT STRIP_TAC THEN EXISTS_TAC
2321 `{ UNIONS { ball(x:real^N,inv(&n + &1)) | x IN s} | n IN (:num)}` THEN
2322 SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE; NUM_COUNTABLE] THEN
2323 SIMP_TAC[FORALL_IN_IMAGE; OPEN_UNIONS; OPEN_BALL] THEN
2324 MATCH_MP_TAC(SET_RULE
2325 `closure s = s /\ s SUBSET t /\ t SUBSET closure s
2327 ASM_REWRITE_TAC[CLOSURE_EQ] THEN CONJ_TAC THENL
2328 [REWRITE_TAC[SUBSET_INTERS; FORALL_IN_IMAGE; IN_UNIV] THEN
2329 X_GEN_TAC `n:num` THEN REWRITE_TAC[UNIONS_IMAGE; SUBSET; IN_ELIM_THM] THEN
2330 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `x:real^N` THEN
2331 ASM_REWRITE_TAC[CENTRE_IN_BALL; REAL_LT_INV_EQ] THEN REAL_ARITH_TAC;
2332 REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE; INTERS_IMAGE; IN_UNIV] THEN
2333 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM; UNIONS_IMAGE] THEN
2334 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2335 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN
2336 DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN
2337 FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[IN_BALL] THEN
2338 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN
2339 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
2340 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LT_TRANS) THEN
2341 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
2342 REAL_LT_TRANS)) THEN
2343 MATCH_MP_TAC REAL_LT_INV2 THEN
2344 REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]);;
2346 (* ------------------------------------------------------------------------- *)
2347 (* Compactness (the definition is the one based on convegent subsequences). *)
2348 (* ------------------------------------------------------------------------- *)
2350 let compact = new_definition
2354 ==> ?l r. l IN s /\ (!m n:num. m < n ==> r(m) < r(n)) /\
2355 ((f o r) --> l) sequentially`;;
2357 let MONOTONE_BIGGER = prove
2358 (`!r. (!m n. m < n ==> r(m) < r(n)) ==> !n:num. n <= r(n)`,
2359 GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THEN
2360 ASM_MESON_TAC[LE_0; ARITH_RULE `n <= m /\ m < p ==> SUC n <= p`; LT]);;
2362 let LIM_SUBSEQUENCE = prove
2363 (`!s r l. (!m n. m < n ==> r(m) < r(n)) /\ (s --> l) sequentially
2364 ==> (s o r --> l) sequentially`,
2365 REWRITE_TAC[LIM_SEQUENTIALLY; o_THM] THEN
2366 MESON_TAC[MONOTONE_BIGGER; LE_TRANS]);;
2368 let MONOTONE_SUBSEQUENCE = prove
2369 (`!s:num->real. ?r:num->num.
2370 (!m n. m < n ==> r(m) < r(n)) /\
2371 ((!m n. m <= n ==> s(r(m)) <= s(r(n))) \/
2372 (!m n. m <= n ==> s(r(n)) <= s(r(m))))`,
2374 ASM_CASES_TAC `!n:num. ?p. n < p /\ !m. p <= m ==> s(m) <= s(p)` THEN
2375 POP_ASSUM MP_TAC THEN
2376 REWRITE_TAC[NOT_FORALL_THM; NOT_EXISTS_THM; NOT_IMP; DE_MORGAN_THM] THEN
2377 REWRITE_TAC[RIGHT_OR_EXISTS_THM; SKOLEM_THM; REAL_NOT_LE; REAL_NOT_LT] THENL
2378 [ABBREV_TAC `N = 0`; DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC)] THEN
2379 DISCH_THEN(X_CHOOSE_THEN `next:num->num` STRIP_ASSUME_TAC) THEN
2380 (MP_TAC o prove_recursive_functions_exist num_RECURSION)
2381 `(r 0 = next(SUC N)) /\ (!n. r(SUC n) = next(r n))` THEN
2382 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THENL
2383 [SUBGOAL_THEN `!m:num n:num. r n <= m ==> s(m) <= s(r n):real`
2384 ASSUME_TAC THEN TRY CONJ_TAC THEN TRY DISJ2_TAC THEN
2385 GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[LT; LE] THEN
2386 ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL; LT_IMP_LE; LT_TRANS];
2387 SUBGOAL_THEN `!n. N < (r:num->num) n` ASSUME_TAC THEN
2388 TRY(CONJ_TAC THENL [GEN_TAC; DISJ1_TAC THEN GEN_TAC]) THEN
2389 INDUCT_TAC THEN ASM_REWRITE_TAC[LT; LE] THEN
2390 TRY STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
2391 ASM_MESON_TAC[REAL_LT_REFL; LT_LE; LTE_TRANS; REAL_LE_REFL;
2392 REAL_LT_LE; REAL_LE_TRANS; LT]]);;
2394 let CONVERGENT_BOUNDED_INCREASING = prove
2395 (`!s:num->real b. (!m n. m <= n ==> s m <= s n) /\ (!n. abs(s n) <= b)
2396 ==> ?l. !e. &0 < e ==> ?N. !n. N <= n ==> abs(s n - l) < e`,
2397 REPEAT STRIP_TAC THEN
2398 MP_TAC(SPEC `\x. ?n. (s:num->real) n = x` REAL_COMPLETE) THEN
2399 REWRITE_TAC[] THEN ANTS_TAC THENL
2400 [ASM_MESON_TAC[REAL_ARITH `abs(x) <= b ==> x <= b`]; ALL_TAC] THEN
2401 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real` THEN STRIP_TAC THEN
2402 X_GEN_TAC `e:real` THEN STRIP_TAC THEN
2403 FIRST_X_ASSUM(MP_TAC o SPEC `l - e`) THEN
2404 ASM_MESON_TAC[REAL_ARITH `&0 < e ==> ~(l <= l - e)`;
2405 REAL_ARITH `x <= y /\ y <= l /\ ~(x <= l - e) ==> abs(y - l) < e`]);;
2407 let CONVERGENT_BOUNDED_MONOTONE = prove
2408 (`!s:num->real b. (!n. abs(s n) <= b) /\
2409 ((!m n. m <= n ==> s m <= s n) \/
2410 (!m n. m <= n ==> s n <= s m))
2411 ==> ?l. !e. &0 < e ==> ?N. !n. N <= n ==> abs(s n - l) < e`,
2412 REPEAT STRIP_TAC THENL
2413 [ASM_MESON_TAC[CONVERGENT_BOUNDED_INCREASING]; ALL_TAC] THEN
2414 MP_TAC(SPEC `\n. --((s:num->real) n)` CONVERGENT_BOUNDED_INCREASING) THEN
2415 ASM_REWRITE_TAC[REAL_LE_NEG2; REAL_ABS_NEG] THEN
2416 ASM_MESON_TAC[REAL_ARITH `abs(x - --l) = abs(--x - l)`]);;
2418 let COMPACT_REAL_LEMMA = prove
2419 (`!s b. (!n:num. abs(s n) <= b)
2420 ==> ?l r. (!m n:num. m < n ==> r(m) < r(n)) /\
2421 !e. &0 < e ==> ?N. !n. N <= n ==> abs(s(r n) - l) < e`,
2422 REPEAT GEN_TAC THEN DISCH_TAC THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
2423 MP_TAC(SPEC `s:num->real` MONOTONE_SUBSEQUENCE) THEN
2424 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[] THEN
2425 MATCH_MP_TAC CONVERGENT_BOUNDED_MONOTONE THEN ASM_MESON_TAC[]);;
2427 let COMPACT_LEMMA = prove
2428 (`!s. bounded s /\ (!n. (x:num->real^N) n IN s)
2429 ==> !d. d <= dimindex(:N)
2430 ==> ?l:real^N r. (!m n. m < n ==> r m < (r:num->num) n) /\
2432 ==> ?N. !n i. 1 <= i /\ i <= d
2434 ==> abs(x(r n)$i - l$i) < e`,
2435 GEN_TAC THEN REWRITE_TAC[bounded] THEN
2436 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `b:real`) ASSUME_TAC) THEN
2438 [REWRITE_TAC[ARITH_RULE `1 <= i /\ i <= 0 <=> F`; CONJ_ASSOC] THEN
2439 DISCH_TAC THEN EXISTS_TAC `\n:num. n` THEN REWRITE_TAC[];
2441 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
2442 ASM_SIMP_TAC[ARITH_RULE `SUC d <= n ==> d <= n`] THEN STRIP_TAC THEN
2443 MP_TAC(SPECL [`\n:num. (x:num->real^N) (r n) $ (SUC d)`; `b:real`]
2444 COMPACT_REAL_LEMMA) THEN
2445 REWRITE_TAC[] THEN ANTS_TAC THENL
2446 [ASM_MESON_TAC[REAL_LE_TRANS; COMPONENT_LE_NORM; ARITH_RULE `1 <= SUC n`];
2448 DISCH_THEN(X_CHOOSE_THEN `y:real` (X_CHOOSE_THEN `s:num->num`
2449 STRIP_ASSUME_TAC)) THEN
2450 MAP_EVERY EXISTS_TAC
2451 [`(lambda k. if k = SUC d then y else (l:real^N)$k):real^N`;
2452 `(r:num->num) o (s:num->num)`] THEN
2453 ASM_SIMP_TAC[o_THM] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2454 REPEAT(FIRST_ASSUM(C UNDISCH_THEN (MP_TAC o SPEC `e:real`) o concl)) THEN
2455 ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `N1:num`) THEN
2456 DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN EXISTS_TAC `N1 + N2:num` THEN
2457 FIRST_ASSUM(fun th -> SIMP_TAC[LAMBDA_BETA; MATCH_MP(ARITH_RULE
2458 `SUC d <= n ==> !i. 1 <= i /\ i <= SUC d ==> 1 <= i /\ i <= n`) th]) THEN
2459 REWRITE_TAC[LE] THEN REPEAT STRIP_TAC THEN
2460 ASM_REWRITE_TAC[] THEN TRY COND_CASES_TAC THEN
2461 ASM_MESON_TAC[MONOTONE_BIGGER; LE_TRANS;
2462 ARITH_RULE `N1 + N2 <= n ==> N2 <= n:num /\ N1 <= n`;
2463 ARITH_RULE `1 <= i /\ i <= d /\ SUC d <= n
2464 ==> ~(i = SUC d) /\ 1 <= SUC d /\ d <= n /\ i <= n`]);;
2466 let BOUNDED_CLOSED_IMP_COMPACT = prove
2467 (`!s:real^N->bool. bounded s /\ closed s ==> compact s`,
2468 REPEAT STRIP_TAC THEN REWRITE_TAC[compact] THEN
2469 X_GEN_TAC `x:num->real^N` THEN DISCH_TAC THEN
2470 MP_TAC(ISPEC `s:real^N->bool` COMPACT_LEMMA) THEN
2471 ASM_REWRITE_TAC[] THEN
2472 DISCH_THEN(MP_TAC o SPEC `dimindex(:N)`) THEN
2473 REWRITE_TAC[LE_REFL] THEN
2474 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN
2475 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN ASM_SIMP_TAC[] THEN
2476 STRIP_TAC THEN MATCH_MP_TAC(TAUT `(b ==> a) /\ b ==> a /\ b`) THEN
2477 REPEAT STRIP_TAC THENL
2478 [FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[CLOSED_SEQUENTIAL_LIMITS]) THEN
2479 EXISTS_TAC `(x:num->real^N) o (r:num->num)` THEN
2480 ASM_REWRITE_TAC[o_THM];
2482 REWRITE_TAC[LIM_SEQUENTIALLY] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2483 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2 / &(dimindex(:N))`) THEN
2484 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; DIMINDEX_NONZERO;
2485 REAL_HALF; ARITH_RULE `0 < n <=> ~(n = 0)`] THEN
2486 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
2487 REWRITE_TAC[dist] THEN REPEAT STRIP_TAC THEN
2488 MATCH_MP_TAC(MATCH_MP (REAL_ARITH `a <= b ==> b < e ==> a < e`)
2489 (SPEC_ALL NORM_LE_L1)) THEN
2490 MATCH_MP_TAC REAL_LET_TRANS THEN
2491 EXISTS_TAC `sum (1..dimindex(:N))
2492 (\k. e / &2 / &(dimindex(:N)))` THEN
2494 [MATCH_MP_TAC SUM_LE_NUMSEG THEN
2495 SIMP_TAC[o_THM; LAMBDA_BETA; vector_sub] THEN
2496 ASM_MESON_TAC[REAL_LT_IMP_LE; LE_TRANS];
2497 ASM_SIMP_TAC[SUM_CONST_NUMSEG; ADD_SUB; REAL_DIV_LMUL; REAL_OF_NUM_EQ;
2498 DIMINDEX_NONZERO; REAL_LE_REFL; REAL_LT_LDIV_EQ; ARITH;
2499 REAL_OF_NUM_LT; REAL_ARITH `x < x * &2 <=> &0 < x`]]);;
2501 (* ------------------------------------------------------------------------- *)
2503 (* ------------------------------------------------------------------------- *)
2505 let cauchy = new_definition
2506 `cauchy (s:num->real^N) <=>
2507 !e. &0 < e ==> ?N. !m n. m >= N /\ n >= N ==> dist(s m,s n) < e`;;
2509 let complete = new_definition
2511 !f:num->real^N. (!n. f n IN s) /\ cauchy f
2512 ==> ?l. l IN s /\ (f --> l) sequentially`;;
2516 cauchy s <=> !e. &0 < e ==> ?N. !n. n >= N ==> dist(s n,s N) < e`,
2517 REPEAT GEN_TAC THEN REWRITE_TAC[cauchy; GE] THEN EQ_TAC THENL
2518 [MESON_TAC[LE_REFL]; DISCH_TAC] THEN
2519 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2520 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
2521 MESON_TAC[DIST_TRIANGLE_HALF_L]);;
2523 let CONVERGENT_IMP_CAUCHY = prove
2524 (`!s l. (s --> l) sequentially ==> cauchy s`,
2525 REWRITE_TAC[LIM_SEQUENTIALLY; cauchy] THEN
2526 REPEAT GEN_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2527 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
2528 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
2529 ASM_MESON_TAC[GE; LE_REFL; DIST_TRIANGLE_HALF_L]);;
2531 let CAUCHY_IMP_BOUNDED = prove
2532 (`!s:num->real^N. cauchy s ==> bounded {y | ?n. y = s n}`,
2533 REWRITE_TAC[cauchy; bounded; IN_ELIM_THM] THEN GEN_TAC THEN
2534 DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[REAL_LT_01] THEN
2535 DISCH_THEN(X_CHOOSE_THEN `N:num` (MP_TAC o SPEC `N:num`)) THEN
2536 REWRITE_TAC[GE_REFL] THEN DISCH_TAC THEN
2537 SUBGOAL_THEN `!n:num. N <= n ==> norm(s n :real^N) <= norm(s N) + &1`
2539 [ASM_MESON_TAC[GE; dist; DIST_SYM; NORM_TRIANGLE_SUB;
2540 REAL_ARITH `a <= b + c /\ c < &1 ==> a <= b + &1`];
2541 MP_TAC(ISPECL [`\n:num. norm(s n :real^N)`; `0..N`]
2542 UPPER_BOUND_FINITE_SET_REAL) THEN
2543 SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; LE_0; LEFT_IMP_EXISTS_THM] THEN
2544 ASM_MESON_TAC[LE_CASES;
2545 REAL_ARITH `x <= a \/ x <= b ==> x <= abs a + abs b`]]);;
2547 let COMPACT_IMP_COMPLETE = prove
2548 (`!s:real^N->bool. compact s ==> complete s`,
2549 GEN_TAC THEN REWRITE_TAC[complete; compact] THEN
2550 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `f:num->real^N` THEN
2551 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
2552 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
2553 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
2554 FIRST_X_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_ADD)) THEN
2555 DISCH_THEN(MP_TAC o SPEC `\n. (f:num->real^N)(n) - f(r n)`) THEN
2556 DISCH_THEN(MP_TAC o SPEC `vec 0: real^N`) THEN ASM_REWRITE_TAC[o_THM] THEN
2557 REWRITE_TAC[VECTOR_ADD_RID; VECTOR_SUB_ADD2; ETA_AX] THEN
2558 DISCH_THEN MATCH_MP_TAC THEN
2559 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [cauchy]) THEN
2560 REWRITE_TAC[GE; LIM; SEQUENTIALLY; dist; VECTOR_SUB_RZERO] THEN
2561 SUBGOAL_THEN `!n:num. n <= r(n)` MP_TAC THENL [INDUCT_TAC; ALL_TAC] THEN
2562 ASM_MESON_TAC[ LE_TRANS; LE_REFL; LT; LET_TRANS; LE_0; LE_SUC_LT]);;
2564 let COMPLETE_UNIV = prove
2565 (`complete(:real^N)`,
2566 REWRITE_TAC[complete; IN_UNIV] THEN X_GEN_TAC `x:num->real^N` THEN
2567 DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CAUCHY_IMP_BOUNDED) THEN
2568 DISCH_THEN(ASSUME_TAC o MATCH_MP BOUNDED_CLOSURE) THEN
2569 MP_TAC(ISPEC `closure {y:real^N | ?n:num. y = x n}`
2570 COMPACT_IMP_COMPLETE) THEN
2571 ASM_SIMP_TAC[BOUNDED_CLOSED_IMP_COMPACT; CLOSED_CLOSURE; complete] THEN
2572 DISCH_THEN(MP_TAC o SPEC `x:num->real^N`) THEN
2573 ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
2574 ASM_REWRITE_TAC[closure; IN_ELIM_THM; IN_UNION] THEN MESON_TAC[]);;
2576 let COMPLETE_EQ_CLOSED = prove
2577 (`!s:real^N->bool. complete s <=> closed s`,
2578 GEN_TAC THEN EQ_TAC THENL
2579 [REWRITE_TAC[complete; CLOSED_LIMPT; LIMPT_SEQUENTIAL] THEN
2580 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN GEN_TAC THEN
2581 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MATCH_MP_TAC MONO_FORALL THEN
2582 MESON_TAC[CONVERGENT_IMP_CAUCHY; IN_DELETE; LIM_UNIQUE;
2583 TRIVIAL_LIMIT_SEQUENTIALLY];
2584 REWRITE_TAC[complete; CLOSED_SEQUENTIAL_LIMITS] THEN DISCH_TAC THEN
2585 X_GEN_TAC `f:num->real^N` THEN STRIP_TAC THEN
2586 MP_TAC(REWRITE_RULE[complete] COMPLETE_UNIV) THEN
2587 DISCH_THEN(MP_TAC o SPEC `f:num->real^N`) THEN
2588 ASM_REWRITE_TAC[IN_UNIV] THEN ASM_MESON_TAC[]]);;
2590 let CONVERGENT_EQ_CAUCHY = prove
2591 (`!s. (?l. (s --> l) sequentially) <=> cauchy s`,
2592 GEN_TAC THEN EQ_TAC THENL
2593 [REWRITE_TAC[LEFT_IMP_EXISTS_THM; CONVERGENT_IMP_CAUCHY];
2594 REWRITE_TAC[REWRITE_RULE[complete; IN_UNIV] COMPLETE_UNIV]]);;
2596 let CONVERGENT_IMP_BOUNDED = prove
2597 (`!s l. (s --> l) sequentially ==> bounded (IMAGE s (:num))`,
2598 REWRITE_TAC[LEFT_FORALL_IMP_THM; CONVERGENT_EQ_CAUCHY] THEN
2599 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CAUCHY_IMP_BOUNDED) THEN
2600 REWRITE_TAC[IMAGE; IN_UNIV]);;
2602 (* ------------------------------------------------------------------------- *)
2603 (* Total boundedness. *)
2604 (* ------------------------------------------------------------------------- *)
2606 let COMPACT_IMP_TOTALLY_BOUNDED = prove
2609 ==> !e. &0 < e ==> ?k. FINITE k /\ k SUBSET s /\
2610 s SUBSET (UNIONS(IMAGE (\x. ball(x,e)) k))`,
2611 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
2612 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN
2613 REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`; SUBSET] THEN
2614 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
2616 `?x:num->real^N. !n. x(n) IN s /\ !m. m < n ==> ~(dist(x(m),x(n)) < e)`
2620 !n. x(n) = @y. y IN s /\ !m. m < n ==> ~(dist(x(m),y) < e)`
2622 [MATCH_MP_TAC(MATCH_MP WF_REC WF_num) THEN SIMP_TAC[]; ALL_TAC] THEN
2623 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:num->real^N` THEN
2624 DISCH_TAC THEN MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN
2625 FIRST_X_ASSUM(SUBST1_TAC o SPEC `n:num`) THEN STRIP_TAC THEN
2626 CONV_TAC SELECT_CONV THEN
2627 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (x:num->real^N) {m | m < n}`) THEN
2628 SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT; NOT_FORALL_THM; NOT_IMP] THEN
2629 REWRITE_TAC[IN_UNIONS; IN_IMAGE; IN_ELIM_THM] THEN ASM_MESON_TAC[IN_BALL];
2631 REWRITE_TAC[compact; NOT_FORALL_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
2632 X_GEN_TAC `x:num->real^N` THEN REWRITE_TAC[NOT_IMP; FORALL_AND_THM] THEN
2633 STRIP_TAC THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN REPEAT STRIP_TAC THEN
2634 FIRST_X_ASSUM(MP_TAC o MATCH_MP CONVERGENT_IMP_CAUCHY) THEN
2635 REWRITE_TAC[cauchy] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN
2636 ASM_REWRITE_TAC[o_THM; NOT_EXISTS_THM; NOT_IMP; NOT_FORALL_THM; NOT_IMP] THEN
2637 X_GEN_TAC `N:num` THEN MAP_EVERY EXISTS_TAC [`N:num`; `SUC N`] THEN
2638 CONJ_TAC THENL [ARITH_TAC; ASM_MESON_TAC[LT]]);;
2640 (* ------------------------------------------------------------------------- *)
2641 (* Heine-Borel theorem (following Burkill & Burkill vol. 2) *)
2642 (* ------------------------------------------------------------------------- *)
2644 let HEINE_BOREL_LEMMA = prove
2647 ==> !t. s SUBSET (UNIONS t) /\ (!b. b IN t ==> open b)
2649 !x. x IN s ==> ?b. b IN t /\ ball(x,e) SUBSET b`,
2650 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
2651 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN
2652 DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2653 DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&1 / (&n + &1)`) THEN
2654 SIMP_TAC[REAL_LT_DIV; REAL_LT_01; REAL_ARITH `x <= y ==> x < y + &1`;
2655 FORALL_AND_THM; REAL_POS; NOT_FORALL_THM; NOT_IMP; SKOLEM_THM; compact] THEN
2656 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REWRITE_TAC[NOT_EXISTS_THM] THEN
2657 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN
2658 DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`l:real^N`; `r:num->num`] THEN
2660 SUBGOAL_THEN `?b:real^N->bool. l IN b /\ b IN t` STRIP_ASSUME_TAC THENL
2661 [ASM_MESON_TAC[SUBSET; IN_UNIONS]; ALL_TAC] THEN
2662 SUBGOAL_THEN `?e. &0 < e /\ !z:real^N. dist(z,l) < e ==> z IN b`
2663 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[open_def]; ALL_TAC] THEN
2664 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
2665 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
2666 SUBGOAL_THEN `&0 < e / &2` (fun th ->
2667 REWRITE_TAC[th; o_THM] THEN MP_TAC(GEN_REWRITE_RULE I [REAL_ARCH_INV] th))
2668 THENL [ASM_REWRITE_TAC[REAL_HALF]; ALL_TAC] THEN
2669 DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
2670 DISCH_THEN(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC) THEN
2671 FIRST_X_ASSUM(MP_TAC o SPECL
2672 [`(r:num->num)(N1 + N2)`; `b:real^N->bool`]) THEN
2673 ASM_REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
2674 FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC DIST_TRIANGLE_HALF_R THEN
2675 EXISTS_TAC `(f:num->real^N)(r(N1 + N2:num))` THEN CONJ_TAC THENL
2676 [ALL_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC] THEN
2677 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_BALL]) THEN
2678 MATCH_MP_TAC(REAL_ARITH `a <= b ==> x < a ==> x < b`) THEN
2679 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&N1)` THEN
2680 ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN REWRITE_TAC[real_div; REAL_MUL_LID] THEN
2681 MATCH_MP_TAC REAL_LE_INV2 THEN
2682 REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN
2683 ASM_MESON_TAC[ARITH_RULE `(~(n = 0) ==> 0 < n)`; LE_ADD; MONOTONE_BIGGER;
2684 LT_IMP_LE; LE_TRANS]);;
2686 let COMPACT_IMP_HEINE_BOREL = prove
2687 (`!s. compact (s:real^N->bool)
2688 ==> !f. (!t. t IN f ==> open t) /\ s SUBSET (UNIONS f)
2689 ==> ?f'. f' SUBSET f /\ FINITE f' /\ s SUBSET (UNIONS f')`,
2690 REPEAT STRIP_TAC THEN
2691 FIRST_ASSUM(MP_TAC o SPEC `f:(real^N->bool)->bool` o
2692 MATCH_MP HEINE_BOREL_LEMMA) THEN ASM_REWRITE_TAC[] THEN
2693 DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
2694 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
2695 REWRITE_TAC[SKOLEM_THM; SUBSET; IN_BALL] THEN
2696 DISCH_THEN(X_CHOOSE_TAC `B:real^N->real^N->bool`) THEN
2697 FIRST_ASSUM(MP_TAC o SPEC `e:real` o
2698 MATCH_MP COMPACT_IMP_TOTALLY_BOUNDED) THEN
2699 ASM_REWRITE_TAC[UNIONS_IMAGE; SUBSET; IN_ELIM_THM] THEN
2700 REWRITE_TAC[IN_UNIONS; IN_BALL] THEN
2701 DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool` STRIP_ASSUME_TAC) THEN
2702 EXISTS_TAC `IMAGE (B:real^N->real^N->bool) k` THEN
2703 ASM_SIMP_TAC[FINITE_IMAGE; SUBSET; IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
2704 ASM_MESON_TAC[IN_BALL]);;
2706 (* ------------------------------------------------------------------------- *)
2707 (* Bolzano-Weierstrass property. *)
2708 (* ------------------------------------------------------------------------- *)
2710 let HEINE_BOREL_IMP_BOLZANO_WEIERSTRASS = prove
2712 (!f. (!t. t IN f ==> open t) /\ s SUBSET (UNIONS f)
2713 ==> ?f'. f' SUBSET f /\ FINITE f' /\ s SUBSET (UNIONS f'))
2714 ==> !t. INFINITE t /\ t SUBSET s ==> ?x. x IN s /\ x limit_point_of t`,
2715 REWRITE_TAC[RIGHT_IMP_FORALL_THM; limit_point_of] THEN REPEAT GEN_TAC THEN
2716 ONCE_REWRITE_TAC[TAUT `a ==> b /\ c ==> d <=> c ==> ~d ==> a ==> ~b`] THEN
2717 REWRITE_TAC[NOT_FORALL_THM; NOT_EXISTS_THM; RIGHT_AND_FORALL_THM] THEN
2718 DISCH_TAC THEN REWRITE_TAC[SKOLEM_THM] THEN
2719 DISCH_THEN(X_CHOOSE_TAC `f:real^N->real^N->bool`) THEN
2720 DISCH_THEN(MP_TAC o SPEC
2721 `{t:real^N->bool | ?x:real^N. x IN s /\ (t = f x)}`) THEN
2722 REWRITE_TAC[INFINITE; SUBSET; IN_ELIM_THM; IN_UNIONS; NOT_IMP] THEN
2723 ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
2724 DISCH_THEN(X_CHOOSE_THEN `g:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
2725 MATCH_MP_TAC FINITE_SUBSET THEN
2726 EXISTS_TAC `{x:real^N | x IN t /\ (f(x):real^N->bool) IN g}` THEN
2728 [MATCH_MP_TAC FINITE_IMAGE_INJ_GENERAL THEN ASM_MESON_TAC[SUBSET];
2729 SIMP_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `u:real^N` THEN
2730 DISCH_TAC THEN SUBGOAL_THEN `(u:real^N) IN s` ASSUME_TAC THEN
2731 ASM_MESON_TAC[SUBSET]]);;
2733 (* ------------------------------------------------------------------------- *)
2734 (* Complete the chain of compactness variants. *)
2735 (* ------------------------------------------------------------------------- *)
2737 let BOLZANO_WEIERSTRASS_IMP_BOUNDED = prove
2739 (!t. INFINITE t /\ t SUBSET s ==> ?x. x limit_point_of t)
2741 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
2742 SIMP_TAC[compact; bounded] THEN
2743 REWRITE_TAC[NOT_FORALL_THM; NOT_EXISTS_THM; SKOLEM_THM; NOT_IMP] THEN
2744 REWRITE_TAC[REAL_NOT_LE] THEN
2745 DISCH_THEN(X_CHOOSE_TAC `beyond:real->real^N`) THEN
2746 (MP_TAC o prove_recursive_functions_exist num_RECURSION)
2747 `(f(0) = beyond(&0)) /\
2748 (!n. f(SUC n) = beyond(norm(f n) + &1):real^N)` THEN
2749 DISCH_THEN(X_CHOOSE_THEN `x:num->real^N` STRIP_ASSUME_TAC) THEN
2750 EXISTS_TAC `IMAGE (x:num->real^N) UNIV` THEN
2752 `!m n. m < n ==> norm((x:num->real^N) m) + &1 < norm(x n)`
2754 [GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[LT] THEN
2755 ASM_MESON_TAC[REAL_LT_TRANS; REAL_ARITH `b < b + &1`];
2757 SUBGOAL_THEN `!m n. ~(m = n) ==> &1 < dist((x:num->real^N) m,x n)`
2759 [REPEAT GEN_TAC THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC
2760 (SPECL [`m:num`; `n:num`] LT_CASES) THEN
2761 ASM_MESON_TAC[dist; LT_CASES; NORM_TRIANGLE_SUB; NORM_SUB;
2762 REAL_ARITH `x + &1 < y /\ y <= x + d ==> &1 < d`];
2764 REPEAT CONJ_TAC THENL
2765 [ASM_MESON_TAC[INFINITE_IMAGE_INJ; num_INFINITE; DIST_REFL;
2766 REAL_ARITH `~(&1 < &0)`];
2767 REWRITE_TAC[SUBSET; IN_IMAGE; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN
2768 GEN_TAC THEN INDUCT_TAC THEN ASM_MESON_TAC[];
2770 X_GEN_TAC `l:real^N` THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN
2771 REWRITE_TAC[IN_IMAGE; IN_UNIV; LEFT_AND_EXISTS_THM] THEN
2772 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN
2773 STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `&1 / &2`) THEN
2774 CONV_TAC REAL_RAT_REDUCE_CONV THEN
2775 DISCH_THEN(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN
2776 FIRST_X_ASSUM(MP_TAC o SPEC `dist((x:num->real^N) k,l)`) THEN
2777 ASM_SIMP_TAC[DIST_POS_LT] THEN
2778 DISCH_THEN(X_CHOOSE_THEN `m:num` STRIP_ASSUME_TAC) THEN
2779 ASM_CASES_TAC `m:num = k` THEN
2780 ASM_MESON_TAC[DIST_TRIANGLE_HALF_L; REAL_LT_TRANS; REAL_LT_REFL]);;
2782 let SEQUENCE_INFINITE_LEMMA = prove
2783 (`!f l. (!n. ~(f(n) = l)) /\ (f --> l) sequentially
2784 ==> INFINITE {y:real^N | ?n. y = f n}`,
2785 REWRITE_TAC[INFINITE] THEN REPEAT STRIP_TAC THEN MP_TAC(ISPEC
2786 `IMAGE (\y:real^N. dist(y,l)) {y | ?n:num. y = f n}` INF_FINITE) THEN
2787 ASM_SIMP_TAC[GSYM MEMBER_NOT_EMPTY; IN_IMAGE; FINITE_IMAGE; IN_ELIM_THM] THEN
2788 ASM_MESON_TAC[LIM_SEQUENTIALLY; LE_REFL; REAL_NOT_LE; DIST_POS_LT]);;
2790 let LIMPT_OF_SEQUENCE_SUBSEQUENCE = prove
2792 l limit_point_of (IMAGE f (:num))
2793 ==> ?r. (!m n. m < n ==> r(m) < r(n)) /\ ((f o r) --> l) sequentially`,
2794 REPEAT STRIP_TAC THEN
2795 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIMPT_APPROACHABLE]) THEN
2796 DISCH_THEN(MP_TAC o GEN `n:num` o SPEC
2797 `inf((inv(&n + &1)) INSERT
2798 IMAGE (\k. dist((f:num->real^N) k,l))
2799 {k | k IN 0..n /\ ~(f k = l)})`) THEN
2800 SIMP_TAC[REAL_LT_INF_FINITE; FINITE_INSERT; NOT_INSERT_EMPTY;
2801 FINITE_RESTRICT; FINITE_NUMSEG; FINITE_IMAGE] THEN
2802 REWRITE_TAC[FORALL_IN_INSERT; EXISTS_IN_IMAGE; FORALL_IN_IMAGE; IN_UNIV] THEN
2803 REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
2804 SIMP_TAC[FORALL_AND_THM; FORALL_IN_GSPEC; GSYM DIST_NZ; SKOLEM_THM] THEN
2805 DISCH_THEN(X_CHOOSE_THEN `nn:num->num` STRIP_ASSUME_TAC) THEN
2806 (MP_TAC o prove_recursive_functions_exist num_RECURSION)
2807 `r 0 = nn 0 /\ (!n. r (SUC n) = nn(r n))` THEN
2808 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN
2810 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
2811 [MATCH_MP_TAC TRANSITIVE_STEPWISE_LT THEN REWRITE_TAC[LT_TRANS] THEN
2812 X_GEN_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN
2813 FIRST_X_ASSUM(MP_TAC o SPECL
2814 [`(r:num->num) n`; `(nn:num->num)(r(n:num))`]) THEN
2815 ASM_REWRITE_TAC[IN_NUMSEG; LE_0; REAL_LT_REFL] THEN ARITH_TAC;
2816 DISCH_THEN(ASSUME_TAC o MATCH_MP MONOTONE_BIGGER)] THEN
2817 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
2818 X_GEN_TAC `e:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN
2819 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN
2820 MATCH_MP_TAC num_INDUCTION THEN ASM_REWRITE_TAC[CONJUNCT1 LE] THEN
2821 X_GEN_TAC `n:num` THEN DISCH_THEN(K ALL_TAC) THEN DISCH_TAC THEN
2822 ASM_REWRITE_TAC[o_THM] THEN MATCH_MP_TAC REAL_LT_TRANS THEN
2823 EXISTS_TAC `inv(&((r:num->num) n) + &1)` THEN ASM_REWRITE_TAC[] THEN
2824 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN
2825 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
2826 ASM_SIMP_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT; LE_1; REAL_OF_NUM_ADD] THEN
2827 MATCH_MP_TAC(ARITH_RULE `N <= SUC n /\ n <= r n ==> N <= r n + 1`) THEN
2828 ASM_REWRITE_TAC[]);;
2830 let SEQUENCE_UNIQUE_LIMPT = prove
2832 (f --> l) sequentially /\ l' limit_point_of {y | ?n. y = f n}
2834 REWRITE_TAC[SET_RULE `{y | ?n. y = f n} = IMAGE f (:num)`] THEN
2835 REPEAT STRIP_TAC THEN
2836 FIRST_X_ASSUM(MP_TAC o MATCH_MP LIMPT_OF_SEQUENCE_SUBSEQUENCE) THEN
2837 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
2838 MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
2839 EXISTS_TAC `(f:num->real^N) o (r:num->num)` THEN
2840 ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIM_SUBSEQUENCE]);;
2842 let BOLZANO_WEIERSTRASS_IMP_CLOSED = prove
2844 (!t. INFINITE t /\ t SUBSET s ==> ?x. x IN s /\ x limit_point_of t)
2846 REPEAT STRIP_TAC THEN REWRITE_TAC[CLOSED_SEQUENTIAL_LIMITS] THEN
2847 MAP_EVERY X_GEN_TAC [`f:num->real^N`; `l:real^N`] THEN
2849 MAP_EVERY (MP_TAC o ISPECL [`f:num->real^N`; `l:real^N`])
2850 [SEQUENCE_UNIQUE_LIMPT; SEQUENCE_INFINITE_LEMMA] THEN
2852 `(~d ==> a /\ ~(b /\ c)) ==> (a ==> b) ==> c ==> d`) THEN
2853 DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[]; STRIP_TAC] THEN
2854 FIRST_X_ASSUM(MP_TAC o SPEC `{y:real^N | ?n:num. y = f n}`) THEN
2855 ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL
2856 [REWRITE_TAC[SUBSET; IN_ELIM_THM];
2857 ABBREV_TAC `t = {y:real^N | ?n:num. y = f n}`] THEN
2860 (* ------------------------------------------------------------------------- *)
2861 (* Hence express everything as an equivalence. *)
2862 (* ------------------------------------------------------------------------- *)
2864 let COMPACT_EQ_HEINE_BOREL = prove
2867 !f. (!t. t IN f ==> open t) /\ s SUBSET (UNIONS f)
2868 ==> ?f'. f' SUBSET f /\ FINITE f' /\ s SUBSET (UNIONS f')`,
2869 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[COMPACT_IMP_HEINE_BOREL] THEN
2870 DISCH_THEN(MP_TAC o MATCH_MP HEINE_BOREL_IMP_BOLZANO_WEIERSTRASS) THEN
2871 DISCH_TAC THEN MATCH_MP_TAC BOUNDED_CLOSED_IMP_COMPACT THEN
2872 ASM_MESON_TAC[BOLZANO_WEIERSTRASS_IMP_BOUNDED;
2873 BOLZANO_WEIERSTRASS_IMP_CLOSED]);;
2875 let COMPACT_EQ_BOLZANO_WEIERSTRASS = prove
2878 !t. INFINITE t /\ t SUBSET s ==> ?x. x IN s /\ x limit_point_of t`,
2879 GEN_TAC THEN EQ_TAC THENL
2880 [SIMP_TAC[COMPACT_EQ_HEINE_BOREL; HEINE_BOREL_IMP_BOLZANO_WEIERSTRASS];
2881 MESON_TAC[BOLZANO_WEIERSTRASS_IMP_BOUNDED; BOLZANO_WEIERSTRASS_IMP_CLOSED;
2882 BOUNDED_CLOSED_IMP_COMPACT]]);;
2884 let COMPACT_EQ_BOUNDED_CLOSED = prove
2885 (`!s:real^N->bool. compact s <=> bounded s /\ closed s`,
2886 GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[BOUNDED_CLOSED_IMP_COMPACT] THEN
2887 MESON_TAC[COMPACT_EQ_BOLZANO_WEIERSTRASS; BOLZANO_WEIERSTRASS_IMP_BOUNDED;
2888 BOLZANO_WEIERSTRASS_IMP_CLOSED]);;
2890 let COMPACT_IMP_BOUNDED = prove
2891 (`!s. compact s ==> bounded s`,
2892 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED]);;
2894 let COMPACT_IMP_CLOSED = prove
2895 (`!s. compact s ==> closed s`,
2896 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED]);;
2898 let COMPACT_SEQUENCE_WITH_LIMIT = prove
2900 (f --> l) sequentially ==> compact (l INSERT IMAGE f (:num))`,
2901 REPEAT STRIP_TAC THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN
2902 REWRITE_TAC[BOUNDED_INSERT] THEN CONJ_TAC THENL
2903 [ASM_MESON_TAC[CONVERGENT_IMP_BOUNDED];
2904 SIMP_TAC[CLOSED_LIMPT; LIMPT_INSERT; IN_INSERT] THEN
2905 REWRITE_TAC[IMAGE; IN_UNIV] THEN REPEAT STRIP_TAC THEN DISJ1_TAC THEN
2906 MATCH_MP_TAC SEQUENCE_UNIQUE_LIMPT THEN ASM_MESON_TAC[]]);;
2908 let CLOSED_IN_COMPACT = prove
2909 (`!s t:real^N->bool.
2910 compact s /\ closed_in (subtopology euclidean s) t
2912 SIMP_TAC[IMP_CONJ; COMPACT_EQ_BOUNDED_CLOSED; CLOSED_IN_CLOSED_EQ] THEN
2913 MESON_TAC[BOUNDED_SUBSET]);;
2915 let CLOSED_IN_COMPACT_EQ = prove
2917 ==> (closed_in (subtopology euclidean s) t <=>
2918 compact t /\ t SUBSET s)`,
2919 MESON_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET]);;
2921 (* ------------------------------------------------------------------------- *)
2922 (* A version of Heine-Borel for subtopology. *)
2923 (* ------------------------------------------------------------------------- *)
2925 let COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY = prove
2928 (!f. (!t. t IN f ==> open_in(subtopology euclidean s) t) /\
2930 ==> ?f'. f' SUBSET f /\ FINITE f' /\ s SUBSET UNIONS f')`,
2931 GEN_TAC THEN REWRITE_TAC[COMPACT_EQ_HEINE_BOREL] THEN EQ_TAC THEN
2932 DISCH_TAC THEN X_GEN_TAC `f:(real^N->bool)->bool` THENL
2933 [REWRITE_TAC[OPEN_IN_OPEN] THEN
2934 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
2935 REWRITE_TAC[SKOLEM_THM] THEN
2936 DISCH_THEN(CONJUNCTS_THEN2
2937 (X_CHOOSE_TAC `m:(real^N->bool)->(real^N->bool)`) ASSUME_TAC) THEN
2938 FIRST_X_ASSUM(MP_TAC o SPEC
2939 `IMAGE (m:(real^N->bool)->(real^N->bool)) f`) THEN
2940 ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN
2941 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2942 DISCH_THEN(X_CHOOSE_THEN `f':(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
2943 EXISTS_TAC `IMAGE (\t:real^N->bool. s INTER t) f'` THEN
2944 ASM_SIMP_TAC[FINITE_IMAGE; UNIONS_IMAGE; SUBSET; FORALL_IN_IMAGE] THEN
2945 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
2946 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET_IMAGE]) THEN
2947 STRIP_TAC THEN ASM_REWRITE_TAC[FORALL_IN_IMAGE] THEN ASM_MESON_TAC[SUBSET];
2949 FIRST_X_ASSUM(MP_TAC o SPEC `{s INTER t:real^N->bool | t IN f}`) THEN
2950 REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; OPEN_IN_OPEN; UNIONS_IMAGE] THEN
2951 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
2952 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN
2953 REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE; UNIONS_IMAGE] THEN
2954 MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]]);;
2956 (* ------------------------------------------------------------------------- *)
2957 (* More easy lemmas. *)
2958 (* ------------------------------------------------------------------------- *)
2960 let COMPACT_CLOSURE = prove
2961 (`!s. compact(closure s) <=> bounded s`,
2962 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_CLOSURE; BOUNDED_CLOSURE_EQ]);;
2964 let BOLZANO_WEIERSTRASS_CONTRAPOS = prove
2965 (`!s t:real^N->bool.
2966 compact s /\ t SUBSET s /\
2967 (!x. x IN s ==> ~(x limit_point_of t))
2969 REWRITE_TAC[COMPACT_EQ_BOLZANO_WEIERSTRASS; INFINITE] THEN MESON_TAC[]);;
2971 let DISCRETE_BOUNDED_IMP_FINITE = prove
2972 (`!s:real^N->bool e.
2974 (!x y. x IN s /\ y IN s /\ norm(y - x) < e ==> y = x) /\
2977 REPEAT STRIP_TAC THEN
2978 SUBGOAL_THEN `compact(s:real^N->bool)` MP_TAC THENL
2979 [ASM_REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN
2980 ASM_MESON_TAC[DISCRETE_IMP_CLOSED];
2981 DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_HEINE_BOREL)] THEN
2982 DISCH_THEN(MP_TAC o SPEC `IMAGE (\x:real^N. ball(x,e)) s`) THEN
2983 REWRITE_TAC[FORALL_IN_IMAGE; OPEN_BALL; UNIONS_IMAGE; IN_ELIM_THM] THEN
2985 [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN ASM_MESON_TAC[CENTRE_IN_BALL];
2986 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`]] THEN
2987 REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN
2988 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
2989 SUBGOAL_THEN `s:real^N->bool = t` (fun th -> ASM_REWRITE_TAC[th]) THEN
2990 MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[] THEN
2991 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
2992 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [UNIONS_IMAGE]) THEN
2993 DISCH_THEN(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [SUBSET]) THEN
2994 ASM_REWRITE_TAC[IN_ELIM_THM; IN_BALL; dist] THEN ASM_MESON_TAC[SUBSET]);;
2996 let BOLZANO_WEIERSTRASS = prove
2997 (`!s:real^N->bool. bounded s /\ INFINITE s ==> ?x. x limit_point_of s`,
2998 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
2999 FIRST_ASSUM(ASSUME_TAC o MATCH_MP NO_LIMIT_POINT_IMP_CLOSED) THEN
3001 MP_TAC(ISPEC `s:real^N->bool` COMPACT_EQ_BOLZANO_WEIERSTRASS) THEN
3002 ASM_REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN
3003 DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN
3004 ASM_REWRITE_TAC[SUBSET_REFL] THEN ASM_MESON_TAC[]);;
3006 let BOUNDED_EQ_BOLZANO_WEIERSTRASS = prove
3008 bounded s <=> !t. t SUBSET s /\ INFINITE t ==> ?x. x limit_point_of t`,
3009 MESON_TAC[BOLZANO_WEIERSTRASS_IMP_BOUNDED; BOLZANO_WEIERSTRASS;
3012 (* ------------------------------------------------------------------------- *)
3013 (* In particular, some common special cases. *)
3014 (* ------------------------------------------------------------------------- *)
3016 let COMPACT_EMPTY = prove
3018 REWRITE_TAC[compact; NOT_IN_EMPTY]);;
3020 let COMPACT_UNION = prove
3021 (`!s t. compact s /\ compact t ==> compact (s UNION t)`,
3022 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_UNION; CLOSED_UNION]);;
3024 let COMPACT_INTER = prove
3025 (`!s t. compact s /\ compact t ==> compact (s INTER t)`,
3026 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_INTER; CLOSED_INTER]);;
3028 let COMPACT_INTER_CLOSED = prove
3029 (`!s t. compact s /\ closed t ==> compact (s INTER t)`,
3030 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_INTER] THEN
3031 MESON_TAC[BOUNDED_SUBSET; INTER_SUBSET]);;
3033 let CLOSED_INTER_COMPACT = prove
3034 (`!s t. closed s /\ compact t ==> compact (s INTER t)`,
3035 MESON_TAC[COMPACT_INTER_CLOSED; INTER_COMM]);;
3037 let COMPACT_INTERS = prove
3038 (`!f:(real^N->bool)->bool.
3039 (!s. s IN f ==> compact s) /\ ~(f = {})
3040 ==> compact(INTERS f)`,
3041 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_INTERS] THEN
3042 REPEAT STRIP_TAC THEN MATCH_MP_TAC BOUNDED_INTERS THEN ASM SET_TAC[]);;
3044 let FINITE_IMP_CLOSED = prove
3045 (`!s. FINITE s ==> closed s`,
3046 MESON_TAC[BOLZANO_WEIERSTRASS_IMP_CLOSED; INFINITE; FINITE_SUBSET]);;
3048 let FINITE_IMP_CLOSED_IN = prove
3049 (`!s t. FINITE s /\ s SUBSET t ==> closed_in (subtopology euclidean t) s`,
3050 SIMP_TAC[CLOSED_SUBSET_EQ; FINITE_IMP_CLOSED]);;
3052 let FINITE_IMP_COMPACT = prove
3053 (`!s. FINITE s ==> compact s`,
3054 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; FINITE_IMP_CLOSED; FINITE_IMP_BOUNDED]);;
3056 let COMPACT_SING = prove
3058 SIMP_TAC[FINITE_IMP_COMPACT; FINITE_RULES]);;
3060 let COMPACT_INSERT = prove
3061 (`!a s. compact s ==> compact(a INSERT s)`,
3062 ONCE_REWRITE_TAC[SET_RULE `a INSERT s = {a} UNION s`] THEN
3063 SIMP_TAC[COMPACT_UNION; COMPACT_SING]);;
3065 let CLOSED_SING = prove
3067 MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; COMPACT_SING]);;
3069 let CLOSED_IN_SING = prove
3070 (`!u x:real^N. closed_in (subtopology euclidean u) {x} <=> x IN u`,
3071 SIMP_TAC[CLOSED_SUBSET_EQ; CLOSED_SING] THEN SET_TAC[]);;
3073 let CLOSURE_SING = prove
3074 (`!x:real^N. closure {x} = {x}`,
3075 SIMP_TAC[CLOSURE_CLOSED; CLOSED_SING]);;
3077 let CLOSED_INSERT = prove
3078 (`!a s. closed s ==> closed(a INSERT s)`,
3079 ONCE_REWRITE_TAC[SET_RULE `a INSERT s = {a} UNION s`] THEN
3080 SIMP_TAC[CLOSED_UNION; CLOSED_SING]);;
3082 let COMPACT_CBALL = prove
3083 (`!x e. compact(cball(x,e))`,
3084 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_CBALL; CLOSED_CBALL]);;
3086 let COMPACT_FRONTIER_BOUNDED = prove
3087 (`!s. bounded s ==> compact(frontier s)`,
3088 SIMP_TAC[frontier; COMPACT_EQ_BOUNDED_CLOSED;
3089 CLOSED_DIFF; OPEN_INTERIOR; CLOSED_CLOSURE] THEN
3090 MESON_TAC[SUBSET_DIFF; BOUNDED_SUBSET; BOUNDED_CLOSURE]);;
3092 let COMPACT_FRONTIER = prove
3093 (`!s. compact s ==> compact (frontier s)`,
3094 MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; COMPACT_FRONTIER_BOUNDED]);;
3096 let BOUNDED_FRONTIER = prove
3097 (`!s:real^N->bool. bounded s ==> bounded(frontier s)`,
3098 MESON_TAC[COMPACT_FRONTIER_BOUNDED; COMPACT_IMP_BOUNDED]);;
3100 let FRONTIER_SUBSET_COMPACT = prove
3101 (`!s. compact s ==> frontier s SUBSET s`,
3102 MESON_TAC[FRONTIER_SUBSET_CLOSED; COMPACT_EQ_BOUNDED_CLOSED]);;
3104 let OPEN_DELETE = prove
3105 (`!s x. open s ==> open(s DELETE x)`,
3106 let lemma = prove(`s DELETE x = s DIFF {x}`,SET_TAC[]) in
3107 SIMP_TAC[lemma; OPEN_DIFF; CLOSED_SING]);;
3109 let OPEN_IN_DELETE = prove
3111 open_in (subtopology euclidean u) s
3112 ==> open_in (subtopology euclidean u) (s DELETE a)`,
3113 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THENL
3114 [ONCE_REWRITE_TAC[SET_RULE `s DELETE a = s DIFF {a}`] THEN
3115 MATCH_MP_TAC OPEN_IN_DIFF THEN ASM_REWRITE_TAC[CLOSED_IN_SING] THEN
3116 FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[];
3117 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> s DELETE a = s`]]);;
3119 let CLOSED_INTERS_COMPACT = prove
3121 closed s <=> !e. compact(cball(vec 0,e) INTER s)`,
3122 GEN_TAC THEN EQ_TAC THENL
3123 [SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_INTER; CLOSED_CBALL;
3124 BOUNDED_INTER; BOUNDED_CBALL];
3126 STRIP_TAC THEN REWRITE_TAC[CLOSED_LIMPT] THEN
3127 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
3128 FIRST_X_ASSUM(MP_TAC o SPEC `norm(x:real^N) + &1`) THEN
3129 DISCH_THEN(MP_TAC o MATCH_MP COMPACT_IMP_CLOSED) THEN
3130 REWRITE_TAC[CLOSED_LIMPT] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
3131 REWRITE_TAC[IN_INTER] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
3132 POP_ASSUM MP_TAC THEN REWRITE_TAC[LIMPT_APPROACHABLE] THEN
3133 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
3134 FIRST_X_ASSUM(MP_TAC o SPEC `min e (&1 / &2)`) THEN
3135 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN
3136 X_GEN_TAC `y:real^N` THEN SIMP_TAC[IN_INTER; IN_CBALL] THEN NORM_ARITH_TAC);;
3138 let COMPACT_UNIONS = prove
3139 (`!s. FINITE s /\ (!t. t IN s ==> compact t) ==> compact(UNIONS s)`,
3140 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_UNIONS; BOUNDED_UNIONS]);;
3142 let COMPACT_DIFF = prove
3143 (`!s t. compact s /\ open t ==> compact(s DIFF t)`,
3144 ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN
3145 SIMP_TAC[COMPACT_INTER_CLOSED; GSYM OPEN_CLOSED]);;
3147 let COMPACT_SPHERE = prove
3148 (`!a:real^N r. compact(sphere(a,r))`,
3150 REWRITE_TAC[GSYM FRONTIER_CBALL] THEN MATCH_MP_TAC COMPACT_FRONTIER THEN
3151 REWRITE_TAC[COMPACT_CBALL]);;
3153 let BOUNDED_SPHERE = prove
3154 (`!a:real^N r. bounded(sphere(a,r))`,
3155 SIMP_TAC[COMPACT_SPHERE; COMPACT_IMP_BOUNDED]);;
3157 let CLOSED_SPHERE = prove
3158 (`!a r. closed(sphere(a,r))`,
3159 SIMP_TAC[COMPACT_SPHERE; COMPACT_IMP_CLOSED]);;
3161 let FRONTIER_SING = prove
3162 (`!a:real^N. frontier {a} = {a}`,
3163 REWRITE_TAC[frontier; CLOSURE_SING; INTERIOR_SING; DIFF_EMPTY]);;
3165 (* ------------------------------------------------------------------------- *)
3166 (* Finite intersection property. I could make it an equivalence in fact. *)
3167 (* ------------------------------------------------------------------------- *)
3169 let COMPACT_IMP_FIP = prove
3170 (`!s:real^N->bool f.
3172 (!t. t IN f ==> closed t) /\
3173 (!f'. FINITE f' /\ f' SUBSET f ==> ~(s INTER (INTERS f') = {}))
3174 ==> ~(s INTER (INTERS f) = {})`,
3175 let lemma = prove(`(s = UNIV DIFF t) <=> (UNIV DIFF s = t)`,SET_TAC[]) in
3176 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
3177 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL]) THEN
3178 DISCH_THEN(MP_TAC o SPEC `IMAGE (\t:real^N->bool. UNIV DIFF t) f`) THEN
3179 ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN
3180 DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN
3181 ASM_SIMP_TAC[OPEN_DIFF; CLOSED_DIFF; OPEN_UNIV; CLOSED_UNIV; NOT_IMP] THEN
3183 [UNDISCH_TAC `(s:real^N->bool) INTER INTERS f = {}` THEN
3184 ONCE_REWRITE_TAC[SUBSET; EXTENSION] THEN
3185 REWRITE_TAC[IN_UNIONS; EXISTS_IN_IMAGE] THEN SET_TAC[];
3186 DISCH_THEN(X_CHOOSE_THEN `g:(real^N->bool)->bool` MP_TAC) THEN
3187 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (\t:real^N->bool. UNIV DIFF t) g`) THEN
3188 ASM_CASES_TAC `FINITE(g:(real^N->bool)->bool)` THEN
3189 ASM_SIMP_TAC[FINITE_IMAGE] THEN ONCE_REWRITE_TAC[SUBSET; EXTENSION] THEN
3190 REWRITE_TAC[FORALL_IN_IMAGE; IN_INTER; IN_INTERS; IN_IMAGE; IN_DIFF;
3191 IN_UNIV; NOT_IN_EMPTY; lemma; UNWIND_THM1; IN_UNIONS] THEN
3194 let CLOSED_IMP_FIP = prove
3195 (`!s:real^N->bool f.
3197 (!t. t IN f ==> closed t) /\ (?t. t IN f /\ bounded t) /\
3198 (!f'. FINITE f' /\ f' SUBSET f ==> ~(s INTER (INTERS f') = {}))
3199 ==> ~(s INTER (INTERS f) = {})`,
3200 REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(SET_RULE
3201 `~((s INTER t) INTER u = {}) ==> ~(s INTER u = {})`) THEN
3202 MATCH_MP_TAC COMPACT_IMP_FIP THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
3203 [ASM_MESON_TAC[CLOSED_INTER_COMPACT; COMPACT_EQ_BOUNDED_CLOSED];
3204 REWRITE_TAC[INTER_ASSOC] THEN ONCE_REWRITE_TAC[GSYM INTERS_INSERT]] THEN
3205 GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
3206 ASM_SIMP_TAC[FINITE_INSERT; INSERT_SUBSET]);;
3208 let CLOSED_IMP_FIP_COMPACT = prove
3209 (`!s:real^N->bool f.
3210 closed s /\ (!t. t IN f ==> compact t) /\
3211 (!f'. FINITE f' /\ f' SUBSET f ==> ~(s INTER (INTERS f') = {}))
3212 ==> ~(s INTER (INTERS f) = {})`,
3214 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THEN
3215 ASM_SIMP_TAC[SUBSET_EMPTY; INTERS_0; INTER_UNIV] THENL
3216 [MESON_TAC[FINITE_EMPTY]; ALL_TAC] THEN
3217 STRIP_TAC THEN MATCH_MP_TAC CLOSED_IMP_FIP THEN
3218 ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; MEMBER_NOT_EMPTY]);;
3220 let CLOSED_FIP = prove
3221 (`!f. (!t:real^N->bool. t IN f ==> closed t) /\ (?t. t IN f /\ bounded t) /\
3222 (!f'. FINITE f' /\ f' SUBSET f ==> ~(INTERS f' = {}))
3223 ==> ~(INTERS f = {})`,
3224 GEN_TAC THEN DISCH_TAC THEN
3225 ONCE_REWRITE_TAC[SET_RULE `s = {} <=> UNIV INTER s = {}`] THEN
3226 MATCH_MP_TAC CLOSED_IMP_FIP THEN ASM_REWRITE_TAC[CLOSED_UNIV; INTER_UNIV]);;
3228 let COMPACT_FIP = prove
3229 (`!f. (!t:real^N->bool. t IN f ==> compact t) /\
3230 (!f'. FINITE f' /\ f' SUBSET f ==> ~(INTERS f' = {}))
3231 ==> ~(INTERS f = {})`,
3232 GEN_TAC THEN DISCH_TAC THEN
3233 ONCE_REWRITE_TAC[SET_RULE `s = {} <=> UNIV INTER s = {}`] THEN
3234 MATCH_MP_TAC CLOSED_IMP_FIP_COMPACT THEN
3235 ASM_REWRITE_TAC[CLOSED_UNIV; INTER_UNIV]);;
3237 (* ------------------------------------------------------------------------- *)
3238 (* Bounded closed nest property (proof does not use Heine-Borel). *)
3239 (* ------------------------------------------------------------------------- *)
3241 let BOUNDED_CLOSED_NEST = prove
3242 (`!s. (!n. closed(s n)) /\ (!n. ~(s n = {})) /\
3243 (!m n. m <= n ==> s(n) SUBSET s(m)) /\
3245 ==> ?a:real^N. !n:num. a IN s(n)`,
3246 GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; SKOLEM_THM] THEN
3247 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
3248 DISCH_THEN(CONJUNCTS_THEN2
3249 (X_CHOOSE_TAC `a:num->real^N`) STRIP_ASSUME_TAC) THEN
3250 SUBGOAL_THEN `compact(s 0:real^N->bool)` MP_TAC THENL
3251 [ASM_MESON_TAC[BOUNDED_CLOSED_IMP_COMPACT]; ALL_TAC] THEN
3252 REWRITE_TAC[compact] THEN
3253 DISCH_THEN(MP_TAC o SPEC `a:num->real^N`) THEN
3254 ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; LE_0]; ALL_TAC] THEN
3255 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN
3256 REWRITE_TAC[LIM_SEQUENTIALLY; o_THM] THEN
3257 DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
3258 GEN_REWRITE_TAC I [TAUT `p <=> ~(~p)`] THEN
3259 GEN_REWRITE_TAC RAND_CONV [NOT_FORALL_THM] THEN
3260 DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN
3261 MP_TAC(ISPECL [`l:real^N`; `(s:num->real^N->bool) N`]
3262 CLOSED_APPROACHABLE) THEN
3263 ASM_MESON_TAC[SUBSET; LE_REFL; LE_TRANS; LE_CASES; MONOTONE_BIGGER]);;
3265 (* ------------------------------------------------------------------------- *)
3266 (* Decreasing case does not even need compactness, just completeness. *)
3267 (* ------------------------------------------------------------------------- *)
3269 let DECREASING_CLOSED_NEST = prove
3270 (`!s. (!n. closed(s n)) /\ (!n. ~(s n = {})) /\
3271 (!m n. m <= n ==> s(n) SUBSET s(m)) /\
3272 (!e. &0 < e ==> ?n. !x y. x IN s(n) /\ y IN s(n) ==> dist(x,y) < e)
3273 ==> ?a:real^N. !n:num. a IN s(n)`,
3274 GEN_TAC THEN REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; SKOLEM_THM] THEN
3275 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
3276 DISCH_THEN(CONJUNCTS_THEN2
3277 (X_CHOOSE_TAC `a:num->real^N`) STRIP_ASSUME_TAC) THEN
3278 SUBGOAL_THEN `?l:real^N. (a --> l) sequentially` MP_TAC THENL
3279 [ASM_MESON_TAC[cauchy; GE; SUBSET; LE_TRANS; LE_REFL;
3280 complete; COMPLETE_UNIV; IN_UNIV];
3281 ASM_MESON_TAC[LIM_SEQUENTIALLY; CLOSED_APPROACHABLE;
3282 SUBSET; LE_REFL; LE_TRANS; LE_CASES]]);;
3284 (* ------------------------------------------------------------------------- *)
3285 (* Strengthen it to the intersection actually being a singleton. *)
3286 (* ------------------------------------------------------------------------- *)
3288 let DECREASING_CLOSED_NEST_SING = prove
3289 (`!s. (!n. closed(s n)) /\ (!n. ~(s n = {})) /\
3290 (!m n. m <= n ==> s(n) SUBSET s(m)) /\
3291 (!e. &0 < e ==> ?n. !x y. x IN s(n) /\ y IN s(n) ==> dist(x,y) < e)
3292 ==> ?a:real^N. INTERS {t | ?n:num. t = s n} = {a}`,
3293 GEN_TAC THEN DISCH_TAC THEN
3294 FIRST_ASSUM(MP_TAC o MATCH_MP DECREASING_CLOSED_NEST) THEN
3295 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
3296 DISCH_TAC THEN REWRITE_TAC[EXTENSION; IN_INTERS; IN_SING; IN_ELIM_THM] THEN
3297 ASM_MESON_TAC[DIST_POS_LT; REAL_LT_REFL; SUBSET; LE_CASES]);;
3299 (* ------------------------------------------------------------------------- *)
3300 (* A version for a more general chain, not indexed by N. *)
3301 (* ------------------------------------------------------------------------- *)
3303 let BOUNDED_CLOSED_CHAIN = prove
3304 (`!f b:real^N->bool.
3305 (!s. s IN f ==> closed s /\ ~(s = {})) /\
3306 (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s) /\
3308 ==> ~(INTERS f = {})`,
3309 REPEAT GEN_TAC THEN STRIP_TAC THEN
3310 SUBGOAL_THEN `~(b INTER (INTERS f):real^N->bool = {})` MP_TAC THENL
3311 [ALL_TAC; SET_TAC[]] THEN
3312 MATCH_MP_TAC COMPACT_IMP_FIP THEN
3313 ASM_SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN
3314 X_GEN_TAC `u:(real^N->bool)->bool` THEN STRIP_TAC THEN
3315 SUBGOAL_THEN `?s:real^N->bool. s IN f /\ !t. t IN u ==> s SUBSET t`
3316 MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
3317 UNDISCH_TAC `(u:(real^N->bool)->bool) SUBSET f` THEN
3318 UNDISCH_TAC `FINITE(u:(real^N->bool)->bool)` THEN
3319 SPEC_TAC(`u:(real^N->bool)->bool`,`u:(real^N->bool)->bool`) THEN
3320 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3321 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3322 MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:(real^N->bool)->bool`] THEN
3323 REWRITE_TAC[INSERT_SUBSET] THEN
3324 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
3325 ASM_REWRITE_TAC[] THEN
3326 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
3327 DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
3328 FIRST_X_ASSUM(MP_TAC o SPECL [`s:real^N->bool`; `t:real^N->bool`]) THEN
3331 (* ------------------------------------------------------------------------- *)
3332 (* Analogous things directly for compactness. *)
3333 (* ------------------------------------------------------------------------- *)
3335 let COMPACT_CHAIN = prove
3336 (`!f:(real^N->bool)->bool.
3337 (!s. s IN f ==> compact s /\ ~(s = {})) /\
3338 (!s t. s IN f /\ t IN f ==> s SUBSET t \/ t SUBSET s)
3339 ==> ~(INTERS f = {})`,
3340 GEN_TAC THEN REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN STRIP_TAC THEN
3341 ASM_CASES_TAC `f:(real^N->bool)->bool = {}` THENL
3342 [ASM_REWRITE_TAC[INTERS_0] THEN SET_TAC[];
3343 MATCH_MP_TAC BOUNDED_CLOSED_CHAIN THEN ASM SET_TAC[]]);;
3345 let COMPACT_NEST = prove
3346 (`!s. (!n. compact(s n) /\ ~(s n = {})) /\
3347 (!m n. m <= n ==> s n SUBSET s m)
3348 ==> ~(INTERS {s n | n IN (:num)} = {})`,
3349 GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC COMPACT_CHAIN THEN
3350 ASM_SIMP_TAC[FORALL_IN_GSPEC; IN_UNIV; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3351 MATCH_MP_TAC WLOG_LE THEN ASM_MESON_TAC[]);;
3353 (* ------------------------------------------------------------------------- *)
3354 (* Cauchy-type criteria for *uniform* convergence. *)
3355 (* ------------------------------------------------------------------------- *)
3357 let UNIFORMLY_CONVERGENT_EQ_CAUCHY = prove
3358 (`!P s:num->A->real^N.
3360 ==> ?N. !n x. N <= n /\ P x ==> dist(s n x,l x) < e) <=>
3362 ==> ?N. !m n x. N <= m /\ N <= n /\ P x
3363 ==> dist(s m x,s n x) < e)`,
3364 REPEAT GEN_TAC THEN EQ_TAC THENL
3365 [DISCH_THEN(X_CHOOSE_TAC `l:A->real^N`) THEN X_GEN_TAC `e:real` THEN
3366 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
3367 ASM_REWRITE_TAC[REAL_HALF] THEN MESON_TAC[DIST_TRIANGLE_HALF_L];
3370 SUBGOAL_THEN `!x:A. P x ==> cauchy (\n. s n x :real^N)` MP_TAC THENL
3371 [REWRITE_TAC[cauchy; GE] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
3372 REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY; LIM_SEQUENTIALLY] THEN
3373 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
3374 REWRITE_TAC[SKOLEM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
3375 X_GEN_TAC `l:A->real^N` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
3376 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
3377 ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN
3378 X_GEN_TAC `N:num` THEN STRIP_TAC THEN
3379 MAP_EVERY X_GEN_TAC [`n:num`; `x:A`] THEN STRIP_TAC THEN
3380 FIRST_X_ASSUM(MP_TAC o SPEC `x:A`) THEN ASM_REWRITE_TAC[] THEN
3381 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
3382 DISCH_THEN(X_CHOOSE_TAC `M:num`) THEN
3383 FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `N + M:num`; `x:A`]) THEN
3384 ASM_REWRITE_TAC[LE_ADD] THEN ONCE_REWRITE_TAC[ADD_SYM] THEN
3385 FIRST_X_ASSUM(MP_TAC o SPEC `M + N:num`) THEN REWRITE_TAC[LE_ADD] THEN
3386 ASM_MESON_TAC[DIST_TRIANGLE_HALF_L; DIST_SYM]);;
3388 let UNIFORMLY_CAUCHY_IMP_UNIFORMLY_CONVERGENT = prove
3389 (`!P (s:num->A->real^N) l.
3391 ==> ?N. !m n x. N <= m /\ N <= n /\ P x ==> dist(s m x,s n x) < e) /\
3392 (!x. P x ==> !e. &0 < e ==> ?N. !n. N <= n ==> dist(s n x,l x) < e)
3393 ==> (!e. &0 < e ==> ?N. !n x. N <= n /\ P x ==> dist(s n x,l x) < e)`,
3394 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM UNIFORMLY_CONVERGENT_EQ_CAUCHY] THEN
3395 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `l':A->real^N`) ASSUME_TAC) THEN
3396 SUBGOAL_THEN `!x. P x ==> (l:A->real^N) x = l' x` MP_TAC THENL
3397 [ALL_TAC; ASM_MESON_TAC[]] THEN
3398 REPEAT STRIP_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
3399 EXISTS_TAC `\n. (s:num->A->real^N) n x` THEN
3400 REWRITE_TAC[LIM_SEQUENTIALLY; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
3403 (* ------------------------------------------------------------------------- *)
3404 (* Define continuity over a net to take in restrictions of the set. *)
3405 (* ------------------------------------------------------------------------- *)
3407 parse_as_infix ("continuous",(12,"right"));;
3409 let continuous = new_definition
3410 `f continuous net <=> (f --> f(netlimit net)) net`;;
3412 let CONTINUOUS_TRIVIAL_LIMIT = prove
3413 (`!f net. trivial_limit net ==> f continuous net`,
3414 SIMP_TAC[continuous; LIM]);;
3416 let CONTINUOUS_WITHIN = prove
3417 (`!f x:real^M. f continuous (at x within s) <=> (f --> f(x)) (at x within s)`,
3418 REPEAT GEN_TAC THEN REWRITE_TAC[continuous] THEN
3419 ASM_CASES_TAC `trivial_limit(at (x:real^M) within s)` THENL
3420 [ASM_REWRITE_TAC[LIM]; ASM_SIMP_TAC[NETLIMIT_WITHIN]]);;
3422 let CONTINUOUS_AT = prove
3423 (`!f (x:real^N). f continuous (at x) <=> (f --> f(x)) (at x)`,
3424 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
3425 REWRITE_TAC[CONTINUOUS_WITHIN; IN_UNIV]);;
3427 let CONTINUOUS_AT_WITHIN = prove
3428 (`!f:real^M->real^N x s.
3429 f continuous (at x) ==> f continuous (at x within s)`,
3430 SIMP_TAC[LIM_AT_WITHIN; CONTINUOUS_AT; CONTINUOUS_WITHIN]);;
3432 let CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL = prove
3433 (`!a s. closed s /\ ~(a IN s) ==> f continuous (at a within s)`,
3434 ASM_SIMP_TAC[continuous; LIM; LIM_WITHIN_CLOSED_TRIVIAL]);;
3436 let CONTINUOUS_TRANSFORM_WITHIN = prove
3437 (`!f g:real^M->real^N s x d.
3439 (!x'. x' IN s /\ dist(x',x) < d ==> f(x') = g(x')) /\
3440 f continuous (at x within s)
3441 ==> g continuous (at x within s)`,
3442 REWRITE_TAC[CONTINUOUS_WITHIN] THEN
3443 MESON_TAC[LIM_TRANSFORM_WITHIN; DIST_REFL]);;
3445 let CONTINUOUS_TRANSFORM_AT = prove
3446 (`!f g:real^M->real^N x d.
3447 &0 < d /\ (!x'. dist(x',x) < d ==> f(x') = g(x')) /\
3449 ==> g continuous (at x)`,
3450 REWRITE_TAC[CONTINUOUS_AT] THEN
3451 MESON_TAC[LIM_TRANSFORM_AT; DIST_REFL]);;
3453 let CONTINUOUS_TRANSFORM_WITHIN_OPEN = prove
3454 (`!f g:real^M->real^N s a.
3456 (!x. x IN s ==> f x = g x) /\
3458 ==> g continuous at a`,
3459 MESON_TAC[CONTINUOUS_AT; LIM_TRANSFORM_WITHIN_OPEN]);;
3461 let CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN = prove
3462 (`!f g:real^M->real^N s t a.
3463 open_in (subtopology euclidean t) s /\ a IN s /\
3464 (!x. x IN s ==> f x = g x) /\
3465 f continuous (at a within t)
3466 ==> g continuous (at a within t)`,
3467 MESON_TAC[CONTINUOUS_WITHIN; LIM_TRANSFORM_WITHIN_OPEN_IN]);;
3469 (* ------------------------------------------------------------------------- *)
3470 (* Derive the epsilon-delta forms, which we often use as "definitions" *)
3471 (* ------------------------------------------------------------------------- *)
3473 let continuous_within = prove
3474 (`f continuous (at x within s) <=>
3477 !x'. x' IN s /\ dist(x',x) < d ==> dist(f(x'),f(x)) < e`,
3478 REWRITE_TAC[CONTINUOUS_WITHIN; LIM_WITHIN] THEN
3479 REWRITE_TAC[GSYM DIST_NZ] THEN MESON_TAC[DIST_REFL]);;
3481 let continuous_at = prove
3482 (`f continuous (at x) <=>
3483 !e. &0 < e ==> ?d. &0 < d /\
3484 !x'. dist(x',x) < d ==> dist(f(x'),f(x)) < e`,
3485 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
3486 REWRITE_TAC[continuous_within; IN_UNIV]);;
3488 (* ------------------------------------------------------------------------- *)
3489 (* Versions in terms of open balls. *)
3490 (* ------------------------------------------------------------------------- *)
3492 let CONTINUOUS_WITHIN_BALL = prove
3493 (`!f s x. f continuous (at x within s) <=>
3496 IMAGE f (ball(x,d) INTER s) SUBSET ball(f x,e)`,
3497 SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_BALL; continuous_within; IN_INTER] THEN
3498 MESON_TAC[DIST_SYM]);;
3500 let CONTINUOUS_AT_BALL = prove
3501 (`!f x. f continuous (at x) <=>
3504 IMAGE f (ball(x,d)) SUBSET ball(f x,e)`,
3505 SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_BALL; continuous_at] THEN
3506 MESON_TAC[DIST_SYM]);;
3508 (* ------------------------------------------------------------------------- *)
3509 (* For setwise continuity, just start from the epsilon-delta definitions. *)
3510 (* ------------------------------------------------------------------------- *)
3512 parse_as_infix ("continuous_on",(12,"right"));;
3513 parse_as_infix ("uniformly_continuous_on",(12,"right"));;
3515 let continuous_on = new_definition
3516 `f continuous_on s <=>
3517 !x. x IN s ==> !e. &0 < e
3519 !x'. x' IN s /\ dist(x',x) < d
3520 ==> dist(f(x'),f(x)) < e`;;
3522 let uniformly_continuous_on = new_definition
3523 `f uniformly_continuous_on s <=>
3526 !x x'. x IN s /\ x' IN s /\ dist(x',x) < d
3527 ==> dist(f(x'),f(x)) < e`;;
3529 (* ------------------------------------------------------------------------- *)
3530 (* Some simple consequential lemmas. *)
3531 (* ------------------------------------------------------------------------- *)
3533 let UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS = prove
3534 (`!f s. f uniformly_continuous_on s ==> f continuous_on s`,
3535 REWRITE_TAC[uniformly_continuous_on; continuous_on] THEN MESON_TAC[]);;
3537 let CONTINUOUS_AT_IMP_CONTINUOUS_ON = prove
3538 (`!f s. (!x. x IN s ==> f continuous (at x)) ==> f continuous_on s`,
3539 REWRITE_TAC[continuous_at; continuous_on] THEN MESON_TAC[]);;
3541 let CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN = prove
3542 (`!f s. f continuous_on s <=> !x. x IN s ==> f continuous (at x within s)`,
3543 REWRITE_TAC[continuous_on; continuous_within]);;
3545 let CONTINUOUS_ON = prove
3546 (`!f (s:real^N->bool).
3547 f continuous_on s <=> !x. x IN s ==> (f --> f(x)) (at x within s)`,
3548 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_WITHIN]);;
3550 let CONTINUOUS_ON_EQ_CONTINUOUS_AT = prove
3551 (`!f:real^M->real^N s.
3552 open s ==> (f continuous_on s <=> (!x. x IN s ==> f continuous (at x)))`,
3553 SIMP_TAC[CONTINUOUS_ON; CONTINUOUS_AT; LIM_WITHIN_OPEN]);;
3555 let CONTINUOUS_WITHIN_SUBSET = prove
3556 (`!f s t x. f continuous (at x within s) /\ t SUBSET s
3557 ==> f continuous (at x within t)`,
3558 REWRITE_TAC[CONTINUOUS_WITHIN] THEN MESON_TAC[LIM_WITHIN_SUBSET]);;
3560 let CONTINUOUS_ON_SUBSET = prove
3561 (`!f s t. f continuous_on s /\ t SUBSET s ==> f continuous_on t`,
3562 REWRITE_TAC[CONTINUOUS_ON] THEN MESON_TAC[SUBSET; LIM_WITHIN_SUBSET]);;
3564 let UNIFORMLY_CONTINUOUS_ON_SUBSET = prove
3565 (`!f s t. f uniformly_continuous_on s /\ t SUBSET s
3566 ==> f uniformly_continuous_on t`,
3567 REWRITE_TAC[uniformly_continuous_on] THEN
3568 MESON_TAC[SUBSET; LIM_WITHIN_SUBSET]);;
3570 let CONTINUOUS_ON_INTERIOR = prove
3571 (`!f:real^M->real^N s x.
3572 f continuous_on s /\ x IN interior(s) ==> f continuous at x`,
3573 REWRITE_TAC[interior; IN_ELIM_THM] THEN
3574 MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; CONTINUOUS_ON_SUBSET]);;
3576 let CONTINUOUS_ON_EQ = prove
3577 (`!f g s. (!x. x IN s ==> f(x) = g(x)) /\ f continuous_on s
3578 ==> g continuous_on s`,
3579 SIMP_TAC[continuous_on; IMP_CONJ]);;
3581 let UNIFORMLY_CONTINUOUS_ON_EQ = prove
3583 (!x. x IN s ==> f x = g x) /\ f uniformly_continuous_on s
3584 ==> g uniformly_continuous_on s`,
3585 SIMP_TAC[uniformly_continuous_on; IMP_CONJ]);;
3587 let CONTINUOUS_ON_SING = prove
3588 (`!f:real^M->real^N a. f continuous_on {a}`,
3589 SIMP_TAC[continuous_on; IN_SING; FORALL_UNWIND_THM2; DIST_REFL] THEN
3592 let CONTINUOUS_ON_EMPTY = prove
3593 (`!f:real^M->real^N. f continuous_on {}`,
3594 MESON_TAC[CONTINUOUS_ON_SING; EMPTY_SUBSET; CONTINUOUS_ON_SUBSET]);;
3596 let CONTINUOUS_ON_NO_LIMPT = prove
3597 (`!f:real^M->real^N s.
3598 ~(?x. x limit_point_of s) ==> f continuous_on s`,
3599 REWRITE_TAC[continuous_on; LIMPT_APPROACHABLE] THEN MESON_TAC[DIST_REFL]);;
3601 let CONTINUOUS_ON_FINITE = prove
3602 (`!f:real^M->real^N s. FINITE s ==> f continuous_on s`,
3603 MESON_TAC[CONTINUOUS_ON_NO_LIMPT; LIMIT_POINT_FINITE]);;
3605 let CONTRACTION_IMP_CONTINUOUS_ON = prove
3606 (`!f:real^M->real^N.
3607 (!x y. x IN s /\ y IN s ==> dist(f x,f y) <= dist(x,y))
3608 ==> f continuous_on s`,
3609 SIMP_TAC[continuous_on] THEN MESON_TAC[REAL_LET_TRANS]);;
3611 let ISOMETRY_ON_IMP_CONTINUOUS_ON = prove
3612 (`!f:real^M->real^N.
3613 (!x y. x IN s /\ y IN s ==> dist(f x,f y) = dist(x,y))
3614 ==> f continuous_on s`,
3615 SIMP_TAC[CONTRACTION_IMP_CONTINUOUS_ON; REAL_LE_REFL]);;
3617 (* ------------------------------------------------------------------------- *)
3618 (* Characterization of various kinds of continuity in terms of sequences. *)
3619 (* ------------------------------------------------------------------------- *)
3621 let CONTINUOUS_WITHIN_SEQUENTIALLY = prove
3623 f continuous (at a within s) <=>
3624 !x. (!n. x(n) IN s) /\ (x --> a) sequentially
3625 ==> ((f o x) --> f(a)) sequentially`,
3626 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_within] THEN EQ_TAC THENL
3627 [REWRITE_TAC[LIM_SEQUENTIALLY; o_THM] THEN MESON_TAC[]; ALL_TAC] THEN
3628 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
3629 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN
3630 DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
3631 DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&1 / (&n + &1)`) THEN
3632 SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; REAL_OF_NUM_LE; REAL_POS; ARITH;
3633 REAL_ARITH `&0 <= n ==> &0 < n + &1`; NOT_FORALL_THM; SKOLEM_THM] THEN
3634 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[NOT_IMP; FORALL_AND_THM] THEN
3635 X_GEN_TAC `y:num->real^N` THEN REWRITE_TAC[LIM_SEQUENTIALLY; o_THM] THEN
3636 STRIP_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LE_REFL]] THEN
3637 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FORALL_POS_MONO_1 THEN
3638 CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN
3639 X_GEN_TAC `n:num` THEN EXISTS_TAC `n:num` THEN X_GEN_TAC `m:num` THEN
3640 DISCH_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
3641 EXISTS_TAC `&1 / (&m + &1)` THEN ASM_REWRITE_TAC[] THEN
3642 ASM_SIMP_TAC[REAL_LE_INV2; real_div; REAL_ARITH `&0 <= x ==> &0 < x + &1`;
3643 REAL_POS; REAL_MUL_LID; REAL_LE_RADD; REAL_OF_NUM_LE]);;
3645 let CONTINUOUS_AT_SEQUENTIALLY = prove
3647 f continuous (at a) <=>
3648 !x. (x --> a) sequentially
3649 ==> ((f o x) --> f(a)) sequentially`,
3650 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
3651 REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY; IN_UNIV]);;
3653 let CONTINUOUS_ON_SEQUENTIALLY = prove
3654 (`!f s:real^N->bool.
3655 f continuous_on s <=>
3656 !x a. a IN s /\ (!n. x(n) IN s) /\ (x --> a) sequentially
3657 ==> ((f o x) --> f(a)) sequentially`,
3658 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN;
3659 CONTINUOUS_WITHIN_SEQUENTIALLY] THEN MESON_TAC[]);;
3661 let UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY = prove
3662 (`!f s:real^N->bool.
3663 f uniformly_continuous_on s <=>
3664 !x y. (!n. x(n) IN s) /\ (!n. y(n) IN s) /\
3665 ((\n. x(n) - y(n)) --> vec 0) sequentially
3666 ==> ((\n. f(x(n)) - f(y(n))) --> vec 0) sequentially`,
3667 REPEAT GEN_TAC THEN REWRITE_TAC[uniformly_continuous_on] THEN
3668 REWRITE_TAC[LIM_SEQUENTIALLY; dist; VECTOR_SUB_RZERO] THEN
3669 EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
3670 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
3671 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_EXISTS_THM] THEN
3672 DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
3673 DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&1 / (&n + &1)`) THEN
3674 SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; REAL_OF_NUM_LE; REAL_POS; ARITH;
3675 REAL_ARITH `&0 <= n ==> &0 < n + &1`; NOT_FORALL_THM; SKOLEM_THM] THEN
3676 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:num->real^N` THEN
3677 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:num->real^N` THEN
3678 REWRITE_TAC[NOT_IMP; FORALL_AND_THM] THEN STRIP_TAC THEN
3679 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN CONJ_TAC THENL
3680 [MATCH_MP_TAC FORALL_POS_MONO_1 THEN
3681 CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN
3682 X_GEN_TAC `n:num` THEN EXISTS_TAC `n:num` THEN X_GEN_TAC `m:num` THEN
3683 DISCH_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
3684 EXISTS_TAC `&1 / (&m + &1)` THEN ASM_REWRITE_TAC[] THEN
3685 ASM_SIMP_TAC[REAL_LE_INV2; real_div; REAL_ARITH `&0 <= x ==> &0 < x + &1`;
3686 REAL_POS; REAL_MUL_LID; REAL_LE_RADD; REAL_OF_NUM_LE];
3687 EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN
3688 EXISTS_TAC `\x:num. x` THEN ASM_REWRITE_TAC[LE_REFL]]);;
3690 let LIM_CONTINUOUS_FUNCTION = prove
3692 f continuous (at l) /\ (g --> l) net ==> ((\x. f(g x)) --> f l) net`,
3693 REWRITE_TAC[tendsto; continuous_at; eventually] THEN MESON_TAC[]);;
3695 (* ------------------------------------------------------------------------- *)
3696 (* Combination results for pointwise continuity. *)
3697 (* ------------------------------------------------------------------------- *)
3699 let CONTINUOUS_CONST = prove
3700 (`!net c. (\x. c) continuous net`,
3701 REWRITE_TAC[continuous; LIM_CONST]);;
3703 let CONTINUOUS_CMUL = prove
3704 (`!f c net. f continuous net ==> (\x. c % f(x)) continuous net`,
3705 REWRITE_TAC[continuous; LIM_CMUL]);;
3707 let CONTINUOUS_NEG = prove
3708 (`!f net. f continuous net ==> (\x. --(f x)) continuous net`,
3709 REWRITE_TAC[continuous; LIM_NEG]);;
3711 let CONTINUOUS_ADD = prove
3712 (`!f g net. f continuous net /\ g continuous net
3713 ==> (\x. f(x) + g(x)) continuous net`,
3714 REWRITE_TAC[continuous; LIM_ADD]);;
3716 let CONTINUOUS_SUB = prove
3717 (`!f g net. f continuous net /\ g continuous net
3718 ==> (\x. f(x) - g(x)) continuous net`,
3719 REWRITE_TAC[continuous; LIM_SUB]);;
3721 let CONTINUOUS_ABS = prove
3722 (`!(f:A->real^N) net.
3724 ==> (\x. (lambda i. abs(f(x)$i)):real^N) continuous net`,
3725 REWRITE_TAC[continuous; LIM_ABS]);;
3727 let CONTINUOUS_MAX = prove
3728 (`!(f:A->real^N) (g:A->real^N) net.
3729 f continuous net /\ g continuous net
3730 ==> (\x. (lambda i. max (f(x)$i) (g(x)$i)):real^N) continuous net`,
3731 REWRITE_TAC[continuous; LIM_MAX]);;
3733 let CONTINUOUS_MIN = prove
3734 (`!(f:A->real^N) (g:A->real^N) net.
3735 f continuous net /\ g continuous net
3736 ==> (\x. (lambda i. min (f(x)$i) (g(x)$i)):real^N) continuous net`,
3737 REWRITE_TAC[continuous; LIM_MIN]);;
3739 let CONTINUOUS_VSUM = prove
3740 (`!net f s. FINITE s /\ (!a. a IN s ==> (f a) continuous net)
3741 ==> (\x. vsum s (\a. f a x)) continuous net`,
3742 GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
3743 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3744 SIMP_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; VSUM_CLAUSES;
3745 CONTINUOUS_CONST; CONTINUOUS_ADD; ETA_AX]);;
3747 (* ------------------------------------------------------------------------- *)
3748 (* Same thing for setwise continuity. *)
3749 (* ------------------------------------------------------------------------- *)
3751 let CONTINUOUS_ON_CONST = prove
3752 (`!s c. (\x. c) continuous_on s`,
3753 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_CONST]);;
3755 let CONTINUOUS_ON_CMUL = prove
3756 (`!f c s. f continuous_on s ==> (\x. c % f(x)) continuous_on s`,
3757 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_CMUL]);;
3759 let CONTINUOUS_ON_NEG = prove
3760 (`!f s. f continuous_on s
3761 ==> (\x. --(f x)) continuous_on s`,
3762 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_NEG]);;
3764 let CONTINUOUS_ON_ADD = prove
3765 (`!f g s. f continuous_on s /\ g continuous_on s
3766 ==> (\x. f(x) + g(x)) continuous_on s`,
3767 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_ADD]);;
3769 let CONTINUOUS_ON_SUB = prove
3770 (`!f g s. f continuous_on s /\ g continuous_on s
3771 ==> (\x. f(x) - g(x)) continuous_on s`,
3772 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_SUB]);;
3774 let CONTINUOUS_ON_ABS = prove
3775 (`!f:real^M->real^N s.
3777 ==> (\x. (lambda i. abs(f(x)$i)):real^N) continuous_on s`,
3778 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_ABS]);;
3780 let CONTINUOUS_ON_MAX = prove
3781 (`!f:real^M->real^N g:real^M->real^N s.
3782 f continuous_on s /\ g continuous_on s
3783 ==> (\x. (lambda i. max (f(x)$i) (g(x)$i)):real^N)
3785 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_MAX]);;
3787 let CONTINUOUS_ON_MIN = prove
3788 (`!f:real^M->real^N g:real^M->real^N s.
3789 f continuous_on s /\ g continuous_on s
3790 ==> (\x. (lambda i. min (f(x)$i) (g(x)$i)):real^N)
3792 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_MIN]);;
3794 let CONTINUOUS_ON_VSUM = prove
3795 (`!t f s. FINITE s /\ (!a. a IN s ==> (f a) continuous_on t)
3796 ==> (\x. vsum s (\a. f a x)) continuous_on t`,
3797 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_VSUM]);;
3799 (* ------------------------------------------------------------------------- *)
3800 (* Same thing for uniform continuity, using sequential formulations. *)
3801 (* ------------------------------------------------------------------------- *)
3803 let UNIFORMLY_CONTINUOUS_ON_CONST = prove
3804 (`!s c. (\x. c) uniformly_continuous_on s`,
3805 REWRITE_TAC[UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY; o_DEF;
3806 VECTOR_SUB_REFL; LIM_CONST]);;
3808 let LINEAR_UNIFORMLY_CONTINUOUS_ON = prove
3809 (`!f:real^M->real^N s. linear f ==> f uniformly_continuous_on s`,
3810 REPEAT STRIP_TAC THEN
3811 ASM_SIMP_TAC[uniformly_continuous_on; dist; GSYM LINEAR_SUB] THEN
3812 FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o
3813 MATCH_MP LINEAR_BOUNDED_POS) THEN
3814 X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `e / B:real` THEN
3815 ASM_SIMP_TAC[REAL_LT_DIV] THEN
3816 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN
3817 MATCH_MP_TAC REAL_LET_TRANS THEN
3818 EXISTS_TAC `B * norm(y - x:real^M)` THEN ASM_REWRITE_TAC[] THEN
3819 ASM_MESON_TAC[REAL_LT_RDIV_EQ; REAL_MUL_SYM]);;
3821 let UNIFORMLY_CONTINUOUS_ON_COMPOSE = prove
3822 (`!f g s. f uniformly_continuous_on s /\
3823 g uniformly_continuous_on (IMAGE f s)
3824 ==> (g o f) uniformly_continuous_on s`,
3826 (`(!y. ((?x. (y = f x) /\ P x) /\ Q y ==> R y)) <=>
3827 (!x. P x /\ Q (f x) ==> R (f x))`,
3830 REWRITE_TAC[uniformly_continuous_on; o_THM; IN_IMAGE] THEN
3831 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[lemma] THEN
3832 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN
3833 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[lemma] THEN
3834 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
3835 MATCH_MP_TAC MONO_FORALL THEN
3836 X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
3839 let BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE = prove
3840 (`!f:real^M->real^N g (h:real^N->real^P->real^Q) s.
3841 f uniformly_continuous_on s /\ g uniformly_continuous_on s /\
3842 bilinear h /\ bounded(IMAGE f s) /\ bounded(IMAGE g s)
3843 ==> (\x. h (f x) (g x)) uniformly_continuous_on s`,
3844 REPEAT STRIP_TAC THEN REWRITE_TAC[uniformly_continuous_on; dist] THEN
3845 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
3847 `!a b c d. (h:real^N->real^P->real^Q) a b - h c d =
3848 h (a - c) b + h c (b - d)`
3849 (fun th -> ONCE_REWRITE_TAC[th])
3851 [FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP BILINEAR_LSUB th]) THEN
3852 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP BILINEAR_RSUB th]) THEN
3855 FIRST_X_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o
3856 MATCH_MP BILINEAR_BOUNDED_POS) THEN
3857 UNDISCH_TAC `bounded(IMAGE (g:real^M->real^P) s)` THEN
3858 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
3859 REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE] THEN
3860 DISCH_THEN(X_CHOOSE_THEN `B1:real` STRIP_ASSUME_TAC) THEN
3861 DISCH_THEN(X_CHOOSE_THEN `B2:real` STRIP_ASSUME_TAC) THEN
3862 UNDISCH_TAC `(g:real^M->real^P) uniformly_continuous_on s` THEN
3863 UNDISCH_TAC `(f:real^M->real^N) uniformly_continuous_on s` THEN
3864 REWRITE_TAC[uniformly_continuous_on] THEN
3865 DISCH_THEN(MP_TAC o SPEC `e / &2 / &2 / B / B2`) THEN
3866 ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; dist] THEN
3867 DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
3868 DISCH_THEN(MP_TAC o SPEC `e / &2 / &2 / B / B1`) THEN
3869 ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; dist] THEN
3870 DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
3871 EXISTS_TAC `min d1 d2` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
3872 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN
3873 REPEAT(FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `y:real^M`])) THEN
3874 ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
3875 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC
3876 `B * e / &2 / &2 / B / B2 * B2 + B * B1 * e / &2 / &2 / B / B1` THEN
3878 [MATCH_MP_TAC(NORM_ARITH
3879 `norm(x) <= a /\ norm(y) <= b ==> norm(x + y:real^N) <= a + b`) THEN
3881 FIRST_X_ASSUM(fun th -> W(MP_TAC o PART_MATCH lhand th o lhand o snd)) THEN
3882 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
3883 MATCH_MP_TAC REAL_LE_LMUL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
3884 MATCH_MP_TAC REAL_LE_MUL2 THEN
3885 ASM_SIMP_TAC[REAL_LT_IMP_LE; NORM_POS_LE];
3886 ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_DIV_LMUL; REAL_LT_IMP_NZ] THEN
3887 ASM_REAL_ARITH_TAC]);;
3889 let UNIFORMLY_CONTINUOUS_ON_MUL = prove
3890 (`!f g:real^M->real^N s.
3891 (lift o f) uniformly_continuous_on s /\ g uniformly_continuous_on s /\
3892 bounded(IMAGE (lift o f) s) /\ bounded(IMAGE g s)
3893 ==> (\x. f x % g x) uniformly_continuous_on s`,
3894 REPEAT STRIP_TAC THEN
3896 [`lift o (f:real^M->real)`; `g:real^M->real^N`;
3897 `\c (v:real^N). drop c % v`; `s:real^M->bool`]
3898 BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE) THEN
3899 ASM_REWRITE_TAC[o_THM; LIFT_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN
3900 REWRITE_TAC[bilinear; linear; DROP_ADD; DROP_CMUL] THEN VECTOR_ARITH_TAC);;
3902 let UNIFORMLY_CONTINUOUS_ON_CMUL = prove
3903 (`!f c s. f uniformly_continuous_on s
3904 ==> (\x. c % f(x)) uniformly_continuous_on s`,
3905 REPEAT GEN_TAC THEN REWRITE_TAC[UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY] THEN
3906 REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
3907 DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
3908 ASM_REWRITE_TAC[] THEN
3909 DISCH_THEN(MP_TAC o MATCH_MP LIM_CMUL) THEN
3910 ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_RZERO]);;
3912 let UNIFORMLY_CONTINUOUS_ON_VMUL = prove
3913 (`!s:real^M->bool c v:real^N.
3914 (lift o c) uniformly_continuous_on s
3915 ==> (\x. c x % v) uniformly_continuous_on s`,
3917 DISCH_THEN(MP_TAC o ISPEC `\x. (drop x % v:real^N)` o MATCH_MP
3918 (REWRITE_RULE[IMP_CONJ] UNIFORMLY_CONTINUOUS_ON_COMPOSE)) THEN
3919 REWRITE_TAC[o_DEF; LIFT_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN
3920 MATCH_MP_TAC LINEAR_UNIFORMLY_CONTINUOUS_ON THEN
3921 MATCH_MP_TAC LINEAR_VMUL_DROP THEN REWRITE_TAC[LINEAR_ID]);;
3923 let UNIFORMLY_CONTINUOUS_ON_NEG = prove
3924 (`!f s. f uniformly_continuous_on s
3925 ==> (\x. --(f x)) uniformly_continuous_on s`,
3926 ONCE_REWRITE_TAC[VECTOR_NEG_MINUS1] THEN
3927 REWRITE_TAC[UNIFORMLY_CONTINUOUS_ON_CMUL]);;
3929 let UNIFORMLY_CONTINUOUS_ON_ADD = prove
3930 (`!f g s. f uniformly_continuous_on s /\ g uniformly_continuous_on s
3931 ==> (\x. f(x) + g(x)) uniformly_continuous_on s`,
3932 REPEAT GEN_TAC THEN REWRITE_TAC[UNIFORMLY_CONTINUOUS_ON_SEQUENTIALLY] THEN
3933 REWRITE_TAC[AND_FORALL_THM] THEN
3934 REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
3935 DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
3936 ASM_REWRITE_TAC[o_DEF] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD) THEN
3937 MATCH_MP_TAC EQ_IMP THEN
3938 REWRITE_TAC[VECTOR_ADD_LID] THEN AP_THM_TAC THEN BINOP_TAC THEN
3939 REWRITE_TAC[FUN_EQ_THM] THEN VECTOR_ARITH_TAC);;
3941 let UNIFORMLY_CONTINUOUS_ON_SUB = prove
3942 (`!f g s. f uniformly_continuous_on s /\ g uniformly_continuous_on s
3943 ==> (\x. f(x) - g(x)) uniformly_continuous_on s`,
3944 REWRITE_TAC[VECTOR_SUB] THEN
3945 SIMP_TAC[UNIFORMLY_CONTINUOUS_ON_NEG; UNIFORMLY_CONTINUOUS_ON_ADD]);;
3947 let UNIFORMLY_CONTINUOUS_ON_VSUM = prove
3948 (`!t f s. FINITE s /\ (!a. a IN s ==> (f a) uniformly_continuous_on t)
3949 ==> (\x. vsum s (\a. f a x)) uniformly_continuous_on t`,
3950 GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
3951 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3952 SIMP_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY; VSUM_CLAUSES;
3953 UNIFORMLY_CONTINUOUS_ON_CONST; UNIFORMLY_CONTINUOUS_ON_ADD; ETA_AX]);;
3955 (* ------------------------------------------------------------------------- *)
3956 (* Identity function is continuous in every sense. *)
3957 (* ------------------------------------------------------------------------- *)
3959 let CONTINUOUS_WITHIN_ID = prove
3960 (`!a s. (\x. x) continuous (at a within s)`,
3961 REWRITE_TAC[continuous_within] THEN MESON_TAC[]);;
3963 let CONTINUOUS_AT_ID = prove
3964 (`!a. (\x. x) continuous (at a)`,
3965 REWRITE_TAC[continuous_at] THEN MESON_TAC[]);;
3967 let CONTINUOUS_ON_ID = prove
3968 (`!s. (\x. x) continuous_on s`,
3969 REWRITE_TAC[continuous_on] THEN MESON_TAC[]);;
3971 let UNIFORMLY_CONTINUOUS_ON_ID = prove
3972 (`!s. (\x. x) uniformly_continuous_on s`,
3973 REWRITE_TAC[uniformly_continuous_on] THEN MESON_TAC[]);;
3975 (* ------------------------------------------------------------------------- *)
3976 (* Continuity of all kinds is preserved under composition. *)
3977 (* ------------------------------------------------------------------------- *)
3979 let CONTINUOUS_WITHIN_COMPOSE = prove
3980 (`!f g x s. f continuous (at x within s) /\
3981 g continuous (at (f x) within IMAGE f s)
3982 ==> (g o f) continuous (at x within s)`,
3983 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_within; o_THM; IN_IMAGE] THEN
3984 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
3985 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
3988 let CONTINUOUS_AT_COMPOSE = prove
3989 (`!f g x. f continuous (at x) /\ g continuous (at (f x))
3990 ==> (g o f) continuous (at x)`,
3991 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
3992 MESON_TAC[CONTINUOUS_WITHIN_COMPOSE; IN_IMAGE; CONTINUOUS_WITHIN_SUBSET;
3993 SUBSET_UNIV; IN_UNIV]);;
3995 let CONTINUOUS_ON_COMPOSE = prove
3996 (`!f g s. f continuous_on s /\ g continuous_on (IMAGE f s)
3997 ==> (g o f) continuous_on s`,
3998 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
3999 MESON_TAC[IN_IMAGE; CONTINUOUS_WITHIN_COMPOSE]);;
4001 (* ------------------------------------------------------------------------- *)
4002 (* Continuity in terms of open preimages. *)
4003 (* ------------------------------------------------------------------------- *)
4005 let CONTINUOUS_WITHIN_OPEN = prove
4006 (`!f:real^M->real^N x u.
4007 f continuous (at x within u) <=>
4008 !t. open t /\ f(x) IN t
4009 ==> ?s. open s /\ x IN s /\
4010 !x'. x' IN s /\ x' IN u ==> f(x') IN t`,
4011 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_within] THEN EQ_TAC THENL
4012 [DISCH_TAC THEN X_GEN_TAC `t:real^N->bool` THEN
4013 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4014 GEN_REWRITE_TAC LAND_CONV [open_def] THEN
4015 DISCH_THEN(MP_TAC o SPEC `(f:real^M->real^N) x`) THEN
4016 ASM_MESON_TAC[IN_BALL; DIST_SYM; OPEN_BALL; CENTRE_IN_BALL; DIST_SYM];
4017 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
4018 FIRST_X_ASSUM(MP_TAC o SPEC `ball((f:real^M->real^N) x,e)`) THEN
4019 ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN
4020 MESON_TAC[open_def; IN_BALL; REAL_LT_TRANS; DIST_SYM]]);;
4022 let CONTINUOUS_AT_OPEN = prove
4023 (`!f:real^M->real^N x.
4024 f continuous (at x) <=>
4025 !t. open t /\ f(x) IN t
4026 ==> ?s. open s /\ x IN s /\
4027 !x'. x' IN s ==> f(x') IN t`,
4028 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_at] THEN EQ_TAC THENL
4029 [DISCH_TAC THEN X_GEN_TAC `t:real^N->bool` THEN
4030 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4031 GEN_REWRITE_TAC LAND_CONV [open_def] THEN
4032 DISCH_THEN(MP_TAC o SPEC `(f:real^M->real^N) x`) THEN
4033 ASM_MESON_TAC[IN_BALL; DIST_SYM; OPEN_BALL; CENTRE_IN_BALL];
4034 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
4035 FIRST_X_ASSUM(MP_TAC o SPEC `ball((f:real^M->real^N) x,e)`) THEN
4036 ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN
4037 MESON_TAC[open_def; IN_BALL; REAL_LT_TRANS; DIST_SYM]]);;
4039 let CONTINUOUS_ON_OPEN_GEN = prove
4040 (`!f:real^M->real^N s t.
4042 ==> (f continuous_on s <=>
4043 !u. open_in (subtopology euclidean t) u
4044 ==> open_in (subtopology euclidean s) {x | x IN s /\ f x IN u})`,
4045 REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_on] THEN EQ_TAC THENL
4046 [REWRITE_TAC[open_in; SUBSET; IN_ELIM_THM] THEN
4047 DISCH_TAC THEN X_GEN_TAC `u:real^N->bool` THEN STRIP_TAC THEN
4048 CONJ_TAC THENL [ASM_MESON_TAC[DIST_REFL]; ALL_TAC] THEN
4049 X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
4050 FIRST_X_ASSUM(MP_TAC o SPEC `(f:real^M->real^N) x`) THEN ASM SET_TAC[];
4051 DISCH_TAC THEN X_GEN_TAC `x:real^M` THEN
4052 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
4053 FIRST_X_ASSUM(MP_TAC o
4054 SPEC `ball((f:real^M->real^N) x,e) INTER t`) THEN
4056 [ASM_MESON_TAC[OPEN_IN_OPEN; INTER_COMM; OPEN_BALL]; ALL_TAC] THEN
4057 REWRITE_TAC[open_in; SUBSET; IN_INTER; IN_ELIM_THM; IN_BALL; IN_IMAGE] THEN
4058 REWRITE_TAC[AND_FORALL_THM] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
4059 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN
4060 ASM_MESON_TAC[DIST_REFL; DIST_SYM]]);;
4062 let CONTINUOUS_ON_OPEN = prove
4063 (`!f:real^M->real^N s.
4064 f continuous_on s <=>
4065 !t. open_in (subtopology euclidean (IMAGE f s)) t
4066 ==> open_in (subtopology euclidean s) {x | x IN s /\ f(x) IN t}`,
4067 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_OPEN_GEN THEN
4068 REWRITE_TAC[SUBSET_REFL]);;
4070 let CONTINUOUS_OPEN_IN_PREIMAGE_GEN = prove
4071 (`!f:real^M->real^N s t u.
4072 f continuous_on s /\ IMAGE f s SUBSET t /\
4073 open_in (subtopology euclidean t) u
4074 ==> open_in (subtopology euclidean s) {x | x IN s /\ f x IN u}`,
4075 MESON_TAC[CONTINUOUS_ON_OPEN_GEN]);;
4077 let CONTINUOUS_ON_IMP_OPEN_IN = prove
4078 (`!f:real^M->real^N s t.
4079 f continuous_on s /\
4080 open_in (subtopology euclidean (IMAGE f s)) t
4081 ==> open_in (subtopology euclidean s) {x | x IN s /\ f x IN t}`,
4082 MESON_TAC[CONTINUOUS_ON_OPEN]);;
4084 (* ------------------------------------------------------------------------- *)
4085 (* Similarly in terms of closed sets. *)
4086 (* ------------------------------------------------------------------------- *)
4088 let CONTINUOUS_ON_CLOSED_GEN = prove
4089 (`!f:real^M->real^N s t.
4091 ==> (f continuous_on s <=>
4092 !u. closed_in (subtopology euclidean t) u
4093 ==> closed_in (subtopology euclidean s)
4094 {x | x IN s /\ f x IN u})`,
4095 REPEAT STRIP_TAC THEN FIRST_ASSUM(fun th ->
4096 ONCE_REWRITE_TAC[MATCH_MP CONTINUOUS_ON_OPEN_GEN th]) THEN
4097 EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `u:real^N->bool` THEN
4098 FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF u:real^N->bool`) THENL
4099 [REWRITE_TAC[closed_in]; REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ]] THEN
4100 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
4101 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
4102 ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN
4103 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
4105 let CONTINUOUS_ON_CLOSED = prove
4106 (`!f:real^M->real^N s.
4107 f continuous_on s <=>
4108 !t. closed_in (subtopology euclidean (IMAGE f s)) t
4109 ==> closed_in (subtopology euclidean s) {x | x IN s /\ f(x) IN t}`,
4110 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_CLOSED_GEN THEN
4111 REWRITE_TAC[SUBSET_REFL]);;
4113 let CONTINUOUS_CLOSED_IN_PREIMAGE_GEN = prove
4114 (`!f:real^M->real^N s t u.
4115 f continuous_on s /\ IMAGE f s SUBSET t /\
4116 closed_in (subtopology euclidean t) u
4117 ==> closed_in (subtopology euclidean s) {x | x IN s /\ f x IN u}`,
4118 MESON_TAC[CONTINUOUS_ON_CLOSED_GEN]);;
4120 let CONTINUOUS_ON_IMP_CLOSED_IN = prove
4121 (`!f:real^M->real^N s t.
4122 f continuous_on s /\
4123 closed_in (subtopology euclidean (IMAGE f s)) t
4124 ==> closed_in (subtopology euclidean s) {x | x IN s /\ f x IN t}`,
4125 MESON_TAC[CONTINUOUS_ON_CLOSED]);;
4127 (* ------------------------------------------------------------------------- *)
4128 (* Half-global and completely global cases. *)
4129 (* ------------------------------------------------------------------------- *)
4131 let CONTINUOUS_OPEN_IN_PREIMAGE = prove
4133 f continuous_on s /\ open t
4134 ==> open_in (subtopology euclidean s) {x | x IN s /\ f x IN t}`,
4135 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE
4136 `x IN s /\ f x IN t <=> x IN s /\ f x IN (t INTER IMAGE f s)`] THEN
4137 FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[CONTINUOUS_ON_OPEN]) THEN
4138 ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN
4139 ASM_REWRITE_TAC[]);;
4141 let CONTINUOUS_CLOSED_IN_PREIMAGE = prove
4143 f continuous_on s /\ closed t
4144 ==> closed_in (subtopology euclidean s) {x | x IN s /\ f x IN t}`,
4145 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[SET_RULE
4146 `x IN s /\ f x IN t <=> x IN s /\ f x IN (t INTER IMAGE f s)`] THEN
4147 FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[CONTINUOUS_ON_CLOSED]) THEN
4148 ONCE_REWRITE_TAC[INTER_COMM] THEN MATCH_MP_TAC CLOSED_IN_CLOSED_INTER THEN
4149 ASM_REWRITE_TAC[]);;
4151 let CONTINUOUS_OPEN_PREIMAGE = prove
4152 (`!f:real^M->real^N s t.
4153 f continuous_on s /\ open s /\ open t
4154 ==> open {x | x IN s /\ f(x) IN t}`,
4155 REPEAT STRIP_TAC THEN
4156 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_OPEN]) THEN
4157 REWRITE_TAC [OPEN_IN_OPEN] THEN
4158 DISCH_THEN(MP_TAC o SPEC `IMAGE (f:real^M->real^N) s INTER t`) THEN
4160 [EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC [];
4162 SUBGOAL_THEN `{x | x IN s /\ (f:real^M->real^N) x IN t} =
4163 s INTER t'` SUBST1_TAC THENL
4164 [ASM SET_TAC []; ASM_MESON_TAC [OPEN_INTER]]]);;
4166 let CONTINUOUS_CLOSED_PREIMAGE = prove
4167 (`!f:real^M->real^N s t.
4168 f continuous_on s /\ closed s /\ closed t
4169 ==> closed {x | x IN s /\ f(x) IN t}`,
4170 REPEAT STRIP_TAC THEN
4171 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_CLOSED]) THEN
4172 REWRITE_TAC [CLOSED_IN_CLOSED] THEN
4173 DISCH_THEN(MP_TAC o SPEC `IMAGE (f:real^M->real^N) s INTER t`) THEN
4175 [EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC [];
4177 SUBGOAL_THEN `{x | x IN s /\ (f:real^M->real^N) x IN t} =
4178 s INTER t'` SUBST1_TAC THENL
4179 [ASM SET_TAC []; ASM_MESON_TAC [CLOSED_INTER]]]);;
4181 let CONTINUOUS_OPEN_PREIMAGE_UNIV = prove
4182 (`!f:real^M->real^N s.
4183 (!x. f continuous (at x)) /\ open s ==> open {x | f(x) IN s}`,
4184 REPEAT STRIP_TAC THEN
4185 MP_TAC(SPECL [`f:real^M->real^N`; `(:real^M)`; `s:real^N->bool`]
4186 CONTINUOUS_OPEN_PREIMAGE) THEN
4187 ASM_SIMP_TAC[OPEN_UNIV; IN_UNIV; CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
4189 let CONTINUOUS_CLOSED_PREIMAGE_UNIV = prove
4190 (`!f:real^M->real^N s.
4191 (!x. f continuous (at x)) /\ closed s ==> closed {x | f(x) IN s}`,
4192 REPEAT STRIP_TAC THEN
4193 MP_TAC(SPECL [`f:real^M->real^N`; `(:real^M)`; `s:real^N->bool`]
4194 CONTINUOUS_CLOSED_PREIMAGE) THEN
4195 ASM_SIMP_TAC[CLOSED_UNIV; IN_UNIV; CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
4197 let CONTINUOUS_OPEN_IN_PREIMAGE_EQ = prove
4198 (`!f:real^M->real^N s.
4199 f continuous_on s <=>
4200 !t. open t ==> open_in (subtopology euclidean s) {x | x IN s /\ f x IN t}`,
4201 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONTINUOUS_OPEN_IN_PREIMAGE] THEN
4202 REWRITE_TAC[CONTINUOUS_ON_OPEN] THEN DISCH_TAC THEN
4203 X_GEN_TAC `t:real^N->bool` THEN GEN_REWRITE_TAC LAND_CONV [OPEN_IN_OPEN] THEN
4204 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4205 FIRST_X_ASSUM(MP_TAC o SPEC `u:real^N->bool`) THEN
4206 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]);;
4208 let CONTINUOUS_CLOSED_IN_PREIMAGE_EQ = prove
4209 (`!f:real^M->real^N s.
4210 f continuous_on s <=>
4212 ==> closed_in (subtopology euclidean s) {x | x IN s /\ f x IN t}`,
4213 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CONTINUOUS_CLOSED_IN_PREIMAGE] THEN
4214 REWRITE_TAC[CONTINUOUS_ON_CLOSED] THEN DISCH_TAC THEN
4215 X_GEN_TAC `t:real^N->bool` THEN
4216 GEN_REWRITE_TAC LAND_CONV [CLOSED_IN_CLOSED] THEN
4217 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4218 FIRST_X_ASSUM(MP_TAC o SPEC `u:real^N->bool`) THEN
4219 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN SET_TAC[]);;
4221 (* ------------------------------------------------------------------------- *)
4222 (* Linear functions are (uniformly) continuous on any set. *)
4223 (* ------------------------------------------------------------------------- *)
4225 let LINEAR_LIM_0 = prove
4226 (`!f. linear f ==> (f --> vec 0) (at (vec 0))`,
4227 REPEAT STRIP_TAC THEN REWRITE_TAC[LIM_AT] THEN
4228 FIRST_X_ASSUM(MP_TAC o MATCH_MP LINEAR_BOUNDED_POS) THEN
4229 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
4230 X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `e / B` THEN
4231 ASM_SIMP_TAC[REAL_LT_DIV] THEN REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN
4232 ASM_MESON_TAC[REAL_MUL_SYM; REAL_LET_TRANS; REAL_LT_RDIV_EQ]);;
4234 let LINEAR_CONTINUOUS_AT = prove
4235 (`!f:real^M->real^N a. linear f ==> f continuous (at a)`,
4236 REPEAT STRIP_TAC THEN
4237 MP_TAC(ISPEC `\x. (f:real^M->real^N) (a + x) - f(a)` LINEAR_LIM_0) THEN
4239 [POP_ASSUM MP_TAC THEN SIMP_TAC[linear] THEN
4240 REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC;
4242 REWRITE_TAC[GSYM LIM_NULL; CONTINUOUS_AT] THEN
4243 GEN_REWRITE_TAC RAND_CONV [LIM_AT_ZERO] THEN SIMP_TAC[]);;
4245 let LINEAR_CONTINUOUS_WITHIN = prove
4246 (`!f:real^M->real^N s x. linear f ==> f continuous (at x within s)`,
4247 SIMP_TAC[CONTINUOUS_AT_WITHIN; LINEAR_CONTINUOUS_AT]);;
4249 let LINEAR_CONTINUOUS_ON = prove
4250 (`!f:real^M->real^N s. linear f ==> f continuous_on s`,
4251 MESON_TAC[LINEAR_CONTINUOUS_AT; CONTINUOUS_AT_IMP_CONTINUOUS_ON]);;
4253 let LINEAR_CONTINUOUS_COMPOSE = prove
4254 (`!net f:A->real^N g:real^N->real^P.
4255 f continuous net /\ linear g ==> (\x. g(f x)) continuous net`,
4256 REWRITE_TAC[continuous; LIM_LINEAR]);;
4258 let LINEAR_CONTINUOUS_ON_COMPOSE = prove
4259 (`!f:real^M->real^N g:real^N->real^P s.
4260 f continuous_on s /\ linear g ==> (\x. g(f x)) continuous_on s`,
4261 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN;
4262 LINEAR_CONTINUOUS_COMPOSE]);;
4264 let CONTINUOUS_LIFT_COMPONENT_COMPOSE = prove
4265 (`!net f:A->real^N i. f continuous net ==> (\x. lift(f x$i)) continuous net`,
4267 SUBGOAL_THEN `linear(\x:real^N. lift (x$i))` MP_TAC THENL
4268 [REWRITE_TAC[LINEAR_LIFT_COMPONENT]; REWRITE_TAC[GSYM IMP_CONJ_ALT]] THEN
4269 REWRITE_TAC[LINEAR_CONTINUOUS_COMPOSE]);;
4271 let CONTINUOUS_ON_LIFT_COMPONENT_COMPOSE = prove
4272 (`!f:real^M->real^N s.
4274 ==> (\x. lift (f x$i)) continuous_on s`,
4275 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN;
4276 CONTINUOUS_LIFT_COMPONENT_COMPOSE]);;
4278 (* ------------------------------------------------------------------------- *)
4279 (* Also bilinear functions, in composition form. *)
4280 (* ------------------------------------------------------------------------- *)
4282 let BILINEAR_CONTINUOUS_COMPOSE = prove
4283 (`!net f:A->real^M g:A->real^N h:real^M->real^N->real^P.
4284 f continuous net /\ g continuous net /\ bilinear h
4285 ==> (\x. h (f x) (g x)) continuous net`,
4286 REWRITE_TAC[continuous; LIM_BILINEAR]);;
4288 let BILINEAR_CONTINUOUS_ON_COMPOSE = prove
4289 (`!f g h s. f continuous_on s /\ g continuous_on s /\ bilinear h
4290 ==> (\x. h (f x) (g x)) continuous_on s`,
4291 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN;
4292 BILINEAR_CONTINUOUS_COMPOSE]);;
4294 let BILINEAR_DOT = prove
4295 (`bilinear (\x y:real^N. lift(x dot y))`,
4296 REWRITE_TAC[bilinear; linear; DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL] THEN
4297 REWRITE_TAC[LIFT_ADD; LIFT_CMUL]);;
4299 let CONTINUOUS_LIFT_DOT2 = prove
4300 (`!net f g:A->real^N.
4301 f continuous net /\ g continuous net
4302 ==> (\x. lift(f x dot g x)) continuous net`,
4303 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (MATCH_MP (REWRITE_RULE
4304 [TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`]
4305 BILINEAR_CONTINUOUS_COMPOSE) BILINEAR_DOT)) THEN REWRITE_TAC[]);;
4307 let CONTINUOUS_ON_LIFT_DOT2 = prove
4308 (`!f:real^M->real^N g s.
4309 f continuous_on s /\ g continuous_on s
4310 ==> (\x. lift(f x dot g x)) continuous_on s`,
4311 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (MATCH_MP (REWRITE_RULE
4312 [TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`]
4313 BILINEAR_CONTINUOUS_ON_COMPOSE) BILINEAR_DOT)) THEN REWRITE_TAC[]);;
4315 (* ------------------------------------------------------------------------- *)
4316 (* Preservation of compactness and connectedness under continuous function. *)
4317 (* ------------------------------------------------------------------------- *)
4319 let COMPACT_CONTINUOUS_IMAGE = prove
4320 (`!f:real^M->real^N s.
4321 f continuous_on s /\ compact s ==> compact(IMAGE f s)`,
4322 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_on; compact] THEN
4323 STRIP_TAC THEN X_GEN_TAC `y:num->real^N` THEN
4324 REWRITE_TAC[IN_IMAGE; SKOLEM_THM; FORALL_AND_THM] THEN
4325 DISCH_THEN(X_CHOOSE_THEN `x:num->real^M` STRIP_ASSUME_TAC) THEN
4326 FIRST_X_ASSUM(MP_TAC o SPEC `x:num->real^M`) THEN ASM_REWRITE_TAC[] THEN
4327 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
4328 X_GEN_TAC `r:num->num` THEN
4329 DISCH_THEN(X_CHOOSE_THEN `l:real^M` STRIP_ASSUME_TAC) THEN
4330 EXISTS_TAC `(f:real^M->real^N) l` THEN ASM_REWRITE_TAC[] THEN
4331 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
4332 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
4333 FIRST_X_ASSUM(MP_TAC o SPEC `l:real^M`) THEN
4334 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
4335 DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
4336 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
4337 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
4338 DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[o_THM] THEN
4341 let COMPACT_TRANSLATION = prove
4342 (`!s a:real^N. compact s ==> compact (IMAGE (\x. a + x) s)`,
4343 SIMP_TAC[COMPACT_CONTINUOUS_IMAGE; CONTINUOUS_ON_ADD;
4344 CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);;
4346 let COMPACT_TRANSLATION_EQ = prove
4347 (`!a s. compact (IMAGE (\x:real^N. a + x) s) <=> compact s`,
4348 REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[COMPACT_TRANSLATION] THEN
4349 DISCH_THEN(MP_TAC o ISPEC `--a:real^N` o MATCH_MP COMPACT_TRANSLATION) THEN
4350 REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
4351 VECTOR_ARITH `--a + a + x:real^N = x`]);;
4353 add_translation_invariants [COMPACT_TRANSLATION_EQ];;
4355 let COMPACT_LINEAR_IMAGE = prove
4356 (`!f:real^M->real^N s. compact s /\ linear f ==> compact(IMAGE f s)`,
4357 SIMP_TAC[LINEAR_CONTINUOUS_ON; COMPACT_CONTINUOUS_IMAGE]);;
4359 let COMPACT_LINEAR_IMAGE_EQ = prove
4360 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
4361 ==> (compact (IMAGE f s) <=> compact s)`,
4362 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COMPACT_LINEAR_IMAGE));;
4364 add_linear_invariants [COMPACT_LINEAR_IMAGE_EQ];;
4366 let CONNECTED_CONTINUOUS_IMAGE = prove
4367 (`!f:real^M->real^N s.
4368 f continuous_on s /\ connected s ==> connected(IMAGE f s)`,
4369 REPEAT GEN_TAC THEN REWRITE_TAC[CONTINUOUS_ON_OPEN] THEN
4370 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4371 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
4372 REWRITE_TAC[CONNECTED_CLOPEN; NOT_FORALL_THM; NOT_IMP; DE_MORGAN_THM] THEN
4373 REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
4374 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
4375 FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `t:real^N->bool` th) THEN
4376 MP_TAC(SPEC `IMAGE (f:real^M->real^N) s DIFF t` th)) THEN
4377 ASM_REWRITE_TAC[] THEN
4378 SUBGOAL_THEN `{x | x IN s /\ (f:real^M->real^N) x IN IMAGE f s DIFF t} =
4379 s DIFF {x | x IN s /\ f x IN t}`
4381 [UNDISCH_TAC `t SUBSET IMAGE (f:real^M->real^N) s` THEN
4382 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_DIFF; IN_ELIM_THM; SUBSET] THEN
4384 REPEAT STRIP_TAC THEN
4385 EXISTS_TAC `{x | x IN s /\ (f:real^M->real^N) x IN t}` THEN
4386 ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
4387 REWRITE_TAC[IN_IMAGE; SUBSET; IN_ELIM_THM; NOT_IN_EMPTY; EXTENSION] THEN
4390 let CONNECTED_TRANSLATION = prove
4391 (`!a s. connected s ==> connected (IMAGE (\x:real^N. a + x) s)`,
4392 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN
4393 ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST]);;
4395 let CONNECTED_TRANSLATION_EQ = prove
4396 (`!a s. connected (IMAGE (\x:real^N. a + x) s) <=> connected s`,
4397 REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[CONNECTED_TRANSLATION] THEN
4398 DISCH_THEN(MP_TAC o ISPEC `--a:real^N` o MATCH_MP CONNECTED_TRANSLATION) THEN
4399 REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID;
4400 VECTOR_ARITH `--a + a + x:real^N = x`]);;
4402 add_translation_invariants [CONNECTED_TRANSLATION_EQ];;
4404 let CONNECTED_LINEAR_IMAGE = prove
4405 (`!f:real^M->real^N s. connected s /\ linear f ==> connected(IMAGE f s)`,
4406 SIMP_TAC[LINEAR_CONTINUOUS_ON; CONNECTED_CONTINUOUS_IMAGE]);;
4408 let CONNECTED_LINEAR_IMAGE_EQ = prove
4409 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
4410 ==> (connected (IMAGE f s) <=> connected s)`,
4411 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE CONNECTED_LINEAR_IMAGE));;
4413 add_linear_invariants [CONNECTED_LINEAR_IMAGE_EQ];;
4415 (* ------------------------------------------------------------------------- *)
4416 (* Preservation properties for pasted sets (Cartesian products). *)
4417 (* ------------------------------------------------------------------------- *)
4419 let BOUNDED_PCROSS_EQ = prove
4420 (`!s:real^M->bool t:real^N->bool.
4421 bounded (s PCROSS t) <=>
4422 s = {} \/ t = {} \/ bounded s /\ bounded t`,
4423 REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS] THEN
4424 ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
4425 ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
4426 REWRITE_TAC[SET_RULE `{f x y |x,y| F} = {}`; BOUNDED_EMPTY] THEN
4427 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
4428 REWRITE_TAC[bounded; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
4429 ASM_MESON_TAC[NORM_LE_PASTECART; REAL_LE_TRANS; NORM_PASTECART_LE;
4432 let BOUNDED_PCROSS = prove
4433 (`!s:real^M->bool t:real^N->bool.
4434 bounded s /\ bounded t ==> bounded (s PCROSS t)`,
4435 SIMP_TAC[BOUNDED_PCROSS_EQ]);;
4437 let CLOSED_PCROSS_EQ = prove
4438 (`!s:real^M->bool t:real^N->bool.
4439 closed (s PCROSS t) <=>
4440 s = {} \/ t = {} \/ closed s /\ closed t`,
4441 REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS] THEN MAP_EVERY ASM_CASES_TAC
4442 [`s:real^M->bool = {}`; `t:real^N->bool = {}`] THEN
4443 ASM_REWRITE_TAC[NOT_IN_EMPTY; CLOSED_EMPTY; SET_RULE
4444 `{f x y |x,y| F} = {}`] THEN
4445 REWRITE_TAC[CLOSED_SEQUENTIAL_LIMITS; LIM_SEQUENTIALLY] THEN
4446 REWRITE_TAC[FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
4447 REWRITE_TAC[IN_ELIM_THM; SKOLEM_THM; FORALL_AND_THM] THEN
4448 ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN
4449 REWRITE_TAC[LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN
4450 SIMP_TAC[TAUT `((p /\ q) /\ r) /\ s ==> t <=> r ==> p /\ q /\ s ==> t`] THEN
4451 ONCE_REWRITE_TAC[MESON[]
4452 `(!a b c d e. P a b c d e) <=> (!d e b c a. P a b c d e)`] THEN
4453 REWRITE_TAC[FORALL_UNWIND_THM2] THEN
4454 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN EQ_TAC THENL
4455 [GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
4456 [TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`; FORALL_AND_THM] THEN
4457 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
4458 [ALL_TAC; GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM]] THEN
4459 MATCH_MP_TAC MONO_FORALL THEN REPEAT STRIP_TAC THEN
4460 FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC(MESON[]
4461 `(?x. P x (\n. x)) ==> (?s x. P x s)`) THEN
4462 ASM_MESON_TAC[DIST_PASTECART_CANCEL];
4463 ONCE_REWRITE_TAC[MESON[]
4464 `(!x l. P x l) /\ (!y m. Q y m) <=> (!x y l m. P x l /\ Q y m)`] THEN
4465 REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
4466 REWRITE_TAC[dist; PASTECART_SUB] THEN
4467 ASM_MESON_TAC[NORM_LE_PASTECART; REAL_LET_TRANS]]);;
4469 let CLOSED_PCROSS = prove
4470 (`!s:real^M->bool t:real^N->bool.
4471 closed s /\ closed t ==> closed (s PCROSS t)`,
4472 SIMP_TAC[CLOSED_PCROSS_EQ]);;
4474 let COMPACT_PCROSS_EQ = prove
4475 (`!s:real^M->bool t:real^N->bool.
4476 compact (s PCROSS t) <=>
4477 s = {} \/ t = {} \/ compact s /\ compact t`,
4478 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_PCROSS_EQ;
4479 BOUNDED_PCROSS_EQ] THEN
4482 let COMPACT_PCROSS = prove
4483 (`!s:real^M->bool t:real^N->bool.
4484 compact s /\ compact t ==> compact (s PCROSS t)`,
4485 SIMP_TAC[COMPACT_PCROSS_EQ]);;
4487 let OPEN_PCROSS_EQ = prove
4488 (`!s:real^M->bool t:real^N->bool.
4489 open (s PCROSS t) <=>
4490 s = {} \/ t = {} \/ open s /\ open t`,
4491 REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS] THEN
4492 ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
4493 ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
4494 REWRITE_TAC[SET_RULE `{f x y |x,y| F} = {}`; OPEN_EMPTY] THEN
4495 RULE_ASSUM_TAC(REWRITE_RULE[GSYM MEMBER_NOT_EMPTY]) THEN
4497 [REWRITE_TAC[open_def; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
4498 ASM_MESON_TAC[DIST_PASTECART_CANCEL];
4499 REWRITE_TAC[OPEN_CLOSED] THEN STRIP_TAC THEN
4501 `UNIV DIFF {pastecart x y | x IN s /\ y IN t} =
4502 {pastecart x y | x IN ((:real^M) DIFF s) /\ y IN (:real^N)} UNION
4503 {pastecart x y | x IN (:real^M) /\ y IN ((:real^N) DIFF t)}`
4505 [REWRITE_TAC[EXTENSION; IN_DIFF; IN_UNION; FORALL_PASTECART; IN_UNIV] THEN
4506 REWRITE_TAC[IN_ELIM_THM; PASTECART_EQ; FSTCART_PASTECART;
4507 SNDCART_PASTECART] THEN MESON_TAC[];
4508 SIMP_TAC[GSYM PCROSS] THEN MATCH_MP_TAC CLOSED_UNION THEN CONJ_TAC THEN
4509 MATCH_MP_TAC CLOSED_PCROSS THEN ASM_REWRITE_TAC[CLOSED_UNIV]]]);;
4511 let OPEN_PCROSS = prove
4512 (`!s:real^M->bool t:real^N->bool.
4513 open s /\ open t ==> open (s PCROSS t)`,
4514 SIMP_TAC[OPEN_PCROSS_EQ]);;
4516 let OPEN_IN_PCROSS = prove
4517 (`!s s':real^M->bool t t':real^N->bool.
4518 open_in (subtopology euclidean s) s' /\
4519 open_in (subtopology euclidean t) t'
4520 ==> open_in (subtopology euclidean (s PCROSS t)) (s' PCROSS t')`,
4521 REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN] THEN DISCH_THEN(CONJUNCTS_THEN2
4522 (X_CHOOSE_THEN `s'':real^M->bool` STRIP_ASSUME_TAC)
4523 (X_CHOOSE_THEN `t'':real^N->bool` STRIP_ASSUME_TAC)) THEN
4524 EXISTS_TAC `(s'':real^M->bool) PCROSS (t'':real^N->bool)` THEN
4525 ASM_SIMP_TAC[OPEN_PCROSS; EXTENSION; FORALL_PASTECART] THEN
4526 REWRITE_TAC[IN_INTER; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]);;
4528 let PASTECART_IN_INTERIOR_SUBTOPOLOGY = prove
4529 (`!s t u x:real^M y:real^N.
4530 pastecart x y IN u /\ open_in (subtopology euclidean (s PCROSS t)) u
4531 ==> ?v w. open_in (subtopology euclidean s) v /\ x IN v /\
4532 open_in (subtopology euclidean t) w /\ y IN w /\
4533 (v PCROSS w) SUBSET u`,
4534 REWRITE_TAC[open_in; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN
4535 REPEAT STRIP_TAC THEN
4536 FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `y:real^N`]) THEN
4537 ASM_REWRITE_TAC[] THEN
4538 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
4539 EXISTS_TAC `ball(x:real^M,e / &2) INTER s` THEN
4540 EXISTS_TAC `ball(y:real^N,e / &2) INTER t` THEN
4541 SUBGOAL_THEN `(x:real^M) IN s /\ (y:real^N) IN t` STRIP_ASSUME_TAC THENL
4542 [ASM_MESON_TAC[SUBSET; PASTECART_IN_PCROSS]; ALL_TAC] THEN
4543 ASM_SIMP_TAC[INTER_SUBSET; IN_INTER; CENTRE_IN_BALL; REAL_HALF] THEN
4544 REWRITE_TAC[IN_BALL] THEN REPEAT(CONJ_TAC THENL
4545 [MESON_TAC[REAL_SUB_LT; NORM_ARITH
4546 `dist(x,y) < e /\ dist(z,y) < e - dist(x,y)
4547 ==> dist(x:real^N,z) < e`];
4549 REWRITE_TAC[SUBSET; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN
4550 REWRITE_TAC[IN_BALL; IN_INTER] THEN REPEAT STRIP_TAC THEN
4551 FIRST_X_ASSUM MATCH_MP_TAC THEN
4552 ASM_REWRITE_TAC[dist; PASTECART_SUB] THEN
4553 W(MP_TAC o PART_MATCH lhand NORM_PASTECART_LE o lhand o snd) THEN
4554 REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[DIST_SYM] dist)] THEN
4555 ASM_REAL_ARITH_TAC);;
4557 let OPEN_IN_PCROSS_EQ = prove
4558 (`!s s':real^M->bool t t':real^N->bool.
4559 open_in (subtopology euclidean (s PCROSS t)) (s' PCROSS t') <=>
4560 s' = {} \/ t' = {} \/
4561 open_in (subtopology euclidean s) s' /\
4562 open_in (subtopology euclidean t) t'`,
4564 ASM_CASES_TAC `s':real^M->bool = {}` THEN
4565 ASM_REWRITE_TAC[PCROSS_EMPTY; OPEN_IN_EMPTY] THEN
4566 ASM_CASES_TAC `t':real^N->bool = {}` THEN
4567 ASM_REWRITE_TAC[PCROSS_EMPTY; OPEN_IN_EMPTY] THEN
4568 EQ_TAC THEN REWRITE_TAC[OPEN_IN_PCROSS] THEN REPEAT STRIP_TAC THENL
4569 [ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN
4570 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
4571 UNDISCH_TAC `~(t':real^N->bool = {})` THEN
4572 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
4573 DISCH_THEN(X_CHOOSE_TAC `y:real^N`);
4574 ONCE_REWRITE_TAC[OPEN_IN_SUBOPEN] THEN
4575 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
4576 UNDISCH_TAC `~(s':real^M->bool = {})` THEN
4577 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
4578 DISCH_THEN(X_CHOOSE_TAC `x:real^M`)] THEN
4580 [`s:real^M->bool`; `t:real^N->bool`;
4581 `(s':real^M->bool) PCROSS (t':real^N->bool)`;
4582 `x:real^M`; `y:real^N`] PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN
4583 ASM_REWRITE_TAC[SUBSET; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN
4586 let INTERIOR_PCROSS = prove
4587 (`!s:real^M->bool t:real^N->bool.
4588 interior (s PCROSS t) = (interior s) PCROSS (interior t)`,
4589 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
4590 [REWRITE_TAC[SUBSET; FORALL_PASTECART; PASTECART_IN_PCROSS] THEN
4591 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN DISCH_TAC THEN
4592 MP_TAC(ISPECL [`(:real^M)`; `(:real^N)`;
4593 `interior((s:real^M->bool) PCROSS (t:real^N->bool))`;
4594 `x:real^M`; `y:real^N`] PASTECART_IN_INTERIOR_SUBTOPOLOGY) THEN
4595 REWRITE_TAC[UNIV_PCROSS_UNIV; SUBTOPOLOGY_UNIV; GSYM OPEN_IN] THEN
4596 ASM_REWRITE_TAC[OPEN_INTERIOR] THEN STRIP_TAC THEN
4597 FIRST_ASSUM(MP_TAC o MATCH_MP (MESON[INTERIOR_SUBSET; SUBSET_TRANS]
4598 `s SUBSET interior t ==> s SUBSET t`)) THEN
4599 REWRITE_TAC[SUBSET_PCROSS] THEN
4600 ASM_MESON_TAC[NOT_IN_EMPTY; INTERIOR_MAXIMAL; SUBSET];
4601 MATCH_MP_TAC INTERIOR_MAXIMAL THEN
4602 SIMP_TAC[OPEN_PCROSS; OPEN_INTERIOR; PCROSS_MONO; INTERIOR_SUBSET]]);;
4604 (* ------------------------------------------------------------------------- *)
4605 (* Quotient maps are occasionally useful. *)
4606 (* ------------------------------------------------------------------------- *)
4608 let QUASICOMPACT_OPEN_CLOSED = prove
4609 (`!f:real^M->real^N s t.
4611 ==> ((!u. u SUBSET t
4612 ==> (open_in (subtopology euclidean s)
4613 {x | x IN s /\ f x IN u}
4614 ==> open_in (subtopology euclidean t) u)) <=>
4616 ==> (closed_in (subtopology euclidean s)
4617 {x | x IN s /\ f x IN u}
4618 ==> closed_in (subtopology euclidean t) u)))`,
4619 SIMP_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
4620 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN
4621 X_GEN_TAC `u:real^N->bool` THEN
4622 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF u:real^N->bool`) THEN
4623 ASM_SIMP_TAC[SET_RULE `u SUBSET t ==> t DIFF (t DIFF u) = u`] THEN
4624 (ANTS_TAC THENL [SET_TAC[]; REPEAT STRIP_TAC]) THEN
4625 FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[SUBSET_RESTRICT] THEN
4626 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[]
4627 `open_in top x ==> x = y ==> open_in top y`)) THEN
4630 let QUOTIENT_MAP_IMP_CONTINUOUS_OPEN = prove
4631 (`!f:real^M->real^N s t.
4632 IMAGE f s SUBSET t /\
4634 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4635 open_in (subtopology euclidean t) u))
4636 ==> f continuous_on s`,
4637 MESON_TAC[OPEN_IN_IMP_SUBSET; CONTINUOUS_ON_OPEN_GEN]);;
4639 let QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED = prove
4640 (`!f:real^M->real^N s t.
4641 IMAGE f s SUBSET t /\
4643 ==> (closed_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4644 closed_in (subtopology euclidean t) u))
4645 ==> f continuous_on s`,
4646 MESON_TAC[CLOSED_IN_IMP_SUBSET; CONTINUOUS_ON_CLOSED_GEN]);;
4648 let OPEN_MAP_IMP_QUOTIENT_MAP = prove
4649 (`!f:real^M->real^N s.
4650 f continuous_on s /\
4651 (!t. open_in (subtopology euclidean s) t
4652 ==> open_in (subtopology euclidean (IMAGE f s)) (IMAGE f t))
4653 ==> !t. t SUBSET IMAGE f s
4654 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN t} <=>
4655 open_in (subtopology euclidean (IMAGE f s)) t)`,
4656 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
4658 `t = IMAGE f {x | x IN s /\ (f:real^M->real^N) x IN t}`
4659 SUBST1_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[]];
4660 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_OPEN]) THEN
4663 let CLOSED_MAP_IMP_QUOTIENT_MAP = prove
4664 (`!f:real^M->real^N s.
4665 f continuous_on s /\
4666 (!t. closed_in (subtopology euclidean s) t
4667 ==> closed_in (subtopology euclidean (IMAGE f s)) (IMAGE f t))
4668 ==> !t. t SUBSET IMAGE f s
4669 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN t} <=>
4670 open_in (subtopology euclidean (IMAGE f s)) t)`,
4671 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
4672 [FIRST_X_ASSUM(MP_TAC o SPEC
4673 `s DIFF {x | x IN s /\ (f:real^M->real^N) x IN t}`) THEN
4675 [MATCH_MP_TAC CLOSED_IN_DIFF THEN
4676 ASM_SIMP_TAC[CLOSED_IN_SUBTOPOLOGY_REFL;
4677 TOPSPACE_EUCLIDEAN; SUBSET_UNIV];
4678 REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
4679 DISCH_THEN(MP_TAC o CONJUNCT2) THEN MATCH_MP_TAC EQ_IMP THEN
4680 AP_TERM_TAC THEN ASM SET_TAC[]];
4681 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONTINUOUS_ON_OPEN]) THEN
4684 let CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP = prove
4685 (`!f:real^M->real^N g s t.
4686 f continuous_on s /\ IMAGE f s SUBSET t /\
4687 g continuous_on t /\ IMAGE g t SUBSET s /\
4688 (!y. y IN t ==> f(g y) = y)
4690 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4691 open_in (subtopology euclidean t) u))`,
4692 REWRITE_TAC[CONTINUOUS_ON_OPEN] THEN REPEAT STRIP_TAC THEN EQ_TAC THENL
4693 [DISCH_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `(IMAGE (g:real^N->real^M) t)
4695 {x | x IN s /\ (f:real^M->real^N) x IN u}`) THEN
4697 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
4698 REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN
4700 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]];
4701 DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
4702 SUBGOAL_THEN `IMAGE (f:real^M->real^N) s = t`
4703 (fun th -> ASM_REWRITE_TAC[th]) THEN
4706 let CONTINUOUS_LEFT_INVERSE_IMP_QUOTIENT_MAP = prove
4707 (`!f:real^M->real^N g s.
4708 f continuous_on s /\ g continuous_on (IMAGE f s) /\
4709 (!x. x IN s ==> g(f x) = x)
4710 ==> (!u. u SUBSET (IMAGE f s)
4711 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4712 open_in (subtopology euclidean (IMAGE f s)) u))`,
4713 REPEAT GEN_TAC THEN STRIP_TAC THEN
4714 MATCH_MP_TAC CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP THEN
4715 EXISTS_TAC `g:real^N->real^M` THEN
4716 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
4718 let QUOTIENT_MAP_OPEN_CLOSED = prove
4719 (`!f:real^M->real^N s t.
4721 ==> ((!u. u SUBSET t
4722 ==> (open_in (subtopology euclidean s)
4723 {x | x IN s /\ f x IN u} <=>
4724 open_in (subtopology euclidean t) u)) <=>
4726 ==> (closed_in (subtopology euclidean s)
4727 {x | x IN s /\ f x IN u} <=>
4728 closed_in (subtopology euclidean t) u)))`,
4729 SIMP_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
4730 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN
4731 X_GEN_TAC `u:real^N->bool` THEN
4732 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF u:real^N->bool`) THEN
4733 ASM_SIMP_TAC[SET_RULE `u SUBSET t ==> t DIFF (t DIFF u) = u`] THEN
4734 (ANTS_TAC THENL [SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)]) THEN
4735 REWRITE_TAC[SUBSET_RESTRICT] THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
4737 let CONTINUOUS_ON_COMPOSE_QUOTIENT = prove
4738 (`!f:real^M->real^N g:real^N->real^P s t u.
4739 IMAGE f s SUBSET t /\ IMAGE g t SUBSET u /\
4741 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN v} <=>
4742 open_in (subtopology euclidean t) v)) /\
4743 (g o f) continuous_on s
4744 ==> g continuous_on t`,
4746 REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
4747 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP CONTINUOUS_ON_OPEN_GEN th]) THEN
4749 `IMAGE ((g:real^N->real^P) o (f:real^M->real^N)) s SUBSET u`
4750 (fun th -> REWRITE_TAC[MATCH_MP CONTINUOUS_ON_OPEN_GEN th])
4751 THENL [REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[]; DISCH_TAC] THEN
4752 X_GEN_TAC `v:real^P->bool` THEN DISCH_TAC THEN
4753 FIRST_X_ASSUM(MP_TAC o SPEC `v:real^P->bool`) THEN
4754 ASM_REWRITE_TAC[o_THM] THEN DISCH_TAC THEN
4755 FIRST_X_ASSUM(MP_TAC o SPEC `{x | x IN t /\ (g:real^N->real^P) x IN v}`) THEN
4756 ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
4757 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[]
4758 `open_in top s ==> s = t ==> open_in top t`)) THEN
4761 let LIFT_TO_QUOTIENT_SPACE = prove
4762 (`!f:real^M->real^N h:real^M->real^P s t u.
4765 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN v} <=>
4766 open_in (subtopology euclidean t) v)) /\
4767 h continuous_on s /\ IMAGE h s = u /\
4768 (!x y. x IN s /\ y IN s /\ f x = f y ==> h x = h y)
4769 ==> ?g. g continuous_on t /\ IMAGE g t = u /\
4770 !x. x IN s ==> h(x) = g(f x)`,
4772 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
4773 REWRITE_TAC[FUNCTION_FACTORS_LEFT_GEN] THEN
4774 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^P` THEN
4775 DISCH_TAC THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
4776 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE_QUOTIENT THEN MAP_EVERY EXISTS_TAC
4777 [`f:real^M->real^N`; `s:real^M->bool`; `u:real^P->bool`] THEN
4778 ASM_SIMP_TAC[SUBSET_REFL] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
4779 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
4780 CONTINUOUS_ON_EQ)) THEN
4781 ASM_REWRITE_TAC[o_THM]);;
4783 let QUOTIENT_MAP_COMPOSE = prove
4784 (`!f:real^M->real^N g:real^N->real^P s t u.
4785 IMAGE f s SUBSET t /\
4787 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN v} <=>
4788 open_in (subtopology euclidean t) v)) /\
4790 ==> (open_in (subtopology euclidean t) {x | x IN t /\ g x IN v} <=>
4791 open_in (subtopology euclidean u) v))
4793 ==> (open_in (subtopology euclidean s)
4794 {x | x IN s /\ (g o f) x IN v} <=>
4795 open_in (subtopology euclidean u) v)`,
4796 REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN
4798 `{x | x IN s /\ (g:real^N->real^P) ((f:real^M->real^N) x) IN v} =
4799 {x | x IN s /\ f x IN {x | x IN t /\ g x IN v}}`
4800 SUBST1_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[SUBSET_RESTRICT]]);;
4802 let QUOTIENT_MAP_FROM_COMPOSITION = prove
4803 (`!f:real^M->real^N g:real^N->real^P s t u.
4804 f continuous_on s /\ IMAGE f s SUBSET t /\
4805 g continuous_on t /\ IMAGE g t SUBSET u /\
4807 ==> (open_in (subtopology euclidean s)
4808 {x | x IN s /\ (g o f) x IN v} <=>
4809 open_in (subtopology euclidean u) v))
4811 ==> (open_in (subtopology euclidean t)
4812 {x | x IN t /\ g x IN v} <=>
4813 open_in (subtopology euclidean u) v)`,
4814 REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THENL
4815 [FIRST_X_ASSUM(MP_TAC o SPEC `v:real^P->bool`) THEN
4816 ASM_REWRITE_TAC[o_THM] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
4818 `{x | x IN s /\ (g:real^N->real^P) ((f:real^M->real^N) x) IN v} =
4819 {x | x IN s /\ f x IN {x | x IN t /\ g x IN v}}`
4820 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
4821 MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN
4822 EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[];
4823 MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN
4824 EXISTS_TAC `u:real^P->bool` THEN ASM_REWRITE_TAC[]]);;
4826 let QUOTIENT_MAP_FROM_SUBSET = prove
4827 (`!f:real^M->real^N s t u.
4828 f continuous_on t /\ IMAGE f t SUBSET u /\
4829 s SUBSET t /\ IMAGE f s = u /\
4831 ==> (open_in (subtopology euclidean s)
4832 {x | x IN s /\ f x IN v} <=>
4833 open_in (subtopology euclidean u) v))
4835 ==> (open_in (subtopology euclidean t)
4836 {x | x IN t /\ f x IN v} <=>
4837 open_in (subtopology euclidean u) v)`,
4838 REPEAT GEN_TAC THEN STRIP_TAC THEN
4839 MATCH_MP_TAC QUOTIENT_MAP_FROM_COMPOSITION THEN
4840 MAP_EVERY EXISTS_TAC [`\x:real^M. x`; `s:real^M->bool`] THEN
4841 ASM_REWRITE_TAC[CONTINUOUS_ON_ID; IMAGE_ID; o_THM]);;
4843 let QUOTIENT_MAP_RESTRICT = prove
4844 (`!f:real^M->real^N s t c.
4845 IMAGE f s SUBSET t /\
4847 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4848 open_in (subtopology euclidean t) u)) /\
4849 (open_in (subtopology euclidean t) c \/
4850 closed_in (subtopology euclidean t) c)
4852 ==> (open_in (subtopology euclidean {x | x IN s /\ f x IN c})
4853 {x | x IN {x | x IN s /\ f x IN c} /\ f x IN u} <=>
4854 open_in (subtopology euclidean c) u)`,
4856 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
4857 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
4858 DISCH_THEN(fun th -> MP_TAC th THEN MP_TAC (MATCH_MP
4859 (REWRITE_RULE[IMP_CONJ_ALT] QUOTIENT_MAP_IMP_CONTINUOUS_OPEN) th)) THEN
4860 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
4861 SUBGOAL_THEN `IMAGE (f:real^M->real^N) {x | x IN s /\ f x IN c} SUBSET c`
4862 ASSUME_TAC THENL [SET_TAC[]; ALL_TAC] THEN
4863 FIRST_X_ASSUM DISJ_CASES_TAC THENL
4864 [FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET);
4865 ASM_SIMP_TAC[QUOTIENT_MAP_OPEN_CLOSED] THEN
4866 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET)] THEN
4867 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `u:real^N->bool` THEN
4868 DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
4869 (ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
4870 (MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL
4871 [MATCH_MP_TAC(MESON[] `t = s /\ (P s <=> Q s) ==> (P s <=> Q t)`) THEN
4872 CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[IN_ELIM_THM]];
4875 [MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] OPEN_IN_SUBSET_TRANS) ORELSE
4876 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] CLOSED_IN_SUBSET_TRANS);
4877 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] OPEN_IN_TRANS) ORELSE
4878 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CLOSED_IN_TRANS)]) THEN
4879 (MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN ORELSE
4880 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN ORELSE ASM_SIMP_TAC[]) THEN
4883 let CONNECTED_MONOTONE_QUOTIENT_PREIMAGE = prove
4884 (`!f:real^M->real^N s t.
4885 f continuous_on s /\ IMAGE f s = t /\
4887 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4888 open_in (subtopology euclidean t) u)) /\
4889 (!y. y IN t ==> connected {x | x IN s /\ f x = y}) /\
4892 REPEAT STRIP_TAC THEN REWRITE_TAC[connected; NOT_EXISTS_THM] THEN
4893 MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `v:real^M->bool`] THEN STRIP_TAC THEN
4894 UNDISCH_TAC `connected(t:real^N->bool)` THEN SIMP_TAC[CONNECTED_OPEN_IN] THEN
4895 MAP_EVERY EXISTS_TAC
4896 [`IMAGE (f:real^M->real^N) (s INTER u)`;
4897 `IMAGE (f:real^M->real^N) (s INTER v)`] THEN
4898 ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN
4900 `IMAGE (f:real^M->real^N) (s INTER u) INTER IMAGE f (s INTER v) = {}`
4902 [REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
4903 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
4904 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN
4905 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[connected]] THEN
4906 MAP_EVERY EXISTS_TAC [`u:real^M->bool`; `v:real^M->bool`] THEN
4909 ONCE_REWRITE_TAC[CONJ_ASSOC] THEN
4910 CONJ_TAC THENL [CONJ_TAC; ASM SET_TAC[]] THEN
4911 FIRST_X_ASSUM(fun th ->
4912 W(MP_TAC o PART_MATCH (rand o rand) th o snd)) THEN
4913 (ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)]) THEN
4914 MATCH_MP_TAC(MESON[]
4915 `{x | x IN s /\ f x IN IMAGE f u} = u /\ open_in top u
4916 ==> open_in top {x | x IN s /\ f x IN IMAGE f u}`) THEN
4917 ASM_SIMP_TAC[OPEN_IN_OPEN_INTER] THEN ASM SET_TAC[]);;
4919 let CONNECTED_MONOTONE_QUOTIENT_PREIMAGE_GEN = prove
4920 (`!f:real^M->real^N s t c.
4923 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4924 open_in (subtopology euclidean t) u)) /\
4925 (!y. y IN t ==> connected {x | x IN s /\ f x = y}) /\
4926 (open_in (subtopology euclidean t) c \/
4927 closed_in (subtopology euclidean t) c) /\
4929 ==> connected {x | x IN s /\ f x IN c}`,
4931 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
4932 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ]
4933 (REWRITE_RULE[CONJ_ASSOC] CONNECTED_MONOTONE_QUOTIENT_PREIMAGE)) THEN
4934 SUBGOAL_THEN `(c:real^N->bool) SUBSET t` ASSUME_TAC THENL
4935 [ASM_MESON_TAC[OPEN_IN_IMP_SUBSET; CLOSED_IN_IMP_SUBSET]; ALL_TAC] THEN
4936 EXISTS_TAC `f:real^M->real^N` THEN REPEAT CONJ_TAC THENL
4937 [FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
4938 QUOTIENT_MAP_IMP_CONTINUOUS_OPEN)) THEN
4939 ASM_REWRITE_TAC[SUBSET_REFL] THEN
4940 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET) THEN
4941 REWRITE_TAC[SUBSET_RESTRICT];
4943 MATCH_MP_TAC QUOTIENT_MAP_RESTRICT THEN
4944 ASM_MESON_TAC[SUBSET_REFL];
4945 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
4946 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN
4947 ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP] THEN
4948 AP_TERM_TAC THEN ASM SET_TAC[]]);;
4950 (* ------------------------------------------------------------------------- *)
4951 (* More properties of open and closed maps. *)
4952 (* ------------------------------------------------------------------------- *)
4954 let OPEN_MAP_RESTRICT = prove
4955 (`!f:real^M->real^N s t t'.
4956 (!u. open_in (subtopology euclidean s) u
4957 ==> open_in (subtopology euclidean t) (IMAGE f u)) /\
4959 ==> !u. open_in (subtopology euclidean {x | x IN s /\ f x IN t'}) u
4960 ==> open_in (subtopology euclidean t') (IMAGE f u)`,
4961 REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN] THEN
4962 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN
4963 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
4964 REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
4965 REPEAT DISCH_TAC THEN X_GEN_TAC `c:real^M->bool` THEN
4966 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN
4967 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);;
4969 let CLOSED_MAP_RESTRICT = prove
4970 (`!f:real^M->real^N s t t'.
4971 (!u. closed_in (subtopology euclidean s) u
4972 ==> closed_in (subtopology euclidean t) (IMAGE f u)) /\
4974 ==> !u. closed_in (subtopology euclidean {x | x IN s /\ f x IN t'}) u
4975 ==> closed_in (subtopology euclidean t') (IMAGE f u)`,
4976 REPEAT GEN_TAC THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN
4977 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN
4978 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
4979 REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
4980 REPEAT DISCH_TAC THEN X_GEN_TAC `c:real^M->bool` THEN
4981 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^M->bool`) THEN
4982 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]);;
4984 let QUOTIENT_MAP_OPEN_MAP_EQ = prove
4985 (`!f:real^M->real^N s t.
4986 IMAGE f s SUBSET t /\
4988 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
4989 open_in (subtopology euclidean t) u))
4990 ==> ((!k. open_in (subtopology euclidean s) k
4991 ==> open_in (subtopology euclidean t) (IMAGE f k)) <=>
4992 (!k. open_in (subtopology euclidean s) k
4993 ==> open_in (subtopology euclidean s)
4994 {x | x IN s /\ f x IN IMAGE f k}))`,
4995 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN
4996 X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN
4997 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
4998 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (f:real^M->real^N) k`) THEN
4999 ASM_SIMP_TAC[IMAGE_SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]);;
5001 let QUOTIENT_MAP_CLOSED_MAP_EQ = prove
5002 (`!f:real^M->real^N s t.
5003 IMAGE f s SUBSET t /\
5005 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
5006 open_in (subtopology euclidean t) u))
5007 ==> ((!k. closed_in (subtopology euclidean s) k
5008 ==> closed_in (subtopology euclidean t) (IMAGE f k)) <=>
5009 (!k. closed_in (subtopology euclidean s) k
5010 ==> closed_in (subtopology euclidean s)
5011 {x | x IN s /\ f x IN IMAGE f k}))`,
5012 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5013 ASM_SIMP_TAC[QUOTIENT_MAP_OPEN_CLOSED] THEN
5014 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN
5015 X_GEN_TAC `k:real^M->bool` THEN STRIP_TAC THEN
5016 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
5017 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (f:real^M->real^N) k`) THEN
5018 ASM_SIMP_TAC[IMAGE_SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]);;
5020 let CLOSED_MAP_IMP_OPEN_MAP = prove
5021 (`!f:real^M->real^N s t.
5023 (!u. closed_in (subtopology euclidean s) u
5024 ==> closed_in (subtopology euclidean t) (IMAGE f u)) /\
5025 (!u. open_in (subtopology euclidean s) u
5026 ==> open_in (subtopology euclidean s)
5027 {x | x IN s /\ f x IN IMAGE f u})
5028 ==> (!u. open_in (subtopology euclidean s) u
5029 ==> open_in (subtopology euclidean t) (IMAGE f u))`,
5030 REPEAT STRIP_TAC THEN
5032 `IMAGE (f:real^M->real^N) u =
5033 t DIFF IMAGE f (s DIFF {x | x IN s /\ f x IN IMAGE f u})`
5035 [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN ASM SET_TAC[];
5036 MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN
5037 FIRST_X_ASSUM MATCH_MP_TAC THEN
5038 MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN
5039 ASM_SIMP_TAC[CLOSED_IN_REFL]]);;
5041 let OPEN_MAP_IMP_CLOSED_MAP = prove
5042 (`!f:real^M->real^N s t.
5044 (!u. open_in (subtopology euclidean s) u
5045 ==> open_in (subtopology euclidean t) (IMAGE f u)) /\
5046 (!u. closed_in (subtopology euclidean s) u
5047 ==> closed_in (subtopology euclidean s)
5048 {x | x IN s /\ f x IN IMAGE f u})
5049 ==> (!u. closed_in (subtopology euclidean s) u
5050 ==> closed_in (subtopology euclidean t) (IMAGE f u))`,
5051 REPEAT STRIP_TAC THEN
5053 `IMAGE (f:real^M->real^N) u =
5054 t DIFF IMAGE f (s DIFF {x | x IN s /\ f x IN IMAGE f u})`
5056 [FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM SET_TAC[];
5057 MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN
5058 FIRST_X_ASSUM MATCH_MP_TAC THEN
5059 MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN
5060 ASM_SIMP_TAC[OPEN_IN_REFL]]);;
5062 let OPEN_MAP_FROM_COMPOSITION_SURJECTIVE = prove
5063 (`!f:real^M->real^N g:real^N->real^P s t u.
5064 f continuous_on s /\ IMAGE f s = t /\ IMAGE g t SUBSET u /\
5065 (!k. open_in (subtopology euclidean s) k
5066 ==> open_in (subtopology euclidean u) (IMAGE (g o f) k))
5067 ==> (!k. open_in (subtopology euclidean t) k
5068 ==> open_in (subtopology euclidean u) (IMAGE g k))`,
5069 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5070 `IMAGE g k = IMAGE ((g:real^N->real^P) o (f:real^M->real^N))
5071 {x | x IN s /\ f(x) IN k}`
5073 [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
5074 REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[];
5075 FIRST_X_ASSUM MATCH_MP_TAC THEN
5076 MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN
5077 EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL]]);;
5079 let CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE = prove
5080 (`!f:real^M->real^N g:real^N->real^P s t u.
5081 f continuous_on s /\ IMAGE f s = t /\ IMAGE g t SUBSET u /\
5082 (!k. closed_in (subtopology euclidean s) k
5083 ==> closed_in (subtopology euclidean u) (IMAGE (g o f) k))
5084 ==> (!k. closed_in (subtopology euclidean t) k
5085 ==> closed_in (subtopology euclidean u) (IMAGE g k))`,
5086 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5087 `IMAGE g k = IMAGE ((g:real^N->real^P) o (f:real^M->real^N))
5088 {x | x IN s /\ f(x) IN k}`
5090 [FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
5091 REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[];
5092 FIRST_X_ASSUM MATCH_MP_TAC THEN
5093 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN
5094 EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[SUBSET_REFL]]);;
5096 let OPEN_MAP_FROM_COMPOSITION_INJECTIVE = prove
5097 (`!f:real^M->real^N g:real^N->real^P s t u.
5098 IMAGE f s SUBSET t /\ IMAGE g t SUBSET u /\
5099 g continuous_on t /\ (!x y. x IN t /\ y IN t /\ g x = g y ==> x = y) /\
5100 (!k. open_in (subtopology euclidean s) k
5101 ==> open_in (subtopology euclidean u) (IMAGE (g o f) k))
5102 ==> (!k. open_in (subtopology euclidean s) k
5103 ==> open_in (subtopology euclidean t) (IMAGE f k))`,
5104 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5105 `IMAGE f k = {x | x IN t /\
5106 g(x) IN IMAGE ((g:real^N->real^P) o (f:real^M->real^N)) k}`
5108 [FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
5109 REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[];
5110 MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE_GEN THEN
5111 EXISTS_TAC `u:real^P->bool` THEN ASM_SIMP_TAC[]]);;
5113 let CLOSED_MAP_FROM_COMPOSITION_INJECTIVE = prove
5114 (`!f:real^M->real^N g:real^N->real^P s t u.
5115 IMAGE f s SUBSET t /\ IMAGE g t SUBSET u /\
5116 g continuous_on t /\ (!x y. x IN t /\ y IN t /\ g x = g y ==> x = y) /\
5117 (!k. closed_in (subtopology euclidean s) k
5118 ==> closed_in (subtopology euclidean u) (IMAGE (g o f) k))
5119 ==> (!k. closed_in (subtopology euclidean s) k
5120 ==> closed_in (subtopology euclidean t) (IMAGE f k))`,
5121 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5122 `IMAGE f k = {x | x IN t /\
5123 g(x) IN IMAGE ((g:real^N->real^P) o (f:real^M->real^N)) k}`
5125 [FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
5126 REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[];
5127 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN
5128 EXISTS_TAC `u:real^P->bool` THEN ASM_SIMP_TAC[]]);;
5130 let OPEN_MAP_CLOSED_SUPERSET_PREIMAGE = prove
5131 (`!f:real^M->real^N s t u w.
5132 (!k. open_in (subtopology euclidean s) k
5133 ==> open_in (subtopology euclidean t) (IMAGE f k)) /\
5134 closed_in (subtopology euclidean s) u /\
5135 w SUBSET t /\ {x | x IN s /\ f(x) IN w} SUBSET u
5136 ==> ?v. closed_in (subtopology euclidean t) v /\
5138 {x | x IN s /\ f(x) IN v} SUBSET u`,
5139 REPEAT STRIP_TAC THEN
5140 EXISTS_TAC `t DIFF IMAGE (f:real^M->real^N) (s DIFF u)` THEN
5141 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
5142 MATCH_MP_TAC CLOSED_IN_DIFF THEN REWRITE_TAC[CLOSED_IN_REFL] THEN
5143 FIRST_X_ASSUM MATCH_MP_TAC THEN
5144 ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL]);;
5146 let OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ = prove
5147 (`!f:real^M->real^N s t.
5149 ==> ((!k. open_in (subtopology euclidean s) k
5150 ==> open_in (subtopology euclidean t) (IMAGE f k)) <=>
5151 (!u w. closed_in (subtopology euclidean s) u /\
5152 w SUBSET t /\ {x | x IN s /\ f(x) IN w} SUBSET u
5153 ==> ?v. closed_in (subtopology euclidean t) v /\
5154 w SUBSET v /\ {x | x IN s /\ f(x) IN v} SUBSET u))`,
5155 REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN
5156 ASM_SIMP_TAC[OPEN_MAP_CLOSED_SUPERSET_PREIMAGE] THEN
5157 FIRST_X_ASSUM(MP_TAC o SPECL
5158 [`s DIFF k:real^M->bool`; `t DIFF IMAGE (f:real^M->real^N) k`]) THEN
5159 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
5160 ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN
5161 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5162 DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN
5163 SUBGOAL_THEN `IMAGE (f:real^M->real^N) k = t DIFF v` SUBST1_TAC THENL
5164 [ASM SET_TAC[]; ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL]]);;
5166 let CLOSED_MAP_OPEN_SUPERSET_PREIMAGE = prove
5167 (`!f:real^M->real^N s t u w.
5168 (!k. closed_in (subtopology euclidean s) k
5169 ==> closed_in (subtopology euclidean t) (IMAGE f k)) /\
5170 open_in (subtopology euclidean s) u /\
5171 w SUBSET t /\ {x | x IN s /\ f(x) IN w} SUBSET u
5172 ==> ?v. open_in (subtopology euclidean t) v /\
5174 {x | x IN s /\ f(x) IN v} SUBSET u`,
5175 REPEAT STRIP_TAC THEN
5176 EXISTS_TAC `t DIFF IMAGE (f:real^M->real^N) (s DIFF u)` THEN
5177 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
5178 MATCH_MP_TAC OPEN_IN_DIFF THEN REWRITE_TAC[OPEN_IN_REFL] THEN
5179 FIRST_X_ASSUM MATCH_MP_TAC THEN
5180 ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL]);;
5182 let CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_EQ = prove
5183 (`!f:real^M->real^N s t.
5185 ==> ((!k. closed_in (subtopology euclidean s) k
5186 ==> closed_in (subtopology euclidean t) (IMAGE f k)) <=>
5187 (!u w. open_in (subtopology euclidean s) u /\
5188 w SUBSET t /\ {x | x IN s /\ f(x) IN w} SUBSET u
5189 ==> ?v. open_in (subtopology euclidean t) v /\
5190 w SUBSET v /\ {x | x IN s /\ f(x) IN v} SUBSET u))`,
5191 REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN
5192 ASM_SIMP_TAC[CLOSED_MAP_OPEN_SUPERSET_PREIMAGE] THEN
5193 FIRST_X_ASSUM(MP_TAC o SPECL
5194 [`s DIFF k:real^M->bool`; `t DIFF IMAGE (f:real^M->real^N) k`]) THEN
5195 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
5196 ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN
5197 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5198 DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN
5199 SUBGOAL_THEN `IMAGE (f:real^M->real^N) k = t DIFF v` SUBST1_TAC THENL
5200 [ASM SET_TAC[]; ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL]]);;
5202 let CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_POINT = prove
5203 (`!f:real^M->real^N s t.
5205 ==> ((!k. closed_in (subtopology euclidean s) k
5206 ==> closed_in (subtopology euclidean t) (IMAGE f k)) <=>
5207 (!u y. open_in (subtopology euclidean s) u /\
5208 y IN t /\ {x | x IN s /\ f(x) = y} SUBSET u
5209 ==> ?v. open_in (subtopology euclidean t) v /\
5210 y IN v /\ {x | x IN s /\ f(x) IN v} SUBSET u))`,
5211 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CLOSED_MAP_OPEN_SUPERSET_PREIMAGE_EQ] THEN
5212 EQ_TAC THEN DISCH_TAC THENL
5213 [MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `y:real^N`] THEN
5215 FIRST_X_ASSUM(MP_TAC o SPECL [`u:real^M->bool`; `{y:real^N}`]) THEN
5216 ASM_REWRITE_TAC[SING_SUBSET; IN_SING];
5217 MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `w:real^N->bool`] THEN
5218 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M->bool`) THEN
5219 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
5220 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
5221 X_GEN_TAC `vv:real^N->real^N->bool` THEN DISCH_TAC THEN
5222 EXISTS_TAC `UNIONS {(vv:real^N->real^N->bool) y | y IN w}` THEN
5224 [MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
5226 REWRITE_TAC[UNIONS_GSPEC] THEN
5227 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5228 REWRITE_TAC[SUBSET; IN_ELIM_THM; RIGHT_AND_EXISTS_THM;
5229 LEFT_IMP_EXISTS_THM] THEN
5230 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN STRIP_TAC THEN
5231 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM SET_TAC[]]]);;
5233 let CONNECTED_OPEN_MONOTONE_PREIMAGE = prove
5234 (`!f:real^M->real^N s t.
5235 f continuous_on s /\ IMAGE f s = t /\
5236 (!c. open_in (subtopology euclidean s) c
5237 ==> open_in (subtopology euclidean t) (IMAGE f c)) /\
5238 (!y. y IN t ==> connected {x | x IN s /\ f x = y})
5239 ==> !c. connected c /\ c SUBSET t
5240 ==> connected {x | x IN s /\ f x IN c}`,
5241 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `c:real^N->bool` o MATCH_MP
5242 (ONCE_REWRITE_RULE[IMP_CONJ] OPEN_MAP_RESTRICT)) THEN
5243 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL
5244 [`f:real^M->real^N`; `{x | x IN s /\ (f:real^M->real^N) x IN c}`]
5245 OPEN_MAP_IMP_QUOTIENT_MAP) THEN
5246 SUBGOAL_THEN `IMAGE f {x | x IN s /\ (f:real^M->real^N) x IN c} = c`
5247 ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN ANTS_TAC THENL
5248 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
5249 CONTINUOUS_ON_SUBSET)) THEN SET_TAC[];
5251 MATCH_MP_TAC CONNECTED_MONOTONE_QUOTIENT_PREIMAGE THEN
5252 MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `c:real^N->bool`] THEN
5253 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
5254 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
5255 CONTINUOUS_ON_SUBSET)) THEN SET_TAC[];
5257 `y IN c ==> {x | x IN {x | x IN s /\ f x IN c} /\ f x = y} =
5258 {x | x IN s /\ f x = y}`] THEN
5261 let CONNECTED_CLOSED_MONOTONE_PREIMAGE = prove
5262 (`!f:real^M->real^N s t.
5263 f continuous_on s /\ IMAGE f s = t /\
5264 (!c. closed_in (subtopology euclidean s) c
5265 ==> closed_in (subtopology euclidean t) (IMAGE f c)) /\
5266 (!y. y IN t ==> connected {x | x IN s /\ f x = y})
5267 ==> !c. connected c /\ c SUBSET t
5268 ==> connected {x | x IN s /\ f x IN c}`,
5269 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o SPEC `c:real^N->bool` o MATCH_MP
5270 (ONCE_REWRITE_RULE[IMP_CONJ] CLOSED_MAP_RESTRICT)) THEN
5271 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPECL
5272 [`f:real^M->real^N`; `{x | x IN s /\ (f:real^M->real^N) x IN c}`]
5273 CLOSED_MAP_IMP_QUOTIENT_MAP) THEN
5274 SUBGOAL_THEN `IMAGE f {x | x IN s /\ (f:real^M->real^N) x IN c} = c`
5275 ASSUME_TAC THENL [ASM SET_TAC[]; ASM_REWRITE_TAC[]] THEN ANTS_TAC THENL
5276 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
5277 CONTINUOUS_ON_SUBSET)) THEN SET_TAC[];
5279 MATCH_MP_TAC CONNECTED_MONOTONE_QUOTIENT_PREIMAGE THEN
5280 MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `c:real^N->bool`] THEN
5281 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
5282 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
5283 CONTINUOUS_ON_SUBSET)) THEN SET_TAC[];
5285 `y IN c ==> {x | x IN {x | x IN s /\ f x IN c} /\ f x = y} =
5286 {x | x IN s /\ f x = y}`] THEN
5289 (* ------------------------------------------------------------------------- *)
5290 (* Proper maps, including projections out of compact sets. *)
5291 (* ------------------------------------------------------------------------- *)
5293 let PROPER_MAP = prove
5294 (`!f:real^M->real^N s t.
5296 ==> ((!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}) <=>
5297 (!k. closed_in (subtopology euclidean s) k
5298 ==> closed_in (subtopology euclidean t) (IMAGE f k)) /\
5299 (!a. a IN t ==> compact {x | x IN s /\ f x = a}))`,
5300 REPEAT STRIP_TAC THEN EQ_TAC THENL
5301 [REPEAT STRIP_TAC THENL
5303 ONCE_REWRITE_TAC[SET_RULE `x = a <=> x IN {a}`] THEN
5304 FIRST_X_ASSUM MATCH_MP_TAC THEN
5305 ASM_REWRITE_TAC[SING_SUBSET; COMPACT_SING]] THEN
5306 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
5307 REWRITE_TAC[CLOSED_IN_LIMPT] THEN
5308 CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `y:real^N`] THEN
5309 REWRITE_TAC[LIMPT_SEQUENTIAL_INJ; IN_DELETE] THEN
5310 REWRITE_TAC[IN_IMAGE; LEFT_AND_EXISTS_THM; SKOLEM_THM] THEN
5311 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
5312 REWRITE_TAC[GSYM CONJ_ASSOC; FORALL_AND_THM] THEN
5313 ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN
5314 REWRITE_TAC[UNWIND_THM2; FUN_EQ_THM] THEN
5315 DISCH_THEN(X_CHOOSE_THEN `x:num->real^M` STRIP_ASSUME_TAC) THEN
5317 `~(INTERS {{a | a IN k /\
5318 (f:real^M->real^N) a IN
5319 (y INSERT IMAGE (\i. f(x(n + i))) (:num))} |
5322 [MATCH_MP_TAC COMPACT_FIP THEN CONJ_TAC THENL
5323 [REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `n:num` THEN
5324 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSED_IN_CLOSED]) THEN
5325 DISCH_THEN(X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC) THEN
5326 ASM_REWRITE_TAC[SET_RULE
5327 `{x | x IN s INTER k /\ P x} = k INTER {x | x IN s /\ P x}`] THEN
5328 MATCH_MP_TAC CLOSED_INTER_COMPACT THEN ASM_REWRITE_TAC[] THEN
5329 FIRST_X_ASSUM MATCH_MP_TAC THEN
5330 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5331 MATCH_MP_TAC COMPACT_SEQUENCE_WITH_LIMIT THEN
5332 FIRST_ASSUM(MP_TAC o SPEC `n:num` o MATCH_MP SEQ_OFFSET) THEN
5333 REWRITE_TAC[ADD_SYM];
5334 REWRITE_TAC[SIMPLE_IMAGE; FORALL_FINITE_SUBSET_IMAGE] THEN
5335 X_GEN_TAC `i:num->bool` THEN STRIP_TAC THEN
5336 FIRST_ASSUM(MP_TAC o ISPEC `\n:num. n` o MATCH_MP
5337 UPPER_BOUND_FINITE_SET) THEN
5338 REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `m:num`) THEN
5339 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; INTERS_IMAGE; IN_ELIM_THM] THEN
5340 EXISTS_TAC `(x:num->real^M) m` THEN
5341 X_GEN_TAC `p:num` THEN DISCH_TAC THEN
5342 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5343 REWRITE_TAC[IN_INSERT; IN_IMAGE; IN_UNIV] THEN DISJ2_TAC THEN
5344 EXISTS_TAC `m - p:num` THEN
5345 ASM_MESON_TAC[ARITH_RULE `p <= m ==> p + m - p:num = m`]];
5346 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN MATCH_MP_TAC MONO_EXISTS THEN
5347 X_GEN_TAC `x:real^M` THEN
5348 REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN
5349 DISCH_THEN(fun th -> LABEL_TAC "*" th THEN MP_TAC(SPEC `0` th)) THEN
5350 REWRITE_TAC[ADD_CLAUSES; IN_INSERT; IN_IMAGE; IN_UNIV] THEN
5351 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (DISJ_CASES_THEN MP_TAC)) THEN
5352 ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `i:num`) THEN
5353 REMOVE_THEN "*" (MP_TAC o SPEC `i + 1`) THEN
5354 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
5355 ASM_REWRITE_TAC[IN_INSERT; IN_IMAGE; IN_UNIV] THEN ARITH_TAC];
5356 STRIP_TAC THEN X_GEN_TAC `k:real^N->bool` THEN STRIP_TAC THEN
5357 REWRITE_TAC[COMPACT_EQ_HEINE_BOREL] THEN
5358 X_GEN_TAC `c:(real^M->bool)->bool` THEN STRIP_TAC THEN
5361 ==> ?g. g SUBSET c /\ FINITE g /\
5362 {x | x IN s /\ (f:real^M->real^N) x = a} SUBSET UNIONS g`
5364 [X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN UNDISCH_THEN
5365 `!a. a IN t ==> compact {x | x IN s /\ (f:real^M->real^N) x = a}`
5366 (MP_TAC o SPEC `a:real^N`) THEN
5367 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[COMPACT_EQ_HEINE_BOREL]] THEN
5368 DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
5369 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
5370 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
5371 X_GEN_TAC `uu:real^N->(real^M->bool)->bool` THEN
5372 DISCH_THEN(LABEL_TAC "*")] THEN
5375 ==> ?v. open v /\ a IN v /\
5376 {x | x IN s /\ (f:real^M->real^N) x IN v} SUBSET UNIONS(uu a)`
5378 [REPEAT STRIP_TAC THEN
5380 `!k. closed_in (subtopology euclidean s) k
5381 ==> closed_in (subtopology euclidean t)
5382 (IMAGE (f:real^M->real^N) k)`
5383 (MP_TAC o SPEC `(s:real^M->bool) DIFF UNIONS(uu(a:real^N))`) THEN
5384 SIMP_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ANTS_TAC THENL
5385 [CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
5386 REWRITE_TAC[SET_RULE `s DIFF (s DIFF t) = s INTER t`] THEN
5387 MATCH_MP_TAC OPEN_IN_OPEN_INTER THEN
5388 MATCH_MP_TAC OPEN_UNIONS THEN ASM SET_TAC[];
5389 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5390 REWRITE_TAC[OPEN_IN_OPEN] THEN MATCH_MP_TAC MONO_EXISTS THEN
5391 X_GEN_TAC `v:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5392 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N`)) THEN
5393 ASM_REWRITE_TAC[] THEN REPEAT
5394 ((ANTS_TAC THENL [ASM SET_TAC[]; DISCH_TAC]) ORELSE STRIP_TAC)
5395 THENL [ALL_TAC; ASM SET_TAC[]] THEN
5396 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
5397 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM SET_TAC[]];
5398 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
5399 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
5400 X_GEN_TAC `vv:real^N->(real^N->bool)` THEN
5401 DISCH_THEN(LABEL_TAC "+")] THEN
5402 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL]) THEN
5403 DISCH_THEN(MP_TAC o SPEC `IMAGE (vv:real^N->(real^N->bool)) k`) THEN
5404 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
5405 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r ==> s <=> q /\ p ==> r ==> s`] THEN
5406 REWRITE_TAC[FORALL_FINITE_SUBSET_IMAGE] THEN
5407 X_GEN_TAC `j:real^N->bool` THEN REPEAT STRIP_TAC THEN
5408 EXISTS_TAC `UNIONS(IMAGE (uu:real^N->(real^M->bool)->bool) j)` THEN
5409 REPEAT CONJ_TAC THENL
5411 ASM_SIMP_TAC[FINITE_UNIONS; FORALL_IN_IMAGE; FINITE_IMAGE] THEN
5413 REWRITE_TAC[UNIONS_IMAGE; SUBSET; IN_UNIONS; IN_ELIM_THM] THEN
5416 let COMPACT_CONTINUOUS_IMAGE_EQ = prove
5417 (`!f:real^M->real^N s.
5418 (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
5419 ==> (f continuous_on s <=>
5420 !t. compact t /\ t SUBSET s ==> compact(IMAGE f t))`,
5421 REPEAT STRIP_TAC THEN EQ_TAC THENL
5422 [MESON_TAC[COMPACT_CONTINUOUS_IMAGE; CONTINUOUS_ON_SUBSET]; DISCH_TAC] THEN
5423 FIRST_X_ASSUM(X_CHOOSE_TAC `g:real^N->real^M` o
5424 GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
5425 REWRITE_TAC[CONTINUOUS_ON_CLOSED] THEN
5426 X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN
5427 MP_TAC(ISPECL [`g:real^N->real^M`; `IMAGE (f:real^M->real^N) s`;
5428 `s:real^M->bool`] PROPER_MAP) THEN
5429 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5430 MATCH_MP_TAC(TAUT `(q ==> s) /\ p ==> (p <=> q /\ r) ==> s`) THEN
5431 REPEAT STRIP_TAC THENL
5433 `{x | x IN s /\ (f:real^M->real^N) x IN u} = IMAGE g u`
5434 (fun th -> ASM_MESON_TAC[th]);
5436 `{x | x IN IMAGE f s /\ (g:real^N->real^M) x IN k} = IMAGE f k`
5437 (fun th -> ASM_SIMP_TAC[th])] THEN
5438 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN ASM SET_TAC[]);;
5440 let PROPER_MAP_FROM_COMPACT = prove
5441 (`!f:real^M->real^N s k.
5442 f continuous_on s /\ IMAGE f s SUBSET t /\ compact s /\
5443 closed_in (subtopology euclidean t) k
5444 ==> compact {x | x IN s /\ f x IN k}`,
5445 REPEAT STRIP_TAC THEN
5446 MATCH_MP_TAC CLOSED_IN_COMPACT THEN EXISTS_TAC `s:real^M->bool` THEN
5447 ASM_MESON_TAC[CONTINUOUS_CLOSED_IN_PREIMAGE_GEN]);;
5449 let PROPER_MAP_COMPOSE = prove
5450 (`!f:real^M->real^N g:real^N->real^P s t u.
5451 IMAGE f s SUBSET t /\
5452 (!k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}) /\
5453 (!k. k SUBSET u /\ compact k ==> compact {x | x IN t /\ g x IN k})
5454 ==> !k. k SUBSET u /\ compact k
5455 ==> compact {x | x IN s /\ (g o f) x IN k}`,
5456 REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN
5457 FIRST_X_ASSUM(MP_TAC o SPEC `k:real^P->bool`) THEN
5458 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
5459 FIRST_X_ASSUM(MP_TAC o SPEC `{x | x IN t /\ (g:real^N->real^P) x IN k}`) THEN
5460 ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP] THEN
5461 AP_TERM_TAC THEN ASM SET_TAC[]);;
5463 let PROPER_MAP_FROM_COMPOSITION_LEFT = prove
5464 (`!f:real^M->real^N g:real^N->real^P s t u.
5465 f continuous_on s /\ IMAGE f s = t /\
5466 g continuous_on t /\ IMAGE g t SUBSET u /\
5467 (!k. k SUBSET u /\ compact k
5468 ==> compact {x | x IN s /\ (g o f) x IN k})
5469 ==> !k. k SUBSET u /\ compact k ==> compact {x | x IN t /\ g x IN k}`,
5470 REWRITE_TAC[o_THM] THEN REPEAT STRIP_TAC THEN
5471 FIRST_X_ASSUM(MP_TAC o SPEC `k:real^P->bool`) THEN ASM_REWRITE_TAC[] THEN
5472 DISCH_THEN(MP_TAC o ISPEC `f:real^M->real^N` o MATCH_MP
5473 (REWRITE_RULE[IMP_CONJ_ALT] COMPACT_CONTINUOUS_IMAGE)) THEN
5475 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
5476 CONTINUOUS_ON_SUBSET)) THEN SET_TAC[];
5477 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]]);;
5479 let PROPER_MAP_FROM_COMPOSITION_RIGHT = prove
5480 (`!f:real^M->real^N g:real^N->real^P s t u.
5481 f continuous_on s /\ IMAGE f s SUBSET t /\
5482 g continuous_on t /\ IMAGE g t SUBSET u /\
5483 (!k. k SUBSET u /\ compact k
5484 ==> compact {x | x IN s /\ (g o f) x IN k})
5485 ==> !k. k SUBSET t /\ compact k ==> compact {x | x IN s /\ f x IN k}`,
5487 (`!s t. closed_in (subtopology euclidean s) t ==> compact s ==> compact t`,
5488 MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET;
5489 CLOSED_IN_CLOSED_EQ]) in
5490 REWRITE_TAC[o_THM] THEN REPEAT STRIP_TAC THEN
5491 FIRST_X_ASSUM(MP_TAC o SPEC `IMAGE (g:real^N->real^P) k`) THEN
5493 [CONJ_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE] THEN
5494 ASM_MESON_TAC[CONTINUOUS_ON_SUBSET];
5495 MATCH_MP_TAC lemma THEN
5496 MATCH_MP_TAC CLOSED_IN_SUBSET_TRANS THEN
5497 EXISTS_TAC `s:real^M->bool` THEN
5498 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
5499 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_GEN THEN
5500 EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN
5501 MATCH_MP_TAC CLOSED_SUBSET THEN ASM_SIMP_TAC[COMPACT_IMP_CLOSED]]);;
5503 let PROPER_MAP_FSTCART = prove
5504 (`!s:real^M->bool t:real^N->bool k.
5505 compact t /\ k SUBSET s /\ compact k
5506 ==> compact {z | z IN s PCROSS t /\ fstcart z IN k}`,
5507 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5508 `{z | z IN s PCROSS t /\ fstcart z IN k} =
5509 (k:real^M->bool) PCROSS (t:real^N->bool)`
5510 (fun th -> ASM_SIMP_TAC[th; COMPACT_PCROSS]) THEN
5511 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM;
5512 PASTECART_IN_PCROSS; FSTCART_PASTECART] THEN
5515 let CLOSED_MAP_FSTCART = prove
5516 (`!s:real^M->bool t:real^N->bool c.
5517 compact t /\ closed_in (subtopology euclidean (s PCROSS t)) c
5518 ==> closed_in (subtopology euclidean s) (IMAGE fstcart c)`,
5519 REPEAT STRIP_TAC THEN
5520 MP_TAC(ISPECL [`fstcart:real^(M,N)finite_sum->real^M`;
5521 `(s:real^M->bool) PCROSS (t:real^N->bool)`;
5524 ASM_SIMP_TAC[PROPER_MAP_FSTCART; IMAGE_FSTCART_PCROSS] THEN
5527 let PROPER_MAP_SNDCART = prove
5528 (`!s:real^M->bool t:real^N->bool k.
5529 compact s /\ k SUBSET t /\ compact k
5530 ==> compact {z | z IN s PCROSS t /\ sndcart z IN k}`,
5531 REPEAT STRIP_TAC THEN SUBGOAL_THEN
5532 `{z | z IN s PCROSS t /\ sndcart z IN k} =
5533 (s:real^M->bool) PCROSS (k:real^N->bool)`
5534 (fun th -> ASM_SIMP_TAC[th; COMPACT_PCROSS]) THEN
5535 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM;
5536 PASTECART_IN_PCROSS; SNDCART_PASTECART] THEN
5539 let CLOSED_MAP_SNDCART = prove
5540 (`!s:real^M->bool t:real^N->bool c.
5541 compact s /\ closed_in (subtopology euclidean (s PCROSS t)) c
5542 ==> closed_in (subtopology euclidean t) (IMAGE sndcart c)`,
5543 REPEAT STRIP_TAC THEN
5544 MP_TAC(ISPECL [`sndcart:real^(M,N)finite_sum->real^N`;
5545 `(s:real^M->bool) PCROSS (t:real^N->bool)`;
5548 ASM_SIMP_TAC[PROPER_MAP_SNDCART; IMAGE_SNDCART_PCROSS] THEN
5551 let CLOSED_IN_COMPACT_PROJECTION = prove
5552 (`!s:real^M->bool t:real^N->bool u.
5553 compact s /\ closed_in (subtopology euclidean (s PCROSS t)) u
5554 ==> closed_in (subtopology euclidean t)
5555 {y | ?x. x IN s /\ pastecart x y IN u}`,
5556 REPEAT GEN_TAC THEN DISCH_TAC THEN
5557 FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_MAP_SNDCART) THEN
5558 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
5559 FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET o CONJUNCT2) THEN
5560 REWRITE_TAC[EXTENSION; SUBSET; IN_IMAGE; FORALL_PASTECART; EXISTS_PASTECART;
5561 PASTECART_IN_PCROSS; IN_ELIM_THM; SNDCART_PASTECART] THEN
5564 let CLOSED_COMPACT_PROJECTION = prove
5565 (`!s:real^M->bool t:real^(M,N)finite_sum->bool.
5566 compact s /\ closed t ==> closed {y | ?x. x IN s /\ pastecart x y IN t}`,
5567 REPEAT STRIP_TAC THEN
5569 `{y | ?x:real^M. x IN s /\ pastecart x y IN t} =
5570 {y | ?x. x IN s /\ pastecart x y IN ((s PCROSS (:real^N)) INTER t)}`
5572 [REWRITE_TAC[PASTECART_IN_PCROSS; IN_UNIV; IN_INTER] THEN SET_TAC[];
5573 MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN
5574 EXISTS_TAC `(:real^N)` THEN REWRITE_TAC[CLOSED_UNIV] THEN
5575 MATCH_MP_TAC CLOSED_IN_COMPACT_PROJECTION THEN
5576 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_SUBSET THEN
5577 ASM_SIMP_TAC[CLOSED_INTER; CLOSED_UNIV; CLOSED_PCROSS; COMPACT_IMP_CLOSED;
5580 let TUBE_LEMMA = prove
5581 (`!s:real^M->bool t:real^N->bool u a.
5582 compact s /\ ~(s = {}) /\ {pastecart x a | x IN s} SUBSET u /\
5583 open_in(subtopology euclidean (s PCROSS t)) u
5584 ==> ?v. open_in (subtopology euclidean t) v /\ a IN v /\
5585 (s PCROSS v) SUBSET u`,
5586 REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS] THEN
5587 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ] THEN
5588 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
5589 REPEAT STRIP_TAC THEN
5590 FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT; PCROSS]
5591 CLOSED_IN_COMPACT_PROJECTION)) THEN
5592 ASM_REWRITE_TAC[IN_ELIM_PASTECART_THM; IN_DIFF] THEN
5593 REWRITE_TAC[GSYM CONJ_ASSOC] THEN MATCH_MP_TAC(MESON[]
5594 `(closed_in top t ==> s DIFF (s DIFF t) = t) /\
5595 s DIFF t SUBSET s /\ P(s DIFF t)
5597 ==> ?v. v SUBSET s /\ closed_in top (s DIFF v) /\ P v`) THEN
5598 REWRITE_TAC[SET_RULE `s DIFF (s DIFF t) = t <=> t SUBSET s`] THEN
5599 REWRITE_TAC[SUBSET_DIFF] THEN
5600 SIMP_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
5601 REWRITE_TAC[IN_DIFF; IN_ELIM_THM] THEN
5602 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
5603 CONJ_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
5604 REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET])) THEN
5605 REWRITE_TAC[FORALL_IN_GSPEC; IN_SING; FORALL_PASTECART] THEN
5606 REWRITE_TAC[IN_ELIM_PASTECART_THM] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY]);;
5608 let TUBE_LEMMA_GEN = prove
5609 (`!s t t' u:real^(M,N)finite_sum->bool.
5610 compact s /\ ~(s = {}) /\ t SUBSET t' /\
5611 s PCROSS t SUBSET u /\
5612 open_in (subtopology euclidean (s PCROSS t')) u
5613 ==> ?v. open_in (subtopology euclidean t') v /\
5615 s PCROSS v SUBSET u`,
5616 REPEAT STRIP_TAC THEN
5618 `!a. a IN t ==> ?v. open_in (subtopology euclidean t') v /\ a IN v /\
5619 (s:real^M->bool) PCROSS (v:real^N->bool) SUBSET u`
5621 [REPEAT STRIP_TAC THEN MATCH_MP_TAC TUBE_LEMMA THEN
5622 ASM_REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN REPEAT STRIP_TAC THEN
5623 REPEAT STRIP_TAC THEN
5624 FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
5625 ASM_REWRITE_TAC[PASTECART_IN_PCROSS];
5626 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
5627 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
5628 X_GEN_TAC `vv:real^N->real^N->bool` THEN DISCH_TAC THEN
5629 EXISTS_TAC `UNIONS (IMAGE (vv:real^N->real^N->bool) t)` THEN
5630 ASM_SIMP_TAC[OPEN_IN_UNIONS; FORALL_IN_IMAGE] THEN
5631 REWRITE_TAC[SUBSET; UNIONS_IMAGE; IN_ELIM_THM; FORALL_IN_PCROSS] THEN
5632 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5633 MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^N`] THEN
5634 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_TAC `c:real^N`)) THEN
5635 FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N`) THEN
5636 ASM_REWRITE_TAC[SUBSET; FORALL_IN_PCROSS] THEN ASM SET_TAC[]]);;
5638 (* ------------------------------------------------------------------------- *)
5639 (* Pasting functions together on open sets. *)
5640 (* ------------------------------------------------------------------------- *)
5642 let PASTING_LEMMA = prove
5643 (`!f:A->real^M->real^N g t s k.
5645 ==> open_in (subtopology euclidean s) (t i) /\
5646 (f i) continuous_on (t i)) /\
5647 (!i j x. i IN k /\ j IN k /\ x IN s INTER t i INTER t j
5648 ==> f i x = f j x) /\
5649 (!x. x IN s ==> ?j. j IN k /\ x IN t j /\ g x = f j x)
5650 ==> g continuous_on s`,
5651 REPEAT GEN_TAC THEN REWRITE_TAC[CONTINUOUS_OPEN_IN_PREIMAGE_EQ] THEN
5652 STRIP_TAC THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN
5654 `{x | x IN s /\ g x IN u} =
5655 UNIONS {{x | x IN (t i) /\ ((f:A->real^M->real^N) i x) IN u} |
5658 [SUBGOAL_THEN `!i. i IN k ==> ((t:A->real^M->bool) i) SUBSET s`
5660 [ASM_MESON_TAC[OPEN_IN_SUBSET; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY];
5661 REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]];
5662 MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
5663 ASM_MESON_TAC[OPEN_IN_TRANS]]);;
5665 let PASTING_LEMMA_EXISTS = prove
5666 (`!f:A->real^M->real^N t s k.
5667 s SUBSET UNIONS {t i | i IN k} /\
5669 ==> open_in (subtopology euclidean s) (t i) /\
5670 (f i) continuous_on (t i)) /\
5671 (!i j x. i IN k /\ j IN k /\ x IN s INTER t i INTER t j
5673 ==> ?g. g continuous_on s /\
5674 (!x i. i IN k /\ x IN s INTER t i ==> g x = f i x)`,
5675 REPEAT STRIP_TAC THEN
5676 EXISTS_TAC `\x. (f:A->real^M->real^N)(@i. i IN k /\ x IN t i) x` THEN
5677 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC PASTING_LEMMA THEN
5678 MAP_EVERY EXISTS_TAC
5679 [`f:A->real^M->real^N`; `t:A->real^M->bool`; `k:A->bool`] THEN
5682 let CONTINUOUS_ON_UNION_LOCAL_OPEN = prove
5683 (`!f:real^M->real^N s.
5684 open_in (subtopology euclidean (s UNION t)) s /\
5685 open_in (subtopology euclidean (s UNION t)) t /\
5686 f continuous_on s /\ f continuous_on t
5687 ==> f continuous_on (s UNION t)`,
5688 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
5689 [`\i:(real^M->bool). (f:real^M->real^N)`; `f:real^M->real^N`;
5690 `\i:(real^M->bool). i`; `s UNION t:real^M->bool`; `{s:real^M->bool,t}`]
5691 PASTING_LEMMA) THEN DISCH_THEN MATCH_MP_TAC THEN
5692 ASM_REWRITE_TAC[FORALL_IN_INSERT; EXISTS_IN_INSERT; NOT_IN_EMPTY] THEN
5693 REWRITE_TAC[IN_UNION]);;
5695 let CONTINUOUS_ON_UNION_OPEN = prove
5696 (`!f s t. open s /\ open t /\ f continuous_on s /\ f continuous_on t
5697 ==> f continuous_on (s UNION t)`,
5698 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_UNION_LOCAL_OPEN THEN
5699 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC OPEN_OPEN_IN_TRANS THEN
5700 ASM_SIMP_TAC[OPEN_UNION] THEN SET_TAC[]);;
5702 let CONTINUOUS_ON_CASES_LOCAL_OPEN = prove
5703 (`!P f g:real^M->real^N s t.
5704 open_in (subtopology euclidean (s UNION t)) s /\
5705 open_in (subtopology euclidean (s UNION t)) t /\
5706 f continuous_on s /\ g continuous_on t /\
5707 (!x. x IN s /\ ~P x \/ x IN t /\ P x ==> f x = g x)
5708 ==> (\x. if P x then f x else g x) continuous_on (s UNION t)`,
5709 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_UNION_LOCAL_OPEN THEN
5710 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL
5711 [EXISTS_TAC `f:real^M->real^N`; EXISTS_TAC `g:real^M->real^N`] THEN
5712 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);;
5714 let CONTINUOUS_ON_CASES_OPEN = prove
5718 f continuous_on s /\
5719 g continuous_on t /\
5720 (!x. x IN s /\ ~P x \/ x IN t /\ P x ==> f x = g x)
5721 ==> (\x. if P x then f x else g x) continuous_on s UNION t`,
5722 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL_OPEN THEN
5723 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC OPEN_OPEN_IN_TRANS THEN
5724 ASM_SIMP_TAC[OPEN_UNION] THEN SET_TAC[]);;
5726 (* ------------------------------------------------------------------------- *)
5727 (* Likewise on closed sets, with a finiteness assumption. *)
5728 (* ------------------------------------------------------------------------- *)
5730 let PASTING_LEMMA_CLOSED = prove
5731 (`!f:A->real^M->real^N g t s k.
5734 ==> closed_in (subtopology euclidean s) (t i) /\
5735 (f i) continuous_on (t i)) /\
5736 (!i j x. i IN k /\ j IN k /\ x IN s INTER t i INTER t j
5737 ==> f i x = f j x) /\
5738 (!x. x IN s ==> ?j. j IN k /\ x IN t j /\ g x = f j x)
5739 ==> g continuous_on s`,
5740 REPEAT GEN_TAC THEN REWRITE_TAC[CONTINUOUS_CLOSED_IN_PREIMAGE_EQ] THEN
5741 STRIP_TAC THEN X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN
5743 `{x | x IN s /\ g x IN u} =
5744 UNIONS {{x | x IN (t i) /\ ((f:A->real^M->real^N) i x) IN u} |
5747 [SUBGOAL_THEN `!i. i IN k ==> ((t:A->real^M->bool) i) SUBSET s`
5749 [ASM_MESON_TAC[CLOSED_IN_SUBSET; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY];
5750 REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[]];
5751 MATCH_MP_TAC CLOSED_IN_UNIONS THEN
5752 ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN
5753 ASM_MESON_TAC[CLOSED_IN_TRANS]]);;
5755 let PASTING_LEMMA_EXISTS_CLOSED = prove
5756 (`!f:A->real^M->real^N t s k.
5758 s SUBSET UNIONS {t i | i IN k} /\
5760 ==> closed_in (subtopology euclidean s) (t i) /\
5761 (f i) continuous_on (t i)) /\
5762 (!i j x. i IN k /\ j IN k /\ x IN s INTER t i INTER t j
5764 ==> ?g. g continuous_on s /\
5765 (!x i. i IN k /\ x IN s INTER t i ==> g x = f i x)`,
5766 REPEAT STRIP_TAC THEN
5767 EXISTS_TAC `\x. (f:A->real^M->real^N)(@i. i IN k /\ x IN t i) x` THEN
5768 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
5769 MATCH_MP_TAC PASTING_LEMMA_CLOSED THEN
5770 MAP_EVERY EXISTS_TAC
5771 [`f:A->real^M->real^N`; `t:A->real^M->bool`; `k:A->bool`] THEN
5774 (* ------------------------------------------------------------------------- *)
5775 (* Closure of halflines, halfspaces and hyperplanes. *)
5776 (* ------------------------------------------------------------------------- *)
5778 let LIM_LIFT_DOT = prove
5779 (`!f:real^M->real^N a.
5780 (f --> l) net ==> ((lift o (\y. a dot f(y))) --> lift(a dot l)) net`,
5781 REPEAT GEN_TAC THEN ASM_CASES_TAC `a = vec 0:real^N` THENL
5782 [ASM_REWRITE_TAC[DOT_LZERO; LIFT_NUM; o_DEF; LIM_CONST]; ALL_TAC] THEN
5783 REWRITE_TAC[LIM] THEN MATCH_MP_TAC MONO_OR THEN REWRITE_TAC[] THEN
5784 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
5785 FIRST_X_ASSUM(MP_TAC o SPEC `e / norm(a:real^N)`) THEN
5786 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_LT_RDIV_EQ] THEN
5787 REWRITE_TAC[dist; o_THM; GSYM LIFT_SUB; GSYM DOT_RSUB; NORM_LIFT] THEN
5788 ONCE_REWRITE_TAC[DOT_SYM] THEN
5789 MESON_TAC[NORM_CAUCHY_SCHWARZ_ABS; REAL_MUL_SYM; REAL_LET_TRANS]);;
5791 let CONTINUOUS_AT_LIFT_DOT = prove
5792 (`!a:real^N x. (lift o (\y. a dot y)) continuous at x`,
5793 REPEAT GEN_TAC THEN REWRITE_TAC[CONTINUOUS_AT; o_THM] THEN
5794 MATCH_MP_TAC LIM_LIFT_DOT THEN REWRITE_TAC[LIM_AT] THEN MESON_TAC[]);;
5796 let CONTINUOUS_ON_LIFT_DOT = prove
5797 (`!s. (lift o (\y. a dot y)) continuous_on s`,
5798 SIMP_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON; CONTINUOUS_AT_LIFT_DOT]);;
5800 let CLOSED_INTERVAL_LEFT = prove
5803 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> x$i <= b$i}`,
5804 REWRITE_TAC[CLOSED_LIMPT; LIMPT_APPROACHABLE; IN_ELIM_THM] THEN
5805 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
5806 FIRST_X_ASSUM(MP_TAC o SPEC `(x:real^N)$i - (b:real^N)$i`) THEN
5807 ASM_REWRITE_TAC[REAL_SUB_LT] THEN
5808 DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN
5809 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5810 REWRITE_TAC[dist; REAL_NOT_LT] THEN
5811 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs((z - x :real^N)$i)` THEN
5812 ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN
5813 ASM_SIMP_TAC[VECTOR_SUB_COMPONENT] THEN
5814 ASM_SIMP_TAC[REAL_ARITH `z <= b /\ b < x ==> x - b <= abs(z - x)`]);;
5816 let CLOSED_INTERVAL_RIGHT = prove
5819 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= x$i}`,
5820 REWRITE_TAC[CLOSED_LIMPT; LIMPT_APPROACHABLE; IN_ELIM_THM] THEN
5821 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
5822 FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N)$i - (x:real^N)$i`) THEN
5823 ASM_REWRITE_TAC[REAL_SUB_LT] THEN
5824 DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN
5825 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5826 REWRITE_TAC[dist; REAL_NOT_LT] THEN
5827 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs((z - x :real^N)$i)` THEN
5828 ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN
5829 ASM_SIMP_TAC[VECTOR_SUB_COMPONENT] THEN
5830 ASM_SIMP_TAC[REAL_ARITH `x < a /\ a <= z ==> a - x <= abs(z - x)`]);;
5832 let CLOSED_HALFSPACE_LE = prove
5833 (`!a:real^N b. closed {x | a dot x <= b}`,
5835 MP_TAC(ISPEC `(:real^N)` CONTINUOUS_ON_LIFT_DOT) THEN
5836 REWRITE_TAC[CONTINUOUS_ON_CLOSED; GSYM CLOSED_IN; SUBTOPOLOGY_UNIV] THEN
5837 DISCH_THEN(MP_TAC o SPEC
5838 `IMAGE lift {r | ?x:real^N. (a dot x = r) /\ r <= b}`) THEN
5841 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
5842 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; IN_UNIV] THEN
5843 REWRITE_TAC[o_DEF] THEN MESON_TAC[LIFT_DROP]] THEN
5844 REWRITE_TAC[CLOSED_IN_CLOSED] THEN
5845 EXISTS_TAC `{x | !i. 1 <= i /\ i <= dimindex(:1)
5846 ==> (x:real^1)$i <= (lift b)$i}` THEN
5847 REWRITE_TAC[CLOSED_INTERVAL_LEFT] THEN
5848 SIMP_TAC[EXTENSION; IN_IMAGE; IN_UNIV; IN_ELIM_THM; IN_INTER;
5849 VEC_COMPONENT; DIMINDEX_1; LAMBDA_BETA; o_THM] THEN
5850 SIMP_TAC[ARITH_RULE `1 <= i /\ i <= 1 <=> (i = 1)`] THEN
5851 REWRITE_TAC[GSYM drop; LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
5852 MESON_TAC[LIFT_DROP]);;
5854 let CLOSED_HALFSPACE_GE = prove
5855 (`!a:real^N b. closed {x | a dot x >= b}`,
5856 REWRITE_TAC[REAL_ARITH `a >= b <=> --a <= --b`] THEN
5857 REWRITE_TAC[GSYM DOT_LNEG; CLOSED_HALFSPACE_LE]);;
5859 let CLOSED_HYPERPLANE = prove
5860 (`!a b. closed {x | a dot x = b}`,
5861 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN
5862 REWRITE_TAC[REAL_ARITH `b <= a dot x <=> a dot x >= b`] THEN
5863 REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN
5864 SIMP_TAC[CLOSED_INTER; CLOSED_HALFSPACE_LE; CLOSED_HALFSPACE_GE]);;
5866 let CLOSED_STANDARD_HYPERPLANE = prove
5867 (`!k a. closed {x:real^N | x$k = a}`,
5869 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5871 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5872 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CLOSED_HYPERPLANE) THEN
5873 ASM_SIMP_TAC[DOT_BASIS]);;
5875 let CLOSED_HALFSPACE_COMPONENT_LE = prove
5876 (`!a k. closed {x:real^N | x$k <= a}`,
5878 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5880 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5881 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CLOSED_HALFSPACE_LE) THEN
5882 ASM_SIMP_TAC[DOT_BASIS]);;
5884 let CLOSED_HALFSPACE_COMPONENT_GE = prove
5885 (`!a k. closed {x:real^N | x$k >= a}`,
5887 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5889 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5890 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CLOSED_HALFSPACE_GE) THEN
5891 ASM_SIMP_TAC[DOT_BASIS]);;
5893 (* ------------------------------------------------------------------------- *)
5894 (* Openness of halfspaces. *)
5895 (* ------------------------------------------------------------------------- *)
5897 let OPEN_HALFSPACE_LT = prove
5898 (`!a b. open {x | a dot x < b}`,
5899 REWRITE_TAC[GSYM REAL_NOT_LE] THEN
5900 REWRITE_TAC[SET_RULE `{x | ~p x} = UNIV DIFF {x | p x}`] THEN
5901 REWRITE_TAC[GSYM closed; GSYM real_ge; CLOSED_HALFSPACE_GE]);;
5903 let OPEN_HALFSPACE_COMPONENT_LT = prove
5904 (`!a k. open {x:real^N | x$k < a}`,
5906 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5908 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5909 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] OPEN_HALFSPACE_LT) THEN
5910 ASM_SIMP_TAC[DOT_BASIS]);;
5912 let OPEN_HALFSPACE_GT = prove
5913 (`!a b. open {x | a dot x > b}`,
5914 REWRITE_TAC[REAL_ARITH `x > y <=> ~(x <= y)`] THEN
5915 REWRITE_TAC[SET_RULE `{x | ~p x} = UNIV DIFF {x | p x}`] THEN
5916 REWRITE_TAC[GSYM closed; CLOSED_HALFSPACE_LE]);;
5918 let OPEN_HALFSPACE_COMPONENT_GT = prove
5919 (`!a k. open {x:real^N | x$k > a}`,
5921 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5923 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5924 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] OPEN_HALFSPACE_GT) THEN
5925 ASM_SIMP_TAC[DOT_BASIS]);;
5927 let OPEN_POSITIVE_MULTIPLES = prove
5928 (`!s:real^N->bool. open s ==> open {c % x | &0 < c /\ x IN s}`,
5929 REWRITE_TAC[open_def; FORALL_IN_GSPEC] THEN GEN_TAC THEN DISCH_TAC THEN
5930 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`] THEN STRIP_TAC THEN
5931 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
5932 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
5933 EXISTS_TAC `c * e:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
5934 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
5935 FIRST_X_ASSUM(MP_TAC o SPEC `inv(c) % y:real^N`) THEN ANTS_TAC THENL
5936 [SUBGOAL_THEN `x:real^N = inv c % c % x` SUBST1_TAC THENL
5937 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID;
5939 ASM_SIMP_TAC[DIST_MUL; real_abs; REAL_LT_INV_EQ; REAL_LT_IMP_LE] THEN
5940 ONCE_REWRITE_TAC[REAL_ARITH `inv c * x:real = x / c`] THEN
5941 ASM_MESON_TAC[REAL_LT_LDIV_EQ; REAL_MUL_SYM]];
5942 DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
5943 EXISTS_TAC `c:real` THEN EXISTS_TAC `inv(c) % y:real^N` THEN
5944 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN
5945 VECTOR_ARITH_TAC]);;
5947 (* ------------------------------------------------------------------------- *)
5948 (* Closures and interiors of halfspaces. *)
5949 (* ------------------------------------------------------------------------- *)
5951 let INTERIOR_HALFSPACE_LE = prove
5953 ~(a = vec 0) ==> interior {x | a dot x <= b} = {x | a dot x < b}`,
5954 REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERIOR_UNIQUE THEN
5955 SIMP_TAC[OPEN_HALFSPACE_LT; SUBSET; IN_ELIM_THM; REAL_LT_IMP_LE] THEN
5956 X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN
5957 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN ASM_SIMP_TAC[REAL_LT_LE] THEN
5959 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
5960 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
5961 DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5962 REWRITE_TAC[SUBSET; IN_CBALL] THEN
5963 DISCH_THEN(MP_TAC o SPEC `x + e / norm(a) % a:real^N`) THEN
5964 REWRITE_TAC[NORM_ARITH `dist(x:real^N,x + y) = norm y`] THEN
5965 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; REAL_DIV_RMUL;
5966 NORM_EQ_0; REAL_ARITH `&0 < x ==> abs x <= x`] THEN
5968 FIRST_X_ASSUM(MP_TAC o SPEC `x + e / norm(a) % a:real^N`) THEN
5969 ASM_REWRITE_TAC[DOT_RADD; DOT_RMUL] THEN
5970 MATCH_MP_TAC(REAL_ARITH `&0 < e ==> ~(b + e <= b)`) THEN
5971 ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; NORM_POS_LT; DOT_POS_LT]);;
5973 let INTERIOR_HALFSPACE_GE = prove
5975 ~(a = vec 0) ==> interior {x | a dot x >= b} = {x | a dot x > b}`,
5976 REPEAT STRIP_TAC THEN
5977 ONCE_REWRITE_TAC[REAL_ARITH `a >= b <=> --a <= --b`;
5978 REAL_ARITH `a > b <=> --a < --b`] THEN
5979 ASM_SIMP_TAC[GSYM DOT_LNEG; INTERIOR_HALFSPACE_LE; VECTOR_NEG_EQ_0]);;
5981 let INTERIOR_HALFSPACE_COMPONENT_LE = prove
5982 (`!a k. interior {x:real^N | x$k <= a} = {x | x$k < a}`,
5984 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5986 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5987 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] INTERIOR_HALFSPACE_LE) THEN
5988 ASM_SIMP_TAC[DOT_BASIS; BASIS_NONZERO]);;
5990 let INTERIOR_HALFSPACE_COMPONENT_GE = prove
5991 (`!a k. interior {x:real^N | x$k >= a} = {x | x$k > a}`,
5993 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
5995 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
5996 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] INTERIOR_HALFSPACE_GE) THEN
5997 ASM_SIMP_TAC[DOT_BASIS; BASIS_NONZERO]);;
5999 let CLOSURE_HALFSPACE_LT = prove
6001 ~(a = vec 0) ==> closure {x | a dot x < b} = {x | a dot x <= b}`,
6002 REPEAT STRIP_TAC THEN REWRITE_TAC[CLOSURE_INTERIOR] THEN
6003 REWRITE_TAC[SET_RULE `UNIV DIFF {x | P x} = {x | ~P x}`] THEN
6004 ASM_SIMP_TAC[REAL_ARITH `~(x < b) <=> x >= b`; INTERIOR_HALFSPACE_GE] THEN
6005 REWRITE_TAC[EXTENSION; IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN REAL_ARITH_TAC);;
6007 let CLOSURE_HALFSPACE_GT = prove
6009 ~(a = vec 0) ==> closure {x | a dot x > b} = {x | a dot x >= b}`,
6010 REPEAT STRIP_TAC THEN
6011 ONCE_REWRITE_TAC[REAL_ARITH `a >= b <=> --a <= --b`;
6012 REAL_ARITH `a > b <=> --a < --b`] THEN
6013 ASM_SIMP_TAC[GSYM DOT_LNEG; CLOSURE_HALFSPACE_LT; VECTOR_NEG_EQ_0]);;
6015 let CLOSURE_HALFSPACE_COMPONENT_LT = prove
6016 (`!a k. closure {x:real^N | x$k < a} = {x | x$k <= a}`,
6018 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
6020 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
6021 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CLOSURE_HALFSPACE_LT) THEN
6022 ASM_SIMP_TAC[DOT_BASIS; BASIS_NONZERO]);;
6024 let CLOSURE_HALFSPACE_COMPONENT_GT = prove
6025 (`!a k. closure {x:real^N | x$k > a} = {x | x$k >= a}`,
6027 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
6029 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
6030 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] CLOSURE_HALFSPACE_GT) THEN
6031 ASM_SIMP_TAC[DOT_BASIS; BASIS_NONZERO]);;
6033 let INTERIOR_HYPERPLANE = prove
6034 (`!a b. ~(a = vec 0) ==> interior {x | a dot x = b} = {}`,
6035 REWRITE_TAC[REAL_ARITH `x = y <=> x <= y /\ x >= y`] THEN
6036 REWRITE_TAC[SET_RULE `{x | p x /\ q x} = {x | p x} INTER {x | q x}`] THEN
6037 REWRITE_TAC[INTERIOR_INTER] THEN
6038 ASM_SIMP_TAC[INTERIOR_HALFSPACE_LE; INTERIOR_HALFSPACE_GE] THEN
6039 REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; NOT_IN_EMPTY] THEN
6042 let FRONTIER_HALFSPACE_LE = prove
6043 (`!a:real^N b. ~(a = vec 0 /\ b = &0)
6044 ==> frontier {x | a dot x <= b} = {x | a dot x = b}`,
6045 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
6046 ASM_SIMP_TAC[DOT_LZERO] THENL
6047 [ASM_CASES_TAC `&0 <= b` THEN
6048 ASM_REWRITE_TAC[UNIV_GSPEC; FRONTIER_UNIV; EMPTY_GSPEC; FRONTIER_EMPTY];
6049 ASM_SIMP_TAC[frontier; INTERIOR_HALFSPACE_LE; CLOSURE_CLOSED;
6050 CLOSED_HALFSPACE_LE] THEN
6051 REWRITE_TAC[EXTENSION; IN_DIFF; IN_ELIM_THM] THEN REAL_ARITH_TAC]);;
6053 let FRONTIER_HALFSPACE_GE = prove
6054 (`!a:real^N b. ~(a = vec 0 /\ b = &0)
6055 ==> frontier {x | a dot x >= b} = {x | a dot x = b}`,
6056 REPEAT STRIP_TAC THEN
6057 MP_TAC(ISPECL [`--a:real^N`; `--b:real`] FRONTIER_HALFSPACE_LE) THEN
6058 ASM_REWRITE_TAC[VECTOR_NEG_EQ_0; REAL_NEG_EQ_0; DOT_LNEG] THEN
6059 REWRITE_TAC[REAL_LE_NEG2; REAL_EQ_NEG2; real_ge]);;
6061 let FRONTIER_HALFSPACE_LT = prove
6062 (`!a:real^N b. ~(a = vec 0 /\ b = &0)
6063 ==> frontier {x | a dot x < b} = {x | a dot x = b}`,
6064 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
6065 ASM_SIMP_TAC[DOT_LZERO] THENL
6066 [ASM_CASES_TAC `&0 < b` THEN
6067 ASM_REWRITE_TAC[UNIV_GSPEC; FRONTIER_UNIV; EMPTY_GSPEC; FRONTIER_EMPTY];
6068 ASM_SIMP_TAC[frontier; CLOSURE_HALFSPACE_LT; INTERIOR_OPEN;
6069 OPEN_HALFSPACE_LT] THEN
6070 REWRITE_TAC[EXTENSION; IN_DIFF; IN_ELIM_THM] THEN REAL_ARITH_TAC]);;
6072 let FRONTIER_HALFSPACE_GT = prove
6073 (`!a:real^N b. ~(a = vec 0 /\ b = &0)
6074 ==> frontier {x | a dot x > b} = {x | a dot x = b}`,
6075 REPEAT STRIP_TAC THEN
6076 MP_TAC(ISPECL [`--a:real^N`; `--b:real`] FRONTIER_HALFSPACE_LT) THEN
6077 ASM_REWRITE_TAC[VECTOR_NEG_EQ_0; REAL_NEG_EQ_0; DOT_LNEG] THEN
6078 REWRITE_TAC[REAL_LT_NEG2; REAL_EQ_NEG2; real_gt]);;
6080 let INTERIOR_STANDARD_HYPERPLANE = prove
6081 (`!k a. interior {x:real^N | x$k = a} = {}`,
6083 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !x:real^N. x$k = x$i`
6085 [ASM_REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
6086 MP_TAC(ISPECL [`basis i:real^N`; `a:real`] INTERIOR_HYPERPLANE) THEN
6087 ASM_SIMP_TAC[DOT_BASIS; BASIS_NONZERO]);;
6089 let EMPTY_INTERIOR_LOWDIM = prove
6090 (`!s:real^N->bool. dim(s) < dimindex(:N) ==> interior s = {}`,
6091 GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP LOWDIM_SUBSET_HYPERPLANE) THEN
6092 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
6093 MATCH_MP_TAC(SET_RULE
6094 `!t u. s SUBSET t /\ t SUBSET u /\ u = {} ==> s = {}`) THEN
6095 MAP_EVERY EXISTS_TAC
6096 [`interior(span(s):real^N->bool)`;
6097 `interior({x:real^N | a dot x = &0})`] THEN
6098 ASM_SIMP_TAC[SUBSET_INTERIOR; SPAN_INC; INTERIOR_HYPERPLANE]);;
6100 (* ------------------------------------------------------------------------- *)
6101 (* Unboundedness of halfspaces. *)
6102 (* ------------------------------------------------------------------------- *)
6104 let UNBOUNDED_HALFSPACE_COMPONENT_LE = prove
6105 (`!a k. ~bounded {x:real^N | x$k <= a}`,
6107 SUBGOAL_THEN `?i. 1 <= i /\ i <= dimindex(:N) /\ !z:real^N. z$k = z$i`
6108 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
6109 ASM_REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN
6110 DISCH_THEN(X_CHOOSE_THEN `B:real` MP_TAC) THEN
6111 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
6112 EXISTS_TAC `--(&1 + max (abs B) (abs a)) % basis i:real^N` THEN
6113 ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; BASIS_COMPONENT;
6114 VECTOR_MUL_COMPONENT] THEN
6117 let UNBOUNDED_HALFSPACE_COMPONENT_GE = prove
6118 (`!a k. ~bounded {x:real^N | x$k >= a}`,
6119 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_NEGATIONS) THEN
6120 MP_TAC(SPECL [`--a:real`; `k:num`] UNBOUNDED_HALFSPACE_COMPONENT_LE) THEN
6121 REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN
6122 AP_TERM_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN CONJ_TAC THENL
6123 [MESON_TAC[VECTOR_NEG_NEG];
6124 REWRITE_TAC[IN_ELIM_THM; VECTOR_NEG_COMPONENT] THEN REAL_ARITH_TAC]);;
6126 let UNBOUNDED_HALFSPACE_COMPONENT_LT = prove
6127 (`!a k. ~bounded {x:real^N | x$k < a}`,
6128 ONCE_REWRITE_TAC[GSYM BOUNDED_CLOSURE_EQ] THEN
6129 REWRITE_TAC[CLOSURE_HALFSPACE_COMPONENT_LT;
6130 UNBOUNDED_HALFSPACE_COMPONENT_LE]);;
6132 let UNBOUNDED_HALFSPACE_COMPONENT_GT = prove
6133 (`!a k. ~bounded {x:real^N | x$k > a}`,
6134 ONCE_REWRITE_TAC[GSYM BOUNDED_CLOSURE_EQ] THEN
6135 REWRITE_TAC[CLOSURE_HALFSPACE_COMPONENT_GT;
6136 UNBOUNDED_HALFSPACE_COMPONENT_GE]);;
6138 let BOUNDED_HALFSPACE_LE = prove
6139 (`!a:real^N b. bounded {x | a dot x <= b} <=> a = vec 0 /\ b < &0`,
6140 GEOM_BASIS_MULTIPLE_TAC 1 `a:real^N` THEN
6141 SIMP_TAC[DOT_LMUL; DOT_BASIS; VECTOR_MUL_EQ_0; DIMINDEX_GE_1; LE_REFL;
6143 X_GEN_TAC `a:real` THEN ASM_CASES_TAC `a = &0` THEN ASM_REWRITE_TAC[] THEN
6144 DISCH_TAC THEN X_GEN_TAC `b:real` THENL
6145 [REWRITE_TAC[REAL_MUL_LZERO; DOT_LZERO; GSYM REAL_NOT_LE] THEN
6146 ASM_CASES_TAC `&0 <= b` THEN
6147 ASM_REWRITE_TAC[BOUNDED_EMPTY; NOT_BOUNDED_UNIV;
6148 SET_RULE `{x | T} = UNIV`; EMPTY_GSPEC];
6149 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
6150 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_LT_LE;
6151 UNBOUNDED_HALFSPACE_COMPONENT_LE]]);;
6153 let BOUNDED_HALFSPACE_GE = prove
6154 (`!a:real^N b. bounded {x | a dot x >= b} <=> a = vec 0 /\ &0 < b`,
6155 REWRITE_TAC[REAL_ARITH `a >= b <=> --a <= --b`] THEN
6156 REWRITE_TAC[GSYM DOT_LNEG; BOUNDED_HALFSPACE_LE] THEN
6157 REWRITE_TAC[VECTOR_NEG_EQ_0; REAL_ARITH `--b < &0 <=> &0 < b`]);;
6159 let BOUNDED_HALFSPACE_LT = prove
6160 (`!a:real^N b. bounded {x | a dot x < b} <=> a = vec 0 /\ b <= &0`,
6161 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
6162 ASM_REWRITE_TAC[] THENL
6163 [REWRITE_TAC[DOT_LZERO; GSYM REAL_NOT_LE] THEN ASM_CASES_TAC `b <= &0` THEN
6164 ASM_REWRITE_TAC[BOUNDED_EMPTY; NOT_BOUNDED_UNIV;
6165 SET_RULE `{x | T} = UNIV`; EMPTY_GSPEC];
6166 ONCE_REWRITE_TAC[GSYM BOUNDED_CLOSURE_EQ] THEN
6167 ASM_SIMP_TAC[CLOSURE_HALFSPACE_LT; BOUNDED_HALFSPACE_LE]]);;
6169 let BOUNDED_HALFSPACE_GT = prove
6170 (`!a:real^N b. bounded {x | a dot x > b} <=> a = vec 0 /\ &0 <= b`,
6171 REWRITE_TAC[REAL_ARITH `a > b <=> --a < --b`] THEN
6172 REWRITE_TAC[GSYM DOT_LNEG; BOUNDED_HALFSPACE_LT] THEN
6173 REWRITE_TAC[VECTOR_NEG_EQ_0; REAL_ARITH `--b <= &0 <=> &0 <= b`]);;
6175 (* ------------------------------------------------------------------------- *)
6176 (* Equality of continuous functions on closure and related results. *)
6177 (* ------------------------------------------------------------------------- *)
6179 let FORALL_IN_CLOSURE = prove
6180 (`!f:real^M->real^N s t.
6181 closed t /\ f continuous_on (closure s) /\
6182 (!x. x IN s ==> f x IN t)
6183 ==> (!x. x IN closure s ==> f x IN t)`,
6184 REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x IN t) <=>
6185 s SUBSET {x | x IN s /\ f x IN t}`] THEN
6186 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_MINIMAL THEN
6187 ASM_REWRITE_TAC[CLOSED_CLOSURE] THEN CONJ_TAC THENL
6188 [MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[];
6189 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
6190 ASM_REWRITE_TAC[CLOSED_CLOSURE]]);;
6192 let FORALL_IN_CLOSURE_EQ = prove
6194 closed t /\ f continuous_on closure s
6195 ==> ((!x. x IN closure s ==> f x IN t) <=>
6196 (!x. x IN s ==> f x IN t))`,
6197 MESON_TAC[FORALL_IN_CLOSURE; CLOSURE_SUBSET; SUBSET]);;
6199 let SUP_CLOSURE = prove
6200 (`!s. sup(IMAGE drop (closure s)) = sup(IMAGE drop s)`,
6201 GEN_TAC THEN MATCH_MP_TAC SUP_EQ THEN
6202 REWRITE_TAC[FORALL_IN_IMAGE] THEN GEN_TAC THEN
6203 ONCE_REWRITE_TAC[SET_RULE `drop x <= b <=> x IN {x | drop x <= b}`] THEN
6204 MATCH_MP_TAC FORALL_IN_CLOSURE_EQ THEN
6205 REWRITE_TAC[CONTINUOUS_ON_ID; drop; CLOSED_HALFSPACE_COMPONENT_LE]);;
6207 let INF_CLOSURE = prove
6208 (`!s. inf(IMAGE drop (closure s)) = inf(IMAGE drop s)`,
6209 GEN_TAC THEN MATCH_MP_TAC INF_EQ THEN
6210 REWRITE_TAC[FORALL_IN_IMAGE] THEN GEN_TAC THEN
6211 ONCE_REWRITE_TAC[SET_RULE `b <= drop x <=> x IN {x | b <= drop x}`] THEN
6212 MATCH_MP_TAC FORALL_IN_CLOSURE_EQ THEN
6213 REWRITE_TAC[CONTINUOUS_ON_ID; drop; CLOSED_HALFSPACE_COMPONENT_GE;
6216 let CONTINUOUS_LE_ON_CLOSURE = prove
6217 (`!f:real^M->real s a.
6218 (lift o f) continuous_on closure(s) /\ (!x. x IN s ==> f(x) <= a)
6219 ==> !x. x IN closure(s) ==> f(x) <= a`,
6221 (`x IN s ==> f x <= a <=> x IN s ==> (lift o f) x IN {y | y$1 <= a}`,
6222 REWRITE_TAC[IN_ELIM_THM; o_THM; GSYM drop; LIFT_DROP]) in
6223 REWRITE_TAC[lemma] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
6224 MATCH_MP_TAC FORALL_IN_CLOSURE THEN
6225 ASM_REWRITE_TAC[ETA_AX; CLOSED_HALFSPACE_COMPONENT_LE]);;
6227 let CONTINUOUS_GE_ON_CLOSURE = prove
6228 (`!f:real^M->real s a.
6229 (lift o f) continuous_on closure(s) /\ (!x. x IN s ==> a <= f(x))
6230 ==> !x. x IN closure(s) ==> a <= f(x)`,
6232 (`x IN s ==> a <= f x <=> x IN s ==> (lift o f) x IN {y | y$1 >= a}`,
6233 REWRITE_TAC[IN_ELIM_THM; o_THM; GSYM drop; real_ge; LIFT_DROP]) in
6234 REWRITE_TAC[lemma] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
6235 MATCH_MP_TAC FORALL_IN_CLOSURE THEN
6236 ASM_REWRITE_TAC[ETA_AX; CLOSED_HALFSPACE_COMPONENT_GE]);;
6238 let CONTINUOUS_CONSTANT_ON_CLOSURE = prove
6239 (`!f:real^M->real^N s a.
6240 f continuous_on closure(s) /\ (!x. x IN s ==> f(x) = a)
6241 ==> !x. x IN closure(s) ==> f(x) = a`,
6242 REWRITE_TAC[SET_RULE
6243 `x IN s ==> f x = a <=> x IN s ==> f x IN {a}`] THEN
6244 REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC FORALL_IN_CLOSURE THEN
6245 ASM_REWRITE_TAC[CLOSED_SING]);;
6247 let CONTINUOUS_AGREE_ON_CLOSURE = prove
6248 (`!g h:real^M->real^N.
6249 g continuous_on closure s /\ h continuous_on closure s /\
6250 (!x. x IN s ==> g x = h x)
6251 ==> !x. x IN closure s ==> g x = h x`,
6252 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN STRIP_TAC THEN
6253 MATCH_MP_TAC CONTINUOUS_CONSTANT_ON_CLOSURE THEN
6254 ASM_SIMP_TAC[CONTINUOUS_ON_SUB]);;
6256 let CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT = prove
6257 (`!f:real^M->real^N s a.
6259 ==> closed_in (subtopology euclidean s) {x | x IN s /\ f x = a}`,
6260 REPEAT STRIP_TAC THEN
6261 ONCE_REWRITE_TAC[SET_RULE
6262 `{x | x IN s /\ f(x) = a} = {x | x IN s /\ f(x) IN {a}}`] THEN
6263 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN
6264 ASM_REWRITE_TAC[CLOSED_SING]);;
6266 let CONTINUOUS_CLOSED_PREIMAGE_CONSTANT = prove
6267 (`!f:real^M->real^N s.
6268 f continuous_on s /\ closed s ==> closed {x | x IN s /\ f(x) = a}`,
6269 REPEAT STRIP_TAC THEN
6270 ASM_CASES_TAC `{x | x IN s /\ (f:real^M->real^N)(x) = a} = {}` THEN
6271 ASM_REWRITE_TAC[CLOSED_EMPTY] THEN ONCE_REWRITE_TAC[SET_RULE
6272 `{x | x IN s /\ f(x) = a} = {x | x IN s /\ f(x) IN {a}}`] THEN
6273 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
6274 ASM_REWRITE_TAC[CLOSED_SING] THEN ASM SET_TAC[]);;
6276 (* ------------------------------------------------------------------------- *)
6277 (* Theorems relating continuity and uniform continuity to closures. *)
6278 (* ------------------------------------------------------------------------- *)
6280 let CONTINUOUS_ON_CLOSURE = prove
6281 (`!f:real^M->real^N s.
6282 f continuous_on closure s <=>
6283 !x e. x IN closure s /\ &0 < e
6285 !y. y IN s /\ dist(y,x) < d ==> dist(f y,f x) < e`,
6286 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_on] THEN
6287 EQ_TAC THENL [MESON_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET]; ALL_TAC] THEN
6288 DISCH_TAC THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
6289 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
6290 FIRST_ASSUM(MP_TAC o SPECL [`x:real^M`; `e / &2`]) THEN
6291 ANTS_TAC THENL [ASM_REWRITE_TAC[REAL_HALF]; ALL_TAC] THEN
6292 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
6293 EXISTS_TAC `d / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN
6294 X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
6295 FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^M`; `e / &2`]) THEN
6296 ASM_REWRITE_TAC[REAL_HALF] THEN
6297 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
6298 MP_TAC(ISPECL [`y:real^M`; `s:real^M->bool`] CLOSURE_APPROACHABLE) THEN
6299 ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `min k (d / &2)`) THEN
6300 ASM_REWRITE_TAC[REAL_HALF; REAL_LT_MIN] THEN
6301 ASM_MESON_TAC[DIST_SYM; NORM_ARITH
6302 `dist(a,b) < e / &2 /\ dist(b,c) < e / &2 ==> dist(a,c) < e`]);;
6304 let CONTINUOUS_ON_CLOSURE_SEQUENTIALLY = prove
6305 (`!f:real^M->real^N s.
6306 f continuous_on closure s <=>
6307 !x a. a IN closure s /\ (!n. x n IN s) /\ (x --> a) sequentially
6308 ==> ((f o x) --> f a) sequentially`,
6309 REWRITE_TAC[CONTINUOUS_ON_CLOSURE] THEN
6310 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
6311 REWRITE_TAC[IMP_IMP; GSYM continuous_within] THEN
6312 REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY] THEN MESON_TAC[]);;
6314 let UNIFORMLY_CONTINUOUS_ON_CLOSURE = prove
6315 (`!f:real^M->real^N s.
6316 f uniformly_continuous_on s /\ f continuous_on closure s
6317 ==> f uniformly_continuous_on closure s`,
6319 REWRITE_TAC[uniformly_continuous_on] THEN STRIP_TAC THEN
6320 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
6321 FIRST_X_ASSUM(MP_TAC o SPEC `e / &3`) THEN
6322 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6323 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
6324 EXISTS_TAC `d / &3` THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6325 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN
6326 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_on]) THEN
6327 DISCH_THEN(fun th ->
6328 MP_TAC(SPEC `y:real^M` th) THEN MP_TAC(SPEC `x:real^M` th)) THEN
6329 ASM_REWRITE_TAC[] THEN
6330 DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
6331 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6332 DISCH_THEN(X_CHOOSE_THEN `d1:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
6333 MP_TAC(ISPECL [`x:real^M`; `s:real^M->bool`] CLOSURE_APPROACHABLE) THEN
6334 ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `min d1 (d / &3)`) THEN
6335 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_LT_MIN]] THEN
6336 DISCH_THEN(X_CHOOSE_THEN `x':real^M` STRIP_ASSUME_TAC) THEN
6337 DISCH_THEN(MP_TAC o SPEC `x':real^M`) THEN
6338 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN DISCH_TAC THEN
6339 DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
6340 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
6341 DISCH_THEN(X_CHOOSE_THEN `d2:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
6342 MP_TAC(ISPECL [`y:real^M`; `s:real^M->bool`] CLOSURE_APPROACHABLE) THEN
6343 ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `min d2 (d / &3)`) THEN
6344 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[REAL_LT_MIN]] THEN
6345 DISCH_THEN(X_CHOOSE_THEN `y':real^M` STRIP_ASSUME_TAC) THEN
6346 DISCH_THEN(MP_TAC o SPEC `y':real^M`) THEN
6347 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN DISCH_TAC THEN
6348 FIRST_X_ASSUM(MP_TAC o SPECL [`x':real^M`; `y':real^M`]) THEN
6349 ASM_MESON_TAC[DIST_SYM; NORM_ARITH
6350 `dist(y,x) < d / &3 /\ dist(x',x) < d / &3 /\ dist(y',y) < d / &3
6351 ==> dist(y',x') < d`]);;
6353 (* ------------------------------------------------------------------------- *)
6354 (* Continuity properties for square roots. We get other forms of this *)
6355 (* later (transcendentals.ml and realanalysis.ml) but it's nice to have *)
6356 (* them around earlier. *)
6357 (* ------------------------------------------------------------------------- *)
6359 let CONTINUOUS_AT_SQRT = prove
6360 (`!a s. &0 < drop a ==> (lift o sqrt o drop) continuous (at a)`,
6361 REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at; o_THM; DIST_LIFT] THEN
6362 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
6363 EXISTS_TAC `min (drop a) (e * sqrt(drop a))` THEN
6364 ASM_SIMP_TAC[REAL_LT_MIN; SQRT_POS_LT; REAL_LT_MUL; DIST_REAL] THEN
6365 X_GEN_TAC `b:real^1` THEN REWRITE_TAC[GSYM drop] THEN STRIP_TAC THEN
6366 FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REAL_ARITH
6367 `abs(b - a) < a ==> &0 < b`)) THEN
6369 `sqrt(drop b) - sqrt(drop a) =
6370 (drop b - drop a) / (sqrt(drop a) + sqrt(drop b))`
6372 [MATCH_MP_TAC(REAL_FIELD
6373 `sa pow 2 = a /\ sb pow 2 = b /\ &0 < sa /\ &0 < sb
6374 ==> sb - sa = (b - a) / (sa + sb)`) THEN
6375 ASM_SIMP_TAC[SQRT_POS_LT; SQRT_POW_2; REAL_LT_IMP_LE];
6376 ASM_SIMP_TAC[REAL_ABS_DIV; SQRT_POS_LT; REAL_LT_ADD; REAL_LT_LDIV_EQ;
6377 REAL_ARITH `&0 < x ==> abs x = x`] THEN
6378 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
6379 REAL_LTE_TRANS)) THEN
6380 ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LE_ADDR; SQRT_POS_LE;
6383 let CONTINUOUS_WITHIN_LIFT_SQRT = prove
6384 (`!a s. (!x. x IN s ==> &0 <= drop x)
6385 ==> (lift o sqrt o drop) continuous (at a within s)`,
6386 REPEAT STRIP_TAC THEN
6387 REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC
6388 (REAL_ARITH `drop a < &0 \/ drop a = &0 \/ &0 < drop a`)
6390 [MATCH_MP_TAC CONTINUOUS_WITHIN_SUBSET THEN
6391 EXISTS_TAC `{x | &0 <= drop x}` THEN
6392 ASM_SIMP_TAC[SUBSET; IN_ELIM_THM] THEN
6393 MATCH_MP_TAC CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL THEN
6394 ASM_REWRITE_TAC[IN_ELIM_THM; REAL_NOT_LE] THEN
6395 REWRITE_TAC[drop; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_COMPONENT_GE];
6396 RULE_ASSUM_TAC(REWRITE_RULE[GSYM LIFT_EQ; LIFT_DROP; LIFT_NUM]) THEN
6397 ASM_REWRITE_TAC[continuous_within; o_THM; DROP_VEC; SQRT_0; LIFT_NUM] THEN
6398 REWRITE_TAC[DIST_0; NORM_LIFT; NORM_REAL; GSYM drop] THEN
6399 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
6400 EXISTS_TAC `(e:real) pow 2` THEN ASM_SIMP_TAC[REAL_POW_LT] THEN
6401 X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN
6402 ASM_SIMP_TAC[real_abs; SQRT_POS_LE] THEN
6403 SUBGOAL_THEN `e = sqrt(e pow 2)` SUBST1_TAC THENL
6404 [ASM_SIMP_TAC[POW_2_SQRT; REAL_LT_IMP_LE];
6405 MATCH_MP_TAC SQRT_MONO_LT THEN ASM_SIMP_TAC[] THEN ASM_REAL_ARITH_TAC];
6406 MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
6407 MATCH_MP_TAC CONTINUOUS_AT_SQRT THEN ASM_REWRITE_TAC[]]);;
6409 let CONTINUOUS_WITHIN_SQRT_COMPOSE = prove
6411 (\x. lift(f x)) continuous (at a within s) /\
6412 (&0 < f a \/ !x. x IN s ==> &0 <= f x)
6413 ==> (\x. lift(sqrt(f x))) continuous (at a within s)`,
6416 `(\x:real^N. lift(sqrt(f x))) = (lift o sqrt o drop) o (lift o f)`
6417 SUBST1_TAC THENL [REWRITE_TAC[o_DEF; LIFT_DROP]; ALL_TAC] THEN
6418 REPEAT STRIP_TAC THEN
6419 (MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN
6420 CONJ_TAC THENL [ASM_REWRITE_TAC[o_DEF]; ALL_TAC])
6422 [MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
6423 MATCH_MP_TAC CONTINUOUS_AT_SQRT THEN ASM_REWRITE_TAC[o_DEF; LIFT_DROP];
6424 MATCH_MP_TAC CONTINUOUS_WITHIN_LIFT_SQRT THEN
6425 ASM_REWRITE_TAC[FORALL_IN_IMAGE; o_DEF; LIFT_DROP]]);;
6427 let CONTINUOUS_AT_SQRT_COMPOSE = prove
6429 (\x. lift(f x)) continuous (at a) /\ (&0 < f a \/ !x. &0 <= f x)
6430 ==> (\x. lift(sqrt(f x))) continuous (at a)`,
6432 MP_TAC(ISPECL [`f:real^N->real`; `(:real^N)`; `a:real^N`]
6433 CONTINUOUS_WITHIN_SQRT_COMPOSE) THEN
6434 REWRITE_TAC[WITHIN_UNIV; IN_UNIV]);;
6436 let CONTINUOUS_ON_LIFT_SQRT = prove
6437 (`!s. (!x. x IN s ==> &0 <= drop x)
6438 ==> (lift o sqrt o drop) continuous_on s`,
6439 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_WITHIN_LIFT_SQRT]);;
6441 let CONTINUOUS_ON_LIFT_SQRT_COMPOSE = prove
6442 (`!f:real^N->real s.
6443 (lift o f) continuous_on s /\ (!x. x IN s ==> &0 <= f x)
6444 ==> (\x. lift(sqrt(f x))) continuous_on s`,
6445 REPEAT STRIP_TAC THEN
6447 `(\x:real^N. lift(sqrt(f x))) = (lift o sqrt o drop) o (lift o f)`
6449 [REWRITE_TAC[o_DEF; LIFT_DROP];
6450 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN
6451 MATCH_MP_TAC CONTINUOUS_ON_LIFT_SQRT THEN
6452 ASM_REWRITE_TAC[FORALL_IN_IMAGE; o_THM; LIFT_DROP]]);;
6454 (* ------------------------------------------------------------------------- *)
6455 (* Cauchy continuity, and the extension of functions to closures. *)
6456 (* ------------------------------------------------------------------------- *)
6458 let UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS = prove
6459 (`!f:real^M->real^N s.
6460 f uniformly_continuous_on s
6461 ==> (!x. cauchy x /\ (!n. (x n) IN s) ==> cauchy(f o x))`,
6462 REPEAT GEN_TAC THEN REWRITE_TAC[uniformly_continuous_on; cauchy; o_DEF] THEN
6465 let CONTINUOUS_CLOSED_IMP_CAUCHY_CONTINUOUS = prove
6466 (`!f:real^M->real^N s.
6467 f continuous_on s /\ closed s
6468 ==> (!x. cauchy x /\ (!n. (x n) IN s) ==> cauchy(f o x))`,
6469 REWRITE_TAC[GSYM COMPLETE_EQ_CLOSED; CONTINUOUS_ON_SEQUENTIALLY] THEN
6470 REWRITE_TAC[complete] THEN MESON_TAC[CONVERGENT_IMP_CAUCHY]);;
6472 let CAUCHY_CONTINUOUS_UNIQUENESS_LEMMA = prove
6473 (`!f:real^M->real^N s.
6474 (!x. cauchy x /\ (!n. (x n) IN s) ==> cauchy(f o x))
6475 ==> !a x. (!n. (x n) IN s) /\ (x --> a) sequentially
6476 ==> ?l. ((f o x) --> l) sequentially /\
6477 !y. (!n. (y n) IN s) /\ (y --> a) sequentially
6478 ==> ((f o y) --> l) sequentially`,
6479 REPEAT STRIP_TAC THEN
6480 FIRST_ASSUM(MP_TAC o SPEC `x:num->real^M`) THEN
6481 ANTS_TAC THENL [ASM_MESON_TAC[CONVERGENT_IMP_CAUCHY]; ALL_TAC] THEN
6482 REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY] THEN MATCH_MP_TAC MONO_EXISTS THEN
6483 X_GEN_TAC `l:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
6484 X_GEN_TAC `y:num->real^M` THEN STRIP_TAC THEN
6485 FIRST_ASSUM(MP_TAC o SPEC `y:num->real^M`) THEN
6486 ANTS_TAC THENL [ASM_MESON_TAC[CONVERGENT_IMP_CAUCHY]; ALL_TAC] THEN
6487 REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY] THEN
6488 DISCH_THEN(X_CHOOSE_THEN `m:real^N` STRIP_ASSUME_TAC) THEN
6489 SUBGOAL_THEN `l:real^N = m` (fun th -> ASM_REWRITE_TAC[th]) THEN
6490 ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
6491 MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
6492 EXISTS_TAC `\n:num. (f:real^M->real^N)(x n) - f(y n)` THEN
6493 RULE_ASSUM_TAC(REWRITE_RULE[o_DEF]) THEN
6494 ASM_SIMP_TAC[LIM_SUB; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
6495 FIRST_X_ASSUM(MP_TAC o SPEC
6496 `\n. if EVEN n then x(n DIV 2):real^M else y(n DIV 2)`) THEN
6497 REWRITE_TAC[cauchy; o_THM; LIM_SEQUENTIALLY] THEN ANTS_TAC THENL
6498 [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
6499 X_GEN_TAC `e:real` THEN DISCH_TAC THEN MAP_EVERY UNDISCH_TAC
6500 [`((y:num->real^M) --> a) sequentially`;
6501 `((x:num->real^M) --> a) sequentially`] THEN
6502 REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl))) THEN
6503 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
6504 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
6505 DISCH_THEN(X_CHOOSE_TAC `N1:num`) THEN
6506 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
6507 DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN
6508 EXISTS_TAC `2 * (N1 + N2)` THEN
6509 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN
6510 REPEAT(FIRST_X_ASSUM(fun th ->
6511 MP_TAC(SPEC `m DIV 2` th) THEN MP_TAC(SPEC `n DIV 2` th))) THEN
6512 REPEAT(ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC]) THEN
6513 REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
6514 REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_NOT_LE])) THEN
6515 CONV_TAC NORM_ARITH;
6516 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
6517 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
6518 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
6519 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
6520 FIRST_X_ASSUM(MP_TAC o SPECL [`2 * n`; `2 * n + 1`]) THEN
6521 ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
6522 REWRITE_TAC[EVEN_ADD; EVEN_MULT; ARITH_EVEN] THEN
6523 REWRITE_TAC[ARITH_RULE `(2 * n) DIV 2 = n /\ (2 * n + 1) DIV 2 = n`] THEN
6524 REWRITE_TAC[dist; VECTOR_SUB_RZERO]]);;
6526 let CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE = prove
6527 (`!f:real^M->real^N s.
6528 (!x. cauchy x /\ (!n. (x n) IN s) ==> cauchy(f o x))
6529 ==> ?g. g continuous_on closure s /\ (!x. x IN s ==> g x = f x)`,
6530 REPEAT STRIP_TAC THEN
6533 a IN closure s ==> (!n. x n IN s) /\ (x --> a) sequentially`
6534 MP_TAC THENL [MESON_TAC[CLOSURE_SEQUENTIAL]; ALL_TAC] THEN
6535 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
6536 X_GEN_TAC `X:real^M->num->real^M` THEN DISCH_TAC THEN
6537 FIRST_ASSUM(MP_TAC o MATCH_MP CAUCHY_CONTINUOUS_UNIQUENESS_LEMMA) THEN
6538 DISCH_THEN(MP_TAC o GEN `a:real^M` o
6539 SPECL [`a:real^M`; `(X:real^M->num->real^M) a`]) THEN
6540 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[]
6541 `(!a. P a ==> Q a) ==> ((!a. P a ==> R a) ==> p)
6542 ==> ((!a. Q a ==> R a) ==> p)`)) THEN
6543 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
6544 REWRITE_TAC[SKOLEM_THM] THEN
6545 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^N` THEN
6547 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
6548 [X_GEN_TAC `a:real^M` THEN DISCH_TAC THEN
6549 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M`) THEN
6550 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET] THEN
6551 DISCH_THEN(MP_TAC o SPEC `(\n. a):num->real^M` o CONJUNCT2) THEN
6552 ASM_SIMP_TAC[LIM_CONST_EQ; o_DEF; TRIVIAL_LIMIT_SEQUENTIALLY];
6554 ASM_SIMP_TAC[CONTINUOUS_ON_CLOSURE_SEQUENTIALLY] THEN
6555 MAP_EVERY X_GEN_TAC [`x:num->real^M`; `a:real^M`] THEN STRIP_TAC THEN
6556 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
6557 EXISTS_TAC `(f:real^M->real^N) o (x:num->real^M)` THEN ASM_SIMP_TAC[] THEN
6558 MATCH_MP_TAC ALWAYS_EVENTUALLY THEN ASM_SIMP_TAC[o_THM]);;
6560 let UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE = prove
6561 (`!f:real^M->real^N s.
6562 f uniformly_continuous_on s
6563 ==> ?g. g uniformly_continuous_on closure s /\ (!x. x IN s ==> g x = f x) /\
6564 !h. h continuous_on closure s /\ (!x. x IN s ==> h x = f x)
6565 ==> !x. x IN closure s ==> h x = g x`,
6566 REPEAT STRIP_TAC THEN
6567 FIRST_ASSUM(MP_TAC o MATCH_MP CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE o
6568 MATCH_MP UNIFORMLY_CONTINUOUS_IMP_CAUCHY_CONTINUOUS) THEN
6569 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^N` THEN
6570 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
6571 [ASM_MESON_TAC[UNIFORMLY_CONTINUOUS_ON_CLOSURE; UNIFORMLY_CONTINUOUS_ON_EQ];
6572 ASM_MESON_TAC[CONTINUOUS_AGREE_ON_CLOSURE]]);;
6574 let CAUCHY_CONTINUOUS_IMP_CONTINUOUS = prove
6575 (`!f:real^M->real^N s.
6576 (!x. cauchy x /\ (!n. (x n) IN s) ==> cauchy(f o x))
6577 ==> f continuous_on s`,
6578 REPEAT STRIP_TAC THEN
6579 FIRST_ASSUM(CHOOSE_TAC o MATCH_MP CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE) THEN
6580 ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; CLOSURE_SUBSET; CONTINUOUS_ON_EQ]);;
6582 let BOUNDED_UNIFORMLY_CONTINUOUS_IMAGE = prove
6583 (`!f:real^M->real^N s.
6584 f uniformly_continuous_on s /\ bounded s ==> bounded(IMAGE f s)`,
6585 REPEAT STRIP_TAC THEN FIRST_ASSUM
6586 (MP_TAC o MATCH_MP UNIFORMLY_CONTINUOUS_EXTENDS_TO_CLOSURE) THEN
6587 DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN
6588 MATCH_MP_TAC BOUNDED_SUBSET THEN
6589 EXISTS_TAC `IMAGE (g:real^M->real^N) (closure s)` THEN CONJ_TAC THENL
6590 [ASM_MESON_TAC[COMPACT_CLOSURE; UNIFORMLY_CONTINUOUS_IMP_CONTINUOUS;
6591 COMPACT_IMP_BOUNDED; COMPACT_CONTINUOUS_IMAGE];
6592 MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]]);;
6594 (* ------------------------------------------------------------------------- *)
6595 (* Occasionally useful invariance properties. *)
6596 (* ------------------------------------------------------------------------- *)
6598 let CONTINUOUS_AT_COMPOSE_EQ = prove
6599 (`!f:real^M->real^N g:real^M->real^M h:real^M->real^M.
6600 g continuous at x /\ h continuous at (g x) /\
6601 (!y. g(h y) = y) /\ h(g x) = x
6602 ==> (f continuous at (g x) <=> (\x. f(g x)) continuous at x)`,
6603 REPEAT STRIP_TAC THEN EQ_TAC THEN
6604 ASM_SIMP_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_AT_COMPOSE] THEN
6607 `((f:real^M->real^N) o (g:real^M->real^M) o (h:real^M->real^M))
6608 continuous at (g(x:real^M))`
6610 [REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN
6611 ASM_REWRITE_TAC[o_DEF];
6613 ASM_REWRITE_TAC[o_DEF; ETA_AX]]);;
6615 let CONTINUOUS_AT_TRANSLATION = prove
6616 (`!a z f:real^M->real^N.
6617 f continuous at (a + z) <=> (\x. f(a + x)) continuous at z`,
6618 REPEAT GEN_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE_EQ THEN
6619 EXISTS_TAC `\x:real^M. x - a` THEN
6620 SIMP_TAC[CONTINUOUS_ADD; CONTINUOUS_SUB;
6621 CONTINUOUS_AT_ID; CONTINUOUS_CONST] THEN
6624 add_translation_invariants [CONTINUOUS_AT_TRANSLATION];;
6626 let CONTINUOUS_AT_LINEAR_IMAGE = prove
6627 (`!h:real^M->real^M z f:real^M->real^N.
6628 linear h /\ (!x. norm(h x) = norm x)
6629 ==> (f continuous at (h z) <=> (\x. f(h x)) continuous at z)`,
6630 REPEAT GEN_TAC THEN DISCH_TAC THEN
6631 FIRST_ASSUM(ASSUME_TAC o GEN_REWRITE_RULE I
6632 [GSYM ORTHOGONAL_TRANSFORMATION]) THEN
6633 FIRST_ASSUM(X_CHOOSE_TAC `g:real^M->real^M` o MATCH_MP
6634 ORTHOGONAL_TRANSFORMATION_INVERSE) THEN
6635 MATCH_MP_TAC CONTINUOUS_AT_COMPOSE_EQ THEN
6636 EXISTS_TAC `g:real^M->real^M` THEN
6637 RULE_ASSUM_TAC(REWRITE_RULE[ORTHOGONAL_TRANSFORMATION]) THEN
6638 ASM_SIMP_TAC[LINEAR_CONTINUOUS_AT]);;
6640 add_linear_invariants [CONTINUOUS_AT_LINEAR_IMAGE];;
6642 (* ------------------------------------------------------------------------- *)
6643 (* Interior of an injective image. *)
6644 (* ------------------------------------------------------------------------- *)
6646 let INTERIOR_IMAGE_SUBSET = prove
6647 (`!f:real^M->real^N s.
6648 (!x. f continuous at x) /\ (!x y. f x = f y ==> x = y)
6649 ==> interior(IMAGE f s) SUBSET IMAGE f (interior s)`,
6650 REPEAT STRIP_TAC THEN REWRITE_TAC[SUBSET] THEN
6651 REWRITE_TAC[interior; IN_ELIM_THM] THEN
6652 X_GEN_TAC `y:real^N` THEN
6653 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
6654 REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN
6655 SUBGOAL_THEN `y IN IMAGE (f:real^M->real^N) s` MP_TAC THENL
6656 [ASM SET_TAC[]; ALL_TAC] THEN
6657 REWRITE_TAC[IN_IMAGE] THEN
6658 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
6659 ASM_REWRITE_TAC[IN_ELIM_THM] THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN
6660 EXISTS_TAC `{x | (f:real^M->real^N)(x) IN t}` THEN
6661 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN CONJ_TAC THENL
6662 [MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE_UNIV THEN ASM_MESON_TAC[];
6665 (* ------------------------------------------------------------------------- *)
6666 (* Making a continuous function avoid some value in a neighbourhood. *)
6667 (* ------------------------------------------------------------------------- *)
6669 let CONTINUOUS_WITHIN_AVOID = prove
6670 (`!f:real^M->real^N x s a.
6671 f continuous (at x within s) /\ x IN s /\ ~(f x = a)
6672 ==> ?e. &0 < e /\ !y. y IN s /\ dist(x,y) < e ==> ~(f y = a)`,
6673 REPEAT STRIP_TAC THEN
6674 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_within]) THEN
6675 DISCH_THEN(MP_TAC o SPEC `norm((f:real^M->real^N) x - a)`) THEN
6676 ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN
6677 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN
6678 REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN
6679 GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN SIMP_TAC[] THEN NORM_ARITH_TAC);;
6681 let CONTINUOUS_AT_AVOID = prove
6682 (`!f:real^M->real^N x a.
6683 f continuous (at x) /\ ~(f x = a)
6684 ==> ?e. &0 < e /\ !y. dist(x,y) < e ==> ~(f y = a)`,
6685 MP_TAC CONTINUOUS_WITHIN_AVOID THEN
6686 REPLICATE_TAC 2 (MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
6687 DISCH_THEN(MP_TAC o SPEC `(:real^M)`) THEN
6688 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
6689 REWRITE_TAC[WITHIN_UNIV; IN_UNIV]);;
6691 let CONTINUOUS_ON_AVOID = prove
6692 (`!f:real^M->real^N x s a.
6693 f continuous_on s /\ x IN s /\ ~(f x = a)
6694 ==> ?e. &0 < e /\ !y. y IN s /\ dist(x,y) < e ==> ~(f y = a)`,
6695 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
6696 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_WITHIN_AVOID THEN
6699 let CONTINUOUS_ON_OPEN_AVOID = prove
6700 (`!f:real^M->real^N x s a.
6701 f continuous_on s /\ open s /\ x IN s /\ ~(f x = a)
6702 ==> ?e. &0 < e /\ !y. dist(x,y) < e ==> ~(f y = a)`,
6703 REPEAT GEN_TAC THEN ASM_CASES_TAC `open(s:real^M->bool)` THEN
6704 ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT] THEN
6705 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_AVOID THEN
6708 (* ------------------------------------------------------------------------- *)
6709 (* Proving a function is constant by proving open-ness of level set. *)
6710 (* ------------------------------------------------------------------------- *)
6712 let CONTINUOUS_LEVELSET_OPEN_IN_CASES = prove
6713 (`!f:real^M->real^N s a.
6715 f continuous_on s /\
6716 open_in (subtopology euclidean s) {x | x IN s /\ f x = a}
6717 ==> (!x. x IN s ==> ~(f x = a)) \/ (!x. x IN s ==> f x = a)`,
6718 REWRITE_TAC[SET_RULE `(!x. x IN s ==> ~(f x = a)) <=>
6719 {x | x IN s /\ f x = a} = {}`;
6720 SET_RULE `(!x. x IN s ==> f x = a) <=>
6721 {x | x IN s /\ f x = a} = s`] THEN
6722 REWRITE_TAC[CONNECTED_CLOPEN] THEN REPEAT STRIP_TAC THEN
6723 FIRST_X_ASSUM MATCH_MP_TAC THEN
6724 ASM_SIMP_TAC[CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT]);;
6726 let CONTINUOUS_LEVELSET_OPEN_IN = prove
6727 (`!f:real^M->real^N s a.
6729 f continuous_on s /\
6730 open_in (subtopology euclidean s) {x | x IN s /\ f x = a} /\
6731 (?x. x IN s /\ f x = a)
6732 ==> (!x. x IN s ==> f x = a)`,
6733 MESON_TAC[CONTINUOUS_LEVELSET_OPEN_IN_CASES]);;
6735 let CONTINUOUS_LEVELSET_OPEN = prove
6736 (`!f:real^M->real^N s a.
6738 f continuous_on s /\
6739 open {x | x IN s /\ f x = a} /\
6740 (?x. x IN s /\ f x = a)
6741 ==> (!x. x IN s ==> f x = a)`,
6742 REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
6743 MATCH_MP_TAC CONTINUOUS_LEVELSET_OPEN_IN THEN
6744 ASM_REWRITE_TAC[OPEN_IN_OPEN] THEN
6745 EXISTS_TAC `{x | x IN s /\ (f:real^M->real^N) x = a}` THEN
6746 ASM_REWRITE_TAC[] THEN SET_TAC[]);;
6748 (* ------------------------------------------------------------------------- *)
6749 (* Some arithmetical combinations (more to prove). *)
6750 (* ------------------------------------------------------------------------- *)
6752 let OPEN_SCALING = prove
6753 (`!s:real^N->bool c. ~(c = &0) /\ open s ==> open(IMAGE (\x. c % x) s)`,
6754 REPEAT GEN_TAC THEN REWRITE_TAC[open_def; FORALL_IN_IMAGE] THEN
6755 STRIP_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
6756 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
6757 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
6758 EXISTS_TAC `e * abs(c)` THEN ASM_SIMP_TAC[REAL_LT_MUL; GSYM REAL_ABS_NZ] THEN
6759 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
6760 EXISTS_TAC `inv(c) % y:real^N` THEN
6761 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID] THEN
6762 FIRST_X_ASSUM MATCH_MP_TAC THEN
6763 SUBGOAL_THEN `x = inv(c) % c % x:real^N` SUBST1_TAC THENL
6764 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID];
6765 REWRITE_TAC[dist; GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN
6766 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[REAL_ABS_INV] THEN
6767 ASM_SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ; GSYM REAL_ABS_NZ] THEN
6768 ASM_REWRITE_TAC[GSYM dist]]);;
6770 let OPEN_NEGATIONS = prove
6771 (`!s:real^N->bool. open s ==> open (IMAGE (--) s)`,
6772 SUBGOAL_THEN `(--) = \x:real^N. --(&1) % x`
6773 (fun th -> SIMP_TAC[th; OPEN_SCALING; REAL_ARITH `~(--(&1) = &0)`]) THEN
6774 REWRITE_TAC[FUN_EQ_THM] THEN VECTOR_ARITH_TAC);;
6776 let OPEN_TRANSLATION = prove
6777 (`!s a:real^N. open s ==> open(IMAGE (\x. a + x) s)`,
6778 REPEAT STRIP_TAC THEN
6779 MP_TAC(ISPECL [`\x:real^N. x - a`; `s:real^N->bool`]
6780 CONTINUOUS_OPEN_PREIMAGE_UNIV) THEN
6781 ASM_SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_AT_ID; CONTINUOUS_CONST] THEN
6782 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
6783 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; IN_UNIV] THEN
6784 ASM_MESON_TAC[VECTOR_ARITH `(a + x) - a = x:real^N`;
6785 VECTOR_ARITH `a + (x - a) = x:real^N`]);;
6787 let OPEN_TRANSLATION_EQ = prove
6788 (`!a s. open (IMAGE (\x:real^N. a + x) s) <=> open s`,
6789 REWRITE_TAC[open_def] THEN GEOM_TRANSLATE_TAC[]);;
6791 add_translation_invariants [OPEN_TRANSLATION_EQ];;
6793 let OPEN_AFFINITY = prove
6795 open s /\ ~(c = &0) ==> open (IMAGE (\x. a + c % x) s)`,
6796 REPEAT STRIP_TAC THEN
6797 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
6798 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
6799 ASM_SIMP_TAC[IMAGE_o; OPEN_TRANSLATION; OPEN_SCALING]);;
6801 let INTERIOR_TRANSLATION = prove
6803 interior (IMAGE (\x. a + x) s) = IMAGE (\x. a + x) (interior s)`,
6804 REWRITE_TAC[interior] THEN GEOM_TRANSLATE_TAC[]);;
6806 add_translation_invariants [INTERIOR_TRANSLATION];;
6808 let OPEN_SUMS = prove
6809 (`!s t:real^N->bool.
6810 open s \/ open t ==> open {x + y | x IN s /\ y IN t}`,
6811 REPEAT GEN_TAC THEN REWRITE_TAC[open_def] THEN STRIP_TAC THEN
6812 REWRITE_TAC[FORALL_IN_GSPEC] THEN
6813 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THENL
6814 [FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`);
6815 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`)] THEN
6816 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
6817 X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
6818 X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
6819 ASM_MESON_TAC[VECTOR_ADD_SYM; VECTOR_ARITH `(z - y) + y:real^N = z`;
6820 NORM_ARITH `dist(z:real^N,x + y) < e ==> dist(z - y,x) < e`]);;
6822 (* ------------------------------------------------------------------------- *)
6823 (* Upper and lower hemicontinuous functions, relation in the case of *)
6824 (* preimage map to open and closed maps, and fact that upper and lower *)
6825 (* hemicontinuity together imply continuity in the sense of the Hausdorff *)
6826 (* metric (at points where the function gives a bounded and nonempty set). *)
6827 (* ------------------------------------------------------------------------- *)
6829 let UPPER_HEMICONTINUOUS = prove
6830 (`!f:real^M->real^N->bool t s.
6831 (!x. x IN s ==> f(x) SUBSET t)
6832 ==> ((!u. open_in (subtopology euclidean t) u
6833 ==> open_in (subtopology euclidean s)
6834 {x | x IN s /\ f(x) SUBSET u}) <=>
6835 (!u. closed_in (subtopology euclidean t) u
6836 ==> closed_in (subtopology euclidean s)
6837 {x | x IN s /\ ~(f(x) INTER u = {})}))`,
6838 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN GEN_TAC THEN
6839 FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF u:real^N->bool`) THEN
6840 MATCH_MP_TAC MONO_IMP THEN
6841 SIMP_TAC[OPEN_IN_DIFF; CLOSED_IN_DIFF; OPEN_IN_REFL; CLOSED_IN_REFL] THENL
6842 [REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ]; REWRITE_TAC[closed_in]] THEN
6843 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET_RESTRICT] THEN
6844 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
6846 let LOWER_HEMICONTINUOUS = prove
6847 (`!f:real^M->real^N->bool t s.
6848 (!x. x IN s ==> f(x) SUBSET t)
6849 ==> ((!u. closed_in (subtopology euclidean t) u
6850 ==> closed_in (subtopology euclidean s)
6851 {x | x IN s /\ f(x) SUBSET u}) <=>
6852 (!u. open_in (subtopology euclidean t) u
6853 ==> open_in (subtopology euclidean s)
6854 {x | x IN s /\ ~(f(x) INTER u = {})}))`,
6855 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN GEN_TAC THEN
6856 FIRST_X_ASSUM(MP_TAC o SPEC `t DIFF u:real^N->bool`) THEN
6857 MATCH_MP_TAC MONO_IMP THEN
6858 SIMP_TAC[OPEN_IN_DIFF; CLOSED_IN_DIFF; OPEN_IN_REFL; CLOSED_IN_REFL] THENL
6859 [REWRITE_TAC[closed_in]; REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ]] THEN
6860 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET_RESTRICT] THEN
6861 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
6863 let OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE = prove
6864 (`!f:real^M->real^N s t.
6866 ==> ((!u. open_in (subtopology euclidean s) u
6867 ==> open_in (subtopology euclidean t) (IMAGE f u)) <=>
6868 (!u. closed_in (subtopology euclidean s) u
6869 ==> closed_in (subtopology euclidean t)
6871 {x | x IN s /\ f x = y} SUBSET u}))`,
6872 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
6873 [X_GEN_TAC `v:real^M->bool` THEN DISCH_TAC THEN
6874 FIRST_X_ASSUM(MP_TAC o SPEC `s DIFF v:real^M->bool`) THEN
6875 ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN
6876 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
6877 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6878 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
6879 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[];
6880 X_GEN_TAC `v:real^M->bool` THEN DISCH_TAC THEN
6881 FIRST_X_ASSUM(MP_TAC o SPEC `s DIFF v:real^M->bool`) THEN
6882 ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN
6883 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
6884 REWRITE_TAC[OPEN_IN_CLOSED_IN_EQ; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
6885 DISCH_THEN(fun th -> CONJ_TAC THENL [ASM SET_TAC[]; MP_TAC th]) THEN
6886 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]]);;
6888 let CLOSED_MAP_IFF_UPPER_HEMICONTINUOUS_PREIMAGE = prove
6889 (`!f:real^M->real^N s t.
6891 ==> ((!u. closed_in (subtopology euclidean s) u
6892 ==> closed_in (subtopology euclidean t) (IMAGE f u)) <=>
6893 (!u. open_in (subtopology euclidean s) u
6894 ==> open_in (subtopology euclidean t)
6896 {x | x IN s /\ f x = y} SUBSET u}))`,
6897 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
6898 [X_GEN_TAC `v:real^M->bool` THEN DISCH_TAC THEN
6899 FIRST_X_ASSUM(MP_TAC o SPEC `s DIFF v:real^M->bool`) THEN
6900 ASM_SIMP_TAC[CLOSED_IN_DIFF; CLOSED_IN_REFL] THEN
6901 REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
6902 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6903 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
6904 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[];
6905 X_GEN_TAC `v:real^M->bool` THEN DISCH_TAC THEN
6906 FIRST_X_ASSUM(MP_TAC o SPEC `s DIFF v:real^M->bool`) THEN
6907 ASM_SIMP_TAC[OPEN_IN_DIFF; OPEN_IN_REFL] THEN
6908 FIRST_ASSUM(ASSUME_TAC o MATCH_MP CLOSED_IN_IMP_SUBSET) THEN
6909 REWRITE_TAC[closed_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
6910 DISCH_THEN(fun th -> CONJ_TAC THENL [ASM SET_TAC[]; MP_TAC th]) THEN
6911 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]]);;
6913 let UPPER_LOWER_HEMICONTINUOUS_EXPLICIT = prove
6914 (`!f:real^M->real^N->bool t s.
6915 (!x. x IN s ==> f(x) SUBSET t) /\
6916 (!u. open_in (subtopology euclidean t) u
6917 ==> open_in (subtopology euclidean s)
6918 {x | x IN s /\ f(x) SUBSET u}) /\
6919 (!u. closed_in (subtopology euclidean t) u
6920 ==> closed_in (subtopology euclidean s)
6921 {x | x IN s /\ f(x) SUBSET u})
6922 ==> !x e. x IN s /\ &0 < e /\ bounded(f x) /\ ~(f x = {})
6924 !x'. x' IN s /\ dist(x,x') < d
6926 ==> ?y'. y' IN f x' /\ dist(y,y') < e) /\
6928 ==> ?y. y IN f x /\ dist(y',y) < e)`,
6929 REPEAT STRIP_TAC THEN
6931 `!u. open_in (subtopology euclidean t) u
6932 ==> open_in (subtopology euclidean s)
6933 {x | x IN s /\ (f:real^M->real^N->bool)(x) SUBSET u}` THEN
6934 DISCH_THEN(MP_TAC o SPEC
6936 {a + b | a IN (f:real^M->real^N->bool) x /\ b IN ball(vec 0,e)}`) THEN
6937 SIMP_TAC[OPEN_SUMS; OPEN_BALL; OPEN_IN_OPEN_INTER] THEN
6938 REWRITE_TAC[open_in; SUBSET_RESTRICT] THEN
6939 DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
6940 ASM_SIMP_TAC[IN_ELIM_THM; SUBSET_INTER] THEN ANTS_TAC THENL
6941 [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
6942 ASM_MESON_TAC[CENTRE_IN_BALL; VECTOR_ADD_RID];
6943 DISCH_THEN(X_CHOOSE_THEN `d1:real`
6944 (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "1")))] THEN
6946 `!u. closed_in (subtopology euclidean t) u
6947 ==> closed_in (subtopology euclidean s)
6948 {x | x IN s /\ (f:real^M->real^N->bool)(x) SUBSET u}` THEN
6949 ASM_SIMP_TAC[LOWER_HEMICONTINUOUS] THEN DISCH_THEN(MP_TAC o
6950 GEN `a:real^N` o SPEC `t INTER ball(a:real^N,e / &2)`) THEN
6951 SIMP_TAC[OPEN_BALL; OPEN_IN_OPEN_INTER] THEN
6953 MP_TAC(SPEC `closure((f:real^M->real^N->bool) x)`
6954 COMPACT_EQ_HEINE_BOREL) THEN
6955 ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN DISCH_THEN(MP_TAC o SPEC
6956 `{ball(a:real^N,e / &2) | a IN (f:real^M->real^N->bool) x}`) THEN
6957 REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; OPEN_BALL] THEN
6958 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
6959 REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN ANTS_TAC THENL
6960 [REWRITE_TAC[CLOSURE_APPROACHABLE; SUBSET; UNIONS_IMAGE; IN_ELIM_THM] THEN
6961 REWRITE_TAC[IN_BALL] THEN ASM_SIMP_TAC[REAL_HALF];
6963 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
6964 DISCH_TAC THEN FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP
6965 (MESON[CLOSURE_SUBSET; SUBSET_TRANS]
6966 `closure s SUBSET t ==> s SUBSET t`)) THEN
6968 `open_in (subtopology euclidean s)
6969 (INTERS {{x | x IN s /\
6970 ~((f:real^M->real^N->bool) x INTER t INTER ball(a,e / &2) = {})} |
6973 [MATCH_MP_TAC OPEN_IN_INTERS THEN
6974 ASM_SIMP_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; FINITE_IMAGE] THEN
6975 ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ASM SET_TAC[];
6977 REWRITE_TAC[open_in] THEN
6978 DISCH_THEN(MP_TAC o SPEC `x:real^M` o CONJUNCT2) THEN ANTS_TAC THENL
6979 [REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM] THEN
6980 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
6981 ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
6982 EXISTS_TAC `a:real^N` THEN
6983 ASM_REWRITE_TAC[IN_INTER; CENTRE_IN_BALL; REAL_HALF] THEN
6985 DISCH_THEN(X_CHOOSE_THEN `d2:real`
6986 (CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "2")))] THEN
6987 EXISTS_TAC `min d1 d2:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
6988 X_GEN_TAC `x':real^M` THEN STRIP_TAC THEN CONJ_TAC THENL
6990 REMOVE_THEN "1" (MP_TAC o SPEC `x':real^M`) THEN
6991 ASM_REWRITE_TAC[] THEN
6992 ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN
6993 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_BALL] THEN
6994 REWRITE_TAC[VECTOR_ARITH `x:real^N = a + b <=> x - a = b`;
6995 DIST_0; ONCE_REWRITE_RULE[CONJ_SYM] UNWIND_THM1] THEN
6996 REWRITE_TAC[dist]] THEN
6997 REMOVE_THEN "2" (MP_TAC o SPEC `x':real^M`) THEN
6998 ASM_REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM] THEN
6999 ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN
7000 DISCH_THEN(LABEL_TAC "3") THEN
7001 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
7002 UNDISCH_TAC `(f:real^M->real^N->bool) x SUBSET
7003 UNIONS (IMAGE (\a. ball (a,e / &2)) c)` THEN
7004 REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN
7005 ASM_REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; IN_BALL] THEN
7006 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
7007 REMOVE_THEN "3" (MP_TAC o SPEC `a:real^N`) THEN
7008 ASM_REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_BALL] THEN
7009 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:real^N` THEN
7010 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
7011 ASM_MESON_TAC[DIST_TRIANGLE_HALF_L; DIST_SYM]);;
7013 (* ------------------------------------------------------------------------- *)
7014 (* Connected components, considered as a "connectedness" relation or a set. *)
7015 (* ------------------------------------------------------------------------- *)
7017 let connected_component = new_definition
7018 `connected_component s x y <=>
7019 ?t. connected t /\ t SUBSET s /\ x IN t /\ y IN t`;;
7021 let CONNECTED_COMPONENT_IN = prove
7022 (`!s x y. connected_component s x y ==> x IN s /\ y IN s`,
7023 REWRITE_TAC[connected_component] THEN SET_TAC[]);;
7025 let CONNECTED_COMPONENT_REFL = prove
7026 (`!s x:real^N. x IN s ==> connected_component s x x`,
7027 REWRITE_TAC[connected_component] THEN REPEAT STRIP_TAC THEN
7028 EXISTS_TAC `{x:real^N}` THEN REWRITE_TAC[CONNECTED_SING] THEN
7031 let CONNECTED_COMPONENT_REFL_EQ = prove
7032 (`!s x:real^N. connected_component s x x <=> x IN s`,
7033 REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[CONNECTED_COMPONENT_REFL] THEN
7034 REWRITE_TAC[connected_component] THEN SET_TAC[]);;
7036 let CONNECTED_COMPONENT_SYM = prove
7037 (`!s x y:real^N. connected_component s x y ==> connected_component s y x`,
7038 REWRITE_TAC[connected_component] THEN MESON_TAC[]);;
7040 let CONNECTED_COMPONENT_TRANS = prove
7042 connected_component s x y /\ connected_component s y z
7043 ==> connected_component s x z`,
7044 REPEAT GEN_TAC THEN REWRITE_TAC[connected_component] THEN
7045 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `t:real^N->bool`)
7046 (X_CHOOSE_TAC `u:real^N->bool`)) THEN
7047 EXISTS_TAC `t UNION u:real^N->bool` THEN
7048 ASM_REWRITE_TAC[IN_UNION; UNION_SUBSET] THEN
7049 MATCH_MP_TAC CONNECTED_UNION THEN ASM SET_TAC[]);;
7051 let CONNECTED_COMPONENT_OF_SUBSET = prove
7052 (`!s t x. s SUBSET t /\ connected_component s x y
7053 ==> connected_component t x y`,
7054 REWRITE_TAC[connected_component] THEN SET_TAC[]);;
7056 let CONNECTED_COMPONENT_SET = prove
7057 (`!s x. connected_component s x =
7058 { y | ?t. connected t /\ t SUBSET s /\ x IN t /\ y IN t}`,
7059 REWRITE_TAC[IN_ELIM_THM; EXTENSION] THEN
7060 REWRITE_TAC[IN; connected_component] THEN MESON_TAC[]);;
7062 let CONNECTED_COMPONENT_UNIONS = prove
7063 (`!s x. connected_component s x =
7064 UNIONS {t | connected t /\ x IN t /\ t SUBSET s}`,
7065 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN SET_TAC[]);;
7067 let CONNECTED_COMPONENT_SUBSET = prove
7068 (`!s x. (connected_component s x) SUBSET s`,
7069 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN SET_TAC[]);;
7071 let CONNECTED_CONNECTED_COMPONENT_SET = prove
7072 (`!s. connected s <=> !x:real^N. x IN s ==> connected_component s x = s`,
7073 GEN_TAC THEN REWRITE_TAC[CONNECTED_COMPONENT_UNIONS] THEN EQ_TAC THENL
7074 [SET_TAC[]; ALL_TAC] THEN
7075 ASM_CASES_TAC `s:real^N->bool = {}` THEN
7076 ASM_REWRITE_TAC[CONNECTED_EMPTY] THEN
7077 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7078 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
7079 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN
7080 DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC CONNECTED_UNIONS THEN
7083 let CONNECTED_COMPONENT_EQ_SELF = prove
7084 (`!s x. connected s /\ x IN s ==> connected_component s x = s`,
7085 MESON_TAC[CONNECTED_CONNECTED_COMPONENT_SET]);;
7087 let CONNECTED_IFF_CONNECTED_COMPONENT = prove
7088 (`!s. connected s <=>
7089 !x y. x IN s /\ y IN s ==> connected_component s x y`,
7090 REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT_SET] THEN
7091 REWRITE_TAC[EXTENSION] THEN MESON_TAC[IN; CONNECTED_COMPONENT_IN]);;
7093 let CONNECTED_COMPONENT_MAXIMAL = prove
7095 x IN t /\ connected t /\ t SUBSET s
7096 ==> t SUBSET (connected_component s x)`,
7097 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN SET_TAC[]);;
7099 let CONNECTED_COMPONENT_MONO = prove
7100 (`!s t x. s SUBSET t
7101 ==> (connected_component s x) SUBSET (connected_component t x)`,
7102 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN SET_TAC[]);;
7104 let CONNECTED_CONNECTED_COMPONENT = prove
7105 (`!s x. connected(connected_component s x)`,
7106 REWRITE_TAC[CONNECTED_COMPONENT_UNIONS] THEN
7107 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_UNIONS THEN SET_TAC[]);;
7109 let CONNECTED_COMPONENT_EQ_EMPTY = prove
7110 (`!s x:real^N. connected_component s x = {} <=> ~(x IN s)`,
7111 REPEAT GEN_TAC THEN EQ_TAC THENL
7112 [REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN
7113 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
7114 REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ];
7115 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN SET_TAC[]]);;
7117 let CONNECTED_COMPONENT_EMPTY = prove
7118 (`!x. connected_component {} x = {}`,
7119 REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY; NOT_IN_EMPTY]);;
7121 let CONNECTED_COMPONENT_EQ = prove
7122 (`!s x y. y IN connected_component s x
7123 ==> (connected_component s y = connected_component s x)`,
7124 REWRITE_TAC[EXTENSION; IN] THEN
7125 MESON_TAC[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS]);;
7127 let CLOSED_CONNECTED_COMPONENT = prove
7128 (`!s x:real^N. closed s ==> closed(connected_component s x)`,
7129 REPEAT STRIP_TAC THEN
7130 ASM_CASES_TAC `(x:real^N) IN s` THENL
7131 [ALL_TAC; ASM_MESON_TAC[CONNECTED_COMPONENT_EQ_EMPTY; CLOSED_EMPTY]] THEN
7132 REWRITE_TAC[GSYM CLOSURE_EQ] THEN
7133 MATCH_MP_TAC SUBSET_ANTISYM THEN REWRITE_TAC[CLOSURE_SUBSET] THEN
7134 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
7135 SIMP_TAC[CONNECTED_CLOSURE; CONNECTED_CONNECTED_COMPONENT] THEN
7137 [MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN
7138 ASM_REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ];
7139 MATCH_MP_TAC CLOSURE_MINIMAL THEN
7140 ASM_REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]]);;
7142 let CONNECTED_COMPONENT_DISJOINT = prove
7143 (`!s a b. DISJOINT (connected_component s a) (connected_component s b) <=>
7144 ~(a IN connected_component s b)`,
7145 REWRITE_TAC[DISJOINT; EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
7146 REWRITE_TAC[IN] THEN
7147 MESON_TAC[CONNECTED_COMPONENT_SYM; CONNECTED_COMPONENT_TRANS]);;
7149 let CONNECTED_COMPONENT_NONOVERLAP = prove
7151 (connected_component s a) INTER (connected_component s b) = {} <=>
7152 ~(a IN s) \/ ~(b IN s) \/
7153 ~(connected_component s a = connected_component s b)`,
7155 ASM_CASES_TAC `(a:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
7156 RULE_ASSUM_TAC(REWRITE_RULE[GSYM CONNECTED_COMPONENT_EQ_EMPTY]) THEN
7157 ASM_REWRITE_TAC[INTER_EMPTY] THEN
7158 ASM_CASES_TAC `(b:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
7159 RULE_ASSUM_TAC(REWRITE_RULE[GSYM CONNECTED_COMPONENT_EQ_EMPTY]) THEN
7160 ASM_REWRITE_TAC[INTER_EMPTY] THEN ASM_CASES_TAC
7161 `connected_component s (a:real^N) = connected_component s b` THEN
7162 ASM_REWRITE_TAC[INTER_IDEMPOT; CONNECTED_COMPONENT_EQ_EMPTY] THEN
7163 FIRST_X_ASSUM(MP_TAC o check(is_neg o concl)) THEN
7164 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
7165 REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN
7166 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM DISJOINT]) THEN
7167 REWRITE_TAC[CONNECTED_COMPONENT_DISJOINT]);;
7169 let CONNECTED_COMPONENT_OVERLAP = prove
7171 ~((connected_component s a) INTER (connected_component s b) = {}) <=>
7173 connected_component s a = connected_component s b`,
7174 REWRITE_TAC[CONNECTED_COMPONENT_NONOVERLAP; DE_MORGAN_THM]);;
7176 let CONNECTED_COMPONENT_SYM_EQ = prove
7177 (`!s x y. connected_component s x y <=> connected_component s y x`,
7178 MESON_TAC[CONNECTED_COMPONENT_SYM]);;
7180 let CONNECTED_COMPONENT_EQ_EQ = prove
7182 connected_component s x = connected_component s y <=>
7183 ~(x IN s) /\ ~(y IN s) \/
7184 x IN s /\ y IN s /\ connected_component s x y`,
7185 REPEAT GEN_TAC THEN ASM_CASES_TAC `(y:real^N) IN s` THENL
7186 [ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THENL
7187 [REWRITE_TAC[FUN_EQ_THM] THEN
7188 ASM_MESON_TAC[CONNECTED_COMPONENT_TRANS; CONNECTED_COMPONENT_REFL;
7189 CONNECTED_COMPONENT_SYM];
7190 ASM_MESON_TAC[CONNECTED_COMPONENT_EQ_EMPTY]];
7191 RULE_ASSUM_TAC(REWRITE_RULE[GSYM CONNECTED_COMPONENT_EQ_EMPTY]) THEN
7192 ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY] THEN
7193 ONCE_REWRITE_TAC[CONNECTED_COMPONENT_SYM_EQ] THEN
7194 ASM_REWRITE_TAC[EMPTY] THEN ASM_MESON_TAC[CONNECTED_COMPONENT_EQ_EMPTY]]);;
7196 let CONNECTED_EQ_CONNECTED_COMPONENT_EQ = prove
7197 (`!s. connected s <=>
7198 !x y. x IN s /\ y IN s
7199 ==> connected_component s x = connected_component s y`,
7200 SIMP_TAC[CONNECTED_COMPONENT_EQ_EQ] THEN
7201 REWRITE_TAC[CONNECTED_IFF_CONNECTED_COMPONENT]);;
7203 let CONNECTED_COMPONENT_IDEMP = prove
7204 (`!s x:real^N. connected_component (connected_component s x) x =
7205 connected_component s x`,
7206 REWRITE_TAC[FUN_EQ_THM; connected_component] THEN
7207 REPEAT GEN_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN EQ_TAC THEN
7208 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
7209 ASM_MESON_TAC[CONNECTED_COMPONENT_MAXIMAL; SUBSET_TRANS;
7210 CONNECTED_COMPONENT_SUBSET]);;
7212 let CONNECTED_COMPONENT_UNIQUE = prove
7214 x IN c /\ c SUBSET s /\ connected c /\
7215 (!c'. x IN c' /\ c' SUBSET s /\ connected c'
7217 ==> connected_component s x = c`,
7218 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
7219 [FIRST_X_ASSUM MATCH_MP_TAC THEN
7220 REWRITE_TAC[CONNECTED_COMPONENT_SUBSET; CONNECTED_CONNECTED_COMPONENT] THEN
7221 REWRITE_TAC[IN] THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN
7223 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[]]);;
7225 let JOINABLE_CONNECTED_COMPONENT_EQ = prove
7227 connected t /\ t SUBSET s /\
7228 ~(connected_component s x INTER t = {}) /\
7229 ~(connected_component s y INTER t = {})
7230 ==> connected_component s x = connected_component s y`,
7232 REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
7233 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN DISCH_THEN(CONJUNCTS_THEN2
7234 (X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC)
7235 (X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC)) THEN
7236 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_EQ THEN
7237 REWRITE_TAC[IN] THEN
7238 MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN
7239 EXISTS_TAC `z:real^N` THEN CONJ_TAC THENL [ASM_MESON_TAC[IN]; ALL_TAC] THEN
7240 MATCH_MP_TAC CONNECTED_COMPONENT_TRANS THEN
7241 EXISTS_TAC `w:real^N` THEN CONJ_TAC THENL
7242 [REWRITE_TAC[connected_component] THEN
7243 EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[];
7244 ASM_MESON_TAC[IN; CONNECTED_COMPONENT_SYM]]);;
7246 let CONNECTED_COMPONENT_TRANSLATION = prove
7247 (`!a s x. connected_component (IMAGE (\x. a + x) s) (a + x) =
7248 IMAGE (\x. a + x) (connected_component s x)`,
7249 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN GEOM_TRANSLATE_TAC[]);;
7251 add_translation_invariants [CONNECTED_COMPONENT_TRANSLATION];;
7253 let CONNECTED_COMPONENT_LINEAR_IMAGE = prove
7254 (`!f s x. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
7255 ==> connected_component (IMAGE f s) (f x) =
7256 IMAGE f (connected_component s x)`,
7257 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN
7258 GEOM_TRANSFORM_TAC[]);;
7260 add_linear_invariants [CONNECTED_COMPONENT_LINEAR_IMAGE];;
7262 let UNIONS_CONNECTED_COMPONENT = prove
7263 (`!s:real^N->bool. UNIONS {connected_component s x |x| x IN s} = s`,
7264 GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
7265 REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC; CONNECTED_COMPONENT_SUBSET] THEN
7266 REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM] THEN
7267 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXISTS_TAC `x:real^N` THEN
7268 ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN] THEN
7269 ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ]);;
7271 let COMPLEMENT_CONNECTED_COMPONENT_UNIONS = prove
7273 s DIFF connected_component s x =
7274 UNIONS({connected_component s y | y | y IN s} DELETE
7275 (connected_component s x))`,
7277 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV)
7278 [GSYM UNIONS_CONNECTED_COMPONENT] THEN
7279 MATCH_MP_TAC(SET_RULE
7280 `(!x. x IN s DELETE a ==> DISJOINT a x)
7281 ==> UNIONS s DIFF a = UNIONS (s DELETE a)`) THEN
7282 REWRITE_TAC[IMP_CONJ; FORALL_IN_GSPEC; IN_DELETE] THEN
7283 SIMP_TAC[CONNECTED_COMPONENT_DISJOINT; CONNECTED_COMPONENT_EQ_EQ] THEN
7284 MESON_TAC[IN; SUBSET; CONNECTED_COMPONENT_SUBSET]);;
7286 let CLOSED_IN_CONNECTED_COMPONENT = prove
7287 (`!s x:real^N. closed_in (subtopology euclidean s) (connected_component s x)`,
7289 ASM_CASES_TAC `connected_component s (x:real^N) = {}` THEN
7290 ASM_REWRITE_TAC[CLOSED_IN_EMPTY] THEN
7291 RULE_ASSUM_TAC(REWRITE_RULE[CONNECTED_COMPONENT_EQ_EMPTY]) THEN
7292 REWRITE_TAC[CLOSED_IN_CLOSED] THEN
7293 EXISTS_TAC `closure(connected_component s x):real^N->bool` THEN
7294 REWRITE_TAC[CLOSED_CLOSURE] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
7295 REWRITE_TAC[SUBSET_INTER; CONNECTED_COMPONENT_SUBSET; CLOSURE_SUBSET] THEN
7296 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN REWRITE_TAC[INTER_SUBSET] THEN
7298 [ASM_REWRITE_TAC[IN_INTER] THEN
7299 MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN
7300 ASM_REWRITE_TAC[IN; CONNECTED_COMPONENT_REFL_EQ];
7301 MATCH_MP_TAC CONNECTED_INTERMEDIATE_CLOSURE THEN
7302 EXISTS_TAC `connected_component s (x:real^N)` THEN
7303 REWRITE_TAC[INTER_SUBSET; CONNECTED_CONNECTED_COMPONENT;
7304 SUBSET_INTER; CONNECTED_COMPONENT_SUBSET; CLOSURE_SUBSET]]);;
7306 let OPEN_IN_CONNECTED_COMPONENT = prove
7308 FINITE {connected_component s x |x| x IN s}
7309 ==> open_in (subtopology euclidean s) (connected_component s x)`,
7310 REPEAT STRIP_TAC THEN
7312 `connected_component s (x:real^N) =
7313 s DIFF (UNIONS {connected_component s y |y| y IN s} DIFF
7314 connected_component s x)`
7316 [REWRITE_TAC[UNIONS_CONNECTED_COMPONENT] THEN
7317 MATCH_MP_TAC(SET_RULE `t SUBSET s ==> t = s DIFF (s DIFF t)`) THEN
7318 REWRITE_TAC[CONNECTED_COMPONENT_SUBSET];
7319 MATCH_MP_TAC OPEN_IN_DIFF THEN
7320 REWRITE_TAC[OPEN_IN_SUBTOPOLOGY_REFL; TOPSPACE_EUCLIDEAN; SUBSET_UNIV] THEN
7321 REWRITE_TAC[UNIONS_DIFF] THEN
7322 MATCH_MP_TAC CLOSED_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
7323 ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN
7324 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
7326 `connected_component s y DIFF connected_component s x =
7327 connected_component s y \/
7328 connected_component s (y:real^N) DIFF connected_component s x = {}`
7329 (DISJ_CASES_THEN SUBST1_TAC)
7331 [MATCH_MP_TAC(SET_RULE
7332 `(~(s INTER t = {}) ==> s = t) ==> s DIFF t = s \/ s DIFF t = {}`) THEN
7333 SIMP_TAC[CONNECTED_COMPONENT_OVERLAP];
7334 REWRITE_TAC[CLOSED_IN_CONNECTED_COMPONENT];
7335 REWRITE_TAC[CLOSED_IN_EMPTY]]]);;
7337 let CONNECTED_COMPONENT_EQUIVALENCE_RELATION = prove
7338 (`!R s:real^N->bool.
7339 (!x y. R x y ==> R y x) /\
7340 (!x y z. R x y /\ R y z ==> R x z) /\
7342 ==> ?t. open_in (subtopology euclidean s) t /\ a IN t /\
7343 !x. x IN t ==> R a x)
7344 ==> !a b. connected_component s a b ==> R a b`,
7345 REPEAT STRIP_TAC THEN
7346 MP_TAC(ISPECL [`R:real^N->real^N->bool`; `connected_component s (a:real^N)`]
7347 CONNECTED_EQUIVALENCE_RELATION) THEN
7348 ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN ANTS_TAC THENL
7349 [X_GEN_TAC `c:real^N` THEN DISCH_TAC THEN
7350 FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N`) THEN ANTS_TAC THENL
7351 [ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET; SUBSET]; ALL_TAC] THEN
7352 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
7353 EXISTS_TAC `t INTER connected_component s (a:real^N)` THEN
7354 ASM_SIMP_TAC[IN_INTER; OPEN_IN_OPEN] THEN
7355 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN]) THEN
7356 MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN
7357 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`]
7358 CONNECTED_COMPONENT_SUBSET) THEN
7360 DISCH_THEN MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN] THEN
7361 REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN
7362 ASM_MESON_TAC[CONNECTED_COMPONENT_IN]]);;
7364 let CONNECTED_COMPONENT_INTERMEDIATE_SUBSET = prove
7366 connected_component u a SUBSET t /\ t SUBSET u
7367 ==> connected_component t a = connected_component u a`,
7368 REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN u` THENL
7369 [REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_UNIQUE THEN
7370 ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN
7371 CONJ_TAC THENL [ASM_MESON_TAC[CONNECTED_COMPONENT_REFL; IN]; ALL_TAC] THEN
7372 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
7374 ASM_MESON_TAC[CONNECTED_COMPONENT_EQ_EMPTY; SUBSET]]);;
7376 (* ------------------------------------------------------------------------- *)
7377 (* The set of connected components of a set. *)
7378 (* ------------------------------------------------------------------------- *)
7380 let components = new_definition
7381 `components s = {connected_component s x | x | x:real^N IN s}`;;
7383 let COMPONENTS_TRANSLATION = prove
7384 (`!a s. components(IMAGE (\x. a + x) s) =
7385 IMAGE (IMAGE (\x. a + x)) (components s)`,
7386 REWRITE_TAC[components] THEN GEOM_TRANSLATE_TAC[] THEN SET_TAC[]);;
7388 add_translation_invariants [COMPONENTS_TRANSLATION];;
7390 let COMPONENTS_LINEAR_IMAGE = prove
7391 (`!f s. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
7392 ==> components(IMAGE f s) = IMAGE (IMAGE f) (components s)`,
7393 REWRITE_TAC[components] THEN GEOM_TRANSFORM_TAC[] THEN SET_TAC[]);;
7395 add_linear_invariants [COMPONENTS_LINEAR_IMAGE];;
7397 let IN_COMPONENTS = prove
7398 (`!u:real^N->bool s. s IN components u
7399 <=> ?x. x IN u /\ s = connected_component u x`,
7400 REPEAT GEN_TAC THEN REWRITE_TAC[components] THEN EQ_TAC
7401 THENL [SET_TAC[];STRIP_TAC THEN ASM_SIMP_TAC[] THEN
7402 UNDISCH_TAC `x:real^N IN u` THEN SET_TAC[]]);;
7404 let UNIONS_COMPONENTS = prove
7405 (`!u:real^N->bool. u = UNIONS (components u)`,
7406 REWRITE_TAC[EXTENSION] THEN REPEAT GEN_TAC THEN EQ_TAC
7407 THENL[DISCH_TAC THEN REWRITE_TAC[IN_UNIONS] THEN
7408 EXISTS_TAC `connected_component (u:real^N->bool) x` THEN CONJ_TAC THENL
7409 [REWRITE_TAC[components] THEN SET_TAC[ASSUME `x:real^N IN u`];
7410 REWRITE_TAC[CONNECTED_COMPONENT_SET] THEN SUBGOAL_THEN
7411 `?s:real^N->bool. connected s /\ s SUBSET u /\ x IN s` MP_TAC
7412 THENL[EXISTS_TAC `{x:real^N}` THEN ASM_REWRITE_TAC[CONNECTED_SING] THEN
7413 POP_ASSUM MP_TAC THEN SET_TAC[]; SET_TAC[]]];
7414 REWRITE_TAC[IN_UNIONS] THEN STRIP_TAC THEN
7415 MATCH_MP_TAC (SET_RULE `!x:real^N s u. x IN s /\ s SUBSET u ==> x IN u`) THEN
7416 EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC[] THEN STRIP_ASSUME_TAC
7417 (MESON[IN_COMPONENTS;ASSUME `t:real^N->bool IN components u`]
7418 `?y. t:real^N->bool = connected_component u y`) THEN
7419 ASM_REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]]);;
7421 let PAIRWISE_DISJOINT_COMPONENTS = prove
7422 (`!u:real^N->bool. pairwise DISJOINT (components u)`,
7423 GEN_TAC THEN REWRITE_TAC[pairwise;DISJOINT] THEN
7424 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`] THEN STRIP_TAC THEN
7425 ASSERT_TAC `(?a. s:real^N->bool = connected_component u a) /\
7426 ?b. t:real^N->bool = connected_component u b`
7427 THENL [ASM_MESON_TAC[IN_COMPONENTS];
7428 ASM_MESON_TAC[CONNECTED_COMPONENT_NONOVERLAP]]);;
7430 let IN_COMPONENTS_NONEMPTY = prove
7431 (`!s c. c IN components s ==> ~(c = {})`,
7432 REPEAT GEN_TAC THEN REWRITE_TAC[components; IN_ELIM_THM] THEN
7433 STRIP_TAC THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY]);;
7435 let IN_COMPONENTS_SUBSET = prove
7436 (`!s c. c IN components s ==> c SUBSET s`,
7437 REPEAT GEN_TAC THEN REWRITE_TAC[components; IN_ELIM_THM] THEN
7438 STRIP_TAC THEN ASM_REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]);;
7440 let IN_COMPONENTS_CONNECTED = prove
7441 (`!s c. c IN components s ==> connected c`,
7442 REPEAT GEN_TAC THEN REWRITE_TAC[components; IN_ELIM_THM] THEN
7443 STRIP_TAC THEN ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT]);;
7445 let IN_COMPONENTS_MAXIMAL = prove
7446 (`!s c:real^N->bool.
7447 c IN components s <=>
7448 ~(c = {}) /\ c SUBSET s /\ connected c /\
7449 !c'. ~(c' = {}) /\ c SUBSET c' /\ c' SUBSET s /\ connected c'
7451 REPEAT GEN_TAC THEN REWRITE_TAC[components; IN_ELIM_THM] THEN EQ_TAC THENL
7452 [DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
7453 ASM_REWRITE_TAC[CONNECTED_COMPONENT_EQ_EMPTY; CONNECTED_COMPONENT_SUBSET;
7454 CONNECTED_CONNECTED_COMPONENT] THEN
7455 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
7456 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
7457 ASM_MESON_TAC[CONNECTED_COMPONENT_REFL; IN; SUBSET];
7459 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7460 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN
7461 DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
7462 MATCH_MP_TAC(GSYM CONNECTED_COMPONENT_UNIQUE) THEN
7463 ASM_REWRITE_TAC[] THEN X_GEN_TAC `c':real^N->bool` THEN STRIP_TAC THEN
7464 REWRITE_TAC[SET_RULE `c' SUBSET c <=> c' UNION c = c`] THEN
7465 FIRST_X_ASSUM MATCH_MP_TAC THEN
7466 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
7467 MATCH_MP_TAC CONNECTED_UNION THEN ASM SET_TAC[]]);;
7469 let JOINABLE_COMPONENTS_EQ = prove
7471 connected t /\ t SUBSET s /\
7472 c1 IN components s /\ c2 IN components s /\
7473 ~(c1 INTER t = {}) /\ ~(c2 INTER t = {})
7475 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; components; FORALL_IN_GSPEC] THEN
7476 MESON_TAC[JOINABLE_CONNECTED_COMPONENT_EQ]);;
7478 let CLOSED_IN_COMPONENT = prove
7479 (`!s c:real^N->bool.
7480 c IN components s ==> closed_in (subtopology euclidean s) c`,
7481 REWRITE_TAC[components; FORALL_IN_GSPEC; CLOSED_IN_CONNECTED_COMPONENT]);;
7483 let CLOSED_COMPONENTS = prove
7484 (`!s c. closed s /\ c IN components s ==> closed c`,
7485 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; components; FORALL_IN_GSPEC] THEN
7486 SIMP_TAC[CLOSED_CONNECTED_COMPONENT]);;
7488 let COMPACT_COMPONENTS = prove
7489 (`!s c:real^N->bool. compact s /\ c IN components s ==> compact c`,
7490 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN
7491 MESON_TAC[CLOSED_COMPONENTS; IN_COMPONENTS_SUBSET; BOUNDED_SUBSET]);;
7493 let CONTINUOUS_ON_COMPONENTS_GEN = prove
7494 (`!f:real^M->real^N s.
7495 (!c. c IN components s
7496 ==> open_in (subtopology euclidean s) c /\ f continuous_on c)
7497 ==> f continuous_on s`,
7498 REPEAT GEN_TAC THEN REWRITE_TAC[CONTINUOUS_OPEN_IN_PREIMAGE_EQ] THEN
7499 DISCH_TAC THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_TAC THEN
7501 `{x | x IN s /\ (f:real^M->real^N) x IN t} =
7502 UNIONS {{x | x IN c /\ f x IN t} | c IN components s}`
7504 [CONV_TAC(LAND_CONV(SUBS_CONV
7505 [ISPEC `s:real^M->bool` UNIONS_COMPONENTS])) THEN
7506 REWRITE_TAC[UNIONS_GSPEC; IN_UNIONS] THEN SET_TAC[];
7507 MATCH_MP_TAC OPEN_IN_UNIONS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
7508 ASM_MESON_TAC[OPEN_IN_TRANS]]);;
7510 let CONTINUOUS_ON_COMPONENTS_FINITE = prove
7511 (`!f:real^M->real^N s.
7512 FINITE(components s) /\
7513 (!c. c IN components s ==> f continuous_on c)
7514 ==> f continuous_on s`,
7515 REPEAT GEN_TAC THEN REWRITE_TAC[CONTINUOUS_CLOSED_IN_PREIMAGE_EQ] THEN
7516 DISCH_TAC THEN X_GEN_TAC `t:real^N->bool` THEN DISCH_TAC THEN
7518 `{x | x IN s /\ (f:real^M->real^N) x IN t} =
7519 UNIONS {{x | x IN c /\ f x IN t} | c IN components s}`
7521 [CONV_TAC(LAND_CONV(SUBS_CONV
7522 [ISPEC `s:real^M->bool` UNIONS_COMPONENTS])) THEN
7523 REWRITE_TAC[UNIONS_GSPEC; IN_UNIONS] THEN SET_TAC[];
7524 MATCH_MP_TAC CLOSED_IN_UNIONS THEN
7525 ASM_SIMP_TAC[SIMPLE_IMAGE; FINITE_IMAGE; FORALL_IN_IMAGE] THEN
7526 ASM_MESON_TAC[CLOSED_IN_TRANS; CLOSED_IN_COMPONENT]]);;
7528 let COMPONENTS_NONOVERLAP = prove
7529 (`!s c c'. c IN components s /\ c' IN components s
7530 ==> (c INTER c' = {} <=> ~(c = c'))`,
7531 REWRITE_TAC[components; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
7532 ASM_SIMP_TAC[CONNECTED_COMPONENT_NONOVERLAP]);;
7534 let COMPONENTS_EQ = prove
7535 (`!s c c'. c IN components s /\ c' IN components s
7536 ==> (c = c' <=> ~(c INTER c' = {}))`,
7537 MESON_TAC[COMPONENTS_NONOVERLAP]);;
7539 let COMPONENTS_EQ_EMPTY = prove
7540 (`!s. components s = {} <=> s = {}`,
7541 GEN_TAC THEN REWRITE_TAC[EXTENSION] THEN
7542 REWRITE_TAC[components; connected_component; IN_ELIM_THM] THEN
7545 let COMPONENTS_EMPTY = prove
7546 (`components {} = {}`,
7547 REWRITE_TAC[COMPONENTS_EQ_EMPTY]);;
7549 let CONNECTED_EQ_CONNECTED_COMPONENTS_EQ = prove
7550 (`!s. connected s <=>
7551 !c c'. c IN components s /\ c' IN components s ==> c = c'`,
7552 REWRITE_TAC[components; IN_ELIM_THM] THEN
7553 MESON_TAC[CONNECTED_EQ_CONNECTED_COMPONENT_EQ]);;
7555 let COMPONENTS_EQ_SING,COMPONENTS_EQ_SING_EXISTS = (CONJ_PAIR o prove)
7556 (`(!s:real^N->bool. components s = {s} <=> connected s /\ ~(s = {})) /\
7557 (!s:real^N->bool. (?a. components s = {a}) <=> connected s /\ ~(s = {}))`,
7558 REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `s:real^N->bool` THEN
7559 MATCH_MP_TAC(TAUT `(p ==> q) /\ (q ==> r) /\ (r ==> p)
7560 ==> (p <=> r) /\ (q <=> r)`) THEN
7561 REPEAT CONJ_TAC THENL
7563 STRIP_TAC THEN ASM_REWRITE_TAC[CONNECTED_EQ_CONNECTED_COMPONENTS_EQ] THEN
7564 ASM_MESON_TAC[IN_SING; COMPONENTS_EQ_EMPTY; NOT_INSERT_EMPTY];
7565 STRIP_TAC THEN ONCE_REWRITE_TAC[EXTENSION] THEN
7566 REWRITE_TAC[IN_SING] THEN
7567 REWRITE_TAC[components; IN_ELIM_THM] THEN
7568 ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT_SET; MEMBER_NOT_EMPTY]]);;
7570 let CONNECTED_EQ_COMPONENTS_SUBSET_SING = prove
7571 (`!s:real^N->bool. connected s <=> components s SUBSET {s}`,
7572 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
7573 ASM_REWRITE_TAC[COMPONENTS_EMPTY; CONNECTED_EMPTY; EMPTY_SUBSET] THEN
7574 REWRITE_TAC[SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN
7575 ASM_REWRITE_TAC[COMPONENTS_EQ_EMPTY; COMPONENTS_EQ_SING]);;
7577 let CONNECTED_EQ_COMPONENTS_SUBSET_SING_EXISTS = prove
7578 (`!s:real^N->bool. connected s <=> ?a. components s SUBSET {a}`,
7579 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
7580 ASM_REWRITE_TAC[COMPONENTS_EMPTY; CONNECTED_EMPTY; EMPTY_SUBSET] THEN
7581 REWRITE_TAC[SET_RULE `s SUBSET {a} <=> s = {} \/ s = {a}`] THEN
7582 ASM_REWRITE_TAC[COMPONENTS_EQ_EMPTY; COMPONENTS_EQ_SING_EXISTS]);;
7584 let IN_COMPONENTS_SELF = prove
7585 (`!s:real^N->bool. s IN components s <=> connected s /\ ~(s = {})`,
7586 GEN_TAC THEN EQ_TAC THENL
7587 [MESON_TAC[IN_COMPONENTS_NONEMPTY; IN_COMPONENTS_CONNECTED];
7588 SIMP_TAC[GSYM COMPONENTS_EQ_SING; IN_SING]]);;
7590 let COMPONENTS_MAXIMAL = prove
7591 (`!s t c:real^N->bool.
7592 c IN components s /\ connected t /\ t SUBSET s /\ ~(c INTER t = {})
7594 REWRITE_TAC[IMP_CONJ; components; FORALL_IN_GSPEC] THEN
7595 REPEAT STRIP_TAC THEN
7596 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7597 REWRITE_TAC[IN_INTER; LEFT_IMP_EXISTS_THM] THEN
7598 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
7599 FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP CONNECTED_COMPONENT_EQ) THEN
7600 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN ASM_REWRITE_TAC[]);;
7602 let COMPONENTS_UNIQUE = prove
7603 (`!s:real^N->bool k.
7606 ==> connected c /\ ~(c = {}) /\
7607 !c'. connected c' /\ c SUBSET c' /\ c' SUBSET s ==> c' = c)
7608 ==> components s = k`,
7609 REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
7610 X_GEN_TAC `c:real^N->bool` THEN REWRITE_TAC[IN_COMPONENTS] THEN
7612 [DISCH_THEN(X_CHOOSE_THEN `x:real^N`
7613 (CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC)) THEN
7614 FIRST_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [EXTENSION]) THEN
7615 REWRITE_TAC[IN_UNIONS] THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
7616 X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN
7617 SUBGOAL_THEN `connected_component s (x:real^N) = c`
7618 (fun th -> ASM_REWRITE_TAC[th]) THEN
7619 MATCH_MP_TAC CONNECTED_COMPONENT_UNIQUE THEN
7620 FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN
7621 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
7622 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
7623 X_GEN_TAC `c':real^N->bool` THEN STRIP_TAC THEN
7624 REWRITE_TAC[SET_RULE `c' SUBSET c <=> c' UNION c = c`] THEN
7625 FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
7626 [MATCH_MP_TAC CONNECTED_UNION; ASM SET_TAC[]] THEN
7628 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN
7629 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
7630 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7631 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
7632 CONJ_TAC THENL [ASM SET_TAC[]; CONV_TAC SYM_CONV] THEN
7633 FIRST_X_ASSUM MATCH_MP_TAC THEN
7634 REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT; CONNECTED_COMPONENT_SUBSET] THEN
7635 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
7636 ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
7638 let COMPONENTS_UNIQUE_EQ = prove
7639 (`!s:real^N->bool k.
7640 components s = k <=>
7643 ==> connected c /\ ~(c = {}) /\
7644 !c'. connected c' /\ c SUBSET c' /\ c' SUBSET s ==> c' = c)`,
7645 REPEAT GEN_TAC THEN EQ_TAC THENL
7646 [DISCH_THEN(SUBST1_TAC o SYM); REWRITE_TAC[COMPONENTS_UNIQUE]] THEN
7647 REWRITE_TAC[GSYM UNIONS_COMPONENTS] THEN
7648 X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN REPEAT CONJ_TAC THENL
7649 [ASM_MESON_TAC[IN_COMPONENTS_CONNECTED];
7650 ASM_MESON_TAC[IN_COMPONENTS_NONEMPTY];
7651 RULE_ASSUM_TAC(REWRITE_RULE[IN_COMPONENTS_MAXIMAL]) THEN
7652 ASM_MESON_TAC[SUBSET_EMPTY]]);;
7654 let EXISTS_COMPONENT_SUPERSET = prove
7655 (`!s t:real^N->bool.
7656 t SUBSET s /\ ~(s = {}) /\ connected t
7657 ==> ?c. c IN components s /\ t SUBSET c`,
7658 REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL
7659 [ASM_REWRITE_TAC[EMPTY_SUBSET] THEN
7660 ASM_MESON_TAC[COMPONENTS_EQ_EMPTY; MEMBER_NOT_EMPTY];
7661 FIRST_X_ASSUM(X_CHOOSE_TAC `a:real^N` o
7662 GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7663 EXISTS_TAC `connected_component s (a:real^N)` THEN
7664 REWRITE_TAC[IN_COMPONENTS] THEN CONJ_TAC THENL
7665 [ASM SET_TAC[]; ASM_MESON_TAC[CONNECTED_COMPONENT_MAXIMAL]]]);;
7667 let COMPONENTS_INTERMEDIATE_SUBSET = prove
7668 (`!s t u:real^N->bool.
7669 s IN components u /\ s SUBSET t /\ t SUBSET u
7670 ==> s IN components t`,
7671 REPEAT GEN_TAC THEN REWRITE_TAC[IN_COMPONENTS; LEFT_AND_EXISTS_THM] THEN
7672 MESON_TAC[CONNECTED_COMPONENT_INTERMEDIATE_SUBSET; SUBSET;
7673 CONNECTED_COMPONENT_REFL; IN; CONNECTED_COMPONENT_SUBSET]);;
7675 let IN_COMPONENTS_UNIONS_COMPLEMENT = prove
7676 (`!s c:real^N->bool.
7678 ==> s DIFF c = UNIONS(components s DELETE c)`,
7679 REWRITE_TAC[components; FORALL_IN_GSPEC;
7680 COMPLEMENT_CONNECTED_COMPONENT_UNIONS]);;
7682 let CONNECTED_SUBSET_CLOPEN = prove
7683 (`!u s c:real^N->bool.
7684 closed_in (subtopology euclidean u) s /\
7685 open_in (subtopology euclidean u) s /\
7686 connected c /\ c SUBSET u /\ ~(c INTER s = {})
7688 REPEAT STRIP_TAC THEN
7689 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOSED_IN]) THEN
7690 REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o
7691 SPECL [`c INTER s:real^N->bool`; `c DIFF s:real^N->bool`]) THEN
7692 ASM_REWRITE_TAC[CONJ_ASSOC; SET_RULE `c DIFF s = {} <=> c SUBSET s`] THEN
7693 MATCH_MP_TAC(TAUT `p ==> ~(p /\ ~q) ==> q`) THEN
7694 REPLICATE_TAC 2 (CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]]) THEN
7696 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSED_IN_CLOSED]);
7697 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_IN_OPEN])] THEN
7698 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
7699 REWRITE_TAC[OPEN_IN_OPEN; CLOSED_IN_CLOSED] THENL
7700 [EXISTS_TAC `t:real^N->bool`; EXISTS_TAC `(:real^N) DIFF t`] THEN
7701 ASM_REWRITE_TAC[GSYM OPEN_CLOSED] THEN ASM SET_TAC[]);;
7703 let CLOPEN_UNIONS_COMPONENTS = prove
7704 (`!u s:real^N->bool.
7705 closed_in (subtopology euclidean u) s /\
7706 open_in (subtopology euclidean u) s
7707 ==> ?k. k SUBSET components u /\ s = UNIONS k`,
7708 REPEAT STRIP_TAC THEN
7709 EXISTS_TAC `{c:real^N->bool | c IN components u /\ ~(c INTER s = {})}` THEN
7710 REWRITE_TAC[SUBSET_RESTRICT] THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
7712 [MP_TAC(ISPEC `u:real^N->bool` UNIONS_COMPONENTS) THEN
7713 FIRST_ASSUM(MP_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN SET_TAC[];
7714 REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN
7715 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONNECTED_SUBSET_CLOPEN THEN
7716 EXISTS_TAC `u:real^N->bool` THEN
7717 ASM_MESON_TAC[IN_COMPONENTS_CONNECTED; IN_COMPONENTS_SUBSET]]);;
7719 let CLOPEN_IN_COMPONENTS = prove
7720 (`!u s:real^N->bool.
7721 closed_in (subtopology euclidean u) s /\
7722 open_in (subtopology euclidean u) s /\
7723 connected s /\ ~(s = {})
7724 ==> s IN components u`,
7725 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN
7726 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
7727 FIRST_ASSUM(MP_TAC o MATCH_MP CLOPEN_UNIONS_COMPONENTS) THEN
7728 DISCH_THEN(X_CHOOSE_THEN `k:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
7729 ASM_CASES_TAC `k:(real^N->bool)->bool = {}` THEN
7730 ASM_REWRITE_TAC[UNIONS_0] THEN
7731 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
7732 DISCH_THEN(X_CHOOSE_TAC `c:real^N->bool`) THEN
7733 ASM_CASES_TAC `k = {c:real^N->bool}` THENL
7734 [ASM_MESON_TAC[UNIONS_1; GSYM SING_SUBSET]; ALL_TAC] THEN
7735 MATCH_MP_TAC(TAUT `~p ==> p /\ q ==> r`) THEN
7736 SUBGOAL_THEN `?c':real^N->bool. c' IN k /\ ~(c = c')` STRIP_ASSUME_TAC THENL
7737 [ASM_MESON_TAC[SET_RULE
7738 `a IN s /\ ~(s = {a}) ==> ?b. b IN s /\ ~(b = a)`];
7739 REWRITE_TAC[CONNECTED_EQ_CONNECTED_COMPONENTS_EQ] THEN
7740 DISCH_THEN(MP_TAC o SPECL [`c:real^N->bool`; `c':real^N->bool`]) THEN
7741 ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THEN
7742 MATCH_MP_TAC COMPONENTS_INTERMEDIATE_SUBSET THEN
7743 EXISTS_TAC `u:real^N->bool` THEN
7744 MP_TAC(ISPEC `u:real^N->bool` UNIONS_COMPONENTS) THEN ASM SET_TAC[]]);;
7746 (* ------------------------------------------------------------------------- *)
7747 (* Continuity implies uniform continuity on a compact domain. *)
7748 (* ------------------------------------------------------------------------- *)
7750 let COMPACT_UNIFORMLY_EQUICONTINUOUS = prove
7751 (`!(fs:(real^M->real^N)->bool) s.
7752 (!x e. x IN s /\ &0 < e
7754 (!f x'. f IN fs /\ x' IN s /\ dist (x',x) < d
7755 ==> dist (f x',f x) < e)) /\
7759 !f x x'. f IN fs /\ x IN s /\ x' IN s /\ dist (x',x) < d
7760 ==> dist(f x',f x) < e`,
7761 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
7762 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
7763 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
7764 X_GEN_TAC `d:real^M->real->real` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
7765 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP HEINE_BOREL_LEMMA) THEN
7766 DISCH_THEN(MP_TAC o SPEC
7767 `{ ball(x:real^M,d x (e / &2)) | x IN s}`) THEN
7768 SIMP_TAC[FORALL_IN_GSPEC; OPEN_BALL; UNIONS_GSPEC; SUBSET; IN_ELIM_THM] THEN
7769 ANTS_TAC THENL [ASM_MESON_TAC[CENTRE_IN_BALL; REAL_HALF]; ALL_TAC] THEN
7770 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN
7771 ASM_REWRITE_TAC[] THEN
7772 MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `u:real^M`; `v:real^M`] THEN
7773 STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `v:real^M` th) THEN
7774 ASM_REWRITE_TAC[] THEN DISCH_THEN(CHOOSE_THEN MP_TAC)) THEN
7775 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
7776 DISCH_THEN(fun th ->
7777 MP_TAC(SPEC `u:real^M` th) THEN MP_TAC(SPEC `v:real^M` th)) THEN
7778 ASM_REWRITE_TAC[DIST_REFL] THEN
7779 FIRST_X_ASSUM(X_CHOOSE_THEN `w:real^M` (CONJUNCTS_THEN2 ASSUME_TAC
7780 SUBST_ALL_TAC)) THEN
7781 ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN ASM_REWRITE_TAC[IN_BALL] THEN
7782 ONCE_REWRITE_TAC[DIST_SYM] THEN REPEAT STRIP_TAC THEN
7783 FIRST_X_ASSUM(MP_TAC o SPECL [`w:real^M`; `e / &2`]) THEN
7784 ASM_REWRITE_TAC[REAL_HALF] THEN
7785 DISCH_THEN(MP_TAC o SPEC `f:real^M->real^N` o CONJUNCT2) THEN
7786 DISCH_THEN(fun th -> MP_TAC(SPEC `u:real^M` th) THEN
7787 MP_TAC(SPEC `v:real^M` th)) THEN
7788 ASM_REWRITE_TAC[] THEN CONV_TAC NORM_ARITH);;
7790 let COMPACT_UNIFORMLY_CONTINUOUS = prove
7791 (`!f:real^M->real^N s.
7792 f continuous_on s /\ compact s ==> f uniformly_continuous_on s`,
7793 REPEAT GEN_TAC THEN REWRITE_TAC[continuous_on; uniformly_continuous_on] THEN
7795 MP_TAC(ISPECL [`{f:real^M->real^N}`; `s:real^M->bool`]
7796 COMPACT_UNIFORMLY_EQUICONTINUOUS) THEN
7797 REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; IN_SING; FORALL_UNWIND_THM2] THEN
7800 (* ------------------------------------------------------------------------- *)
7801 (* A uniformly convergent limit of continuous functions is continuous. *)
7802 (* ------------------------------------------------------------------------- *)
7804 let CONTINUOUS_UNIFORM_LIMIT = prove
7805 (`!net f:A->real^M->real^N g s.
7806 ~(trivial_limit net) /\
7807 eventually (\n. (f n) continuous_on s) net /\
7809 ==> eventually (\n. !x. x IN s ==> norm(f n x - g x) < e) net)
7810 ==> g continuous_on s`,
7811 REWRITE_TAC[continuous_on] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
7812 X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
7813 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
7814 FIRST_X_ASSUM(MP_TAC o SPEC `e / &3`) THEN
7815 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
7816 FIRST_X_ASSUM(fun th -> MP_TAC th THEN REWRITE_TAC[IMP_IMP] THEN
7817 GEN_REWRITE_TAC LAND_CONV [GSYM EVENTUALLY_AND]) THEN
7818 DISCH_THEN(MP_TAC o MATCH_MP EVENTUALLY_HAPPENS) THEN
7819 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:A` THEN
7820 DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `x:real^M`) ASSUME_TAC) THEN
7821 ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
7822 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
7823 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
7824 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
7825 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^M` THEN
7826 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
7827 FIRST_X_ASSUM(fun th ->
7828 MP_TAC(SPEC `x:real^M` th) THEN MP_TAC(SPEC `y:real^M` th)) THEN
7829 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
7831 ==> x < e / &3 ==> y < e / &3 ==> z < e / &3 ==> w < e`) THEN
7832 REWRITE_TAC[dist] THEN
7833 SUBST1_TAC(VECTOR_ARITH
7834 `(g:real^M->real^N) y - g x =
7835 --(f (a:A) y - g y) + (f a x - g x) + (f a y - f a x)`) THEN
7836 MATCH_MP_TAC NORM_TRIANGLE_LE THEN REWRITE_TAC[NORM_NEG; REAL_LE_LADD] THEN
7837 MATCH_MP_TAC NORM_TRIANGLE_LE THEN REWRITE_TAC[NORM_NEG; REAL_LE_REFL]);;
7839 (* ------------------------------------------------------------------------- *)
7840 (* Topological stuff lifted from and dropped to R *)
7841 (* ------------------------------------------------------------------------- *)
7843 let OPEN_LIFT = prove
7844 (`!s. open(IMAGE lift s) <=>
7845 !x. x IN s ==> ?e. &0 < e /\ !x'. abs(x' - x) < e ==> x' IN s`,
7846 REWRITE_TAC[open_def; FORALL_LIFT; LIFT_IN_IMAGE_LIFT; DIST_LIFT]);;
7848 let LIMPT_APPROACHABLE_LIFT = prove
7849 (`!x s. (lift x) limit_point_of (IMAGE lift s) <=>
7850 !e. &0 < e ==> ?x'. x' IN s /\ ~(x' = x) /\ abs(x' - x) < e`,
7851 REWRITE_TAC[LIMPT_APPROACHABLE; EXISTS_LIFT; LIFT_IN_IMAGE_LIFT;
7852 LIFT_EQ; DIST_LIFT]);;
7854 let CLOSED_LIFT = prove
7855 (`!s. closed (IMAGE lift s) <=>
7856 !x. (!e. &0 < e ==> ?x'. x' IN s /\ ~(x' = x) /\ abs(x' - x) < e)
7858 GEN_TAC THEN REWRITE_TAC[CLOSED_LIMPT; LIMPT_APPROACHABLE] THEN
7859 ONCE_REWRITE_TAC[FORALL_LIFT] THEN
7860 REWRITE_TAC[LIMPT_APPROACHABLE_LIFT; LIFT_EQ; DIST_LIFT;
7861 EXISTS_LIFT; LIFT_IN_IMAGE_LIFT]);;
7863 let CONTINUOUS_AT_LIFT_RANGE = prove
7864 (`!f x. (lift o f) continuous (at x) <=>
7867 (!x'. norm(x' - x) < d
7868 ==> abs(f x' - f x) < e)`,
7869 REWRITE_TAC[continuous_at; o_THM; DIST_LIFT] THEN REWRITE_TAC[dist]);;
7871 let CONTINUOUS_ON_LIFT_RANGE = prove
7872 (`!f s. (lift o f) continuous_on s <=>
7876 (!x'. x' IN s /\ norm(x' - x) < d
7877 ==> abs(f x' - f x) < e)`,
7878 REWRITE_TAC[continuous_on; o_THM; DIST_LIFT] THEN REWRITE_TAC[dist]);;
7880 let CONTINUOUS_LIFT_NORM_COMPOSE = prove
7883 ==> (\x. lift(norm(f x))) continuous net`,
7884 REPEAT GEN_TAC THEN REWRITE_TAC[continuous; tendsto] THEN
7885 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
7887 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
7888 REWRITE_TAC[DIST_REAL; GSYM drop; LIFT_DROP] THEN
7891 let CONTINUOUS_ON_LIFT_NORM_COMPOSE = prove
7892 (`!f:real^M->real^N s.
7894 ==> (\x. lift(norm(f x))) continuous_on s`,
7895 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_LIFT_NORM_COMPOSE]);;
7897 let CONTINUOUS_AT_LIFT_NORM = prove
7898 (`!x. (lift o norm) continuous (at x)`,
7899 REWRITE_TAC[CONTINUOUS_AT_LIFT_RANGE; NORM_LIFT] THEN
7900 MESON_TAC[REAL_ABS_SUB_NORM; REAL_LET_TRANS]);;
7902 let CONTINUOUS_ON_LIFT_NORM = prove
7903 (`!s. (lift o norm) continuous_on s`,
7904 REWRITE_TAC[CONTINUOUS_ON_LIFT_RANGE; NORM_LIFT] THEN
7905 MESON_TAC[REAL_ABS_SUB_NORM; REAL_LET_TRANS]);;
7907 let CONTINUOUS_AT_LIFT_COMPONENT = prove
7908 (`!i a. 1 <= i /\ i <= dimindex(:N)
7909 ==> (\x:real^N. lift(x$i)) continuous (at a)`,
7910 SIMP_TAC[continuous_at; DIST_LIFT; GSYM VECTOR_SUB_COMPONENT] THEN
7911 MESON_TAC[dist; REAL_LET_TRANS; COMPONENT_LE_NORM]);;
7913 let CONTINUOUS_ON_LIFT_COMPONENT = prove
7914 (`!i s. 1 <= i /\ i <= dimindex(:N)
7915 ==> (\x:real^N. lift(x$i)) continuous_on s`,
7916 SIMP_TAC[continuous_on; DIST_LIFT; GSYM VECTOR_SUB_COMPONENT] THEN
7917 MESON_TAC[dist; REAL_LET_TRANS; COMPONENT_LE_NORM]);;
7919 let CONTINUOUS_AT_LIFT_INFNORM = prove
7920 (`!x:real^N. (lift o infnorm) continuous (at x)`,
7921 REWRITE_TAC[CONTINUOUS_AT; LIM_AT; o_THM; DIST_LIFT] THEN
7922 MESON_TAC[REAL_LET_TRANS; dist; REAL_ABS_SUB_INFNORM; INFNORM_LE_NORM]);;
7924 let CONTINUOUS_AT_LIFT_DIST = prove
7925 (`!a:real^N x. (lift o (\x. dist(a,x))) continuous (at x)`,
7926 REWRITE_TAC[CONTINUOUS_AT_LIFT_RANGE] THEN
7927 MESON_TAC[NORM_ARITH `abs(dist(a:real^N,x) - dist(a,y)) <= norm(x - y)`;
7930 let CONTINUOUS_ON_LIFT_DIST = prove
7931 (`!a s. (lift o (\x. dist(a,x))) continuous_on s`,
7932 REWRITE_TAC[CONTINUOUS_ON_LIFT_RANGE] THEN
7933 MESON_TAC[NORM_ARITH `abs(dist(a:real^N,x) - dist(a,y)) <= norm(x - y)`;
7936 (* ------------------------------------------------------------------------- *)
7937 (* Hence some handy theorems on distance, diameter etc. of/from a set. *)
7938 (* ------------------------------------------------------------------------- *)
7940 let COMPACT_ATTAINS_SUP = prove
7941 (`!s. compact (IMAGE lift s) /\ ~(s = {})
7942 ==> ?x. x IN s /\ !y. y IN s ==> y <= x`,
7943 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN REPEAT STRIP_TAC THEN
7944 MP_TAC(SPEC `s:real->bool` BOUNDED_HAS_SUP) THEN ASM_REWRITE_TAC[] THEN
7945 STRIP_TAC THEN EXISTS_TAC `sup s` THEN ASM_REWRITE_TAC[] THEN
7946 ASM_MESON_TAC[CLOSED_LIFT; REAL_ARITH `s <= s - e <=> ~(&0 < e)`;
7947 REAL_ARITH `x <= s /\ ~(x <= s - e) ==> abs(x - s) < e`]);;
7949 let COMPACT_ATTAINS_INF = prove
7950 (`!s. compact (IMAGE lift s) /\ ~(s = {})
7951 ==> ?x. x IN s /\ !y. y IN s ==> x <= y`,
7952 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN REPEAT STRIP_TAC THEN
7953 MP_TAC(SPEC `s:real->bool` BOUNDED_HAS_INF) THEN ASM_REWRITE_TAC[] THEN
7954 STRIP_TAC THEN EXISTS_TAC `inf s` THEN ASM_REWRITE_TAC[] THEN
7955 ASM_MESON_TAC[CLOSED_LIFT; REAL_ARITH `s + e <= s <=> ~(&0 < e)`;
7956 REAL_ARITH `s <= x /\ ~(s + e <= x) ==> abs(x - s) < e`]);;
7958 let CONTINUOUS_ATTAINS_SUP = prove
7959 (`!f:real^N->real s.
7960 compact s /\ ~(s = {}) /\ (lift o f) continuous_on s
7961 ==> ?x. x IN s /\ !y. y IN s ==> f(y) <= f(x)`,
7962 REPEAT STRIP_TAC THEN
7963 MP_TAC(SPEC `IMAGE (f:real^N->real) s` COMPACT_ATTAINS_SUP) THEN
7964 ASM_SIMP_TAC[GSYM IMAGE_o; COMPACT_CONTINUOUS_IMAGE; IMAGE_EQ_EMPTY] THEN
7965 MESON_TAC[IN_IMAGE]);;
7967 let CONTINUOUS_ATTAINS_INF = prove
7968 (`!f:real^N->real s.
7969 compact s /\ ~(s = {}) /\ (lift o f) continuous_on s
7970 ==> ?x. x IN s /\ !y. y IN s ==> f(x) <= f(y)`,
7971 REPEAT STRIP_TAC THEN
7972 MP_TAC(SPEC `IMAGE (f:real^N->real) s` COMPACT_ATTAINS_INF) THEN
7973 ASM_SIMP_TAC[GSYM IMAGE_o; COMPACT_CONTINUOUS_IMAGE; IMAGE_EQ_EMPTY] THEN
7974 MESON_TAC[IN_IMAGE]);;
7976 let DISTANCE_ATTAINS_SUP = prove
7977 (`!s a. compact s /\ ~(s = {})
7978 ==> ?x. x IN s /\ !y. y IN s ==> dist(a,y) <= dist(a,x)`,
7979 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ATTAINS_SUP THEN
7980 ASM_REWRITE_TAC[CONTINUOUS_ON_LIFT_RANGE] THEN REWRITE_TAC[dist] THEN
7981 ASM_MESON_TAC[REAL_LET_TRANS; REAL_ABS_SUB_NORM; NORM_NEG;
7982 VECTOR_ARITH `(a - x) - (a - y) = --(x - y):real^N`]);;
7984 (* ------------------------------------------------------------------------- *)
7985 (* For *minimal* distance, we only need closure, not compactness. *)
7986 (* ------------------------------------------------------------------------- *)
7988 let DISTANCE_ATTAINS_INF = prove
7990 closed s /\ ~(s = {})
7991 ==> ?x. x IN s /\ !y. y IN s ==> dist(a,x) <= dist(a,y)`,
7992 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
7993 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
7994 DISCH_THEN(X_CHOOSE_TAC `b:real^N`) THEN
7995 MP_TAC(ISPECL [`\x:real^N. dist(a,x)`; `cball(a:real^N,dist(b,a)) INTER s`]
7996 CONTINUOUS_ATTAINS_INF) THEN
7998 [ASM_SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_INTER; BOUNDED_INTER;
7999 BOUNDED_CBALL; CLOSED_CBALL; GSYM MEMBER_NOT_EMPTY] THEN
8000 REWRITE_TAC[dist; CONTINUOUS_ON_LIFT_RANGE; IN_INTER; IN_CBALL] THEN
8001 ASM_MESON_TAC[REAL_LET_TRANS; REAL_ABS_SUB_NORM; NORM_NEG; REAL_LE_REFL;
8002 NORM_SUB; VECTOR_ARITH `(a - x) - (a - y) = --(x - y):real^N`];
8003 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[IN_INTER; IN_CBALL] THEN
8004 ASM_MESON_TAC[DIST_SYM; REAL_LE_TOTAL; REAL_LE_TRANS]]);;
8006 (* ------------------------------------------------------------------------- *)
8007 (* We can now extend limit compositions to consider the scalar multiplier. *)
8008 (* ------------------------------------------------------------------------- *)
8011 (`!net:(A)net f l:real^N c d.
8012 ((lift o c) --> lift d) net /\ (f --> l) net
8013 ==> ((\x. c(x) % f(x)) --> (d % l)) net`,
8014 REPEAT STRIP_TAC THEN
8015 MP_TAC(ISPECL [`net:(A)net`; `\x (y:real^N). drop x % y`;
8016 `lift o (c:A->real)`; `f:A->real^N`; `lift d`; `l:real^N`] LIM_BILINEAR) THEN
8017 ASM_REWRITE_TAC[LIFT_DROP; o_THM] THEN DISCH_THEN MATCH_MP_TAC THEN
8018 REWRITE_TAC[bilinear; linear; DROP_ADD; DROP_CMUL] THEN
8019 REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
8021 let LIM_VMUL = prove
8022 (`!net:(A)net c d v:real^N.
8023 ((lift o c) --> lift d) net ==> ((\x. c(x) % v) --> d % v) net`,
8024 REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_MUL THEN ASM_REWRITE_TAC[LIM_CONST]);;
8026 let CONTINUOUS_VMUL = prove
8027 (`!net c v. (lift o c) continuous net ==> (\x. c(x) % v) continuous net`,
8028 REWRITE_TAC[continuous; LIM_VMUL; o_THM]);;
8030 let CONTINUOUS_MUL = prove
8031 (`!net f c. (lift o c) continuous net /\ f continuous net
8032 ==> (\x. c(x) % f(x)) continuous net`,
8033 REWRITE_TAC[continuous; LIM_MUL; o_THM]);;
8035 let CONTINUOUS_ON_VMUL = prove
8036 (`!s c v. (lift o c) continuous_on s ==> (\x. c(x) % v) continuous_on s`,
8037 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
8038 SIMP_TAC[CONTINUOUS_VMUL]);;
8040 let CONTINUOUS_ON_MUL = prove
8041 (`!s c f. (lift o c) continuous_on s /\ f continuous_on s
8042 ==> (\x. c(x) % f(x)) continuous_on s`,
8043 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
8044 SIMP_TAC[CONTINUOUS_MUL]);;
8046 let CONTINUOUS_LIFT_POW = prove
8048 (\x. lift(f x)) continuous net
8049 ==> (\x. lift(f x pow n)) continuous net`,
8050 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN
8051 INDUCT_TAC THEN ASM_REWRITE_TAC[LIFT_CMUL; real_pow; CONTINUOUS_CONST] THEN
8052 MATCH_MP_TAC CONTINUOUS_MUL THEN ASM_REWRITE_TAC[o_DEF]);;
8054 let CONTINUOUS_ON_LIFT_POW = prove
8055 (`!f:real^N->real s n.
8056 (\x. lift(f x)) continuous_on s
8057 ==> (\x. lift(f x pow n)) continuous_on s`,
8058 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN
8059 DISCH_TAC THEN INDUCT_TAC THEN
8060 ASM_REWRITE_TAC[LIFT_CMUL; real_pow; CONTINUOUS_ON_CONST] THEN
8061 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN ASM_REWRITE_TAC[o_DEF]);;
8063 let CONTINUOUS_LIFT_PRODUCT = prove
8064 (`!net:(A)net f (t:B->bool).
8066 (!i. i IN t ==> (\x. lift(f x i)) continuous net)
8067 ==> (\x. lift(product t (f x))) continuous net`,
8068 GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
8069 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[PRODUCT_CLAUSES] THEN
8070 REWRITE_TAC[CONTINUOUS_CONST; LIFT_CMUL; FORALL_IN_INSERT] THEN
8071 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
8072 ASM_SIMP_TAC[o_DEF]);;
8074 let CONTINUOUS_ON_LIFT_PRODUCT = prove
8075 (`!f:real^N->A->real s t.
8078 (!i. i IN t ==> (\x. lift(f x i)) continuous_on s)
8079 ==> (\x. lift(product t (f x))) continuous_on s`,
8080 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_LIFT_PRODUCT]);;
8082 (* ------------------------------------------------------------------------- *)
8083 (* And so we have continuity of inverse. *)
8084 (* ------------------------------------------------------------------------- *)
8088 ((lift o f) --> lift l) net /\ ~(l = &0)
8089 ==> ((lift o inv o f) --> lift(inv l)) net`,
8090 REPEAT GEN_TAC THEN REWRITE_TAC[LIM] THEN
8091 ASM_CASES_TAC `trivial_limit(net:(A)net)` THEN ASM_REWRITE_TAC[] THEN
8092 REWRITE_TAC[o_THM; DIST_LIFT] THEN STRIP_TAC THEN
8093 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
8094 FIRST_X_ASSUM(MP_TAC o SPEC `min (abs(l) / &2) ((l pow 2 * e) / &2)`) THEN
8095 REWRITE_TAC[REAL_LT_MIN] THEN ANTS_TAC THENL
8096 [ASM_SIMP_TAC[GSYM REAL_ABS_NZ; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
8097 MATCH_MP_TAC REAL_LT_DIV THEN REWRITE_TAC[REAL_OF_NUM_LT; ARITH] THEN
8098 ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN
8099 ASM_SIMP_TAC[REAL_LT_MUL; GSYM REAL_ABS_NZ; REAL_POW_LT];
8101 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:A` THEN
8102 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
8103 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `b:A` THEN
8104 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
8105 SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN STRIP_TAC THEN
8106 FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REAL_ARITH
8107 `abs(x - l) * &2 < abs l ==> ~(x = &0)`)) THEN
8108 ASM_SIMP_TAC[REAL_SUB_INV; REAL_ABS_DIV; REAL_LT_LDIV_EQ;
8109 GSYM REAL_ABS_NZ; REAL_ENTIRE] THEN
8110 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
8111 `abs(x - y) * &2 < b * c ==> c * b <= d * &2 ==> abs(y - x) < d`)) THEN
8112 ASM_SIMP_TAC[GSYM REAL_MUL_ASSOC; REAL_LE_LMUL_EQ] THEN
8113 ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
8114 ASM_SIMP_TAC[REAL_ABS_MUL; REAL_POW_2; REAL_MUL_ASSOC; GSYM REAL_ABS_NZ;
8115 REAL_LE_RMUL_EQ] THEN
8116 ASM_SIMP_TAC[REAL_ARITH `abs(x - y) * &2 < abs y ==> abs y <= &2 * abs x`]);;
8118 let CONTINUOUS_INV = prove
8119 (`!net f. (lift o f) continuous net /\ ~(f(netlimit net) = &0)
8120 ==> (lift o inv o f) continuous net`,
8121 REWRITE_TAC[continuous; LIM_INV; o_THM]);;
8123 let CONTINUOUS_AT_WITHIN_INV = prove
8125 (lift o f) continuous (at a within s) /\ ~(f a = &0)
8126 ==> (lift o inv o f) continuous (at a within s)`,
8128 ASM_CASES_TAC `trivial_limit (at (a:real^N) within s)` THENL
8129 [ASM_REWRITE_TAC[continuous; LIM];
8130 ASM_SIMP_TAC[NETLIMIT_WITHIN; CONTINUOUS_INV]]);;
8132 let CONTINUOUS_AT_INV = prove
8133 (`!f a. (lift o f) continuous at a /\ ~(f a = &0)
8134 ==> (lift o inv o f) continuous at a`,
8135 ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
8136 REWRITE_TAC[CONTINUOUS_AT_WITHIN_INV]);;
8138 let CONTINUOUS_ON_INV = prove
8139 (`!f s. (lift o f) continuous_on s /\ (!x. x IN s ==> ~(f x = &0))
8140 ==> (lift o inv o f) continuous_on s`,
8141 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_AT_WITHIN_INV]);;
8143 (* ------------------------------------------------------------------------- *)
8144 (* More preservation properties for pasted sets (Cartesian products). *)
8145 (* ------------------------------------------------------------------------- *)
8147 let LIM_PASTECART = prove
8148 (`!net f:A->real^M g:A->real^N.
8149 (f --> a) net /\ (g --> b) net
8150 ==> ((\x. pastecart (f x) (g x)) --> pastecart a b) net`,
8151 REPEAT GEN_TAC THEN REWRITE_TAC[LIM] THEN
8152 ASM_CASES_TAC `trivial_limit(net:(A)net)` THEN ASM_REWRITE_TAC[] THEN
8153 REWRITE_TAC[AND_FORALL_THM] THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
8154 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
8155 ASM_REWRITE_TAC[REAL_HALF] THEN
8156 DISCH_THEN(MP_TAC o MATCH_MP NET_DILEMMA) THEN
8157 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN
8158 REWRITE_TAC[] THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
8159 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
8160 REWRITE_TAC[dist; PASTECART_SUB] THEN
8161 MATCH_MP_TAC(REAL_ARITH
8162 `z <= x + y ==> x < e / &2 /\ y < e / &2 ==> z < e`) THEN
8163 REWRITE_TAC[NORM_PASTECART_LE]);;
8165 let LIM_PASTECART_EQ = prove
8166 (`!net f:A->real^M g:A->real^N.
8167 ((\x. pastecart (f x) (g x)) --> pastecart a b) net <=>
8168 (f --> a) net /\ (g --> b) net`,
8169 REPEAT GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[LIM_PASTECART] THEN
8170 REPEAT STRIP_TAC THENL
8171 [FIRST_ASSUM(MP_TAC o ISPEC `fstcart:real^(M,N)finite_sum->real^M` o
8172 MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_LINEAR)) THEN
8173 REWRITE_TAC[LINEAR_FSTCART; FSTCART_PASTECART; ETA_AX];
8174 FIRST_ASSUM(MP_TAC o ISPEC `sndcart:real^(M,N)finite_sum->real^N` o
8175 MATCH_MP (REWRITE_RULE[IMP_CONJ] LIM_LINEAR)) THEN
8176 REWRITE_TAC[LINEAR_SNDCART; SNDCART_PASTECART; ETA_AX]]);;
8178 let CONTINUOUS_PASTECART = prove
8179 (`!net f:A->real^M g:A->real^N.
8180 f continuous net /\ g continuous net
8181 ==> (\x. pastecart (f x) (g x)) continuous net`,
8182 REWRITE_TAC[continuous; LIM_PASTECART]);;
8184 let CONTINUOUS_ON_PASTECART = prove
8185 (`!f:real^M->real^N g:real^M->real^P s.
8186 f continuous_on s /\ g continuous_on s
8187 ==> (\x. pastecart (f x) (g x)) continuous_on s`,
8188 SIMP_TAC[CONTINUOUS_ON; LIM_PASTECART]);;
8190 let CONNECTED_PCROSS = prove
8191 (`!s:real^M->bool t:real^N->bool.
8192 connected s /\ connected t
8193 ==> connected (s PCROSS t)`,
8195 REWRITE_TAC[PCROSS; CONNECTED_IFF_CONNECTED_COMPONENT] THEN
8196 DISCH_TAC THEN REWRITE_TAC[FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
8197 MAP_EVERY X_GEN_TAC [`x1:real^M`; `y1:real^N`; `x2:real^M`; `y2:real^N`] THEN
8198 STRIP_TAC THEN FIRST_X_ASSUM(CONJUNCTS_THEN2
8199 (MP_TAC o SPECL [`x1:real^M`; `x2:real^M`])
8200 (MP_TAC o SPECL [`y1:real^N`; `y2:real^N`])) THEN
8201 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; connected_component] THEN
8202 X_GEN_TAC `c2:real^N->bool` THEN STRIP_TAC THEN
8203 X_GEN_TAC `c1:real^M->bool` THEN STRIP_TAC THEN
8205 `IMAGE (\x:real^M. pastecart x y1) c1 UNION
8206 IMAGE (\y:real^N. pastecart x2 y) c2` THEN
8207 REWRITE_TAC[IN_UNION] THEN REPEAT CONJ_TAC THENL
8208 [MATCH_MP_TAC CONNECTED_UNION THEN
8209 ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE; CONTINUOUS_ON_PASTECART;
8210 CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN
8211 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER; EXISTS_IN_IMAGE] THEN
8212 EXISTS_TAC `x2:real^M` THEN ASM SET_TAC[];
8213 REWRITE_TAC[SUBSET; IN_UNION; FORALL_AND_THM; FORALL_IN_IMAGE;
8214 TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
8219 let CONNECTED_PCROSS_EQ = prove
8220 (`!s:real^M->bool t:real^N->bool.
8221 connected (s PCROSS t) <=>
8222 s = {} \/ t = {} \/ connected s /\ connected t`,
8224 ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
8225 ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
8226 REWRITE_TAC[PCROSS_EMPTY; CONNECTED_EMPTY] THEN
8227 EQ_TAC THEN SIMP_TAC[CONNECTED_PCROSS] THEN
8228 REWRITE_TAC[PCROSS] THEN REPEAT STRIP_TAC THENL
8229 [SUBGOAL_THEN `connected (IMAGE fstcart
8230 {pastecart (x:real^M) (y:real^N) | x IN s /\ y IN t})`
8231 MP_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE; ALL_TAC];
8232 SUBGOAL_THEN `connected (IMAGE sndcart
8233 {pastecart (x:real^M) (y:real^N) | x IN s /\ y IN t})`
8234 MP_TAC THENL [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE; ALL_TAC]] THEN
8235 ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; LINEAR_SNDCART] THEN
8236 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
8237 REWRITE_TAC[EXTENSION; IN_IMAGE; EXISTS_PASTECART; IN_ELIM_PASTECART_THM;
8238 FSTCART_PASTECART; SNDCART_PASTECART] THEN
8241 let CLOSURE_PCROSS = prove
8242 (`!s:real^M->bool t:real^N->bool.
8243 closure (s PCROSS t) = (closure s) PCROSS (closure t)`,
8244 REWRITE_TAC[EXTENSION; PCROSS; FORALL_PASTECART] THEN REPEAT GEN_TAC THEN
8245 REWRITE_TAC[CLOSURE_APPROACHABLE; EXISTS_PASTECART; FORALL_PASTECART] THEN
8246 REWRITE_TAC[IN_ELIM_PASTECART_THM; PASTECART_INJ] THEN
8247 REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN
8248 REWRITE_TAC[dist; PASTECART_SUB] THEN EQ_TAC THENL
8249 [MESON_TAC[NORM_LE_PASTECART; REAL_LET_TRANS]; DISCH_TAC] THEN
8250 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
8251 FIRST_X_ASSUM(CONJUNCTS_THEN (MP_TAC o SPEC `e / &2`)) THEN
8252 ASM_MESON_TAC[REAL_HALF; NORM_PASTECART_LE; REAL_ARITH
8253 `z <= x + y /\ x < e / &2 /\ y < e / &2 ==> z < e`]);;
8255 let LIMPT_PCROSS = prove
8256 (`!s:real^M->bool t:real^N->bool x y.
8257 x limit_point_of s /\ y limit_point_of t
8258 ==> (pastecart x y) limit_point_of (s PCROSS t)`,
8260 REWRITE_TAC[PCROSS; LIMPT_APPROACHABLE; EXISTS_PASTECART] THEN
8261 REWRITE_TAC[IN_ELIM_PASTECART_THM; PASTECART_INJ; dist; PASTECART_SUB] THEN
8262 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
8263 FIRST_X_ASSUM(CONJUNCTS_THEN (MP_TAC o SPEC `e / &2`)) THEN
8264 ASM_MESON_TAC[REAL_HALF; NORM_PASTECART_LE; REAL_ARITH
8265 `z <= x + y /\ x < e / &2 /\ y < e / &2 ==> z < e`]);;
8267 let CLOSED_IN_PCROSS = prove
8268 (`!s:real^M->bool s' t:real^N->bool t'.
8269 closed_in (subtopology euclidean s) s' /\
8270 closed_in (subtopology euclidean t) t'
8271 ==> closed_in (subtopology euclidean (s PCROSS t)) (s' PCROSS t')`,
8272 REPEAT GEN_TAC THEN REWRITE_TAC[CLOSED_IN_CLOSED] THEN
8273 DISCH_THEN(CONJUNCTS_THEN2
8274 (X_CHOOSE_THEN `s'':real^M->bool` STRIP_ASSUME_TAC)
8275 (X_CHOOSE_THEN `t'':real^N->bool` STRIP_ASSUME_TAC)) THEN
8276 EXISTS_TAC `(s'':real^M->bool) PCROSS (t'':real^N->bool)` THEN
8277 ASM_SIMP_TAC[CLOSED_PCROSS; EXTENSION; FORALL_PASTECART] THEN
8278 REWRITE_TAC[IN_INTER; PASTECART_IN_PCROSS] THEN ASM SET_TAC[]);;
8280 let CLOSED_IN_PCROSS_EQ = prove
8281 (`!s s':real^M->bool t t':real^N->bool.
8282 closed_in (subtopology euclidean (s PCROSS t)) (s' PCROSS t') <=>
8283 s' = {} \/ t' = {} \/
8284 closed_in (subtopology euclidean s) s' /\
8285 closed_in (subtopology euclidean t) t'`,
8287 ASM_CASES_TAC `s':real^M->bool = {}` THEN
8288 ASM_REWRITE_TAC[PCROSS_EMPTY; CLOSED_IN_EMPTY] THEN
8289 ASM_CASES_TAC `t':real^N->bool = {}` THEN
8290 ASM_REWRITE_TAC[PCROSS_EMPTY; CLOSED_IN_EMPTY] THEN
8291 EQ_TAC THEN REWRITE_TAC[CLOSED_IN_PCROSS] THEN
8292 ASM_REWRITE_TAC[CLOSED_IN_INTER_CLOSURE; CLOSURE_PCROSS; INTER_PCROSS;
8293 PCROSS_EQ; PCROSS_EQ_EMPTY]);;
8295 let FRONTIER_PCROSS = prove
8296 (`!s:real^M->bool t:real^N->bool.
8297 frontier(s PCROSS t) = frontier s PCROSS closure t UNION
8298 closure s PCROSS frontier t`,
8300 REWRITE_TAC[frontier; CLOSURE_PCROSS; INTERIOR_PCROSS; PCROSS_DIFF] THEN
8301 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_DIFF; IN_UNION;
8302 PASTECART_IN_PCROSS] THEN
8305 (* ------------------------------------------------------------------------- *)
8306 (* Hence some useful properties follow quite easily. *)
8307 (* ------------------------------------------------------------------------- *)
8309 let CONNECTED_SCALING = prove
8310 (`!s:real^N->bool c. connected s ==> connected (IMAGE (\x. c % x) s)`,
8311 REPEAT STRIP_TAC THEN
8312 MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN
8313 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
8314 REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
8315 REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);;
8317 let CONNECTED_NEGATIONS = prove
8318 (`!s:real^N->bool. connected s ==> connected (IMAGE (--) s)`,
8319 REPEAT STRIP_TAC THEN
8320 MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN
8321 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
8322 REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
8323 REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);;
8325 let CONNECTED_SUMS = prove
8326 (`!s t:real^N->bool.
8327 connected s /\ connected t ==> connected {x + y | x IN s /\ y IN t}`,
8328 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CONNECTED_PCROSS) THEN
8329 DISCH_THEN(MP_TAC o ISPEC
8330 `\z. (fstcart z + sndcart z:real^N)` o
8331 MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] CONNECTED_CONTINUOUS_IMAGE)) THEN
8332 SIMP_TAC[CONTINUOUS_ON_ADD; LINEAR_CONTINUOUS_ON; LINEAR_FSTCART;
8333 LINEAR_SNDCART; PCROSS] THEN
8334 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
8335 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; EXISTS_PASTECART] THEN
8336 REWRITE_TAC[PASTECART_INJ; FSTCART_PASTECART; SNDCART_PASTECART] THEN
8339 let COMPACT_SCALING = prove
8340 (`!s:real^N->bool c. compact s ==> compact (IMAGE (\x. c % x) s)`,
8341 REPEAT STRIP_TAC THEN
8342 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN
8343 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
8344 REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
8345 REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);;
8347 let COMPACT_NEGATIONS = prove
8348 (`!s:real^N->bool. compact s ==> compact (IMAGE (--) s)`,
8349 REPEAT STRIP_TAC THEN
8350 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN
8351 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
8352 REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
8353 REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);;
8355 let COMPACT_SUMS = prove
8356 (`!s:real^N->bool t.
8357 compact s /\ compact t ==> compact {x + y | x IN s /\ y IN t}`,
8358 REPEAT STRIP_TAC THEN
8359 SUBGOAL_THEN `{x + y | x IN s /\ y IN t} =
8360 IMAGE (\z. fstcart z + sndcart z :real^N) (s PCROSS t)`
8362 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE; PCROSS] THEN
8363 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
8364 ASM_MESON_TAC[FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_FST_SND];
8366 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
8367 ASM_SIMP_TAC[COMPACT_PCROSS] THEN
8368 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
8369 REPEAT STRIP_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
8370 REWRITE_TAC[linear; FSTCART_ADD; FSTCART_CMUL; SNDCART_ADD;
8372 CONJ_TAC THEN VECTOR_ARITH_TAC);;
8374 let COMPACT_DIFFERENCES = prove
8375 (`!s:real^N->bool t.
8376 compact s /\ compact t ==> compact {x - y | x IN s /\ y IN t}`,
8377 REPEAT STRIP_TAC THEN
8378 SUBGOAL_THEN `{x - y | x:real^N IN s /\ y IN t} =
8379 {x + y | x IN s /\ y IN (IMAGE (--) t)}`
8380 (fun th -> ASM_SIMP_TAC[th; COMPACT_SUMS; COMPACT_NEGATIONS]) THEN
8381 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN
8382 ONCE_REWRITE_TAC[VECTOR_ARITH `(x:real^N = --y) <=> (y = --x)`] THEN
8383 SIMP_TAC[VECTOR_SUB; GSYM CONJ_ASSOC; UNWIND_THM2] THEN
8384 MESON_TAC[VECTOR_NEG_NEG]);;
8386 let COMPACT_AFFINITY = prove
8388 compact s ==> compact (IMAGE (\x. a + c % x) s)`,
8389 REPEAT STRIP_TAC THEN
8390 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
8391 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
8392 ASM_SIMP_TAC[IMAGE_o; COMPACT_TRANSLATION; COMPACT_SCALING]);;
8394 (* ------------------------------------------------------------------------- *)
8395 (* Hence we get the following. *)
8396 (* ------------------------------------------------------------------------- *)
8398 let COMPACT_SUP_MAXDISTANCE = prove
8400 compact s /\ ~(s = {})
8401 ==> ?x y. x IN s /\ y IN s /\
8402 !u v. u IN s /\ v IN s ==> norm(u - v) <= norm(x - y)`,
8403 REPEAT STRIP_TAC THEN
8404 MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN s}`; `vec 0:real^N`]
8405 DISTANCE_ATTAINS_SUP) THEN
8407 [ASM_SIMP_TAC[COMPACT_DIFFERENCES] THEN
8408 REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN
8409 ASM_MESON_TAC[MEMBER_NOT_EMPTY];
8410 REWRITE_TAC[IN_ELIM_THM; dist; VECTOR_SUB_RZERO; VECTOR_SUB_LZERO;
8414 (* ------------------------------------------------------------------------- *)
8415 (* We can state this in terms of diameter of a set. *)
8416 (* ------------------------------------------------------------------------- *)
8418 let diameter = new_definition
8421 else sup {norm(x - y) | x IN s /\ y IN s}`;;
8423 let DIAMETER_BOUNDED = prove
8425 ==> (!x:real^N y. x IN s /\ y IN s ==> norm(x - y) <= diameter s) /\
8426 (!d. &0 <= d /\ d < diameter s
8427 ==> ?x y. x IN s /\ y IN s /\ norm(x - y) > d)`,
8428 GEN_TAC THEN DISCH_TAC THEN
8429 ASM_CASES_TAC `s:real^N->bool = {}` THEN
8430 ASM_REWRITE_TAC[diameter; NOT_IN_EMPTY; REAL_LET_ANTISYM] THEN
8431 MP_TAC(SPEC `{norm(x - y:real^N) | x IN s /\ y IN s}` SUP) THEN
8432 ABBREV_TAC `b = sup {norm(x - y:real^N) | x IN s /\ y IN s}` THEN
8433 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
8434 REWRITE_TAC[NOT_IN_EMPTY; real_gt] THEN ANTS_TAC THENL
8435 [CONJ_TAC THENL [ASM_MESON_TAC[MEMBER_NOT_EMPTY]; ALL_TAC];
8436 MESON_TAC[REAL_NOT_LE]] THEN
8437 SIMP_TAC[VECTOR_SUB; LEFT_IMP_EXISTS_THM] THEN
8438 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
8439 MESON_TAC[REAL_ARITH `x <= y + z /\ y <= b /\ z<= b ==> x <= b + b`;
8440 NORM_TRIANGLE; NORM_NEG]);;
8442 let DIAMETER_BOUNDED_BOUND = prove
8443 (`!s x y. bounded s /\ x IN s /\ y IN s ==> norm(x - y) <= diameter s`,
8444 MESON_TAC[DIAMETER_BOUNDED]);;
8446 let DIAMETER_COMPACT_ATTAINED = prove
8448 compact s /\ ~(s = {})
8449 ==> ?x y. x IN s /\ y IN s /\ (norm(x - y) = diameter s)`,
8450 GEN_TAC THEN DISCH_TAC THEN
8451 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_SUP_MAXDISTANCE) THEN
8452 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
8453 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
8454 MP_TAC(SPEC `s:real^N->bool` DIAMETER_BOUNDED) THEN
8455 RULE_ASSUM_TAC(REWRITE_RULE[COMPACT_EQ_BOUNDED_CLOSED]) THEN
8456 ASM_REWRITE_TAC[real_gt] THEN STRIP_TAC THEN
8457 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN
8458 ASM_MESON_TAC[NORM_POS_LE; REAL_NOT_LT]);;
8460 let DIAMETER_TRANSLATION = prove
8461 (`!a s. diameter (IMAGE (\x. a + x) s) = diameter s`,
8462 REWRITE_TAC[diameter] THEN GEOM_TRANSLATE_TAC[]);;
8464 add_translation_invariants [DIAMETER_TRANSLATION];;
8466 let DIAMETER_LINEAR_IMAGE = prove
8467 (`!f:real^M->real^N s.
8468 linear f /\ (!x. norm(f x) = norm x)
8469 ==> diameter(IMAGE f s) = diameter s`,
8470 REWRITE_TAC[diameter] THEN
8471 REPEAT STRIP_TAC THEN REWRITE_TAC[diameter; IMAGE_EQ_EMPTY] THEN
8472 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN AP_TERM_TAC THEN
8473 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
8474 REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; EXISTS_IN_IMAGE] THEN
8475 ASM_MESON_TAC[LINEAR_SUB]);;
8477 add_linear_invariants [DIAMETER_LINEAR_IMAGE];;
8479 let DIAMETER_EMPTY = prove
8480 (`diameter {} = &0`,
8481 REWRITE_TAC[diameter]);;
8483 let DIAMETER_SING = prove
8484 (`!a. diameter {a} = &0`,
8485 REWRITE_TAC[diameter; NOT_INSERT_EMPTY; IN_SING] THEN
8486 REWRITE_TAC[SET_RULE `{f x y | x = a /\ y = a} = {f a a }`] THEN
8487 REWRITE_TAC[SUP_SING; VECTOR_SUB_REFL; NORM_0]);;
8489 let DIAMETER_POS_LE = prove
8490 (`!s:real^N->bool. bounded s ==> &0 <= diameter s`,
8491 REPEAT STRIP_TAC THEN REWRITE_TAC[diameter] THEN
8492 COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN
8493 MP_TAC(SPEC `{norm(x - y:real^N) | x IN s /\ y IN s}` SUP) THEN
8494 REWRITE_TAC[FORALL_IN_GSPEC] THEN ANTS_TAC THENL
8495 [CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
8496 FIRST_X_ASSUM(X_CHOOSE_TAC `B:real` o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
8497 EXISTS_TAC `&2 * B` THEN
8498 ASM_SIMP_TAC[NORM_ARITH
8499 `norm x <= B /\ norm y <= B ==> norm(x - y) <= &2 * B`];
8500 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
8501 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
8502 DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `a:real^N`] o CONJUNCT1) THEN
8503 ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0]]);;
8505 let DIAMETER_SUBSET = prove
8506 (`!s t:real^N->bool. s SUBSET t /\ bounded t ==> diameter s <= diameter t`,
8507 REPEAT STRIP_TAC THEN
8508 ASM_CASES_TAC `s:real^N->bool = {}` THEN
8509 ASM_SIMP_TAC[DIAMETER_EMPTY; DIAMETER_POS_LE] THEN
8510 ASM_REWRITE_TAC[diameter] THEN
8511 COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
8512 MATCH_MP_TAC REAL_SUP_LE_SUBSET THEN
8513 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
8514 REWRITE_TAC[FORALL_IN_GSPEC] THEN
8515 FIRST_X_ASSUM(X_CHOOSE_TAC `B:real` o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
8516 EXISTS_TAC `&2 * B` THEN
8517 ASM_SIMP_TAC[NORM_ARITH
8518 `norm x <= B /\ norm y <= B ==> norm(x - y) <= &2 * B`]);;
8520 let DIAMETER_CLOSURE = prove
8521 (`!s:real^N->bool. bounded s ==> diameter(closure s) = diameter s`,
8522 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN REPEAT STRIP_TAC THEN
8523 ASM_SIMP_TAC[DIAMETER_SUBSET; BOUNDED_CLOSURE; CLOSURE_SUBSET] THEN
8524 REWRITE_TAC[GSYM REAL_NOT_LT] THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
8525 DISCH_TAC THEN MP_TAC(ISPEC `closure s:real^N->bool` DIAMETER_BOUNDED) THEN
8526 ABBREV_TAC `d = diameter(closure s) - diameter(s:real^N->bool)` THEN
8527 ASM_SIMP_TAC[BOUNDED_CLOSURE] THEN DISCH_THEN(MP_TAC o
8528 SPEC `diameter(closure(s:real^N->bool)) - d / &2` o CONJUNCT2) THEN
8529 REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; NOT_EXISTS_THM] THEN
8530 FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIAMETER_POS_LE) THEN
8531 REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
8532 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
8533 REWRITE_TAC[CLOSURE_APPROACHABLE; CONJ_ASSOC; AND_FORALL_THM] THEN
8534 DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `d / &4`) ASSUME_TAC) THEN
8535 ASM_REWRITE_TAC[REAL_ARITH `&0 < d / &4 <=> &0 < d`] THEN
8536 DISCH_THEN(CONJUNCTS_THEN2
8537 (X_CHOOSE_THEN `u:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))
8538 (X_CHOOSE_THEN `v:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))) THEN
8539 FIRST_ASSUM(MP_TAC o MATCH_MP DIAMETER_BOUNDED) THEN
8540 DISCH_THEN(MP_TAC o SPECL [`u:real^N`; `v:real^N`] o CONJUNCT1) THEN
8541 ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);;
8543 let DIAMETER_SUBSET_CBALL_NONEMPTY = prove
8545 bounded s /\ ~(s = {}) ==> ?z. z IN s /\ s SUBSET cball(z,diameter s)`,
8546 REPEAT STRIP_TAC THEN
8547 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
8548 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
8549 DISCH_TAC THEN ASM_REWRITE_TAC[SUBSET] THEN X_GEN_TAC `b:real^N` THEN
8550 DISCH_TAC THEN REWRITE_TAC[IN_CBALL; dist] THEN
8551 ASM_MESON_TAC[DIAMETER_BOUNDED]);;
8553 let DIAMETER_SUBSET_CBALL = prove
8554 (`!s:real^N->bool. bounded s ==> ?z. s SUBSET cball(z,diameter s)`,
8555 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
8556 ASM_MESON_TAC[DIAMETER_SUBSET_CBALL_NONEMPTY; EMPTY_SUBSET]);;
8558 let DIAMETER_EQ_0 = prove
8560 bounded s ==> (diameter s = &0 <=> s = {} \/ ?a. s = {a})`,
8561 REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THEN
8562 ASM_REWRITE_TAC[DIAMETER_EMPTY; DIAMETER_SING] THEN
8563 REWRITE_TAC[SET_RULE
8564 `s = {} \/ (?a. s = {a}) <=> !a b. a IN s /\ b IN s ==> a = b`] THEN
8565 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN
8566 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`]
8567 DIAMETER_BOUNDED_BOUND) THEN
8568 ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC);;
8570 let DIAMETER_LE = prove
8572 (~(s = {}) \/ &0 <= d) /\
8573 (!x y. x IN s /\ y IN s ==> norm(x - y) <= d) ==> diameter s <= d`,
8574 GEN_TAC THEN REWRITE_TAC[diameter] THEN
8575 COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
8576 STRIP_TAC THEN MATCH_MP_TAC REAL_SUP_LE THEN
8577 CONJ_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[FORALL_IN_GSPEC]]);;
8579 let DIAMETER_CBALL = prove
8580 (`!a:real^N r. diameter(cball(a,r)) = if r < &0 then &0 else &2 * r`,
8581 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
8582 [ASM_MESON_TAC[CBALL_EQ_EMPTY; DIAMETER_EMPTY]; ALL_TAC] THEN
8583 RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LT]) THEN
8584 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL
8585 [MATCH_MP_TAC DIAMETER_LE THEN
8586 ASM_SIMP_TAC[CBALL_EQ_EMPTY; REAL_LE_MUL; REAL_POS; REAL_NOT_LT] THEN
8587 REWRITE_TAC[IN_CBALL] THEN NORM_ARITH_TAC;
8588 MATCH_MP_TAC REAL_LE_TRANS THEN
8589 EXISTS_TAC `norm((a + r % basis 1) - (a - r % basis 1):real^N)` THEN
8591 [REWRITE_TAC[VECTOR_ARITH `(a + r % b) - (a - r % b:real^N) =
8593 SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN
8595 MATCH_MP_TAC DIAMETER_BOUNDED_BOUND THEN
8596 REWRITE_TAC[BOUNDED_CBALL; IN_CBALL] THEN
8597 REWRITE_TAC[NORM_ARITH
8598 `dist(a:real^N,a + b) = norm b /\ dist(a,a - b) = norm b`] THEN
8599 SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN
8600 ASM_REAL_ARITH_TAC]]);;
8602 let DIAMETER_BALL = prove
8603 (`!a:real^N r. diameter(ball(a,r)) = if r < &0 then &0 else &2 * r`,
8604 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
8605 [ASM_SIMP_TAC[BALL_EMPTY; REAL_LT_IMP_LE; DIAMETER_EMPTY]; ALL_TAC] THEN
8606 ASM_CASES_TAC `r = &0` THEN
8607 ASM_SIMP_TAC[BALL_EMPTY; REAL_LE_REFL; DIAMETER_EMPTY; REAL_MUL_RZERO] THEN
8608 MATCH_MP_TAC EQ_TRANS THEN
8609 EXISTS_TAC `diameter(cball(a:real^N,r))` THEN CONJ_TAC THENL
8610 [SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
8611 ASM_SIMP_TAC[GSYM CLOSURE_BALL; DIAMETER_CLOSURE; BOUNDED_BALL];
8612 ASM_SIMP_TAC[DIAMETER_CBALL]]);;
8614 let DIAMETER_SUMS = prove
8615 (`!s t:real^N->bool.
8616 bounded s /\ bounded t
8617 ==> diameter {x + y | x IN s /\ y IN t} <= diameter s + diameter t`,
8618 REPEAT STRIP_TAC THEN
8619 ASM_CASES_TAC `s:real^N->bool = {}` THEN
8620 ASM_SIMP_TAC[NOT_IN_EMPTY; SET_RULE `{f x y |x,y| F} = {}`;
8621 DIAMETER_EMPTY; REAL_ADD_LID; DIAMETER_POS_LE] THEN
8622 ASM_CASES_TAC `t:real^N->bool = {}` THEN
8623 ASM_SIMP_TAC[NOT_IN_EMPTY; SET_RULE `{f x y |x,y| F} = {}`;
8624 DIAMETER_EMPTY; REAL_ADD_RID; DIAMETER_POS_LE] THEN
8625 MATCH_MP_TAC DIAMETER_LE THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
8626 REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_GSPEC] THEN
8627 REPEAT STRIP_TAC THEN MATCH_MP_TAC(NORM_ARITH
8628 `norm(x - x') <= s /\ norm(y - y') <= t
8629 ==> norm((x + y) - (x' + y'):real^N) <= s + t`) THEN
8630 ASM_SIMP_TAC[DIAMETER_BOUNDED_BOUND]);;
8632 let LEBESGUE_COVERING_LEMMA = prove
8633 (`!s:real^N->bool c.
8634 compact s /\ ~(c = {}) /\ s SUBSET UNIONS c /\ (!b. b IN c ==> open b)
8636 !t. t SUBSET s /\ diameter t <= d
8637 ==> ?b. b IN c /\ t SUBSET b`,
8638 REPEAT STRIP_TAC THEN
8639 FIRST_ASSUM(MP_TAC o MATCH_MP HEINE_BOREL_LEMMA) THEN
8640 DISCH_THEN(MP_TAC o SPEC `c:(real^N->bool)->bool`) THEN ASM_SIMP_TAC[] THEN
8641 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `e:real` THEN
8642 STRIP_TAC THEN EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN
8643 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
8644 ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN
8645 MP_TAC(ISPEC `t:real^N->bool` DIAMETER_SUBSET_CBALL_NONEMPTY) THEN
8647 [ASM_MESON_TAC[BOUNDED_SUBSET; COMPACT_IMP_BOUNDED]; ALL_TAC] THEN
8648 DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
8649 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
8650 ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN
8651 X_GEN_TAC `b:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
8652 MATCH_MP_TAC SUBSET_TRANS THEN
8653 EXISTS_TAC `cball(x:real^N,diameter(t:real^N->bool))` THEN
8654 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
8655 EXISTS_TAC `ball(x:real^N,e)` THEN ASM_REWRITE_TAC[] THEN
8656 REWRITE_TAC[SUBSET; IN_CBALL; IN_BALL] THEN
8657 MAP_EVERY UNDISCH_TAC [`&0 < e`; `diameter(t:real^N->bool) <= e / &2`] THEN
8660 (* ------------------------------------------------------------------------- *)
8661 (* Related results with closure as the conclusion. *)
8662 (* ------------------------------------------------------------------------- *)
8664 let CLOSED_SCALING = prove
8665 (`!s:real^N->bool c. closed s ==> closed (IMAGE (\x. c % x) s)`,
8667 ASM_CASES_TAC `s :real^N->bool = {}` THEN
8668 ASM_REWRITE_TAC[CLOSED_EMPTY; IMAGE_CLAUSES] THEN
8669 ASM_CASES_TAC `c = &0` THENL
8670 [SUBGOAL_THEN `IMAGE (\x:real^N. c % x) s = {(vec 0)}`
8671 (fun th -> REWRITE_TAC[th; CLOSED_SING]) THEN
8672 ASM_REWRITE_TAC[EXTENSION; IN_IMAGE; IN_SING; VECTOR_MUL_LZERO] THEN
8673 ASM_MESON_TAC[MEMBER_NOT_EMPTY];
8675 REWRITE_TAC[CLOSED_SEQUENTIAL_LIMITS; IN_IMAGE; SKOLEM_THM] THEN
8676 STRIP_TAC THEN X_GEN_TAC `x:num->real^N` THEN X_GEN_TAC `l:real^N` THEN
8677 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
8678 DISCH_THEN(X_CHOOSE_THEN `y:num->real^N` MP_TAC) THEN
8679 REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN
8680 EXISTS_TAC `inv(c) % l :real^N` THEN
8681 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID] THEN
8682 FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `\n:num. inv(c) % x n:real^N` THEN
8683 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
8684 [ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID];
8685 MATCH_MP_TAC LIM_CMUL THEN
8686 FIRST_ASSUM(fun th -> REWRITE_TAC[SYM(SPEC_ALL th)]) THEN
8687 ASM_REWRITE_TAC[ETA_AX]]);;
8689 let CLOSED_NEGATIONS = prove
8690 (`!s:real^N->bool. closed s ==> closed (IMAGE (--) s)`,
8692 SUBGOAL_THEN `IMAGE (--) s = IMAGE (\x:real^N. --(&1) % x) s`
8693 SUBST1_TAC THEN SIMP_TAC[CLOSED_SCALING] THEN
8694 REWRITE_TAC[VECTOR_ARITH `--(&1) % x = --x`] THEN REWRITE_TAC[ETA_AX]);;
8696 let COMPACT_CLOSED_SUMS = prove
8697 (`!s:real^N->bool t.
8698 compact s /\ closed t ==> closed {x + y | x IN s /\ y IN t}`,
8700 REWRITE_TAC[compact; IN_ELIM_THM; CLOSED_SEQUENTIAL_LIMITS] THEN
8701 STRIP_TAC THEN X_GEN_TAC `f:num->real^N` THEN X_GEN_TAC `l:real^N` THEN
8702 REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM] THEN
8703 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
8704 DISCH_THEN(X_CHOOSE_THEN `a:num->real^N` MP_TAC) THEN
8705 DISCH_THEN(X_CHOOSE_THEN `b:num->real^N` STRIP_ASSUME_TAC) THEN
8706 FIRST_X_ASSUM(MP_TAC o check(is_imp o concl) o SPEC `a:num->real^N`) THEN
8707 ASM_REWRITE_TAC[] THEN
8708 DISCH_THEN(X_CHOOSE_THEN `la:real^N` (X_CHOOSE_THEN `sub:num->num`
8709 STRIP_ASSUME_TAC)) THEN
8710 MAP_EVERY EXISTS_TAC [`la:real^N`; `l - la:real^N`] THEN
8711 ASM_REWRITE_TAC[VECTOR_ARITH `a + (b - a) = b:real^N`] THEN
8712 FIRST_X_ASSUM MATCH_MP_TAC THEN
8713 EXISTS_TAC `\n. (f o (sub:num->num)) n - (a o sub) n:real^N` THEN
8714 CONJ_TAC THENL [ASM_REWRITE_TAC[VECTOR_ADD_SUB; o_THM]; ALL_TAC] THEN
8715 MATCH_MP_TAC LIM_SUB THEN ASM_SIMP_TAC[LIM_SUBSEQUENCE; ETA_AX]);;
8717 let CLOSED_COMPACT_SUMS = prove
8718 (`!s:real^N->bool t.
8719 closed s /\ compact t ==> closed {x + y | x IN s /\ y IN t}`,
8721 SUBGOAL_THEN `{x + y:real^N | x IN s /\ y IN t} = {y + x | y IN t /\ x IN s}`
8722 SUBST1_TAC THEN SIMP_TAC[COMPACT_CLOSED_SUMS] THEN
8723 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_SYM]);;
8725 let CLOSURE_SUMS = prove
8726 (`!s t:real^N->bool.
8727 bounded s \/ bounded t
8728 ==> closure {x + y | x IN s /\ y IN t} =
8729 {x + y | x IN closure s /\ y IN closure t}`,
8730 REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
8731 REWRITE_TAC[FORALL_AND_THM] THEN
8732 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SUMS_SYM] THEN
8733 MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN
8735 REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; CLOSURE_SEQUENTIAL] THEN
8736 X_GEN_TAC `z:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN EQ_TAC THENL
8737 [REWRITE_TAC[IN_ELIM_THM; IN_DELETE; SKOLEM_THM; LEFT_AND_EXISTS_THM] THEN
8738 REWRITE_TAC[FORALL_AND_THM] THEN
8739 ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN
8740 ONCE_REWRITE_TAC[MESON[] `(?f x y. P f x y) <=> (?x y f. P f x y)`] THEN
8741 ONCE_REWRITE_TAC[GSYM FUN_EQ_THM] THEN
8742 REWRITE_TAC[ETA_AX; UNWIND_THM2] THEN
8743 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
8744 MAP_EVERY X_GEN_TAC [`a:num->real^N`; `b:num->real^N`] THEN
8746 MP_TAC(ISPEC `closure s:real^N->bool` compact) THEN
8747 ASM_REWRITE_TAC[COMPACT_CLOSURE] THEN
8748 DISCH_THEN(MP_TAC o SPEC `a:num->real^N`) THEN
8749 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; LEFT_IMP_EXISTS_THM] THEN
8750 MAP_EVERY X_GEN_TAC [`u:real^N`; `r:num->num`] THEN STRIP_TAC THEN
8751 EXISTS_TAC `z - u:real^N` THEN
8752 EXISTS_TAC `(a:num->real^N) o (r:num->num)` THEN EXISTS_TAC `u:real^N` THEN
8753 ASM_REWRITE_TAC[o_THM] THEN
8754 CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
8755 EXISTS_TAC `(\n. ((\n. a n + b n) o (r:num->num)) n - (a o r) n)
8758 [ASM_REWRITE_TAC[o_DEF; VECTOR_ARITH `(a + b) - a:real^N = b`];
8759 MATCH_MP_TAC LIM_SUB THEN ASM_REWRITE_TAC[ETA_AX] THEN
8760 MATCH_MP_TAC LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[]];
8761 REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
8762 REWRITE_TAC[LEFT_IMP_EXISTS_THM; LEFT_AND_EXISTS_THM;
8763 RIGHT_AND_EXISTS_THM] THEN
8765 [`x:real^N`; `y:real^N`; `a:num->real^N`; `b:num->real^N`] THEN
8766 STRIP_TAC THEN EXISTS_TAC `(\n. a n + b n):num->real^N` THEN
8767 ASM_SIMP_TAC[LIM_ADD] THEN ASM_MESON_TAC[]]);;
8769 let COMPACT_CLOSED_DIFFERENCES = prove
8770 (`!s:real^N->bool t.
8771 compact s /\ closed t ==> closed {x - y | x IN s /\ y IN t}`,
8772 REPEAT STRIP_TAC THEN
8773 SUBGOAL_THEN `{x - y | x:real^N IN s /\ y IN t} =
8774 {x + y | x IN s /\ y IN (IMAGE (--) t)}`
8775 (fun th -> ASM_SIMP_TAC[th; COMPACT_CLOSED_SUMS; CLOSED_NEGATIONS]) THEN
8776 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN
8777 ONCE_REWRITE_TAC[VECTOR_ARITH `(x:real^N = --y) <=> (y = --x)`] THEN
8778 SIMP_TAC[VECTOR_SUB; GSYM CONJ_ASSOC; UNWIND_THM2] THEN
8779 MESON_TAC[VECTOR_NEG_NEG]);;
8781 let CLOSED_COMPACT_DIFFERENCES = prove
8782 (`!s:real^N->bool t.
8783 closed s /\ compact t ==> closed {x - y | x IN s /\ y IN t}`,
8784 REPEAT STRIP_TAC THEN
8785 SUBGOAL_THEN `{x - y | x:real^N IN s /\ y IN t} =
8786 {x + y | x IN s /\ y IN (IMAGE (--) t)}`
8787 (fun th -> ASM_SIMP_TAC[th; CLOSED_COMPACT_SUMS; COMPACT_NEGATIONS]) THEN
8788 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN
8789 ONCE_REWRITE_TAC[VECTOR_ARITH `(x:real^N = --y) <=> (y = --x)`] THEN
8790 SIMP_TAC[VECTOR_SUB; GSYM CONJ_ASSOC; UNWIND_THM2] THEN
8791 MESON_TAC[VECTOR_NEG_NEG]);;
8793 let CLOSED_TRANSLATION_EQ = prove
8794 (`!a s. closed (IMAGE (\x:real^N. a + x) s) <=> closed s`,
8795 REWRITE_TAC[closed] THEN GEOM_TRANSLATE_TAC[]);;
8797 let CLOSED_TRANSLATION = prove
8798 (`!s a:real^N. closed s ==> closed (IMAGE (\x. a + x) s)`,
8799 REWRITE_TAC[CLOSED_TRANSLATION_EQ]);;
8801 add_translation_invariants [CLOSED_TRANSLATION_EQ];;
8803 let COMPLETE_TRANSLATION_EQ = prove
8804 (`!a s. complete(IMAGE (\x:real^N. a + x) s) <=> complete s`,
8805 REWRITE_TAC[COMPLETE_EQ_CLOSED; CLOSED_TRANSLATION_EQ]);;
8807 add_translation_invariants [COMPLETE_TRANSLATION_EQ];;
8809 let TRANSLATION_UNIV = prove
8810 (`!a. IMAGE (\x. a + x) (:real^N) = (:real^N)`,
8811 CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN GEOM_TRANSLATE_TAC[]);;
8813 let TRANSLATION_DIFF = prove
8814 (`!s t:real^N->bool.
8815 IMAGE (\x. a + x) (s DIFF t) =
8816 (IMAGE (\x. a + x) s) DIFF (IMAGE (\x. a + x) t)`,
8817 REWRITE_TAC[EXTENSION; IN_DIFF; IN_IMAGE] THEN
8818 ONCE_REWRITE_TAC[VECTOR_ARITH `x:real^N = a + y <=> y = x - a`] THEN
8819 REWRITE_TAC[UNWIND_THM2]);;
8821 let CLOSURE_TRANSLATION = prove
8822 (`!a s. closure(IMAGE (\x:real^N. a + x) s) = IMAGE (\x. a + x) (closure s)`,
8823 REWRITE_TAC[CLOSURE_INTERIOR] THEN GEOM_TRANSLATE_TAC[]);;
8825 add_translation_invariants [CLOSURE_TRANSLATION];;
8827 let FRONTIER_TRANSLATION = prove
8828 (`!a s. frontier(IMAGE (\x:real^N. a + x) s) = IMAGE (\x. a + x) (frontier s)`,
8829 REWRITE_TAC[frontier] THEN GEOM_TRANSLATE_TAC[]);;
8831 add_translation_invariants [FRONTIER_TRANSLATION];;
8833 (* ------------------------------------------------------------------------- *)
8834 (* Separation between points and sets. *)
8835 (* ------------------------------------------------------------------------- *)
8837 let SEPARATE_POINT_CLOSED = prove
8839 closed s /\ ~(a IN s)
8840 ==> ?d. &0 < d /\ !x. x IN s ==> d <= dist(a,x)`,
8841 REPEAT STRIP_TAC THEN
8842 ASM_CASES_TAC `s:real^N->bool = {}` THENL
8843 [EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY; REAL_LT_01];
8845 MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`] DISTANCE_ATTAINS_INF) THEN
8846 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `b:real^N` THEN
8847 STRIP_TAC THEN EXISTS_TAC `dist(a:real^N,b)` THEN
8848 ASM_MESON_TAC[DIST_POS_LT]);;
8850 let SEPARATE_COMPACT_CLOSED = prove
8851 (`!s t:real^N->bool.
8852 compact s /\ closed t /\ s INTER t = {}
8853 ==> ?d. &0 < d /\ !x y. x IN s /\ y IN t ==> d <= dist(x,y)`,
8854 REPEAT STRIP_TAC THEN
8855 MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0:real^N`]
8856 SEPARATE_POINT_CLOSED) THEN
8857 ASM_SIMP_TAC[COMPACT_CLOSED_DIFFERENCES; IN_ELIM_THM] THEN
8858 REWRITE_TAC[VECTOR_ARITH `vec 0 = x - y <=> x = y`] THEN
8859 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
8860 MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN
8861 MESON_TAC[NORM_ARITH `dist(vec 0,x - y) = dist(x,y)`]);;
8863 let SEPARATE_CLOSED_COMPACT = prove
8864 (`!s t:real^N->bool.
8865 closed s /\ compact t /\ s INTER t = {}
8866 ==> ?d. &0 < d /\ !x y. x IN s /\ y IN t ==> d <= dist(x,y)`,
8867 ONCE_REWRITE_TAC[DIST_SYM; INTER_COMM] THEN
8868 MESON_TAC[SEPARATE_COMPACT_CLOSED]);;
8870 (* ------------------------------------------------------------------------- *)
8871 (* Representing sets as the union of a chain of compact sets. *)
8872 (* ------------------------------------------------------------------------- *)
8874 let CLOSED_UNION_COMPACT_SUBSETS = prove
8876 ==> ?f:num->real^N->bool.
8877 (!n. compact(f n)) /\
8878 (!n. (f n) SUBSET s) /\
8879 (!n. (f n) SUBSET f(n + 1)) /\
8880 UNIONS {f n | n IN (:num)} = s /\
8881 (!k. compact k /\ k SUBSET s
8882 ==> ?N. !n. n >= N ==> k SUBSET (f n))`,
8883 REPEAT STRIP_TAC THEN
8884 EXISTS_TAC `\n. s INTER cball(vec 0:real^N,&n)` THEN
8885 ASM_SIMP_TAC[INTER_SUBSET; COMPACT_CBALL; CLOSED_INTER_COMPACT] THEN
8886 REPEAT CONJ_TAC THENL
8887 [GEN_TAC THEN MATCH_MP_TAC(SET_RULE
8888 `t SUBSET u ==> s INTER t SUBSET s INTER u`) THEN
8889 REWRITE_TAC[SUBSET_BALLS; DIST_REFL; GSYM REAL_OF_NUM_ADD] THEN
8891 REWRITE_TAC[EXTENSION; UNIONS_GSPEC; IN_ELIM_THM; IN_UNIV; IN_INTER] THEN
8892 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_CBALL_0] THEN
8893 MESON_TAC[REAL_ARCH_SIMPLE];
8894 X_GEN_TAC `k:real^N->bool` THEN SIMP_TAC[SUBSET_INTER] THEN
8895 REPEAT STRIP_TAC THEN
8896 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN DISCH_THEN
8897 (MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_CBALL) THEN
8898 DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
8899 MP_TAC(ISPEC `r:real` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN
8900 X_GEN_TAC `N:num` THEN REWRITE_TAC[GSYM REAL_OF_NUM_GE] THEN
8902 REPEAT STRIP_TAC THEN
8903 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
8905 REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN ASM_REAL_ARITH_TAC]);;
8907 let OPEN_UNION_COMPACT_SUBSETS = prove
8909 ==> ?f:num->real^N->bool.
8910 (!n. compact(f n)) /\
8911 (!n. (f n) SUBSET s) /\
8912 (!n. (f n) SUBSET interior(f(n + 1))) /\
8913 UNIONS {f n | n IN (:num)} = s /\
8914 (!k. compact k /\ k SUBSET s
8915 ==> ?N. !n. n >= N ==> k SUBSET (f n))`,
8916 GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THENL
8917 [DISCH_TAC THEN EXISTS_TAC `(\n. {}):num->real^N->bool` THEN
8918 ASM_SIMP_TAC[EMPTY_SUBSET; SUBSET_EMPTY; COMPACT_EMPTY] THEN
8919 REWRITE_TAC[EXTENSION; UNIONS_GSPEC; IN_ELIM_THM; NOT_IN_EMPTY];
8920 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
8921 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN STRIP_TAC] THEN
8922 MATCH_MP_TAC(MESON[]
8923 `(!f. p1 f /\ p3 f /\ p4 f ==> p5 f) /\
8924 (?f. p1 f /\ p2 f /\ p3 f /\ (p2 f ==> p4 f))
8925 ==> ?f. p1 f /\ p2 f /\ p3 f /\ p4 f /\ p5 f`) THEN
8927 [X_GEN_TAC `f:num->real^N->bool` THEN STRIP_TAC THEN
8928 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
8929 X_GEN_TAC `k:real^N->bool` THEN STRIP_TAC THEN
8930 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL]) THEN
8931 DISCH_THEN(MP_TAC o SPEC `{interior(f n):real^N->bool | n IN (:num)}`) THEN
8932 REWRITE_TAC[FORALL_IN_GSPEC; OPEN_INTERIOR] THEN ANTS_TAC THENL
8933 [FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
8935 REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM] THEN ASM SET_TAC[];
8936 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
8937 REWRITE_TAC[SIMPLE_IMAGE; EXISTS_FINITE_SUBSET_IMAGE] THEN
8938 REWRITE_TAC[SUBSET_UNIV] THEN
8939 DISCH_THEN(X_CHOOSE_THEN `i:num->bool` STRIP_ASSUME_TAC) THEN
8940 FIRST_ASSUM(MP_TAC o SPEC `\n:num. n` o
8941 MATCH_MP UPPER_BOUND_FINITE_SET) THEN
8942 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
8943 REWRITE_TAC[GE] THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
8944 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
8946 REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_IMAGE] THEN
8947 X_GEN_TAC `m:num` THEN DISCH_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
8948 EXISTS_TAC `(f:num->real^N->bool) m` THEN
8949 REWRITE_TAC[INTERIOR_SUBSET] THEN
8950 SUBGOAL_THEN `!m n. m <= n ==> (f:num->real^N->bool) m SUBSET f n`
8951 (fun th -> ASM_MESON_TAC[th; LE_TRANS]) THEN
8952 MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN
8953 ASM_MESON_TAC[SUBSET; ADD1; INTERIOR_SUBSET]];
8955 `\n. cball(a,&n) DIFF
8956 {x + e | x IN (:real^N) DIFF s /\ e IN ball(vec 0,inv(&n + &1))}` THEN
8957 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
8958 [X_GEN_TAC `n:num` THEN MATCH_MP_TAC COMPACT_DIFF THEN
8959 SIMP_TAC[COMPACT_CBALL; OPEN_SUMS; OPEN_BALL];
8960 GEN_TAC THEN MATCH_MP_TAC(SET_RULE
8961 `(UNIV DIFF s) SUBSET t ==> c DIFF t SUBSET s`) THEN
8962 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
8963 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
8964 MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
8965 ASM_REWRITE_TAC[VECTOR_ADD_RID; CENTRE_IN_BALL; REAL_LT_INV_EQ] THEN
8967 GEN_TAC THEN REWRITE_TAC[INTERIOR_DIFF] THEN MATCH_MP_TAC(SET_RULE
8968 `s SUBSET s' /\ t' SUBSET t ==> (s DIFF t) SUBSET (s' DIFF t')`) THEN
8970 [REWRITE_TAC[INTERIOR_CBALL; SUBSET; IN_BALL; IN_CBALL] THEN
8971 REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC;
8972 MATCH_MP_TAC SUBSET_TRANS THEN
8973 EXISTS_TAC `{x + e | x IN (:real^N) DIFF s /\
8974 e IN cball(vec 0,inv(&n + &2))}` THEN
8976 [MATCH_MP_TAC CLOSURE_MINIMAL THEN
8977 ASM_SIMP_TAC[CLOSED_COMPACT_SUMS; COMPACT_CBALL;
8978 GSYM OPEN_CLOSED] THEN
8979 MATCH_MP_TAC(SET_RULE
8981 ==> {f x y | x IN s /\ y IN t} SUBSET
8982 {f x y | x IN s /\ y IN t'}`) THEN
8983 REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL; GSYM REAL_OF_NUM_ADD] THEN
8985 MATCH_MP_TAC(SET_RULE
8987 ==> {f x y | x IN s /\ y IN t} SUBSET
8988 {f x y | x IN s /\ y IN t'}`) THEN
8989 REWRITE_TAC[SUBSET; IN_BALL; IN_CBALL; GSYM REAL_OF_NUM_ADD] THEN
8990 GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH
8991 `a < b ==> x <= a ==> x < b`) THEN
8992 MATCH_MP_TAC REAL_LT_INV2 THEN REAL_ARITH_TAC]];
8993 DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
8994 ASM_REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC] THEN
8995 REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_UNIV; IN_ELIM_THM] THEN
8996 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_DIFF] THEN
8997 REWRITE_TAC[IN_ELIM_THM; IN_UNIV; IN_BALL_0] THEN
8998 REWRITE_TAC[VECTOR_ARITH `x:real^N = y + e <=> e = x - y`] THEN
8999 REWRITE_TAC[TAUT `(p /\ q) /\ r <=> r /\ p /\ q`; UNWIND_THM2] THEN
9000 REWRITE_TAC[MESON[] `~(?x. ~P x /\ Q x) <=> !x. Q x ==> P x`] THEN
9001 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
9002 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
9003 ASM_REWRITE_TAC[SUBSET; IN_BALL; dist] THEN
9004 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9005 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN
9006 DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
9007 MP_TAC(ISPEC `norm(x - a:real^N)` REAL_ARCH_SIMPLE) THEN
9008 DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN EXISTS_TAC `N1 + N2:num` THEN
9010 [REWRITE_TAC[IN_CBALL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
9011 UNDISCH_TAC `norm(x - a:real^N) <= &N2` THEN
9012 REWRITE_TAC[dist; GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC;
9013 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
9014 SUBGOAL_THEN `inv(&(N1 + N2) + &1) <= inv(&N1)` MP_TAC THENL
9015 [MATCH_MP_TAC REAL_LE_INV2 THEN
9016 ASM_SIMP_TAC[REAL_OF_NUM_LT; LE_1] THEN
9017 REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN REAL_ARITH_TAC;
9018 ASM_REAL_ARITH_TAC]]]]);;
9020 (* ------------------------------------------------------------------------- *)
9021 (* Closed-graph characterization of continuity. *)
9022 (* ------------------------------------------------------------------------- *)
9024 let CONTINUOUS_CLOSED_GRAPH_GEN = prove
9025 (`!f:real^M->real^N s t.
9026 f continuous_on s /\ IMAGE f s SUBSET t
9027 ==> closed_in (subtopology euclidean (s PCROSS t))
9028 {pastecart x (f x) | x IN s}`,
9029 REPEAT STRIP_TAC THEN
9031 `{pastecart (x:real^M) (f x:real^N) | x IN s} =
9032 {z | z IN s PCROSS t /\ f(fstcart z) - sndcart z IN {vec 0}}`
9034 [REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_THM; IN_SING;
9035 PASTECART_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART;
9036 PASTECART_INJ; VECTOR_SUB_EQ] THEN
9038 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN
9039 REWRITE_TAC[CLOSED_SING] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN
9040 SIMP_TAC[GSYM o_DEF; LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN
9041 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
9042 SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART; IMAGE_FSTCART_PCROSS] THEN
9043 ASM_MESON_TAC[CONTINUOUS_ON_EMPTY]]);;
9045 let CONTINUOUS_CLOSED_GRAPH_EQ = prove
9046 (`!f:real^M->real^N s t.
9047 compact t /\ IMAGE f s SUBSET t
9048 ==> (f continuous_on s <=>
9049 closed_in (subtopology euclidean (s PCROSS t))
9050 {pastecart x (f x) | x IN s})`,
9051 REPEAT STRIP_TAC THEN EQ_TAC THEN
9052 ASM_SIMP_TAC[CONTINUOUS_CLOSED_GRAPH_GEN] THEN DISCH_TAC THEN
9053 FIRST_ASSUM(fun th ->
9054 REWRITE_TAC[MATCH_MP CONTINUOUS_ON_CLOSED_GEN th]) THEN
9055 X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN
9057 `{x | x IN s /\ (f:real^M->real^N) x IN c} =
9058 IMAGE fstcart ({pastecart x (f x) | x IN s} INTER
9061 [REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM; EXISTS_PASTECART;
9062 FSTCART_PASTECART; IN_INTER; IN_ELIM_PASTECART_THM;
9063 PASTECART_IN_PCROSS; PASTECART_INJ] THEN
9065 MATCH_MP_TAC CLOSED_MAP_FSTCART THEN EXISTS_TAC `t:real^N->bool` THEN
9066 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_IN_INTER THEN
9067 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CLOSED_IN_PCROSS THEN
9068 ASM_REWRITE_TAC[CLOSED_IN_REFL]]);;
9070 let CONTINUOUS_CLOSED_GRAPH = prove
9071 (`!f:real^M->real^N s.
9072 closed s /\ f continuous_on s ==> closed {pastecart x (f x) | x IN s}`,
9073 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN
9074 EXISTS_TAC `(s:real^M->bool) PCROSS (:real^N)` THEN
9075 ASM_SIMP_TAC[CLOSED_PCROSS; CLOSED_UNIV] THEN
9076 MATCH_MP_TAC CONTINUOUS_CLOSED_GRAPH_GEN THEN
9077 ASM_REWRITE_TAC[SUBSET_UNIV]);;
9079 let CONTINUOUS_FROM_CLOSED_GRAPH = prove
9080 (`!f:real^M->real^N s t.
9081 compact t /\ IMAGE f s SUBSET t /\
9082 closed {pastecart x (f x) | x IN s}
9083 ==> f continuous_on s`,
9084 REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN
9085 DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
9086 FIRST_ASSUM(SUBST1_TAC o MATCH_MP CONTINUOUS_CLOSED_GRAPH_EQ) THEN
9087 MATCH_MP_TAC CLOSED_SUBSET THEN ASM_REWRITE_TAC[] THEN
9088 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; PASTECART_IN_PCROSS] THEN
9091 (* ------------------------------------------------------------------------- *)
9092 (* A cute way of denoting open and closed intervals using overloading. *)
9093 (* ------------------------------------------------------------------------- *)
9095 let open_interval = new_definition
9096 `open_interval(a:real^N,b:real^N) =
9097 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
9098 ==> a$i < x$i /\ x$i < b$i}`;;
9100 let closed_interval = new_definition
9101 `closed_interval(l:(real^N#real^N)list) =
9102 {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
9103 ==> FST(HD l)$i <= x$i /\ x$i <= SND(HD l)$i}`;;
9105 make_overloadable "interval" `:A`;;
9107 overload_interface("interval",`open_interval`);;
9108 overload_interface("interval",`closed_interval`);;
9110 let interval = prove
9111 (`(interval (a,b) = {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
9112 ==> a$i < x$i /\ x$i < b$i}) /\
9113 (interval [a,b] = {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
9114 ==> a$i <= x$i /\ x$i <= b$i})`,
9115 REWRITE_TAC[open_interval; closed_interval; HD; FST; SND]);;
9117 let IN_INTERVAL = prove
9119 x IN interval (a,b) <=>
9120 !i. 1 <= i /\ i <= dimindex(:N)
9121 ==> a$i < x$i /\ x$i < b$i) /\
9123 x IN interval [a,b] <=>
9124 !i. 1 <= i /\ i <= dimindex(:N)
9125 ==> a$i <= x$i /\ x$i <= b$i)`,
9126 REWRITE_TAC[interval; IN_ELIM_THM]);;
9128 let IN_INTERVAL_REFLECT = prove
9129 (`(!a b x. (--x) IN interval[--b,--a] <=> x IN interval[a,b]) /\
9130 (!a b x. (--x) IN interval(--b,--a) <=> x IN interval(a,b))`,
9131 SIMP_TAC[IN_INTERVAL; REAL_LT_NEG2; REAL_LE_NEG2; VECTOR_NEG_COMPONENT] THEN
9134 let REFLECT_INTERVAL = prove
9135 (`(!a b:real^N. IMAGE (--) (interval[a,b]) = interval[--b,--a]) /\
9136 (!a b:real^N. IMAGE (--) (interval(a,b)) = interval(--b,--a))`,
9137 REPEAT STRIP_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
9138 REWRITE_TAC[IN_INTERVAL_REFLECT] THEN MESON_TAC[VECTOR_NEG_NEG]);;
9140 let INTERVAL_EQ_EMPTY = prove
9141 (`((interval [a:real^N,b] = {}) <=>
9142 ?i. 1 <= i /\ i <= dimindex(:N) /\ b$i < a$i) /\
9143 ((interval (a:real^N,b) = {}) <=>
9144 ?i. 1 <= i /\ i <= dimindex(:N) /\ b$i <= a$i)`,
9145 REWRITE_TAC[EXTENSION; IN_INTERVAL; NOT_IN_EMPTY] THEN
9146 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; GSYM CONJ_ASSOC] THEN
9147 CONJ_TAC THEN EQ_TAC THENL
9148 [MESON_TAC[REAL_LE_REFL; REAL_NOT_LE];
9149 MESON_TAC[REAL_LE_TRANS; REAL_NOT_LE];
9151 MESON_TAC[REAL_LT_TRANS; REAL_NOT_LT]] THEN
9152 SUBGOAL_THEN `!a b. ?c. a < b ==> a < c /\ c < b`
9153 (MP_TAC o REWRITE_RULE[SKOLEM_THM]) THENL
9154 [MESON_TAC[REAL_LT_BETWEEN]; ALL_TAC] THEN
9155 DISCH_THEN(X_CHOOSE_TAC `mid:real->real->real`) THEN
9156 DISCH_THEN(MP_TAC o SPEC
9157 `(lambda i. mid ((a:real^N)$i) ((b:real^N)$i)):real^N`) THEN
9158 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> ~(a /\ b ==> ~c)`] THEN
9159 SIMP_TAC[LAMBDA_BETA] THEN ASM_MESON_TAC[REAL_NOT_LT]);;
9161 let INTERVAL_NE_EMPTY = prove
9162 (`(~(interval [a:real^N,b] = {}) <=>
9163 !i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= b$i) /\
9164 (~(interval (a:real^N,b) = {}) <=>
9165 !i. 1 <= i /\ i <= dimindex(:N) ==> a$i < b$i)`,
9166 REWRITE_TAC[INTERVAL_EQ_EMPTY] THEN MESON_TAC[REAL_NOT_LE]);;
9168 let SUBSET_INTERVAL_IMP = prove
9169 (`((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)
9170 ==> interval[c,d] SUBSET interval[a:real^N,b]) /\
9171 ((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < c$i /\ d$i < b$i)
9172 ==> interval[c,d] SUBSET interval(a:real^N,b)) /\
9173 ((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)
9174 ==> interval(c,d) SUBSET interval[a:real^N,b]) /\
9175 ((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)
9176 ==> interval(c,d) SUBSET interval(a:real^N,b))`,
9177 REWRITE_TAC[SUBSET; IN_INTERVAL] THEN REPEAT CONJ_TAC THEN
9178 DISCH_TAC THEN GEN_TAC THEN POP_ASSUM MP_TAC THEN
9179 REWRITE_TAC[IMP_IMP; AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN
9180 GEN_TAC THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
9181 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
9183 let INTERVAL_SING = prove
9184 (`interval[a,a] = {a} /\ interval(a,a) = {}`,
9185 REWRITE_TAC[EXTENSION; IN_SING; NOT_IN_EMPTY; IN_INTERVAL] THEN
9186 REWRITE_TAC[REAL_LE_ANTISYM; REAL_LT_ANTISYM; CART_EQ; EQ_SYM_EQ] THEN
9187 MESON_TAC[DIMINDEX_GE_1; LE_REFL]);;
9189 let SUBSET_INTERVAL = prove
9190 (`(interval[c,d] SUBSET interval[a:real^N,b] <=>
9191 (!i. 1 <= i /\ i <= dimindex(:N) ==> c$i <= d$i)
9192 ==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)) /\
9193 (interval[c,d] SUBSET interval(a:real^N,b) <=>
9194 (!i. 1 <= i /\ i <= dimindex(:N) ==> c$i <= d$i)
9195 ==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < c$i /\ d$i < b$i)) /\
9196 (interval(c,d) SUBSET interval[a:real^N,b] <=>
9197 (!i. 1 <= i /\ i <= dimindex(:N) ==> c$i < d$i)
9198 ==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)) /\
9199 (interval(c,d) SUBSET interval(a:real^N,b) <=>
9200 (!i. 1 <= i /\ i <= dimindex(:N) ==> c$i < d$i)
9201 ==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i))`,
9203 (`(!x:real^N. (!i. 1 <= i /\ i <= dimindex(:N) ==> Q i (x$i))
9204 ==> (!i. 1 <= i /\ i <= dimindex(:N) ==> R i (x$i)))
9205 ==> (!i. 1 <= i /\ i <= dimindex(:N) ==> ?y. Q i y)
9206 ==> !i y. 1 <= i /\ i <= dimindex(:N) /\ Q i y ==> R i y`,
9207 DISCH_TAC THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN
9208 DISCH_THEN(X_CHOOSE_THEN `f:num->real` STRIP_ASSUME_TAC) THEN
9209 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o
9210 SPEC `(lambda j. if j = i then y else f j):real^N`) THEN
9211 SIMP_TAC[LAMBDA_BETA] THEN ASM_MESON_TAC[]) in
9212 REPEAT STRIP_TAC THEN
9214 `(~q ==> p) /\ (q ==> (p <=> r)) ==> (p <=> q ==> r)`) THEN
9216 [DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `s = {} ==> s SUBSET t`) THEN
9217 REWRITE_TAC[INTERVAL_EQ_EMPTY] THEN ASM_MESON_TAC[REAL_NOT_LT];
9219 DISCH_TAC THEN EQ_TAC THEN REWRITE_TAC[SUBSET_INTERVAL_IMP] THEN
9220 REWRITE_TAC[SUBSET; IN_INTERVAL] THEN
9221 DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN ANTS_TAC THENL
9222 [ASM_MESON_TAC[REAL_LT_BETWEEN; REAL_LE_BETWEEN]; ALL_TAC] THEN
9223 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN
9224 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
9225 FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
9226 ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(K ALL_TAC) THEN STRIP_TAC)
9228 [ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL];
9229 ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL];
9230 ALL_TAC; ALL_TAC] THEN
9231 (REPEAT STRIP_TAC THENL
9232 [FIRST_X_ASSUM(MP_TAC o SPEC
9233 `((c:real^N)$i + min ((a:real^N)$i) ((d:real^N)$i)) / &2`) THEN
9234 POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
9235 FIRST_X_ASSUM(MP_TAC o SPEC
9236 `(max ((b:real^N)$i) ((c:real^N)$i) + (d:real^N)$i) / &2`) THEN
9237 POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]));;
9239 let DISJOINT_INTERVAL = prove
9241 (interval[a,b] INTER interval[c,d] = {} <=>
9242 ?i. 1 <= i /\ i <= dimindex(:N) /\
9243 (b$i < a$i \/ d$i < c$i \/ b$i < c$i \/ d$i < a$i)) /\
9244 (interval[a,b] INTER interval(c,d) = {} <=>
9245 ?i. 1 <= i /\ i <= dimindex(:N) /\
9246 (b$i < a$i \/ d$i <= c$i \/ b$i <= c$i \/ d$i <= a$i)) /\
9247 (interval(a,b) INTER interval[c,d] = {} <=>
9248 ?i. 1 <= i /\ i <= dimindex(:N) /\
9249 (b$i <= a$i \/ d$i < c$i \/ b$i <= c$i \/ d$i <= a$i)) /\
9250 (interval(a,b) INTER interval(c,d) = {} <=>
9251 ?i. 1 <= i /\ i <= dimindex(:N) /\
9252 (b$i <= a$i \/ d$i <= c$i \/ b$i <= c$i \/ d$i <= a$i))`,
9253 REWRITE_TAC[EXTENSION; IN_INTER; IN_INTERVAL; NOT_IN_EMPTY] THEN
9254 REWRITE_TAC[AND_FORALL_THM; NOT_FORALL_THM] THEN
9255 REWRITE_TAC[TAUT `~((p ==> q) /\ (p ==> r)) <=> p /\ (~q \/ ~r)`] THEN
9256 REWRITE_TAC[DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN
9258 [DISCH_THEN(MP_TAC o SPEC
9259 `(lambda i. (max ((a:real^N)$i) ((c:real^N)$i) +
9260 min ((b:real^N)$i) ((d:real^N)$i)) / &2):real^N`) THEN
9261 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
9262 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
9263 ASM_SIMP_TAC[LAMBDA_BETA] THEN REAL_ARITH_TAC;
9264 DISCH_THEN(fun th -> GEN_TAC THEN MP_TAC th) THEN
9265 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN SIMP_TAC[] THEN
9268 let ENDS_IN_INTERVAL = prove
9269 (`(!a b. a IN interval[a,b] <=> ~(interval[a,b] = {})) /\
9270 (!a b. b IN interval[a,b] <=> ~(interval[a,b] = {})) /\
9271 (!a b. ~(a IN interval(a,b))) /\
9272 (!a b. ~(b IN interval(a,b)))`,
9273 REWRITE_TAC[IN_INTERVAL; INTERVAL_NE_EMPTY] THEN
9274 REWRITE_TAC[REAL_LE_REFL; REAL_LT_REFL] THEN
9275 MESON_TAC[DIMINDEX_GE_1; LE_REFL]);;
9277 let ENDS_IN_UNIT_INTERVAL = prove
9278 (`vec 0 IN interval[vec 0,vec 1] /\
9279 vec 1 IN interval[vec 0,vec 1] /\
9280 ~(vec 0 IN interval(vec 0,vec 1)) /\
9281 ~(vec 1 IN interval(vec 0,vec 1))`,
9282 REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_NE_EMPTY; VEC_COMPONENT] THEN
9283 REWRITE_TAC[REAL_POS]);;
9285 let INTER_INTERVAL = prove
9286 (`interval[a,b] INTER interval[c,d] =
9287 interval[(lambda i. max (a$i) (c$i)),(lambda i. min (b$i) (d$i))]`,
9288 REWRITE_TAC[EXTENSION; IN_INTER; IN_INTERVAL] THEN
9289 SIMP_TAC[LAMBDA_BETA; REAL_MAX_LE; REAL_LE_MIN] THEN MESON_TAC[]);;
9291 let INTERVAL_OPEN_SUBSET_CLOSED = prove
9292 (`!a b. interval(a,b) SUBSET interval[a,b]`,
9293 REWRITE_TAC[SUBSET; IN_INTERVAL] THEN MESON_TAC[REAL_LT_IMP_LE]);;
9295 let OPEN_INTERVAL_LEMMA = prove
9296 (`!a b x. a < x /\ x < b
9297 ==> ?d. &0 < d /\ !x'. abs(x' - x) < d ==> a < x' /\ x' < b`,
9298 REPEAT STRIP_TAC THEN
9299 EXISTS_TAC `min (x - a) (b - x)` THEN REWRITE_TAC[REAL_LT_MIN] THEN
9300 ASM_REAL_ARITH_TAC);;
9302 let OPEN_INTERVAL = prove
9303 (`!a:real^N b. open(interval (a,b))`,
9304 REPEAT GEN_TAC THEN REWRITE_TAC[open_def; interval; IN_ELIM_THM] THEN
9305 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
9306 SUBGOAL_THEN `!i. 1 <= i /\ i <= dimindex(:N)
9308 !x'. abs(x' - (x:real^N)$i) < d
9309 ==> (a:real^N)$i < x' /\ x' < (b:real^N)$i`
9310 MP_TAC THENL [ASM_SIMP_TAC[OPEN_INTERVAL_LEMMA]; ALL_TAC] THEN
9311 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
9312 REWRITE_TAC[SKOLEM_THM] THEN
9313 DISCH_THEN(X_CHOOSE_THEN `d:num->real` STRIP_ASSUME_TAC) THEN
9314 EXISTS_TAC `inf (IMAGE d (1..dimindex(:N)))` THEN
9315 SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; FINITE_NUMSEG;
9316 IMAGE_EQ_EMPTY; NOT_INSERT_EMPTY; NUMSEG_EMPTY;
9317 ARITH_RULE `n < 1 <=> (n = 0)`; DIMINDEX_NONZERO] THEN
9318 REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG; dist] THEN
9319 ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LET_TRANS; VECTOR_SUB_COMPONENT]);;
9321 let CLOSED_INTERVAL = prove
9322 (`!a:real^N b. closed(interval [a,b])`,
9323 REWRITE_TAC[CLOSED_LIMPT; LIMPT_APPROACHABLE; IN_INTERVAL] THEN
9324 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THENL
9325 [FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N)$i - (x:real^N)$i`);
9326 FIRST_X_ASSUM(MP_TAC o SPEC `(x:real^N)$i - (b:real^N)$i`)] THEN
9327 ASM_REWRITE_TAC[REAL_SUB_LT] THEN
9328 DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN
9329 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
9330 REWRITE_TAC[dist; REAL_NOT_LT] THEN
9331 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs((z - x :real^N)$i)` THEN
9332 ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN
9333 ASM_SIMP_TAC[VECTOR_SUB_COMPONENT] THEN
9334 ASM_SIMP_TAC[REAL_ARITH `x < a /\ a <= z ==> a - x <= abs(z - x)`;
9335 REAL_ARITH `z <= b /\ b < x ==> x - b <= abs(z - x)`]);;
9337 let INTERIOR_CLOSED_INTERVAL = prove
9338 (`!a:real^N b. interior(interval [a,b]) = interval (a,b)`,
9339 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
9341 MATCH_MP_TAC INTERIOR_MAXIMAL THEN
9342 REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED; OPEN_INTERVAL]] THEN
9343 REWRITE_TAC[interior; SUBSET; IN_INTERVAL; IN_ELIM_THM] THEN
9344 X_GEN_TAC `x:real^N` THEN
9345 DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
9346 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
9347 ASM_SIMP_TAC[REAL_LT_LE] THEN REPEAT STRIP_TAC THEN
9348 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [open_def]) THEN
9349 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
9350 DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL
9351 [(let t = `x - (e / &2) % basis i :real^N` in
9352 DISCH_THEN(MP_TAC o SPEC t) THEN FIRST_X_ASSUM(MP_TAC o SPEC t));
9353 (let t = `x + (e / &2) % basis i :real^N` in
9354 DISCH_THEN(MP_TAC o SPEC t) THEN FIRST_X_ASSUM(MP_TAC o SPEC t))] THEN
9355 REWRITE_TAC[dist; VECTOR_ADD_SUB; VECTOR_ARITH `x - y - x = --y:real^N`] THEN
9356 ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; NORM_NEG; REAL_MUL_RID;
9357 REAL_ARITH `&0 < e ==> abs(e / &2) < e`] THEN
9358 MATCH_MP_TAC(TAUT `~b ==> (a ==> b) ==> ~a`) THEN
9359 REWRITE_TAC[NOT_FORALL_THM] THEN EXISTS_TAC `i:num` THEN
9360 ASM_SIMP_TAC[DE_MORGAN_THM; VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT] THENL
9361 [DISJ1_TAC THEN REWRITE_TAC[REAL_ARITH `a <= a - b <=> ~(&0 < b)`];
9362 DISJ2_TAC THEN REWRITE_TAC[REAL_ARITH `a + b <= a <=> ~(&0 < b)`]] THEN
9363 ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; basis; LAMBDA_BETA; REAL_MUL_RID] THEN
9364 ASM_REWRITE_TAC[REAL_HALF]);;
9366 let INTERIOR_INTERVAL = prove
9367 (`(!a b. interior(interval[a,b]) = interval(a,b)) /\
9368 (!a b. interior(interval(a,b)) = interval(a,b))`,
9369 SIMP_TAC[INTERIOR_CLOSED_INTERVAL; INTERIOR_OPEN; OPEN_INTERVAL]);;
9371 let BOUNDED_CLOSED_INTERVAL = prove
9372 (`!a b:real^N. bounded (interval [a,b])`,
9373 REPEAT STRIP_TAC THEN REWRITE_TAC[bounded; interval] THEN
9374 EXISTS_TAC `sum(1..dimindex(:N))
9375 (\i. abs((a:real^N)$i) + abs((b:real^N)$i))` THEN
9376 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_ELIM_THM] THEN
9377 STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
9378 EXISTS_TAC `sum(1..dimindex(:N)) (\i. abs((x:real^N)$i))` THEN
9379 REWRITE_TAC[NORM_LE_L1] THEN MATCH_MP_TAC SUM_LE THEN
9380 ASM_SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; REAL_ARITH
9381 `a <= x /\ x <= b ==> abs(x) <= abs(a) + abs(b)`]);;
9383 let BOUNDED_INTERVAL = prove
9384 (`(!a b. bounded (interval [a,b])) /\ (!a b. bounded (interval (a,b)))`,
9385 MESON_TAC[BOUNDED_CLOSED_INTERVAL; BOUNDED_SUBSET;
9386 INTERVAL_OPEN_SUBSET_CLOSED]);;
9388 let NOT_INTERVAL_UNIV = prove
9389 (`(!a b. ~(interval[a,b] = UNIV)) /\
9390 (!a b. ~(interval(a,b) = UNIV))`,
9391 MESON_TAC[BOUNDED_INTERVAL; NOT_BOUNDED_UNIV]);;
9393 let COMPACT_INTERVAL = prove
9394 (`!a b. compact (interval [a,b])`,
9395 SIMP_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_INTERVAL; CLOSED_INTERVAL]);;
9397 let OPEN_INTERVAL_MIDPOINT = prove
9399 ~(interval(a,b) = {}) ==> (inv(&2) % (a + b)) IN interval(a,b)`,
9400 REWRITE_TAC[INTERVAL_NE_EMPTY; IN_INTERVAL] THEN
9401 SIMP_TAC[VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT] THEN
9402 REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
9403 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN REAL_ARITH_TAC);;
9405 let OPEN_CLOSED_INTERVAL_CONVEX = prove
9406 (`!a b x y:real^N e.
9407 x IN interval(a,b) /\ y IN interval[a,b] /\ &0 < e /\ e <= &1
9408 ==> (e % x + (&1 - e) % y) IN interval(a,b)`,
9409 REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT
9410 `(c /\ d ==> a /\ b ==> e) ==> a /\ b /\ c /\ d ==> e`) THEN
9411 STRIP_TAC THEN REWRITE_TAC[IN_INTERVAL; AND_FORALL_THM] THEN
9412 SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
9413 MATCH_MP_TAC MONO_FORALL THEN
9414 GEN_TAC THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
9415 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
9416 SUBST1_TAC(REAL_ARITH `(a:real^N)$i = e * a$i + (&1 - e) * a$i`) THEN
9417 SUBST1_TAC(REAL_ARITH `(b:real^N)$i = e * b$i + (&1 - e) * b$i`) THEN
9418 CONJ_TAC THEN MATCH_MP_TAC REAL_LTE_ADD2 THEN
9419 ASM_SIMP_TAC[REAL_LT_LMUL_EQ; REAL_LE_LMUL; REAL_SUB_LE]);;
9421 let CLOSURE_OPEN_INTERVAL = prove
9423 ~(interval(a,b) = {}) ==> closure(interval(a,b)) = interval[a,b]`,
9424 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
9425 [MATCH_MP_TAC CLOSURE_MINIMAL THEN
9426 REWRITE_TAC[INTERVAL_OPEN_SUBSET_CLOSED; CLOSED_INTERVAL];
9428 REWRITE_TAC[SUBSET; closure; IN_UNION] THEN X_GEN_TAC `x:real^N` THEN
9429 DISCH_TAC THEN MATCH_MP_TAC(TAUT `(~b ==> c) ==> b \/ c`) THEN DISCH_TAC THEN
9430 REWRITE_TAC[IN_ELIM_THM; LIMPT_SEQUENTIAL] THEN
9431 ABBREV_TAC `(c:real^N) = inv(&2) % (a + b)` THEN
9432 EXISTS_TAC `\n. (x:real^N) + inv(&n + &1) % (c - x)` THEN CONJ_TAC THENL
9433 [X_GEN_TAC `n:num` THEN REWRITE_TAC[IN_DELETE] THEN
9434 REWRITE_TAC[VECTOR_ARITH `x + a = x <=> a = vec 0`] THEN
9435 REWRITE_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0] THEN
9436 REWRITE_TAC[VECTOR_SUB_EQ; REAL_ARITH `~(&n + &1 = &0)`] THEN
9437 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[OPEN_INTERVAL_MIDPOINT]] THEN
9438 REWRITE_TAC[VECTOR_ARITH `x + a % (y - x) = a % y + (&1 - a) % x`] THEN
9439 MATCH_MP_TAC OPEN_CLOSED_INTERVAL_CONVEX THEN
9440 CONJ_TAC THENL [ASM_MESON_TAC[OPEN_INTERVAL_MIDPOINT]; ALL_TAC] THEN
9441 ASM_REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
9442 MATCH_MP_TAC REAL_INV_LE_1 THEN REAL_ARITH_TAC;
9444 GEN_REWRITE_TAC LAND_CONV [VECTOR_ARITH `x:real^N = x + &0 % (c - x)`] THEN
9445 MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
9446 MATCH_MP_TAC LIM_VMUL THEN REWRITE_TAC[LIM_CONST] THEN
9447 REWRITE_TAC[LIM_SEQUENTIALLY; o_THM; DIST_LIFT; REAL_SUB_RZERO] THEN
9448 X_GEN_TAC `e:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN
9449 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
9450 STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
9451 REWRITE_TAC[REAL_ABS_INV] THEN MATCH_MP_TAC REAL_LET_TRANS THEN
9452 EXISTS_TAC `inv(&N)` THEN ASM_REWRITE_TAC[] THEN
9453 MATCH_MP_TAC REAL_LE_INV2 THEN UNDISCH_TAC `N:num <= n` THEN
9454 UNDISCH_TAC `~(N = 0)` THEN
9455 REWRITE_TAC[GSYM LT_NZ; GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_LT] THEN
9458 let CLOSURE_INTERVAL = prove
9459 (`(!a b. closure(interval[a,b]) = interval[a,b]) /\
9460 (!a b. closure(interval(a,b)) =
9461 if interval(a,b) = {} then {} else interval[a,b])`,
9462 SIMP_TAC[CLOSURE_CLOSED; CLOSED_INTERVAL] THEN REPEAT GEN_TAC THEN
9463 COND_CASES_TAC THEN ASM_SIMP_TAC[CLOSURE_OPEN_INTERVAL; CLOSURE_EMPTY]);;
9465 let BOUNDED_SUBSET_OPEN_INTERVAL_SYMMETRIC = prove
9466 (`!s:real^N->bool. bounded s ==> ?a. s SUBSET interval(--a,a)`,
9467 REWRITE_TAC[BOUNDED_POS; LEFT_IMP_EXISTS_THM] THEN
9468 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `B:real`] THEN STRIP_TAC THEN
9469 EXISTS_TAC `(lambda i. B + &1):real^N` THEN
9470 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
9471 SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; REAL_BOUNDS_LT; VECTOR_NEG_COMPONENT] THEN
9472 ASM_MESON_TAC[COMPONENT_LE_NORM;
9473 REAL_ARITH `x <= y ==> a <= x ==> a < y + &1`]);;
9475 let BOUNDED_SUBSET_OPEN_INTERVAL = prove
9476 (`!s:real^N->bool. bounded s ==> ?a b. s SUBSET interval(a,b)`,
9477 MESON_TAC[BOUNDED_SUBSET_OPEN_INTERVAL_SYMMETRIC]);;
9479 let BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC = prove
9480 (`!s:real^N->bool. bounded s ==> ?a. s SUBSET interval[--a,a]`,
9482 DISCH_THEN(MP_TAC o MATCH_MP BOUNDED_SUBSET_OPEN_INTERVAL_SYMMETRIC) THEN
9483 MATCH_MP_TAC MONO_EXISTS THEN
9484 SIMP_TAC[IN_BALL; IN_INTERVAL; SUBSET; REAL_LT_IMP_LE]);;
9486 let BOUNDED_SUBSET_CLOSED_INTERVAL = prove
9487 (`!s:real^N->bool. bounded s ==> ?a b. s SUBSET interval[a,b]`,
9488 MESON_TAC[BOUNDED_SUBSET_CLOSED_INTERVAL_SYMMETRIC]);;
9490 let FRONTIER_CLOSED_INTERVAL = prove
9491 (`!a b. frontier(interval[a,b]) = interval[a,b] DIFF interval(a,b)`,
9492 SIMP_TAC[frontier; INTERIOR_CLOSED_INTERVAL; CLOSURE_CLOSED;
9495 let FRONTIER_OPEN_INTERVAL = prove
9496 (`!a b. frontier(interval(a,b)) =
9497 if interval(a,b) = {} then {}
9498 else interval[a,b] DIFF interval(a,b)`,
9499 REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[FRONTIER_EMPTY] THEN
9500 ASM_SIMP_TAC[frontier; CLOSURE_OPEN_INTERVAL; INTERIOR_OPEN;
9503 let INTER_INTERVAL_MIXED_EQ_EMPTY = prove
9505 ~(interval(c,d) = {})
9506 ==> (interval(a,b) INTER interval[c,d] = {} <=>
9507 interval(a,b) INTER interval(c,d) = {})`,
9508 SIMP_TAC[GSYM CLOSURE_OPEN_INTERVAL; OPEN_INTER_CLOSURE_EQ_EMPTY;
9511 let INTERVAL_TRANSLATION = prove
9512 (`(!c a b. interval[c + a,c + b] = IMAGE (\x. c + x) (interval[a,b])) /\
9513 (!c a b. interval(c + a,c + b) = IMAGE (\x. c + x) (interval(a,b)))`,
9514 REWRITE_TAC[interval] THEN CONJ_TAC THEN GEOM_TRANSLATE_TAC[] THEN
9515 REWRITE_TAC[VECTOR_ADD_COMPONENT; REAL_LT_LADD; REAL_LE_LADD]);;
9517 add_translation_invariants
9518 [CONJUNCT1 INTERVAL_TRANSLATION; CONJUNCT2 INTERVAL_TRANSLATION];;
9520 let EMPTY_AS_INTERVAL = prove
9521 (`{} = interval[vec 1,vec 0]`,
9522 SIMP_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTERVAL; VEC_COMPONENT] THEN
9523 GEN_TAC THEN DISCH_THEN(MP_TAC o SPEC `1`) THEN
9524 REWRITE_TAC[LE_REFL; DIMINDEX_GE_1] THEN REAL_ARITH_TAC);;
9526 let UNIT_INTERVAL_NONEMPTY = prove
9527 (`~(interval[vec 0:real^N,vec 1] = {}) /\
9528 ~(interval(vec 0:real^N,vec 1) = {})`,
9529 SIMP_TAC[INTERVAL_NE_EMPTY; VEC_COMPONENT; REAL_LT_01; REAL_POS]);;
9531 let IMAGE_STRETCH_INTERVAL = prove
9533 IMAGE (\x. lambda k. m(k) * x$k) (interval[a,b]) =
9534 if interval[a,b] = {} then {}
9535 else interval[(lambda k. min (m(k) * a$k) (m(k) * b$k)):real^N,
9536 (lambda k. max (m(k) * a$k) (m(k) * b$k))]`,
9537 REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[IMAGE_CLAUSES] THEN
9538 ASM_SIMP_TAC[EXTENSION; IN_IMAGE; CART_EQ; IN_INTERVAL; AND_FORALL_THM;
9539 TAUT `(a ==> b) /\ (a ==> c) <=> a ==> b /\ c`;
9540 LAMBDA_BETA; GSYM LAMBDA_SKOLEM] THEN
9541 X_GEN_TAC `x:real^N` THEN MATCH_MP_TAC(MESON[]
9542 `(!x. p x ==> (q x <=> r x))
9543 ==> ((!x. p x ==> q x) <=> (!x. p x ==> r x))`) THEN
9544 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INTERVAL_NE_EMPTY]) THEN
9545 MATCH_MP_TAC MONO_FORALL THEN
9546 X_GEN_TAC `k:num` THEN ASM_CASES_TAC `1 <= k /\ k <= dimindex(:N)` THEN
9547 ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `(m:num->real) k = &0` THENL
9548 [ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MAX_ACI; REAL_MIN_ACI] THEN
9549 ASM_MESON_TAC[REAL_LE_ANTISYM; REAL_LE_REFL];
9551 ASM_SIMP_TAC[REAL_FIELD `~(m = &0) ==> (x = m * y <=> y = x / m)`] THEN
9552 REWRITE_TAC[UNWIND_THM2] THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP
9553 (REAL_ARITH `~(z = &0) ==> &0 < z \/ &0 < --z`))
9556 ONCE_REWRITE_TAC[GSYM REAL_LE_NEG2] THEN
9557 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
9558 REWRITE_TAC[REAL_ARITH `--(max a b) = min (--a) (--b)`;
9559 REAL_ARITH `--(min a b) = max (--a) (--b)`; real_div;
9560 GSYM REAL_MUL_RNEG; GSYM REAL_INV_NEG] THEN
9561 REWRITE_TAC[GSYM real_div]] THEN
9562 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ] THEN
9563 ASM_SIMP_TAC[real_min; real_max; REAL_LE_LMUL_EQ; REAL_LE_RMUL_EQ] THEN
9566 let INTERVAL_IMAGE_STRETCH_INTERVAL = prove
9567 (`!a b:real^N m. ?u v:real^N.
9568 IMAGE (\x. lambda k. m k * x$k) (interval[a,b]) = interval[u,v]`,
9569 REWRITE_TAC[IMAGE_STRETCH_INTERVAL] THEN MESON_TAC[EMPTY_AS_INTERVAL]);;
9571 let CLOSED_INTERVAL_IMAGE_UNIT_INTERVAL = prove
9573 ~(interval[a,b] = {})
9574 ==> interval[a,b] = IMAGE (\x:real^N. a + x)
9575 (IMAGE (\x. (lambda i. (b$i - a$i) * x$i))
9576 (interval[vec 0:real^N,vec 1]))`,
9577 REWRITE_TAC[INTERVAL_NE_EMPTY] THEN REPEAT STRIP_TAC THEN
9578 REWRITE_TAC[IMAGE_STRETCH_INTERVAL; UNIT_INTERVAL_NONEMPTY] THEN
9579 REWRITE_TAC[GSYM INTERVAL_TRANSLATION] THEN
9580 REWRITE_TAC[EXTENSION; IN_INTERVAL] THEN
9581 SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT; VEC_COMPONENT] THEN
9582 GEN_TAC THEN REWRITE_TAC[REAL_MUL_RZERO; REAL_MUL_RID] THEN
9583 MATCH_MP_TAC(MESON[] `(!x. P x <=> Q x) ==> ((!x. P x) <=> (!x. Q x))`) THEN
9584 POP_ASSUM MP_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN
9585 ASM_CASES_TAC `1 <= i /\ i <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN
9588 let SUMS_INTERVALS = prove
9590 ~(interval[a,b] = {}) /\ ~(interval[c,d] = {})
9591 ==> {x + y | x IN interval[a,b] /\ y IN interval[c,d]} =
9592 interval[a+c,b+d]) /\
9594 ~(interval(a,b) = {}) /\ ~(interval(c,d) = {})
9595 ==> {x + y | x IN interval(a,b) /\ y IN interval(c,d)} =
9596 interval(a+c,b+d))`,
9597 CONJ_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[INTERVAL_NE_EMPTY] THEN
9598 STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_INTERVAL; IN_ELIM_THM] THEN
9599 REWRITE_TAC[TAUT `(a /\ b) /\ c <=> c /\ a /\ b`] THEN
9600 REWRITE_TAC[VECTOR_ARITH `x:real^N = y + z <=> z = x - y`] THEN
9601 REWRITE_TAC[UNWIND_THM2; VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT] THEN
9602 (X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
9603 [DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC);
9605 REWRITE_TAC[AND_FORALL_THM; GSYM LAMBDA_SKOLEM;
9606 TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN
9607 REWRITE_TAC[REAL_ARITH
9608 `((a <= y /\ y <= b) /\ c <= x - y /\ x - y <= d <=>
9609 max a (x - d) <= y /\ y <= min b (x - c)) /\
9610 ((a < y /\ y < b) /\ c < x - y /\ x - y < d <=>
9611 max a (x - d) < y /\ y < min b (x - c))`] THEN
9612 REWRITE_TAC[GSYM REAL_LE_BETWEEN; GSYM REAL_LT_BETWEEN]] THEN
9613 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
9614 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN
9615 ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC));;
9617 let PCROSS_INTERVAL = prove
9618 (`!a b:real^M c d:real^N.
9619 interval[a,b] PCROSS interval[c,d] =
9620 interval[pastecart a c,pastecart b d]`,
9621 REPEAT GEN_TAC THEN REWRITE_TAC[PCROSS] THEN
9622 REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM] THEN
9623 SIMP_TAC[IN_INTERVAL; pastecart; LAMBDA_BETA; DIMINDEX_FINITE_SUM] THEN
9624 MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN EQ_TAC THEN STRIP_TAC THENL
9625 [X_GEN_TAC `i:num` THEN STRIP_TAC THEN
9626 COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
9627 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
9628 CONJ_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THENL
9629 [FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
9630 DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC;
9631 FIRST_X_ASSUM(MP_TAC o SPEC `i + dimindex(:M)`) THEN
9632 COND_CASES_TAC THEN ASM_REWRITE_TAC[ADD_SUB] THENL
9634 DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC]]]);;
9636 let OPEN_CONTAINS_INTERVAL,OPEN_CONTAINS_OPEN_INTERVAL = (CONJ_PAIR o prove)
9639 !x. x IN s ==> ?a b. x IN interval(a,b) /\ interval[a,b] SUBSET s) /\
9642 !x. x IN s ==> ?a b. x IN interval(a,b) /\ interval(a,b) SUBSET s)`,
9643 REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN
9645 `(q ==> r) /\ (r ==> p) /\ (p ==> q) ==> (p <=> q) /\ (p <=> r)`) THEN
9646 REPEAT CONJ_TAC THENL
9647 [MESON_TAC[SUBSET_TRANS; INTERVAL_OPEN_SUBSET_CLOSED];
9648 DISCH_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL] THEN
9649 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
9650 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
9651 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
9652 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN
9653 MP_TAC(ISPEC `interval(a:real^N,b)` OPEN_CONTAINS_BALL) THEN
9654 REWRITE_TAC[OPEN_INTERVAL] THEN
9655 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
9656 MATCH_MP_TAC MONO_EXISTS THEN
9657 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
9658 ASM_MESON_TAC[SUBSET_TRANS; INTERVAL_OPEN_SUBSET_CLOSED];
9659 DISCH_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
9660 FIRST_ASSUM(MP_TAC o SPEC `x:real^N` o
9661 GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
9662 ASM_REWRITE_TAC[] THEN
9663 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
9664 EXISTS_TAC `x - e / &(dimindex(:N)) % vec 1:real^N` THEN
9665 EXISTS_TAC `x + e / &(dimindex(:N)) % vec 1:real^N` THEN
9666 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
9667 `b SUBSET s ==> x IN i /\ j SUBSET b ==> x IN i /\ j SUBSET s`)) THEN
9668 SIMP_TAC[IN_INTERVAL; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; IN_CBALL;
9669 VEC_COMPONENT; VECTOR_ADD_COMPONENT; SUBSET; REAL_MUL_RID] THEN
9670 REWRITE_TAC[REAL_ARITH `x - e < x /\ x < x + e <=> &0 < e`;
9671 REAL_ARITH `x - e <= y /\ y <= x + e <=> abs(x - y) <= e`] THEN
9672 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN
9673 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT] THEN
9674 DISCH_TAC THEN REWRITE_TAC[dist] THEN
9675 MATCH_MP_TAC REAL_LE_TRANS THEN
9676 EXISTS_TAC `sum(1..dimindex(:N)) (\i. abs((x - y:real^N)$i))` THEN
9677 REWRITE_TAC[NORM_LE_L1] THEN MATCH_MP_TAC SUM_BOUND_GEN THEN
9678 ASM_SIMP_TAC[CARD_NUMSEG_1; IN_NUMSEG; FINITE_NUMSEG] THEN
9679 REWRITE_TAC[NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1]]);;
9681 let DIAMETER_INTERVAL = prove
9683 diameter(interval[a,b]) =
9684 if interval[a,b] = {} then &0 else norm(b - a)) /\
9686 diameter(interval(a,b)) =
9687 if interval(a,b) = {} then &0 else norm(b - a))`,
9688 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN
9689 ASM_CASES_TAC `interval[a:real^N,b] = {}` THENL
9690 [ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET_EMPTY; DIAMETER_EMPTY];
9691 ASM_REWRITE_TAC[]] THEN
9692 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
9693 [REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN
9694 ASM_SIMP_TAC[DIAMETER_BOUNDED_BOUND;
9695 ENDS_IN_INTERVAL; BOUNDED_INTERVAL] THEN
9696 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
9697 `diameter(cball(inv(&2) % (a + b):real^N,norm(b - a) / &2))` THEN
9699 [MATCH_MP_TAC DIAMETER_SUBSET THEN REWRITE_TAC[BOUNDED_CBALL] THEN
9700 REWRITE_TAC[SUBSET; IN_INTERVAL; IN_CBALL] THEN
9701 GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[dist] THEN
9702 REWRITE_TAC[GSYM NORM_MUL; REAL_ARITH `x / &2 = abs(inv(&2)) * x`] THEN
9703 MATCH_MP_TAC NORM_LE_COMPONENTWISE THEN
9704 X_GEN_TAC `i:num` THEN DISCH_TAC THEN
9705 FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
9706 ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT;
9707 VECTOR_MUL_COMPONENT] THEN
9709 REWRITE_TAC[DIAMETER_CBALL] THEN NORM_ARITH_TAC];
9710 DISCH_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[DIAMETER_EMPTY] THEN
9711 SUBGOAL_THEN `interval[a:real^N,b] = closure(interval(a,b))`
9712 SUBST_ALL_TAC THEN ASM_REWRITE_TAC[CLOSURE_INTERVAL] THEN
9713 ASM_MESON_TAC[DIAMETER_CLOSURE; BOUNDED_INTERVAL]]);;
9715 (* ------------------------------------------------------------------------- *)
9716 (* Some special cases for intervals in R^1. *)
9717 (* ------------------------------------------------------------------------- *)
9719 let INTERVAL_CASES_1 = prove
9720 (`!x:real^1. x IN interval[a,b] ==> x IN interval(a,b) \/ (x = a) \/ (x = b)`,
9721 REWRITE_TAC[CART_EQ; IN_INTERVAL; FORALL_DIMINDEX_1] THEN REAL_ARITH_TAC);;
9723 let IN_INTERVAL_1 = prove
9725 (x IN interval[a,b] <=> drop a <= drop x /\ drop x <= drop b) /\
9726 (x IN interval(a,b) <=> drop a < drop x /\ drop x < drop b)`,
9727 REWRITE_TAC[IN_INTERVAL; drop; CONJ_ASSOC; DIMINDEX_1; LE_ANTISYM] THEN
9730 let INTERVAL_EQ_EMPTY_1 = prove
9732 (interval[a,b] = {} <=> drop b < drop a) /\
9733 (interval(a,b) = {} <=> drop b <= drop a)`,
9734 REWRITE_TAC[INTERVAL_EQ_EMPTY; drop; CONJ_ASSOC; DIMINDEX_1; LE_ANTISYM] THEN
9737 let INTERVAL_NE_EMPTY_1 = prove
9738 (`(!a b:real^1. ~(interval[a,b] = {}) <=> drop a <= drop b) /\
9739 (!a b:real^1. ~(interval(a,b) = {}) <=> drop a < drop b)`,
9740 REWRITE_TAC[INTERVAL_EQ_EMPTY_1] THEN REAL_ARITH_TAC);;
9742 let SUBSET_INTERVAL_1 = prove
9744 (interval[a,b] SUBSET interval[c,d] <=>
9746 drop c <= drop a /\ drop a <= drop b /\ drop b <= drop d) /\
9747 (interval[a,b] SUBSET interval(c,d) <=>
9749 drop c < drop a /\ drop a <= drop b /\ drop b < drop d) /\
9750 (interval(a,b) SUBSET interval[c,d] <=>
9752 drop c <= drop a /\ drop a < drop b /\ drop b <= drop d) /\
9753 (interval(a,b) SUBSET interval(c,d) <=>
9755 drop c <= drop a /\ drop a < drop b /\ drop b <= drop d)`,
9756 REWRITE_TAC[SUBSET_INTERVAL; FORALL_1; DIMINDEX_1; drop] THEN
9759 let EQ_INTERVAL_1 = prove
9761 (interval[a,b] = interval[c,d] <=>
9762 drop b < drop a /\ drop d < drop c \/
9763 drop a = drop c /\ drop b = drop d)`,
9764 REWRITE_TAC[SET_RULE `s = t <=> s SUBSET t /\ t SUBSET s`] THEN
9765 REWRITE_TAC[SUBSET_INTERVAL_1] THEN REAL_ARITH_TAC);;
9767 let DISJOINT_INTERVAL_1 = prove
9769 (interval[a,b] INTER interval[c,d] = {} <=>
9770 drop b < drop a \/ drop d < drop c \/
9771 drop b < drop c \/ drop d < drop a) /\
9772 (interval[a,b] INTER interval(c,d) = {} <=>
9773 drop b < drop a \/ drop d <= drop c \/
9774 drop b <= drop c \/ drop d <= drop a) /\
9775 (interval(a,b) INTER interval[c,d] = {} <=>
9776 drop b <= drop a \/ drop d < drop c \/
9777 drop b <= drop c \/ drop d <= drop a) /\
9778 (interval(a,b) INTER interval(c,d) = {} <=>
9779 drop b <= drop a \/ drop d <= drop c \/
9780 drop b <= drop c \/ drop d <= drop a)`,
9781 REWRITE_TAC[DISJOINT_INTERVAL; CONJ_ASSOC; DIMINDEX_1; LE_ANTISYM;
9782 UNWIND_THM1; drop]);;
9784 let OPEN_CLOSED_INTERVAL_1 = prove
9785 (`!a b:real^1. interval(a,b) = interval[a,b] DIFF {a,b}`,
9786 REWRITE_TAC[EXTENSION; IN_INTERVAL_1; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN
9787 REWRITE_TAC[GSYM DROP_EQ] THEN REAL_ARITH_TAC);;
9789 let CLOSED_OPEN_INTERVAL_1 = prove
9790 (`!a b:real^1. drop a <= drop b ==> interval[a,b] = interval(a,b) UNION {a,b}`,
9791 REWRITE_TAC[EXTENSION; IN_INTERVAL_1; IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN
9792 REWRITE_TAC[GSYM DROP_EQ] THEN REAL_ARITH_TAC);;
9795 (`!x:real^1 r. cball(x,r) = interval[x - lift r,x + lift r] /\
9796 ball(x,r) = interval(x - lift r,x + lift r)`,
9797 REWRITE_TAC[EXTENSION; IN_BALL; IN_CBALL; IN_INTERVAL_1] THEN
9798 REWRITE_TAC[dist; NORM_REAL; GSYM drop; DROP_SUB; LIFT_DROP; DROP_ADD] THEN
9801 let SPHERE_1 = prove
9802 (`!a:real^1 r. sphere(a,r) = if r < &0 then {} else {a - lift r,a + lift r}`,
9803 REPEAT GEN_TAC THEN REWRITE_TAC[sphere] THEN COND_CASES_TAC THEN
9804 REWRITE_TAC[DIST_REAL; GSYM drop; FORALL_DROP] THEN
9805 REWRITE_TAC[EXTENSION; IN_INSERT; NOT_IN_EMPTY; IN_ELIM_THM] THEN
9806 REWRITE_TAC[GSYM DROP_EQ; DROP_ADD; DROP_SUB; LIFT_DROP] THEN
9807 ASM_REAL_ARITH_TAC);;
9809 let FINITE_SPHERE_1 = prove
9810 (`!a:real^1 r. FINITE(sphere(a,r))`,
9811 REPEAT GEN_TAC THEN REWRITE_TAC[SPHERE_1] THEN
9812 MESON_TAC[FINITE_INSERT; FINITE_EMPTY]);;
9814 let FINITE_INTERVAL_1 = prove
9815 (`(!a b. FINITE(interval[a,b]) <=> drop b <= drop a) /\
9816 (!a b. FINITE(interval(a,b)) <=> drop b <= drop a)`,
9817 REWRITE_TAC[OPEN_CLOSED_INTERVAL_1] THEN
9818 REWRITE_TAC[SET_RULE `s DIFF {a,b} = s DELETE a DELETE b`] THEN
9819 REWRITE_TAC[FINITE_DELETE] THEN REPEAT GEN_TAC THEN
9820 SUBGOAL_THEN `interval[a,b] = IMAGE lift {x | drop a <= x /\ x <= drop b}`
9822 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
9823 CONJ_TAC THENL [MESON_TAC[LIFT_DROP]; ALL_TAC] THEN
9824 REWRITE_TAC[IN_INTERVAL_1; IN_ELIM_THM; LIFT_DROP];
9825 SIMP_TAC[FINITE_IMAGE_INJ_EQ; LIFT_EQ; FINITE_REAL_INTERVAL]]);;
9827 let BALL_INTERVAL = prove
9828 (`!x:real^1 e. ball(x,e) = interval(x - lift e,x + lift e)`,
9829 REWRITE_TAC[EXTENSION; IN_BALL; IN_INTERVAL_1; DIST_REAL] THEN
9830 REWRITE_TAC[GSYM drop; DROP_SUB; DROP_ADD; LIFT_DROP] THEN REAL_ARITH_TAC);;
9832 let CBALL_INTERVAL = prove
9833 (`!x:real^1 e. cball(x,e) = interval[x - lift e,x + lift e]`,
9834 REWRITE_TAC[EXTENSION; IN_CBALL; IN_INTERVAL_1; DIST_REAL] THEN
9835 REWRITE_TAC[GSYM drop; DROP_SUB; DROP_ADD; LIFT_DROP] THEN REAL_ARITH_TAC);;
9837 let BALL_INTERVAL_0 = prove
9838 (`!e. ball(vec 0:real^1,e) = interval(--lift e,lift e)`,
9839 GEN_TAC THEN REWRITE_TAC[BALL_INTERVAL] THEN AP_TERM_TAC THEN
9840 BINOP_TAC THEN VECTOR_ARITH_TAC);;
9842 let CBALL_INTERVAL_0 = prove
9843 (`!e. cball(vec 0:real^1,e) = interval[--lift e,lift e]`,
9844 GEN_TAC THEN REWRITE_TAC[CBALL_INTERVAL] THEN AP_TERM_TAC THEN
9845 AP_THM_TAC THEN AP_TERM_TAC THEN BINOP_TAC THEN VECTOR_ARITH_TAC);;
9847 let INTER_INTERVAL_1 = prove
9849 interval[a,b] INTER interval[c,d] =
9850 interval[lift(max (drop a) (drop c)),lift(min (drop b) (drop d))]`,
9851 REWRITE_TAC[EXTENSION; IN_INTER; IN_INTERVAL_1; real_max; real_min] THEN
9853 REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[LIFT_DROP]) THEN
9854 ASM_REAL_ARITH_TAC);;
9856 let CLOSED_DIFF_OPEN_INTERVAL_1 = prove
9858 interval[a,b] DIFF interval(a,b) =
9859 if interval[a,b] = {} then {} else {a,b}`,
9860 REWRITE_TAC[EXTENSION; IN_DIFF; INTERVAL_EQ_EMPTY_1; IN_INTERVAL_1] THEN
9861 REPEAT GEN_TAC THEN COND_CASES_TAC THEN
9862 ASM_REWRITE_TAC[NOT_IN_EMPTY; IN_INSERT; NOT_IN_EMPTY] THEN
9863 REWRITE_TAC[GSYM DROP_EQ] THEN ASM_REAL_ARITH_TAC);;
9865 (* ------------------------------------------------------------------------- *)
9866 (* Intervals in general, including infinite and mixtures of open and closed. *)
9867 (* ------------------------------------------------------------------------- *)
9869 let is_interval = new_definition
9870 `is_interval(s:real^N->bool) <=>
9871 !a b x. a IN s /\ b IN s /\
9872 (!i. 1 <= i /\ i <= dimindex(:N)
9873 ==> (a$i <= x$i /\ x$i <= b$i) \/
9874 (b$i <= x$i /\ x$i <= a$i))
9877 let IS_INTERVAL_INTERVAL = prove
9878 (`!a:real^N b. is_interval(interval (a,b)) /\ is_interval(interval [a,b])`,
9879 REWRITE_TAC[is_interval; IN_INTERVAL] THEN
9880 MESON_TAC[REAL_LT_TRANS; REAL_LE_TRANS; REAL_LET_TRANS; REAL_LTE_TRANS]);;
9882 let IS_INTERVAL_EMPTY = prove
9884 REWRITE_TAC[is_interval; NOT_IN_EMPTY]);;
9886 let IS_INTERVAL_UNIV = prove
9887 (`is_interval(UNIV:real^N->bool)`,
9888 REWRITE_TAC[is_interval; IN_UNIV]);;
9890 let IS_INTERVAL_TRANSLATION_EQ = prove
9891 (`!a:real^N s. is_interval(IMAGE (\x. a + x) s) <=> is_interval s`,
9892 REWRITE_TAC[is_interval] THEN GEOM_TRANSLATE_TAC[] THEN
9893 REWRITE_TAC[VECTOR_ADD_COMPONENT; REAL_LT_LADD; REAL_LE_LADD]);;
9895 add_translation_invariants [IS_INTERVAL_TRANSLATION_EQ];;
9897 let IS_INTERVAL_TRANSLATION = prove
9898 (`!s a:real^N. is_interval s ==> is_interval(IMAGE (\x. a + x) s)`,
9899 REWRITE_TAC[IS_INTERVAL_TRANSLATION_EQ]);;
9901 let IS_INTERVAL_POINTWISE = prove
9902 (`!s:real^N->bool x.
9904 (!i. 1 <= i /\ i <= dimindex(:N) ==> ?a. a IN s /\ a$i = x$i)
9906 REWRITE_TAC[is_interval] THEN REPEAT STRIP_TAC THEN
9908 `!n. ?y:real^N. (!i. 1 <= i /\ i <= n ==> y$i = (x:real^N)$i) /\ y IN s`
9910 [INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `~(1 <= i /\ i <= 0)`] THENL
9911 [ASM_MESON_TAC[DIMINDEX_GE_1; LE_REFL]; ALL_TAC] THEN
9912 FIRST_X_ASSUM(X_CHOOSE_TAC `y:real^N`) THEN
9913 ASM_CASES_TAC `SUC n <= dimindex(:N)` THENL
9914 [FIRST_X_ASSUM(MP_TAC o SPEC `SUC n`) THEN
9915 ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
9916 DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
9918 `(lambda i. if i <= n then (y:real^N)$i else (z:real^N)$i):real^N` THEN
9920 [X_GEN_TAC `i:num` THEN STRIP_TAC THEN
9921 SUBGOAL_THEN `i <= dimindex(:N)` ASSUME_TAC THENL
9922 [ASM_ARITH_TAC; ASM_SIMP_TAC[LAMBDA_BETA]] THEN
9923 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
9924 SUBGOAL_THEN `i = SUC n` (fun th -> ASM_REWRITE_TAC[th]) THEN
9926 FIRST_X_ASSUM(ASSUME_TAC o CONJUNCT2) THEN
9927 FIRST_X_ASSUM MATCH_MP_TAC THEN
9928 MAP_EVERY EXISTS_TAC [`y:real^N`; `z:real^N`] THEN
9929 ASM_SIMP_TAC[LAMBDA_BETA] THEN REAL_ARITH_TAC];
9930 EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN
9931 SUBGOAL_THEN `y:real^N = x` (fun th -> REWRITE_TAC[th]) THEN
9932 REWRITE_TAC[CART_EQ] THEN
9933 ASM_MESON_TAC[ARITH_RULE `i <= N /\ ~(SUC n <= N) ==> i <= n`]];
9934 DISCH_THEN(MP_TAC o SPEC `dimindex(:N)`) THEN
9935 REWRITE_TAC[GSYM CART_EQ] THEN MESON_TAC[]]);;
9937 let IS_INTERVAL_COMPACT = prove
9938 (`!s:real^N->bool. is_interval s /\ compact s <=> ?a b. s = interval[a,b]`,
9939 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN
9940 ASM_SIMP_TAC[IS_INTERVAL_INTERVAL; COMPACT_INTERVAL] THEN
9941 ASM_CASES_TAC `s:real^N->bool = {}` THENL
9942 [ASM_MESON_TAC[EMPTY_AS_INTERVAL]; ALL_TAC] THEN
9943 EXISTS_TAC `(lambda i. inf { (x:real^N)$i | x IN s}):real^N` THEN
9944 EXISTS_TAC `(lambda i. sup { (x:real^N)$i | x IN s}):real^N` THEN
9945 SIMP_TAC[EXTENSION; IN_INTERVAL; LAMBDA_BETA] THEN X_GEN_TAC `x:real^N` THEN
9947 [DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
9948 MP_TAC(ISPEC `{ (x:real^N)$i | x IN s}` INF) THEN
9949 MP_TAC(ISPEC `{ (x:real^N)$i | x IN s}` SUP) THEN
9950 ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
9951 ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
9952 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_IMP_BOUNDED) THEN
9953 REWRITE_TAC[bounded] THEN
9954 ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS; MEMBER_NOT_EMPTY;
9955 REAL_ARITH `abs(x) <= B ==> --B <= x /\ x <= B`];
9956 DISCH_TAC THEN MATCH_MP_TAC IS_INTERVAL_POINTWISE THEN
9957 ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
9959 `?a b:real^N. a IN s /\ b IN s /\ a$i <= (x:real^N)$i /\ x$i <= b$i`
9960 STRIP_ASSUME_TAC THENL
9961 [MP_TAC(ISPECL [`\x:real^N. x$i`; `s:real^N->bool`]
9962 CONTINUOUS_ATTAINS_INF) THEN
9963 ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT; o_DEF] THEN
9964 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN
9965 MP_TAC(ISPECL [`\x:real^N. x$i`; `s:real^N->bool`]
9966 CONTINUOUS_ATTAINS_SUP) THEN
9967 ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT; o_DEF] THEN
9968 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN
9969 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THENL
9970 [EXISTS_TAC `inf {(x:real^N)$i | x IN s}` THEN ASM_SIMP_TAC[] THEN
9971 MATCH_MP_TAC REAL_LE_INF THEN ASM SET_TAC[];
9972 EXISTS_TAC `sup {(x:real^N)$i | x IN s}` THEN ASM_SIMP_TAC[] THEN
9973 MATCH_MP_TAC REAL_SUP_LE THEN ASM SET_TAC[]];
9975 `(lambda j. if j = i then (x:real^N)$i else (a:real^N)$j):real^N` THEN
9976 ASM_SIMP_TAC[LAMBDA_BETA] THEN
9977 FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[is_interval]) THEN
9978 MAP_EVERY EXISTS_TAC
9980 `(lambda j. if j = i then (b:real^N)$i else (a:real^N)$j):real^N`] THEN
9981 ASM_SIMP_TAC[LAMBDA_BETA] THEN CONJ_TAC THENL
9982 [FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[is_interval]) THEN
9983 MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`] THEN
9984 ASM_SIMP_TAC[LAMBDA_BETA];
9986 GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
9987 ASM_REAL_ARITH_TAC]]);;
9989 let IS_INTERVAL_1 = prove
9992 !a b x. a IN s /\ b IN s /\ drop a <= drop x /\ drop x <= drop b
9994 REWRITE_TAC[is_interval; DIMINDEX_1; FORALL_1; GSYM drop] THEN
9995 REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN MESON_TAC[]);;
9997 let IS_INTERVAL_1_CASES = prove
10002 (?a. s = {x | a < drop x}) \/
10003 (?a. s = {x | a <= drop x}) \/
10004 (?b. s = {x | drop x <= b}) \/
10005 (?b. s = {x | drop x < b}) \/
10006 (?a b. s = {x | a < drop x /\ drop x < b}) \/
10007 (?a b. s = {x | a < drop x /\ drop x <= b}) \/
10008 (?a b. s = {x | a <= drop x /\ drop x < b}) \/
10009 (?a b. s = {x | a <= drop x /\ drop x <= b})`,
10010 GEN_TAC THEN REWRITE_TAC[IS_INTERVAL_1] THEN EQ_TAC THENL
10012 STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM; IN_UNIV; NOT_IN_EMPTY] THEN
10013 REAL_ARITH_TAC] THEN
10014 ASM_CASES_TAC `s:real^1->bool = {}` THEN ASM_REWRITE_TAC[] THEN
10015 MP_TAC(ISPEC `IMAGE drop s` SUP) THEN
10016 MP_TAC(ISPEC `IMAGE drop s` INF) THEN
10017 ASM_REWRITE_TAC[IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
10018 ASM_CASES_TAC `?a. !x. x IN s ==> a <= drop x` THEN
10019 ASM_CASES_TAC `?b. !x. x IN s ==> drop x <= b` THEN
10020 ASM_REWRITE_TAC[] THENL
10021 [STRIP_TAC THEN STRIP_TAC THEN
10022 MAP_EVERY ASM_CASES_TAC
10023 [`inf(IMAGE drop s) IN IMAGE drop s`; `sup(IMAGE drop s) IN IMAGE drop s`]
10025 [REPLICATE_TAC 8 DISJ2_TAC;
10026 REPLICATE_TAC 7 DISJ2_TAC THEN DISJ1_TAC;
10027 REPLICATE_TAC 6 DISJ2_TAC THEN DISJ1_TAC;
10028 REPLICATE_TAC 5 DISJ2_TAC THEN DISJ1_TAC] THEN
10029 MAP_EVERY EXISTS_TAC [`inf(IMAGE drop s)`; `sup(IMAGE drop s)`];
10030 STRIP_TAC THEN ASM_CASES_TAC `inf(IMAGE drop s) IN IMAGE drop s` THENL
10031 [REPLICATE_TAC 2 DISJ2_TAC THEN DISJ1_TAC;
10032 DISJ2_TAC THEN DISJ1_TAC] THEN
10033 EXISTS_TAC `inf(IMAGE drop s)`;
10034 STRIP_TAC THEN ASM_CASES_TAC `sup(IMAGE drop s) IN IMAGE drop s` THENL
10035 [REPLICATE_TAC 3 DISJ2_TAC THEN DISJ1_TAC;
10036 REPLICATE_TAC 4 DISJ2_TAC THEN DISJ1_TAC] THEN
10037 EXISTS_TAC `sup(IMAGE drop s)`;
10039 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_UNIV] THEN
10040 RULE_ASSUM_TAC(REWRITE_RULE[IN_IMAGE]) THEN
10041 REWRITE_TAC[GSYM REAL_NOT_LE] THEN
10042 ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_TOTAL; REAL_LE_ANTISYM]);;
10044 let IS_INTERVAL_PCROSS = prove
10045 (`!s:real^M->bool t:real^N->bool.
10046 is_interval s /\ is_interval t ==> is_interval(s PCROSS t)`,
10047 REWRITE_TAC[is_interval; DIMINDEX_FINITE_SUM] THEN
10048 REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS] THEN
10049 REPEAT GEN_TAC THEN
10050 MATCH_MP_TAC(MESON[]
10051 `(!a b a' b' x x'. P a b x /\ Q a' b' x' ==> R a b x a' b' x')
10052 ==> (!a b x. P a b x) /\ (!a' b' x'. Q a' b' x')
10053 ==> (!a a' b b' x x'. R a b x a' b' x')`) THEN
10054 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
10055 ASM_REWRITE_TAC[] THEN X_GEN_TAC `i:num` THEN STRIP_TAC THENL
10056 [FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
10057 ASM_SIMP_TAC[pastecart; LAMBDA_BETA; DIMINDEX_FINITE_SUM;
10058 ARITH_RULE `x:num <= m ==> x <= m + n`];
10059 FIRST_X_ASSUM(MP_TAC o SPEC `dimindex(:M) + i`) THEN
10060 ASM_SIMP_TAC[pastecart; LAMBDA_BETA; DIMINDEX_FINITE_SUM;
10061 ARITH_RULE `x:num <= n ==> m + x <= m + n`;
10062 ARITH_RULE `1 <= x ==> 1 <= m + x`] THEN
10063 COND_CASES_TAC THEN ASM_REWRITE_TAC[ADD_SUB2] THEN ASM_ARITH_TAC]);;
10065 let IS_INTERVAL_PCROSS_EQ = prove
10066 (`!s:real^M->bool t:real^N->bool.
10067 is_interval(s PCROSS t) <=>
10068 s = {} \/ t = {} \/ is_interval s /\ is_interval t`,
10069 REPEAT GEN_TAC THEN
10070 ASM_CASES_TAC `s:real^M->bool = {}` THEN
10071 ASM_REWRITE_TAC[PCROSS_EMPTY; IS_INTERVAL_EMPTY] THEN
10072 ASM_CASES_TAC `t:real^N->bool = {}` THEN
10073 ASM_REWRITE_TAC[PCROSS_EMPTY; IS_INTERVAL_EMPTY] THEN
10074 EQ_TAC THEN REWRITE_TAC[IS_INTERVAL_PCROSS] THEN
10075 REWRITE_TAC[is_interval] THEN
10076 REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS] THEN
10077 STRIP_TAC THEN CONJ_TAC THENL
10078 [MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `x:real^M`] THEN
10079 STRIP_TAC THEN UNDISCH_TAC `~(t:real^N->bool = {})` THEN
10080 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
10081 DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN
10082 FIRST_X_ASSUM(MP_TAC o SPECL
10083 [`a:real^M`; `y:real^N`; `b:real^M`;
10084 `y:real^N`; `x:real^M`; `y:real^N`]);
10085 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `x:real^N`] THEN
10086 STRIP_TAC THEN UNDISCH_TAC `~(s:real^M->bool = {})` THEN
10087 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
10088 DISCH_THEN(X_CHOOSE_TAC `w:real^M`) THEN
10089 FIRST_X_ASSUM(MP_TAC o SPECL
10090 [`w:real^M`; `a:real^N`; `w:real^M`;
10091 `b:real^N`; `w:real^M`; `x:real^N`])] THEN
10092 ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
10093 SIMP_TAC[pastecart; LAMBDA_BETA] THEN
10094 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN
10095 ASM_MESON_TAC[DIMINDEX_FINITE_SUM; ARITH_RULE
10096 `1 <= i /\ i <= m + n /\ ~(i <= m) ==> 1 <= i - m /\ i - m <= n`]);;
10098 let IS_INTERVAL_INTER = prove
10099 (`!s t:real^N->bool.
10100 is_interval s /\ is_interval t ==> is_interval(s INTER t)`,
10101 REWRITE_TAC[is_interval; IN_INTER] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
10102 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `x:real^N`] THEN
10103 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
10104 MAP_EVERY EXISTS_TAC [`a:real^N`; `b:real^N`] THEN ASM_REWRITE_TAC[]);;
10106 let INTERVAL_SUBSET_IS_INTERVAL = prove
10109 ==> (interval[a,b] SUBSET s <=> interval[a,b] = {} \/ a IN s /\ b IN s)`,
10110 REWRITE_TAC[is_interval] THEN REPEAT STRIP_TAC THEN
10111 ASM_CASES_TAC `interval[a:real^N,b] = {}` THEN
10112 ASM_REWRITE_TAC[EMPTY_SUBSET] THEN
10113 EQ_TAC THENL [ASM_MESON_TAC[ENDS_IN_INTERVAL; SUBSET]; ALL_TAC] THEN
10114 REWRITE_TAC[SUBSET; IN_INTERVAL] THEN ASM_MESON_TAC[]);;
10116 let INTERVAL_CONTAINS_COMPACT_NEIGHBOURHOOD = prove
10118 is_interval s /\ x IN s
10119 ==> ?a b d. &0 < d /\ x IN interval[a,b] /\
10120 interval[a,b] SUBSET s /\
10121 ball(x,d) INTER s SUBSET interval[a,b]`,
10122 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERVAL_SUBSET_IS_INTERVAL] THEN
10124 `!i. 1 <= i /\ i <= dimindex(:N)
10125 ==> ?a. (?y. y IN s /\ y$i = a) /\
10126 (a < x$i \/ a = (x:real^N)$i /\
10127 !y:real^N. y IN s ==> a <= y$i)`
10128 MP_TAC THENL [ASM_MESON_TAC[REAL_NOT_LT]; REWRITE_TAC[LAMBDA_SKOLEM]] THEN
10129 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN
10131 `!i. 1 <= i /\ i <= dimindex(:N)
10132 ==> ?b. (?y. y IN s /\ y$i = b) /\
10133 (x$i < b \/ b = (x:real^N)$i /\
10134 !y:real^N. y IN s ==> y$i <= b)`
10135 MP_TAC THENL [ASM_MESON_TAC[REAL_NOT_LT]; REWRITE_TAC[LAMBDA_SKOLEM]] THEN
10136 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN STRIP_TAC THEN
10137 EXISTS_TAC `min (inf (IMAGE (\i. if a$i < x$i
10138 then (x:real^N)$i - (a:real^N)$i else &1)
10139 (1..dimindex(:N))))
10140 (inf (IMAGE (\i. if x$i < b$i
10141 then (b:real^N)$i - x$i else &1)
10142 (1..dimindex(:N))))` THEN
10143 REWRITE_TAC[REAL_LT_MIN; SUBSET; IN_BALL; IN_INTER] THEN
10144 SIMP_TAC[REAL_LT_INF_FINITE; IMAGE_EQ_EMPTY; FINITE_IMAGE;
10145 FINITE_NUMSEG; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1] THEN
10146 REWRITE_TAC[FORALL_IN_IMAGE; IN_INTERVAL] THEN REPEAT CONJ_TAC THENL
10147 [MESON_TAC[REAL_SUB_LT; REAL_LT_01];
10148 MESON_TAC[REAL_SUB_LT; REAL_LT_01];
10149 ASM_MESON_TAC[REAL_LE_LT];
10150 DISJ2_TAC THEN CONJ_TAC THEN MATCH_MP_TAC IS_INTERVAL_POINTWISE THEN
10152 X_GEN_TAC `y:real^N` THEN
10153 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
10154 REWRITE_TAC[AND_FORALL_THM] THEN MATCH_MP_TAC MONO_FORALL THEN
10155 X_GEN_TAC `i:num` THEN DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
10156 ASM_REWRITE_TAC[IN_NUMSEG] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
10157 (COND_CASES_TAC THENL [REWRITE_TAC[dist]; ASM_MESON_TAC[]]) THEN
10158 DISCH_TAC THEN MP_TAC(ISPECL [`x - y:real^N`; `i:num`]
10159 COMPONENT_LE_NORM) THEN
10160 ASM_REWRITE_TAC[VECTOR_SUB_COMPONENT] THEN ASM_REAL_ARITH_TAC]);;
10162 let IS_INTERVAL_SUMS = prove
10163 (`!s t:real^N->bool.
10164 is_interval s /\ is_interval t
10165 ==> is_interval {x + y | x IN s /\ y IN t}`,
10166 REPEAT GEN_TAC THEN REWRITE_TAC[is_interval] THEN
10167 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
10168 REWRITE_TAC[FORALL_IN_GSPEC] THEN
10169 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN
10170 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
10171 MAP_EVERY X_GEN_TAC
10172 [`a:real^N`; `a':real^N`; `b:real^N`; `b':real^N`; `y:real^N`] THEN
10173 DISCH_THEN(CONJUNCTS_THEN2
10174 (MP_TAC o SPECL [`a:real^N`; `b:real^N`]) MP_TAC) THEN
10175 DISCH_THEN(CONJUNCTS_THEN2
10176 (MP_TAC o SPECL [`a':real^N`; `b':real^N`]) STRIP_ASSUME_TAC) THEN
10177 ASM_REWRITE_TAC[IMP_IMP; IN_ELIM_THM] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
10178 ONCE_REWRITE_TAC[VECTOR_ARITH `z:real^N = x + y <=> y = z - x`] THEN
10179 REWRITE_TAC[UNWIND_THM2] THEN MATCH_MP_TAC(MESON[]
10180 `(?x. P x /\ Q(f x))
10181 ==> (!x. P x ==> x IN s) /\ (!x. Q x ==> x IN t)
10182 ==> ?x. x IN s /\ f x IN t`) THEN
10183 REWRITE_TAC[VECTOR_SUB_COMPONENT; AND_FORALL_THM;
10184 TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN
10185 REWRITE_TAC[GSYM LAMBDA_SKOLEM] THEN
10186 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
10187 FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
10188 ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT] THEN
10189 REWRITE_TAC[REAL_ARITH
10190 `c <= y - x /\ y - x <= d <=> y - d <= x /\ x <= y - c`] THEN
10191 REWRITE_TAC[REAL_ARITH
10192 `a <= x /\ x <= b \/ b <= x /\ x <= a <=> min a b <= x /\ x <= max a b`] THEN
10193 ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ (r /\ s) <=> (p /\ r) /\ (q /\ s)`] THEN
10194 REWRITE_TAC[GSYM REAL_LE_MIN; GSYM REAL_MAX_LE] THEN
10195 REWRITE_TAC[GSYM REAL_LE_BETWEEN] THEN REAL_ARITH_TAC);;
10197 let IS_INTERVAL_SING = prove
10198 (`!a:real^N. is_interval {a}`,
10199 SIMP_TAC[is_interval; IN_SING; IMP_CONJ; CART_EQ; REAL_LE_ANTISYM]);;
10201 let IS_INTERVAL_SCALING = prove
10202 (`!s:real^N->bool c. is_interval s ==> is_interval(IMAGE (\x. c % x) s)`,
10203 REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THENL
10204 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
10205 SUBGOAL_THEN `IMAGE ((\x. vec 0):real^N->real^N) s = {} \/
10206 IMAGE ((\x. vec 0):real^N->real^N) s = {vec 0}`
10207 STRIP_ASSUME_TAC THENL
10209 ASM_REWRITE_TAC[IS_INTERVAL_EMPTY];
10210 ASM_REWRITE_TAC[IS_INTERVAL_SING]];
10211 REWRITE_TAC[is_interval; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
10212 REWRITE_TAC[FORALL_IN_IMAGE] THEN
10213 GEN_REWRITE_TAC (BINOP_CONV o REDEPTH_CONV) [RIGHT_IMP_FORALL_THM] THEN
10214 REWRITE_TAC[IMP_IMP; VECTOR_MUL_COMPONENT] THEN
10215 MAP_EVERY (fun t -> MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC t)
10216 [`a:real^N`; `b:real^N`] THEN
10217 DISCH_THEN(fun th -> X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
10218 MP_TAC(SPEC `inv(c) % x:real^N` th)) THEN
10219 ASM_REWRITE_TAC[VECTOR_MUL_COMPONENT; IN_IMAGE] THEN ANTS_TAC THENL
10220 [X_GEN_TAC `i:num` THEN STRIP_TAC THEN
10221 FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
10222 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN
10223 FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
10224 `~(c = &0) ==> &0 < c \/ &0 < --c`)) THEN
10225 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ] THEN
10226 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM REAL_LE_NEG2] THEN
10227 ASM_SIMP_TAC[GSYM REAL_MUL_RNEG; GSYM REAL_LE_RDIV_EQ; GSYM
10228 REAL_LE_LDIV_EQ] THEN
10229 REWRITE_TAC[real_div; REAL_INV_NEG] THEN REAL_ARITH_TAC;
10230 DISCH_TAC THEN EXISTS_TAC `inv c % x:real^N` THEN
10231 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; VECTOR_MUL_LID]]]);;
10233 let IS_INTERVAL_SCALING_EQ = prove
10234 (`!s:real^N->bool c.
10235 is_interval(IMAGE (\x. c % x) s) <=> c = &0 \/ is_interval s`,
10236 REPEAT GEN_TAC THEN ASM_CASES_TAC `c = &0` THENL
10237 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO] THEN
10238 SUBGOAL_THEN `IMAGE ((\x. vec 0):real^N->real^N) s = {} \/
10239 IMAGE ((\x. vec 0):real^N->real^N) s = {vec 0}`
10240 STRIP_ASSUME_TAC THENL
10242 ASM_REWRITE_TAC[IS_INTERVAL_EMPTY];
10243 ASM_REWRITE_TAC[IS_INTERVAL_SING]];
10244 ASM_REWRITE_TAC[] THEN EQ_TAC THEN REWRITE_TAC[IS_INTERVAL_SCALING] THEN
10245 DISCH_THEN(MP_TAC o SPEC `inv c:real` o MATCH_MP IS_INTERVAL_SCALING) THEN
10246 ASM_SIMP_TAC[GSYM IMAGE_o; VECTOR_MUL_ASSOC; o_DEF; REAL_MUL_LINV;
10247 VECTOR_MUL_LID; IMAGE_ID]]);;
10251 ==> !s:real^N->bool. is_interval(IMAGE (\x. c % x) s) <=>
10253 SIMP_TAC[IS_INTERVAL_SCALING_EQ; REAL_LT_IMP_NZ]) in
10254 add_scaling_theorems [lemma];;
10256 (* ------------------------------------------------------------------------- *)
10257 (* Line segments, with same open/closed overloading as for intervals. *)
10258 (* ------------------------------------------------------------------------- *)
10260 let closed_segment = define
10261 `closed_segment[a,b] = {(&1 - u) % a + u % b | &0 <= u /\ u <= &1}`;;
10263 let open_segment = new_definition
10264 `open_segment(a,b) = closed_segment[a,b] DIFF {a,b}`;;
10266 let OPEN_SEGMENT_ALT = prove
10269 ==> open_segment(a,b) = {(&1 - u) % a + u % b | &0 < u /\ u < &1}`,
10270 REPEAT STRIP_TAC THEN REWRITE_TAC[open_segment; closed_segment] THEN
10271 REWRITE_TAC[EXTENSION; IN_DIFF; IN_INSERT; NOT_IN_EMPTY; IN_ELIM_THM] THEN
10272 X_GEN_TAC `x:real^N` THEN REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN
10273 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
10274 X_GEN_TAC `u:real` THEN ASM_CASES_TAC `x:real^N = (&1 - u) % a + u % b` THEN
10275 ASM_REWRITE_TAC[REAL_LE_LT;
10276 VECTOR_ARITH `(&1 - u) % a + u % b = a <=> u % (b - a) = vec 0`;
10277 VECTOR_ARITH `(&1 - u) % a + u % b = b <=> (&1 - u) % (b - a) = vec 0`;
10278 VECTOR_MUL_EQ_0; REAL_SUB_0; VECTOR_SUB_EQ] THEN
10281 make_overloadable "segment" `:A`;;
10283 overload_interface("segment",`open_segment`);;
10284 overload_interface("segment",`closed_segment`);;
10286 let segment = prove
10287 (`segment[a,b] = {(&1 - u) % a + u % b | &0 <= u /\ u <= &1} /\
10288 segment(a,b) = segment[a,b] DIFF {a,b}`,
10289 REWRITE_TAC[open_segment; closed_segment]);;
10291 let SEGMENT_REFL = prove
10292 (`(!a. segment[a,a] = {a}) /\
10293 (!a. segment(a,a) = {})`,
10294 REWRITE_TAC[segment; VECTOR_ARITH `(&1 - u) % a + u % a = a`] THEN
10295 SET_TAC[REAL_POS]);;
10297 let IN_SEGMENT = prove
10299 (x IN segment[a,b] <=>
10300 ?u. &0 <= u /\ u <= &1 /\ x = (&1 - u) % a + u % b) /\
10301 (x IN segment(a,b) <=>
10302 ~(a = b) /\ ?u. &0 < u /\ u < &1 /\ x = (&1 - u) % a + u % b)`,
10303 REPEAT STRIP_TAC THENL
10304 [REWRITE_TAC[segment; IN_ELIM_THM; CONJ_ASSOC]; ALL_TAC] THEN
10305 ASM_CASES_TAC `a:real^N = b` THEN
10306 ASM_REWRITE_TAC[SEGMENT_REFL; NOT_IN_EMPTY] THEN
10307 ASM_SIMP_TAC[OPEN_SEGMENT_ALT; IN_ELIM_THM; CONJ_ASSOC]);;
10309 let SEGMENT_SYM = prove
10310 (`(!a b:real^N. segment[a,b] = segment[b,a]) /\
10311 (!a b:real^N. segment(a,b) = segment(b,a))`,
10312 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN
10313 SIMP_TAC[open_segment] THEN
10314 CONJ_TAC THENL [ALL_TAC; SIMP_TAC[INSERT_AC]] THEN
10315 REWRITE_TAC[EXTENSION; IN_SEGMENT] THEN REPEAT GEN_TAC THEN EQ_TAC THEN
10316 DISCH_THEN(X_CHOOSE_TAC `u:real`) THEN EXISTS_TAC `&1 - u` THEN
10317 ASM_REWRITE_TAC[] THEN
10318 REPEAT CONJ_TAC THEN TRY ASM_ARITH_TAC THEN VECTOR_ARITH_TAC);;
10320 let ENDS_IN_SEGMENT = prove
10321 (`!a b. a IN segment[a,b] /\ b IN segment[a,b]`,
10322 REPEAT STRIP_TAC THEN REWRITE_TAC[segment; IN_ELIM_THM] THENL
10323 [EXISTS_TAC `&0`; EXISTS_TAC `&1`] THEN
10324 (CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]));;
10326 let ENDS_NOT_IN_SEGMENT = prove
10327 (`!a b. ~(a IN segment(a,b)) /\ ~(b IN segment(a,b))`,
10328 REWRITE_TAC[open_segment] THEN SET_TAC[]);;
10330 let SEGMENT_CLOSED_OPEN = prove
10331 (`!a b. segment[a,b] = segment(a,b) UNION {a,b}`,
10332 REPEAT GEN_TAC THEN REWRITE_TAC[open_segment] THEN MATCH_MP_TAC(SET_RULE
10333 `a IN s /\ b IN s ==> s = (s DIFF {a,b}) UNION {a,b}`) THEN
10334 REWRITE_TAC[ENDS_IN_SEGMENT]);;
10336 let MIDPOINT_IN_SEGMENT = prove
10337 (`(!a b:real^N. midpoint(a,b) IN segment[a,b]) /\
10338 (!a b:real^N. midpoint(a,b) IN segment(a,b) <=> ~(a = b))`,
10339 REWRITE_TAC[IN_SEGMENT] THEN REPEAT STRIP_TAC THENL
10340 [ALL_TAC; ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[]] THEN
10341 EXISTS_TAC `&1 / &2` THEN REWRITE_TAC[midpoint] THEN
10342 CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC);;
10344 let BETWEEN_IN_SEGMENT = prove
10345 (`!x a b:real^N. between x (a,b) <=> x IN segment[a,b]`,
10346 REPEAT GEN_TAC THEN REWRITE_TAC[between] THEN
10347 ASM_CASES_TAC `a:real^N = b` THEN
10348 ASM_REWRITE_TAC[SEGMENT_REFL; IN_SING] THENL [NORM_ARITH_TAC; ALL_TAC] THEN
10349 REWRITE_TAC[segment; IN_ELIM_THM] THEN EQ_TAC THENL
10350 [DISCH_THEN(ASSUME_TAC o SYM) THEN
10351 EXISTS_TAC `dist(a:real^N,x) / dist(a,b)` THEN
10352 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; DIST_POS_LT] THEN CONJ_TAC
10353 THENL [FIRST_ASSUM(SUBST1_TAC o SYM) THEN NORM_ARITH_TAC; ALL_TAC] THEN
10354 MATCH_MP_TAC VECTOR_MUL_LCANCEL_IMP THEN EXISTS_TAC `dist(a:real^N,b)` THEN
10355 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; VECTOR_ADD_LDISTRIB; REAL_SUB_LDISTRIB;
10356 REAL_DIV_LMUL; DIST_EQ_0] THEN
10357 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DIST_TRIANGLE_EQ] o SYM) THEN
10358 FIRST_ASSUM(SUBST1_TAC o SYM) THEN
10359 REWRITE_TAC[dist; REAL_ARITH `(a + b) * &1 - a = b`] THEN
10361 STRIP_TAC THEN ASM_REWRITE_TAC[dist] THEN
10362 REWRITE_TAC[VECTOR_ARITH `a - ((&1 - u) % a + u % b) = u % (a - b)`;
10363 VECTOR_ARITH `((&1 - u) % a + u % b) - b = (&1 - u) % (a - b)`;
10364 NORM_MUL; GSYM REAL_ADD_LDISTRIB] THEN
10365 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]);;
10367 let IN_SEGMENT_COMPONENT = prove
10369 x IN segment[a,b] /\ 1 <= i /\ i <= dimindex(:N)
10370 ==> min (a$i) (b$i) <= x$i /\ x$i <= max (a$i) (b$i)`,
10371 REPEAT STRIP_TAC THEN
10372 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_SEGMENT]) THEN
10373 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
10374 FIRST_X_ASSUM(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN
10375 ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
10376 SIMP_TAC[REAL_ARITH `c <= u * a + t * b <=> u * --a + t * --b <= --c`] THEN
10377 MATCH_MP_TAC REAL_CONVEX_BOUND_LE THEN ASM_REAL_ARITH_TAC);;
10379 let SEGMENT_1 = prove
10380 (`(!a b. segment[a,b] =
10381 if drop a <= drop b then interval[a,b] else interval[b,a]) /\
10382 (!a b. segment(a,b) =
10383 if drop a <= drop b then interval(a,b) else interval(b,a))`,
10384 CONJ_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[open_segment] THEN
10385 COND_CASES_TAC THEN
10386 REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY;
10387 EXTENSION; GSYM BETWEEN_IN_SEGMENT; between; IN_INTERVAL_1] THEN
10388 REWRITE_TAC[GSYM DROP_EQ; DIST_REAL; GSYM drop] THEN ASM_REAL_ARITH_TAC);;
10390 let OPEN_SEGMENT_1 = prove
10391 (`!a b:real^1. open(segment(a,b))`,
10392 REPEAT GEN_TAC THEN REWRITE_TAC[SEGMENT_1] THEN
10393 COND_CASES_TAC THEN REWRITE_TAC[OPEN_INTERVAL]);;
10395 let SEGMENT_TRANSLATION = prove
10396 (`(!c a b. segment[c + a,c + b] = IMAGE (\x. c + x) (segment[a,b])) /\
10397 (!c a b. segment(c + a,c + b) = IMAGE (\x. c + x) (segment(a,b)))`,
10398 REWRITE_TAC[EXTENSION; IN_SEGMENT; IN_IMAGE] THEN
10399 REWRITE_TAC[VECTOR_ARITH `(&1 - u) % (c + a) + u % (c + b) =
10400 c + (&1 - u) % a + u % b`] THEN
10401 REWRITE_TAC[VECTOR_ARITH `c + a:real^N = c + b <=> a = b`] THEN
10404 add_translation_invariants
10405 [CONJUNCT1 SEGMENT_TRANSLATION; CONJUNCT2 SEGMENT_TRANSLATION];;
10407 let CLOSED_SEGMENT_LINEAR_IMAGE = prove
10409 ==> segment[f a,f b] = IMAGE f (segment[a,b])`,
10410 REPEAT STRIP_TAC THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_SEGMENT] THEN
10411 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN
10412 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_ADD th)]) THEN
10415 add_linear_invariants [CLOSED_SEGMENT_LINEAR_IMAGE];;
10417 let OPEN_SEGMENT_LINEAR_IMAGE = prove
10418 (`!f:real^M->real^N a b.
10419 linear f /\ (!x y. f x = f y ==> x = y)
10420 ==> segment(f a,f b) = IMAGE f (segment(a,b))`,
10421 REWRITE_TAC[open_segment] THEN GEOM_TRANSFORM_TAC[]);;
10423 add_linear_invariants [OPEN_SEGMENT_LINEAR_IMAGE];;
10425 let IN_OPEN_SEGMENT = prove
10427 x IN segment(a,b) <=> x IN segment[a,b] /\ ~(x = a) /\ ~(x = b)`,
10428 REPEAT GEN_TAC THEN REWRITE_TAC[open_segment; IN_DIFF] THEN SET_TAC[]);;
10430 let IN_OPEN_SEGMENT_ALT = prove
10432 x IN segment(a,b) <=>
10433 x IN segment[a,b] /\ ~(x = a) /\ ~(x = b) /\ ~(a = b)`,
10434 REPEAT GEN_TAC THEN ASM_CASES_TAC `a:real^N = b` THEN
10435 ASM_REWRITE_TAC[SEGMENT_REFL; IN_SING; NOT_IN_EMPTY] THEN
10436 ASM_MESON_TAC[IN_OPEN_SEGMENT]);;
10438 let COLLINEAR_DIST_IN_CLOSED_SEGMENT = prove
10439 (`!a b x. collinear {x,a,b} /\
10440 dist(x,a) <= dist(a,b) /\ dist(x,b) <= dist(a,b)
10441 ==> x IN segment[a,b]`,
10442 REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; COLLINEAR_DIST_BETWEEN]);;
10444 let COLLINEAR_DIST_IN_OPEN_SEGMENT = prove
10445 (`!a b x. collinear {x,a,b} /\
10446 dist(x,a) < dist(a,b) /\ dist(x,b) < dist(a,b)
10447 ==> x IN segment(a,b)`,
10448 REWRITE_TAC[IN_OPEN_SEGMENT] THEN
10449 MESON_TAC[COLLINEAR_DIST_IN_CLOSED_SEGMENT; REAL_LT_LE; DIST_SYM]);;
10451 let SEGMENT_SCALAR_MULTIPLE = prove
10452 (`(!a b v. segment[a % v,b % v] =
10453 {x % v:real^N | a <= x /\ x <= b \/ b <= x /\ x <= a}) /\
10454 (!a b v. ~(v = vec 0)
10455 ==> segment(a % v,b % v) =
10456 {x % v:real^N | a < x /\ x < b \/ b < x /\ x < a})`,
10457 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN REPEAT STRIP_TAC THENL
10458 [REPEAT GEN_TAC THEN
10459 MP_TAC(SPECL [`a % basis 1:real^1`; `b % basis 1:real^1`]
10460 (CONJUNCT1 SEGMENT_1)) THEN
10461 REWRITE_TAC[segment; VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_RDISTRIB] THEN
10462 REWRITE_TAC[SET_RULE `{f x % b | p x} = IMAGE (\a. a % b) {f x | p x}`] THEN
10463 DISCH_TAC THEN AP_TERM_TAC THEN
10464 FIRST_X_ASSUM(MP_TAC o AP_TERM `IMAGE drop`) THEN
10465 REWRITE_TAC[GSYM IMAGE_o; o_DEF; DROP_CMUL] THEN
10466 SIMP_TAC[drop; BASIS_COMPONENT; DIMINDEX_GE_1; LE_REFL] THEN
10467 REWRITE_TAC[REAL_MUL_RID; IMAGE_ID] THEN DISCH_THEN SUBST1_TAC THEN
10468 MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
10469 CONJ_TAC THENL [MESON_TAC[LIFT_DROP]; ALL_TAC] THEN
10470 REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN GEN_TAC THEN
10471 COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP] THEN
10472 SIMP_TAC[drop; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_GE_1;
10473 LE_REFL; IN_ELIM_THM] THEN ASM_REAL_ARITH_TAC;
10474 ASM_REWRITE_TAC[open_segment] THEN
10475 ASM_SIMP_TAC[VECTOR_MUL_RCANCEL; SET_RULE
10476 `(!x y. x % v = y % v <=> x = y)
10477 ==> {x % v | P x} DIFF {a % v,b % v} =
10478 {x % v | P x /\ ~(x = a) /\ ~(x = b)}`] THEN
10479 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN AP_TERM_TAC THEN
10480 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN REAL_ARITH_TAC]);;
10482 let FINITE_INTER_COLLINEAR_OPEN_SEGMENTS = prove
10485 ==> (FINITE(segment(a,b) INTER segment(c,d)) <=>
10486 segment(a,b) INTER segment(c,d) = {})`,
10487 REPEAT GEN_TAC THEN ABBREV_TAC `m:real^N = b - a` THEN POP_ASSUM MP_TAC THEN
10488 GEOM_NORMALIZE_TAC `m:real^N` THEN
10489 SIMP_TAC[VECTOR_SUB_EQ; SEGMENT_REFL; INTER_EMPTY; FINITE_EMPTY] THEN
10490 X_GEN_TAC `m:real^N` THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
10491 DISCH_THEN(SUBST_ALL_TAC o SYM) THEN POP_ASSUM MP_TAC THEN
10492 GEOM_ORIGIN_TAC `a:real^N` THEN GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN
10493 X_GEN_TAC `b:real` THEN DISCH_TAC THEN
10494 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
10495 SIMP_TAC[VECTOR_SUB_RZERO; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN
10496 ASM_REWRITE_TAC[real_abs; REAL_MUL_RID] THEN DISCH_THEN SUBST_ALL_TAC THEN
10497 POP_ASSUM(K ALL_TAC) THEN
10498 ASM_CASES_TAC `collinear{vec 0:real^N,&1 % basis 1,y}` THENL
10499 [POP_ASSUM MP_TAC THEN
10500 SIMP_TAC[COLLINEAR_LEMMA_ALT; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL] THEN
10502 `~a /\ (b ==> c ==> d) ==> a \/ b ==> a \/ c ==> d`) THEN
10504 [SIMP_TAC[VECTOR_MUL_LID; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL];
10505 REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
10506 X_GEN_TAC `b:real` THEN DISCH_THEN SUBST_ALL_TAC THEN
10507 X_GEN_TAC `a:real` THEN DISCH_THEN SUBST_ALL_TAC THEN
10508 REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RID] THEN
10509 SUBST1_TAC(VECTOR_ARITH `vec 0:real^N = &0 % basis 1`) THEN
10510 SIMP_TAC[SEGMENT_SCALAR_MULTIPLE; BASIS_NONZERO; DIMINDEX_GE_1; LE_REFL;
10511 VECTOR_MUL_RCANCEL; IMAGE_EQ_EMPTY; FINITE_IMAGE_INJ_EQ; SET_RULE
10512 `(!x y. x % v = y % v <=> x = y)
10513 ==> {x % v | P x} INTER {x % v | Q x} =
10514 IMAGE (\x. x % v) {x | P x /\ Q x}`] THEN
10515 REWRITE_TAC[REAL_ARITH `(&0 < x /\ x < &1 \/ &1 < x /\ x < &0) /\
10516 (b < x /\ x < a \/ a < x /\ x < b) <=>
10517 max (&0) (min a b) < x /\ x < min (&1) (max a b)`] THEN
10518 SIMP_TAC[FINITE_REAL_INTERVAL; EXTENSION; NOT_IN_EMPTY; IN_ELIM_THM] THEN
10519 SIMP_TAC[GSYM REAL_LT_BETWEEN; GSYM NOT_EXISTS_THM] THEN REAL_ARITH_TAC;
10520 DISCH_TAC THEN ASM_CASES_TAC
10521 `segment(vec 0:real^N,&1 % basis 1) INTER segment (x,y) = {}` THEN
10522 ASM_REWRITE_TAC[FINITE_EMPTY] THEN DISCH_THEN(K ALL_TAC) THEN
10523 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
10524 REWRITE_TAC[open_segment; IN_DIFF; NOT_IN_EMPTY;
10525 DE_MORGAN_THM; IN_INTER; IN_INSERT] THEN
10526 DISCH_THEN(X_CHOOSE_THEN `p:real^N` STRIP_ASSUME_TAC) THEN
10527 UNDISCH_TAC `~collinear{vec 0:real^N,&1 % basis 1, y}` THEN
10528 RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_MUL_LID]) THEN
10529 REWRITE_TAC[VECTOR_MUL_LID] THEN
10530 MATCH_MP_TAC COLLINEAR_SUBSET THEN
10531 EXISTS_TAC `{p,x:real^N, y, vec 0, basis 1}` THEN
10532 CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN
10533 MP_TAC(ISPECL [`{y:real^N,vec 0,basis 1}`; `p:real^N`; `x:real^N`]
10534 COLLINEAR_TRIPLES) THEN
10535 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
10536 REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY] THEN CONJ_TAC THENL
10537 [ONCE_REWRITE_TAC[SET_RULE `{p,x,y} = {x,p,y}`] THEN
10538 MATCH_MP_TAC BETWEEN_IMP_COLLINEAR THEN
10539 ASM_REWRITE_TAC[BETWEEN_IN_SEGMENT];
10541 ASM_SIMP_TAC[GSYM COLLINEAR_4_3] THEN
10542 ONCE_REWRITE_TAC[SET_RULE `{p,x,z,w} = {w,z,p,x}`] THEN
10543 SIMP_TAC[COLLINEAR_4_3; BASIS_NONZERO; DIMINDEX_GE_1; ARITH] THEN
10544 REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP BETWEEN_IMP_COLLINEAR o
10545 GEN_REWRITE_RULE I [GSYM BETWEEN_IN_SEGMENT])) THEN
10546 REPEAT(POP_ASSUM MP_TAC) THEN SIMP_TAC[INSERT_AC]]);;
10548 let DIST_IN_CLOSED_SEGMENT,DIST_IN_OPEN_SEGMENT = (CONJ_PAIR o prove)
10550 x IN segment[a,b] ==> dist(x,a) <= dist(a,b) /\ dist(x,b) <= dist(a,b)) /\
10552 x IN segment(a,b) ==> dist(x,a) < dist(a,b) /\ dist(x,b) < dist(a,b))`,
10553 SIMP_TAC[IN_SEGMENT; RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM; dist;
10555 `((&1 - u) % a + u % b) - a:real^N = u % (b - a) /\
10556 ((&1 - u) % a + u % b) - b = --(&1 - u) % (b - a)`] THEN
10557 REWRITE_TAC[NORM_MUL; REAL_ABS_NEG; NORM_SUB] THEN CONJ_TAC THEN
10558 REPEAT GEN_TAC THEN STRIP_TAC THENL
10559 [REWRITE_TAC[REAL_ARITH `x * y <= y <=> x * y <= &1 * y`] THEN
10560 CONJ_TAC THEN MATCH_MP_TAC REAL_LE_RMUL THEN
10561 REWRITE_TAC[NORM_POS_LE] THEN ASM_REAL_ARITH_TAC;
10562 REWRITE_TAC[REAL_ARITH `x * y < y <=> x * y < &1 * y`] THEN
10563 ASM_SIMP_TAC[REAL_LT_RMUL_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
10564 ASM_REAL_ARITH_TAC]);;
10566 let DIST_DECREASES_OPEN_SEGMENT = prove
10568 x IN segment(a,b) ==> dist(c,x) < dist(c,a) \/ dist(c,x) < dist(c,b)`,
10569 GEOM_ORIGIN_TAC `a:real^N` THEN GEOM_NORMALIZE_TAC `b:real^N` THEN
10570 REWRITE_TAC[SEGMENT_REFL; NOT_IN_EMPTY] THEN X_GEN_TAC `b:real^N` THEN
10571 GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN X_GEN_TAC `b:real` THEN
10572 SIMP_TAC[NORM_MUL; NORM_BASIS; real_abs; DIMINDEX_GE_1; LE_REFL;
10573 REAL_MUL_RID; VECTOR_MUL_LID] THEN
10574 REPEAT(DISCH_THEN(K ALL_TAC)) THEN REPEAT GEN_TAC THEN
10575 REWRITE_TAC[IN_SEGMENT; dist] THEN STRIP_TAC THEN
10576 ASM_REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
10578 `norm((c$1 - u) % basis 1:real^N) < norm((c:real^N)$1 % basis 1:real^N) \/
10579 norm((c$1 - u) % basis 1:real^N) < norm((c$1 - &1) % basis 1:real^N)`
10581 [SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL] THEN
10582 ASM_REAL_ARITH_TAC;
10583 ASM_SIMP_TAC[NORM_LT; DOT_LMUL; DOT_RMUL; DOT_BASIS; DIMINDEX_GE_1;
10584 DOT_LSUB; DOT_RSUB; LE_REFL; VECTOR_MUL_COMPONENT; VEC_COMPONENT;
10585 BASIS_COMPONENT; DOT_LZERO; DOT_RZERO; VECTOR_SUB_COMPONENT] THEN
10586 ASM_REAL_ARITH_TAC]);;
10588 let DIST_DECREASES_CLOSED_SEGMENT = prove
10590 x IN segment[a,b] ==> dist(c,x) <= dist(c,a) \/ dist(c,x) <= dist(c,b)`,
10591 REWRITE_TAC[SEGMENT_CLOSED_OPEN; IN_UNION; IN_INSERT; NOT_IN_EMPTY] THEN
10592 ASM_MESON_TAC[DIST_DECREASES_OPEN_SEGMENT; REAL_LE_REFL; REAL_LT_IMP_LE]);;
10594 (* ------------------------------------------------------------------------- *)
10595 (* Limit component bounds. *)
10596 (* ------------------------------------------------------------------------- *)
10598 let LIM_COMPONENT_UBOUND = prove
10599 (`!net:(A)net f (l:real^N) b k.
10600 ~(trivial_limit net) /\ (f --> l) net /\
10601 eventually (\x. (f x)$k <= b) net /\
10602 1 <= k /\ k <= dimindex(:N)
10604 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
10605 [`net:(A)net`; `f:A->real^N`; `{y:real^N | basis k dot y <= b}`; `l:real^N`]
10606 LIM_IN_CLOSED_SET) THEN
10607 ASM_SIMP_TAC[CLOSED_HALFSPACE_LE; IN_ELIM_THM; DOT_BASIS]);;
10609 let LIM_COMPONENT_LBOUND = prove
10610 (`!net:(A)net f (l:real^N) b k.
10611 ~(trivial_limit net) /\ (f --> l) net /\
10612 eventually (\x. b <= (f x)$k) net /\
10613 1 <= k /\ k <= dimindex(:N)
10615 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
10616 [`net:(A)net`; `f:A->real^N`; `{y:real^N | b <= basis k dot y}`; `l:real^N`]
10617 LIM_IN_CLOSED_SET) THEN
10618 ASM_SIMP_TAC[REWRITE_RULE[real_ge] CLOSED_HALFSPACE_GE;
10619 IN_ELIM_THM; DOT_BASIS]);;
10621 let LIM_COMPONENT_EQ = prove
10622 (`!net f:A->real^N i l b.
10623 (f --> l) net /\ 1 <= i /\ i <= dimindex(:N) /\
10624 ~(trivial_limit net) /\ eventually (\x. f(x)$i = b) net
10626 REWRITE_TAC[GSYM REAL_LE_ANTISYM; EVENTUALLY_AND] THEN
10627 MESON_TAC[LIM_COMPONENT_UBOUND; LIM_COMPONENT_LBOUND]);;
10629 let LIM_COMPONENT_LE = prove
10630 (`!net:(A)net f:A->real^N g:A->real^N k l m.
10631 ~(trivial_limit net) /\ (f --> l) net /\ (g --> m) net /\
10632 eventually (\x. (f x)$k <= (g x)$k) net /\
10633 1 <= k /\ k <= dimindex(:N)
10635 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN
10636 REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT; LIM_COMPONENT_LBOUND] THEN
10637 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
10638 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> b /\ a ==> c ==> d`] THEN
10639 DISCH_THEN(MP_TAC o MATCH_MP LIM_SUB) THEN POP_ASSUM MP_TAC THEN
10640 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; LIM_COMPONENT_LBOUND]);;
10642 let LIM_DROP_LE = prove
10643 (`!net:(A)net f g l m.
10644 ~(trivial_limit net) /\ (f --> l) net /\ (g --> m) net /\
10645 eventually (\x. drop(f x) <= drop(g x)) net
10646 ==> drop l <= drop m`,
10647 REWRITE_TAC[drop] THEN REPEAT STRIP_TAC THEN
10648 MATCH_MP_TAC(ISPEC `net:(A)net` LIM_COMPONENT_LE) THEN
10649 MAP_EVERY EXISTS_TAC [`f:A->real^1`; `g:A->real^1`] THEN
10650 ASM_REWRITE_TAC[DIMINDEX_1; LE_REFL]);;
10652 let LIM_DROP_UBOUND = prove
10653 (`!net f:A->real^1 l b.
10655 ~(trivial_limit net) /\ eventually (\x. drop(f x) <= b) net
10657 SIMP_TAC[drop] THEN REPEAT STRIP_TAC THEN
10658 MATCH_MP_TAC LIM_COMPONENT_UBOUND THEN
10659 REWRITE_TAC[LE_REFL; DIMINDEX_1] THEN ASM_MESON_TAC[]);;
10661 let LIM_DROP_LBOUND = prove
10662 (`!net f:A->real^1 l b.
10664 ~(trivial_limit net) /\ eventually (\x. b <= drop(f x)) net
10666 SIMP_TAC[drop] THEN REPEAT STRIP_TAC THEN
10667 MATCH_MP_TAC LIM_COMPONENT_LBOUND THEN
10668 REWRITE_TAC[LE_REFL; DIMINDEX_1] THEN ASM_MESON_TAC[]);;
10670 (* ------------------------------------------------------------------------- *)
10671 (* Also extending closed bounds to closures. *)
10672 (* ------------------------------------------------------------------------- *)
10674 let IMAGE_CLOSURE_SUBSET = prove
10675 (`!f (s:real^N->bool) (t:real^M->bool).
10676 f continuous_on closure s /\ closed t /\ IMAGE f s SUBSET t
10677 ==> IMAGE f (closure s) SUBSET t`,
10678 REPEAT STRIP_TAC THEN
10679 SUBGOAL_THEN `closure s SUBSET {x | (f:real^N->real^M) x IN t}` MP_TAC
10680 THENL [MATCH_MP_TAC SUBSET_TRANS; SET_TAC []] THEN
10681 EXISTS_TAC `{x | x IN closure s /\ (f:real^N->real^M) x IN t}` THEN
10683 [MATCH_MP_TAC CLOSURE_MINIMAL; SET_TAC[]] THEN
10684 ASM_SIMP_TAC[CONTINUOUS_CLOSED_PREIMAGE; CLOSED_CLOSURE] THEN
10685 MP_TAC (ISPEC `s:real^N->bool` CLOSURE_SUBSET) THEN ASM SET_TAC[]);;
10687 let CLOSURE_IMAGE_CLOSURE = prove
10688 (`!f:real^M->real^N s.
10689 f continuous_on closure s
10690 ==> closure(IMAGE f (closure s)) = closure(IMAGE f s)`,
10691 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
10692 SIMP_TAC[SUBSET_CLOSURE; IMAGE_SUBSET; CLOSURE_SUBSET] THEN
10693 SIMP_TAC[CLOSURE_MINIMAL_EQ; CLOSED_CLOSURE] THEN
10694 MATCH_MP_TAC IMAGE_CLOSURE_SUBSET THEN
10695 ASM_REWRITE_TAC[CLOSED_CLOSURE; CLOSURE_SUBSET]);;
10697 let CLOSURE_IMAGE_BOUNDED = prove
10698 (`!f:real^M->real^N s.
10699 f continuous_on closure s /\ bounded s
10700 ==> closure(IMAGE f s) = IMAGE f (closure s)`,
10701 REPEAT STRIP_TAC THEN
10702 TRANS_TAC EQ_TRANS `closure(IMAGE (f:real^M->real^N) (closure s))` THEN
10703 CONJ_TAC THENL [ASM_MESON_TAC[CLOSURE_IMAGE_CLOSURE]; ALL_TAC] THEN
10704 MATCH_MP_TAC CLOSURE_CLOSED THEN MATCH_MP_TAC COMPACT_IMP_CLOSED THEN
10705 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
10706 ASM_REWRITE_TAC[COMPACT_CLOSURE]);;
10708 let CONTINUOUS_ON_CLOSURE_NORM_LE = prove
10709 (`!f:real^N->real^M s x b.
10710 f continuous_on (closure s) /\
10711 (!y. y IN s ==> norm(f y) <= b) /\
10713 ==> norm(f x) <= b`,
10714 REWRITE_TAC [GSYM IN_CBALL_0] THEN REPEAT STRIP_TAC THEN
10715 SUBGOAL_THEN `IMAGE (f:real^N->real^M) (closure s) SUBSET cball(vec 0,b)`
10717 [MATCH_MP_TAC IMAGE_CLOSURE_SUBSET; ASM SET_TAC []] THEN
10718 ASM_REWRITE_TAC [CLOSED_CBALL] THEN ASM SET_TAC []);;
10720 let CONTINUOUS_ON_CLOSURE_COMPONENT_LE = prove
10721 (`!f:real^N->real^M s x b k.
10722 f continuous_on (closure s) /\
10723 (!y. y IN s ==> (f y)$k <= b) /\
10726 REWRITE_TAC [GSYM IN_CBALL_0] THEN REPEAT STRIP_TAC THEN
10727 SUBGOAL_THEN `IMAGE (f:real^N->real^M) (closure s) SUBSET {x | x$k <= b}`
10729 [MATCH_MP_TAC IMAGE_CLOSURE_SUBSET; ASM SET_TAC []] THEN
10730 ASM_REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE] THEN ASM SET_TAC[]);;
10732 let CONTINUOUS_ON_CLOSURE_COMPONENT_GE = prove
10733 (`!f:real^N->real^M s x b k.
10734 f continuous_on (closure s) /\
10735 (!y. y IN s ==> b <= (f y)$k) /\
10738 REWRITE_TAC [GSYM IN_CBALL_0] THEN REPEAT STRIP_TAC THEN
10739 SUBGOAL_THEN `IMAGE (f:real^N->real^M) (closure s) SUBSET {x | x$k >= b}`
10741 [MATCH_MP_TAC IMAGE_CLOSURE_SUBSET; ASM SET_TAC [real_ge]] THEN
10742 ASM_REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM SET_TAC[real_ge]);;
10744 (* ------------------------------------------------------------------------- *)
10745 (* Limits relative to a union. *)
10746 (* ------------------------------------------------------------------------- *)
10748 let LIM_WITHIN_UNION = prove
10749 (`(f --> l) (at x within (s UNION t)) <=>
10750 (f --> l) (at x within s) /\ (f --> l) (at x within t)`,
10751 REWRITE_TAC[LIM_WITHIN; IN_UNION; AND_FORALL_THM] THEN
10752 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
10753 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
10754 EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_THEN
10755 (CONJUNCTS_THEN2 (X_CHOOSE_TAC `d:real`) (X_CHOOSE_TAC `k:real`)) THEN
10756 EXISTS_TAC `min d k` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
10759 let CONTINUOUS_ON_UNION = prove
10760 (`!f s t. closed s /\ closed t /\ f continuous_on s /\ f continuous_on t
10761 ==> f continuous_on (s UNION t)`,
10762 REWRITE_TAC[CONTINUOUS_ON; CLOSED_LIMPT; IN_UNION; LIM_WITHIN_UNION] THEN
10763 MESON_TAC[LIM; TRIVIAL_LIMIT_WITHIN]);;
10765 let CONTINUOUS_ON_CASES = prove
10766 (`!P f g:real^M->real^N s t.
10767 closed s /\ closed t /\ f continuous_on s /\ g continuous_on t /\
10768 (!x. x IN s /\ ~P x \/ x IN t /\ P x ==> f x = g x)
10769 ==> (\x. if P x then f x else g x) continuous_on (s UNION t)`,
10770 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_UNION THEN
10771 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL
10772 [EXISTS_TAC `f:real^M->real^N`; EXISTS_TAC `g:real^M->real^N`] THEN
10773 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);;
10775 let CONTINUOUS_ON_UNION_LOCAL = prove
10776 (`!f:real^M->real^N s.
10777 closed_in (subtopology euclidean (s UNION t)) s /\
10778 closed_in (subtopology euclidean (s UNION t)) t /\
10779 f continuous_on s /\ f continuous_on t
10780 ==> f continuous_on (s UNION t)`,
10781 REWRITE_TAC[CONTINUOUS_ON; CLOSED_IN_LIMPT; IN_UNION; LIM_WITHIN_UNION] THEN
10782 MESON_TAC[LIM; TRIVIAL_LIMIT_WITHIN]);;
10784 let CONTINUOUS_ON_CASES_LOCAL = prove
10785 (`!P f g:real^M->real^N s t.
10786 closed_in (subtopology euclidean (s UNION t)) s /\
10787 closed_in (subtopology euclidean (s UNION t)) t /\
10788 f continuous_on s /\ g continuous_on t /\
10789 (!x. x IN s /\ ~P x \/ x IN t /\ P x ==> f x = g x)
10790 ==> (\x. if P x then f x else g x) continuous_on (s UNION t)`,
10791 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_UNION_LOCAL THEN
10792 ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THENL
10793 [EXISTS_TAC `f:real^M->real^N`; EXISTS_TAC `g:real^M->real^N`] THEN
10794 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);;
10796 let CONTINUOUS_ON_CASES_LE = prove
10797 (`!f g:real^M->real^N h s a.
10798 f continuous_on {t | t IN s /\ h t <= a} /\
10799 g continuous_on {t | t IN s /\ a <= h t} /\
10800 (lift o h) continuous_on s /\
10801 (!t. t IN s /\ h t = a ==> f t = g t)
10802 ==> (\t. if h t <= a then f(t) else g(t)) continuous_on s`,
10803 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC
10804 `{t | t IN s /\ (h:real^M->real) t <= a} UNION
10805 {t | t IN s /\ a <= h t}` THEN
10807 [ALL_TAC; SIMP_TAC[SUBSET; IN_UNION; IN_ELIM_THM; REAL_LE_TOTAL]] THEN
10808 MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN ASM_REWRITE_TAC[] THEN
10809 REWRITE_TAC[IN_ELIM_THM; GSYM CONJ_ASSOC; REAL_LE_ANTISYM] THEN
10810 REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
10811 [ALL_TAC; ASM_MESON_TAC[]] THEN
10814 `{t | t IN s /\ (h:real^M->real) t <= a} =
10815 {t | t IN ({t | t IN s /\ h t <= a} UNION {t | t IN s /\ a <= h t}) /\
10816 (lift o h) t IN {x | x$1 <= a}}`
10817 (fun th -> GEN_REWRITE_TAC RAND_CONV [th])
10819 [REWRITE_TAC[GSYM drop; o_THM; IN_ELIM_THM; LIFT_DROP; EXTENSION;
10821 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
10822 ASM_REAL_ARITH_TAC;
10823 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN
10824 ASM_REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; ETA_AX] THEN
10825 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10826 CONTINUOUS_ON_SUBSET)) THEN
10829 `{t | t IN s /\ a <= (h:real^M->real) t} =
10830 {t | t IN ({t | t IN s /\ h t <= a} UNION {t | t IN s /\ a <= h t}) /\
10831 (lift o h) t IN {x | x$1 >= a}}`
10832 (fun th -> GEN_REWRITE_TAC RAND_CONV [th])
10834 [REWRITE_TAC[GSYM drop; o_THM; IN_ELIM_THM; LIFT_DROP; EXTENSION;
10836 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
10837 ASM_REAL_ARITH_TAC;
10838 MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE THEN
10839 ASM_REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_GE; ETA_AX] THEN
10840 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
10841 CONTINUOUS_ON_SUBSET)) THEN
10844 let CONTINUOUS_ON_CASES_1 = prove
10845 (`!f g:real^1->real^N s a.
10846 f continuous_on {t | t IN s /\ drop t <= a} /\
10847 g continuous_on {t | t IN s /\ a <= drop t} /\
10848 (lift a IN s ==> f(lift a) = g(lift a))
10849 ==> (\t. if drop t <= a then f(t) else g(t)) continuous_on s`,
10850 REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LE THEN
10851 ASM_REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID] THEN
10852 REWRITE_TAC[GSYM LIFT_EQ; LIFT_DROP] THEN ASM_MESON_TAC[]);;
10854 let EXTENSION_FROM_CLOPEN = prove
10855 (`!f:real^M->real^N s t u.
10856 open_in (subtopology euclidean s) t /\
10857 closed_in (subtopology euclidean s) t /\
10858 f continuous_on t /\ IMAGE f t SUBSET u /\ (u = {} ==> s = {})
10859 ==> ?g. g continuous_on s /\ IMAGE g s SUBSET u /\
10860 !x. x IN t ==> g x = f x`,
10861 REPEAT GEN_TAC THEN ASM_CASES_TAC `u:real^N->bool = {}` THEN
10862 ASM_SIMP_TAC[CONTINUOUS_ON_EMPTY; IMAGE_CLAUSES; SUBSET_EMPTY;
10863 IMAGE_EQ_EMPTY; NOT_IN_EMPTY] THEN
10865 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
10866 DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
10867 EXISTS_TAC `\x. if x IN t then (f:real^M->real^N) x else a` THEN
10868 SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN
10869 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
10870 FIRST_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
10871 SUBGOAL_THEN `s:real^M->bool = t UNION (s DIFF t)` SUBST1_TAC THENL
10872 [ASM SET_TAC[]; MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL] THEN
10873 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> t UNION (s DIFF t) = s`] THEN
10874 REWRITE_TAC[CONTINUOUS_ON_CONST; IN_DIFF] THEN
10875 CONJ_TAC THENL [MATCH_MP_TAC CLOSED_IN_DIFF; MESON_TAC[]] THEN
10876 ASM_REWRITE_TAC[CLOSED_IN_REFL]);;
10878 (* ------------------------------------------------------------------------- *)
10879 (* Componentwise limits and continuity. *)
10880 (* ------------------------------------------------------------------------- *)
10882 let LIM_COMPONENTWISE_LIFT = prove
10883 (`!net f:A->real^N.
10885 !i. 1 <= i /\ i <= dimindex(:N)
10886 ==> ((\x. lift((f x)$i)) --> lift(l$i)) net`,
10887 REPEAT GEN_TAC THEN REWRITE_TAC[tendsto] THEN EQ_TAC THENL
10888 [DISCH_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
10889 X_GEN_TAC `e:real` THEN
10890 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN
10891 ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT] THEN
10892 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
10893 GEN_TAC THEN REWRITE_TAC[dist] THEN MATCH_MP_TAC(REAL_ARITH
10894 `y <= x ==> x < e ==> y < e`) THEN
10895 ASM_SIMP_TAC[COMPONENT_LE_NORM; GSYM LIFT_SUB; NORM_LIFT;
10896 GSYM VECTOR_SUB_COMPONENT];
10897 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_FORALL_THM] THEN
10898 ONCE_REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN
10899 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
10900 REWRITE_TAC[GSYM IN_NUMSEG; RIGHT_FORALL_IMP_THM] THEN
10901 SIMP_TAC[FORALL_EVENTUALLY; FINITE_NUMSEG; NUMSEG_EMPTY;
10902 GSYM NOT_LE; DIMINDEX_GE_1] THEN
10903 REWRITE_TAC[DIST_LIFT; GSYM VECTOR_SUB_COMPONENT] THEN
10904 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
10905 FIRST_X_ASSUM(MP_TAC o SPEC `e / &(dimindex(:N))`) THEN
10906 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1] THEN
10907 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EVENTUALLY_MONO) THEN
10908 X_GEN_TAC `x:A` THEN SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; dist] THEN
10909 DISCH_TAC THEN W(MP_TAC o PART_MATCH lhand NORM_LE_L1 o lhand o snd) THEN
10910 MATCH_MP_TAC(REAL_ARITH `s < e ==> n <= s ==> n < e`) THEN
10911 MATCH_MP_TAC SUM_BOUND_LT_GEN THEN
10912 ASM_SIMP_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; GSYM NOT_LE; DIMINDEX_GE_1;
10913 CARD_NUMSEG_1; GSYM IN_NUMSEG]]);;
10915 let CONTINUOUS_COMPONENTWISE_LIFT = prove
10916 (`!net f:A->real^N.
10917 f continuous net <=>
10918 !i. 1 <= i /\ i <= dimindex(:N)
10919 ==> (\x. lift((f x)$i)) continuous net`,
10920 REWRITE_TAC[continuous; GSYM LIM_COMPONENTWISE_LIFT]);;
10922 let CONTINUOUS_ON_COMPONENTWISE_LIFT = prove
10923 (`!f:real^M->real^N s.
10924 f continuous_on s <=>
10925 !i. 1 <= i /\ i <= dimindex(:N)
10926 ==> (\x. lift((f x)$i)) continuous_on s`,
10927 REPEAT GEN_TAC THEN
10928 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
10929 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
10930 [CONTINUOUS_COMPONENTWISE_LIFT] THEN
10933 (* ------------------------------------------------------------------------- *)
10934 (* Some more convenient intermediate-value theorem formulations. *)
10935 (* ------------------------------------------------------------------------- *)
10937 let CONNECTED_IVT_HYPERPLANE = prove
10938 (`!s x y:real^N a b.
10940 x IN s /\ y IN s /\ a dot x <= b /\ b <= a dot y
10941 ==> ?z. z IN s /\ a dot z = b`,
10942 REPEAT STRIP_TAC THEN
10943 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [connected]) THEN
10944 REWRITE_TAC[NOT_EXISTS_THM] THEN DISCH_THEN(MP_TAC o SPECL
10945 [`{x:real^N | a dot x < b}`; `{x:real^N | a dot x > b}`]) THEN
10946 REWRITE_TAC[OPEN_HALFSPACE_LT; OPEN_HALFSPACE_GT] THEN
10947 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN STRIP_TAC THEN
10948 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER; NOT_IN_EMPTY; SUBSET;
10949 IN_UNION; REAL_LT_LE; real_gt] THEN
10950 ASM_MESON_TAC[REAL_LE_TOTAL; REAL_LE_ANTISYM]);;
10952 let CONNECTED_IVT_COMPONENT = prove
10953 (`!s x y:real^N a k.
10954 connected s /\ x IN s /\ y IN s /\
10955 1 <= k /\ k <= dimindex(:N) /\ x$k <= a /\ a <= y$k
10956 ==> ?z. z IN s /\ z$k = a`,
10957 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
10958 [`s:real^N->bool`; `x:real^N`; `y:real^N`; `(basis k):real^N`;
10959 `a:real`] CONNECTED_IVT_HYPERPLANE) THEN
10960 ASM_SIMP_TAC[DOT_BASIS]);;
10962 (* ------------------------------------------------------------------------- *)
10963 (* Rather trivial observation that we can map any connected set on segment. *)
10964 (* ------------------------------------------------------------------------- *)
10966 let MAPPING_CONNECTED_ONTO_SEGMENT = prove
10967 (`!s:real^M->bool a b:real^N.
10968 connected s /\ ~(?a. s SUBSET {a})
10969 ==> ?f. f continuous_on s /\ IMAGE f s = segment[a,b]`,
10970 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
10971 `~(?a. s SUBSET {a}) ==> ?a b. a IN s /\ b IN s /\ ~(a = b)`)) THEN
10972 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
10973 MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN STRIP_TAC THEN EXISTS_TAC
10974 `\x:real^M. a + dist(u,x) / (dist(u,x) + dist(v,x)) % (b - a:real^N)` THEN
10976 [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
10977 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN SIMP_TAC[o_DEF; CONTINUOUS_ON_CONST];
10978 REWRITE_TAC[segment; VECTOR_ARITH
10979 `(&1 - u) % a + u % b:real^N = a + u % (b - a)`] THEN
10980 MATCH_MP_TAC(SET_RULE
10981 `IMAGE f s = {x | P x}
10982 ==> IMAGE (\x. a + f x % b) s = {a + u % b:real^N | P u}`) THEN
10983 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_IMAGE] THEN
10984 ASM_SIMP_TAC[IN_ELIM_THM; REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ;
10985 NORM_ARITH `~(u:real^N = v) ==> &0 < dist(u,x) + dist(v,x)`] THEN
10986 CONJ_TAC THENL [CONV_TAC NORM_ARITH; REWRITE_TAC[IN_IMAGE]] THEN
10987 X_GEN_TAC `t:real` THEN STRIP_TAC THEN
10989 [`IMAGE (\x:real^M. lift(dist(u,x) / (dist(u,x) + dist(v,x)))) s`;
10990 `vec 0:real^1`; `vec 1:real^1`; `t:real`; `1`]
10991 CONNECTED_IVT_COMPONENT) THEN
10992 ASM_SIMP_TAC[VEC_COMPONENT; DIMINDEX_1; ARITH_LE] THEN
10993 REWRITE_TAC[EXISTS_IN_IMAGE; GSYM drop; LIFT_DROP] THEN
10994 ANTS_TAC THENL [REWRITE_TAC[IN_IMAGE]; MESON_TAC[]] THEN
10995 REPEAT CONJ_TAC THENL
10996 [MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[];
10997 EXISTS_TAC `u:real^M` THEN ASM_REWRITE_TAC[DIST_REFL; real_div] THEN
10998 REWRITE_TAC[GSYM LIFT_NUM; LIFT_EQ] THEN REAL_ARITH_TAC;
10999 EXISTS_TAC `v:real^M` THEN ASM_REWRITE_TAC[DIST_REFL] THEN
11000 ASM_SIMP_TAC[REAL_DIV_REFL; DIST_EQ_0; REAL_ADD_RID] THEN
11001 REWRITE_TAC[GSYM LIFT_NUM; LIFT_EQ]]] THEN
11002 REWRITE_TAC[real_div; LIFT_CMUL] THEN
11003 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
11004 REWRITE_TAC[CONTINUOUS_ON_LIFT_DIST] THEN
11005 MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
11006 ASM_SIMP_TAC[LIFT_ADD; NORM_ARITH
11007 `~(u:real^N = v) ==> ~(dist(u,x) + dist(v,x) = &0)`] THEN
11008 MATCH_MP_TAC CONTINUOUS_ON_ADD THEN
11009 REWRITE_TAC[REWRITE_RULE[o_DEF] CONTINUOUS_ON_LIFT_DIST]);;
11011 (* ------------------------------------------------------------------------- *)
11012 (* Also more convenient formulations of monotone convergence. *)
11013 (* ------------------------------------------------------------------------- *)
11015 let BOUNDED_INCREASING_CONVERGENT = prove
11017 bounded {s n | n IN (:num)} /\ (!n. drop(s n) <= drop(s(SUC n)))
11018 ==> ?l. (s --> l) sequentially`,
11020 REWRITE_TAC[bounded; IN_ELIM_THM; ABS_DROP; LIM_SEQUENTIALLY; dist;
11021 DROP_SUB; IN_UNIV; GSYM EXISTS_DROP] THEN
11022 DISCH_TAC THEN MATCH_MP_TAC CONVERGENT_BOUNDED_MONOTONE THEN
11023 REWRITE_TAC[LEFT_EXISTS_AND_THM] THEN
11024 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
11025 DISJ1_TAC THEN MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN
11026 ASM_REWRITE_TAC[REAL_LE_TRANS; REAL_LE_REFL]);;
11028 let BOUNDED_DECREASING_CONVERGENT = prove
11030 bounded {s n | n IN (:num)} /\ (!n. drop(s(SUC n)) <= drop(s(n)))
11031 ==> ?l. (s --> l) sequentially`,
11032 GEN_TAC THEN REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN
11033 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
11034 MP_TAC(ISPEC `\n. --((s:num->real^1) n)` BOUNDED_INCREASING_CONVERGENT) THEN
11035 ASM_SIMP_TAC[bounded; FORALL_IN_GSPEC; NORM_NEG; DROP_NEG; REAL_LE_NEG2] THEN
11036 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [GSYM LIM_NEG_EQ] THEN
11037 REWRITE_TAC[VECTOR_NEG_NEG; ETA_AX] THEN MESON_TAC[]);;
11039 (* ------------------------------------------------------------------------- *)
11040 (* Since we'll use some cardinality reasoning, add invariance theorems. *)
11041 (* ------------------------------------------------------------------------- *)
11043 let card_translation_invariants = (CONJUNCTS o prove)
11044 (`(!a (s:real^N->bool) (t:A->bool).
11045 IMAGE (\x. a + x) s =_c t <=> s =_c t) /\
11046 (!a (s:A->bool) (t:real^N->bool).
11047 s =_c IMAGE (\x. a + x) t <=> s =_c t) /\
11048 (!a (s:real^N->bool) (t:A->bool).
11049 IMAGE (\x. a + x) s <_c t <=> s <_c t) /\
11050 (!a (s:A->bool) (t:real^N->bool).
11051 s <_c IMAGE (\x. a + x) t <=> s <_c t) /\
11052 (!a (s:real^N->bool) (t:A->bool).
11053 IMAGE (\x. a + x) s <=_c t <=> s <=_c t) /\
11054 (!a (s:A->bool) (t:real^N->bool).
11055 s <=_c IMAGE (\x. a + x) t <=> s <=_c t) /\
11056 (!a (s:real^N->bool) (t:A->bool).
11057 IMAGE (\x. a + x) s >_c t <=> s >_c t) /\
11058 (!a (s:A->bool) (t:real^N->bool).
11059 s >_c IMAGE (\x. a + x) t <=> s >_c t) /\
11060 (!a (s:real^N->bool) (t:A->bool).
11061 IMAGE (\x. a + x) s >=_c t <=> s >=_c t) /\
11062 (!a (s:A->bool) (t:real^N->bool).
11063 s >=_c IMAGE (\x. a + x) t <=> s >=_c t)`,
11064 REWRITE_TAC[gt_c; ge_c] THEN REPEAT STRIP_TAC THENL
11065 [MATCH_MP_TAC CARD_EQ_CONG;
11066 MATCH_MP_TAC CARD_EQ_CONG;
11067 MATCH_MP_TAC CARD_LT_CONG;
11068 MATCH_MP_TAC CARD_LT_CONG;
11069 MATCH_MP_TAC CARD_LE_CONG;
11070 MATCH_MP_TAC CARD_LE_CONG;
11071 MATCH_MP_TAC CARD_LT_CONG;
11072 MATCH_MP_TAC CARD_LT_CONG;
11073 MATCH_MP_TAC CARD_LE_CONG;
11074 MATCH_MP_TAC CARD_LE_CONG] THEN
11075 REWRITE_TAC[CARD_EQ_REFL] THEN MATCH_MP_TAC CARD_EQ_IMAGE THEN
11076 SIMP_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`]) in
11077 add_translation_invariants card_translation_invariants;;
11079 let card_linear_invariants = (CONJUNCTS o prove)
11080 (`(!(f:real^M->real^N) s (t:A->bool).
11081 linear f /\ (!x y. f x = f y ==> x = y)
11082 ==> (IMAGE f s =_c t <=> s =_c t)) /\
11083 (!(f:real^M->real^N) (s:A->bool) t.
11084 linear f /\ (!x y. f x = f y ==> x = y)
11085 ==> (s =_c IMAGE f t <=> s =_c t)) /\
11086 (!(f:real^M->real^N) s (t:A->bool).
11087 linear f /\ (!x y. f x = f y ==> x = y)
11088 ==> (IMAGE f s <_c t <=> s <_c t)) /\
11089 (!(f:real^M->real^N) (s:A->bool) t.
11090 linear f /\ (!x y. f x = f y ==> x = y)
11091 ==> (s <_c IMAGE f t <=> s <_c t)) /\
11092 (!(f:real^M->real^N) s (t:A->bool).
11093 linear f /\ (!x y. f x = f y ==> x = y)
11094 ==> (IMAGE f s <=_c t <=> s <=_c t)) /\
11095 (!(f:real^M->real^N) (s:A->bool) t.
11096 linear f /\ (!x y. f x = f y ==> x = y)
11097 ==> (s <=_c IMAGE f t <=> s <=_c t)) /\
11098 (!(f:real^M->real^N) s (t:A->bool).
11099 linear f /\ (!x y. f x = f y ==> x = y)
11100 ==> (IMAGE f s >_c t <=> s >_c t)) /\
11101 (!(f:real^M->real^N) (s:A->bool) t.
11102 linear f /\ (!x y. f x = f y ==> x = y)
11103 ==> (s >_c IMAGE f t <=> s >_c t)) /\
11104 (!(f:real^M->real^N) s (t:A->bool).
11105 linear f /\ (!x y. f x = f y ==> x = y)
11106 ==> (IMAGE f s >=_c t <=> s >=_c t)) /\
11107 (!(f:real^M->real^N) (s:A->bool) t.
11108 linear f /\ (!x y. f x = f y ==> x = y)
11109 ==> (s >=_c IMAGE f t <=> s >=_c t))`,
11110 REWRITE_TAC[gt_c; ge_c] THEN REPEAT STRIP_TAC THENL
11111 [MATCH_MP_TAC CARD_EQ_CONG;
11112 MATCH_MP_TAC CARD_EQ_CONG;
11113 MATCH_MP_TAC CARD_LT_CONG;
11114 MATCH_MP_TAC CARD_LT_CONG;
11115 MATCH_MP_TAC CARD_LE_CONG;
11116 MATCH_MP_TAC CARD_LE_CONG;
11117 MATCH_MP_TAC CARD_LT_CONG;
11118 MATCH_MP_TAC CARD_LT_CONG;
11119 MATCH_MP_TAC CARD_LE_CONG;
11120 MATCH_MP_TAC CARD_LE_CONG] THEN
11121 REWRITE_TAC[CARD_EQ_REFL] THEN MATCH_MP_TAC CARD_EQ_IMAGE THEN
11122 ASM_MESON_TAC[]) in
11123 add_linear_invariants card_linear_invariants;;
11125 (* ------------------------------------------------------------------------- *)
11126 (* Basic homeomorphism definitions. *)
11127 (* ------------------------------------------------------------------------- *)
11129 let homeomorphism = new_definition
11130 `homeomorphism (s,t) (f,g) <=>
11131 (!x. x IN s ==> (g(f(x)) = x)) /\ (IMAGE f s = t) /\ f continuous_on s /\
11132 (!y. y IN t ==> (f(g(y)) = y)) /\ (IMAGE g t = s) /\ g continuous_on t`;;
11134 parse_as_infix("homeomorphic",(12,"right"));;
11136 let homeomorphic = new_definition
11137 `s homeomorphic t <=> ?f g. homeomorphism (s,t) (f,g)`;;
11139 let HOMEOMORPHISM = prove
11140 (`!s:real^M->bool t:real^N->bool f g.
11141 homeomorphism (s,t) (f,g) <=>
11142 f continuous_on s /\ IMAGE f s SUBSET t /\
11143 g continuous_on t /\ IMAGE g t SUBSET s /\
11144 (!x. x IN s ==> g (f x) = x) /\
11145 (!y. y IN t ==> f (g y) = y)`,
11146 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphism] THEN
11147 EQ_TAC THEN SIMP_TAC[] THEN SET_TAC[]);;
11149 let HOMEOMORPHISM_OF_SUBSETS = prove
11151 homeomorphism (s,t) (f,g) /\ s' SUBSET s /\ t' SUBSET t /\ IMAGE f s' = t'
11152 ==> homeomorphism (s',t') (f,g)`,
11153 REWRITE_TAC[homeomorphism] THEN
11154 REPEAT STRIP_TAC THEN
11155 TRY(MATCH_MP_TAC CONTINUOUS_ON_SUBSET) THEN ASM SET_TAC[]);;
11157 let HOMEOMORPHISM_ID = prove
11158 (`!s:real^N->bool. homeomorphism (s,s) ((\x. x),(\x. x))`,
11159 REWRITE_TAC[homeomorphism; IMAGE_ID; CONTINUOUS_ON_ID]);;
11161 let HOMEOMORPHISM_I = prove
11162 (`!s:real^N->bool. homeomorphism (s,s) (I,I)`,
11163 REWRITE_TAC[I_DEF; HOMEOMORPHISM_ID]);;
11165 let HOMEOMORPHIC_REFL = prove
11166 (`!s:real^N->bool. s homeomorphic s`,
11167 REWRITE_TAC[homeomorphic] THEN MESON_TAC[HOMEOMORPHISM_I]);;
11169 let HOMEOMORPHISM_SYM = prove
11170 (`!f:real^M->real^N g s t.
11171 homeomorphism (s,t) (f,g) <=> homeomorphism (t,s) (g,f)`,
11172 REWRITE_TAC[homeomorphism] THEN MESON_TAC[]);;
11174 let HOMEOMORPHIC_SYM = prove
11175 (`!s t. s homeomorphic t <=> t homeomorphic s`,
11176 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN
11177 GEN_REWRITE_TAC RAND_CONV [SWAP_EXISTS_THM] THEN
11178 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN CONV_TAC TAUT);;
11180 let HOMEOMORPHISM_COMPOSE = prove
11181 (`!f:real^M->real^N g h:real^N->real^P k s t u.
11182 homeomorphism (s,t) (f,g) /\ homeomorphism (t,u) (h,k)
11183 ==> homeomorphism (s,u) (h o f,g o k)`,
11184 SIMP_TAC[homeomorphism; CONTINUOUS_ON_COMPOSE; IMAGE_o; o_THM] THEN
11187 let HOMEOMORPHIC_TRANS = prove
11188 (`!s:real^M->bool t:real^N->bool u:real^P->bool.
11189 s homeomorphic t /\ t homeomorphic u ==> s homeomorphic u`,
11190 REWRITE_TAC[homeomorphic] THEN MESON_TAC[HOMEOMORPHISM_COMPOSE]);;
11192 let HOMEOMORPHIC_IMP_CARD_EQ = prove
11193 (`!s:real^M->bool t:real^N->bool. s homeomorphic t ==> s =_c t`,
11194 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism; eq_c] THEN
11195 MATCH_MP_TAC MONO_EXISTS THEN SET_TAC[]);;
11197 let HOMEOMORPHIC_EMPTY = prove
11198 (`(!s. (s:real^N->bool) homeomorphic ({}:real^M->bool) <=> s = {}) /\
11199 (!s. ({}:real^M->bool) homeomorphic (s:real^N->bool) <=> s = {})`,
11200 REWRITE_TAC[homeomorphic; homeomorphism; IMAGE_CLAUSES; IMAGE_EQ_EMPTY] THEN
11201 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
11202 ASM_REWRITE_TAC[continuous_on; NOT_IN_EMPTY]);;
11204 let HOMEOMORPHIC_MINIMAL = prove
11205 (`!s t. s homeomorphic t <=>
11206 ?f g. (!x. x IN s ==> f(x) IN t /\ (g(f(x)) = x)) /\
11207 (!y. y IN t ==> g(y) IN s /\ (f(g(y)) = y)) /\
11208 f continuous_on s /\ g continuous_on t`,
11209 REWRITE_TAC[homeomorphic; homeomorphism; EXTENSION; IN_IMAGE] THEN
11210 REPEAT GEN_TAC THEN REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN MESON_TAC[]);;
11212 let HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF = prove
11213 (`!f:real^M->real^N s.
11214 linear f /\ (!x y. f x = f y ==> x = y)
11215 ==> (IMAGE f s) homeomorphic s`,
11216 REPEAT STRIP_TAC THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11217 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_LEFT_INVERSE]) THEN
11218 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN
11219 EXISTS_TAC `f:real^M->real^N` THEN
11220 ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; FORALL_IN_IMAGE; FUN_IN_IMAGE] THEN
11221 ASM_SIMP_TAC[continuous_on; IMP_CONJ; FORALL_IN_IMAGE] THEN
11222 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
11223 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
11224 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_BOUNDED_BELOW_POS) THEN
11225 ASM_REWRITE_TAC[] THEN
11226 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
11227 EXISTS_TAC `e * B:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
11228 X_GEN_TAC `y:real^M` THEN ASM_SIMP_TAC[dist; GSYM LINEAR_SUB] THEN
11229 DISCH_TAC THEN ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ] THEN
11230 MATCH_MP_TAC(REAL_ARITH `a <= b ==> b < x ==> a < x`) THEN
11231 ASM_SIMP_TAC[REAL_LE_RDIV_EQ]);;
11233 let HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ = prove
11234 (`!f:real^M->real^N s t.
11235 linear f /\ (!x y. f x = f y ==> x = y)
11236 ==> ((IMAGE f s) homeomorphic t <=> s homeomorphic t)`,
11237 REPEAT GEN_TAC THEN DISCH_THEN(ASSUME_TAC o SPEC `s:real^M->bool` o
11238 MATCH_MP HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_SELF) THEN
11240 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HOMEOMORPHIC_SYM]);
11241 POP_ASSUM MP_TAC] THEN
11242 REWRITE_TAC[IMP_IMP; HOMEOMORPHIC_TRANS]);;
11244 let HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ = prove
11245 (`!f:real^M->real^N s t.
11246 linear f /\ (!x y. f x = f y ==> x = y)
11247 ==> (s homeomorphic (IMAGE f t) <=> s homeomorphic t)`,
11248 ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
11249 REWRITE_TAC[HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ]);;
11251 add_linear_invariants
11252 [HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ;
11253 HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ];;
11255 let HOMEOMORPHIC_TRANSLATION_SELF = prove
11256 (`!a:real^N s. (IMAGE (\x. a + x) s) homeomorphic s`,
11257 REPEAT GEN_TAC THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11258 EXISTS_TAC `\x:real^N. x - a` THEN
11259 EXISTS_TAC `\x:real^N. a + x` THEN
11260 SIMP_TAC[FORALL_IN_IMAGE; CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID;
11261 CONTINUOUS_ON_CONST; CONTINUOUS_ON_ADD; VECTOR_ADD_SUB] THEN
11262 REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[]);;
11264 let HOMEOMORPHIC_TRANSLATION_LEFT_EQ = prove
11266 (IMAGE (\x. a + x) s) homeomorphic t <=> s homeomorphic t`,
11267 MESON_TAC[HOMEOMORPHIC_TRANSLATION_SELF;
11268 HOMEOMORPHIC_SYM; HOMEOMORPHIC_TRANS]);;
11270 let HOMEOMORPHIC_TRANSLATION_RIGHT_EQ = prove
11272 s homeomorphic (IMAGE (\x. a + x) t) <=> s homeomorphic t`,
11273 ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
11274 REWRITE_TAC[HOMEOMORPHIC_TRANSLATION_LEFT_EQ]);;
11276 add_translation_invariants
11277 [HOMEOMORPHIC_TRANSLATION_LEFT_EQ;
11278 HOMEOMORPHIC_TRANSLATION_RIGHT_EQ];;
11280 let HOMEOMORPHISM_IMP_QUOTIENT_MAP = prove
11281 (`!f:real^M->real^N g s t.
11282 homeomorphism (s,t) (f,g)
11284 ==> (open_in (subtopology euclidean s) {x | x IN s /\ f x IN u} <=>
11285 open_in (subtopology euclidean t) u)`,
11286 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphism] THEN
11287 STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP THEN
11288 EXISTS_TAC `g:real^N->real^M` THEN ASM_REWRITE_TAC[SUBSET_REFL]);;
11290 let HOMEOMORPHIC_PCROSS = prove
11291 (`!s:real^M->bool t:real^N->bool s':real^P->bool t':real^Q->bool.
11292 s homeomorphic s' /\ t homeomorphic t'
11293 ==> (s PCROSS t) homeomorphic (s' PCROSS t')`,
11294 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN
11295 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
11296 DISCH_THEN(CONJUNCTS_THEN2
11297 (X_CHOOSE_THEN `f:real^M->real^P`
11298 (X_CHOOSE_THEN `f':real^P->real^M` STRIP_ASSUME_TAC))
11299 (X_CHOOSE_THEN `g:real^N->real^Q`
11300 (X_CHOOSE_THEN `g':real^Q->real^N` STRIP_ASSUME_TAC))) THEN
11301 MAP_EVERY EXISTS_TAC
11302 [`(\z. pastecart (f(fstcart z)) (g(sndcart z)))
11303 :real^(M,N)finite_sum->real^(P,Q)finite_sum`;
11304 `(\z. pastecart (f'(fstcart z)) (g'(sndcart z)))
11305 :real^(P,Q)finite_sum->real^(M,N)finite_sum`] THEN
11306 ASM_SIMP_TAC[FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART;
11307 SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS] THEN
11308 CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN
11309 CONJ_TAC THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN
11310 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
11311 SIMP_TAC[LINEAR_FSTCART; LINEAR_SNDCART; LINEAR_CONTINUOUS_ON] THEN
11312 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
11313 CONTINUOUS_ON_SUBSET)) THEN
11314 REWRITE_TAC[FORALL_IN_IMAGE; FORALL_IN_PCROSS; SUBSET] THEN
11315 SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART]);;
11317 let HOMEOMORPHIC_PCROSS_SYM = prove
11318 (`!s:real^M->bool t:real^N->bool. (s PCROSS t) homeomorphic (t PCROSS s)`,
11319 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN
11320 EXISTS_TAC `(\z. pastecart (sndcart z) (fstcart z))
11321 :real^(M,N)finite_sum->real^(N,M)finite_sum` THEN
11322 EXISTS_TAC `(\z. pastecart (sndcart z) (fstcart z))
11323 :real^(N,M)finite_sum->real^(M,N)finite_sum` THEN
11324 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET; FORALL_IN_IMAGE] THEN
11325 SIMP_TAC[CONTINUOUS_ON_PASTECART; LINEAR_CONTINUOUS_ON;
11326 LINEAR_FSTCART; LINEAR_SNDCART] THEN
11327 REWRITE_TAC[FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART;
11328 IN_IMAGE; EXISTS_PASTECART; PASTECART_INJ; PASTECART_IN_PCROSS] THEN
11331 let HOMEOMORPHIC_PCROSS_ASSOC = prove
11332 (`!s:real^M->bool t:real^N->bool u:real^P->bool.
11333 (s PCROSS (t PCROSS u)) homeomorphic ((s PCROSS t) PCROSS u)`,
11334 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN
11335 MAP_EVERY EXISTS_TAC
11336 [`\z:real^(M,(N,P)finite_sum)finite_sum.
11337 pastecart (pastecart (fstcart z) (fstcart(sndcart z)))
11338 (sndcart(sndcart z))`;
11339 `\z:real^((M,N)finite_sum,P)finite_sum.
11340 pastecart (fstcart(fstcart z))
11341 (pastecart (sndcart(fstcart z)) (sndcart z))`] THEN
11342 REWRITE_TAC[FORALL_IN_PCROSS; SUBSET; FORALL_IN_IMAGE;
11343 RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
11344 SIMP_TAC[FSTCART_PASTECART; SNDCART_PASTECART; PASTECART_IN_PCROSS] THEN
11345 CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN
11346 REPEAT(MATCH_MP_TAC LINEAR_PASTECART THEN CONJ_TAC) THEN
11347 TRY(GEN_REWRITE_TAC RAND_CONV [GSYM o_DEF] THEN
11348 MATCH_MP_TAC LINEAR_COMPOSE) THEN
11349 REWRITE_TAC[LINEAR_FSTCART; LINEAR_SNDCART]);;
11351 let HOMEOMORPHIC_SCALING_LEFT = prove
11353 ==> !s t. (IMAGE (\x. c % x) s) homeomorphic t <=> s homeomorphic t`,
11354 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN
11355 MATCH_MP_TAC HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ THEN
11356 ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; LINEAR_SCALING]);;
11358 let HOMEOMORPHIC_SCALING_RIGHT = prove
11360 ==> !s t. s homeomorphic (IMAGE (\x. c % x) t) <=> s homeomorphic t`,
11361 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN
11362 MATCH_MP_TAC HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ THEN
11363 ASM_SIMP_TAC[VECTOR_MUL_LCANCEL; REAL_LT_IMP_NZ; LINEAR_SCALING]);;
11365 let HOMEOMORPHIC_SUBSPACES = prove
11366 (`!s:real^M->bool t:real^N->bool.
11367 subspace s /\ subspace t /\ dim s = dim t ==> s homeomorphic t`,
11368 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN
11369 DISCH_THEN(MP_TAC o MATCH_MP ISOMETRIES_SUBSPACES) THEN
11370 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:real^M->real^N` THEN
11371 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN
11372 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_CBALL_0] THEN
11373 SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN ASM SET_TAC[]);;
11375 let HOMEOMORPHIC_FINITE = prove
11376 (`!s:real^M->bool t:real^N->bool.
11377 FINITE s /\ FINITE t ==> (s homeomorphic t <=> CARD s = CARD t)`,
11378 REPEAT STRIP_TAC THEN EQ_TAC THENL
11379 [DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_IMP_CARD_EQ) THEN
11380 ASM_SIMP_TAC[CARD_EQ_CARD];
11381 STRIP_TAC THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN
11382 MP_TAC(ISPECL [`s:real^M->bool`; `t:real^N->bool`]
11383 CARD_EQ_BIJECTIONS) THEN
11384 ASM_REWRITE_TAC[] THEN
11385 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
11386 ASM_SIMP_TAC[CONTINUOUS_ON_FINITE] THEN ASM SET_TAC[]]);;
11388 let HOMEOMORPHIC_FINITE_STRONG = prove
11389 (`!s:real^M->bool t:real^N->bool.
11390 FINITE s \/ FINITE t
11391 ==> (s homeomorphic t <=> FINITE s /\ FINITE t /\ CARD s = CARD t)`,
11392 REPEAT GEN_TAC THEN DISCH_TAC THEN EQ_TAC THEN
11393 SIMP_TAC[HOMEOMORPHIC_FINITE] THEN DISCH_TAC THEN
11394 FIRST_ASSUM(MP_TAC o MATCH_MP CARD_FINITE_CONG o MATCH_MP
11395 HOMEOMORPHIC_IMP_CARD_EQ) THEN
11396 FIRST_X_ASSUM DISJ_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
11397 ASM_MESON_TAC[HOMEOMORPHIC_FINITE]);;
11399 let HOMEOMORPHIC_SING = prove
11400 (`!a:real^M b:real^N. {a} homeomorphic {b}`,
11401 SIMP_TAC[HOMEOMORPHIC_FINITE; FINITE_SING; CARD_SING]);;
11403 let HOMEOMORPHIC_PCROSS_SING = prove
11404 (`(!s:real^M->bool a:real^N. s homeomorphic (s PCROSS {a})) /\
11405 (!s:real^M->bool a:real^N. s homeomorphic ({a} PCROSS s))`,
11406 MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN CONJ_TAC THENL
11407 [MESON_TAC[HOMEOMORPHIC_PCROSS_SYM; HOMEOMORPHIC_TRANS]; ALL_TAC] THEN
11408 REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN
11409 EXISTS_TAC `\x. (pastecart x a:real^(M,N)finite_sum)` THEN
11410 EXISTS_TAC `fstcart:real^(M,N)finite_sum->real^M` THEN
11411 SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN
11412 SIMP_TAC[LINEAR_FSTCART; LINEAR_CONTINUOUS_ON; SUBSET; FORALL_IN_IMAGE] THEN
11413 REWRITE_TAC[FORALL_IN_PCROSS; PASTECART_IN_PCROSS; IN_SING] THEN
11414 SIMP_TAC[FSTCART_PASTECART]);;
11416 (* ------------------------------------------------------------------------- *)
11417 (* Inverse function property for open/closed maps. *)
11418 (* ------------------------------------------------------------------------- *)
11420 let CONTINUOUS_ON_INVERSE_OPEN_MAP = prove
11421 (`!f:real^M->real^N g s t.
11422 f continuous_on s /\ IMAGE f s = t /\ (!x. x IN s ==> g(f x) = x) /\
11423 (!u. open_in (subtopology euclidean s) u
11424 ==> open_in (subtopology euclidean t) (IMAGE f u))
11425 ==> g continuous_on t`,
11426 REPEAT STRIP_TAC THEN
11427 MP_TAC(ISPECL [`g:real^N->real^M`; `t:real^N->bool`; `s:real^M->bool`]
11428 CONTINUOUS_ON_OPEN_GEN) THEN
11429 ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN
11430 X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN
11431 FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN
11432 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
11433 FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN
11436 let CONTINUOUS_ON_INVERSE_CLOSED_MAP = prove
11437 (`!f:real^M->real^N g s t.
11438 f continuous_on s /\ IMAGE f s = t /\ (!x. x IN s ==> g(f x) = x) /\
11439 (!u. closed_in (subtopology euclidean s) u
11440 ==> closed_in (subtopology euclidean t) (IMAGE f u))
11441 ==> g continuous_on t`,
11442 REPEAT STRIP_TAC THEN
11443 MP_TAC(ISPECL [`g:real^N->real^M`; `t:real^N->bool`; `s:real^M->bool`]
11444 CONTINUOUS_ON_CLOSED_GEN) THEN
11445 ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN
11446 X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN
11447 FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN
11448 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
11449 FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [closed_in]) THEN
11450 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM SET_TAC[]);;
11452 let HOMEOMORPHISM_INJECTIVE_OPEN_MAP = prove
11453 (`!f:real^M->real^N s t.
11454 f continuous_on s /\ IMAGE f s = t /\
11455 (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
11456 (!u. open_in (subtopology euclidean s) u
11457 ==> open_in (subtopology euclidean t) (IMAGE f u))
11458 ==> ?g. homeomorphism (s,t) (f,g)`,
11459 REPEAT STRIP_TAC THEN
11460 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
11461 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN
11462 DISCH_TAC THEN ASM_SIMP_TAC[homeomorphism] THEN
11463 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
11464 MATCH_MP_TAC CONTINUOUS_ON_INVERSE_OPEN_MAP THEN ASM_MESON_TAC[]);;
11466 let HOMEOMORPHISM_INJECTIVE_CLOSED_MAP = prove
11467 (`!f:real^M->real^N s t.
11468 f continuous_on s /\ IMAGE f s = t /\
11469 (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
11470 (!u. closed_in (subtopology euclidean s) u
11471 ==> closed_in (subtopology euclidean t) (IMAGE f u))
11472 ==> ?g. homeomorphism (s,t) (f,g)`,
11473 REPEAT STRIP_TAC THEN
11474 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
11475 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^N->real^M` THEN
11476 DISCH_TAC THEN ASM_SIMP_TAC[homeomorphism] THEN
11477 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
11478 MATCH_MP_TAC CONTINUOUS_ON_INVERSE_CLOSED_MAP THEN ASM_MESON_TAC[]);;
11480 let HOMEOMORPHISM_IMP_OPEN_MAP = prove
11481 (`!f:real^M->real^N g s t u.
11482 homeomorphism (s,t) (f,g) /\ open_in (subtopology euclidean s) u
11483 ==> open_in (subtopology euclidean t) (IMAGE f u)`,
11484 REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN
11485 SUBGOAL_THEN `IMAGE (f:real^M->real^N) u =
11486 {y | y IN t /\ g(y) IN u}`
11488 [FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [open_in]) THEN
11490 MATCH_MP_TAC CONTINUOUS_ON_IMP_OPEN_IN THEN ASM_REWRITE_TAC[]]);;
11492 let HOMEOMORPHISM_IMP_CLOSED_MAP = prove
11493 (`!f:real^M->real^N g s t u.
11494 homeomorphism (s,t) (f,g) /\ closed_in (subtopology euclidean s) u
11495 ==> closed_in (subtopology euclidean t) (IMAGE f u)`,
11496 REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN
11497 SUBGOAL_THEN `IMAGE (f:real^M->real^N) u =
11498 {y | y IN t /\ g(y) IN u}`
11500 [FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [closed_in]) THEN
11501 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM SET_TAC[];
11502 MATCH_MP_TAC CONTINUOUS_ON_IMP_CLOSED_IN THEN ASM_REWRITE_TAC[]]);;
11504 let HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ = prove
11505 (`!f:real^M->real^N s t.
11506 f continuous_on s /\ IMAGE f s = t /\
11507 (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
11508 ==> ((?g. homeomorphism (s,t) (f,g)) <=>
11509 !u. open_in (subtopology euclidean s) u
11510 ==> open_in (subtopology euclidean t) (IMAGE f u))`,
11511 REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
11512 [MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN ASM_MESON_TAC[];
11513 MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN
11514 ASM_REWRITE_TAC[]]);;
11516 let HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ = prove
11517 (`!f:real^M->real^N s t.
11518 f continuous_on s /\ IMAGE f s = t /\
11519 (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
11520 ==> ((?g. homeomorphism (s,t) (f,g)) <=>
11521 !u. closed_in (subtopology euclidean s) u
11522 ==> closed_in (subtopology euclidean t) (IMAGE f u))`,
11523 REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
11524 [MATCH_MP_TAC HOMEOMORPHISM_IMP_CLOSED_MAP THEN ASM_MESON_TAC[];
11525 MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_CLOSED_MAP THEN
11526 ASM_REWRITE_TAC[]]);;
11528 let INJECTIVE_MAP_OPEN_IFF_CLOSED = prove
11529 (`!f:real^M->real^N s t.
11530 f continuous_on s /\ IMAGE f s = t /\
11531 (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y)
11532 ==> ((!u. open_in (subtopology euclidean s) u
11533 ==> open_in (subtopology euclidean t) (IMAGE f u)) <=>
11534 (!u. closed_in (subtopology euclidean s) u
11535 ==> closed_in (subtopology euclidean t) (IMAGE f u)))`,
11536 REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
11537 EXISTS_TAC `?g:real^N->real^M. homeomorphism (s,t) (f,g)` THEN
11539 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ;
11540 MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ] THEN
11541 ASM_REWRITE_TAC[]);;
11543 (* ------------------------------------------------------------------------- *)
11544 (* Relatively weak hypotheses if the domain of the function is compact. *)
11545 (* ------------------------------------------------------------------------- *)
11547 let CONTINUOUS_IMP_CLOSED_MAP = prove
11548 (`!f:real^M->real^N s t.
11549 f continuous_on s /\ IMAGE f s = t /\ compact s
11550 ==> !u. closed_in (subtopology euclidean s) u
11551 ==> closed_in (subtopology euclidean t) (IMAGE f u)`,
11552 SIMP_TAC[CLOSED_IN_CLOSED_EQ; COMPACT_IMP_CLOSED] THEN
11553 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_SUBSET THEN
11554 EXPAND_TAC "t" THEN ASM_SIMP_TAC[IMAGE_SUBSET] THEN
11555 MATCH_MP_TAC COMPACT_IMP_CLOSED THEN
11556 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[] THEN
11557 ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_IN_CLOSED_TRANS;
11558 BOUNDED_SUBSET; CONTINUOUS_ON_SUBSET]);;
11560 let CONTINUOUS_IMP_QUOTIENT_MAP = prove
11561 (`!f:real^M->real^N s t.
11562 f continuous_on s /\ IMAGE f s = t /\ compact s
11564 ==> (open_in (subtopology euclidean s)
11565 {x | x IN s /\ f x IN u} <=>
11566 open_in (subtopology euclidean t) u)`,
11567 REPEAT GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
11568 MATCH_MP_TAC CLOSED_MAP_IMP_QUOTIENT_MAP THEN
11569 ASM_REWRITE_TAC[] THEN
11570 MATCH_MP_TAC CONTINUOUS_IMP_CLOSED_MAP THEN
11571 ASM_REWRITE_TAC[]);;
11573 let CONTINUOUS_ON_INVERSE = prove
11574 (`!f:real^M->real^N g s.
11575 f continuous_on s /\ compact s /\ (!x. x IN s ==> (g(f(x)) = x))
11576 ==> g continuous_on (IMAGE f s)`,
11577 REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_ON_CLOSED] THEN
11578 SUBGOAL_THEN `IMAGE g (IMAGE (f:real^M->real^N) s) = s` SUBST1_TAC THENL
11579 [REWRITE_TAC[EXTENSION; IN_IMAGE] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
11580 X_GEN_TAC `t:real^M->bool` THEN DISCH_TAC THEN
11581 REWRITE_TAC[CLOSED_IN_CLOSED] THEN
11582 EXISTS_TAC `IMAGE (f:real^M->real^N) t` THEN CONJ_TAC THENL
11583 [MATCH_MP_TAC COMPACT_IMP_CLOSED THEN
11584 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
11585 FIRST_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET) THEN
11586 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
11587 ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_IN_CLOSED_TRANS;
11588 BOUNDED_SUBSET; CONTINUOUS_ON_SUBSET];
11589 REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; IN_IMAGE] THEN
11590 ASM_MESON_TAC[CLOSED_IN_SUBSET; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY; SUBSET]]);;
11592 let HOMEOMORPHISM_COMPACT = prove
11593 (`!s f t. compact s /\ f continuous_on s /\ (IMAGE f s = t) /\
11594 (!x y. x IN s /\ y IN s /\ (f x = f y) ==> (x = y))
11595 ==> ?g. homeomorphism(s,t) (f,g)`,
11596 REWRITE_TAC[INJECTIVE_ON_LEFT_INVERSE] THEN REPEAT GEN_TAC THEN
11597 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
11598 MATCH_MP_TAC MONO_EXISTS THEN ASM_SIMP_TAC[EXTENSION; homeomorphism] THEN
11599 FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
11600 ASM_MESON_TAC[CONTINUOUS_ON_INVERSE; IN_IMAGE]);;
11602 let HOMEOMORPHIC_COMPACT = prove
11603 (`!s f t. compact s /\ f continuous_on s /\ (IMAGE f s = t) /\
11604 (!x y. x IN s /\ y IN s /\ (f x = f y) ==> (x = y))
11605 ==> s homeomorphic t`,
11606 REWRITE_TAC[homeomorphic] THEN MESON_TAC[HOMEOMORPHISM_COMPACT]);;
11608 (* ------------------------------------------------------------------------- *)
11609 (* Lemmas about composition of homeomorphisms. *)
11610 (* ------------------------------------------------------------------------- *)
11612 let HOMEOMORPHISM_FROM_COMPOSITION_SURJECTIVE = prove
11613 (`!f:real^M->real^N g:real^N->real^P s t u.
11614 f continuous_on s /\ IMAGE f s = t /\
11615 g continuous_on t /\ IMAGE g t SUBSET u /\
11616 (?h. homeomorphism (s,u) (g o f,h))
11617 ==> (?f'. homeomorphism (s,t) (f,f')) /\
11618 (?g'. homeomorphism (t,u) (g,g'))`,
11619 REPEAT GEN_TAC THEN STRIP_TAC THEN
11620 RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism; o_THM]) THEN
11621 MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL
11622 [MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN
11623 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
11624 MATCH_MP_TAC OPEN_MAP_FROM_COMPOSITION_SURJECTIVE THEN
11625 MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `s:real^M->bool`] THEN
11626 ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
11627 MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN
11628 MAP_EVERY EXISTS_TAC [`h:real^P->real^M`; `s:real^M->bool`] THEN
11629 ASM_REWRITE_TAC[homeomorphism; o_THM];
11630 REWRITE_TAC[homeomorphism; o_THM] THEN
11631 DISCH_THEN(X_CHOOSE_THEN `g':real^P->real^N` STRIP_ASSUME_TAC) THEN
11632 EXISTS_TAC `(h:real^P->real^M) o (g:real^N->real^P)` THEN
11633 ASM_SIMP_TAC[o_THM; IMAGE_o] THEN
11634 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
11635 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
11636 ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]]);;
11638 let HOMEOMORPHISM_FROM_COMPOSITION_INJECTIVE = prove
11639 (`!f:real^M->real^N g:real^N->real^P s t u.
11640 f continuous_on s /\ IMAGE f s SUBSET t /\
11641 g continuous_on t /\ IMAGE g t SUBSET u /\
11642 (!x y. x IN t /\ y IN t /\ g x = g y ==> x = y) /\
11643 (?h. homeomorphism (s,u) (g o f,h))
11644 ==> (?f'. homeomorphism (s,t) (f,f')) /\
11645 (?g'. homeomorphism (t,u) (g,g'))`,
11646 REPEAT GEN_TAC THEN STRIP_TAC THEN
11647 RULE_ASSUM_TAC(REWRITE_RULE[homeomorphism; o_THM]) THEN
11648 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
11649 [MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN
11650 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
11651 MATCH_MP_TAC OPEN_MAP_FROM_COMPOSITION_INJECTIVE THEN
11652 MAP_EVERY EXISTS_TAC [`g:real^N->real^P`; `u:real^P->bool`] THEN
11653 ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
11654 MATCH_MP_TAC HOMEOMORPHISM_IMP_OPEN_MAP THEN
11655 MAP_EVERY EXISTS_TAC [`h:real^P->real^M`; `s:real^M->bool`] THEN
11656 ASM_REWRITE_TAC[homeomorphism; o_THM];
11657 REWRITE_TAC[homeomorphism; o_THM] THEN
11658 DISCH_THEN(X_CHOOSE_THEN `f':real^N->real^M` STRIP_ASSUME_TAC) THEN
11659 EXISTS_TAC `(f:real^M->real^N) o (h:real^P->real^M)` THEN
11660 ASM_SIMP_TAC[o_THM; IMAGE_o] THEN
11661 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
11662 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
11663 ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]]);;
11665 (* ------------------------------------------------------------------------- *)
11666 (* Preservation of topological properties. *)
11667 (* ------------------------------------------------------------------------- *)
11669 let HOMEOMORPHIC_COMPACTNESS = prove
11670 (`!s t. s homeomorphic t ==> (compact s <=> compact t)`,
11671 REWRITE_TAC[homeomorphic; homeomorphism] THEN
11672 MESON_TAC[COMPACT_CONTINUOUS_IMAGE]);;
11674 let HOMEOMORPHIC_CONNECTEDNESS = prove
11675 (`!s t. s homeomorphic t ==> (connected s <=> connected t)`,
11676 REWRITE_TAC[homeomorphic; homeomorphism] THEN
11677 MESON_TAC[CONNECTED_CONTINUOUS_IMAGE]);;
11679 (* ------------------------------------------------------------------------- *)
11680 (* Results on translation, scaling etc. *)
11681 (* ------------------------------------------------------------------------- *)
11683 let HOMEOMORPHIC_SCALING = prove
11684 (`!s:real^N->bool c. ~(c = &0) ==> s homeomorphic (IMAGE (\x. c % x) s)`,
11685 REPEAT STRIP_TAC THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11686 MAP_EVERY EXISTS_TAC [`\x:real^N. c % x`; `\x:real^N. inv(c) % x`] THEN
11687 ASM_SIMP_TAC[CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID; FORALL_IN_IMAGE] THEN
11688 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_MUL_RINV] THEN
11689 SIMP_TAC[VECTOR_MUL_LID; IN_IMAGE; REAL_MUL_LID] THEN MESON_TAC[]);;
11691 let HOMEOMORPHIC_TRANSLATION = prove
11692 (`!s a:real^N. s homeomorphic (IMAGE (\x. a + x) s)`,
11693 REPEAT STRIP_TAC THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11694 MAP_EVERY EXISTS_TAC [`\x:real^N. a + x`; `\x:real^N. --a + x`] THEN
11695 ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN
11696 SIMP_TAC[VECTOR_ADD_ASSOC; VECTOR_ADD_LINV; VECTOR_ADD_RINV;
11697 FORALL_IN_IMAGE; VECTOR_ADD_LID] THEN
11698 REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[]);;
11700 let HOMEOMORPHIC_AFFINITY = prove
11701 (`!s a:real^N c. ~(c = &0) ==> s homeomorphic (IMAGE (\x. a + c % x) s)`,
11702 REPEAT STRIP_TAC THEN
11703 MATCH_MP_TAC HOMEOMORPHIC_TRANS THEN
11704 EXISTS_TAC `IMAGE (\x:real^N. c % x) s` THEN
11705 ASM_SIMP_TAC[HOMEOMORPHIC_SCALING] THEN
11706 SUBGOAL_THEN `(\x:real^N. a + c % x) = (\x. a + x) o (\x. c % x)`
11707 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
11708 REWRITE_TAC[IMAGE_o; HOMEOMORPHIC_TRANSLATION]);;
11710 let [HOMEOMORPHIC_BALLS; HOMEOMORPHIC_CBALLS; HOMEOMORPHIC_SPHERES] =
11711 (CONJUNCTS o prove)
11712 (`(!a:real^N b:real^N d e.
11713 &0 < d /\ &0 < e ==> ball(a,d) homeomorphic ball(b,e)) /\
11714 (!a:real^N b:real^N d e.
11715 &0 < d /\ &0 < e ==> cball(a,d) homeomorphic cball(b,e)) /\
11716 (!a:real^N b:real^N d e.
11717 &0 < d /\ &0 < e ==> sphere(a,d) homeomorphic sphere(b,e))`,
11718 REPEAT STRIP_TAC THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11719 EXISTS_TAC `\x:real^N. b + (e / d) % (x - a)` THEN
11720 EXISTS_TAC `\x:real^N. a + (d / e) % (x - b)` THEN
11721 ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_SUB; CONTINUOUS_ON_CMUL;
11722 CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID; IN_BALL; IN_CBALL; IN_SPHERE] THEN
11723 REWRITE_TAC[dist; VECTOR_ARITH `a - (a + b) = --b:real^N`; NORM_NEG] THEN
11724 REWRITE_TAC[real_div; VECTOR_ARITH
11725 `a + d % ((b + e % (x - a)) - b) = (&1 - d * e) % a + (d * e) % x`] THEN
11726 ONCE_REWRITE_TAC[REAL_ARITH
11727 `(e * d') * (d * e') = (d * d') * (e * e')`] THEN
11728 ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_MUL_LID; REAL_SUB_REFL] THEN
11729 REWRITE_TAC[NORM_MUL; VECTOR_MUL_LZERO; VECTOR_MUL_LID; VECTOR_ADD_LID] THEN
11730 ASM_SIMP_TAC[REAL_ABS_MUL; REAL_ABS_INV; REAL_ARITH
11731 `&0 < x ==> (abs x = x)`] THEN
11732 GEN_REWRITE_TAC(BINOP_CONV o BINDER_CONV o funpow 2 RAND_CONV)
11733 [GSYM REAL_MUL_RID] THEN
11734 ONCE_REWRITE_TAC[AC REAL_MUL_AC `(a * b) * c = (a * c) * b`] THEN
11735 ASM_SIMP_TAC[REAL_LE_LMUL_EQ; GSYM real_div; REAL_LE_LDIV_EQ; REAL_MUL_LID;
11736 GSYM REAL_MUL_ASSOC; REAL_LT_LMUL_EQ; REAL_LT_LDIV_EQ; NORM_SUB] THEN
11737 ASM_SIMP_TAC[REAL_DIV_REFL; REAL_LT_IMP_NZ; REAL_MUL_RID]);;
11739 (* ------------------------------------------------------------------------- *)
11740 (* Homeomorphism of one-point compactifications. *)
11741 (* ------------------------------------------------------------------------- *)
11743 let HOMEOMORPHIC_ONE_POINT_COMPACTIFICATIONS = prove
11744 (`!s:real^M->bool t:real^N->bool a b.
11745 compact s /\ compact t /\ a IN s /\ b IN t /\
11746 (s DELETE a) homeomorphic (t DELETE b)
11747 ==> s homeomorphic t`,
11748 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHIC_COMPACT THEN
11749 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [homeomorphic]) THEN
11750 REWRITE_TAC[HOMEOMORPHISM; LEFT_IMP_EXISTS_THM] THEN
11751 MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN
11753 EXISTS_TAC `\x. if x = a then b else (f:real^M->real^N) x` THEN
11754 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
11755 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
11756 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
11757 ASM_CASES_TAC `x:real^M = a` THEN ASM_REWRITE_TAC[] THENL
11758 [REWRITE_TAC[continuous_within] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
11759 MP_TAC(ISPECL [`b:real^N`; `e:real`] CENTRE_IN_BALL) THEN
11760 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
11762 `closed_in (subtopology euclidean s)
11763 { x | x IN (s DELETE a) /\
11764 (f:real^M->real^N)(x) IN t DIFF ball(b,e)}`
11766 [MATCH_MP_TAC CLOSED_SUBSET THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
11767 MATCH_MP_TAC COMPACT_IMP_CLOSED THEN SUBGOAL_THEN
11768 `{x | x IN s DELETE a /\ f x IN t DIFF ball(b,e)} =
11769 IMAGE (g:real^N->real^M) (t DIFF ball (b,e))`
11770 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
11771 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
11772 ASM_SIMP_TAC[COMPACT_DIFF; OPEN_BALL] THEN
11773 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
11774 CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[];
11775 REWRITE_TAC[closed_in; open_in; TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN
11776 DISCH_THEN(MP_TAC o SPEC `a:real^M` o last o CONJUNCTS) THEN
11777 ASM_REWRITE_TAC[IN_ELIM_THM; IN_DIFF; IN_DELETE] THEN
11778 SIMP_TAC[IMP_CONJ; DE_MORGAN_THM] THEN
11779 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
11780 ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN
11781 ASM_REWRITE_TAC[DIST_REFL] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
11782 RULE_ASSUM_TAC(REWRITE_RULE[IN_BALL]) THEN ASM SET_TAC[]];
11783 UNDISCH_TAC `(f:real^M->real^N) continuous_on (s DELETE a)` THEN
11784 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
11785 DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[IN_DELETE] THEN
11786 REWRITE_TAC[continuous_within] THEN
11787 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
11788 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[IN_DELETE] THEN
11789 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
11790 EXISTS_TAC `min d (dist(a:real^M,x))` THEN
11791 ASM_REWRITE_TAC[REAL_LT_MIN; GSYM DIST_NZ] THEN
11792 ASM_MESON_TAC[REAL_LT_REFL]]);;
11794 (* ------------------------------------------------------------------------- *)
11795 (* Homeomorphisms between open intervals in real^1 and then in real^N. *)
11796 (* Could prove similar things for closed intervals, but they drop out of *)
11797 (* later stuff in "convex.ml" even more easily. *)
11798 (* ------------------------------------------------------------------------- *)
11800 let HOMEOMORPHIC_OPEN_INTERVALS_1 = prove
11802 drop a < drop b /\ drop c < drop d
11803 ==> interval(a,b) homeomorphic interval(c,d)`,
11805 `!a b. drop a < drop b
11806 ==> interval(vec 0:real^1,vec 1) homeomorphic interval(a,b)`
11808 [REPEAT STRIP_TAC THEN REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11809 EXISTS_TAC `(\x. a + drop x % (b - a)):real^1->real^1` THEN
11810 EXISTS_TAC `(\x. inv(drop b - drop a) % (x - a)):real^1->real^1` THEN
11811 ASM_REWRITE_TAC[IN_INTERVAL_1; GSYM DROP_EQ] THEN
11812 REWRITE_TAC[DROP_ADD; DROP_CMUL; DROP_NEG; DROP_VEC; DROP_SUB] THEN
11813 REWRITE_TAC[REAL_ARITH `inv b * a:real = a / b`] THEN
11814 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ; REAL_SUB_LT;
11815 REAL_LT_ADDR; REAL_EQ_LDIV_EQ; REAL_DIV_RMUL; REAL_LT_IMP_NZ;
11816 REAL_LT_MUL; REAL_MUL_LZERO; REAL_ADD_SUB; REAL_LT_RMUL_EQ;
11817 REAL_ARITH `a + x < b <=> x < &1 * (b - a)`] THEN
11818 REPEAT CONJ_TAC THENL
11820 MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
11821 MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN
11822 REWRITE_TAC[o_DEF; LIFT_DROP; CONTINUOUS_ON_ID];
11823 MATCH_MP_TAC CONTINUOUS_ON_CMUL THEN
11824 ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]];
11825 REPEAT STRIP_TAC THEN
11826 FIRST_ASSUM(MP_TAC o SPECL [`a:real^1`; `b:real^1`]) THEN
11827 FIRST_X_ASSUM(MP_TAC o SPECL [`c:real^1`; `d:real^1`]) THEN
11828 ASM_REWRITE_TAC[GSYM IMP_CONJ_ALT] THEN
11829 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [HOMEOMORPHIC_SYM] THEN
11830 REWRITE_TAC[HOMEOMORPHIC_TRANS]]);;
11832 let HOMEOMORPHIC_OPEN_INTERVAL_UNIV_1 = prove
11833 (`!a b. drop a < drop b ==> interval(a,b) homeomorphic (:real^1)`,
11834 REPEAT STRIP_TAC THEN
11835 MP_TAC(SPECL [`a:real^1`; `b:real^1`; `--vec 1:real^1`; `vec 1:real^1`]
11836 HOMEOMORPHIC_OPEN_INTERVALS_1) THEN
11837 ASM_REWRITE_TAC[DROP_VEC; DROP_NEG] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
11838 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] HOMEOMORPHIC_TRANS) THEN
11839 POP_ASSUM_LIST(K ALL_TAC) THEN
11840 REWRITE_TAC[HOMEOMORPHIC_MINIMAL; IN_UNIV] THEN
11841 EXISTS_TAC `\x:real^1. inv(&1 - norm x) % x` THEN
11842 EXISTS_TAC `\y. if &0 <= drop y then inv(&1 + drop y) % y
11843 else inv(&1 - drop y) % y` THEN
11844 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
11845 [X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1] THEN
11846 REWRITE_TAC[DROP_NEG; DROP_VEC; DROP_CMUL; NORM_REAL; GSYM drop] THEN
11847 SIMP_TAC[REAL_LE_MUL_EQ; REAL_LT_INV_EQ; REAL_LE_MUL_EQ; REAL_ARITH
11848 `--a < x /\ x < a ==> &0 < a - abs x`] THEN
11849 SIMP_TAC[real_abs; VECTOR_MUL_ASSOC] THEN
11850 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
11851 GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN
11852 AP_THM_TAC THEN AP_TERM_TAC THEN
11853 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
11854 X_GEN_TAC `y:real^1` THEN COND_CASES_TAC THEN
11855 ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; DROP_VEC; REAL_BOUNDS_LT] THEN
11856 REWRITE_TAC[DROP_CMUL; REAL_ABS_MUL; REAL_ABS_INV] THEN
11857 REWRITE_TAC[GSYM(ONCE_REWRITE_RULE[REAL_MUL_SYM] real_div)] THEN
11858 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_ARITH `&0 <= x ==> &0 < abs(&1 + x)`;
11859 REAL_ARITH `~(&0 <= x) ==> &0 < abs(&1 - x)`] THEN
11860 (CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
11861 REWRITE_TAC[NORM_REAL; VECTOR_MUL_ASSOC] THEN
11862 REWRITE_TAC[GSYM drop; DROP_CMUL; REAL_ABS_MUL] THEN
11863 ASM_REWRITE_TAC[real_abs; REAL_LE_INV_EQ] THEN
11864 ASM_SIMP_TAC[REAL_ARITH `&0 <= x ==> &0 <= &1 + x`;
11865 REAL_ARITH `~(&0 <= x) ==> &0 <= &1 - x`] THEN
11866 GEN_REWRITE_TAC RAND_CONV [GSYM VECTOR_MUL_LID] THEN
11867 AP_THM_TAC THEN AP_TERM_TAC THEN
11868 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD;
11869 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
11870 X_GEN_TAC `x:real^1` THEN
11871 REWRITE_TAC[IN_INTERVAL_1; DROP_NEG; DROP_VEC] THEN
11872 DISCH_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
11873 REWRITE_TAC[CONTINUOUS_AT_ID] THEN
11874 ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_INV THEN
11875 REWRITE_TAC[NETLIMIT_AT; o_DEF; LIFT_SUB; LIFT_DROP] THEN
11877 [MATCH_MP_TAC CONTINUOUS_SUB THEN
11878 SIMP_TAC[CONTINUOUS_CONST; REWRITE_RULE[o_DEF] CONTINUOUS_AT_LIFT_NORM];
11879 REWRITE_TAC[NORM_REAL; GSYM drop] THEN ASM_REAL_ARITH_TAC];
11880 SUBGOAL_THEN `(:real^1) = {x | x$1 >= &0} UNION {x | x$1 <= &0}`
11882 [REWRITE_TAC[EXTENSION; IN_UNION; IN_UNION; IN_ELIM_THM; IN_UNIV] THEN
11884 MATCH_MP_TAC CONTINUOUS_ON_CASES THEN
11885 REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE;
11887 REWRITE_TAC[GSYM drop; REAL_NOT_LE; real_ge; REAL_LET_ANTISYM] THEN
11888 SIMP_TAC[REAL_LE_ANTISYM; REAL_SUB_RZERO; REAL_ADD_RID] THEN
11889 CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
11890 X_GEN_TAC `y:real^1` THEN REWRITE_TAC[IN_ELIM_THM; real_ge] THEN
11891 DISCH_TAC THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
11892 REWRITE_TAC[CONTINUOUS_AT_ID] THEN
11893 ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_INV THEN
11894 REWRITE_TAC[NETLIMIT_AT; o_DEF; LIFT_ADD; LIFT_SUB; LIFT_DROP] THEN
11895 ASM_SIMP_TAC[CONTINUOUS_ADD; CONTINUOUS_AT_ID; CONTINUOUS_SUB;
11896 CONTINUOUS_CONST] THEN
11897 ASM_REAL_ARITH_TAC]]);;
11899 let HOMEOMORPHIC_OPEN_INTERVALS = prove
11900 (`!a b:real^N c d:real^N.
11901 (interval(a,b) = {} <=> interval(c,d) = {})
11902 ==> interval(a,b) homeomorphic interval(c,d)`,
11903 REPEAT GEN_TAC THEN ASM_CASES_TAC `interval(c:real^N,d) = {}` THEN
11904 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN ASM_REWRITE_TAC[HOMEOMORPHIC_REFL] THEN
11906 `!i. 1 <= i /\ i <= dimindex(:N)
11907 ==> interval(lift((a:real^N)$i),lift((b:real^N)$i)) homeomorphic
11908 interval(lift((c:real^N)$i),lift((d:real^N)$i))`
11910 [RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN
11911 ASM_SIMP_TAC[HOMEOMORPHIC_OPEN_INTERVALS_1; LIFT_DROP];
11913 REWRITE_TAC[HOMEOMORPHIC_MINIMAL; IN_INTERVAL_1; LIFT_DROP] THEN
11914 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
11915 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
11916 MAP_EVERY X_GEN_TAC [`f:num->real^1->real^1`; `g:num->real^1->real^1`] THEN
11920 drop((f:num->real^1->real^1) i (lift(x$i)))):real^N->real^N` THEN
11923 drop((g:num->real^1->real^1) i (lift(x$i)))):real^N->real^N` THEN
11924 ASM_SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; CART_EQ; LIFT_DROP] THEN
11925 ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN
11926 SIMP_TAC[LAMBDA_BETA; LIFT_DROP] THEN CONJ_TAC THEN REPEAT STRIP_TAC THEN
11927 ONCE_REWRITE_TAC[GSYM o_DEF] THEN
11928 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
11929 ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT] THEN
11930 MATCH_MP_TAC CONTINUOUS_ON_SUBSET THENL
11931 [EXISTS_TAC `interval(lift((a:real^N)$i),lift((b:real^N)$i))`;
11932 EXISTS_TAC `interval(lift((c:real^N)$i),lift((d:real^N)$i))`] THEN
11933 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1] THEN
11934 ASM_SIMP_TAC[LIFT_DROP; IN_INTERVAL]);;
11936 let HOMEOMORPHIC_OPEN_INTERVAL_UNIV = prove
11938 ~(interval(a,b) = {})
11939 ==> interval(a,b) homeomorphic (:real^N)`,
11940 REPEAT STRIP_TAC THEN
11942 `!i. 1 <= i /\ i <= dimindex(:N)
11943 ==> interval(lift((a:real^N)$i),lift((b:real^N)$i)) homeomorphic
11946 [RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN
11947 ASM_SIMP_TAC[HOMEOMORPHIC_OPEN_INTERVAL_UNIV_1; LIFT_DROP];
11949 REWRITE_TAC[HOMEOMORPHIC_MINIMAL; IN_INTERVAL_1; LIFT_DROP] THEN
11950 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
11951 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; IN_UNIV] THEN
11952 MAP_EVERY X_GEN_TAC [`f:num->real^1->real^1`; `g:num->real^1->real^1`] THEN
11956 drop((f:num->real^1->real^1) i (lift(x$i)))):real^N->real^N` THEN
11959 drop((g:num->real^1->real^1) i (lift(x$i)))):real^N->real^N` THEN
11960 ASM_SIMP_TAC[IN_INTERVAL; LAMBDA_BETA; CART_EQ; LIFT_DROP; IN_UNIV] THEN
11961 ONCE_REWRITE_TAC[CONTINUOUS_ON_COMPONENTWISE_LIFT] THEN
11962 SIMP_TAC[LAMBDA_BETA; LIFT_DROP] THEN CONJ_TAC THEN REPEAT STRIP_TAC THEN
11963 ONCE_REWRITE_TAC[GSYM o_DEF] THEN
11964 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
11965 ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_COMPONENT] THEN
11966 MATCH_MP_TAC CONTINUOUS_ON_SUBSET THENL
11967 [EXISTS_TAC `interval(lift((a:real^N)$i),lift((b:real^N)$i))`;
11968 EXISTS_TAC `(:real^1)`] THEN
11969 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; IN_UNIV] THEN
11970 ASM_SIMP_TAC[LIFT_DROP; IN_INTERVAL]);;
11972 let HOMEOMORPHIC_BALL_UNIV = prove
11973 (`!a:real^N r. &0 < r ==> ball(a,r) homeomorphic (:real^N)`,
11974 REPEAT GEN_TAC THEN GEOM_ORIGIN_TAC `a:real^N` THEN REPEAT STRIP_TAC THEN
11975 SUBGOAL_THEN `?y:real^N. r = norm(y)` (CHOOSE_THEN SUBST_ALL_TAC) THENL
11976 [ASM_MESON_TAC[VECTOR_CHOOSE_SIZE; REAL_LT_IMP_LE]; POP_ASSUM MP_TAC] THEN
11977 REWRITE_TAC[NORM_POS_LT] THEN GEOM_NORMALIZE_TAC `y:real^N` THEN
11978 SIMP_TAC[] THEN GEN_TAC THEN REPEAT(DISCH_THEN(K ALL_TAC)) THEN
11979 REWRITE_TAC[HOMEOMORPHIC_MINIMAL] THEN
11980 EXISTS_TAC `\z:real^N. inv(&1 - norm(z)) % z` THEN
11981 EXISTS_TAC `\z:real^N. inv(&1 + norm(z)) % z` THEN
11982 REWRITE_TAC[IN_BALL; IN_UNIV; DIST_0; VECTOR_MUL_ASSOC; VECTOR_MUL_EQ_0;
11983 VECTOR_ARITH `a % x:real^N = x <=> (a - &1) % x = vec 0`] THEN
11984 REPEAT CONJ_TAC THENL
11985 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN DISJ1_TAC THEN
11986 REWRITE_TAC[GSYM REAL_INV_MUL; REAL_SUB_0; REAL_INV_EQ_1] THEN
11987 REWRITE_TAC[NORM_MUL; REAL_ABS_INV] THEN
11988 ASM_SIMP_TAC[REAL_ARITH `x < &1 ==> abs(&1 - x) = &1 - x`] THEN
11989 POP_ASSUM MP_TAC THEN CONV_TAC REAL_FIELD;
11990 X_GEN_TAC `y:real^N` THEN REWRITE_TAC[NORM_MUL; REAL_ABS_INV] THEN
11991 ASM_SIMP_TAC[NORM_POS_LE; REAL_ARITH
11992 `&0 <= y ==> inv(abs(&1 + y)) * z = z / (&1 + y)`] THEN
11993 ASM_SIMP_TAC[NORM_POS_LE; REAL_LT_LDIV_EQ; REAL_ARITH
11994 `&0 <= y ==> &0 < &1 + y`] THEN
11995 CONJ_TAC THENL [REAL_ARITH_TAC; DISJ1_TAC] THEN
11996 REWRITE_TAC[GSYM REAL_INV_MUL; REAL_SUB_0; REAL_INV_EQ_1] THEN
11997 MP_TAC(ISPEC `y:real^N` NORM_POS_LE) THEN CONV_TAC REAL_FIELD;
11998 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN
11999 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
12000 MATCH_MP_TAC CONTINUOUS_ON_INV THEN
12001 SIMP_TAC[IN_BALL_0; REAL_SUB_0; REAL_ARITH `x < &1 ==> ~(&1 = x)`] THEN
12002 REWRITE_TAC[o_DEF; LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN
12003 REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
12004 MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN
12005 REWRITE_TAC[CONTINUOUS_ON_ID];
12006 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN REWRITE_TAC[CONTINUOUS_ON_ID] THEN
12007 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
12008 MATCH_MP_TAC CONTINUOUS_ON_INV THEN
12009 SIMP_TAC[NORM_POS_LE; REAL_ARITH `&0 <= x ==> ~(&1 + x = &0)`] THEN
12010 REWRITE_TAC[o_DEF; LIFT_ADD] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN
12011 REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
12012 MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN
12013 REWRITE_TAC[CONTINUOUS_ON_ID]]);;
12015 (* ------------------------------------------------------------------------- *)
12016 (* Cardinalities of various useful sets. *)
12017 (* ------------------------------------------------------------------------- *)
12019 let CARD_EQ_EUCLIDEAN = prove
12020 (`(:real^N) =_c (:real)`,
12021 MATCH_MP_TAC CARD_EQ_CART THEN REWRITE_TAC[real_INFINITE]);;
12023 let UNCOUNTABLE_EUCLIDEAN = prove
12024 (`~COUNTABLE(:real^N)`,
12025 MATCH_MP_TAC CARD_EQ_REAL_IMP_UNCOUNTABLE THEN
12026 REWRITE_TAC[CARD_EQ_EUCLIDEAN]);;
12028 let CARD_EQ_INTERVAL = prove
12029 (`(!a b:real^N. ~(interval(a,b) = {}) ==> interval[a,b] =_c (:real)) /\
12030 (!a b:real^N. ~(interval(a,b) = {}) ==> interval(a,b) =_c (:real))`,
12031 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN
12032 ASM_CASES_TAC `interval(a:real^N,b) = {}` THEN ASM_REWRITE_TAC[] THEN
12033 CONJ_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL
12034 [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN
12035 REWRITE_TAC[CARD_LE_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN
12036 REWRITE_TAC[CARD_EQ_EUCLIDEAN];
12037 TRANS_TAC CARD_LE_TRANS `interval(a:real^N,b)` THEN
12038 SIMP_TAC[CARD_LE_SUBSET; INTERVAL_OPEN_SUBSET_CLOSED];
12039 TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN
12040 REWRITE_TAC[CARD_LE_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN
12041 REWRITE_TAC[CARD_EQ_EUCLIDEAN];
12043 TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN
12044 SIMP_TAC[ONCE_REWRITE_RULE[CARD_EQ_SYM] CARD_EQ_IMP_LE;
12045 CARD_EQ_EUCLIDEAN] THEN
12046 FIRST_X_ASSUM(MP_TAC o MATCH_MP HOMEOMORPHIC_OPEN_INTERVAL_UNIV) THEN
12047 DISCH_THEN(MP_TAC o MATCH_MP HOMEOMORPHIC_IMP_CARD_EQ) THEN
12048 MESON_TAC[CARD_EQ_IMP_LE; CARD_EQ_SYM]);;
12050 let UNCOUNTABLE_INTERVAL = prove
12051 (`(!a b. ~(interval(a,b) = {}) ==> ~COUNTABLE(interval[a,b])) /\
12052 (!a b. ~(interval(a,b) = {}) ==> ~COUNTABLE(interval(a,b)))`,
12053 SIMP_TAC[CARD_EQ_REAL_IMP_UNCOUNTABLE; CARD_EQ_INTERVAL]);;
12055 let COUNTABLE_OPEN_INTERVAL = prove
12056 (`!a b. COUNTABLE(interval(a,b)) <=> interval(a,b) = {}`,
12057 MESON_TAC[COUNTABLE_EMPTY; UNCOUNTABLE_INTERVAL]);;
12059 let CARD_EQ_OPEN = prove
12060 (`!s:real^N->bool. open s /\ ~(s = {}) ==> s =_c (:real)`,
12061 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL
12062 [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN
12063 REWRITE_TAC[CARD_LE_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN
12064 REWRITE_TAC[CARD_EQ_EUCLIDEAN];
12065 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_INTERVAL]) THEN
12066 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
12067 DISCH_THEN(X_CHOOSE_TAC `c:real^N`) THEN
12068 DISCH_THEN(MP_TAC o SPEC `c:real^N`) THEN
12069 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
12070 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
12071 ASM_CASES_TAC `interval(a:real^N,b) = {}` THEN
12072 ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN STRIP_TAC THEN
12073 TRANS_TAC CARD_LE_TRANS `interval[a:real^N,b]` THEN
12074 ASM_SIMP_TAC[CARD_LE_SUBSET] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN
12075 ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN ASM_SIMP_TAC[CARD_EQ_INTERVAL]]);;
12077 let UNCOUNTABLE_OPEN = prove
12078 (`!s:real^N->bool. open s /\ ~(s = {}) ==> ~(COUNTABLE s)`,
12079 SIMP_TAC[CARD_EQ_OPEN; CARD_EQ_REAL_IMP_UNCOUNTABLE]);;
12081 let CARD_EQ_BALL = prove
12082 (`!a:real^N r. &0 < r ==> ball(a,r) =_c (:real)`,
12083 SIMP_TAC[CARD_EQ_OPEN; OPEN_BALL; BALL_EQ_EMPTY; GSYM REAL_NOT_LT]);;
12085 let CARD_EQ_CBALL = prove
12086 (`!a:real^N r. &0 < r ==> cball(a,r) =_c (:real)`,
12087 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL
12088 [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN
12089 REWRITE_TAC[CARD_LE_UNIV] THEN MATCH_MP_TAC CARD_EQ_IMP_LE THEN
12090 REWRITE_TAC[CARD_EQ_EUCLIDEAN];
12091 TRANS_TAC CARD_LE_TRANS `ball(a:real^N,r)` THEN
12092 SIMP_TAC[CARD_LE_SUBSET; BALL_SUBSET_CBALL] THEN
12093 MATCH_MP_TAC CARD_EQ_IMP_LE THEN
12094 ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN ASM_SIMP_TAC[CARD_EQ_BALL]]);;
12096 let FINITE_IMP_NOT_OPEN = prove
12097 (`!s:real^N->bool. FINITE s /\ ~(s = {}) ==> ~(open s)`,
12098 MESON_TAC[UNCOUNTABLE_OPEN; FINITE_IMP_COUNTABLE]);;
12100 let OPEN_IMP_INFINITE = prove
12101 (`!s. open s ==> s = {} \/ INFINITE s`,
12102 MESON_TAC[FINITE_IMP_NOT_OPEN; INFINITE]);;
12104 let EMPTY_INTERIOR_FINITE = prove
12105 (`!s:real^N->bool. FINITE s ==> interior s = {}`,
12106 REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` OPEN_INTERIOR) THEN
12107 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
12108 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] FINITE_IMP_NOT_OPEN) THEN
12109 MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN
12110 ASM_REWRITE_TAC[INTERIOR_SUBSET]);;
12112 let CARD_EQ_CONNECTED = prove
12114 connected s /\ a IN s /\ b IN s /\ ~(a = b) ==> s =_c (:real)`,
12115 GEOM_ORIGIN_TAC `b:real^N` THEN GEOM_NORMALIZE_TAC `a:real^N` THEN
12116 REWRITE_TAC[NORM_EQ_SQUARE; REAL_POS; REAL_POW_ONE] THEN
12117 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL
12118 [TRANS_TAC CARD_LE_TRANS `(:real^N)` THEN
12119 SIMP_TAC[CARD_LE_UNIV; CARD_EQ_EUCLIDEAN; CARD_EQ_IMP_LE];
12120 TRANS_TAC CARD_LE_TRANS `interval[vec 0:real^1,vec 1]` THEN CONJ_TAC THENL
12121 [MATCH_MP_TAC(ONCE_REWRITE_RULE[CARD_EQ_SYM] CARD_EQ_IMP_LE) THEN
12122 SIMP_TAC[UNIT_INTERVAL_NONEMPTY; CARD_EQ_INTERVAL];
12123 REWRITE_TAC[LE_C] THEN EXISTS_TAC `\x:real^N. lift(a dot x)` THEN
12124 SIMP_TAC[FORALL_LIFT; LIFT_EQ; IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN
12125 X_GEN_TAC `t:real` THEN STRIP_TAC THEN
12126 MATCH_MP_TAC CONNECTED_IVT_HYPERPLANE THEN
12127 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `a:real^N`] THEN
12128 ASM_REWRITE_TAC[DOT_RZERO]]]);;
12130 let UNCOUNTABLE_CONNECTED = prove
12132 connected s /\ a IN s /\ b IN s /\ ~(a = b) ==> ~COUNTABLE s`,
12133 REPEAT GEN_TAC THEN STRIP_TAC THEN
12134 MATCH_MP_TAC CARD_EQ_REAL_IMP_UNCOUNTABLE THEN
12135 MATCH_MP_TAC CARD_EQ_CONNECTED THEN
12138 let CARD_LT_IMP_DISCONNECTED = prove
12139 (`!s x:real^N. s <_c (:real) /\ x IN s ==> connected_component s x = {x}`,
12140 REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE
12141 `s = {a} <=> a IN s /\ !a b. a IN s /\ b IN s /\ ~(a = b) ==> F`] THEN
12142 REPEAT STRIP_TAC THEN REWRITE_TAC[IN] THEN
12143 ASM_REWRITE_TAC[CONNECTED_COMPONENT_REFL_EQ] THEN
12144 MP_TAC(ISPECL [`connected_component s (x:real^N)`; `a:real^N`; `b:real^N`]
12145 CARD_EQ_CONNECTED) THEN
12146 ASM_REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN
12147 DISCH_TAC THEN UNDISCH_TAC `(s:real^N->bool) <_c (:real)` THEN
12148 REWRITE_TAC[CARD_NOT_LT] THEN
12149 TRANS_TAC CARD_LE_TRANS `connected_component s (x:real^N)` THEN
12150 ASM_SIMP_TAC[ONCE_REWRITE_RULE[CARD_EQ_SYM] CARD_EQ_IMP_LE] THEN
12151 MATCH_MP_TAC CARD_LE_SUBSET THEN REWRITE_TAC[CONNECTED_COMPONENT_SUBSET]);;
12153 let COUNTABLE_IMP_DISCONNECTED = prove
12154 (`!s x:real^N. COUNTABLE s /\ x IN s ==> connected_component s x = {x}`,
12155 SIMP_TAC[CARD_LT_IMP_DISCONNECTED; COUNTABLE_IMP_CARD_LT_REAL]);;
12157 let CONNECTED_CARD_EQ_IFF_NONTRIVIAL = prove
12159 connected s ==> (s =_c (:real) <=> ~(?a. s SUBSET {a}))`,
12160 REPEAT STRIP_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
12161 [ALL_TAC; MATCH_MP_TAC CARD_EQ_CONNECTED THEN ASM SET_TAC[]] THEN
12162 FIRST_ASSUM(MP_TAC o MATCH_MP(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET)) THEN
12163 REWRITE_TAC[FINITE_SING] THEN
12164 ASM_MESON_TAC[CARD_EQ_REAL_IMP_UNCOUNTABLE; FINITE_IMP_COUNTABLE]);;
12166 (* ------------------------------------------------------------------------- *)
12167 (* "Iff" forms of constancy of function from connected set into a set that *)
12168 (* is smaller than R, or countable, or finite, or disconnected, or discrete. *)
12169 (* ------------------------------------------------------------------------- *)
12171 let [CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ;
12172 CONTINUOUS_DISCRETE_RANGE_CONSTANT_EQ;
12173 CONTINUOUS_FINITE_RANGE_CONSTANT_EQ] = (CONJUNCTS o prove)
12174 (`(!s. connected s <=>
12175 !f:real^M->real^N t.
12176 f continuous_on s /\ IMAGE f s SUBSET t /\
12177 (!y. y IN t ==> connected_component t y = {y})
12178 ==> ?a. !x. x IN s ==> f x = a) /\
12179 (!s. connected s <=>
12181 f continuous_on s /\
12184 !y. y IN s /\ ~(f y = f x) ==> e <= norm(f y - f x))
12185 ==> ?a. !x. x IN s ==> f x = a) /\
12186 (!s. connected s <=>
12188 f continuous_on s /\ FINITE(IMAGE f s)
12189 ==> ?a. !x. x IN s ==> f x = a)`,
12190 REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `s:real^M->bool` THEN
12192 `(s ==> t) /\ (t ==> u) /\ (u ==> v) /\ (v ==> s)
12193 ==> (s <=> t) /\ (s <=> u) /\ (s <=> v)`) THEN
12194 REPEAT CONJ_TAC THENL
12195 [REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^M->bool = {}` THEN
12196 ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
12197 FIRST_X_ASSUM(X_CHOOSE_TAC `x:real^M` o
12198 GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
12199 EXISTS_TAC `(f:real^M->real^N) x` THEN
12200 MATCH_MP_TAC(SET_RULE
12201 `IMAGE f s SUBSET {a} ==> !y. y IN s ==> f y = a`) THEN
12202 FIRST_X_ASSUM(MP_TAC o SPEC `(f:real^M->real^N) x`) THEN
12203 ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN
12204 MATCH_MP_TAC CONNECTED_COMPONENT_MAXIMAL THEN
12205 ASM_SIMP_TAC[CONNECTED_CONTINUOUS_IMAGE] THEN ASM SET_TAC[];
12206 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
12207 EXISTS_TAC `IMAGE (f:real^M->real^N) s` THEN
12208 ASM_REWRITE_TAC[FORALL_IN_IMAGE; SUBSET_REFL] THEN
12209 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
12210 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN
12211 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
12212 MATCH_MP_TAC(SET_RULE
12213 `(!y. y IN s /\ f y IN connected_component (IMAGE f s) a ==> f y = a) /\
12214 connected_component (IMAGE f s) a SUBSET (IMAGE f s) /\
12215 connected_component (IMAGE f s) a a
12216 ==> connected_component (IMAGE f s) a = {a}`) THEN
12217 REWRITE_TAC[CONNECTED_COMPONENT_SUBSET; CONNECTED_COMPONENT_REFL_EQ] THEN
12218 ASM_SIMP_TAC[FUN_IN_IMAGE] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
12219 MP_TAC(ISPEC `connected_component (IMAGE (f:real^M->real^N) s) (f x)`
12220 CONNECTED_CLOSED) THEN
12221 REWRITE_TAC[CONNECTED_CONNECTED_COMPONENT] THEN
12222 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
12223 ASM_REWRITE_TAC[] THEN MAP_EVERY EXISTS_TAC
12224 [`cball((f:real^M->real^N) x,e / &2)`;
12225 `(:real^N) DIFF ball((f:real^M->real^N) x,e)`] THEN
12226 REWRITE_TAC[GSYM OPEN_CLOSED; OPEN_BALL; CLOSED_CBALL] THEN
12227 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN REPEAT CONJ_TAC THENL
12228 [REWRITE_TAC[SUBSET; IN_CBALL; IN_UNION; IN_DIFF; IN_BALL; IN_UNIV] THEN
12229 MATCH_MP_TAC(MESON[SUBSET; CONNECTED_COMPONENT_SUBSET]
12230 `(!x. x IN s ==> P x)
12231 ==> (!x. x IN connected_component s y ==> P x)`) THEN
12232 REWRITE_TAC[FORALL_IN_IMAGE] THEN X_GEN_TAC `z:real^M` THEN
12233 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:real^M`) THEN
12234 ASM_REWRITE_TAC[] THEN CONV_TAC NORM_ARITH;
12235 MATCH_MP_TAC(SET_RULE
12236 `(!x. x IN s /\ x IN t ==> F) ==> s INTER t INTER u = {}`) THEN
12237 REWRITE_TAC[IN_BALL; IN_CBALL; IN_DIFF; IN_UNIV] THEN
12238 UNDISCH_TAC `&0 < e` THEN CONV_TAC NORM_ARITH;
12239 EXISTS_TAC `(f:real^M->real^N) x` THEN
12240 ASM_SIMP_TAC[CENTRE_IN_CBALL; REAL_HALF; REAL_LT_IMP_LE; IN_INTER] THEN
12241 REWRITE_TAC[IN] THEN
12242 ASM_SIMP_TAC[CONNECTED_COMPONENT_REFL_EQ; FUN_IN_IMAGE];
12243 EXISTS_TAC `(f:real^M->real^N) y` THEN
12244 ASM_REWRITE_TAC[IN_INTER; IN_DIFF; IN_UNIV; IN_BALL; REAL_NOT_LT] THEN
12245 ASM_SIMP_TAC[ONCE_REWRITE_RULE[DIST_SYM] dist]];
12246 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `f:real^M->real^N` THEN
12247 DISCH_THEN(fun th -> STRIP_TAC THEN MATCH_MP_TAC th) THEN
12248 ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
12249 ASM_CASES_TAC `IMAGE (f:real^M->real^N) s DELETE (f x) = {}` THENL
12250 [EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN ASM SET_TAC[];
12253 `inf{norm(z - f x) |z| z IN IMAGE (f:real^M->real^N) s DELETE (f x)}` THEN
12254 REWRITE_TAC[SIMPLE_IMAGE] THEN
12255 ASM_SIMP_TAC[REAL_LT_INF_FINITE; REAL_INF_LE_FINITE; FINITE_DELETE;
12256 FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
12257 REWRITE_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN
12258 REWRITE_TAC[IN_DELETE; NORM_POS_LT; VECTOR_SUB_EQ; IN_IMAGE] THEN
12259 MESON_TAC[REAL_LE_REFL];
12260 REWRITE_TAC[CONNECTED_CLOSED_IN_EQ] THEN
12261 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
12262 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
12263 MAP_EVERY X_GEN_TAC [`t:real^M->bool`; `u:real^M->bool`] THEN
12264 STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC
12265 `(\x. if x IN t then vec 0 else basis 1):real^M->real^N`) THEN
12266 REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL
12267 [EXPAND_TAC "s" THEN MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN
12268 ASM_REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ASM SET_TAC[];
12269 MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{vec 0:real^N,basis 1}` THEN
12270 REWRITE_TAC[FINITE_INSERT; FINITE_EMPTY] THEN SET_TAC[];
12271 SUBGOAL_THEN `?a b:real^M. a IN s /\ a IN t /\ b IN s /\ ~(b IN t)`
12272 STRIP_ASSUME_TAC THENL
12273 [ASM SET_TAC[]; DISCH_THEN(CHOOSE_THEN MP_TAC)] THEN
12274 DISCH_THEN(fun th -> MP_TAC(SPEC `a:real^M` th) THEN
12275 MP_TAC(SPEC `b:real^M` th)) THEN
12276 ASM_REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
12277 CONV_TAC(RAND_CONV SYM_CONV) THEN
12278 SIMP_TAC[BASIS_NONZERO; LE_REFL; DIMINDEX_GE_1; REAL_LE_REFL]]]);;
12280 let CONTINUOUS_DISCONNECTED_RANGE_CONSTANT = prove
12281 (`!f:real^M->real^N s.
12283 f continuous_on s /\ IMAGE f s SUBSET t /\
12284 (!y. y IN t ==> connected_component t y = {y})
12285 ==> ?a. !x. x IN s ==> f x = a`,
12286 MESON_TAC[CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ]);;
12288 let CONTINUOUS_DISCRETE_RANGE_CONSTANT = prove
12289 (`!f:real^M->real^N s.
12291 f continuous_on s /\
12294 !y. y IN s /\ ~(f y = f x) ==> e <= norm(f y - f x))
12295 ==> ?a. !x. x IN s ==> f x = a`,
12296 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
12297 REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
12298 REWRITE_TAC[IMP_IMP; GSYM CONTINUOUS_DISCRETE_RANGE_CONSTANT_EQ]);;
12300 let CONTINUOUS_FINITE_RANGE_CONSTANT = prove
12301 (`!f:real^M->real^N s.
12303 f continuous_on s /\
12305 ==> ?a. !x. x IN s ==> f x = a`,
12306 MESON_TAC[CONTINUOUS_FINITE_RANGE_CONSTANT_EQ]);;
12308 let CONTINUOUS_COUNTABLE_RANGE_CONSTANT_EQ = prove
12309 (`!s. connected s <=>
12311 f continuous_on s /\ COUNTABLE(IMAGE f s)
12312 ==> ?a. !x. x IN s ==> f x = a`,
12313 GEN_TAC THEN EQ_TAC THENL
12314 [REWRITE_TAC[CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ];
12315 REWRITE_TAC[CONTINUOUS_FINITE_RANGE_CONSTANT_EQ]] THEN
12316 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
12317 ASM_SIMP_TAC[FINITE_IMP_COUNTABLE] THEN
12318 EXISTS_TAC `IMAGE (f:real^M->real^N) s` THEN
12319 ASM_SIMP_TAC[COUNTABLE_IMP_DISCONNECTED; SUBSET_REFL]);;
12321 let CONTINUOUS_CARD_LT_RANGE_CONSTANT_EQ = prove
12322 (`!s. connected s <=>
12324 f continuous_on s /\ (IMAGE f s) <_c (:real)
12325 ==> ?a. !x. x IN s ==> f x = a`,
12326 GEN_TAC THEN EQ_TAC THENL
12327 [REWRITE_TAC[CONTINUOUS_DISCONNECTED_RANGE_CONSTANT_EQ];
12328 REWRITE_TAC[CONTINUOUS_COUNTABLE_RANGE_CONSTANT_EQ]] THEN
12329 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
12330 ASM_SIMP_TAC[COUNTABLE_IMP_CARD_LT_REAL] THEN
12331 EXISTS_TAC `IMAGE (f:real^M->real^N) s` THEN
12332 ASM_SIMP_TAC[CARD_LT_IMP_DISCONNECTED; SUBSET_REFL]);;
12334 let CONTINUOUS_COUNTABLE_RANGE_CONSTANT = prove
12335 (`!f:real^M->real^N s.
12336 connected s /\ f continuous_on s /\ COUNTABLE(IMAGE f s)
12337 ==> ?a. !x. x IN s ==> f x = a`,
12338 MESON_TAC[CONTINUOUS_COUNTABLE_RANGE_CONSTANT_EQ]);;
12340 let CONTINUOUS_CARD_LT_RANGE_CONSTANT = prove
12341 (`!f:real^M->real^N s.
12342 connected s /\ f continuous_on s /\ (IMAGE f s) <_c (:real)
12343 ==> ?a. !x. x IN s ==> f x = a`,
12344 MESON_TAC[CONTINUOUS_CARD_LT_RANGE_CONSTANT_EQ]);;
12346 (* ------------------------------------------------------------------------- *)
12347 (* Homeomorphism of hyperplanes. *)
12348 (* ------------------------------------------------------------------------- *)
12350 let HOMEOMORPHIC_HYPERPLANES = prove
12351 (`!a:real^N b c:real^N d.
12352 ~(a = vec 0) /\ ~(c = vec 0)
12353 ==> {x | a dot x = b} homeomorphic {x | c dot x = d}`,
12356 ==> {x:real^N | a dot x = b} homeomorphic {x:real^N | x$1 = &0}`,
12357 REPEAT STRIP_TAC THEN SUBGOAL_THEN `?c:real^N. a dot c = b`
12358 STRIP_ASSUME_TAC THENL
12359 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN
12360 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; VEC_COMPONENT] THEN
12361 DISCH_THEN(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN
12362 EXISTS_TAC `b / (a:real^N)$k % basis k:real^N` THEN
12363 ASM_SIMP_TAC[DOT_RMUL; DOT_BASIS; REAL_DIV_RMUL];
12364 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
12365 ABBREV_TAC `p = {x:real^N | x$1 = &0}` THEN
12366 GEOM_ORIGIN_TAC `c:real^N` THEN
12367 REWRITE_TAC[VECTOR_ADD_RID; DOT_RADD; DOT_RZERO; REAL_EQ_ADD_LCANCEL_0;
12369 REPEAT STRIP_TAC THEN UNDISCH_TAC `~(a:real^N = vec 0)` THEN
12370 GEOM_BASIS_MULTIPLE_TAC 1 `a:real^N` THEN
12371 SIMP_TAC[VECTOR_MUL_EQ_0; DE_MORGAN_THM; DOT_LMUL; REAL_ENTIRE] THEN
12372 SIMP_TAC[DOT_BASIS; LE_REFL; DIMINDEX_GE_1] THEN
12373 EXPAND_TAC "p" THEN REWRITE_TAC[HOMEOMORPHIC_REFL]]) in
12374 REPEAT STRIP_TAC THEN
12375 TRANS_TAC HOMEOMORPHIC_TRANS `{x:real^N | x$1 = &0}` THEN
12376 ASM_SIMP_TAC[lemma] THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
12377 ASM_SIMP_TAC[lemma]);;
12379 let HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE = prove
12381 ~(a = vec 0) /\ 1 <= k /\ k <= dimindex(:N)
12382 ==> {x | a dot x = b} homeomorphic {x:real^N | x$k = c}`,
12383 REPEAT STRIP_TAC THEN
12384 SUBGOAL_THEN `{x:real^N | x$k = c} = {x | basis k dot x = c}` SUBST1_TAC
12385 THENL [ASM_SIMP_TAC[DOT_BASIS]; MATCH_MP_TAC HOMEOMORPHIC_HYPERPLANES] THEN
12386 ASM_SIMP_TAC[BASIS_NONZERO]);;
12388 let HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE = prove
12390 ~(a = vec 0) /\ 1 <= k /\ k <= dimindex(:N)
12391 ==> {x:real^N | x$k = c} homeomorphic {x | a dot x = b}`,
12392 ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN
12393 REWRITE_TAC[HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE]);;
12395 let HOMEOMORPHIC_HYPERPLANE_UNIV = prove
12396 (`!a b. ~(a = vec 0) /\ dimindex(:N) = dimindex(:M) + 1
12397 ==> {x:real^N | a dot x = b} homeomorphic (:real^M)`,
12398 REPEAT STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS
12399 `{x:real^N | basis(dimindex(:N)) dot x = &0}` THEN
12400 ASM_SIMP_TAC[HOMEOMORPHIC_HYPERPLANES; BASIS_NONZERO;
12401 LE_REFL; DIMINDEX_GE_1] THEN
12402 REWRITE_TAC[homeomorphic; HOMEOMORPHISM] THEN
12403 EXISTS_TAC `(\x. lambda i. x$i):real^N->real^M` THEN
12404 EXISTS_TAC `(\x. lambda i. if i <= dimindex(:M) then x$i else &0)
12405 :real^M->real^N` THEN
12406 REPEAT CONJ_TAC THENL
12407 [MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN
12408 SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
12409 VECTOR_MUL_COMPONENT];
12410 REWRITE_TAC[SUBSET_UNIV];
12411 MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN
12412 SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
12413 VECTOR_MUL_COMPONENT] THEN
12414 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
12416 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; IN_UNIV] THEN
12417 ASM_SIMP_TAC[DOT_BASIS; LAMBDA_BETA; LE_REFL; ARITH_RULE `1 <= n + 1`;
12418 ARITH_RULE `~(m + 1 <= m)`];
12419 ASM_SIMP_TAC[IN_ELIM_THM; LAMBDA_BETA; DOT_BASIS; LE_REFL; CART_EQ;
12420 ARITH_RULE `1 <= n + 1`] THEN
12421 GEN_TAC THEN DISCH_TAC THEN X_GEN_TAC `i:num` THEN
12422 ASM_CASES_TAC `i = dimindex(:M) + 1` THEN ASM_REWRITE_TAC[COND_ID] THEN
12423 COND_CASES_TAC THEN REWRITE_TAC[] THEN ASM_ARITH_TAC;
12424 ASM_SIMP_TAC[LAMBDA_BETA; CART_EQ; IN_UNIV; LE_REFL;
12425 ARITH_RULE `i <= n ==> i <= n + 1`]]);;
12427 (* ------------------------------------------------------------------------- *)
12428 (* "Isometry" (up to constant bounds) of injective linear map etc. *)
12429 (* ------------------------------------------------------------------------- *)
12431 let CAUCHY_ISOMETRIC = prove
12433 &0 < e /\ subspace s /\
12434 linear f /\ (!x. x IN s ==> norm(f x) >= e * norm(x)) /\
12435 (!n. x(n) IN s) /\ cauchy(f o x)
12437 REPEAT GEN_TAC THEN REWRITE_TAC[real_ge] THEN
12438 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
12439 REWRITE_TAC[CAUCHY; dist; o_THM] THEN
12440 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_SUB th)]) THEN
12441 DISCH_THEN(fun th -> X_GEN_TAC `d:real` THEN DISCH_TAC THEN MP_TAC th) THEN
12442 DISCH_THEN(MP_TAC o SPEC `d * e`) THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
12443 ASM_MESON_TAC[REAL_LE_RDIV_EQ; REAL_MUL_SYM; REAL_LET_TRANS; SUBSPACE_SUB;
12444 REAL_LT_LDIV_EQ]);;
12446 let COMPLETE_ISOMETRIC_IMAGE = prove
12447 (`!f:real^M->real^N s e.
12448 &0 < e /\ subspace s /\
12449 linear f /\ (!x. x IN s ==> norm(f x) >= e * norm(x)) /\
12451 ==> complete(IMAGE f s)`,
12452 REPEAT GEN_TAC THEN REWRITE_TAC[complete; EXISTS_IN_IMAGE] THEN
12453 STRIP_TAC THEN X_GEN_TAC `g:num->real^N` THEN
12454 REWRITE_TAC[IN_IMAGE; SKOLEM_THM; FORALL_AND_THM] THEN
12455 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
12456 DISCH_THEN(X_CHOOSE_THEN `x:num->real^M` MP_TAC) THEN
12457 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM FUN_EQ_THM] THEN
12458 REWRITE_TAC[GSYM o_DEF] THEN
12459 DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
12460 FIRST_X_ASSUM(MP_TAC o SPEC `x:num->real^M`) THEN
12461 ASM_MESON_TAC[CAUCHY_ISOMETRIC; LINEAR_CONTINUOUS_AT;
12462 CONTINUOUS_AT_SEQUENTIALLY]);;
12464 let INJECTIVE_IMP_ISOMETRIC = prove
12465 (`!f:real^M->real^N s.
12466 closed s /\ subspace s /\
12467 linear f /\ (!x. x IN s /\ (f x = vec 0) ==> (x = vec 0))
12468 ==> ?e. &0 < e /\ !x. x IN s ==> norm(f x) >= e * norm(x)`,
12469 REPEAT STRIP_TAC THEN
12470 ASM_CASES_TAC `s SUBSET {vec 0 :real^M}` THENL
12471 [EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; REAL_MUL_LID; real_ge] THEN
12472 ASM_MESON_TAC[SUBSET; IN_SING; NORM_0; LINEAR_0; REAL_LE_REFL];
12474 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [SUBSET]) THEN
12475 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; IN_SING] THEN
12476 DISCH_THEN(X_CHOOSE_THEN `a:real^M` STRIP_ASSUME_TAC) THEN
12478 [`{(f:real^M->real^N) x | x IN s /\ norm(x) = norm(a:real^M)}`;
12479 `vec 0:real^N`] DISTANCE_ATTAINS_INF) THEN
12481 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
12482 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
12483 MATCH_MP_TAC COMPACT_IMP_CLOSED THEN
12484 SUBST1_TAC(SET_RULE
12485 `{f x | x IN s /\ norm(x) = norm(a:real^M)} =
12486 IMAGE (f:real^M->real^N) (s INTER {x | norm x = norm a})`) THEN
12487 MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
12488 ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN
12489 MATCH_MP_TAC CLOSED_INTER_COMPACT THEN ASM_REWRITE_TAC[] THEN
12491 `{x:real^M | norm x = norm(a:real^M)} = frontier(cball(vec 0,norm a))`
12493 [ASM_SIMP_TAC[FRONTIER_CBALL; NORM_POS_LT; dist; VECTOR_SUB_LZERO;
12495 ASM_SIMP_TAC[COMPACT_FRONTIER; COMPACT_CBALL]];
12497 ONCE_REWRITE_TAC[SET_RULE `{f x | P x} = IMAGE f {x | P x}`] THEN
12498 REWRITE_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN
12499 DISCH_THEN(X_CHOOSE_THEN `b:real^M` MP_TAC) THEN
12500 REWRITE_TAC[IN_ELIM_THM; dist; VECTOR_SUB_LZERO; NORM_NEG] THEN
12501 STRIP_TAC THEN REWRITE_TAC[CLOSED_LIMPT; LIMPT_APPROACHABLE] THEN
12502 EXISTS_TAC `norm((f:real^M->real^N) b) / norm(b)` THEN CONJ_TAC THENL
12503 [ASM_MESON_TAC[REAL_LT_DIV; NORM_POS_LT; NORM_EQ_0]; ALL_TAC] THEN
12504 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
12505 ASM_CASES_TAC `x:real^M = vec 0` THENL
12506 [FIRST_ASSUM(fun th -> ASM_REWRITE_TAC[MATCH_MP LINEAR_0 th]) THEN
12507 REWRITE_TAC[NORM_0; REAL_MUL_RZERO; real_ge; REAL_LE_REFL];
12509 FIRST_X_ASSUM(MP_TAC o SPEC `(norm(a:real^M) / norm(x)) % x:real^M`) THEN
12511 [ASM_SIMP_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
12512 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN ASM_MESON_TAC[subspace];
12514 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN
12515 ASM_REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM; real_ge] THEN
12516 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; NORM_POS_LT] THEN
12517 REWRITE_TAC[real_div; REAL_MUL_AC]);;
12519 let CLOSED_INJECTIVE_IMAGE_SUBSPACE = prove
12520 (`!f s. subspace s /\
12522 (!x. x IN s /\ f(x) = vec 0 ==> x = vec 0) /\
12524 ==> closed(IMAGE f s)`,
12525 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM COMPLETE_EQ_CLOSED] THEN
12526 MATCH_MP_TAC COMPLETE_ISOMETRIC_IMAGE THEN
12527 ASM_REWRITE_TAC[COMPLETE_EQ_CLOSED] THEN
12528 MATCH_MP_TAC INJECTIVE_IMP_ISOMETRIC THEN
12529 ASM_REWRITE_TAC[]);;
12531 (* ------------------------------------------------------------------------- *)
12532 (* Relating linear images to open/closed/interior/closure. *)
12533 (* ------------------------------------------------------------------------- *)
12535 let OPEN_SURJECTIVE_LINEAR_IMAGE = prove
12536 (`!f:real^M->real^N.
12537 linear f /\ (!y. ?x. f x = y)
12538 ==> !s. open s ==> open(IMAGE f s)`,
12539 GEN_TAC THEN STRIP_TAC THEN
12540 REWRITE_TAC[open_def; FORALL_IN_IMAGE] THEN
12541 FIRST_ASSUM(MP_TAC o GEN `k:num` o SPEC `basis k:real^N`) THEN
12542 REWRITE_TAC[SKOLEM_THM] THEN
12543 DISCH_THEN(X_CHOOSE_THEN `b:num->real^M` STRIP_ASSUME_TAC) THEN
12544 SUBGOAL_THEN `bounded(IMAGE (b:num->real^M) (1..dimindex(:N)))` MP_TAC THENL
12545 [SIMP_TAC[FINITE_IMP_BOUNDED; FINITE_IMAGE; FINITE_NUMSEG]; ALL_TAC] THEN
12546 REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE; IN_NUMSEG] THEN
12547 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
12548 X_GEN_TAC `s:real^M->bool` THEN MATCH_MP_TAC MONO_FORALL THEN
12549 X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `(x:real^M) IN s` THEN
12550 ASM_REWRITE_TAC[] THEN
12551 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
12552 EXISTS_TAC `e / B / &(dimindex(:N))` THEN
12553 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN
12554 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
12555 ABBREV_TAC `u = y - (f:real^M->real^N) x` THEN
12556 EXISTS_TAC `x + vsum(1..dimindex(:N)) (\i. (u:real^N)$i % b i):real^M` THEN
12557 ASM_SIMP_TAC[LINEAR_ADD; LINEAR_VSUM; FINITE_NUMSEG; o_DEF;
12558 LINEAR_CMUL; BASIS_EXPANSION] THEN
12559 CONJ_TAC THENL [EXPAND_TAC "u" THEN VECTOR_ARITH_TAC; ALL_TAC] THEN
12560 FIRST_X_ASSUM MATCH_MP_TAC THEN
12561 REWRITE_TAC[NORM_ARITH `dist(x + y,x) = norm y`] THEN
12562 MATCH_MP_TAC REAL_LET_TRANS THEN
12563 EXISTS_TAC `(dist(y,(f:real^M->real^N) x) * &(dimindex(:N))) * B` THEN
12564 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; DIMINDEX_GE_1; LE_1] THEN
12565 MATCH_MP_TAC VSUM_NORM_TRIANGLE THEN REWRITE_TAC[FINITE_NUMSEG] THEN
12566 ONCE_REWRITE_TAC[REAL_ARITH `(a * b) * c:real = b * a * c`] THEN
12567 GEN_REWRITE_TAC(RAND_CONV o LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN
12568 MATCH_MP_TAC SUM_BOUND THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
12569 X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[NORM_MUL; dist] THEN
12570 MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[REAL_ABS_POS; NORM_POS_LE] THEN
12571 ASM_SIMP_TAC[COMPONENT_LE_NORM]);;
12573 let OPEN_BIJECTIVE_LINEAR_IMAGE_EQ = prove
12574 (`!f:real^M->real^N s.
12575 linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
12576 ==> (open(IMAGE f s) <=> open s)`,
12577 REPEAT STRIP_TAC THEN EQ_TAC THENL
12578 [DISCH_TAC; ASM_MESON_TAC[OPEN_SURJECTIVE_LINEAR_IMAGE]] THEN
12579 SUBGOAL_THEN `s = {x | (f:real^M->real^N) x IN IMAGE f s}`
12580 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
12581 MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE_UNIV THEN
12582 ASM_SIMP_TAC[LINEAR_CONTINUOUS_AT]);;
12584 add_linear_invariants [OPEN_BIJECTIVE_LINEAR_IMAGE_EQ];;
12586 let CLOSED_INJECTIVE_LINEAR_IMAGE = prove
12587 (`!f:real^M->real^N.
12588 linear f /\ (!x y. f x = f y ==> x = y)
12589 ==> !s. closed s ==> closed(IMAGE f s)`,
12590 REPEAT STRIP_TAC THEN
12591 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
12592 ASM_REWRITE_TAC[] THEN
12593 DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
12594 MATCH_MP_TAC CLOSED_IN_CLOSED_TRANS THEN
12595 EXISTS_TAC `IMAGE (f:real^M->real^N) (:real^M)` THEN
12597 [MP_TAC(ISPECL [`g:real^N->real^M`; `IMAGE (f:real^M->real^N) (:real^M)`;
12598 `IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) s)`]
12599 CONTINUOUS_CLOSED_IN_PREIMAGE) THEN
12600 ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN ANTS_TAC THENL
12601 [ASM_REWRITE_TAC[GSYM IMAGE_o; IMAGE_I]; ALL_TAC] THEN
12602 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
12603 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FUN_EQ_THM]) THEN
12604 REWRITE_TAC[EXTENSION; o_THM; I_THM] THEN SET_TAC[];
12605 MATCH_MP_TAC CLOSED_INJECTIVE_IMAGE_SUBSPACE THEN
12606 ASM_REWRITE_TAC[IN_UNIV; SUBSPACE_UNIV; CLOSED_UNIV] THEN
12607 X_GEN_TAC `x:real^M` THEN
12608 DISCH_THEN(MP_TAC o AP_TERM `g:real^N->real^M`) THEN
12609 RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM; I_THM; o_THM]) THEN
12610 ASM_MESON_TAC[LINEAR_0]]);;
12612 let CLOSED_INJECTIVE_LINEAR_IMAGE_EQ = prove
12613 (`!f:real^M->real^N s.
12614 linear f /\ (!x y. f x = f y ==> x = y)
12615 ==> (closed(IMAGE f s) <=> closed s)`,
12616 REPEAT STRIP_TAC THEN EQ_TAC THENL
12617 [DISCH_TAC; ASM_MESON_TAC[CLOSED_INJECTIVE_LINEAR_IMAGE]] THEN
12618 SUBGOAL_THEN `s = {x | (f:real^M->real^N) x IN IMAGE f s}`
12619 SUBST1_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
12620 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN
12621 ASM_SIMP_TAC[LINEAR_CONTINUOUS_AT]);;
12623 add_linear_invariants [CLOSED_INJECTIVE_LINEAR_IMAGE_EQ];;
12625 let CLOSURE_LINEAR_IMAGE_SUBSET = prove
12626 (`!f:real^M->real^N s.
12627 linear f ==> IMAGE f (closure s) SUBSET closure(IMAGE f s)`,
12628 REPEAT STRIP_TAC THEN
12629 MATCH_MP_TAC IMAGE_CLOSURE_SUBSET THEN
12630 ASM_SIMP_TAC[CLOSED_CLOSURE; CLOSURE_SUBSET; LINEAR_CONTINUOUS_ON]);;
12632 let CLOSURE_INJECTIVE_LINEAR_IMAGE = prove
12633 (`!f:real^M->real^N s.
12634 linear f /\ (!x y. f x = f y ==> x = y)
12635 ==> closure(IMAGE f s) = IMAGE f (closure s)`,
12636 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
12637 ASM_SIMP_TAC[CLOSURE_LINEAR_IMAGE_SUBSET] THEN
12638 MATCH_MP_TAC CLOSURE_MINIMAL THEN
12639 SIMP_TAC[CLOSURE_SUBSET; IMAGE_SUBSET] THEN
12640 ASM_MESON_TAC[CLOSED_INJECTIVE_LINEAR_IMAGE; CLOSED_CLOSURE]);;
12642 add_linear_invariants [CLOSURE_INJECTIVE_LINEAR_IMAGE];;
12644 let CLOSURE_BOUNDED_LINEAR_IMAGE = prove
12645 (`!f:real^M->real^N s.
12646 linear f /\ bounded s
12647 ==> closure(IMAGE f s) = IMAGE f (closure s)`,
12648 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
12649 ASM_SIMP_TAC[CLOSURE_LINEAR_IMAGE_SUBSET] THEN
12650 MATCH_MP_TAC CLOSURE_MINIMAL THEN
12651 SIMP_TAC[CLOSURE_SUBSET; IMAGE_SUBSET] THEN
12652 MATCH_MP_TAC COMPACT_IMP_CLOSED THEN
12653 MATCH_MP_TAC COMPACT_LINEAR_IMAGE THEN
12654 ASM_REWRITE_TAC[COMPACT_CLOSURE]);;
12656 let LINEAR_INTERIOR_IMAGE_SUBSET = prove
12657 (`!f:real^M->real^N s.
12658 linear f /\ (!x y. f x = f y ==> x = y)
12659 ==> interior(IMAGE f s) SUBSET IMAGE f (interior s)`,
12660 MESON_TAC[INTERIOR_IMAGE_SUBSET; LINEAR_CONTINUOUS_AT]);;
12662 let LINEAR_IMAGE_SUBSET_INTERIOR = prove
12663 (`!f:real^M->real^N s.
12664 linear f /\ (!y. ?x. f x = y)
12665 ==> IMAGE f (interior s) SUBSET interior(IMAGE f s)`,
12666 REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERIOR_MAXIMAL THEN
12667 ASM_SIMP_TAC[OPEN_SURJECTIVE_LINEAR_IMAGE; OPEN_INTERIOR;
12668 IMAGE_SUBSET; INTERIOR_SUBSET]);;
12670 let INTERIOR_BIJECTIVE_LINEAR_IMAGE = prove
12671 (`!f:real^M->real^N s.
12672 linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
12673 ==> interior(IMAGE f s) = IMAGE f (interior s)`,
12674 REWRITE_TAC[interior] THEN GEOM_TRANSFORM_TAC[]);;
12676 add_linear_invariants [INTERIOR_BIJECTIVE_LINEAR_IMAGE];;
12678 let FRONTIER_BIJECTIVE_LINEAR_IMAGE = prove
12679 (`!f:real^M->real^N s.
12680 linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
12681 ==> frontier(IMAGE f s) = IMAGE f (frontier s)`,
12682 REWRITE_TAC[frontier] THEN GEOM_TRANSFORM_TAC[]);;
12684 add_linear_invariants [FRONTIER_BIJECTIVE_LINEAR_IMAGE];;
12686 (* ------------------------------------------------------------------------- *)
12687 (* Corollaries, reformulations and special cases for M = N. *)
12688 (* ------------------------------------------------------------------------- *)
12690 let IN_INTERIOR_LINEAR_IMAGE = prove
12691 (`!f:real^M->real^N g s x.
12692 linear f /\ linear g /\ (f o g = I) /\ x IN interior s
12693 ==> (f x) IN interior (IMAGE f s)`,
12694 REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REPEAT STRIP_TAC THEN
12695 MP_TAC(ISPECL [`f:real^M->real^N`; `s:real^M->bool`]
12696 LINEAR_IMAGE_SUBSET_INTERIOR) THEN
12697 ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
12700 let LINEAR_OPEN_MAPPING = prove
12701 (`!f:real^M->real^N g.
12702 linear f /\ linear g /\ (f o g = I)
12703 ==> !s. open s ==> open(IMAGE f s)`,
12704 REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN DISCH_TAC THEN
12705 MATCH_MP_TAC OPEN_SURJECTIVE_LINEAR_IMAGE THEN
12708 let INTERIOR_INJECTIVE_LINEAR_IMAGE = prove
12709 (`!f:real^N->real^N s.
12710 linear f /\ (!x y. f x = f y ==> x = y)
12711 ==> interior(IMAGE f s) = IMAGE f (interior s)`,
12712 REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERIOR_BIJECTIVE_LINEAR_IMAGE THEN
12713 ASM_MESON_TAC[LINEAR_INJECTIVE_IMP_SURJECTIVE]);;
12715 let INTERIOR_SURJECTIVE_LINEAR_IMAGE = prove
12716 (`!f:real^N->real^N s.
12717 linear f /\ (!y. ?x. f x = y)
12718 ==> interior(IMAGE f s) = IMAGE f (interior s)`,
12719 REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERIOR_BIJECTIVE_LINEAR_IMAGE THEN
12720 ASM_MESON_TAC[LINEAR_SURJECTIVE_IMP_INJECTIVE]);;
12722 let CLOSURE_SURJECTIVE_LINEAR_IMAGE = prove
12723 (`!f:real^N->real^N s.
12724 linear f /\ (!y. ?x. f x = y)
12725 ==> closure(IMAGE f s) = IMAGE f (closure s)`,
12726 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_INJECTIVE_LINEAR_IMAGE THEN
12727 ASM_MESON_TAC[LINEAR_SURJECTIVE_IMP_INJECTIVE]);;
12729 let FRONTIER_INJECTIVE_LINEAR_IMAGE = prove
12730 (`!f:real^N->real^N s.
12731 linear f /\ (!x y. f x = f y ==> x = y)
12732 ==> frontier(IMAGE f s) = IMAGE f (frontier s)`,
12733 REPEAT STRIP_TAC THEN MATCH_MP_TAC FRONTIER_BIJECTIVE_LINEAR_IMAGE THEN
12734 ASM_MESON_TAC[LINEAR_INJECTIVE_IMP_SURJECTIVE]);;
12736 let FRONTIER_SURJECTIVE_LINEAR_IMAGE = prove
12737 (`!f:real^N->real^N.
12738 linear f /\ (!y. ?x. f x = y)
12739 ==> frontier(IMAGE f s) = IMAGE f (frontier s)`,
12740 REPEAT STRIP_TAC THEN MATCH_MP_TAC FRONTIER_BIJECTIVE_LINEAR_IMAGE THEN
12741 ASM_MESON_TAC[LINEAR_SURJECTIVE_IMP_INJECTIVE]);;
12743 let COMPLETE_INJECTIVE_LINEAR_IMAGE = prove
12744 (`!f:real^M->real^N.
12745 linear f /\ (!x y. f x = f y ==> x = y)
12746 ==> !s. complete s ==> complete(IMAGE f s)`,
12747 REWRITE_TAC[COMPLETE_EQ_CLOSED; CLOSED_INJECTIVE_LINEAR_IMAGE]);;
12749 let COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ = prove
12750 (`!f:real^M->real^N s.
12751 linear f /\ (!x y. f x = f y ==> x = y)
12752 ==> (complete(IMAGE f s) <=> complete s)`,
12753 REWRITE_TAC[COMPLETE_EQ_CLOSED; CLOSED_INJECTIVE_LINEAR_IMAGE_EQ]);;
12755 add_linear_invariants [COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ];;
12757 let LIMPT_INJECTIVE_LINEAR_IMAGE_EQ = prove
12758 (`!f:real^M->real^N s.
12759 linear f /\ (!x y. f x = f y ==> x = y)
12760 ==> ((f x) limit_point_of (IMAGE f s) <=> x limit_point_of s)`,
12761 REWRITE_TAC[LIMPT_APPROACHABLE; EXISTS_IN_IMAGE] THEN
12762 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
12764 [MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_BOUNDED_BELOW_POS);
12765 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_BOUNDED_POS)] THEN
12766 ASM_REWRITE_TAC[] THEN
12767 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THENL
12768 [FIRST_X_ASSUM(MP_TAC o SPEC `e * B:real`);
12769 FIRST_X_ASSUM(MP_TAC o SPEC `e / B:real`)] THEN
12770 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MUL; dist; GSYM LINEAR_SUB] THEN
12771 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
12772 REPEAT(MATCH_MP_TAC MONO_AND THEN
12773 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN
12774 ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ; REAL_LT_RDIV_EQ] THEN
12775 MATCH_MP_TAC(REAL_ARITH `a <= b ==> b < x ==> a < x`) THEN
12776 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[REAL_LE_RDIV_EQ]);;
12778 add_linear_invariants [LIMPT_INJECTIVE_LINEAR_IMAGE_EQ];;
12780 let LIMPT_TRANSLATION_EQ = prove
12781 (`!a s x. (a + x) limit_point_of (IMAGE (\y. a + y) s) <=> x limit_point_of s`,
12782 REWRITE_TAC[limit_point_of] THEN GEOM_TRANSLATE_TAC[]);;
12784 add_translation_invariants [LIMPT_TRANSLATION_EQ];;
12786 let OPEN_OPEN_LEFT_PROJECTION = prove
12787 (`!s t:real^(M,N)finite_sum->bool.
12788 open s /\ open t ==> open {x | x IN s /\ ?y. pastecart x y IN t}`,
12789 REPEAT STRIP_TAC THEN
12791 `{x | x IN s /\ ?y. (pastecart x y:real^(M,N)finite_sum) IN t} =
12792 s INTER IMAGE fstcart t`
12794 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER; IN_IMAGE] THEN
12795 MESON_TAC[FSTCART_PASTECART; PASTECART_FST_SND];
12796 MATCH_MP_TAC OPEN_INTER THEN ASM_REWRITE_TAC[] THEN
12797 MATCH_MP_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM]
12798 OPEN_SURJECTIVE_LINEAR_IMAGE) THEN
12799 ASM_REWRITE_TAC[LINEAR_FSTCART] THEN MESON_TAC[FSTCART_PASTECART]]);;
12801 let OPEN_OPEN_RIGHT_PROJECTION = prove
12802 (`!s t:real^(M,N)finite_sum->bool.
12803 open s /\ open t ==> open {y | y IN s /\ ?x. pastecart x y IN t}`,
12804 REPEAT STRIP_TAC THEN
12806 `{y | y IN s /\ ?x. (pastecart x y:real^(M,N)finite_sum) IN t} =
12807 s INTER IMAGE sndcart t`
12809 [REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER; IN_IMAGE] THEN
12810 MESON_TAC[SNDCART_PASTECART; PASTECART_FST_SND];
12811 MATCH_MP_TAC OPEN_INTER THEN ASM_REWRITE_TAC[] THEN
12812 MATCH_MP_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM]
12813 OPEN_SURJECTIVE_LINEAR_IMAGE) THEN
12814 ASM_REWRITE_TAC[LINEAR_SNDCART] THEN MESON_TAC[SNDCART_PASTECART]]);;
12816 (* ------------------------------------------------------------------------- *)
12817 (* Even more special cases. *)
12818 (* ------------------------------------------------------------------------- *)
12820 let INTERIOR_NEGATIONS = prove
12821 (`!s. interior(IMAGE (--) s) = IMAGE (--) (interior s)`,
12822 GEN_TAC THEN MATCH_MP_TAC INTERIOR_INJECTIVE_LINEAR_IMAGE THEN
12823 REWRITE_TAC[linear] THEN REPEAT CONJ_TAC THEN VECTOR_ARITH_TAC);;
12825 let SYMMETRIC_INTERIOR = prove
12827 (!x. x IN s ==> --x IN s)
12828 ==> !x. x IN interior s ==> (--x) IN interior s`,
12829 REPEAT GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
12830 DISCH_THEN(MP_TAC o MATCH_MP(ISPEC `(--):real^N->real^N` FUN_IN_IMAGE)) THEN
12831 REWRITE_TAC[GSYM INTERIOR_NEGATIONS] THEN
12832 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
12833 REWRITE_TAC[EXTENSION; IN_IMAGE] THEN ASM_MESON_TAC[VECTOR_NEG_NEG]);;
12835 let CLOSURE_NEGATIONS = prove
12836 (`!s. closure(IMAGE (--) s) = IMAGE (--) (closure s)`,
12837 GEN_TAC THEN MATCH_MP_TAC CLOSURE_INJECTIVE_LINEAR_IMAGE THEN
12838 REWRITE_TAC[linear] THEN REPEAT CONJ_TAC THEN VECTOR_ARITH_TAC);;
12840 let SYMMETRIC_CLOSURE = prove
12842 (!x. x IN s ==> --x IN s)
12843 ==> !x. x IN closure s ==> (--x) IN closure s`,
12844 REPEAT GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
12845 DISCH_THEN(MP_TAC o MATCH_MP(ISPEC `(--):real^N->real^N` FUN_IN_IMAGE)) THEN
12846 REWRITE_TAC[GSYM CLOSURE_NEGATIONS] THEN
12847 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
12848 REWRITE_TAC[EXTENSION; IN_IMAGE] THEN ASM_MESON_TAC[VECTOR_NEG_NEG]);;
12850 (* ------------------------------------------------------------------------- *)
12851 (* Some properties of a canonical subspace. *)
12852 (* ------------------------------------------------------------------------- *)
12854 let SUBSPACE_SUBSTANDARD = prove
12856 {x:real^N | !i. d < i /\ i <= dimindex(:N) ==> x$i = &0}`,
12857 GEN_TAC THEN ASM_CASES_TAC `d <= dimindex(:N)` THENL
12858 [MP_TAC(ARITH_RULE `!i. d < i ==> 1 <= i`) THEN
12859 SIMP_TAC[subspace; IN_ELIM_THM; REAL_MUL_RZERO; REAL_ADD_LID;
12860 VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT];
12861 ASM_SIMP_TAC[ARITH_RULE `~(d:num <= e) ==> (d < i /\ i <= e <=> F)`] THEN
12862 REWRITE_TAC[SET_RULE `{x | T} = UNIV`; SUBSPACE_UNIV]]);;
12864 let CLOSED_SUBSTANDARD = prove
12866 {x:real^N | !i. d < i /\ i <= dimindex(:N) ==> x$i = &0}`,
12869 `{x:real^N | !i. d < i /\ i <= dimindex(:N) ==> x$i = &0} =
12870 INTERS {{x | basis i dot x = &0} | d < i /\ i <= dimindex(:N)}`
12873 SIMP_TAC[CLOSED_INTERS; CLOSED_HYPERPLANE; IN_ELIM_THM;
12874 LEFT_IMP_EXISTS_THM]] THEN
12875 GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_INTERS; IN_ELIM_THM] THEN
12876 SIMP_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
12877 MP_TAC(ARITH_RULE `!i. d < i ==> 1 <= i`) THEN
12878 SIMP_TAC[DOT_BASIS] THEN MESON_TAC[]);;
12880 let DIM_SUBSTANDARD = prove
12881 (`!d. d <= dimindex(:N)
12882 ==> (dim {x:real^N | !i. d < i /\ i <= dimindex(:N)
12885 REPEAT STRIP_TAC THEN MATCH_MP_TAC DIM_UNIQUE THEN
12886 EXISTS_TAC `IMAGE (basis:num->real^N) (1..d)` THEN REPEAT CONJ_TAC THENL
12887 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; IN_NUMSEG] THEN
12888 MESON_TAC[BASIS_COMPONENT; ARITH_RULE `d < i ==> 1 <= i`; NOT_LT];
12890 MATCH_MP_TAC INDEPENDENT_MONO THEN
12891 EXISTS_TAC `{basis i :real^N | 1 <= i /\ i <= dimindex(:N)}` THEN
12892 REWRITE_TAC[INDEPENDENT_STDBASIS]THEN
12893 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; IN_NUMSEG] THEN
12894 ASM_MESON_TAC[LE_TRANS];
12895 MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN REWRITE_TAC[HAS_SIZE_NUMSEG_1] THEN
12896 REWRITE_TAC[IN_NUMSEG] THEN ASM_MESON_TAC[LE_TRANS; BASIS_INJ]] THEN
12897 POP_ASSUM MP_TAC THEN SPEC_TAC(`d:num`,`d:num`) THEN
12899 [REWRITE_TAC[ARITH_RULE `0 < i <=> 1 <= i`; SPAN_STDBASIS] THEN
12900 SUBGOAL_THEN `IMAGE basis (1 .. 0) :real^N->bool = {}` SUBST1_TAC THENL
12901 [REWRITE_TAC[IMAGE_EQ_EMPTY; NUMSEG_EMPTY; ARITH]; ALL_TAC] THEN
12902 DISCH_TAC THEN REWRITE_TAC[SPAN_EMPTY; SUBSET; IN_ELIM_THM; IN_SING] THEN
12903 SIMP_TAC[CART_EQ; VEC_COMPONENT];
12905 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
12906 ASM_SIMP_TAC[ARITH_RULE `SUC d <= n ==> d <= n`] THEN
12907 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN DISCH_TAC THEN
12908 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
12909 FIRST_X_ASSUM(MP_TAC o SPEC `x - (x$(SUC d)) % basis(SUC d) :real^N`) THEN
12911 [X_GEN_TAC `i:num` THEN STRIP_TAC THEN
12912 FIRST_ASSUM(ASSUME_TAC o MATCH_MP(ARITH_RULE `d < i ==> 1 <= i`)) THEN
12913 ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN
12914 ASM_SIMP_TAC[BASIS_COMPONENT] THEN COND_CASES_TAC THEN
12915 ASM_REWRITE_TAC[REAL_MUL_RID; REAL_SUB_REFL] THEN
12916 ASM_REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_RZERO] THEN
12917 ASM_MESON_TAC[ARITH_RULE `d < i /\ ~(i = SUC d) ==> SUC d < i`];
12920 SUBST1_TAC(VECTOR_ARITH
12921 `x = (x - (x$(SUC d)) % basis(SUC d)) +
12922 x$(SUC d) % basis(SUC d) :real^N`) THEN
12923 MATCH_MP_TAC SPAN_ADD THEN CONJ_TAC THENL
12924 [ASM_MESON_TAC[SPAN_MONO; SUBSET_IMAGE; SUBSET; SUBSET_NUMSEG; LE_REFL; LE];
12925 MATCH_MP_TAC SPAN_MUL THEN MATCH_MP_TAC SPAN_SUPERSET THEN
12926 REWRITE_TAC[IN_IMAGE; IN_NUMSEG] THEN
12927 MESON_TAC[LE_REFL; ARITH_RULE `1 <= SUC d`]]);;
12929 (* ------------------------------------------------------------------------- *)
12930 (* Hence closure and completeness of all subspaces. *)
12931 (* ------------------------------------------------------------------------- *)
12933 let CLOSED_SUBSPACE = prove
12934 (`!s:real^N->bool. subspace s ==> closed s`,
12935 REPEAT STRIP_TAC THEN ABBREV_TAC `d = dim(s:real^N->bool)` THEN
12936 MP_TAC(MATCH_MP DIM_SUBSTANDARD
12937 (ISPEC `s:real^N->bool` DIM_SUBSET_UNIV)) THEN
12938 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
12940 [`{x:real^N | !i. d < i /\ i <= dimindex(:N)
12942 `s:real^N->bool`] SUBSPACE_ISOMORPHISM) THEN
12943 ASM_REWRITE_TAC[SUBSPACE_SUBSTANDARD] THEN
12944 DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` MP_TAC) THEN
12945 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
12946 DISCH_THEN(CONJUNCTS_THEN2 (SUBST_ALL_TAC o SYM) STRIP_ASSUME_TAC) THEN
12947 MATCH_MP_TAC(ISPEC `f:real^N->real^N` CLOSED_INJECTIVE_IMAGE_SUBSPACE) THEN
12948 ASM_REWRITE_TAC[SUBSPACE_SUBSTANDARD; CLOSED_SUBSTANDARD] THEN
12949 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
12950 ASM_REWRITE_TAC[] THEN
12951 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LINEAR_0]] THEN
12952 REWRITE_TAC[IN_ELIM_THM] THEN
12953 ASM_MESON_TAC[VEC_COMPONENT; ARITH_RULE `d < i ==> 1 <= i`]);;
12955 let COMPLETE_SUBSPACE = prove
12956 (`!s:real^N->bool. subspace s ==> complete s`,
12957 REWRITE_TAC[COMPLETE_EQ_CLOSED; CLOSED_SUBSPACE]);;
12959 let CLOSED_SPAN = prove
12960 (`!s. closed(span s)`,
12961 SIMP_TAC[CLOSED_SUBSPACE; SUBSPACE_SPAN]);;
12963 let DIM_CLOSURE = prove
12964 (`!s:real^N->bool. dim(closure s) = dim s`,
12965 GEN_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THENL
12966 [GEN_REWRITE_TAC RAND_CONV [GSYM DIM_SPAN]; ALL_TAC] THEN
12967 MATCH_MP_TAC DIM_SUBSET THEN REWRITE_TAC[CLOSURE_SUBSET] THEN
12968 MATCH_MP_TAC CLOSURE_MINIMAL THEN
12969 SIMP_TAC[CLOSED_SUBSPACE; SUBSPACE_SPAN; SPAN_INC]);;
12971 let CLOSED_BOUNDEDPREIM_CONTINUOUS_IMAGE = prove
12972 (`!f:real^M->real^N s.
12973 closed s /\ f continuous_on s /\
12974 (!e. bounded {x | x IN s /\ norm(f x) <= e})
12975 ==> closed(IMAGE f s)`,
12976 REPEAT STRIP_TAC THEN REWRITE_TAC[CLOSED_INTERS_COMPACT] THEN
12977 REWRITE_TAC[SET_RULE
12978 `cball(vec 0,e) INTER IMAGE (f:real^M->real^N) s =
12979 IMAGE f (s INTER {x | x IN s /\ f x IN cball(vec 0,e)})`] THEN
12980 X_GEN_TAC `e:real` THEN MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE THEN
12982 [MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN
12983 ASM_REWRITE_TAC[] THEN SET_TAC[];
12984 MATCH_MP_TAC CLOSED_INTER_COMPACT THEN ASM_REWRITE_TAC[] THEN
12985 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL
12986 [ASM_REWRITE_TAC[IN_CBALL_0];
12987 ASM_SIMP_TAC[CONTINUOUS_CLOSED_PREIMAGE; CLOSED_CBALL]]]);;
12989 let CLOSED_INJECTIVE_IMAGE_SUBSET_SUBSPACE = prove
12990 (`!f:real^M->real^N s t.
12991 closed s /\ s SUBSET t /\ subspace t /\
12993 (!x. x IN t /\ f(x) = vec 0 ==> x = vec 0)
12994 ==> closed(IMAGE f s)`,
12995 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSED_BOUNDEDPREIM_CONTINUOUS_IMAGE THEN
12996 ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON] THEN
12997 MP_TAC(ISPECL [`f:real^M->real^N`; `t:real^M->bool`]
12998 INJECTIVE_IMP_ISOMETRIC) THEN
12999 ASM_SIMP_TAC[CLOSED_SUBSPACE; real_ge] THEN
13000 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
13001 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
13002 X_GEN_TAC `e:real` THEN MATCH_MP_TAC BOUNDED_SUBSET THEN
13003 EXISTS_TAC `cball(vec 0:real^M,e / B)` THEN
13004 REWRITE_TAC[BOUNDED_CBALL] THEN
13005 ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; IN_CBALL_0; REAL_LE_RDIV_EQ] THEN
13006 ASM_MESON_TAC[SUBSET; REAL_LE_TRANS]);;
13008 let BASIS_COORDINATES_LIPSCHITZ = prove
13013 ==> abs(c v) <= B * norm(vsum b (\v. c(v) % v))`,
13014 X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN
13015 FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP INDEPENDENT_BOUND) THEN
13016 FIRST_ASSUM(X_CHOOSE_THEN `b:num->real^N` STRIP_ASSUME_TAC o
13017 GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN
13018 ABBREV_TAC `n = CARD(k:real^N->bool)` THEN
13020 [`(\x. vsum(1..n) (\i. x$i % b i)):real^N->real^N`;
13021 `span(IMAGE basis (1..n)):real^N->bool`]
13022 INJECTIVE_IMP_ISOMETRIC) THEN
13023 REWRITE_TAC[SUBSPACE_SPAN] THEN ANTS_TAC THENL
13024 [CONJ_TAC THENL [SIMP_TAC[CLOSED_SUBSPACE; SUBSPACE_SPAN]; ALL_TAC] THEN
13026 [MATCH_MP_TAC LINEAR_COMPOSE_VSUM THEN
13027 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
13028 MATCH_MP_TAC LINEAR_VMUL_COMPONENT THEN
13029 SIMP_TAC[LINEAR_ID] THEN ASM_ARITH_TAC;
13031 X_GEN_TAC `x:real^N` THEN
13032 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
13033 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_SPAN_IMAGE_BASIS]) THEN
13034 REWRITE_TAC[IN_NUMSEG] THEN DISCH_TAC THEN
13035 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
13036 DISCH_THEN(X_CHOOSE_TAC `c:real^N->num`) THEN
13038 `vsum(1..n) (\i. (x:real^N)$i % b i:real^N) = vsum k (\v. x$(c v) % v)`
13040 [MATCH_MP_TAC VSUM_EQ_GENERAL_INVERSES THEN
13041 MAP_EVERY EXISTS_TAC [`b:num->real^N`; `c:real^N->num`] THEN
13045 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INDEPENDENT_EXPLICIT]) THEN
13046 DISCH_THEN(MP_TAC o SPEC `\v:real^N. (x:real^N)$(c v)` o CONJUNCT2) THEN
13047 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
13048 REWRITE_TAC[CART_EQ; FORALL_IN_IMAGE; VEC_COMPONENT] THEN
13049 ASM_MESON_TAC[IN_NUMSEG];
13051 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
13052 EXISTS_TAC `inv(B:real)` THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
13053 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_NUMSEG] THEN
13054 MAP_EVERY X_GEN_TAC [`c:real^N->real`; `j:num`] THEN STRIP_TAC THEN
13055 ONCE_REWRITE_TAC[REAL_ARITH `inv B * x = x / B`] THEN
13056 ASM_SIMP_TAC[REAL_LE_RDIV_EQ] THEN
13057 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o rand o rand o snd) THEN
13058 ASM_REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN
13059 FIRST_X_ASSUM(MP_TAC o SPEC
13060 `(lambda i. if 1 <= i /\ i <= n then c(b i:real^N) else &0):real^N`) THEN
13061 SIMP_TAC[IN_SPAN_IMAGE_BASIS; LAMBDA_BETA] THEN
13062 ANTS_TAC THENL [MESON_TAC[IN_NUMSEG]; ALL_TAC] THEN
13063 MATCH_MP_TAC(REAL_ARITH `x = v /\ u <= y ==> x >= y ==> u <= v`) THEN
13065 [AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ_NUMSEG THEN
13066 SUBGOAL_THEN `!i. i <= n ==> i <= dimindex(:N)` MP_TAC THENL
13067 [ASM_ARITH_TAC; SIMP_TAC[LAMBDA_BETA] THEN DISCH_THEN(K ALL_TAC)] THEN
13068 REWRITE_TAC[o_THM];
13069 GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN
13070 ASM_SIMP_TAC[REAL_LE_RMUL_EQ] THEN
13072 [`(lambda i. if 1 <= i /\ i <= n then c(b i:real^N) else &0):real^N`;
13073 `j:num`] COMPONENT_LE_NORM) THEN
13074 SUBGOAL_THEN `1 <= j /\ j <= dimindex(:N)` MP_TAC THENL
13075 [ASM_ARITH_TAC; SIMP_TAC[LAMBDA_BETA] THEN ASM_REWRITE_TAC[]]]);;
13077 let BASIS_COORDINATES_CONTINUOUS = prove
13078 (`!b:real^N->bool e.
13079 independent b /\ &0 < e
13081 !c. norm(vsum b (\v. c(v) % v)) < d
13082 ==> !v. v IN b ==> abs(c v) < e`,
13083 REPEAT STRIP_TAC THEN
13084 FIRST_X_ASSUM(MP_TAC o MATCH_MP BASIS_COORDINATES_LIPSCHITZ) THEN
13085 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
13086 EXISTS_TAC `e / B:real` THEN ASM_SIMP_TAC[REAL_LT_DIV] THEN
13087 X_GEN_TAC `c:real^N->real` THEN DISCH_TAC THEN
13088 X_GEN_TAC `v:real^N` THEN DISCH_TAC THEN
13089 MATCH_MP_TAC REAL_LET_TRANS THEN
13090 EXISTS_TAC `B * norm(vsum b (\v:real^N. c v % v))` THEN
13091 ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
13092 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ]);;
13094 (* ------------------------------------------------------------------------- *)
13095 (* Affine transformations of intervals. *)
13096 (* ------------------------------------------------------------------------- *)
13098 let AFFINITY_INVERSES = prove
13100 ==> (\x. m % x + c) o (\x. inv(m) % x + (--(inv(m) % c))) = I /\
13101 (\x. inv(m) % x + (--(inv(m) % c))) o (\x. m % x + c) = I`,
13102 REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
13103 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_RNEG] THEN
13104 SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_MUL_RINV] THEN
13105 REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
13107 let REAL_AFFINITY_LE = prove
13108 (`!m c x y. &0 < m ==> (m * x + c <= y <=> x <= inv(m) * y + --(c / m))`,
13109 REWRITE_TAC[REAL_ARITH `m * x + c <= y <=> x * m <= y - c`] THEN
13110 SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN REAL_ARITH_TAC);;
13112 let REAL_LE_AFFINITY = prove
13113 (`!m c x y. &0 < m ==> (y <= m * x + c <=> inv(m) * y + --(c / m) <= x)`,
13114 REWRITE_TAC[REAL_ARITH `y <= m * x + c <=> y - c <= x * m`] THEN
13115 SIMP_TAC[GSYM REAL_LE_LDIV_EQ] THEN REAL_ARITH_TAC);;
13117 let REAL_AFFINITY_LT = prove
13118 (`!m c x y. &0 < m ==> (m * x + c < y <=> x < inv(m) * y + --(c / m))`,
13119 SIMP_TAC[REAL_LE_AFFINITY; GSYM REAL_NOT_LE]);;
13121 let REAL_LT_AFFINITY = prove
13122 (`!m c x y. &0 < m ==> (y < m * x + c <=> inv(m) * y + --(c / m) < x)`,
13123 SIMP_TAC[REAL_AFFINITY_LE; GSYM REAL_NOT_LE]);;
13125 let REAL_AFFINITY_EQ = prove
13126 (`!m c x y. ~(m = &0) ==> (m * x + c = y <=> x = inv(m) * y + --(c / m))`,
13127 CONV_TAC REAL_FIELD);;
13129 let REAL_EQ_AFFINITY = prove
13130 (`!m c x y. ~(m = &0) ==> (y = m * x + c <=> inv(m) * y + --(c / m) = x)`,
13131 CONV_TAC REAL_FIELD);;
13133 let VECTOR_AFFINITY_EQ = prove
13134 (`!m c x y. ~(m = &0)
13135 ==> (m % x + c = y <=> x = inv(m) % y + --(inv(m) % c))`,
13136 SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
13137 real_div; VECTOR_NEG_COMPONENT; REAL_AFFINITY_EQ] THEN
13138 REWRITE_TAC[REAL_MUL_AC]);;
13140 let VECTOR_EQ_AFFINITY = prove
13141 (`!m c x y. ~(m = &0)
13142 ==> (y = m % x + c <=> inv(m) % y + --(inv(m) % c) = x)`,
13143 SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
13144 real_div; VECTOR_NEG_COMPONENT; REAL_EQ_AFFINITY] THEN
13145 REWRITE_TAC[REAL_MUL_AC]);;
13147 let IMAGE_AFFINITY_INTERVAL = prove
13149 IMAGE (\x. m % x + c) (interval[a,b]) =
13150 if interval[a,b] = {} then {}
13151 else if &0 <= m then interval[m % a + c,m % b + c]
13152 else interval[m % b + c,m % a + c]`,
13153 REPEAT GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[IMAGE_CLAUSES] THEN
13154 ASM_CASES_TAC `m = &0` THEN ASM_REWRITE_TAC[REAL_LE_LT] THENL
13155 [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID; COND_ID] THEN
13156 REWRITE_TAC[INTERVAL_SING] THEN ASM SET_TAC[];
13158 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
13159 `~(x = &0) ==> &0 < x \/ &0 < --x`)) THEN
13160 ASM_SIMP_TAC[EXTENSION; IN_IMAGE; REAL_ARITH `&0 < --x ==> ~(&0 < x)`] THENL
13162 ONCE_REWRITE_TAC[VECTOR_ARITH `x = m % y + c <=> c = (--m) % y + x`]] THEN
13163 ASM_SIMP_TAC[VECTOR_EQ_AFFINITY; REAL_LT_IMP_NZ; UNWIND_THM1] THEN
13164 SIMP_TAC[IN_INTERVAL; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
13165 VECTOR_NEG_COMPONENT] THEN
13166 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_LT_INV_EQ]) THEN
13167 SIMP_TAC[REAL_AFFINITY_LE; REAL_LE_AFFINITY; real_div] THEN
13168 DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[REAL_INV_INV] THEN
13169 REWRITE_TAC[REAL_MUL_LNEG; REAL_NEGNEG] THEN
13170 ASM_SIMP_TAC[REAL_FIELD `&0 < m ==> (inv m * x) * m = x`] THEN
13171 GEN_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN AP_TERM_TAC THEN REAL_ARITH_TAC);;
13173 (* ------------------------------------------------------------------------- *)
13174 (* Existence of eigenvectors. The proof is only in this file because it uses *)
13175 (* a few simple results about continuous functions (at least *)
13176 (* CONTINUOUS_ON_LIFT_DOT2, CONTINUOUS_ATTAINS_SUP and CLOSED_SUBSPACE). *)
13177 (* ------------------------------------------------------------------------- *)
13179 let SELF_ADJOINT_HAS_EIGENVECTOR_IN_SUBSPACE = prove
13180 (`!f:real^N->real^N s.
13181 linear f /\ adjoint f = f /\
13182 subspace s /\ ~(s = {vec 0}) /\ (!x. x IN s ==> f x IN s)
13183 ==> ?v c. v IN s /\ norm(v) = &1 /\ f(v) = c % v`,
13185 (`!a b. (!x. a * x <= b * x pow 2) ==> &0 <= b ==> a = &0`,
13186 REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[REAL_LE_LT] THEN
13187 ASM_CASES_TAC `b = &0` THEN ASM_REWRITE_TAC[] THENL
13188 [FIRST_X_ASSUM(fun t -> MP_TAC(SPEC `&1` t) THEN
13189 MP_TAC(SPEC `-- &1` t)) THEN ASM_REAL_ARITH_TAC;
13190 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `a / &2 / b`) THEN
13191 ASM_SIMP_TAC[REAL_FIELD
13192 `&0 < b ==> (b * (a / b) pow 2) = a pow 2 / b`] THEN
13193 REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN SIMP_TAC[GSYM real_div] THEN
13194 ASM_SIMP_TAC[REAL_LE_DIV2_EQ] THEN
13195 REWRITE_TAC[REAL_LT_SQUARE; REAL_ARITH
13196 `(a * a) / &2 <= (a / &2) pow 2 <=> ~(&0 < a * a)`]]) in
13197 REPEAT STRIP_TAC THEN
13198 MP_TAC(ISPECL [`\x:real^N. (f x) dot x`;
13199 `s INTER sphere(vec 0:real^N,&1)`]
13200 CONTINUOUS_ATTAINS_SUP) THEN
13201 REWRITE_TAC[EXISTS_IN_GSPEC; FORALL_IN_GSPEC; o_DEF] THEN ANTS_TAC THENL
13202 [ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_DOT2; LINEAR_CONTINUOUS_ON;
13203 CONTINUOUS_ON_ID] THEN
13204 ASM_SIMP_TAC[COMPACT_SPHERE; CLOSED_INTER_COMPACT; CLOSED_SUBSPACE] THEN
13205 FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
13206 `~(s = {a}) ==> a IN s ==> ?b. ~(b = a) /\ b IN s`)) THEN
13207 ASM_SIMP_TAC[SUBSPACE_0; IN_SPHERE_0; GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
13208 DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
13209 EXISTS_TAC `inv(norm x) % x:real^N` THEN
13210 ASM_REWRITE_TAC[IN_ELIM_THM; VECTOR_SUB_RZERO; NORM_MUL] THEN
13211 ASM_SIMP_TAC[SUBSPACE_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN
13212 ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0];
13213 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `v:real^N` THEN
13214 REWRITE_TAC[IN_INTER; IN_SPHERE_0] THEN STRIP_TAC THEN
13215 ABBREV_TAC `c = (f:real^N->real^N) v dot v` THEN
13216 EXISTS_TAC `c:real` THEN ASM_REWRITE_TAC[]] THEN
13217 ABBREV_TAC `p = \x y:real^N. c * (x dot y) - (f x) dot y` THEN
13218 SUBGOAL_THEN `!x:real^N. x IN s ==> &0 <= p x x` (LABEL_TAC "POSDEF") THENL
13219 [X_GEN_TAC `x:real^N` THEN EXPAND_TAC "p" THEN REWRITE_TAC[] THEN
13220 ASM_CASES_TAC `x:real^N = vec 0` THEN DISCH_TAC THEN
13221 ASM_REWRITE_TAC[DOT_RZERO; REAL_MUL_RZERO; REAL_SUB_LE; REAL_LE_REFL] THEN
13222 FIRST_X_ASSUM(MP_TAC o SPEC `inv(norm x) % x:real^N`) THEN
13223 ASM_SIMP_TAC[SUBSPACE_MUL] THEN
13224 ASM_SIMP_TAC[LINEAR_CMUL; NORM_MUL; REAL_ABS_INV; DOT_RMUL] THEN
13225 ASM_SIMP_TAC[REAL_ABS_NORM; REAL_MUL_LINV; NORM_EQ_0; DOT_LMUL] THEN
13226 ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; DOT_POS_LT] THEN
13227 REWRITE_TAC[GSYM NORM_POW_2; real_div; REAL_INV_POW] THEN REAL_ARITH_TAC;
13229 SUBGOAL_THEN `!y:real^N. y IN s ==> !a. p v y * a <= p y y * a pow 2`
13231 [REPEAT STRIP_TAC THEN
13232 REMOVE_THEN "POSDEF" (MP_TAC o SPEC `v - (&2 * a) % y:real^N`) THEN
13233 EXPAND_TAC "p" THEN ASM_SIMP_TAC[SUBSPACE_SUB; SUBSPACE_MUL] THEN
13234 ASM_SIMP_TAC[LINEAR_SUB; LINEAR_CMUL] THEN
13235 REWRITE_TAC[DOT_LSUB; DOT_LMUL] THEN
13236 REWRITE_TAC[DOT_RSUB; DOT_RMUL] THEN
13237 SUBGOAL_THEN `f y dot (v:real^N) = f v dot y` SUBST1_TAC THENL
13238 [ASM_MESON_TAC[ADJOINT_CLAUSES; DOT_SYM]; ALL_TAC] THEN
13239 ASM_REWRITE_TAC[GSYM NORM_POW_2] THEN REWRITE_TAC[NORM_POW_2] THEN
13240 MATCH_MP_TAC(REAL_ARITH
13241 `&4 * (z - y) = x ==> &0 <= x ==> y <= z`) THEN
13242 REWRITE_TAC[DOT_SYM] THEN CONV_TAC REAL_RING;
13243 DISCH_THEN(MP_TAC o GEN `y:real^N` o DISCH `(y:real^N) IN s` o
13244 MATCH_MP lemma o C MP (ASSUME `(y:real^N) IN s`) o SPEC `y:real^N`) THEN
13245 ASM_SIMP_TAC[] THEN EXPAND_TAC "p" THEN
13246 REWRITE_TAC[GSYM DOT_LMUL; GSYM DOT_LSUB] THEN
13247 DISCH_THEN(MP_TAC o SPEC `c % v - f v:real^N`) THEN
13248 ASM_SIMP_TAC[SUBSPACE_MUL; SUBSPACE_SUB; DOT_EQ_0; VECTOR_SUB_EQ]]);;
13250 let SELF_ADJOINT_HAS_EIGENVECTOR = prove
13251 (`!f:real^N->real^N.
13252 linear f /\ adjoint f = f ==> ?v c. norm(v) = &1 /\ f(v) = c % v`,
13253 REPEAT STRIP_TAC THEN
13254 MP_TAC(ISPECL [`f:real^N->real^N`; `(:real^N)`]
13255 SELF_ADJOINT_HAS_EIGENVECTOR_IN_SUBSPACE) THEN
13256 ASM_REWRITE_TAC[SUBSPACE_UNIV; IN_UNIV] THEN DISCH_THEN MATCH_MP_TAC THEN
13257 MATCH_MP_TAC(SET_RULE `!a. ~(a IN s) ==> ~(UNIV = s)`) THEN
13258 EXISTS_TAC `vec 1:real^N` THEN
13259 REWRITE_TAC[IN_SING; VEC_EQ; ARITH_EQ]);;
13261 let SELF_ADJOINT_HAS_EIGENVECTOR_BASIS_OF_SUBSPACE = prove
13262 (`!f:real^N->real^N s.
13263 linear f /\ adjoint f = f /\
13264 subspace s /\ (!x. x IN s ==> f x IN s)
13265 ==> ?b. b SUBSET s /\
13266 pairwise orthogonal b /\
13267 (!x. x IN b ==> norm x = &1 /\ ?c. f(x) = c % x) /\
13272 (`!f:real^N->real^N s.
13273 linear f /\ adjoint f = f /\ subspace s /\ (!x. x IN s ==> f x IN s)
13274 ==> ?b. b SUBSET s /\ b HAS_SIZE dim s /\
13275 pairwise orthogonal b /\
13276 (!x. x IN b ==> norm x = &1 /\ ?c. f(x) = c % x)`,
13277 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN REWRITE_TAC[IMP_IMP] THEN
13278 GEN_TAC THEN STRIP_TAC THEN GEN_TAC THEN
13279 WF_INDUCT_TAC `dim(s:real^N->bool)` THEN STRIP_TAC THEN
13280 ASM_CASES_TAC `dim(s:real^N->bool) = 0` THENL
13281 [EXISTS_TAC `{}:real^N->bool` THEN
13282 ASM_SIMP_TAC[HAS_SIZE_CLAUSES; NOT_IN_EMPTY;
13283 PAIRWISE_EMPTY; EMPTY_SUBSET];
13285 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [DIM_EQ_0]) THEN
13286 DISCH_THEN(ASSUME_TAC o MATCH_MP (SET_RULE
13287 `~(s SUBSET {a}) ==> ~(s = {a})`)) THEN
13288 MP_TAC(ISPECL [`f:real^N->real^N`; `s:real^N->bool`]
13289 SELF_ADJOINT_HAS_EIGENVECTOR_IN_SUBSPACE) THEN
13290 ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
13291 DISCH_THEN(X_CHOOSE_THEN `v:real^N` MP_TAC) THEN
13292 ASM_CASES_TAC `v:real^N = vec 0` THEN ASM_REWRITE_TAC[NORM_0] THEN
13293 CONV_TAC REAL_RAT_REDUCE_CONV THEN
13294 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
13295 FIRST_X_ASSUM(MP_TAC o SPEC `{y:real^N | y IN s /\ orthogonal v y}`) THEN
13296 REWRITE_TAC[SUBSPACE_ORTHOGONAL_TO_VECTOR; IN_ELIM_THM] THEN
13297 MP_TAC(ISPECL [`span {v:real^N}`; `s:real^N->bool`]
13298 DIM_SUBSPACE_ORTHOGONAL_TO_VECTORS) THEN
13299 REWRITE_TAC[ONCE_REWRITE_RULE[ORTHOGONAL_SYM] ORTHOGONAL_TO_SPAN_EQ] THEN
13300 ASM_REWRITE_TAC[SUBSPACE_SPAN; IN_SING; FORALL_UNWIND_THM2] THEN
13302 [MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN ASM SET_TAC[];
13303 DISCH_THEN(SUBST1_TAC o SYM)] THEN
13304 ASM_REWRITE_TAC[DIM_SPAN; DIM_SING; ARITH_RULE `n < n + 1`] THEN
13306 [REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN
13307 ASM_SIMP_TAC[SUBSPACE_INTER; SUBSPACE_ORTHOGONAL_TO_VECTOR] THEN
13308 REWRITE_TAC[orthogonal] THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
13309 MATCH_MP_TAC EQ_TRANS THEN
13310 EXISTS_TAC `(f:real^N->real^N) v dot x` THEN CONJ_TAC THENL
13311 [ASM_MESON_TAC[ADJOINT_CLAUSES];
13312 ASM_MESON_TAC[DOT_LMUL; REAL_MUL_RZERO]];
13313 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
13314 EXISTS_TAC `(v:real^N) INSERT b` THEN
13315 ASM_REWRITE_TAC[FORALL_IN_INSERT] THEN
13316 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
13317 ASM_REWRITE_TAC[PAIRWISE_INSERT] THEN
13318 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE; SUBSET; IN_ELIM_THM]) THEN
13320 [ASM_SIMP_TAC[HAS_SIZE; FINITE_INSERT; CARD_CLAUSES] THEN
13321 COND_CASES_TAC THEN ASM_REWRITE_TAC[ADD1] THEN
13322 ASM_MESON_TAC[ORTHOGONAL_REFL];
13323 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_ELIM_THM]) THEN
13324 ASM_MESON_TAC[ORTHOGONAL_SYM]]]) in
13325 REPEAT STRIP_TAC THEN
13326 MP_TAC(ISPECL [`f:real^N->real^N`; `s:real^N->bool`] lemma) THEN
13327 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
13328 X_GEN_TAC `b:real^N->bool` THEN
13329 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
13330 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
13331 [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN
13332 ASM_MESON_TAC[NORM_ARITH `~(norm(vec 0:real^N) = &1)`];
13333 DISCH_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
13334 [ASM_MESON_TAC[SPAN_SUBSET_SUBSPACE];
13335 MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
13336 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
13337 ASM_REWRITE_TAC[LE_REFL]]]);;
13339 let SELF_ADJOINT_HAS_EIGENVECTOR_BASIS = prove
13340 (`!f:real^N->real^N.
13341 linear f /\ adjoint f = f
13342 ==> ?b. pairwise orthogonal b /\
13343 (!x. x IN b ==> norm x = &1 /\ ?c. f(x) = c % x) /\
13345 span b = (:real^N) /\
13346 b HAS_SIZE (dimindex(:N))`,
13347 REPEAT STRIP_TAC THEN
13348 MP_TAC(ISPECL [`f:real^N->real^N`; `(:real^N)`]
13349 SELF_ADJOINT_HAS_EIGENVECTOR_BASIS_OF_SUBSPACE) THEN
13350 ASM_REWRITE_TAC[SUBSPACE_UNIV; IN_UNIV; DIM_UNIV; SUBSET_UNIV]);;
13352 (* ------------------------------------------------------------------------- *)
13353 (* Diagonalization of symmetric matrix. *)
13354 (* ------------------------------------------------------------------------- *)
13356 let SYMMETRIC_MATRIX_DIAGONALIZABLE_EXPLICIT = prove
13359 ==> ?P d. orthogonal_matrix P /\
13360 transp P ** A ** P = (lambda i j. if i = j then d i else &0)`,
13362 (`!A:real^N^N P:real^N^N d.
13363 A ** P = P ** (lambda i j. if i = j then d i else &0) <=>
13364 !i. 1 <= i /\ i <= dimindex(:N)
13365 ==> A ** column i P = d i % column i P`,
13366 SIMP_TAC[CART_EQ; matrix_mul; matrix_vector_mul; LAMBDA_BETA;
13367 column; VECTOR_MUL_COMPONENT] THEN
13368 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[COND_RAND] THEN
13369 SIMP_TAC[REAL_MUL_RZERO; SUM_DELTA; IN_NUMSEG] THEN
13370 EQ_TAC THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN
13371 REWRITE_TAC[REAL_MUL_SYM]) in
13373 (`!A:real^N^N P:real^N^N d.
13374 orthogonal_matrix P /\
13375 transp P ** A ** P = (lambda i j. if i = j then d i else &0) <=>
13376 orthogonal_matrix P /\
13377 !i. 1 <= i /\ i <= dimindex(:N)
13378 ==> A ** column i P = d i % column i P`,
13379 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM lemma1; orthogonal_matrix] THEN
13380 ABBREV_TAC `D:real^N^N = lambda i j. if i = j then d i else &0` THEN
13381 MESON_TAC[MATRIX_MUL_ASSOC; MATRIX_MUL_LID]) in
13382 REPEAT STRIP_TAC THEN
13383 REWRITE_TAC[lemma2] THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
13384 REWRITE_TAC[GSYM SKOLEM_THM] THEN
13385 MP_TAC(ISPEC `\x:real^N. (A:real^N^N) ** x`
13386 SELF_ADJOINT_HAS_EIGENVECTOR_BASIS) THEN
13387 ASM_SIMP_TAC[MATRIX_SELF_ADJOINT; MATRIX_VECTOR_MUL_LINEAR;
13388 MATRIX_OF_MATRIX_VECTOR_MUL] THEN
13389 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` MP_TAC) THEN
13390 REWRITE_TAC[CONJ_ASSOC] THEN ONCE_REWRITE_TAC[IMP_CONJ_ALT] THEN
13391 REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC THEN
13392 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [FINITE_INDEX_NUMSEG]) THEN
13393 ASM_REWRITE_TAC[IN_NUMSEG; TAUT
13394 `p /\ q /\ x = y ==> a = b <=> p /\ q /\ ~(a = b) ==> ~(x = y)`] THEN
13395 DISCH_THEN(X_CHOOSE_THEN `f:num->real^N` STRIP_ASSUME_TAC) THEN
13396 ASM_REWRITE_TAC[PAIRWISE_IMAGE; FORALL_IN_IMAGE] THEN
13397 ASM_SIMP_TAC[pairwise; IN_NUMSEG] THEN STRIP_TAC THEN
13398 EXISTS_TAC `transp(lambda i. f i):real^N^N` THEN
13399 SIMP_TAC[COLUMN_TRANSP; ORTHOGONAL_MATRIX_TRANSP] THEN
13400 SIMP_TAC[ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_INDEXED; row] THEN
13401 SIMP_TAC[LAMBDA_ETA; LAMBDA_BETA; pairwise; IN_NUMSEG] THEN
13404 let SYMMETRIC_MATRIX_IMP_DIAGONALIZABLE = prove
13407 ==> ?P. orthogonal_matrix P /\ diagonal_matrix(transp P ** A ** P)`,
13409 DISCH_THEN(MP_TAC o MATCH_MP SYMMETRIC_MATRIX_DIAGONALIZABLE_EXPLICIT) THEN
13410 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
13411 SIMP_TAC[diagonal_matrix; LAMBDA_BETA]);;
13413 let SYMMETRIC_MATRIX_EQ_DIAGONALIZABLE = prove
13416 ?P. orthogonal_matrix P /\ diagonal_matrix(transp P ** A ** P)`,
13417 GEN_TAC THEN EQ_TAC THEN
13418 REWRITE_TAC[SYMMETRIC_MATRIX_IMP_DIAGONALIZABLE] THEN
13419 REWRITE_TAC[orthogonal_matrix] THEN
13420 DISCH_THEN(X_CHOOSE_THEN `P:real^N^N` STRIP_ASSUME_TAC) THEN
13421 ABBREV_TAC `D:real^N^N = transp P ** (A:real^N^N) ** P` THEN
13422 SUBGOAL_THEN `A:real^N^N = P ** (D:real^N^N) ** transp P` SUBST1_TAC THENL
13423 [EXPAND_TAC "D" THEN REWRITE_TAC[MATRIX_MUL_ASSOC] THEN
13424 ASM_REWRITE_TAC[MATRIX_MUL_LID] THEN
13425 ASM_REWRITE_TAC[GSYM MATRIX_MUL_ASSOC; MATRIX_MUL_RID];
13426 REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_TRANSP; MATRIX_MUL_ASSOC] THEN
13427 ASM_MESON_TAC[TRANSP_DIAGONAL_MATRIX]]);;
13429 (* ------------------------------------------------------------------------- *)
13430 (* Some matrix identities are easier to deduce for invertible matrices. We *)
13431 (* can then extend by continuity, which is why this material needs to be *)
13432 (* here after basic topological notions have been defined. *)
13433 (* ------------------------------------------------------------------------- *)
13435 let CONTINUOUS_LIFT_DET = prove
13436 (`!(A:A->real^N^N) net.
13437 (!i j. 1 <= i /\ i <= dimindex(:N) /\
13438 1 <= j /\ j <= dimindex(:N)
13439 ==> (\x. lift(A x$i$j)) continuous net)
13440 ==> (\x. lift(det(A x))) continuous net`,
13441 REPEAT STRIP_TAC THEN REWRITE_TAC[det] THEN
13442 SIMP_TAC[LIFT_SUM; FINITE_PERMUTATIONS; FINITE_NUMSEG; o_DEF] THEN
13443 MATCH_MP_TAC CONTINUOUS_VSUM THEN
13444 SIMP_TAC[FINITE_PERMUTATIONS; FINITE_NUMSEG; LIFT_CMUL; IN_ELIM_THM] THEN
13445 X_GEN_TAC `p:num->num` THEN DISCH_TAC THEN
13446 MATCH_MP_TAC CONTINUOUS_CMUL THEN
13447 MATCH_MP_TAC CONTINUOUS_LIFT_PRODUCT THEN
13448 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
13449 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
13450 FIRST_ASSUM(MP_TAC o MATCH_MP PERMUTES_IMAGE) THEN
13451 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `s = t ==> s SUBSET t`)) THEN
13452 ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG]);;
13454 let CONTINUOUS_ON_LIFT_DET = prove
13455 (`!A:real^M->real^N^N s.
13456 (!i j. 1 <= i /\ i <= dimindex(:N) /\
13457 1 <= j /\ j <= dimindex(:N)
13458 ==> (\x. lift(A x$i$j)) continuous_on s)
13459 ==> (\x. lift(det(A x))) continuous_on s`,
13460 SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN; CONTINUOUS_LIFT_DET]);;
13462 let NEARBY_INVERTIBLE_MATRIX = prove
13464 ?e. &0 < e /\ !x. ~(x = &0) /\ abs x < e ==> invertible(A + x %% mat 1)`,
13465 GEN_TAC THEN MP_TAC(ISPEC `A:real^N^N` CHARACTERISTIC_POLYNOMIAL) THEN
13466 DISCH_THEN(X_CHOOSE_THEN `a:num->real` STRIP_ASSUME_TAC) THEN
13467 MP_TAC(ISPECL [`dimindex(:N)`; `a:num->real`] REAL_POLYFUN_FINITE_ROOTS) THEN
13468 MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`) THEN CONJ_TAC THENL
13469 [EXISTS_TAC `dimindex(:N)` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC;
13471 DISCH_THEN(MP_TAC o ISPEC `lift` o MATCH_MP FINITE_IMAGE) THEN
13472 DISCH_THEN(MP_TAC o MATCH_MP LIMIT_POINT_FINITE) THEN
13473 DISCH_THEN(MP_TAC o SPEC `lift(&0)`) THEN
13474 REWRITE_TAC[LIMPT_APPROACHABLE; EXISTS_IN_IMAGE; EXISTS_IN_GSPEC] THEN
13475 REWRITE_TAC[DIST_LIFT; LIFT_EQ; REAL_SUB_RZERO; NOT_FORALL_THM; NOT_IMP] THEN
13476 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
13477 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
13478 DISCH_THEN(fun th -> X_GEN_TAC `x:real` THEN STRIP_TAC THEN
13479 MP_TAC(SPEC `--x:real` th)) THEN
13480 FIRST_X_ASSUM(SUBST1_TAC o SYM o SPEC `--x:real`) THEN
13481 ASM_REWRITE_TAC[REAL_NEG_EQ_0; REAL_ABS_NEG] THEN
13482 ONCE_REWRITE_TAC[GSYM INVERTIBLE_NEG] THEN
13483 REWRITE_TAC[INVERTIBLE_DET_NZ; CONTRAPOS_THM] THEN
13484 REWRITE_TAC[MATRIX_SUB; MATRIX_NEG_MINUS1] THEN
13485 ONCE_REWRITE_TAC[REAL_ARITH `--x = -- &1 * x`] THEN
13486 REWRITE_TAC[GSYM MATRIX_CMUL_ADD_LDISTRIB; GSYM MATRIX_CMUL_ASSOC] THEN
13487 REWRITE_TAC[MATRIX_CMUL_LID; MATRIX_ADD_SYM]);;
13489 let MATRIX_WLOG_INVERTIBLE = prove
13490 (`!P. (!A:real^N^N. invertible A ==> P A) /\
13491 (!A:real^N^N. ?d. &0 < d /\
13492 closed {x | x IN cball(vec 0,d) /\
13493 P(A + drop x %% mat 1)})
13494 ==> !A:real^N^N. P A`,
13495 REPEAT GEN_TAC THEN
13496 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
13497 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
13498 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
13499 FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^1` o
13500 GEN_REWRITE_RULE I [CLOSED_LIMPT]) THEN
13501 ASM_SIMP_TAC[IN_ELIM_THM; DROP_VEC; MATRIX_CMUL_LZERO; MATRIX_ADD_RID] THEN
13502 ANTS_TAC THENL [ALL_TAC; CONV_TAC TAUT] THEN
13503 MP_TAC(ISPEC `A:real^N^N` NEARBY_INVERTIBLE_MATRIX) THEN
13504 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
13505 REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `k:real` THEN
13506 DISCH_TAC THEN REWRITE_TAC[EXISTS_LIFT; IN_ELIM_THM] THEN
13507 REWRITE_TAC[GSYM LIFT_NUM; IN_CBALL_0; NORM_LIFT; DIST_LIFT] THEN
13508 REWRITE_TAC[REAL_SUB_RZERO; LIFT_EQ; LIFT_DROP] THEN
13509 EXISTS_TAC `min d ((min e k) / &2)` THEN
13510 CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
13511 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; FIRST_X_ASSUM MATCH_MP_TAC] THEN
13512 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC);;
13514 let SYLVESTER_DETERMINANT_IDENTITY = prove
13515 (`!A:real^N^M B:real^M^N. det(mat 1 + A ** B) = det(mat 1 + B ** A)`,
13517 (`!A:real^N^N B:real^N^N. det(mat 1 + A ** B) = det(mat 1 + B ** A)`,
13518 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN GEN_TAC THEN
13519 MATCH_MP_TAC MATRIX_WLOG_INVERTIBLE THEN CONJ_TAC THENL
13520 [REPEAT STRIP_TAC THEN
13521 SUBGOAL_THEN `det((mat 1 + A ** B) ** A:real^N^N) =
13522 det(A ** (mat 1 + B ** A))`
13524 [REWRITE_TAC[MATRIX_ADD_RDISTRIB; MATRIX_ADD_LDISTRIB] THEN
13525 REWRITE_TAC[MATRIX_MUL_LID; MATRIX_MUL_RID; MATRIX_MUL_ASSOC];
13526 REWRITE_TAC[DET_MUL] THEN
13527 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INVERTIBLE_DET_NZ]) THEN
13528 CONV_TAC REAL_RING];
13529 X_GEN_TAC `A:real^N^N` THEN EXISTS_TAC `&1` THEN
13530 REWRITE_TAC[REAL_LT_01; SET_RULE
13531 `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN
13532 MATCH_MP_TAC CLOSED_INTER THEN REWRITE_TAC[CLOSED_CBALL] THEN
13533 ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
13534 REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN
13535 REWRITE_TAC[SET_RULE `{x | f x = a} = {x | f x IN {a}}`] THEN
13536 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN
13537 REWRITE_TAC[CLOSED_SING; LIFT_SUB] THEN X_GEN_TAC `x:real^1` THEN
13538 REWRITE_TAC[o_DEF; LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_SUB THEN
13539 CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN
13540 MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN
13541 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; LIFT_ADD] THEN
13542 MATCH_MP_TAC CONTINUOUS_ADD THEN
13543 ASM_SIMP_TAC[matrix_mul; LAMBDA_BETA; CONTINUOUS_CONST] THEN
13544 SIMP_TAC[LIFT_SUM; FINITE_NUMSEG; o_DEF] THEN
13545 MATCH_MP_TAC CONTINUOUS_VSUM THEN
13546 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `k:num` THEN
13547 DISCH_TAC THENL [ONCE_REWRITE_TAC[REAL_MUL_SYM]; ALL_TAC] THEN
13548 REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_CMUL THEN
13549 REWRITE_TAC[MATRIX_ADD_COMPONENT; MATRIX_CMUL_COMPONENT; LIFT_ADD] THEN
13550 MATCH_MP_TAC CONTINUOUS_ADD THEN REWRITE_TAC[CONTINUOUS_CONST] THEN
13551 REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] LIFT_CMUL] THEN
13552 MATCH_MP_TAC CONTINUOUS_CMUL THEN
13553 REWRITE_TAC[LIFT_DROP; CONTINUOUS_AT_ID]]) in
13555 (`!A:real^N^M B:real^M^N.
13556 dimindex(:M) <= dimindex(:N)
13557 ==> det(mat 1 + A ** B) = det(mat 1 + B ** A)`,
13558 REPEAT STRIP_TAC THEN
13559 MAP_EVERY ABBREV_TAC
13561 lambda i j. if i <= dimindex(:M) then (A:real^N^M)$i$j
13564 lambda i j. if j <= dimindex(:M) then (B:real^M^N)$i$j
13566 MP_TAC(ISPECL [`A':real^N^N`; `B':real^N^N`] lemma1) THEN
13568 `(B':real^N^N) ** (A':real^N^N) = (B:real^M^N) ** (A:real^N^M)`
13570 [MAP_EVERY EXPAND_TAC ["A'"; "B'"] THEN
13571 SIMP_TAC[CART_EQ; LAMBDA_BETA; matrix_mul] THEN REPEAT STRIP_TAC THEN
13572 MATCH_MP_TAC SUM_EQ_SUPERSET THEN
13573 ASM_SIMP_TAC[IN_NUMSEG; REAL_MUL_LZERO; FINITE_NUMSEG; SUBSET_NUMSEG;
13574 LE_REFL; TAUT `(p /\ q) /\ ~(p /\ r) <=> p /\ q /\ ~r`];
13575 DISCH_THEN(SUBST1_TAC o SYM)] THEN
13576 REWRITE_TAC[det] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
13577 `sum {p | p permutes 1..dimindex(:N) /\ !i. dimindex(:M) < i ==> p i = i}
13578 (\p. sign p * product (1..dimindex(:N))
13579 (\i. (mat 1 + (A':real^N^N) ** (B':real^N^N))$i$p i))` THEN
13582 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_SUPERSET THEN
13583 CONJ_TAC THENL [SET_TAC[]; SIMP_TAC[IN_ELIM_THM; IMP_CONJ]] THEN
13584 X_GEN_TAC `p:num->num` THEN REPEAT STRIP_TAC THEN
13585 REWRITE_TAC[REAL_ENTIRE; PRODUCT_EQ_0_NUMSEG] THEN DISJ2_TAC THEN
13586 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
13587 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:num` THEN
13588 REWRITE_TAC[NOT_IMP] THEN STRIP_TAC THEN
13589 FIRST_ASSUM(MP_TAC o SPEC `k:num` o CONJUNCT1 o
13590 GEN_REWRITE_RULE I [permutes]) THEN
13591 ASM_REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
13592 FIRST_ASSUM(MP_TAC o MATCH_MP PERMUTES_IMAGE) THEN
13593 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `s = t ==> s SUBSET t`)) THEN
13594 ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG] THEN
13595 DISCH_THEN(MP_TAC o SPEC `k:num`) THEN ASM_SIMP_TAC[] THEN STRIP_TAC THEN
13596 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; MAT_COMPONENT; REAL_ADD_LID] THEN
13597 ASM_SIMP_TAC[matrix_mul; LAMBDA_BETA] THEN
13598 MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN REPEAT STRIP_TAC THEN
13599 REWRITE_TAC[REAL_ENTIRE] THEN DISJ1_TAC THEN EXPAND_TAC "A'" THEN
13600 ASM_SIMP_TAC[LAMBDA_BETA; GSYM NOT_LT]] THEN
13601 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_EQ_GENERAL THEN
13602 EXISTS_TAC `\f:num->num. f` THEN REWRITE_TAC[IN_ELIM_THM] THEN
13603 CONJ_TAC THEN X_GEN_TAC `p:num->num` THEN STRIP_TAC THENL
13604 [REWRITE_TAC[MESON[] `(?!x. P x /\ x = y) <=> P y`] THEN CONJ_TAC THENL
13605 [MATCH_MP_TAC PERMUTES_SUBSET THEN
13606 EXISTS_TAC `1..dimindex(:M)` THEN
13607 ASM_REWRITE_TAC[SUBSET_NUMSEG; LE_REFL];
13608 X_GEN_TAC `k:num` THEN DISCH_TAC THEN
13609 FIRST_X_ASSUM(MATCH_MP_TAC o CONJUNCT1 o
13610 GEN_REWRITE_RULE I [permutes]) THEN
13611 ASM_REWRITE_TAC[IN_NUMSEG; DE_MORGAN_THM; NOT_LE]];
13612 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
13613 [MATCH_MP_TAC PERMUTES_SUPERSET THEN
13614 EXISTS_TAC `1..dimindex(:N)` THEN
13615 ASM_REWRITE_TAC[IN_DIFF; IN_NUMSEG] THEN ASM_MESON_TAC[NOT_LE];
13617 AP_TERM_TAC THEN FIRST_ASSUM(SUBST1_TAC o MATCH_MP (ARITH_RULE
13618 `m:num <= n ==> n = m + (n - m)`)) THEN
13619 SIMP_TAC[PRODUCT_ADD_SPLIT; ARITH_RULE `1 <= n + 1`] THEN
13620 MATCH_MP_TAC(REAL_RING `x = y /\ z = &1 ==> x = y * z`) THEN
13622 [MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN
13623 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
13624 SUBGOAL_THEN `i <= dimindex(:N)` ASSUME_TAC THENL
13625 [ASM_ARITH_TAC; ALL_TAC] THEN
13626 MP_TAC(ISPECL [`p:num->num`; `1..dimindex(:M)`] PERMUTES_IMAGE) THEN
13627 ASM_REWRITE_TAC[] THEN
13628 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE `s = t ==> s SUBSET t`)) THEN
13629 ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG] THEN
13630 DISCH_THEN(MP_TAC o SPEC `i:num`) THEN
13631 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
13632 SUBGOAL_THEN `(p:num->num) i <= dimindex(:N)` ASSUME_TAC THENL
13633 [ASM_ARITH_TAC; ALL_TAC] THEN
13634 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; MAT_COMPONENT] THEN
13635 AP_TERM_TAC THEN ASM_SIMP_TAC[matrix_mul; LAMBDA_BETA] THEN
13636 MATCH_MP_TAC SUM_EQ_NUMSEG THEN REPEAT STRIP_TAC THEN
13637 MAP_EVERY EXPAND_TAC ["A'"; "B'"] THEN
13638 ASM_SIMP_TAC[LAMBDA_BETA];
13639 MATCH_MP_TAC PRODUCT_EQ_1_NUMSEG THEN
13640 ASM_SIMP_TAC[ARITH_RULE `n + 1 <= i ==> n < i`] THEN
13641 ASM_SIMP_TAC[ARITH_RULE `m:num <= n ==> m + (n - m) = n`] THEN
13642 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
13643 SUBGOAL_THEN `1 <= i` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
13644 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; MAT_COMPONENT] THEN
13645 ASM_SIMP_TAC[REAL_EQ_ADD_LCANCEL_0; matrix_mul; LAMBDA_BETA] THEN
13646 MATCH_MP_TAC SUM_EQ_0_NUMSEG THEN REPEAT STRIP_TAC THEN
13647 REWRITE_TAC[REAL_ENTIRE] THEN DISJ1_TAC THEN EXPAND_TAC "A'" THEN
13648 ASM_SIMP_TAC[LAMBDA_BETA; ARITH_RULE `m + 1 <= i ==> ~(i <= m)`]]]) in
13649 REPEAT GEN_TAC THEN DISJ_CASES_TAC (ARITH_RULE
13650 `dimindex(:M) <= dimindex(:N) \/ dimindex(:N) <= dimindex(:M)`)
13651 THENL [ALL_TAC; CONV_TAC SYM_CONV] THEN
13652 MATCH_MP_TAC lemma2 THEN ASM_REWRITE_TAC[]);;
13654 let COFACTOR_MATRIX_MUL = prove
13655 (`!A B:real^N^N. cofactor(A ** B) = cofactor(A) ** cofactor(B)`,
13656 MATCH_MP_TAC MATRIX_WLOG_INVERTIBLE THEN CONJ_TAC THENL
13657 [GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC MATRIX_WLOG_INVERTIBLE THEN
13659 [ASM_SIMP_TAC[COFACTOR_MATRIX_INV; GSYM INVERTIBLE_DET_NZ;
13660 INVERTIBLE_MATRIX_MUL] THEN
13661 REWRITE_TAC[DET_MUL; MATRIX_MUL_LMUL] THEN
13662 REWRITE_TAC[MATRIX_MUL_RMUL; MATRIX_CMUL_ASSOC;
13663 GSYM MATRIX_TRANSP_MUL] THEN
13664 ASM_SIMP_TAC[MATRIX_INV_MUL];
13665 GEN_TAC THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01]];
13666 X_GEN_TAC `A:real^N^N` THEN EXISTS_TAC `&1` THEN
13667 REWRITE_TAC[REAL_LT_01] THEN REWRITE_TAC[RIGHT_AND_FORALL_THM] THEN
13668 MATCH_MP_TAC CLOSED_FORALL THEN GEN_TAC] THEN
13669 REWRITE_TAC[SET_RULE
13670 `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN
13671 MATCH_MP_TAC CLOSED_INTER THEN REWRITE_TAC[CLOSED_CBALL] THEN
13672 REWRITE_TAC[CART_EQ] THEN
13673 MATCH_MP_TAC CLOSED_FORALL_IN THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
13674 MATCH_MP_TAC CLOSED_FORALL_IN THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN
13675 ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
13676 REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN
13677 REWRITE_TAC[SET_RULE `{x | f x = a} = {x | f x IN {a}}`] THEN
13678 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN
13679 REWRITE_TAC[CLOSED_SING; LIFT_SUB] THEN X_GEN_TAC `x:real^1` THEN
13680 ASM_SIMP_TAC[matrix_mul; LAMBDA_BETA; cofactor; LIFT_SUM;
13681 FINITE_NUMSEG; o_DEF] THEN
13682 (MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THENL
13684 MATCH_MP_TAC CONTINUOUS_VSUM THEN
13685 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
13686 X_GEN_TAC `k:num` THEN STRIP_TAC THEN
13687 REWRITE_TAC[LIFT_CMUL] THEN MATCH_MP_TAC CONTINUOUS_MUL THEN
13688 REWRITE_TAC[o_DEF] THEN CONJ_TAC]) THEN
13689 MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN
13690 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN
13691 ASM_SIMP_TAC[LAMBDA_BETA; CONTINUOUS_CONST] THEN
13692 REPEAT(W(fun (asl,w) ->
13693 let t = find_term is_cond w in
13694 ASM_CASES_TAC (lhand(rator t)) THEN ASM_REWRITE_TAC[CONTINUOUS_CONST])) THEN
13695 SIMP_TAC[LIFT_SUM; FINITE_NUMSEG; o_DEF] THEN
13696 TRY(MATCH_MP_TAC CONTINUOUS_VSUM THEN REWRITE_TAC[FINITE_NUMSEG] THEN
13697 REWRITE_TAC[IN_NUMSEG] THEN X_GEN_TAC `p:num` THEN STRIP_TAC) THEN
13698 REWRITE_TAC[LIFT_CMUL] THEN
13699 TRY(MATCH_MP_TAC CONTINUOUS_MUL THEN
13700 REWRITE_TAC[o_DEF; CONTINUOUS_CONST]) THEN
13701 REWRITE_TAC[MATRIX_ADD_COMPONENT; LIFT_ADD] THEN
13702 MATCH_MP_TAC CONTINUOUS_ADD THEN REWRITE_TAC[CONTINUOUS_CONST] THEN
13703 REWRITE_TAC[MATRIX_CMUL_COMPONENT; LIFT_CMUL; o_DEF] THEN
13704 MATCH_MP_TAC CONTINUOUS_MUL THEN
13705 REWRITE_TAC[CONTINUOUS_CONST; o_DEF; LIFT_DROP; CONTINUOUS_AT_ID]);;
13707 let DET_COFACTOR = prove
13708 (`!A:real^N^N. det(cofactor A) = det(A) pow (dimindex(:N) - 1)`,
13709 MATCH_MP_TAC MATRIX_WLOG_INVERTIBLE THEN CONJ_TAC THEN
13710 X_GEN_TAC `A:real^N^N` THENL
13711 [REWRITE_TAC[INVERTIBLE_DET_NZ] THEN STRIP_TAC THEN
13712 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_FIELD
13713 `~(a = &0) ==> a * x = a * y ==> x = y`)) THEN
13714 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM DET_TRANSP] THEN
13715 REWRITE_TAC[GSYM DET_MUL; MATRIX_MUL_RIGHT_COFACTOR] THEN
13716 REWRITE_TAC[DET_CMUL; GSYM(CONJUNCT2 real_pow); DET_I; REAL_MUL_RID] THEN
13717 SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> SUC(n - 1) = n`];
13719 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN
13720 REWRITE_TAC[SET_RULE
13721 `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN
13722 MATCH_MP_TAC CLOSED_INTER THEN REWRITE_TAC[CLOSED_CBALL] THEN
13723 ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
13724 REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN
13725 REWRITE_TAC[SET_RULE `{x | f x = a} = {x | f x IN {a}}`] THEN
13726 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN
13727 REWRITE_TAC[CLOSED_SING; LIFT_SUB] THEN X_GEN_TAC `x:real^1` THEN
13728 MATCH_MP_TAC CONTINUOUS_SUB THEN
13729 CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CONTINUOUS_LIFT_POW] THEN
13730 MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN
13731 MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN
13732 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; MATRIX_CMUL_COMPONENT; LIFT_ADD;
13733 LIFT_CMUL; LIFT_DROP; CONTINUOUS_ADD; CONTINUOUS_CONST;
13734 CONTINUOUS_MUL; o_DEF; LIFT_DROP; CONTINUOUS_AT_ID] THEN
13735 ASM_SIMP_TAC[cofactor; LAMBDA_BETA] THEN
13736 MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN
13737 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN
13738 ASM_SIMP_TAC[LAMBDA_BETA] THEN
13739 REPEAT(W(fun (asl,w) ->
13740 let t = find_term is_cond w in
13741 ASM_CASES_TAC (lhand(rator t)) THEN ASM_REWRITE_TAC[CONTINUOUS_CONST])) THEN
13742 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; MATRIX_CMUL_COMPONENT; LIFT_ADD;
13743 LIFT_CMUL; LIFT_DROP; CONTINUOUS_ADD; CONTINUOUS_CONST;
13744 CONTINUOUS_MUL; o_DEF; LIFT_DROP; CONTINUOUS_AT_ID]);;
13746 let INVERTIBLE_COFACTOR = prove
13747 (`!A:real^N^N. invertible(cofactor A) <=> dimindex(:N) = 1 \/ invertible A`,
13748 SIMP_TAC[DET_COFACTOR; INVERTIBLE_DET_NZ; REAL_POW_EQ_0; DE_MORGAN_THM;
13749 DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (n - 1 = 0 <=> n = 1)`;
13752 let COFACTOR_COFACTOR = prove
13755 ==> cofactor(cofactor A) = (det(A) pow (dimindex(:N) - 2)) %% A`,
13756 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN
13757 MATCH_MP_TAC MATRIX_WLOG_INVERTIBLE THEN CONJ_TAC THEN
13758 X_GEN_TAC `A:real^N^N` THENL
13759 [REWRITE_TAC[INVERTIBLE_DET_NZ] THEN DISCH_TAC THEN
13760 MP_TAC(ISPECL [`A:real^N^N`; `transp(cofactor A):real^N^N`]
13761 COFACTOR_MATRIX_MUL) THEN
13762 REWRITE_TAC[MATRIX_MUL_RIGHT_COFACTOR; COFACTOR_CMUL; COFACTOR_I] THEN
13763 REWRITE_TAC[COFACTOR_TRANSP] THEN
13764 DISCH_THEN(MP_TAC o AP_TERM `transp:real^N^N->real^N^N`) THEN
13765 REWRITE_TAC[MATRIX_TRANSP_MUL; TRANSP_TRANSP; TRANSP_MATRIX_CMUL] THEN
13766 REWRITE_TAC[TRANSP_MAT] THEN
13767 DISCH_THEN(MP_TAC o AP_TERM `(\x. x ** A):real^N^N->real^N^N`) THEN
13768 REWRITE_TAC[GSYM MATRIX_MUL_ASSOC; MATRIX_MUL_LEFT_COFACTOR] THEN
13769 REWRITE_TAC[MATRIX_MUL_LMUL; MATRIX_MUL_RMUL] THEN
13770 REWRITE_TAC[MATRIX_MUL_LID; MATRIX_MUL_RID] THEN
13771 DISCH_THEN(MP_TAC o AP_TERM `\x:real^N^N. inv(det(A:real^N^N)) %% x`) THEN
13772 ASM_SIMP_TAC[MATRIX_CMUL_ASSOC; REAL_MUL_LINV; MATRIX_CMUL_LID] THEN
13773 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_THM_TAC THEN AP_TERM_TAC THEN
13774 ASM_SIMP_TAC[REAL_POW_SUB; ARITH_RULE `2 <= n ==> 1 <= n`] THEN
13775 REWRITE_TAC[REAL_POW_2; real_div; REAL_INV_POW] THEN REAL_ARITH_TAC;
13776 POP_ASSUM(K ALL_TAC)] THEN
13777 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN
13778 REWRITE_TAC[SET_RULE
13779 `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN
13780 MATCH_MP_TAC CLOSED_INTER THEN REWRITE_TAC[CLOSED_CBALL] THEN
13781 REWRITE_TAC[CART_EQ] THEN
13782 MATCH_MP_TAC CLOSED_FORALL_IN THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
13783 MATCH_MP_TAC CLOSED_FORALL_IN THEN X_GEN_TAC `j:num` THEN STRIP_TAC THEN
13784 ONCE_REWRITE_TAC[GSYM REAL_SUB_0] THEN
13785 REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM] THEN
13786 REWRITE_TAC[SET_RULE `{x | f x = a} = {x | f x IN {a}}`] THEN
13787 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE_UNIV THEN
13788 REWRITE_TAC[CLOSED_SING; LIFT_SUB] THEN X_GEN_TAC `x:real^1` THEN
13789 MATCH_MP_TAC CONTINUOUS_SUB THEN CONJ_TAC THENL
13791 (ONCE_REWRITE_TAC[cofactor] THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN
13792 MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN REPEAT STRIP_TAC THEN
13793 ASM_SIMP_TAC[LAMBDA_BETA] THEN
13794 REPEAT(W(fun (asl,w) ->
13795 let t = find_term is_cond w in
13796 ASM_CASES_TAC (lhand(rator t)) THEN
13797 ASM_REWRITE_TAC[CONTINUOUS_CONST])));
13798 REWRITE_TAC[MATRIX_CMUL_COMPONENT; LIFT_CMUL] THEN
13799 MATCH_MP_TAC CONTINUOUS_MUL THEN REWRITE_TAC[o_DEF] THEN CONJ_TAC THENL
13800 [MATCH_MP_TAC CONTINUOUS_LIFT_POW THEN
13801 MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN REPEAT STRIP_TAC;
13803 REWRITE_TAC[MATRIX_ADD_COMPONENT; MATRIX_CMUL_COMPONENT] THEN
13804 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
13805 REWRITE_TAC[LIFT_ADD; LIFT_CMUL; LIFT_DROP] THEN
13806 SIMP_TAC[CONTINUOUS_ADD; CONTINUOUS_CONST; CONTINUOUS_CMUL;
13807 CONTINUOUS_AT_ID]);;
13809 let RANK_COFACTOR_EQ_FULL = prove
13810 (`!A:real^N^N. rank(cofactor A) = dimindex(:N) <=>
13811 dimindex(:N) = 1 \/ rank A = dimindex(:N)`,
13812 REWRITE_TAC[RANK_EQ_FULL_DET; DET_COFACTOR; REAL_POW_EQ_0] THEN
13813 SIMP_TAC[DIMINDEX_GE_1; ARITH_RULE `1 <= n ==> (n - 1 = 0 <=> n = 1)`] THEN
13816 let COFACTOR_EQ_0 = prove
13817 (`!A:real^N^N. cofactor A = mat 0 <=> rank(A) < dimindex(:N) - 1`,
13819 (`!A:real^N^N. rank(A) < dimindex(:N) - 1 ==> cofactor A = mat 0`,
13820 GEN_TAC THEN REWRITE_TAC[RANK_ROW] THEN DISCH_TAC THEN
13821 SIMP_TAC[CART_EQ; cofactor; MAT_COMPONENT; LAMBDA_BETA; COND_ID] THEN
13822 X_GEN_TAC `m:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN
13823 REWRITE_TAC[DET_EQ_0_RANK] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
13824 (ARITH_RULE `r < n - 1 ==> s <= r + 1 ==> s < n`)) THEN
13825 REWRITE_TAC[RANK_ROW; rows] THEN MATCH_MP_TAC LE_TRANS THEN
13827 `dim (basis n INSERT
13828 {row i ((lambda k l. if l = n then &0 else (A:real^N^N)$k$l)
13830 | i IN (1..dimindex(:N)) DELETE m})` THEN
13832 [MATCH_MP_TAC DIM_SUBSET THEN REWRITE_TAC[GSYM IN_NUMSEG] THEN
13833 MATCH_MP_TAC(SET_RULE
13834 `m IN s /\ (!i. i IN s DELETE m ==> f i = g i) /\ f m = a
13835 ==> {f i | i IN s} SUBSET a INSERT {g i | i IN s DELETE m}`) THEN
13836 ASM_SIMP_TAC[IN_NUMSEG; IN_DELETE; row; LAMBDA_BETA; basis; LAMBDA_ETA];
13837 REWRITE_TAC[DIM_INSERT] THEN MATCH_MP_TAC(ARITH_RULE
13838 `n <= k ==> (if p then n else n + 1) <= k + 1`) THEN
13839 MATCH_MP_TAC(MESON[DIM_LINEAR_IMAGE_LE; DIM_SUBSET; LE_TRANS]
13840 `(?f. linear f /\ t SUBSET IMAGE f s) ==> dim t <= dim s`) THEN
13841 EXISTS_TAC `(\x. lambda i. if i = n then &0 else x$i)
13842 :real^N->real^N` THEN
13843 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN CONJ_TAC THENL
13844 [SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
13845 VECTOR_MUL_COMPONENT] THEN
13846 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
13848 X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_NUMSEG; IN_DELETE] THEN
13849 STRIP_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
13850 ONCE_REWRITE_TAC[CONJ_SYM] THEN
13851 REWRITE_TAC[EXISTS_IN_GSPEC] THEN EXISTS_TAC `i:num` THEN
13852 ASM_SIMP_TAC[row; CART_EQ; LAMBDA_BETA]]])
13855 rank A < dimindex(:N)
13856 ==> ?n x. 1 <= n /\ n <= dimindex(:N) /\
13858 rank((lambda i. if i = n then x else row i A):real^N^N)`,
13859 REPEAT STRIP_TAC THEN SUBGOAL_THEN
13860 `?n. 1 <= n /\ n <= dimindex(:N) /\
13861 row n (A:real^N^N) IN
13862 span {row j A | j IN (1..dimindex(:N)) DELETE n}`
13864 [MP_TAC(ISPEC `transp A:real^N^N` HOMOGENEOUS_LINEAR_EQUATIONS_DET) THEN
13865 ASM_REWRITE_TAC[DET_EQ_0_RANK; RANK_TRANSP] THEN
13866 DISCH_THEN(X_CHOOSE_THEN `c:real^N` STRIP_ASSUME_TAC) THEN
13867 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN
13868 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; VEC_COMPONENT] THEN
13869 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN
13870 ASM_REWRITE_TAC[] THEN
13871 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
13872 SIMP_TAC[matrix_vector_mul; transp; VEC_COMPONENT; LAMBDA_BETA] THEN
13874 SUBGOAL_THEN `row n A = vsum ((1..dimindex(:N)) DELETE n)
13875 (\i. --((c:real^N)$i / c$n) % row i (A:real^N^N))`
13877 [ASM_SIMP_TAC[VSUM_DELETE; FINITE_NUMSEG; IN_NUMSEG; REAL_DIV_REFL] THEN
13878 REWRITE_TAC[VECTOR_ARITH `n = x - -- &1 % n <=> x:real^N = vec 0`] THEN
13879 SIMP_TAC[VSUM_COMPONENT; row; VECTOR_MUL_COMPONENT; LAMBDA_BETA;
13880 CART_EQ; REAL_ARITH `--(x / y) * z:real = --(inv y) * z * x`] THEN
13881 ASM_SIMP_TAC[SUM_LMUL; VEC_COMPONENT; REAL_MUL_RZERO];
13882 MATCH_MP_TAC SPAN_VSUM THEN SIMP_TAC[FINITE_DELETE; FINITE_NUMSEG] THEN
13883 X_GEN_TAC `i:num` THEN REWRITE_TAC[IN_DELETE; IN_NUMSEG] THEN
13884 STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
13885 MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]];
13886 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN
13887 ASM_REWRITE_TAC[] THEN
13888 SUBGOAL_THEN `span {row j (A:real^N^N) | j IN (1..dimindex(:N)) DELETE n}
13891 [REWRITE_TAC[PSUBSET; SUBSET_UNIV] THEN
13892 DISCH_THEN(MP_TAC o AP_TERM `dim:(real^N->bool)->num`) THEN
13893 REWRITE_TAC[DIM_UNIV] THEN
13894 MATCH_MP_TAC(ARITH_RULE `1 <= n /\ x <= n - 1 ==> ~(x = n)`) THEN
13895 REWRITE_TAC[DIMINDEX_GE_1; DIM_SPAN] THEN
13896 W(MP_TAC o PART_MATCH (lhand o rand) DIM_LE_CARD o lhand o snd) THEN
13897 ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
13898 SIMP_TAC[FINITE_IMAGE; FINITE_DELETE; FINITE_NUMSEG] THEN
13899 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LE_TRANS) THEN
13900 W(MP_TAC o PART_MATCH (lhand o rand) CARD_IMAGE_LE o lhand o snd) THEN
13901 SIMP_TAC[FINITE_DELETE; FINITE_NUMSEG] THEN
13902 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] LE_TRANS) THEN
13903 ASM_SIMP_TAC[CARD_DELETE; IN_NUMSEG; FINITE_NUMSEG] THEN
13904 REWRITE_TAC[CARD_NUMSEG_1; LE_REFL];
13905 DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
13906 `s PSUBSET UNIV ==> ?x. ~(x IN s)`)) THEN
13907 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN
13908 REWRITE_TAC[RANK_ROW] THEN DISCH_TAC THEN
13910 `!A:real^N^N. rows A = row n A INSERT
13911 {row j A | j IN (1..dimindex (:N)) DELETE n}`
13912 (fun th -> REWRITE_TAC[th])
13914 [REWRITE_TAC[rows; IN_DELETE; IN_NUMSEG] THEN ASM SET_TAC[];
13915 ASM_SIMP_TAC[DIM_INSERT]] THEN
13916 COND_CASES_TAC THENL
13917 [FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
13918 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[]
13919 `x IN span s ==> x = y /\ s = t ==> ~(y IN span t) ==> q`)) THEN
13920 ASM_SIMP_TAC[row; LAMBDA_BETA; LAMBDA_ETA];
13921 MATCH_MP_TAC(ARITH_RULE `s = t ==> s < t + 1`) THEN
13922 AP_TERM_TAC THEN REWRITE_TAC[row]] THEN
13923 MATCH_MP_TAC(SET_RULE
13924 `(!x. x IN s ==> f x = g x) ==> {f x | x IN s} = {g x | x IN s}`) THEN
13925 ASM_SIMP_TAC[IN_DELETE; IN_NUMSEG; LAMBDA_BETA; CART_EQ]]]) in
13926 GEN_TAC THEN EQ_TAC THEN REWRITE_TAC[lemma1] THEN DISCH_TAC THEN
13927 MATCH_MP_TAC(ARITH_RULE
13928 `r <= n /\ ~(r = n) /\ ~(r = n - 1) ==> r < n - 1`) THEN
13929 REPEAT CONJ_TAC THENL
13930 [MP_TAC(ISPEC `A:real^N^N` RANK_BOUND) THEN ARITH_TAC;
13931 REWRITE_TAC[RANK_EQ_FULL_DET] THEN
13932 MP_TAC(SYM(ISPEC `A:real^N^N` MATRIX_MUL_LEFT_COFACTOR)) THEN
13933 ASM_REWRITE_TAC[MATRIX_CMUL_EQ_0; TRANSP_MAT; MATRIX_MUL_LZERO] THEN
13934 REWRITE_TAC[MAT_EQ; ARITH_EQ];
13936 MP_TAC(ISPEC `A:real^N^N` lemma2) THEN
13937 ASM_REWRITE_TAC[DIMINDEX_GE_1; ARITH_RULE `n - 1 < n <=> 1 <= n`] THEN
13938 DISCH_THEN(X_CHOOSE_THEN `n:num` (X_CHOOSE_THEN `x:real^N`
13939 STRIP_ASSUME_TAC)) THEN
13940 FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE
13941 `n - 1 < k ==> k <= MIN n n ==> k = n`)) THEN
13942 REWRITE_TAC[RANK_BOUND; RANK_EQ_FULL_DET] THEN
13943 MP_TAC(GEN `A:real^N^N` (ISPECL [`A:real^N^N`; `n:num`]
13944 DET_COFACTOR_EXPANSION)) THEN
13945 ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN MATCH_MP_TAC SUM_EQ_0 THEN
13946 X_GEN_TAC `m:num` THEN SIMP_TAC[IN_NUMSEG; REAL_ENTIRE] THEN STRIP_TAC THEN
13947 DISJ2_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
13948 DISCH_THEN(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[CART_EQ] THEN
13949 DISCH_THEN(MP_TAC o SPEC `m:num`) THEN
13950 ASM_SIMP_TAC[MAT_COMPONENT; COND_ID] THEN
13951 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] EQ_TRANS) THEN
13952 ASM_SIMP_TAC[cofactor; LAMBDA_BETA] THEN AP_TERM_TAC THEN
13953 ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA; row] THEN
13954 REPEAT STRIP_TAC THEN
13955 REPEAT(COND_CASES_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA]) THEN
13958 let RANK_COFACTOR_EQ_1 = prove
13959 (`!A:real^N^N. rank(cofactor A) = 1 <=>
13960 dimindex(:N) = 1 \/ rank A = dimindex(:N) - 1`,
13961 GEN_TAC THEN ASM_CASES_TAC `dimindex(:N) = 1` THENL
13962 [ASM_MESON_TAC[RANK_COFACTOR_EQ_FULL]; ASM_REWRITE_TAC[]] THEN
13964 [ASM_CASES_TAC `cofactor A:real^N^N = mat 0` THEN
13965 ASM_REWRITE_TAC[RANK_0; ARITH_EQ] THEN DISCH_TAC THEN
13966 MATCH_MP_TAC(ARITH_RULE
13967 `~(r < n - 1) /\ ~(r = n) /\ r <= MIN n n ==> r = n - 1`) THEN
13968 ASM_REWRITE_TAC[RANK_BOUND; GSYM COFACTOR_EQ_0] THEN
13969 MP_TAC(ISPEC `A:real^N^N` RANK_COFACTOR_EQ_FULL) THEN ASM_REWRITE_TAC[];
13970 DISCH_TAC THEN MATCH_MP_TAC(ARITH_RULE
13971 `~(n = 0) /\ n <= 1 ==> n = 1`) THEN
13972 ASM_REWRITE_TAC[RANK_EQ_0; COFACTOR_EQ_0; LT_REFL] THEN
13973 MP_TAC(ISPECL [`A:real^N^N`; `transp(cofactor A):real^N^N`]
13974 RANK_SYLVESTER) THEN
13975 ASM_REWRITE_TAC[MATRIX_MUL_RIGHT_COFACTOR; RANK_TRANSP] THEN
13976 FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE
13977 `a = n - 1 ==> 1 <= n ==> a < n`)) THEN
13978 ASM_SIMP_TAC[GSYM DET_EQ_0_RANK; DIMINDEX_GE_1] THEN
13979 DISCH_TAC THEN REWRITE_TAC[MATRIX_CMUL_LZERO; RANK_0] THEN
13982 let RANK_COFACTOR = prove
13984 rank(cofactor A) = if rank(A) = dimindex(:N) then dimindex(:N)
13985 else if rank(A) = dimindex(:N) - 1 then 1
13987 GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[RANK_COFACTOR_EQ_FULL] THEN
13988 COND_CASES_TAC THEN ASM_REWRITE_TAC[RANK_COFACTOR_EQ_1] THEN
13989 REWRITE_TAC[RANK_EQ_0; COFACTOR_EQ_0] THEN
13990 MATCH_MP_TAC(ARITH_RULE
13991 `r <= MIN n n /\ ~(r = n) /\ ~(r = n - 1) ==> r < n - 1`) THEN
13992 ASM_REWRITE_TAC[RANK_BOUND]);;
13994 (* ------------------------------------------------------------------------- *)
13995 (* Not in so many words, but combining this with intermediate value theorem *)
13996 (* implies the determinant is an open map. *)
13997 (* ------------------------------------------------------------------------- *)
13999 let DET_OPEN_MAP = prove
14002 ==> (?B:real^N^N. (!i j. abs(B$i$j - A$i$j) < e) /\ det B < det A) /\
14003 (?C:real^N^N. (!i j. abs(C$i$j - A$i$j) < e) /\ det C > det A)`,
14006 1 <= i /\ i <= dimindex(:N) /\ row i A = vec 0 /\ &0 < e
14007 ==> (?B:real^N^N. (!i j. abs(B$i$j - A$i$j) < e) /\ det B < &0) /\
14008 (?C:real^N^N. (!i j. abs(C$i$j - A$i$j) < e) /\ det C > &0)`,
14009 REPEAT GEN_TAC THEN STRIP_TAC THEN
14010 SUBGOAL_THEN `det(A:real^N^N) = &0` ASSUME_TAC THENL
14011 [ASM_MESON_TAC[DET_ZERO_ROW]; ALL_TAC] THEN
14012 MP_TAC(ISPEC `A:real^N^N` NEARBY_INVERTIBLE_MATRIX) THEN
14013 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
14014 FIRST_X_ASSUM(MP_TAC o SPEC `min d e / &2`) THEN
14015 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[INVERTIBLE_DET_NZ]] THEN
14016 DISCH_THEN(STRIP_ASSUME_TAC o MATCH_MP (REAL_ARITH
14017 `~(x = &0) ==> x < &0 \/ &0 < x`))
14018 THENL [ALL_TAC; ONCE_REWRITE_TAC[CONJ_SYM]] THEN
14020 [EXISTS_TAC `A + min d e / &2 %% mat 1:real^N^N`;
14021 EXISTS_TAC `(lambda j. if j = i then
14022 --(&1) % row i (A + min d e / &2 %% mat 1:real^N^N)
14023 else row j (A + min d e / &2 %% mat 1:real^N^N))
14025 ASM_SIMP_TAC[DET_ROW_MUL; MESON[]
14026 `(if j = i then f i else f j) = f j`] THEN
14027 REWRITE_TAC[row; LAMBDA_ETA] THEN
14028 ASM_REWRITE_TAC[real_gt; GSYM row] THEN
14029 TRY(CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC]) THEN
14030 (MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN
14031 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !A:real^N^N. A$m = A$k`
14032 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
14033 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$n = z$l`
14034 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC]) THEN
14035 ASM_SIMP_TAC[LAMBDA_BETA] THEN
14036 TRY COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
14037 ASM_SIMP_TAC[MATRIX_ADD_COMPONENT; MATRIX_CMUL_COMPONENT; MAT_COMPONENT;
14038 VECTOR_MUL_COMPONENT] THEN
14039 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CART_EQ]) THEN
14040 DISCH_THEN(MP_TAC o SPEC `l:num`) THEN
14041 ASM_SIMP_TAC[row; LAMBDA_BETA; VEC_COMPONENT] THEN
14042 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC)
14044 (`!A:real^N^N x:real^N i.
14045 1 <= i /\ i <= dimindex(:N) /\ x$i = &1
14046 ==> det(lambda k. if k = i then transp A ** x else row k A) = det A`,
14047 REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
14049 `det(lambda k. if k = i
14050 then row i (A:real^N^N) + (transp A ** x - row i A)
14051 else row k A)` THEN
14053 [REWRITE_TAC[VECTOR_ARITH `r + (x - r):real^N = x`]; ALL_TAC] THEN
14054 MATCH_MP_TAC DET_ROW_SPAN THEN
14056 `transp(A:real^N^N) ** x - row i A =
14057 vsum ((1..dimindex(:N)) DELETE i) (\k. x$k % row k A)`
14059 [SIMP_TAC[CART_EQ; VSUM_COMPONENT; VECTOR_SUB_COMPONENT; row; transp;
14060 LAMBDA_BETA; matrix_vector_mul; VECTOR_MUL_COMPONENT] THEN
14061 ASM_SIMP_TAC[SUM_DELETE; IN_NUMSEG; FINITE_NUMSEG; REAL_MUL_LID] THEN
14062 REWRITE_TAC[REAL_MUL_AC];
14063 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_VSUM THEN
14064 REWRITE_TAC[FINITE_DELETE; IN_DELETE; IN_NUMSEG; FINITE_NUMSEG] THEN
14065 X_GEN_TAC `j:num` THEN STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
14066 MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[]]) in
14067 REPEAT GEN_TAC THEN DISCH_TAC THEN
14068 ASM_CASES_TAC `cofactor(A:real^N^N) = mat 0` THENL
14069 [MP_TAC(SYM(ISPEC `A:real^N^N` MATRIX_MUL_LEFT_COFACTOR)) THEN
14070 ASM_REWRITE_TAC[MATRIX_CMUL_EQ_0; TRANSP_MAT; MATRIX_MUL_LZERO] THEN
14071 REWRITE_TAC[MAT_EQ; ARITH_EQ] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
14073 `?c i. 1 <= i /\ i <= dimindex(:N) /\ c$i = &1 /\
14074 transp(A:real^N^N) ** c = vec 0`
14075 STRIP_ASSUME_TAC THENL
14076 [MP_TAC(ISPEC `transp A:real^N^N` HOMOGENEOUS_LINEAR_EQUATIONS_DET) THEN
14077 ASM_REWRITE_TAC[DET_TRANSP] THEN
14078 DISCH_THEN(X_CHOOSE_THEN `c:real^N` STRIP_ASSUME_TAC) THEN
14079 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
14080 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN
14081 REWRITE_TAC[VEC_COMPONENT; NOT_IMP; NOT_FORALL_THM] THEN
14082 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
14083 EXISTS_TAC `inv(c$i) % c:real^N` THEN
14084 ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_MUL_LINV] THEN
14085 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_RMUL; VECTOR_MUL_RZERO];
14088 [`(lambda k. if k = i then transp A ** c else row k (A:real^N^N)):real^N^N`;
14089 `i:num`; `min e (e / &(dimindex(:N)) /
14090 (&1 + norm(&2 % basis i - c:real^N)))`] lemma1) THEN
14091 ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1;
14092 NORM_ARITH `&0 < &1 + norm(x:real^N)`] THEN
14094 [ASM_SIMP_TAC[row; CART_EQ; VEC_COMPONENT; LAMBDA_BETA];
14096 MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
14097 ABBREV_TAC `A':real^N^N =
14098 lambda k. if k = i then vec 0 else row k (A:real^N^N)` THEN
14099 DISCH_THEN(X_CHOOSE_THEN `B:real^N^N` STRIP_ASSUME_TAC) THEN
14100 EXISTS_TAC `(lambda k. if k = i then transp(B:real^N^N) **
14102 else row k B):real^N^N` THEN
14103 ASM_SIMP_TAC[lemma2; BASIS_COMPONENT; VECTOR_MUL_COMPONENT;
14104 VECTOR_SUB_COMPONENT; REAL_ARITH `&2 * x - x = x`] THEN
14105 (MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN
14106 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !A:real^N^N. A$m = A$k`
14107 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
14108 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$n = z$l`
14109 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC]) THEN
14110 EXPAND_TAC "A'" THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN
14111 (COND_CASES_TAC THENL
14113 FIRST_X_ASSUM(MP_TAC o SPECL [`k:num`; `l:num`]) THEN
14114 EXPAND_TAC "A'" THEN ASM_SIMP_TAC[LAMBDA_BETA; row]] THEN
14116 `(A:real^N^N)$k$l = (transp(A':real^N^N) ** (&2 % basis i - c:real^N))$l`
14118 [ASM_SIMP_TAC[matrix_vector_mul; transp; LAMBDA_BETA] THEN
14119 EXPAND_TAC "A'" THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN
14120 REWRITE_TAC[COND_RAND; COND_RATOR] THEN
14121 SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT;
14122 VEC_COMPONENT; REAL_MUL_RZERO; REAL_SUB_LZERO; REAL_MUL_LZERO] THEN
14123 ASM_SIMP_TAC[SUM_CASES; FINITE_NUMSEG; SUM_0; REAL_ADD_LID] THEN
14124 ASM_SIMP_TAC[GSYM DELETE; SUM_DELETE; IN_NUMSEG; FINITE_NUMSEG] THEN
14125 UNDISCH_TAC `transp(A:real^N^N) ** (c:real^N) = vec 0` THEN
14126 ASM_SIMP_TAC[CART_EQ; VEC_COMPONENT; matrix_vector_mul; LAMBDA_BETA;
14128 DISCH_THEN(MP_TAC o SPEC `l:num`) THEN ASM_REWRITE_TAC[] THEN
14129 SIMP_TAC[REAL_MUL_RNEG; SUM_NEG] THEN REAL_ARITH_TAC;
14130 REWRITE_TAC[GSYM VECTOR_SUB_COMPONENT; GSYM TRANSP_MATRIX_SUB;
14131 GSYM MATRIX_VECTOR_MUL_SUB_RDISTRIB]] THEN
14132 ASM_SIMP_TAC[matrix_vector_mul; transp; LAMBDA_BETA] THEN
14133 W(MP_TAC o PART_MATCH lhand SUM_ABS_NUMSEG o lhand o snd) THEN
14134 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN
14135 MATCH_MP_TAC SUM_BOUND_LT_GEN THEN
14136 ASM_SIMP_TAC[FINITE_NUMSEG; NUMSEG_EMPTY;
14137 GSYM NOT_LE; DIMINDEX_GE_1] THEN
14138 X_GEN_TAC `r:num` THEN REWRITE_TAC[CARD_NUMSEG_1; IN_NUMSEG] THEN
14139 STRIP_TAC THEN REWRITE_TAC[REAL_ABS_MUL] THEN
14140 TRANS_TAC REAL_LET_TRANS
14141 `abs((B - A':real^N^N)$r$l) * (&1 + norm(&2 % basis i - c:real^N))` THEN
14143 [MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_ABS_POS] THEN
14144 MATCH_MP_TAC(REAL_ARITH `a <= b ==> a <= &1 + b`) THEN
14145 ASM_SIMP_TAC[COMPONENT_LE_NORM];
14146 ASM_SIMP_TAC[MATRIX_SUB_COMPONENT; GSYM REAL_LT_RDIV_EQ;
14147 NORM_ARITH `&0 < &1 + norm(x:real^N)`]]);
14148 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [CART_EQ]) THEN
14149 SIMP_TAC[CART_EQ; MAT_COMPONENT; COND_ID] THEN
14150 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; real_gt] THEN
14151 DISCH_THEN(X_CHOOSE_THEN `i:num` (CONJUNCTS_THEN2 STRIP_ASSUME_TAC
14152 (X_CHOOSE_THEN `j:num` STRIP_ASSUME_TAC))) THEN
14153 FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (REAL_ARITH
14154 `~(x = &0) ==> &0 < x \/ x < &0`))
14155 THENL [ALL_TAC; ONCE_REWRITE_TAC[CONJ_SYM]] THEN
14157 [EXISTS_TAC `(lambda m n. if m = i /\ n = j
14158 then (A:real^N^N)$i$j -
14159 e / (&1 + abs(cofactor A$i$j))
14160 else A$m$n):real^N^N`;
14161 EXISTS_TAC `(lambda m n. if m = i /\ n = j
14162 then (A:real^N^N)$i$j +
14163 e / (&1 + abs(cofactor A$i$j))
14164 else A$m$n):real^N^N`]) THEN
14166 [MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN
14167 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !A:real^N^N. A$m = A$k`
14168 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
14169 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$n = z$l`
14170 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
14171 ASM_SIMP_TAC[LAMBDA_BETA] THEN
14172 COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_ABS_NUM] THEN
14173 REWRITE_TAC[REAL_ARITH `abs(a - e - a) = abs e`;
14174 REAL_ARITH `abs((a + e) - a) = abs e`] THEN
14175 REWRITE_TAC[REAL_ABS_DIV; REAL_ABS_NUM; REAL_ABS_ABS] THEN
14176 ASM_SIMP_TAC[REAL_ARITH `abs(&1 + abs x) = &1 + abs x`;
14177 REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &1 + abs x`] THEN
14178 MATCH_MP_TAC(REAL_ARITH
14179 `&0 < e /\ &0 < e * x ==> abs e < e * (&1 + x)`) THEN
14180 ASM_SIMP_TAC[REAL_LT_MUL_EQ] THEN ASM_REAL_ARITH_TAC;
14182 MP_TAC(GEN `A:real^N^N` (SPECL [`A:real^N^N`; `i:num`]
14183 DET_COFACTOR_EXPANSION)) THEN
14184 ASM_SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN
14185 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
14186 ASM_SIMP_TAC[GSYM SUM_SUB_NUMSEG; LAMBDA_BETA] THEN
14187 REWRITE_TAC[REAL_ARITH `p - A$i$j * cofactor A$i$j =
14188 --(A$i$j * cofactor A$i$j - p)`] THEN
14189 REWRITE_TAC[SUM_NEG; REAL_ARITH
14190 `a * b - c * d:real = b * (a - c) + c * (b - d)`] THEN
14191 REWRITE_TAC[SUM_ADD_NUMSEG; REAL_NEG_ADD] THEN MATCH_MP_TAC(REAL_ARITH
14192 `b = &0 /\ &0 < a ==> &0 < a + b`) THEN
14194 [REWRITE_TAC[REAL_NEG_EQ_0] THEN
14195 MATCH_MP_TAC SUM_EQ_0 THEN X_GEN_TAC `m:num` THEN
14196 REWRITE_TAC[IN_NUMSEG; REAL_ENTIRE] THEN STRIP_TAC THEN DISJ2_TAC THEN
14197 REWRITE_TAC[REAL_SUB_0] THEN REWRITE_TAC[cofactor] THEN
14198 ASM_SIMP_TAC[LAMBDA_BETA] THEN AP_TERM_TAC THEN
14199 ASM_SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN ASM_MESON_TAC[];
14201 REWRITE_TAC[GSYM SUM_NEG; GSYM REAL_MUL_RNEG] THEN
14202 MATCH_MP_TAC SUM_POS_LT THEN REWRITE_TAC[FINITE_NUMSEG] THEN
14203 MATCH_MP_TAC(MESON[REAL_LT_IMP_LE; REAL_LE_REFL]
14204 `(?i. P i /\ &0 < f i /\ (!j. P j /\ ~(j = i) ==> f j = &0))
14205 ==> (!j. P j ==> &0 <= f j) /\ (?j. P j /\ &0 < f j)`) THEN
14206 EXISTS_TAC `j:num` THEN ASM_REWRITE_TAC[FINITE_NUMSEG] THEN
14207 ASM_SIMP_TAC[REAL_SUB_REFL; REAL_MUL_RZERO; IN_NUMSEG; REAL_NEG_0] THEN
14208 REWRITE_TAC[REAL_ARITH `a - (a + e):real = --e`;
14209 REAL_ARITH `a - (a - e):real = e`; REAL_NEG_NEG] THEN
14210 ASM_SIMP_TAC[REAL_LT_MUL_EQ] THEN
14211 REWRITE_TAC[REAL_ARITH `&0 < a * --b <=> &0 < --a * b`] THEN
14212 ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_NEG_GT0] THEN
14213 MATCH_MP_TAC REAL_LT_DIV THEN ASM_REAL_ARITH_TAC]);;
14215 (* ------------------------------------------------------------------------- *)
14216 (* Infinite sums of vectors. Allow general starting point (and more). *)
14217 (* ------------------------------------------------------------------------- *)
14219 parse_as_infix("sums",(12,"right"));;
14221 let sums = new_definition
14222 `(f sums l) s = ((\n. vsum(s INTER (0..n)) f) --> l) sequentially`;;
14224 let infsum = new_definition
14225 `infsum s f = @l. (f sums l) s`;;
14227 let summable = new_definition
14228 `summable s f = ?l. (f sums l) s`;;
14230 let SUMS_SUMMABLE = prove
14231 (`!f l s. (f sums l) s ==> summable s f`,
14232 REWRITE_TAC[summable] THEN MESON_TAC[]);;
14234 let SUMS_INFSUM = prove
14235 (`!f s. (f sums (infsum s f)) s <=> summable s f`,
14236 REWRITE_TAC[infsum; summable] THEN MESON_TAC[]);;
14238 let SUMS_LIM = prove
14239 (`!f:num->real^N s.
14240 (f sums lim sequentially (\n. vsum (s INTER (0..n)) f)) s
14242 GEN_TAC THEN GEN_TAC THEN EQ_TAC THENL [MESON_TAC[summable];
14243 REWRITE_TAC[summable; sums] THEN STRIP_TAC THEN REWRITE_TAC[lim] THEN
14244 ASM_MESON_TAC[]]);;
14246 let FINITE_INTER_NUMSEG = prove
14247 (`!s m n. FINITE(s INTER (m..n))`,
14248 MESON_TAC[FINITE_SUBSET; FINITE_NUMSEG; INTER_SUBSET]);;
14250 let SERIES_FROM = prove
14251 (`!f l k. (f sums l) (from k) = ((\n. vsum(k..n) f) --> l) sequentially`,
14252 REPEAT GEN_TAC THEN REWRITE_TAC[sums] THEN
14253 AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
14254 AP_THM_TAC THEN AP_TERM_TAC THEN
14255 REWRITE_TAC[EXTENSION; numseg; from; IN_ELIM_THM; IN_INTER] THEN ARITH_TAC);;
14257 let SERIES_UNIQUE = prove
14258 (`!f:num->real^N l l' s. (f sums l) s /\ (f sums l') s ==> (l = l')`,
14259 REWRITE_TAC[sums] THEN MESON_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIM_UNIQUE]);;
14261 let INFSUM_UNIQUE = prove
14262 (`!f:num->real^N l s. (f sums l) s ==> infsum s f = l`,
14263 MESON_TAC[SERIES_UNIQUE; SUMS_INFSUM; summable]);;
14265 let SERIES_TERMS_TOZERO = prove
14266 (`!f l n. (f sums l) (from n) ==> (f --> vec 0) sequentially`,
14267 REPEAT GEN_TAC THEN SIMP_TAC[sums; LIM_SEQUENTIALLY; FROM_INTER_NUMSEG] THEN
14268 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
14269 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
14270 ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
14271 EXISTS_TAC `N + n + 1` THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN
14272 FIRST_X_ASSUM(fun th ->
14273 MP_TAC(SPEC `m - 1` th) THEN MP_TAC(SPEC `m:num` th)) THEN
14274 SUBGOAL_THEN `0 < m /\ n <= m` (fun th -> SIMP_TAC[VSUM_CLAUSES_RIGHT; th])
14275 THENL [ASM_ARITH_TAC; ALL_TAC] THEN
14276 REPEAT(ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC]) THEN
14277 REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);;
14279 let SERIES_FINITE = prove
14280 (`!f s. FINITE s ==> (f sums (vsum s f)) s`,
14281 REPEAT GEN_TAC THEN REWRITE_TAC[num_FINITE; LEFT_IMP_EXISTS_THM] THEN
14282 X_GEN_TAC `n:num` THEN REWRITE_TAC[sums; LIM_SEQUENTIALLY] THEN
14283 DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN EXISTS_TAC `n:num` THEN
14284 X_GEN_TAC `m:num` THEN DISCH_TAC THEN
14285 SUBGOAL_THEN `s INTER (0..m) = s`
14286 (fun th -> ASM_REWRITE_TAC[th; DIST_REFL]) THEN
14287 REWRITE_TAC[EXTENSION; IN_INTER; IN_NUMSEG; LE_0] THEN
14288 ASM_MESON_TAC[LE_TRANS]);;
14290 let SERIES_LINEAR = prove
14291 (`!f h l s. (f sums l) s /\ linear h ==> ((\n. h(f n)) sums h l) s`,
14292 SIMP_TAC[sums; LIM_LINEAR; FINITE_INTER; FINITE_NUMSEG;
14293 GSYM(REWRITE_RULE[o_DEF] LINEAR_VSUM)]);;
14295 let SERIES_0 = prove
14296 (`!s. ((\n. vec 0) sums (vec 0)) s`,
14297 REWRITE_TAC[sums; VSUM_0; LIM_CONST]);;
14299 let SERIES_ADD = prove
14301 (x sums x0) s /\ (y sums y0) s ==> ((\n. x n + y n) sums (x0 + y0)) s`,
14302 SIMP_TAC[sums; FINITE_INTER_NUMSEG; VSUM_ADD; LIM_ADD]);;
14304 let SERIES_SUB = prove
14306 (x sums x0) s /\ (y sums y0) s ==> ((\n. x n - y n) sums (x0 - y0)) s`,
14307 SIMP_TAC[sums; FINITE_INTER_NUMSEG; VSUM_SUB; LIM_SUB]);;
14309 let SERIES_CMUL = prove
14310 (`!x x0 c s. (x sums x0) s ==> ((\n. c % x n) sums (c % x0)) s`,
14311 SIMP_TAC[sums; FINITE_INTER_NUMSEG; VSUM_LMUL; LIM_CMUL]);;
14313 let SERIES_NEG = prove
14314 (`!x x0 s. (x sums x0) s ==> ((\n. --(x n)) sums (--x0)) s`,
14315 SIMP_TAC[sums; FINITE_INTER_NUMSEG; VSUM_NEG; LIM_NEG]);;
14317 let SUMS_IFF = prove
14318 (`!f g k. (!x. x IN k ==> f x = g x) ==> ((f sums l) k <=> (g sums l) k)`,
14319 REPEAT STRIP_TAC THEN REWRITE_TAC[sums] THEN
14320 AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
14321 MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[IN_INTER]);;
14323 let SUMS_EQ = prove
14324 (`!f g k. (!x. x IN k ==> f x = g x) /\ (f sums l) k ==> (g sums l) k`,
14325 MESON_TAC[SUMS_IFF]);;
14328 (`!f:num->real^N s. (!n. n IN s ==> f n = vec 0) ==> (f sums vec 0) s`,
14329 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUMS_EQ THEN
14330 EXISTS_TAC `\n:num. vec 0:real^N` THEN ASM_SIMP_TAC[SERIES_0]);;
14332 let SERIES_FINITE_SUPPORT = prove
14333 (`!f:num->real^N s k.
14334 FINITE (s INTER k) /\ (!x. ~(x IN s INTER k) ==> f x = vec 0)
14335 ==> (f sums vsum (s INTER k) f) k`,
14336 REWRITE_TAC[sums; LIM_SEQUENTIALLY] THEN REPEAT STRIP_TAC THEN
14337 FIRST_ASSUM(MP_TAC o ISPEC `\x:num. x` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN
14338 REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
14339 STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
14340 SUBGOAL_THEN `vsum (k INTER (0..n)) (f:num->real^N) = vsum(s INTER k) f`
14341 (fun th -> ASM_REWRITE_TAC[DIST_REFL; th]) THEN
14342 MATCH_MP_TAC VSUM_SUPERSET THEN
14343 ASM_SIMP_TAC[SUBSET; IN_INTER; IN_NUMSEG; LE_0] THEN
14344 ASM_MESON_TAC[IN_INTER; LE_TRANS]);;
14346 let SERIES_COMPONENT = prove
14347 (`!f s l:real^N k. (f sums l) s /\ 1 <= k /\ k <= dimindex(:N)
14348 ==> ((\i. lift(f(i)$k)) sums lift(l$k)) s`,
14349 REPEAT GEN_TAC THEN REWRITE_TAC[sums] THEN STRIP_TAC THEN
14350 ONCE_REWRITE_TAC[GSYM o_DEF] THEN
14351 ASM_SIMP_TAC[GSYM LIFT_SUM; GSYM VSUM_COMPONENT;
14352 FINITE_INTER; FINITE_NUMSEG] THEN
14353 ASM_SIMP_TAC[o_DEF; LIM_COMPONENT]);;
14355 let SERIES_DIFFS = prove
14356 (`!f:num->real^N k.
14357 (f --> vec 0) sequentially
14358 ==> ((\n. f(n) - f(n + 1)) sums f(k)) (from k)`,
14359 REWRITE_TAC[sums; FROM_INTER_NUMSEG; VSUM_DIFFS] THEN
14360 REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
14361 EXISTS_TAC `\n. (f:num->real^N) k - f(n + 1)` THEN CONJ_TAC THENL
14362 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `k:num` THEN
14364 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_SUB_RZERO] THEN
14365 MATCH_MP_TAC LIM_SUB THEN REWRITE_TAC[LIM_CONST] THEN
14366 MATCH_MP_TAC SEQ_OFFSET THEN ASM_REWRITE_TAC[]]);;
14368 let SERIES_TRIVIAL = prove
14369 (`!f. (f sums vec 0) {}`,
14370 REWRITE_TAC[sums; INTER_EMPTY; VSUM_CLAUSES; LIM_CONST]);;
14372 let SERIES_RESTRICT = prove
14374 ((\n. if n IN k then f(n) else vec 0) sums l) (:num) <=>
14376 REPEAT GEN_TAC THEN REWRITE_TAC[sums] THEN
14377 AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
14378 REWRITE_TAC[FUN_EQ_THM; INTER_UNIV] THEN GEN_TAC THEN
14379 MATCH_MP_TAC(MESON[] `vsum s f = vsum t f /\ vsum t f = vsum t g
14380 ==> vsum s f = vsum t g`) THEN
14382 [MATCH_MP_TAC VSUM_SUPERSET THEN SET_TAC[];
14383 MATCH_MP_TAC VSUM_EQ THEN SIMP_TAC[IN_INTER]]);;
14385 let SERIES_VSUM = prove
14386 (`!f l k s. FINITE s /\ s SUBSET k /\ (!x. ~(x IN s) ==> f x = vec 0) /\
14387 vsum s f = l ==> (f sums l) k`,
14388 REPEAT STRIP_TAC THEN EXPAND_TAC "l" THEN
14389 SUBGOAL_THEN `s INTER k = s:num->bool` ASSUME_TAC THENL
14390 [ASM SET_TAC []; ASM_MESON_TAC [SERIES_FINITE_SUPPORT]]);;
14392 let SUMS_REINDEX = prove
14393 (`!k a l n. ((\x. a(x + k)) sums l) (from n) <=> (a sums l) (from(n + k))`,
14394 REPEAT GEN_TAC THEN REWRITE_TAC[sums; FROM_INTER_NUMSEG] THEN
14395 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM VSUM_OFFSET] THEN
14396 REWRITE_TAC[LIM_SEQUENTIALLY] THEN
14397 ASM_MESON_TAC[ARITH_RULE `N + k:num <= n ==> n = (n - k) + k /\ N <= n - k`;
14398 ARITH_RULE `N + k:num <= n ==> N <= n + k`]);;
14400 (* ------------------------------------------------------------------------- *)
14401 (* Similar combining theorems just for summability. *)
14402 (* ------------------------------------------------------------------------- *)
14404 let SUMMABLE_LINEAR = prove
14405 (`!f h s. summable s f /\ linear h ==> summable s (\n. h(f n))`,
14406 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_LINEAR]);;
14408 let SUMMABLE_0 = prove
14409 (`!s. summable s (\n. vec 0)`,
14410 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_0]);;
14412 let SUMMABLE_ADD = prove
14413 (`!x y s. summable s x /\ summable s y ==> summable s (\n. x n + y n)`,
14414 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_ADD]);;
14416 let SUMMABLE_SUB = prove
14417 (`!x y s. summable s x /\ summable s y ==> summable s (\n. x n - y n)`,
14418 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_SUB]);;
14420 let SUMMABLE_CMUL = prove
14421 (`!s x c. summable s x ==> summable s (\n. c % x n)`,
14422 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_CMUL]);;
14424 let SUMMABLE_NEG = prove
14425 (`!x s. summable s x ==> summable s (\n. --(x n))`,
14426 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_NEG]);;
14428 let SUMMABLE_IFF = prove
14429 (`!f g k. (!x. x IN k ==> f x = g x) ==> (summable k f <=> summable k g)`,
14430 REWRITE_TAC[summable] THEN MESON_TAC[SUMS_IFF]);;
14432 let SUMMABLE_EQ = prove
14433 (`!f g k. (!x. x IN k ==> f x = g x) /\ summable k f ==> summable k g`,
14434 REWRITE_TAC[summable] THEN MESON_TAC[SUMS_EQ]);;
14436 let SUMMABLE_COMPONENT = prove
14437 (`!f:num->real^N s k.
14438 summable s f /\ 1 <= k /\ k <= dimindex(:N)
14439 ==> summable s (\i. lift(f(i)$k))`,
14440 REPEAT STRIP_TAC THEN
14441 FIRST_X_ASSUM(X_CHOOSE_TAC `l:real^N` o REWRITE_RULE[summable]) THEN
14442 REWRITE_TAC[summable] THEN EXISTS_TAC `lift((l:real^N)$k)` THEN
14443 ASM_SIMP_TAC[SERIES_COMPONENT]);;
14445 let SERIES_SUBSET = prove
14448 ((\i. if i IN s then x i else vec 0) sums l) t
14450 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
14451 REWRITE_TAC[sums] THEN MATCH_MP_TAC EQ_IMP THEN
14452 AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
14453 ASM_SIMP_TAC[GSYM VSUM_RESTRICT_SET; FINITE_INTER_NUMSEG] THEN
14454 AP_THM_TAC THEN AP_TERM_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[]);;
14456 let SUMMABLE_SUBSET = prove
14459 summable t (\i. if i IN s then x i else vec 0)
14461 REWRITE_TAC[summable] THEN MESON_TAC[SERIES_SUBSET]);;
14463 let SUMMABLE_TRIVIAL = prove
14464 (`!f:num->real^N. summable {} f`,
14465 GEN_TAC THEN REWRITE_TAC[summable] THEN EXISTS_TAC `vec 0:real^N` THEN
14466 REWRITE_TAC[SERIES_TRIVIAL]);;
14468 let SUMMABLE_RESTRICT = prove
14469 (`!f:num->real^N k.
14470 summable (:num) (\n. if n IN k then f(n) else vec 0) <=>
14472 REWRITE_TAC[summable; SERIES_RESTRICT]);;
14474 let SUMS_FINITE_DIFF = prove
14475 (`!f:num->real^N t s l.
14476 t SUBSET s /\ FINITE t /\ (f sums l) s
14477 ==> (f sums (l - vsum t f)) (s DIFF t)`,
14478 REPEAT GEN_TAC THEN
14479 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
14480 FIRST_ASSUM(MP_TAC o ISPEC `f:num->real^N` o MATCH_MP SERIES_FINITE) THEN
14481 ONCE_REWRITE_TAC[GSYM SERIES_RESTRICT] THEN
14482 REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
14483 DISCH_THEN(MP_TAC o MATCH_MP SERIES_SUB) THEN
14484 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
14485 REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:num` THEN REWRITE_TAC[IN_DIFF] THEN
14486 FIRST_ASSUM(MP_TAC o SPEC `x:num` o GEN_REWRITE_RULE I [SUBSET]) THEN
14487 MAP_EVERY ASM_CASES_TAC [`(x:num) IN s`; `(x:num) IN t`] THEN
14488 ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);;
14490 let SUMS_FINITE_UNION = prove
14491 (`!f:num->real^N s t l.
14492 FINITE t /\ (f sums l) s
14493 ==> (f sums (l + vsum (t DIFF s) f)) (s UNION t)`,
14494 REPEAT GEN_TAC THEN
14495 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
14496 FIRST_ASSUM(MP_TAC o SPEC `s:num->bool` o MATCH_MP FINITE_DIFF) THEN
14497 DISCH_THEN(MP_TAC o ISPEC `f:num->real^N` o MATCH_MP SERIES_FINITE) THEN
14498 ONCE_REWRITE_TAC[GSYM SERIES_RESTRICT] THEN
14499 REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
14500 DISCH_THEN(MP_TAC o MATCH_MP SERIES_ADD) THEN
14501 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
14502 REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `x:num` THEN
14503 REWRITE_TAC[IN_DIFF; IN_UNION] THEN
14504 MAP_EVERY ASM_CASES_TAC [`(x:num) IN s`; `(x:num) IN t`] THEN
14505 ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);;
14507 let SUMS_OFFSET = prove
14508 (`!f:num->real^N l m n.
14509 (f sums l) (from m) /\ m < n
14510 ==> (f sums (l - vsum(m..(n-1)) f)) (from n)`,
14511 REPEAT STRIP_TAC THEN
14512 SUBGOAL_THEN `from n = from m DIFF (m..(n-1))` SUBST1_TAC THENL
14513 [REWRITE_TAC[EXTENSION; IN_FROM; IN_DIFF; IN_NUMSEG] THEN ASM_ARITH_TAC;
14514 MATCH_MP_TAC SUMS_FINITE_DIFF THEN ASM_REWRITE_TAC[FINITE_NUMSEG] THEN
14515 SIMP_TAC[SUBSET; IN_FROM; IN_NUMSEG]]);;
14517 let SUMS_OFFSET_REV = prove
14518 (`!f:num->real^N l m n.
14519 (f sums l) (from m) /\ n < m
14520 ==> (f sums (l + vsum(n..m-1) f)) (from n)`,
14521 REPEAT STRIP_TAC THEN
14522 MP_TAC(ISPECL [`f:num->real^N`; `from m`; `n..m-1`; `l:real^N`]
14523 SUMS_FINITE_UNION) THEN
14524 ASM_REWRITE_TAC[FINITE_NUMSEG] THEN MATCH_MP_TAC EQ_IMP THEN
14525 BINOP_TAC THENL [AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC; ALL_TAC] THEN
14526 REWRITE_TAC[EXTENSION; IN_DIFF; IN_UNION; IN_FROM; IN_NUMSEG] THEN
14529 let SUMMABLE_REINDEX = prove
14530 (`!k a n. summable (from n) (\x. a (x + k)) <=> summable (from(n + k)) a`,
14531 REWRITE_TAC[summable; GSYM SUMS_REINDEX]);;
14533 let SERIES_DROP_LE = prove
14535 (f sums a) s /\ (g sums b) s /\
14536 (!x. x IN s ==> drop(f x) <= drop(g x))
14537 ==> drop a <= drop b`,
14538 REWRITE_TAC[sums] THEN REPEAT STRIP_TAC THEN
14539 MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LE) THEN
14540 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
14541 EXISTS_TAC `\n. vsum (s INTER (0..n)) (f:num->real^1)` THEN
14542 EXISTS_TAC `\n. vsum (s INTER (0..n)) (g:num->real^1)` THEN
14543 ASM_REWRITE_TAC[DROP_VSUM] THEN EXISTS_TAC `0` THEN REPEAT STRIP_TAC THEN
14544 MATCH_MP_TAC SUM_LE THEN
14545 ASM_SIMP_TAC[FINITE_INTER; FINITE_NUMSEG; o_THM; IN_INTER; IN_NUMSEG]);;
14547 let SERIES_DROP_POS = prove
14549 (f sums a) s /\ (!x. x IN s ==> &0 <= drop(f x))
14551 REPEAT STRIP_TAC THEN
14552 MP_TAC(ISPECL [`(\n. vec 0):num->real^1`; `f:num->real^1`; `s:num->bool`;
14553 `vec 0:real^1`; `a:real^1`] SERIES_DROP_LE) THEN
14554 ASM_SIMP_TAC[SUMS_0; DROP_VEC]);;
14556 let SERIES_BOUND = prove
14557 (`!f:num->real^N g s a b.
14558 (f sums a) s /\ ((lift o g) sums (lift b)) s /\
14559 (!i. i IN s ==> norm(f i) <= g i)
14561 REWRITE_TAC[sums] THEN REPEAT STRIP_TAC THEN
14562 MATCH_MP_TAC(ISPEC `sequentially` LIM_NORM_UBOUND) THEN
14563 EXISTS_TAC `\n. vsum (s INTER (0..n)) (f:num->real^N)` THEN
14564 ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
14565 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `0` THEN
14566 X_GEN_TAC `m:num` THEN DISCH_TAC THEN
14567 TRANS_TAC REAL_LE_TRANS `sum (s INTER (0..m)) g` THEN CONJ_TAC THEN
14568 ASM_SIMP_TAC[VSUM_NORM_LE; IN_INTER; FINITE_NUMSEG; FINITE_INTER] THEN
14569 RULE_ASSUM_TAC(REWRITE_RULE[GSYM sums]) THEN
14570 UNDISCH_TAC `((lift o g) sums lift b) s` THEN
14571 GEN_REWRITE_TAC LAND_CONV [GSYM SERIES_RESTRICT] THEN
14572 REWRITE_TAC[GSYM FROM_0] THEN DISCH_THEN(MP_TAC o SPEC `m + 1` o MATCH_MP
14573 (REWRITE_RULE[IMP_CONJ] SUMS_OFFSET)) THEN
14574 REWRITE_TAC[ARITH_RULE `0 < m + 1`; o_DEF; ADD_SUB] THEN
14575 REWRITE_TAC[GSYM VSUM_RESTRICT_SET] THEN
14576 REWRITE_TAC[VSUM_REAL; o_DEF; LIFT_DROP; ETA_AX] THEN
14577 DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] SERIES_DROP_POS)) THEN
14578 REWRITE_TAC[DROP_SUB; LIFT_DROP; ONCE_REWRITE_RULE[INTER_COMM] (GSYM INTER);
14580 DISCH_THEN MATCH_MP_TAC THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
14581 ASM_SIMP_TAC[LIFT_DROP; DROP_VEC; REAL_LE_REFL] THEN
14582 ASM_MESON_TAC[NORM_ARITH `norm(x:real^N) <= y ==> &0 <= y`]);;
14584 (* ------------------------------------------------------------------------- *)
14585 (* Similar combining theorems for infsum. *)
14586 (* ------------------------------------------------------------------------- *)
14588 let INFSUM_LINEAR = prove
14589 (`!f h s. summable s f /\ linear h
14590 ==> infsum s (\n. h(f n)) = h(infsum s f)`,
14591 REPEAT STRIP_TAC THEN MATCH_MP_TAC INFSUM_UNIQUE THEN
14592 MATCH_MP_TAC SERIES_LINEAR THEN ASM_REWRITE_TAC[SUMS_INFSUM]);;
14594 let INFSUM_0 = prove
14595 (`infsum s (\i. vec 0) = vec 0`,
14596 MATCH_MP_TAC INFSUM_UNIQUE THEN REWRITE_TAC[SERIES_0]);;
14598 let INFSUM_ADD = prove
14599 (`!x y s. summable s x /\ summable s y
14600 ==> infsum s (\i. x i + y i) = infsum s x + infsum s y`,
14601 REPEAT STRIP_TAC THEN MATCH_MP_TAC INFSUM_UNIQUE THEN
14602 MATCH_MP_TAC SERIES_ADD THEN ASM_REWRITE_TAC[SUMS_INFSUM]);;
14604 let INFSUM_SUB = prove
14605 (`!x y s. summable s x /\ summable s y
14606 ==> infsum s (\i. x i - y i) = infsum s x - infsum s y`,
14607 REPEAT STRIP_TAC THEN MATCH_MP_TAC INFSUM_UNIQUE THEN
14608 MATCH_MP_TAC SERIES_SUB THEN ASM_REWRITE_TAC[SUMS_INFSUM]);;
14610 let INFSUM_CMUL = prove
14611 (`!s x c. summable s x ==> infsum s (\n. c % x n) = c % infsum s x`,
14612 REPEAT STRIP_TAC THEN MATCH_MP_TAC INFSUM_UNIQUE THEN
14613 MATCH_MP_TAC SERIES_CMUL THEN ASM_REWRITE_TAC[SUMS_INFSUM]);;
14615 let INFSUM_NEG = prove
14616 (`!s x. summable s x ==> infsum s (\n. --(x n)) = --(infsum s x)`,
14617 REPEAT STRIP_TAC THEN MATCH_MP_TAC INFSUM_UNIQUE THEN
14618 MATCH_MP_TAC SERIES_NEG THEN ASM_REWRITE_TAC[SUMS_INFSUM]);;
14620 let INFSUM_EQ = prove
14621 (`!f g k. summable k f /\ summable k g /\ (!x. x IN k ==> f x = g x)
14622 ==> infsum k f = infsum k g`,
14623 REPEAT STRIP_TAC THEN REWRITE_TAC[infsum] THEN
14624 AP_TERM_TAC THEN ABS_TAC THEN ASM_MESON_TAC[SUMS_EQ; SUMS_INFSUM]);;
14626 let INFSUM_RESTRICT = prove
14627 (`!k a:num->real^N.
14628 infsum (:num) (\n. if n IN k then a n else vec 0) = infsum k a`,
14629 REPEAT GEN_TAC THEN
14630 MP_TAC(ISPECL [`a:num->real^N`; `k:num->bool`] SUMMABLE_RESTRICT) THEN
14631 ASM_CASES_TAC `summable k (a:num->real^N)` THEN ASM_REWRITE_TAC[] THEN
14633 [MATCH_MP_TAC INFSUM_UNIQUE THEN
14634 ASM_REWRITE_TAC[SERIES_RESTRICT; SUMS_INFSUM];
14635 RULE_ASSUM_TAC(REWRITE_RULE[summable; NOT_EXISTS_THM]) THEN
14636 ASM_REWRITE_TAC[infsum]]);;
14638 let PARTIAL_SUMS_COMPONENT_LE_INFSUM = prove
14639 (`!f:num->real^N s k n.
14640 1 <= k /\ k <= dimindex(:N) /\
14641 (!i. i IN s ==> &0 <= (f i)$k) /\
14643 ==> (vsum (s INTER (0..n)) f)$k <= (infsum s f)$k`,
14644 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM SUMS_INFSUM] THEN
14645 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
14646 REWRITE_TAC[sums; LIM_SEQUENTIALLY] THEN DISCH_TAC THEN
14647 REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
14648 FIRST_X_ASSUM(MP_TAC o SPEC
14649 `vsum (s INTER (0..n)) (f:num->real^N)$k - (infsum s f)$k`) THEN
14650 ASM_REWRITE_TAC[REAL_SUB_LT] THEN
14651 DISCH_THEN(X_CHOOSE_THEN `N:num` (MP_TAC o SPEC `N + n:num`)) THEN
14652 REWRITE_TAC[LE_ADD; REAL_NOT_LT; dist] THEN
14653 MATCH_MP_TAC REAL_LE_TRANS THEN
14654 EXISTS_TAC `abs((vsum (s INTER (0..N + n)) f - infsum s f:real^N)$k)` THEN
14655 ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN REWRITE_TAC[VECTOR_SUB_COMPONENT] THEN
14656 MATCH_MP_TAC(REAL_ARITH `s < a /\ a <= b ==> a - s <= abs(b - s)`) THEN
14657 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[ADD_SYM] THEN
14658 SIMP_TAC[NUMSEG_ADD_SPLIT; LE_0; UNION_OVER_INTER] THEN
14659 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_UNION o lhand o rand o snd) THEN
14661 [SIMP_TAC[FINITE_INTER; FINITE_NUMSEG; DISJOINT; EXTENSION] THEN
14662 REWRITE_TAC[IN_INTER; NOT_IN_EMPTY; IN_NUMSEG] THEN ARITH_TAC;
14663 DISCH_THEN SUBST1_TAC THEN
14664 REWRITE_TAC[REAL_LE_ADDR; VECTOR_ADD_COMPONENT] THEN
14665 ASM_SIMP_TAC[VSUM_COMPONENT] THEN MATCH_MP_TAC SUM_POS_LE THEN
14666 ASM_SIMP_TAC[FINITE_INTER; IN_INTER; FINITE_NUMSEG]]);;
14668 let PARTIAL_SUMS_DROP_LE_INFSUM = prove
14670 (!i. i IN s ==> &0 <= drop(f i)) /\
14672 ==> drop(vsum (s INTER (0..n)) f) <= drop(infsum s f)`,
14673 REPEAT STRIP_TAC THEN REWRITE_TAC[drop] THEN
14674 MATCH_MP_TAC PARTIAL_SUMS_COMPONENT_LE_INFSUM THEN
14675 ASM_REWRITE_TAC[DIMINDEX_1; LE_REFL; GSYM drop]);;
14677 (* ------------------------------------------------------------------------- *)
14678 (* Cauchy criterion for series. *)
14679 (* ------------------------------------------------------------------------- *)
14681 let SEQUENCE_CAUCHY_WLOG = prove
14682 (`!P s. (!m n:num. P m /\ P n ==> dist(s m,s n) < e) <=>
14683 (!m n. P m /\ P n /\ m <= n ==> dist(s m,s n) < e)`,
14684 MESON_TAC[DIST_SYM; LE_CASES]);;
14686 let VSUM_DIFF_LEMMA = prove
14687 (`!f:num->real^N k m n.
14689 ==> vsum(k INTER (0..n)) f - vsum(k INTER (0..m)) f =
14690 vsum(k INTER (m+1..n)) f`,
14691 REPEAT STRIP_TAC THEN
14692 MP_TAC(ISPECL [`f:num->real^N`; `k INTER (0..n)`; `k INTER (0..m)`]
14695 [SIMP_TAC[FINITE_INTER; FINITE_NUMSEG] THEN MATCH_MP_TAC
14696 (SET_RULE `s SUBSET t ==> (u INTER s SUBSET u INTER t)`) THEN
14697 REWRITE_TAC[SUBSET; IN_NUMSEG] THEN POP_ASSUM MP_TAC THEN ARITH_TAC;
14698 DISCH_THEN(SUBST1_TAC o SYM) THEN AP_THM_TAC THEN AP_TERM_TAC THEN
14699 REWRITE_TAC[SET_RULE
14700 `(k INTER s) DIFF (k INTER t) = k INTER (s DIFF t)`] THEN
14701 AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_DIFF; IN_NUMSEG] THEN
14702 POP_ASSUM MP_TAC THEN ARITH_TAC]);;
14704 let NORM_VSUM_TRIVIAL_LEMMA = prove
14705 (`!e. &0 < e ==> (P ==> norm(vsum(s INTER (m..n)) f) < e <=>
14706 P ==> n < m \/ norm(vsum(s INTER (m..n)) f) < e)`,
14707 REPEAT STRIP_TAC THEN ASM_CASES_TAC `n:num < m` THEN ASM_REWRITE_TAC[] THEN
14708 FIRST_X_ASSUM(SUBST1_TAC o GEN_REWRITE_RULE I [GSYM NUMSEG_EMPTY]) THEN
14709 ASM_REWRITE_TAC[VSUM_CLAUSES; NORM_0; INTER_EMPTY]);;
14711 let SERIES_CAUCHY = prove
14712 (`!f s. (?l. (f sums l) s) =
14714 ==> ?N. !m n. m >= N
14715 ==> norm(vsum(s INTER (m..n)) f) < e`,
14716 REPEAT GEN_TAC THEN REWRITE_TAC[sums; CONVERGENT_EQ_CAUCHY; cauchy] THEN
14717 REWRITE_TAC[SEQUENCE_CAUCHY_WLOG] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
14718 SIMP_TAC[dist; VSUM_DIFF_LEMMA; NORM_VSUM_TRIVIAL_LEMMA] THEN
14719 REWRITE_TAC[GE; TAUT `a ==> b \/ c <=> a /\ ~b ==> c`] THEN
14720 REWRITE_TAC[NOT_LT; ARITH_RULE
14721 `(N <= m /\ N <= n /\ m <= n) /\ m + 1 <= n <=>
14722 N + 1 <= m + 1 /\ m + 1 <= n`] THEN
14723 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
14724 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
14725 EQ_TAC THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THENL
14726 [EXISTS_TAC `N + 1`; EXISTS_TAC `N:num`] THEN
14727 REPEAT STRIP_TAC THEN
14728 ASM_SIMP_TAC[ARITH_RULE `N + 1 <= m + 1 ==> N <= m + 1`] THEN
14729 FIRST_X_ASSUM(MP_TAC o SPECL [`m - 1`; `n:num`]) THEN
14730 SUBGOAL_THEN `m - 1 + 1 = m` SUBST_ALL_TAC THENL
14731 [ALL_TAC; ANTS_TAC THEN SIMP_TAC[]] THEN
14734 let SUMMABLE_CAUCHY = prove
14735 (`!f s. summable s f <=>
14737 ==> ?N. !m n. m >= N ==> norm(vsum(s INTER (m..n)) f) < e`,
14738 REWRITE_TAC[summable; GSYM SERIES_CAUCHY]);;
14740 let SUMMABLE_IFF_EVENTUALLY = prove
14741 (`!f g k. (?N. !n. N <= n /\ n IN k ==> f n = g n)
14742 ==> (summable k f <=> summable k g)`,
14743 REWRITE_TAC[summable; SERIES_CAUCHY] THEN REPEAT GEN_TAC THEN
14744 DISCH_THEN(X_CHOOSE_THEN `N0:num` STRIP_ASSUME_TAC) THEN
14745 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
14746 AP_TERM_TAC THEN EQ_TAC THEN
14747 DISCH_THEN(X_CHOOSE_THEN `N1:num`
14748 (fun th -> EXISTS_TAC `N0 + N1:num` THEN MP_TAC th)) THEN
14749 REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
14750 DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
14751 (ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN
14752 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
14753 MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[IN_INTER; IN_NUMSEG] THEN
14754 REPEAT STRIP_TAC THENL [ALL_TAC; CONV_TAC SYM_CONV] THEN
14755 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
14758 let SUMMABLE_EQ_EVENTUALLY = prove
14759 (`!f g k. (?N. !n. N <= n /\ n IN k ==> f n = g n) /\ summable k f
14761 MESON_TAC[SUMMABLE_IFF_EVENTUALLY]);;
14763 let SUMMABLE_IFF_COFINITE = prove
14764 (`!f s t. FINITE((s DIFF t) UNION (t DIFF s))
14765 ==> (summable s f <=> summable t f)`,
14766 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM SUMMABLE_RESTRICT] THEN
14767 MATCH_MP_TAC SUMMABLE_IFF_EVENTUALLY THEN
14768 FIRST_ASSUM(MP_TAC o ISPEC `\x:num.x` o MATCH_MP UPPER_BOUND_FINITE_SET) THEN
14769 DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN REWRITE_TAC[IN_UNIV] THEN
14770 DISCH_TAC THEN EXISTS_TAC `N + 1` THEN
14771 REWRITE_TAC[ARITH_RULE `N + 1 <= n <=> ~(n <= N)`] THEN ASM SET_TAC[]);;
14773 let SUMMABLE_EQ_COFINITE = prove
14774 (`!f s t. FINITE((s DIFF t) UNION (t DIFF s)) /\ summable s f
14776 MESON_TAC[SUMMABLE_IFF_COFINITE]);;
14778 let SUMMABLE_FROM_ELSEWHERE = prove
14779 (`!f m n. summable (from m) f ==> summable (from n) f`,
14780 REPEAT GEN_TAC THEN
14781 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] SUMMABLE_EQ_COFINITE) THEN
14782 MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `0..(m+n)` THEN
14783 SIMP_TAC[FINITE_NUMSEG; SUBSET; IN_NUMSEG; IN_UNION; IN_DIFF; IN_FROM] THEN
14786 (* ------------------------------------------------------------------------- *)
14787 (* Uniform vesion of Cauchy criterion. *)
14788 (* ------------------------------------------------------------------------- *)
14790 let SERIES_CAUCHY_UNIFORM = prove
14791 (`!P f:A->num->real^N k.
14793 ==> ?N. !n x. N <= n /\ P x
14794 ==> dist(vsum(k INTER (0..n)) (f x),
14796 (!e. &0 < e ==> ?N. !m n x. N <= m /\ P x
14797 ==> norm(vsum(k INTER (m..n)) (f x)) < e)`,
14798 REPEAT GEN_TAC THEN
14799 REWRITE_TAC[sums; UNIFORMLY_CONVERGENT_EQ_CAUCHY; cauchy] THEN
14800 ONCE_REWRITE_TAC[MESON[]
14801 `(!m n:num y. N <= m /\ N <= n /\ P y ==> Q m n y) <=>
14802 (!y. P y ==> !m n. N <= m /\ N <= n ==> Q m n y)`] THEN
14803 REWRITE_TAC[SEQUENCE_CAUCHY_WLOG] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
14804 SIMP_TAC[dist; VSUM_DIFF_LEMMA; NORM_VSUM_TRIVIAL_LEMMA] THEN
14805 REWRITE_TAC[GE; TAUT `a ==> b \/ c <=> a /\ ~b ==> c`] THEN
14806 REWRITE_TAC[NOT_LT; ARITH_RULE
14807 `(N <= m /\ N <= n /\ m <= n) /\ m + 1 <= n <=>
14808 N + 1 <= m + 1 /\ m + 1 <= n`] THEN
14809 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
14810 ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
14811 EQ_TAC THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THENL
14812 [EXISTS_TAC `N + 1`; EXISTS_TAC `N:num`] THEN
14813 REPEAT STRIP_TAC THEN
14814 ASM_SIMP_TAC[ARITH_RULE `N + 1 <= m + 1 ==> N <= m + 1`] THEN
14815 FIRST_X_ASSUM(MP_TAC o SPEC `x:A`) THEN ASM_REWRITE_TAC[] THEN
14816 DISCH_THEN(MP_TAC o SPECL [`m - 1`; `n:num`]) THEN
14817 SUBGOAL_THEN `m - 1 + 1 = m` SUBST_ALL_TAC THENL
14818 [ALL_TAC; ANTS_TAC THEN SIMP_TAC[]] THEN
14821 (* ------------------------------------------------------------------------- *)
14822 (* So trivially, terms of a convergent series go to zero. *)
14823 (* ------------------------------------------------------------------------- *)
14825 let SERIES_GOESTOZERO = prove
14826 (`!s x. summable s x
14828 ==> eventually (\n. n IN s ==> norm(x n) < e) sequentially`,
14829 REPEAT GEN_TAC THEN REWRITE_TAC[summable; SERIES_CAUCHY] THEN
14830 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
14831 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
14832 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
14833 X_GEN_TAC `n:num` THEN REPEAT STRIP_TAC THEN
14834 FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `n:num`]) THEN
14835 ASM_SIMP_TAC[NUMSEG_SING; GE; SET_RULE `n IN s ==> s INTER {n} = {n}`] THEN
14836 REWRITE_TAC[VSUM_SING]);;
14838 let SUMMABLE_IMP_TOZERO = prove
14839 (`!f:num->real^N k.
14841 ==> ((\n. if n IN k then f(n) else vec 0) --> vec 0) sequentially`,
14842 REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM SUMMABLE_RESTRICT] THEN
14843 REWRITE_TAC[summable; LIM_SEQUENTIALLY; INTER_UNIV; sums] THEN
14844 DISCH_THEN(X_CHOOSE_TAC `l:real^N`) THEN X_GEN_TAC `e:real` THEN
14845 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
14846 ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN
14847 X_GEN_TAC `N:num` THEN DISCH_TAC THEN EXISTS_TAC `N + 1` THEN
14848 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
14849 FIRST_X_ASSUM(fun th ->
14850 MP_TAC(SPEC `n - 1` th) THEN MP_TAC(SPEC `n:num` th)) THEN
14851 ASM_SIMP_TAC[ARITH_RULE `N + 1 <= n ==> N <= n /\ N <= n - 1`] THEN
14852 ABBREV_TAC `m = n - 1` THEN
14853 SUBGOAL_THEN `n = SUC m` SUBST1_TAC THENL
14854 [ASM_ARITH_TAC; ALL_TAC] THEN
14855 REWRITE_TAC[VSUM_CLAUSES_NUMSEG; LE_0] THEN
14856 REWRITE_TAC[NORM_ARITH `dist(x,vec 0) = norm x`] THEN
14857 COND_CASES_TAC THEN ASM_REWRITE_TAC[NORM_0] THEN CONV_TAC NORM_ARITH);;
14859 let SUMMABLE_IMP_BOUNDED = prove
14860 (`!f:num->real^N k. summable k f ==> bounded (IMAGE f k)`,
14861 REPEAT GEN_TAC THEN
14862 DISCH_THEN(MP_TAC o MATCH_MP SUMMABLE_IMP_TOZERO) THEN
14863 DISCH_THEN(MP_TAC o MATCH_MP CONVERGENT_IMP_BOUNDED) THEN
14864 REWRITE_TAC[BOUNDED_POS; FORALL_IN_IMAGE; IN_UNIV] THEN
14865 MATCH_MP_TAC MONO_EXISTS THEN MESON_TAC[REAL_LT_IMP_LE; NORM_0]);;
14867 let SUMMABLE_IMP_SUMS_BOUNDED = prove
14868 (`!f:num->real^N k.
14869 summable (from k) f ==> bounded { vsum(k..n) f | n IN (:num) }`,
14870 REWRITE_TAC[summable; sums; LEFT_IMP_EXISTS_THM] THEN REPEAT GEN_TAC THEN
14871 DISCH_THEN(MP_TAC o MATCH_MP CONVERGENT_IMP_BOUNDED) THEN
14872 REWRITE_TAC[FROM_INTER_NUMSEG; SIMPLE_IMAGE]);;
14874 (* ------------------------------------------------------------------------- *)
14875 (* Comparison test. *)
14876 (* ------------------------------------------------------------------------- *)
14878 let SERIES_COMPARISON = prove
14879 (`!f g s. (?l. ((lift o g) sums l) s) /\
14880 (?N. !n. n >= N /\ n IN s ==> norm(f n) <= g n)
14881 ==> ?l:real^N. (f sums l) s`,
14882 REPEAT GEN_TAC THEN REWRITE_TAC[SERIES_CAUCHY] THEN
14883 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `N1:num`)) THEN
14884 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
14885 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
14886 DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN
14887 EXISTS_TAC `N1 + N2:num` THEN
14888 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN
14889 MATCH_MP_TAC REAL_LET_TRANS THEN
14890 EXISTS_TAC `norm (vsum (s INTER (m .. n)) (lift o g))` THEN CONJ_TAC THENL
14891 [SIMP_TAC[GSYM LIFT_SUM; FINITE_INTER_NUMSEG; NORM_LIFT] THEN
14892 MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs(a)`) THEN
14893 MATCH_MP_TAC VSUM_NORM_LE THEN
14894 REWRITE_TAC[FINITE_INTER_NUMSEG; IN_INTER; IN_NUMSEG] THEN
14895 ASM_MESON_TAC[ARITH_RULE `m >= N1 + N2:num /\ m <= x ==> x >= N1`];
14896 ASM_MESON_TAC[ARITH_RULE `m >= N1 + N2:num ==> m >= N2`]]);;
14898 let SUMMABLE_COMPARISON = prove
14899 (`!f g s. summable s (lift o g) /\
14900 (?N. !n. n >= N /\ n IN s ==> norm(f n) <= g n)
14902 REWRITE_TAC[summable; SERIES_COMPARISON]);;
14904 let SERIES_LIFT_ABSCONV_IMP_CONV = prove
14905 (`!x:num->real^N k. summable k (\n. lift(norm(x n))) ==> summable k x`,
14906 REWRITE_TAC[summable] THEN REPEAT STRIP_TAC THEN
14907 MATCH_MP_TAC SERIES_COMPARISON THEN
14908 EXISTS_TAC `\n:num. norm(x n:real^N)` THEN
14909 ASM_REWRITE_TAC[o_DEF; REAL_LE_REFL] THEN ASM_MESON_TAC[]);;
14911 let SUMMABLE_SUBSET_ABSCONV = prove
14912 (`!x:num->real^N s t.
14913 summable s (\n. lift(norm(x n))) /\ t SUBSET s
14914 ==> summable t (\n. lift(norm(x n)))`,
14915 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUMMABLE_SUBSET THEN
14916 EXISTS_TAC `s:num->bool` THEN ASM_REWRITE_TAC[] THEN
14917 REWRITE_TAC[summable] THEN MATCH_MP_TAC SERIES_COMPARISON THEN
14918 EXISTS_TAC `\n:num. norm(x n:real^N)` THEN
14919 ASM_REWRITE_TAC[o_DEF; GSYM summable] THEN
14920 EXISTS_TAC `0` THEN REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
14921 REWRITE_TAC[REAL_LE_REFL; NORM_LIFT; REAL_ABS_NORM; NORM_0; NORM_POS_LE]);;
14923 let SERIES_COMPARISON_BOUND = prove
14924 (`!f:num->real^N g s a.
14925 (g sums a) s /\ (!i. i IN s ==> norm(f i) <= drop(g i))
14926 ==> ?l. (f sums l) s /\ norm(l) <= drop a`,
14927 REPEAT STRIP_TAC THEN
14928 MP_TAC(ISPECL [`f:num->real^N`; `drop o (g:num->real^1)`; `s:num->bool`]
14929 SUMMABLE_COMPARISON) THEN
14930 REWRITE_TAC[o_DEF; LIFT_DROP; GE; ETA_AX; summable] THEN
14931 ANTS_TAC THENL [ASM_MESON_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN
14932 X_GEN_TAC `l:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
14933 RULE_ASSUM_TAC(REWRITE_RULE[FROM_0; INTER_UNIV; sums]) THEN
14934 MATCH_MP_TAC SERIES_BOUND THEN MAP_EVERY EXISTS_TAC
14935 [`f:num->real^N`; `drop o (g:num->real^1)`; `s:num->bool`] THEN
14936 ASM_REWRITE_TAC[sums; o_DEF; LIFT_DROP; ETA_AX]);;
14938 (* ------------------------------------------------------------------------- *)
14939 (* Uniform version of comparison test. *)
14940 (* ------------------------------------------------------------------------- *)
14942 let SERIES_COMPARISON_UNIFORM = prove
14943 (`!f g P s. (?l. ((lift o g) sums l) s) /\
14944 (?N. !n x. N <= n /\ n IN s /\ P x ==> norm(f x n) <= g n)
14947 ==> ?N. !n x. N <= n /\ P x
14948 ==> dist(vsum(s INTER (0..n)) (f x),
14950 REPEAT GEN_TAC THEN SIMP_TAC[GE; SERIES_CAUCHY; SERIES_CAUCHY_UNIFORM] THEN
14951 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `N1:num`)) THEN
14952 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
14953 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
14954 DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN
14955 EXISTS_TAC `N1 + N2:num` THEN
14956 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`; `x:A`] THEN DISCH_TAC THEN
14957 MATCH_MP_TAC REAL_LET_TRANS THEN
14958 EXISTS_TAC `norm (vsum (s INTER (m .. n)) (lift o g))` THEN CONJ_TAC THENL
14959 [SIMP_TAC[GSYM LIFT_SUM; FINITE_INTER_NUMSEG; NORM_LIFT] THEN
14960 MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs(a)`) THEN
14961 MATCH_MP_TAC VSUM_NORM_LE THEN
14962 REWRITE_TAC[FINITE_INTER_NUMSEG; IN_INTER; IN_NUMSEG] THEN
14963 ASM_MESON_TAC[ARITH_RULE `N1 + N2:num <= m /\ m <= x ==> N1 <= x`];
14964 ASM_MESON_TAC[ARITH_RULE `N1 + N2:num <= m ==> N2 <= m`]]);;
14966 (* ------------------------------------------------------------------------- *)
14968 (* ------------------------------------------------------------------------- *)
14970 let SERIES_RATIO = prove
14973 (!n. n >= N ==> norm(a(SUC n)) <= c * norm(a(n)))
14974 ==> ?l:real^N. (a sums l) s`,
14975 REWRITE_TAC[GE] THEN REPEAT STRIP_TAC THEN
14976 MATCH_MP_TAC SERIES_COMPARISON THEN
14977 DISJ_CASES_TAC(REAL_ARITH `c <= &0 \/ &0 < c`) THENL
14978 [EXISTS_TAC `\n:num. &0` THEN REWRITE_TAC[o_DEF; LIFT_NUM] THEN
14979 CONJ_TAC THENL [MESON_TAC[SERIES_0]; ALL_TAC] THEN
14980 EXISTS_TAC `N + 1` THEN REWRITE_TAC[GE] THEN REPEAT STRIP_TAC THEN
14981 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `c * norm(a(n - 1):real^N)` THEN
14983 [ASM_MESON_TAC[ARITH_RULE `N + 1 <= n ==> SUC(n - 1) = n /\ N <= n - 1`];
14985 MATCH_MP_TAC(REAL_ARITH `&0 <= --c * x ==> c * x <= &0`) THEN
14986 MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[NORM_POS_LE] THEN
14987 UNDISCH_TAC `c <= &0` THEN REAL_ARITH_TAC;
14988 ASSUME_TAC(MATCH_MP REAL_LT_IMP_LE (ASSUME `&0 < c`))] THEN
14989 EXISTS_TAC `\n. norm(a(N):real^N) * c pow (n - N)` THEN
14990 REWRITE_TAC[] THEN CONJ_TAC THENL
14992 EXISTS_TAC `N:num` THEN
14993 SIMP_TAC[GE; LE_EXISTS; IMP_CONJ; ADD_SUB2; LEFT_IMP_EXISTS_THM] THEN
14994 SUBGOAL_THEN `!d:num. norm(a(N + d):real^N) <= norm(a N) * c pow d`
14995 (fun th -> MESON_TAC[th]) THEN INDUCT_TAC THEN
14996 REWRITE_TAC[ADD_CLAUSES; real_pow; REAL_MUL_RID; REAL_LE_REFL] THEN
14997 MATCH_MP_TAC REAL_LE_TRANS THEN
14998 EXISTS_TAC `c * norm((a:num->real^N) (N + d))` THEN
14999 ASM_SIMP_TAC[LE_ADD] THEN ASM_MESON_TAC[REAL_LE_LMUL; REAL_MUL_AC]] THEN
15000 GEN_REWRITE_TAC I [SERIES_CAUCHY] THEN X_GEN_TAC `e:real` THEN
15001 SIMP_TAC[GSYM LIFT_SUM; FINITE_INTER; NORM_LIFT; FINITE_NUMSEG] THEN
15002 DISCH_TAC THEN SIMP_TAC[SUM_LMUL; FINITE_INTER; FINITE_NUMSEG] THEN
15003 ASM_CASES_TAC `(a:num->real^N) N = vec 0` THENL
15004 [ASM_REWRITE_TAC[NORM_0; REAL_MUL_LZERO; REAL_ABS_NUM]; ALL_TAC] THEN
15005 MP_TAC(SPECL [`c:real`; `((&1 - c) * e) / norm((a:num->real^N) N)`]
15006 REAL_ARCH_POW_INV) THEN
15007 ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MUL; REAL_SUB_LT; NORM_POS_LT; GE] THEN
15008 DISCH_THEN(X_CHOOSE_TAC `M:num`) THEN EXISTS_TAC `N + M:num` THEN
15009 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN
15010 MATCH_MP_TAC REAL_LET_TRANS THEN
15011 EXISTS_TAC `abs(norm((a:num->real^N) N) *
15012 sum(m..n) (\i. c pow (i - N)))` THEN
15014 [REWRITE_TAC[REAL_ABS_MUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN
15015 REWRITE_TAC[REAL_ABS_POS] THEN
15016 MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x <= y ==> abs x <= abs y`) THEN
15017 ASM_SIMP_TAC[SUM_POS_LE; FINITE_INTER_NUMSEG; REAL_POW_LE] THEN
15018 MATCH_MP_TAC SUM_SUBSET THEN ASM_SIMP_TAC[REAL_POW_LE] THEN
15019 REWRITE_TAC[FINITE_INTER_NUMSEG; FINITE_NUMSEG] THEN
15020 REWRITE_TAC[IN_INTER; IN_DIFF] THEN MESON_TAC[];
15022 REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM] THEN
15023 DISJ_CASES_TAC(ARITH_RULE `n:num < m \/ m <= n`) THENL
15024 [ASM_SIMP_TAC[SUM_TRIV_NUMSEG; REAL_ABS_NUM; REAL_MUL_RZERO]; ALL_TAC] THEN
15025 SUBGOAL_THEN `m = 0 + m /\ n = (n - m) + m` (CONJUNCTS_THEN SUBST1_TAC) THENL
15026 [UNDISCH_TAC `m:num <= n` THEN ARITH_TAC; ALL_TAC] THEN
15027 REWRITE_TAC[SUM_OFFSET] THEN UNDISCH_TAC `N + M:num <= m` THEN
15028 SIMP_TAC[LE_EXISTS] THEN DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST_ALL_TAC) THEN
15029 REWRITE_TAC[ARITH_RULE `(i + (N + M) + d) - N:num = (M + d) + i`] THEN
15030 ONCE_REWRITE_TAC[REAL_POW_ADD] THEN REWRITE_TAC[SUM_LMUL; SUM_GP] THEN
15031 ASM_SIMP_TAC[LT; REAL_LT_IMP_NE] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
15032 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT; REAL_ABS_MUL] THEN
15033 REWRITE_TAC[REAL_ABS_POW] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
15034 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_ABS_DIV; REAL_POW_LT; REAL_ARITH
15035 `&0 < c /\ c < &1 ==> &0 < abs c /\ &0 < abs(&1 - c)`; REAL_LT_LDIV_EQ] THEN
15036 MATCH_MP_TAC(REAL_ARITH
15037 `&0 < x /\ x <= &1 /\ &1 <= e ==> abs(c pow 0 - x) < e`) THEN
15038 ASM_SIMP_TAC[REAL_POW_LT; REAL_POW_1_LE; REAL_LT_IMP_LE] THEN
15039 ASM_SIMP_TAC[REAL_ARITH `c < &1 ==> x * abs(&1 - c) = (&1 - c) * x`] THEN
15040 REWRITE_TAC[real_div; REAL_INV_MUL; REAL_POW_ADD; REAL_MUL_ASSOC] THEN
15041 REWRITE_TAC[REAL_ARITH
15042 `(((a * b) * c) * d) * e = (e * ((a * b) * c)) * d`] THEN
15043 ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ; REAL_POW_LT; REAL_MUL_LID;
15044 REAL_ARITH `&0 < c ==> abs c = c`] THEN
15045 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
15046 `xm < e ==> &0 <= (d - &1) * e ==> xm <= d * e`)) THEN
15047 MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL
15048 [REWRITE_TAC[REAL_SUB_LE; GSYM REAL_POW_INV] THEN
15049 MATCH_MP_TAC REAL_POW_LE_1 THEN
15050 MATCH_MP_TAC REAL_INV_1_LE THEN ASM_SIMP_TAC[REAL_LT_IMP_LE];
15051 MATCH_MP_TAC REAL_LT_IMP_LE THEN
15052 ASM_SIMP_TAC[REAL_SUB_LT; REAL_LT_MUL; REAL_LT_DIV; NORM_POS_LT]]);;
15054 (* ------------------------------------------------------------------------- *)
15055 (* Ostensibly weaker versions of the boundedness of partial sums. *)
15056 (* ------------------------------------------------------------------------- *)
15058 let BOUNDED_PARTIAL_SUMS = prove
15059 (`!f:num->real^N k.
15060 bounded { vsum(k..n) f | n IN (:num) }
15061 ==> bounded { vsum(m..n) f | m IN (:num) /\ n IN (:num) }`,
15062 REPEAT STRIP_TAC THEN
15063 SUBGOAL_THEN `bounded { vsum(0..n) f:real^N | n IN (:num) }` MP_TAC THENL
15064 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
15065 REWRITE_TAC[bounded] THEN
15066 REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; IN_UNIV] THEN
15067 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
15068 EXISTS_TAC `sum { i:num | i < k} (\i. norm(f i:real^N)) + B` THEN
15069 X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num < k` THENL
15070 [MATCH_MP_TAC(REAL_ARITH
15071 `!y. x <= y /\ y <= a /\ &0 < b ==> x <= a + b`) THEN
15072 EXISTS_TAC `sum (0..i) (\i. norm(f i:real^N))` THEN
15073 ASM_SIMP_TAC[VSUM_NORM; FINITE_NUMSEG] THEN
15074 MATCH_MP_TAC SUM_SUBSET THEN
15075 REWRITE_TAC[FINITE_NUMSEG; FINITE_NUMSEG_LT; NORM_POS_LE] THEN
15076 REWRITE_TAC[IN_DIFF; IN_NUMSEG; IN_ELIM_THM] THEN ASM_ARITH_TAC;
15078 ASM_CASES_TAC `k = 0` THENL
15079 [FIRST_X_ASSUM SUBST_ALL_TAC THEN MATCH_MP_TAC(REAL_ARITH
15080 `x <= B /\ &0 <= b ==> x <= b + B`) THEN
15081 ASM_SIMP_TAC[SUM_POS_LE; FINITE_NUMSEG_LT; NORM_POS_LE];
15083 MP_TAC(ISPECL [`f:num->real^N`; `0`; `k:num`; `i:num`]
15084 VSUM_COMBINE_L) THEN
15085 ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
15086 DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[NUMSEG_LT] THEN
15087 MATCH_MP_TAC(NORM_ARITH
15088 `norm(x) <= a /\ norm(y) <= b ==> norm(x + y) <= a + b`) THEN
15089 ASM_SIMP_TAC[VSUM_NORM; FINITE_NUMSEG];
15091 DISCH_THEN(fun th ->
15092 MP_TAC(MATCH_MP BOUNDED_DIFFS (W CONJ th)) THEN MP_TAC th) THEN
15093 REWRITE_TAC[IMP_IMP; GSYM BOUNDED_UNION] THEN
15094 MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b ==> c <=> b ==> a ==> c`]
15095 BOUNDED_SUBSET) THEN
15096 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNION; LEFT_IMP_EXISTS_THM; IN_UNIV] THEN
15097 MAP_EVERY X_GEN_TAC [`x:real^N`; `m:num`; `n:num`] THEN
15098 DISCH_THEN SUBST1_TAC THEN
15099 ASM_CASES_TAC `m = 0` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
15100 ASM_CASES_TAC `n:num < m` THENL
15101 [DISJ2_TAC THEN REPEAT(EXISTS_TAC `vsum(0..0) (f:num->real^N)`) THEN
15102 ASM_SIMP_TAC[VSUM_TRIV_NUMSEG; VECTOR_SUB_REFL] THEN MESON_TAC[];
15104 DISJ2_TAC THEN MAP_EVERY EXISTS_TAC
15105 [`vsum(0..n) (f:num->real^N)`; `vsum(0..(m-1)) (f:num->real^N)`] THEN
15106 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
15107 MP_TAC(ISPECL [`f:num->real^N`; `0`; `m:num`; `n:num`]
15108 VSUM_COMBINE_L) THEN
15109 ANTS_TAC THENL [ASM_ARITH_TAC; VECTOR_ARITH_TAC]);;
15111 (* ------------------------------------------------------------------------- *)
15112 (* General Dirichlet convergence test (could make this uniform on a set). *)
15113 (* ------------------------------------------------------------------------- *)
15115 let SUMMABLE_BILINEAR_PARTIAL_PRE = prove
15116 (`!f g h:real^M->real^N->real^P l k.
15118 ((\n. h (f(n + 1)) (g(n))) --> l) sequentially /\
15119 summable (from k) (\n. h (f(n + 1) - f(n)) (g(n)))
15120 ==> summable (from k) (\n. h (f n) (g(n) - g(n - 1)))`,
15121 REPEAT GEN_TAC THEN
15122 REWRITE_TAC[summable; sums; FROM_INTER_NUMSEG] THEN
15123 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
15124 FIRST_ASSUM(fun th ->
15125 REWRITE_TAC[MATCH_MP BILINEAR_VSUM_PARTIAL_PRE th]) THEN
15126 DISCH_THEN(X_CHOOSE_TAC `l':real^P`) THEN
15127 EXISTS_TAC `l - (h:real^M->real^N->real^P) (f k) (g(k - 1)) - l'` THEN
15128 REWRITE_TAC[LIM_CASES_SEQUENTIALLY] THEN
15129 REPEAT(MATCH_MP_TAC LIM_SUB THEN ASM_REWRITE_TAC[LIM_CONST]));;
15131 let SERIES_DIRICHLET_BILINEAR = prove
15132 (`!f g h:real^M->real^N->real^P k m p l.
15134 bounded { vsum (m..n) f | n IN (:num)} /\
15135 summable (from p) (\n. lift(norm(g(n + 1) - g(n)))) /\
15136 ((\n. h (g(n + 1)) (vsum(1..n) f)) --> l) sequentially
15137 ==> summable (from k) (\n. h (g n) (f n))`,
15138 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUMMABLE_FROM_ELSEWHERE THEN
15139 EXISTS_TAC `1` THEN
15140 FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP BOUNDED_PARTIAL_SUMS) THEN
15141 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
15142 SIMP_TAC[IN_ELIM_THM; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN
15143 REWRITE_TAC[MESON[] `(!x a b. x = f a b ==> p a b) <=> (!a b. p a b)`] THEN
15144 X_GEN_TAC `B:real` THEN STRIP_TAC THEN
15145 FIRST_ASSUM(MP_TAC o MATCH_MP BILINEAR_BOUNDED_POS) THEN
15146 DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN
15147 MATCH_MP_TAC SUMMABLE_EQ THEN
15148 EXISTS_TAC `\n. (h:real^M->real^N->real^P)
15149 (g n) (vsum (1..n) f - vsum (1..n-1) f)` THEN
15150 SIMP_TAC[IN_FROM; GSYM NUMSEG_RREC] THEN
15151 SIMP_TAC[VSUM_CLAUSES; FINITE_NUMSEG; IN_NUMSEG;
15152 ARITH_RULE `1 <= n ==> ~(n <= n - 1)`] THEN
15154 [REPEAT STRIP_TAC THEN ASM_SIMP_TAC[BILINEAR_RADD; BILINEAR_RSUB] THEN
15157 MATCH_MP_TAC SUMMABLE_FROM_ELSEWHERE THEN EXISTS_TAC `p:num` THEN
15158 MP_TAC(ISPECL [`g:num->real^M`; `\n. vsum(1..n) f:real^N`;
15159 `h:real^M->real^N->real^P`; `l:real^P`; `p:num`]
15160 SUMMABLE_BILINEAR_PARTIAL_PRE) THEN
15161 REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
15162 ASM_REWRITE_TAC[] THEN
15164 `summable (from p) (lift o (\n. C * B * norm(g(n + 1) - g(n):real^M)))`
15165 MP_TAC THENL [ASM_SIMP_TAC[o_DEF; LIFT_CMUL; SUMMABLE_CMUL]; ALL_TAC] THEN
15166 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] SUMMABLE_COMPARISON) THEN
15167 EXISTS_TAC `0` THEN REWRITE_TAC[IN_FROM; GE; LE_0] THEN
15168 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
15169 `C * norm(g(n + 1) - g(n):real^M) * norm(vsum (1..n) f:real^N)` THEN
15170 ASM_SIMP_TAC[REAL_LE_LMUL_EQ] THEN
15171 GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN
15172 ASM_SIMP_TAC[REAL_LE_LMUL; NORM_POS_LE]);;
15174 let SERIES_DIRICHLET = prove
15175 (`!f:num->real^N g N k m.
15176 bounded { vsum (m..n) f | n IN (:num)} /\
15177 (!n. N <= n ==> g(n + 1) <= g(n)) /\
15178 ((lift o g) --> vec 0) sequentially
15179 ==> summable (from k) (\n. g(n) % f(n))`,
15180 REPEAT STRIP_TAC THEN
15181 MP_TAC(ISPECL [`f:num->real^N`; `lift o (g:num->real)`;
15182 `\x y:real^N. drop x % y`] SERIES_DIRICHLET_BILINEAR) THEN
15183 REWRITE_TAC[o_THM; LIFT_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN
15184 MAP_EVERY EXISTS_TAC [`m:num`; `N:num`; `vec 0:real^N`] THEN CONJ_TAC THENL
15185 [REWRITE_TAC[bilinear; linear; DROP_ADD; DROP_CMUL] THEN
15186 REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC;
15188 ASM_REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT] THEN
15189 FIRST_ASSUM(MP_TAC o SPEC `1` o MATCH_MP SEQ_OFFSET) THEN
15190 REWRITE_TAC[o_THM] THEN DISCH_TAC THEN CONJ_TAC THENL
15191 [MATCH_MP_TAC SUMMABLE_EQ_EVENTUALLY THEN
15192 EXISTS_TAC `\n. lift(g(n) - g(n + 1))` THEN REWRITE_TAC[] THEN
15194 [ASM_MESON_TAC[REAL_ARITH `b <= a ==> abs(b - a) = a - b`];
15195 REWRITE_TAC[summable; sums; FROM_INTER_NUMSEG; VSUM_DIFFS; LIFT_SUB] THEN
15196 REWRITE_TAC[LIM_CASES_SEQUENTIALLY] THEN
15197 EXISTS_TAC `lift(g(N:num)) - vec 0` THEN
15198 MATCH_MP_TAC LIM_SUB THEN ASM_REWRITE_TAC[LIM_CONST]];
15199 MATCH_MP_TAC LIM_NULL_VMUL_BOUNDED THEN ASM_REWRITE_TAC[o_DEF] THEN
15200 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
15201 FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP BOUNDED_PARTIAL_SUMS) THEN
15202 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BOUNDED_POS]) THEN
15203 SIMP_TAC[IN_ELIM_THM; IN_UNIV] THEN MESON_TAC[]]);;
15205 (* ------------------------------------------------------------------------- *)
15206 (* Rearranging absolutely convergent series. *)
15207 (* ------------------------------------------------------------------------- *)
15209 let SERIES_INJECTIVE_IMAGE_STRONG = prove
15210 (`!x:num->real^N s f.
15211 summable (IMAGE f s) (\n. lift(norm(x n))) /\
15212 (!m n. m IN s /\ n IN s /\ f m = f n ==> m = n)
15213 ==> ((\n. vsum (IMAGE f s INTER (0..n)) x -
15214 vsum (s INTER (0..n)) (x o f)) --> vec 0)
15217 (`!f:A->real^N s t.
15218 FINITE s /\ FINITE t
15219 ==> vsum s f - vsum t f = vsum (s DIFF t) f - vsum (t DIFF s) f`,
15220 REPEAT STRIP_TAC THEN
15221 ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN
15222 ASM_SIMP_TAC[VSUM_DIFF; INTER_SUBSET] THEN
15223 REWRITE_TAC[INTER_COMM] THEN VECTOR_ARITH_TAC) in
15224 REPEAT STRIP_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN
15225 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
15226 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUMMABLE_CAUCHY]) THEN
15227 SIMP_TAC[VSUM_REAL; FINITE_INTER; FINITE_NUMSEG] THEN
15228 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [o_DEF] THEN
15229 REWRITE_TAC[NORM_LIFT; LIFT_DROP] THEN
15230 SIMP_TAC[real_abs; SUM_POS_LE; NORM_POS_LE; FINITE_INTER; FINITE_NUMSEG] THEN
15231 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
15232 ASM_REWRITE_TAC[dist; GE; VECTOR_SUB_RZERO; REAL_HALF] THEN
15233 DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN
15234 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
15235 DISCH_THEN(X_CHOOSE_TAC `g:num->num`) THEN
15236 MP_TAC(ISPECL [`g:num->num`; `0..N`] UPPER_BOUND_FINITE_SET) THEN
15237 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG; LE_0] THEN
15238 DISCH_THEN(X_CHOOSE_TAC `P:num`) THEN
15239 EXISTS_TAC `MAX N P` THEN X_GEN_TAC `n:num` THEN
15240 SIMP_TAC[ARITH_RULE `MAX a b <= c <=> a <= c /\ b <= c`] THEN DISCH_TAC THEN
15241 W(MP_TAC o PART_MATCH (rand o rand) VSUM_IMAGE o rand o
15242 rand o lhand o snd) THEN
15244 [ASM_MESON_TAC[FINITE_INTER; FINITE_NUMSEG; IN_INTER];
15245 DISCH_THEN(SUBST1_TAC o SYM)] THEN
15246 W(MP_TAC o PART_MATCH (lhand o rand) lemma o rand o lhand o snd) THEN
15247 SIMP_TAC[FINITE_INTER; FINITE_IMAGE; FINITE_NUMSEG] THEN
15248 DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(NORM_ARITH
15249 `norm a < e / &2 /\ norm b < e / &2 ==> norm(a - b:real^N) < e`) THEN
15251 W(MP_TAC o PART_MATCH (lhand o rand) VSUM_NORM o lhand o snd) THEN
15252 SIMP_TAC[FINITE_DIFF; FINITE_IMAGE; FINITE_INTER; FINITE_NUMSEG] THEN
15253 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN
15254 MATCH_MP_TAC REAL_LET_TRANS THENL
15256 `sum(IMAGE (f:num->num) s INTER (N..n)) (\i. norm(x i :real^N))` THEN
15257 ASM_SIMP_TAC[LE_REFL] THEN MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
15258 SIMP_TAC[NORM_POS_LE; FINITE_INTER; FINITE_NUMSEG] THEN
15259 MATCH_MP_TAC(SET_RULE
15260 `(!x. x IN s /\ f(x) IN n /\ ~(x IN m) ==> f x IN t)
15261 ==> (IMAGE f s INTER n) DIFF (IMAGE f (s INTER m)) SUBSET
15262 IMAGE f s INTER t`) THEN
15263 ASM_SIMP_TAC[IN_NUMSEG; LE_0; NOT_LE] THEN
15264 X_GEN_TAC `i:num` THEN STRIP_TAC THEN
15265 MATCH_MP_TAC LT_IMP_LE THEN ONCE_REWRITE_TAC[GSYM NOT_LE] THEN
15266 FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE BINDER_CONV
15267 [GSYM CONTRAPOS_THM]) THEN
15268 ASM_SIMP_TAC[] THEN ASM_ARITH_TAC;
15269 MP_TAC(ISPECL [`f:num->num`; `0..n`] UPPER_BOUND_FINITE_SET) THEN
15270 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG; LE_0] THEN
15271 DISCH_THEN(X_CHOOSE_TAC `p:num`) THEN
15273 `sum(IMAGE (f:num->num) s INTER (N..p)) (\i. norm(x i :real^N))` THEN
15274 ASM_SIMP_TAC[LE_REFL] THEN MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN
15275 SIMP_TAC[NORM_POS_LE; FINITE_INTER; FINITE_NUMSEG] THEN
15276 MATCH_MP_TAC(SET_RULE
15277 `(!x. x IN s /\ x IN n /\ ~(f x IN m) ==> f x IN t)
15278 ==> (IMAGE f (s INTER n) DIFF (IMAGE f s) INTER m) SUBSET
15279 (IMAGE f s INTER t)`) THEN
15280 ASM_SIMP_TAC[IN_NUMSEG; LE_0] THEN ASM_ARITH_TAC]);;
15282 let SERIES_INJECTIVE_IMAGE = prove
15283 (`!x:num->real^N s f l.
15284 summable (IMAGE f s) (\n. lift(norm(x n))) /\
15285 (!m n. m IN s /\ n IN s /\ f m = f n ==> m = n)
15286 ==> (((x o f) sums l) s <=> (x sums l) (IMAGE f s))`,
15287 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN REWRITE_TAC[sums] THEN
15288 MATCH_MP_TAC LIM_TRANSFORM_EQ THEN REWRITE_TAC[] THEN
15289 MATCH_MP_TAC SERIES_INJECTIVE_IMAGE_STRONG THEN
15290 ASM_REWRITE_TAC[]);;
15292 let SERIES_REARRANGE_EQ = prove
15293 (`!x:num->real^N s p l.
15294 summable s (\n. lift(norm(x n))) /\ p permutes s
15295 ==> (((x o p) sums l) s <=> (x sums l) s)`,
15296 REPEAT STRIP_TAC THEN
15297 MP_TAC(ISPECL [`x:num->real^N`; `s:num->bool`; `p:num->num`; `l:real^N`]
15298 SERIES_INJECTIVE_IMAGE) THEN
15299 ASM_SIMP_TAC[PERMUTES_IMAGE] THEN
15300 ASM_MESON_TAC[PERMUTES_INJECTIVE]);;
15302 let SERIES_REARRANGE = prove
15303 (`!x:num->real^N s p l.
15304 summable s (\n. lift(norm(x n))) /\ p permutes s /\ (x sums l) s
15305 ==> ((x o p) sums l) s`,
15306 MESON_TAC[SERIES_REARRANGE_EQ]);;
15308 let SUMMABLE_REARRANGE = prove
15310 summable s (\n. lift(norm(x n))) /\ p permutes s
15311 ==> summable s (x o p)`,
15312 MESON_TAC[SERIES_LIFT_ABSCONV_IMP_CONV; summable; SERIES_REARRANGE]);;
15314 (* ------------------------------------------------------------------------- *)
15315 (* Banach fixed point theorem (not really topological...) *)
15316 (* ------------------------------------------------------------------------- *)
15318 let BANACH_FIX = prove
15319 (`!f s c. complete s /\ ~(s = {}) /\
15320 &0 <= c /\ c < &1 /\
15321 (IMAGE f s) SUBSET s /\
15322 (!x y. x IN s /\ y IN s ==> dist(f(x),f(y)) <= c * dist(x,y))
15323 ==> ?!x:real^N. x IN s /\ (f x = x)`,
15324 REPEAT STRIP_TAC THEN REWRITE_TAC[EXISTS_UNIQUE_THM] THEN CONJ_TAC THENL
15326 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
15327 SUBGOAL_THEN `dist((f:real^N->real^N) x,f y) <= c * dist(x,y)` MP_TAC THENL
15328 [ASM_MESON_TAC[]; ALL_TAC] THEN
15329 ASM_REWRITE_TAC[REAL_ARITH `a <= c * a <=> &0 <= --a * (&1 - c)`] THEN
15330 ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_SUB_LT; real_div] THEN
15331 REWRITE_TAC[REAL_MUL_LZERO; REAL_ARITH `&0 <= --x <=> ~(&0 < x)`] THEN
15332 MESON_TAC[DIST_POS_LT]] THEN
15333 STRIP_ASSUME_TAC(prove_recursive_functions_exist num_RECURSION
15334 `(z 0 = @x:real^N. x IN s) /\ (!n. z(SUC n) = f(z n))`) THEN
15335 SUBGOAL_THEN `!n. (z:num->real^N) n IN s` ASSUME_TAC THENL
15336 [INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN
15337 ASM_MESON_TAC[MEMBER_NOT_EMPTY; SUBSET; IN_IMAGE];
15339 UNDISCH_THEN `z 0 = @x:real^N. x IN s` (K ALL_TAC) THEN
15340 SUBGOAL_THEN `?x:real^N. x IN s /\ (z --> x) sequentially` MP_TAC THENL
15342 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
15343 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
15344 ABBREV_TAC `e = dist(f(a:real^N),a)` THEN
15345 SUBGOAL_THEN `~(&0 < e)` (fun th -> ASM_MESON_TAC[th; DIST_POS_LT]) THEN
15347 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
15348 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
15349 ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
15351 `dist(f(z N),a:real^N) < e / &2 /\ dist(f(z(N:num)),f(a)) < e / &2`
15352 (fun th -> ASM_MESON_TAC[th; DIST_TRIANGLE_HALF_R; REAL_LT_REFL]) THEN
15353 CONJ_TAC THENL [ASM_MESON_TAC[ARITH_RULE `N <= SUC N`]; ALL_TAC] THEN
15354 MATCH_MP_TAC REAL_LET_TRANS THEN
15355 EXISTS_TAC `c * dist((z:num->real^N) N,a)` THEN ASM_SIMP_TAC[] THEN
15356 MATCH_MP_TAC(REAL_ARITH `x < y /\ c * x <= &1 * x ==> c * x < y`) THEN
15357 ASM_SIMP_TAC[LE_REFL; REAL_LE_RMUL; DIST_POS_LE; REAL_LT_IMP_LE]] THEN
15358 FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [complete]) THEN
15359 ASM_REWRITE_TAC[CAUCHY] THEN
15360 SUBGOAL_THEN `!n. dist(z(n):real^N,z(SUC n)) <= c pow n * dist(z(0),z(1))`
15363 REWRITE_TAC[real_pow; ARITH; REAL_MUL_LID; REAL_LE_REFL] THEN
15364 MATCH_MP_TAC REAL_LE_TRANS THEN
15365 EXISTS_TAC `c * dist(z(n):real^N,z(SUC n))` THEN
15366 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
15367 REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[REAL_LE_LMUL];
15370 `!m n:num. (&1 - c) * dist(z(m):real^N,z(m+n))
15371 <= c pow m * dist(z(0),z(1)) * (&1 - c pow n)`
15373 [GEN_TAC THEN INDUCT_TAC THENL
15374 [REWRITE_TAC[ADD_CLAUSES; DIST_REFL; REAL_MUL_RZERO] THEN
15375 MATCH_MP_TAC REAL_LE_MUL THEN
15376 ASM_SIMP_TAC[REAL_LE_MUL; REAL_POW_LE; DIST_POS_LE; REAL_SUB_LE;
15377 REAL_POW_1_LE; REAL_LT_IMP_LE];
15379 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
15380 `(&1 - c) * (dist(z m:real^N,z(m + n)) + dist(z(m + n),z(m + SUC n)))` THEN
15381 ASM_SIMP_TAC[REAL_LE_LMUL; REAL_SUB_LE; REAL_LT_IMP_LE; DIST_TRIANGLE] THEN
15382 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
15383 `c * x <= y ==> c * x' + y <= y' ==> c * (x + x') <= y'`)) THEN
15384 REWRITE_TAC[REAL_ARITH
15385 `q + a * b * (&1 - x) <= a * b * (&1 - y) <=> q <= a * b * (x - y)`] THEN
15386 REWRITE_TAC[ADD_CLAUSES; real_pow] THEN
15387 REWRITE_TAC[REAL_ARITH `a * b * (d - c * d) = (&1 - c) * a * d * b`] THEN
15388 MATCH_MP_TAC REAL_LE_LMUL THEN
15389 ASM_SIMP_TAC[REAL_SUB_LE; REAL_LT_IMP_LE] THEN
15390 REWRITE_TAC[GSYM REAL_POW_ADD; REAL_MUL_ASSOC] THEN ASM_MESON_TAC[];
15392 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
15393 ASM_CASES_TAC `(z:num->real^N) 0 = z 1` THENL
15394 [FIRST_X_ASSUM SUBST_ALL_TAC THEN EXISTS_TAC `0` THEN
15395 REWRITE_TAC[GE; LE_0] THEN X_GEN_TAC `n:num` THEN
15396 FIRST_X_ASSUM(MP_TAC o SPECL [`0`; `n:num`]) THEN
15397 REWRITE_TAC[ADD_CLAUSES; DIST_REFL; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN
15398 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
15399 ASM_CASES_TAC `(z:num->real^N) 0 = z n` THEN
15400 ASM_REWRITE_TAC[DIST_REFL; REAL_NOT_LE] THEN
15401 ASM_SIMP_TAC[REAL_LT_MUL; DIST_POS_LT; REAL_SUB_LT];
15403 MP_TAC(SPECL [`c:real`; `e * (&1 - c) / dist((z:num->real^N) 0,z 1)`]
15404 REAL_ARCH_POW_INV) THEN
15405 ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; REAL_SUB_LT; DIST_POS_LT] THEN
15406 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
15407 REWRITE_TAC[real_div; GE; REAL_MUL_ASSOC] THEN
15408 ASM_SIMP_TAC[REAL_LT_RDIV_EQ; GSYM real_div; DIST_POS_LT] THEN
15409 ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ; REAL_SUB_LT] THEN DISCH_TAC THEN
15410 REWRITE_TAC[LE_EXISTS; LEFT_IMP_EXISTS_THM] THEN
15411 GEN_TAC THEN X_GEN_TAC `d:num` THEN DISCH_THEN SUBST_ALL_TAC THEN
15412 ONCE_REWRITE_TAC[DIST_SYM] THEN
15413 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REAL_ARITH
15414 `d < e ==> x <= d ==> x < e`)) THEN
15415 ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN
15416 FIRST_X_ASSUM(MP_TAC o SPECL [`N:num`; `d:num`]) THEN
15417 MATCH_MP_TAC(REAL_ARITH
15418 `(c * d) * e <= (c * d) * &1 ==> x * y <= c * d * e ==> y * x <= c * d`) THEN
15419 MATCH_MP_TAC REAL_LE_LMUL THEN
15420 ASM_SIMP_TAC[REAL_LE_MUL; REAL_POW_LE; DIST_POS_LE; REAL_ARITH
15421 `&0 <= x ==> &1 - x <= &1`]);;
15423 (* ------------------------------------------------------------------------- *)
15424 (* Edelstein fixed point theorem. *)
15425 (* ------------------------------------------------------------------------- *)
15427 let EDELSTEIN_FIX = prove
15428 (`!f s. compact s /\ ~(s = {}) /\ (IMAGE f s) SUBSET s /\
15429 (!x y. x IN s /\ y IN s /\ ~(x = y) ==> dist(f(x),f(y)) < dist(x,y))
15430 ==> ?!x:real^N. x IN s /\ f x = x`,
15431 MAP_EVERY X_GEN_TAC [`g:real^N->real^N`; `s:real^N->bool`] THEN
15432 REPEAT STRIP_TAC THEN REWRITE_TAC[EXISTS_UNIQUE_THM] THEN CONJ_TAC THENL
15433 [ALL_TAC; ASM_MESON_TAC[REAL_LT_REFL]] THEN
15435 `!x y. x IN s /\ y IN s ==> dist((g:real^N->real^N)(x),g(y)) <= dist(x,y)`
15437 [REPEAT STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN
15438 ASM_SIMP_TAC[DIST_REFL; REAL_LE_LT];
15440 ASM_CASES_TAC `?x:real^N. x IN s /\ ~(g x = x)` THENL
15441 [ALL_TAC; ASM SET_TAC[]] THEN
15442 FIRST_X_ASSUM(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
15443 ABBREV_TAC `y = (g:real^N->real^N) x` THEN
15444 SUBGOAL_THEN `(y:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
15445 FIRST_ASSUM(MP_TAC o MATCH_MP COMPACT_PCROSS o W CONJ) THEN
15446 REWRITE_TAC[compact; PCROSS] THEN
15447 (STRIP_ASSUME_TAC o prove_general_recursive_function_exists)
15448 `?f:num->real^N->real^N.
15449 (!z. f 0 z = z) /\ (!z n. f (SUC n) z = g(f n z))` THEN
15450 SUBGOAL_THEN `!n z. z IN s ==> (f:num->real^N->real^N) n z IN s`
15451 STRIP_ASSUME_TAC THENL [INDUCT_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN
15453 `!m n w z. m <= n /\ w IN s /\ z IN s
15454 ==> dist((f:num->real^N->real^N) n w,f n z) <= dist(f m w,f m z)`
15456 [REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
15457 MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN
15458 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE]) THEN
15459 ASM_SIMP_TAC[REAL_LE_REFL] THEN MESON_TAC[REAL_LE_TRANS];
15461 DISCH_THEN(MP_TAC o SPEC
15462 `\n:num. pastecart (f n (x:real^N)) (f n y:real^N)`) THEN
15463 ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
15464 MAP_EVERY X_GEN_TAC [`l:real^(N,N)finite_sum`; `s:num->num`] THEN
15465 REWRITE_TAC[o_DEF; IN_ELIM_THM] THEN
15466 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
15467 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
15468 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
15469 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC SUBST_ALL_TAC) THEN
15471 `(\x:real^(N,N)finite_sum. fstcart x) continuous_on UNIV /\
15472 (\x:real^(N,N)finite_sum. sndcart x) continuous_on UNIV`
15474 [CONJ_TAC THEN MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN
15475 REWRITE_TAC[ETA_AX; LINEAR_FSTCART; LINEAR_SNDCART];
15477 REWRITE_TAC[CONTINUOUS_ON_SEQUENTIALLY; IN_UNIV] THEN
15478 DISCH_THEN(CONJUNCTS_THEN(fun th -> FIRST_ASSUM(MP_TAC o MATCH_MP th))) THEN
15479 REWRITE_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART; IMP_IMP] THEN
15480 ONCE_REWRITE_TAC[CONJ_SYM] THEN
15481 DISCH_THEN(fun th -> CONJUNCTS_THEN2 (LABEL_TAC "A") (LABEL_TAC "B") th THEN
15482 MP_TAC(MATCH_MP LIM_SUB th)) THEN
15483 REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC "AB") THEN
15485 `!n. dist(a:real^N,b) <= dist((f:num->real^N->real^N) n x,f n y)`
15486 STRIP_ASSUME_TAC THENL
15487 [X_GEN_TAC `N:num` THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN
15488 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN DISCH_TAC THEN
15489 USE_THEN "AB" (MP_TAC o REWRITE_RULE[LIM_SEQUENTIALLY]) THEN
15490 DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o MATCH_MP th)) THEN
15491 REWRITE_TAC[NOT_EXISTS_THM] THEN X_GEN_TAC `M:num` THEN
15492 DISCH_THEN(MP_TAC o SPEC `M + N:num`) THEN REWRITE_TAC[LE_ADD] THEN
15493 MATCH_MP_TAC(NORM_ARITH
15494 `dist(fx,fy) <= dist(x,y)
15495 ==> ~(dist(fx - fy,a - b) < dist(a,b) - dist(x,y))`) THEN
15496 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
15497 FIRST_X_ASSUM(MP_TAC o SPEC `M + N:num` o MATCH_MP MONOTONE_BIGGER) THEN
15500 SUBGOAL_THEN `b:real^N = a` SUBST_ALL_TAC THENL
15501 [MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
15502 ABBREV_TAC `e = dist(a,b) - dist((g:real^N->real^N) a,g b)` THEN
15503 SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL
15504 [ASM_MESON_TAC[REAL_SUB_LT]; ALL_TAC] THEN
15506 `?n. dist((f:num->real^N->real^N) n x,a) < e / &2 /\
15507 dist(f n y,b) < e / &2`
15508 STRIP_ASSUME_TAC THENL
15509 [MAP_EVERY (fun s -> USE_THEN s (MP_TAC o SPEC `e / &2` o
15510 REWRITE_RULE[LIM_SEQUENTIALLY])) ["A"; "B"] THEN
15511 ASM_REWRITE_TAC[REAL_HALF] THEN
15512 DISCH_THEN(X_CHOOSE_TAC `M:num`) THEN
15513 DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
15514 EXISTS_TAC `(s:num->num) (M + N)` THEN
15515 CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC;
15517 SUBGOAL_THEN `dist(f (SUC n) x,(g:real^N->real^N) a) +
15518 dist((f:num->real^N->real^N) (SUC n) y,g b) < e`
15520 [ASM_REWRITE_TAC[] THEN
15521 MATCH_MP_TAC(REAL_ARITH `x < e / &2 /\ y < e / &2 ==> x + y < e`) THEN
15522 CONJ_TAC THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
15524 ==> dist(g x,g y) <= dist(x,y) ==> dist(g x,g y) < e`)) THEN
15527 MP_TAC(SPEC `SUC n` (ASSUME
15528 `!n. dist (a:real^N,b) <=
15529 dist ((f:num->real^N->real^N) n x,f n y)`)) THEN
15530 EXPAND_TAC "e" THEN NORM_ARITH_TAC;
15532 EXISTS_TAC `a:real^N` THEN ASM_REWRITE_TAC[] THEN
15533 MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
15534 EXISTS_TAC `\n:num. (f:num->real^N->real^N) (SUC(s n)) x` THEN
15535 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL
15536 [ASM_REWRITE_TAC[] THEN
15537 SUBGOAL_THEN `(g:real^N->real^N) continuous_on s` MP_TAC THENL
15538 [REWRITE_TAC[continuous_on] THEN ASM_MESON_TAC[REAL_LET_TRANS];
15540 REWRITE_TAC[CONTINUOUS_ON_SEQUENTIALLY; o_DEF] THEN
15541 DISCH_THEN MATCH_MP_TAC THEN ASM_SIMP_TAC[];
15542 SUBGOAL_THEN `!n. (f:num->real^N->real^N) (SUC n) x = f n y`
15543 (fun th -> ASM_SIMP_TAC[th]) THEN
15544 INDUCT_TAC THEN ASM_REWRITE_TAC[]]);;
15546 (* ------------------------------------------------------------------------- *)
15547 (* Dini's theorem. *)
15548 (* ------------------------------------------------------------------------- *)
15551 (`!f:num->real^N->real^1 g s.
15552 compact s /\ (!n. (f n) continuous_on s) /\ g continuous_on s /\
15553 (!x. x IN s ==> ((\n. (f n x)) --> g x) sequentially) /\
15554 (!n x. x IN s ==> drop(f n x) <= drop(f (n + 1) x))
15556 ==> eventually (\n. !x. x IN s ==> norm(f n x - g x) < e)
15558 REPEAT STRIP_TAC THEN
15560 `!x:real^N m n:num. x IN s /\ m <= n ==> drop(f m x) <= drop(f n x)`
15562 [GEN_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN
15563 MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_SIMP_TAC[ADD1] THEN
15566 SUBGOAL_THEN `!n:num x:real^N. x IN s ==> drop(f n x) <= drop(g x)`
15568 [REPEAT STRIP_TAC THEN
15569 MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LE) THEN
15570 EXISTS_TAC `\m:num. (f:num->real^N->real^1) n x` THEN
15571 EXISTS_TAC `\m:num. (f:num->real^N->real^1) m x` THEN
15572 ASM_SIMP_TAC[LIM_CONST; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
15573 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN ASM_MESON_TAC[];
15575 RULE_ASSUM_TAC(REWRITE_RULE[LIM_SEQUENTIALLY; dist]) THEN
15576 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I
15577 [COMPACT_EQ_HEINE_BOREL_SUBTOPOLOGY]) THEN
15578 DISCH_THEN(MP_TAC o SPEC
15579 `IMAGE (\n. { x | x IN s /\ norm((f:num->real^N->real^1) n x - g x) < e})
15581 REWRITE_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN
15582 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
15583 REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE; SUBSET_UNION; UNIONS_IMAGE] THEN
15584 REWRITE_TAC[IN_UNIV; IN_ELIM_THM; EVENTUALLY_SEQUENTIALLY] THEN
15585 SIMP_TAC[SUBSET; IN_UNIV; IN_ELIM_THM] THEN ANTS_TAC THENL
15586 [CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[LE_REFL]] THEN
15587 X_GEN_TAC `n:num` THEN REWRITE_TAC[GSYM IN_BALL_0] THEN
15588 MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE THEN
15589 ASM_SIMP_TAC[OPEN_BALL; CONTINUOUS_ON_SUB; ETA_AX];
15591 DISCH_THEN(X_CHOOSE_THEN `k:num->bool` (CONJUNCTS_THEN2
15592 (MP_TAC o SPEC `\n:num. n` o MATCH_MP UPPER_BOUND_FINITE_SET)
15593 (LABEL_TAC "*"))) THEN
15594 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
15595 REWRITE_TAC[] THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN
15596 DISCH_TAC THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
15597 REMOVE_THEN "*" (MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
15598 DISCH_THEN(X_CHOOSE_THEN `m:num` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
15599 REWRITE_TAC[NORM_REAL; GSYM drop; DROP_SUB] THEN MATCH_MP_TAC(REAL_ARITH
15600 `m <= n /\ n <= g ==> abs(m - g) < e ==> abs(n - g) < e`) THEN
15601 ASM_MESON_TAC[LE_TRANS]]);;
15603 (* ------------------------------------------------------------------------- *)
15604 (* Closest point of a (closed) set to a point. *)
15605 (* ------------------------------------------------------------------------- *)
15607 let closest_point = new_definition
15608 `closest_point s a = @x. x IN s /\ !y. y IN s ==> dist(a,x) <= dist(a,y)`;;
15610 let CLOSEST_POINT_EXISTS = prove
15611 (`!s a. closed s /\ ~(s = {})
15612 ==> (closest_point s a) IN s /\
15613 !y. y IN s ==> dist(a,closest_point s a) <= dist(a,y)`,
15614 REWRITE_TAC[closest_point] THEN CONV_TAC(ONCE_DEPTH_CONV SELECT_CONV) THEN
15615 REWRITE_TAC[DISTANCE_ATTAINS_INF]);;
15617 let CLOSEST_POINT_IN_SET = prove
15618 (`!s a. closed s /\ ~(s = {}) ==> (closest_point s a) IN s`,
15619 MESON_TAC[CLOSEST_POINT_EXISTS]);;
15621 let CLOSEST_POINT_LE = prove
15622 (`!s a x. closed s /\ x IN s ==> dist(a,closest_point s a) <= dist(a,x)`,
15623 MESON_TAC[CLOSEST_POINT_EXISTS; MEMBER_NOT_EMPTY]);;
15625 let CLOSEST_POINT_SELF = prove
15626 (`!s x:real^N. x IN s ==> closest_point s x = x`,
15627 REPEAT STRIP_TAC THEN REWRITE_TAC[closest_point] THEN
15628 MATCH_MP_TAC SELECT_UNIQUE THEN REWRITE_TAC[] THEN GEN_TAC THEN EQ_TAC THENL
15629 [STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
15630 ASM_SIMP_TAC[DIST_LE_0; DIST_REFL];
15631 STRIP_TAC THEN ASM_REWRITE_TAC[DIST_REFL; DIST_POS_LE]]);;
15633 let CLOSEST_POINT_REFL = prove
15634 (`!s x:real^N. closed s /\ ~(s = {}) ==> (closest_point s x = x <=> x IN s)`,
15635 MESON_TAC[CLOSEST_POINT_IN_SET; CLOSEST_POINT_SELF]);;
15637 let DIST_CLOSEST_POINT_LIPSCHITZ = prove
15639 closed s /\ ~(s = {})
15640 ==> abs(dist(x,closest_point s x) - dist(y,closest_point s y))
15642 REPEAT GEN_TAC THEN DISCH_TAC THEN
15643 FIRST_ASSUM(MP_TAC o MATCH_MP CLOSEST_POINT_EXISTS) THEN
15644 DISCH_THEN(fun th ->
15645 CONJUNCTS_THEN2 ASSUME_TAC
15646 (MP_TAC o SPEC `closest_point s (y:real^N)`) (SPEC `x:real^N` th) THEN
15647 CONJUNCTS_THEN2 ASSUME_TAC
15648 (MP_TAC o SPEC `closest_point s (x:real^N)`) (SPEC `y:real^N` th)) THEN
15649 ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC);;
15651 let CONTINUOUS_AT_DIST_CLOSEST_POINT = prove
15653 closed s /\ ~(s = {})
15654 ==> (\x. lift(dist(x,closest_point s x))) continuous (at x)`,
15655 REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at; DIST_LIFT] THEN
15656 ASM_MESON_TAC[DIST_CLOSEST_POINT_LIPSCHITZ; REAL_LET_TRANS]);;
15658 let CONTINUOUS_ON_DIST_CLOSEST_POINT = prove
15659 (`!s t. closed s /\ ~(s = {})
15660 ==> (\x. lift(dist(x,closest_point s x))) continuous_on t`,
15661 MESON_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON;
15662 CONTINUOUS_AT_DIST_CLOSEST_POINT]);;
15664 let UNIFORMLY_CONTINUOUS_ON_DIST_CLOSEST_POINT = prove
15665 (`!s t:real^N->bool.
15666 closed s /\ ~(s = {})
15667 ==> (\x. lift(dist(x,closest_point s x))) uniformly_continuous_on t`,
15668 REPEAT STRIP_TAC THEN REWRITE_TAC[uniformly_continuous_on; DIST_LIFT] THEN
15669 ASM_MESON_TAC[DIST_CLOSEST_POINT_LIPSCHITZ; REAL_LET_TRANS]);;
15671 let SEGMENT_TO_CLOSEST_POINT = prove
15673 closed s /\ ~(s = {})
15674 ==> segment(a,closest_point s a) INTER s = {}`,
15675 REPEAT STRIP_TAC THEN
15676 REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN s ==> ~(x IN t)`] THEN
15677 GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP DIST_IN_OPEN_SEGMENT) THEN
15678 MATCH_MP_TAC(TAUT `(r ==> ~p) ==> p /\ q ==> ~r`) THEN
15679 ASM_MESON_TAC[CLOSEST_POINT_EXISTS; REAL_NOT_LT; DIST_SYM]);;
15681 let SEGMENT_TO_POINT_EXISTS = prove
15683 closed s /\ ~(s = {}) ==> ?b. b IN s /\ segment(a,b) INTER s = {}`,
15684 MESON_TAC[SEGMENT_TO_CLOSEST_POINT; CLOSEST_POINT_EXISTS]);;
15686 let CLOSEST_POINT_IN_INTERIOR = prove
15688 closed s /\ ~(s = {})
15689 ==> ((closest_point s x) IN interior s <=> x IN interior s)`,
15690 REPEAT STRIP_TAC THEN ASM_CASES_TAC `(x:real^N) IN s` THEN
15691 ASM_SIMP_TAC[CLOSEST_POINT_SELF] THEN
15692 MATCH_MP_TAC(TAUT `~q /\ ~p ==> (p <=> q)`) THEN
15693 CONJ_TAC THENL [ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET]; STRIP_TAC] THEN
15694 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN
15695 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
15696 SUBGOAL_THEN `closest_point s (x:real^N) IN s` ASSUME_TAC THENL
15697 [ASM_MESON_TAC[INTERIOR_SUBSET; SUBSET]; ALL_TAC] THEN
15698 SUBGOAL_THEN `~(closest_point s (x:real^N) = x)` ASSUME_TAC THENL
15699 [ASM_MESON_TAC[]; ALL_TAC] THEN
15700 MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`;
15701 `closest_point s x -
15702 (min (&1) (e / norm(closest_point s x - x))) %
15703 (closest_point s x - x):real^N`]
15704 CLOSEST_POINT_LE) THEN
15705 ASM_REWRITE_TAC[dist; NOT_IMP; VECTOR_ARITH
15706 `x - (y - e % (y - x)):real^N = (&1 - e) % (x - y)`] THEN
15708 [FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
15709 REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(a:real^N,a - x) = norm x`] THEN
15710 REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
15711 ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
15712 MATCH_MP_TAC(REAL_ARITH `&0 <= a ==> abs(min (&1) a) <= a`) THEN
15713 ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_DIV; NORM_POS_LE];
15714 REWRITE_TAC[NORM_MUL; REAL_ARITH
15715 `~(n <= a * n) <=> &0 < (&1 - a) * n`] THEN
15716 MATCH_MP_TAC REAL_LT_MUL THEN
15717 ASM_SIMP_TAC[NORM_POS_LT; VECTOR_SUB_EQ] THEN
15718 MATCH_MP_TAC(REAL_ARITH
15719 `&0 < e /\ e <= &1 ==> &0 < &1 - abs(&1 - e)`) THEN
15720 REWRITE_TAC[REAL_MIN_LE; REAL_LT_MIN; REAL_LT_01; REAL_LE_REFL] THEN
15721 ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ]]);;
15723 let CLOSEST_POINT_IN_FRONTIER = prove
15725 closed s /\ ~(s = {}) /\ ~(x IN interior s)
15726 ==> (closest_point s x) IN frontier s`,
15727 SIMP_TAC[frontier; IN_DIFF; CLOSEST_POINT_IN_INTERIOR] THEN
15728 SIMP_TAC[CLOSEST_POINT_IN_SET; CLOSURE_CLOSED]);;
15730 (* ------------------------------------------------------------------------- *)
15731 (* More general infimum of distance between two sets. *)
15732 (* ------------------------------------------------------------------------- *)
15734 let setdist = new_definition
15736 if s = {} \/ t = {} then &0
15737 else inf {dist(x,y) | x IN s /\ y IN t}`;;
15739 let SETDIST_EMPTY = prove
15740 (`(!t. setdist({},t) = &0) /\ (!s. setdist(s,{}) = &0)`,
15741 REWRITE_TAC[setdist]);;
15743 let SETDIST_POS_LE = prove
15744 (`!s t. &0 <= setdist(s,t)`,
15745 REPEAT GEN_TAC THEN REWRITE_TAC[setdist] THEN
15746 COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL] THEN
15747 MATCH_MP_TAC REAL_LE_INF THEN
15748 REWRITE_TAC[FORALL_IN_GSPEC; DIST_POS_LE] THEN ASM SET_TAC[]);;
15750 let REAL_LE_SETDIST = prove
15751 (`!s t:real^N->bool d.
15752 ~(s = {}) /\ ~(t = {}) /\
15753 (!x y. x IN s /\ y IN t ==> d <= dist(x,y))
15754 ==> d <= setdist(s,t)`,
15755 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[setdist] THEN
15756 MP_TAC(ISPEC `{dist(x:real^N,y) | x IN s /\ y IN t}` INF) THEN
15757 REWRITE_TAC[FORALL_IN_GSPEC] THEN ANTS_TAC THENL
15758 [CONJ_TAC THENL [ASM SET_TAC[]; MESON_TAC[DIST_POS_LE]]; ALL_TAC] THEN
15761 let SETDIST_LE_DIST = prove
15762 (`!s t x y:real^N. x IN s /\ y IN t ==> setdist(s,t) <= dist(x,y)`,
15763 REPEAT GEN_TAC THEN REWRITE_TAC[setdist] THEN
15764 COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
15765 MP_TAC(ISPEC `{dist(x:real^N,y) | x IN s /\ y IN t}` INF) THEN
15766 REWRITE_TAC[FORALL_IN_GSPEC] THEN ANTS_TAC THENL
15767 [CONJ_TAC THENL [ASM SET_TAC[]; MESON_TAC[DIST_POS_LE]]; ALL_TAC] THEN
15770 let REAL_LE_SETDIST_EQ = prove
15771 (`!d s t:real^N->bool.
15772 d <= setdist(s,t) <=>
15773 (!x y. x IN s /\ y IN t ==> d <= dist(x,y)) /\
15774 (s = {} \/ t = {} ==> d <= &0)`,
15775 REPEAT GEN_TAC THEN MAP_EVERY ASM_CASES_TAC
15776 [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
15777 ASM_REWRITE_TAC[SETDIST_EMPTY; NOT_IN_EMPTY] THEN
15778 ASM_MESON_TAC[REAL_LE_SETDIST; SETDIST_LE_DIST; REAL_LE_TRANS]);;
15780 let REAL_SETDIST_LT_EXISTS = prove
15781 (`!s t:real^N->bool b.
15782 ~(s = {}) /\ ~(t = {}) /\ setdist(s,t) < b
15783 ==> ?x y. x IN s /\ y IN t /\ dist(x,y) < b`,
15784 REWRITE_TAC[GSYM REAL_NOT_LE; REAL_LE_SETDIST_EQ] THEN MESON_TAC[]);;
15786 let SETDIST_REFL = prove
15787 (`!s:real^N->bool. setdist(s,s) = &0`,
15788 GEN_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM; SETDIST_POS_LE] THEN
15789 ASM_CASES_TAC `s:real^N->bool = {}` THENL
15790 [ASM_REWRITE_TAC[setdist; REAL_LE_REFL]; ALL_TAC] THEN
15791 ASM_MESON_TAC[SETDIST_LE_DIST; MEMBER_NOT_EMPTY; DIST_REFL]);;
15793 let SETDIST_SYM = prove
15794 (`!s t. setdist(s,t) = setdist(t,s)`,
15795 REPEAT GEN_TAC THEN REWRITE_TAC[setdist; DISJ_SYM] THEN
15796 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
15797 AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
15798 MESON_TAC[DIST_SYM]);;
15800 let SETDIST_TRIANGLE = prove
15801 (`!s a t:real^N->bool.
15802 setdist(s,t) <= setdist(s,{a}) + setdist({a},t)`,
15803 REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
15804 ASM_REWRITE_TAC[SETDIST_EMPTY; REAL_ADD_LID; SETDIST_POS_LE] THEN
15805 ASM_CASES_TAC `t:real^N->bool = {}` THEN
15806 ASM_REWRITE_TAC[SETDIST_EMPTY; REAL_ADD_RID; SETDIST_POS_LE] THEN
15807 ONCE_REWRITE_TAC[GSYM REAL_LE_SUB_RADD] THEN
15808 MATCH_MP_TAC REAL_LE_SETDIST THEN
15809 ASM_REWRITE_TAC[NOT_INSERT_EMPTY; IN_SING; IMP_CONJ;
15810 RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
15811 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
15812 ONCE_REWRITE_TAC[REAL_ARITH `x - y <= z <=> x - z <= y`] THEN
15813 MATCH_MP_TAC REAL_LE_SETDIST THEN
15814 ASM_REWRITE_TAC[NOT_INSERT_EMPTY; IN_SING; IMP_CONJ;
15815 RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
15816 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
15817 REWRITE_TAC[REAL_LE_SUB_RADD] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
15818 EXISTS_TAC `dist(x:real^N,y)` THEN
15819 ASM_SIMP_TAC[SETDIST_LE_DIST] THEN CONV_TAC NORM_ARITH);;
15821 let SETDIST_SINGS = prove
15822 (`!x y. setdist({x},{y}) = dist(x,y)`,
15823 REWRITE_TAC[setdist; NOT_INSERT_EMPTY] THEN
15824 REWRITE_TAC[SET_RULE `{f x y | x IN {a} /\ y IN {b}} = {f a b}`] THEN
15825 SIMP_TAC[INF_INSERT_FINITE; FINITE_EMPTY]);;
15827 let SETDIST_LIPSCHITZ = prove
15828 (`!s t x y:real^N. abs(setdist({x},s) - setdist({y},s)) <= dist(x,y)`,
15829 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM SETDIST_SINGS] THEN
15830 REWRITE_TAC[REAL_ARITH
15831 `abs(x - y) <= z <=> x <= z + y /\ y <= z + x`] THEN
15832 MESON_TAC[SETDIST_TRIANGLE; SETDIST_SYM]);;
15834 let CONTINUOUS_AT_LIFT_SETDIST = prove
15835 (`!s x:real^N. (\y. lift(setdist({y},s))) continuous (at x)`,
15836 REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at; DIST_LIFT] THEN
15837 ASM_MESON_TAC[SETDIST_LIPSCHITZ; REAL_LET_TRANS]);;
15839 let CONTINUOUS_ON_LIFT_SETDIST = prove
15840 (`!s t:real^N->bool. (\y. lift(setdist({y},s))) continuous_on t`,
15841 MESON_TAC[CONTINUOUS_AT_IMP_CONTINUOUS_ON;
15842 CONTINUOUS_AT_LIFT_SETDIST]);;
15844 let UNIFORMLY_CONTINUOUS_ON_LIFT_SETDIST = prove
15845 (`!s t:real^N->bool.
15846 (\y. lift(setdist({y},s))) uniformly_continuous_on t`,
15847 REPEAT GEN_TAC THEN REWRITE_TAC[uniformly_continuous_on; DIST_LIFT] THEN
15848 ASM_MESON_TAC[SETDIST_LIPSCHITZ; REAL_LET_TRANS]);;
15850 let SETDIST_DIFFERENCES = prove
15851 (`!s t. setdist(s,t) = setdist({vec 0},{x - y:real^N | x IN s /\ y IN t})`,
15852 REPEAT GEN_TAC THEN REWRITE_TAC[setdist; NOT_INSERT_EMPTY;
15853 SET_RULE `{f x y | x IN s /\ y IN t} = {} <=> s = {} \/ t = {}`] THEN
15854 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN AP_TERM_TAC THEN
15855 REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_SING] THEN
15856 REWRITE_TAC[GSYM CONJ_ASSOC; RIGHT_EXISTS_AND_THM; UNWIND_THM2; DIST_0] THEN
15857 REWRITE_TAC[dist] THEN MESON_TAC[]);;
15859 let SETDIST_SUBSET_RIGHT = prove
15860 (`!s t u:real^N->bool.
15861 ~(t = {}) /\ t SUBSET u ==> setdist(s,u) <= setdist(s,t)`,
15862 REPEAT STRIP_TAC THEN
15863 MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `u:real^N->bool = {}`] THEN
15864 ASM_REWRITE_TAC[SETDIST_EMPTY; SETDIST_POS_LE; REAL_LE_REFL] THEN
15865 ASM_REWRITE_TAC[setdist] THEN MATCH_MP_TAC REAL_LE_INF_SUBSET THEN
15866 ASM_REWRITE_TAC[FORALL_IN_GSPEC; SUBSET] THEN
15867 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
15868 MESON_TAC[DIST_POS_LE]);;
15870 let SETDIST_SUBSET_LEFT = prove
15871 (`!s t u:real^N->bool.
15872 ~(s = {}) /\ s SUBSET t ==> setdist(t,u) <= setdist(s,u)`,
15873 MESON_TAC[SETDIST_SUBSET_RIGHT; SETDIST_SYM]);;
15875 let SETDIST_CLOSURE = prove
15876 (`(!s t:real^N->bool. setdist(closure s,t) = setdist(s,t)) /\
15877 (!s t:real^N->bool. setdist(s,closure t) = setdist(s,t))`,
15878 GEN_REWRITE_TAC RAND_CONV [SWAP_FORALL_THM] THEN
15879 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SETDIST_SYM] THEN
15881 REWRITE_TAC[MESON[REAL_LE_ANTISYM]
15882 `x:real = y <=> !d. d <= x <=> d <= y`] THEN
15883 REPEAT GEN_TAC THEN REWRITE_TAC[REAL_LE_SETDIST_EQ] THEN
15884 MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
15885 ASM_REWRITE_TAC[CLOSURE_EQ_EMPTY; CLOSURE_EMPTY; NOT_IN_EMPTY] THEN
15886 MATCH_MP_TAC(SET_RULE
15888 (!y. Q y /\ (!x. x IN s ==> P x y) ==> (!x. x IN c ==> P x y))
15889 ==> ((!x y. x IN c /\ Q y ==> P x y) <=>
15890 (!x y. x IN s /\ Q y ==> P x y))`) THEN
15891 REWRITE_TAC[CLOSURE_SUBSET] THEN GEN_TAC THEN STRIP_TAC THEN
15892 MATCH_MP_TAC CONTINUOUS_GE_ON_CLOSURE THEN
15893 ASM_REWRITE_TAC[o_DEF; dist] THEN
15894 MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN
15895 SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID]);;
15897 let SETDIST_COMPACT_CLOSED = prove
15898 (`!s t:real^N->bool.
15899 compact s /\ closed t /\ ~(s = {}) /\ ~(t = {})
15900 ==> ?x y. x IN s /\ y IN t /\ dist(x,y) = setdist(s,t)`,
15901 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN
15902 MATCH_MP_TAC(MESON[]
15903 `(!x y. P x /\ Q y ==> S x y) /\ (?x y. P x /\ Q y /\ R x y)
15904 ==> ?x y. P x /\ Q y /\ R x y /\ S x y`) THEN
15905 SIMP_TAC[SETDIST_LE_DIST] THEN
15906 ASM_REWRITE_TAC[REAL_LE_SETDIST_EQ] THEN
15907 MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0:real^N`]
15908 DISTANCE_ATTAINS_INF) THEN
15909 ASM_SIMP_TAC[COMPACT_CLOSED_DIFFERENCES; EXISTS_IN_GSPEC; FORALL_IN_GSPEC;
15910 DIST_0; GSYM CONJ_ASSOC] THEN
15911 REWRITE_TAC[dist] THEN DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]);;
15913 let SETDIST_CLOSED_COMPACT = prove
15914 (`!s t:real^N->bool.
15915 closed s /\ compact t /\ ~(s = {}) /\ ~(t = {})
15916 ==> ?x y. x IN s /\ y IN t /\ dist(x,y) = setdist(s,t)`,
15917 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN
15918 MATCH_MP_TAC(MESON[]
15919 `(!x y. P x /\ Q y ==> S x y) /\ (?x y. P x /\ Q y /\ R x y)
15920 ==> ?x y. P x /\ Q y /\ R x y /\ S x y`) THEN
15921 SIMP_TAC[SETDIST_LE_DIST] THEN
15922 ASM_REWRITE_TAC[REAL_LE_SETDIST_EQ] THEN
15923 MP_TAC(ISPECL [`{x - y:real^N | x IN s /\ y IN t}`; `vec 0:real^N`]
15924 DISTANCE_ATTAINS_INF) THEN
15925 ASM_SIMP_TAC[CLOSED_COMPACT_DIFFERENCES; EXISTS_IN_GSPEC; FORALL_IN_GSPEC;
15926 DIST_0; GSYM CONJ_ASSOC] THEN
15927 REWRITE_TAC[dist] THEN DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]);;
15929 let SETDIST_EQ_0_COMPACT_CLOSED = prove
15930 (`!s t:real^N->bool.
15931 compact s /\ closed t
15932 ==> (setdist(s,t) = &0 <=> s = {} \/ t = {} \/ ~(s INTER t = {}))`,
15933 REPEAT STRIP_TAC THEN
15934 MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
15935 ASM_REWRITE_TAC[SETDIST_EMPTY] THEN EQ_TAC THENL
15936 [MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`]
15937 SETDIST_COMPACT_CLOSED) THEN ASM_REWRITE_TAC[] THEN
15938 REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN MESON_TAC[DIST_EQ_0];
15939 REWRITE_TAC[GSYM REAL_LE_ANTISYM; SETDIST_POS_LE] THEN
15940 REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN
15941 MESON_TAC[SETDIST_LE_DIST; DIST_EQ_0]]);;
15943 let SETDIST_EQ_0_CLOSED_COMPACT = prove
15944 (`!s t:real^N->bool.
15945 closed s /\ compact t
15946 ==> (setdist(s,t) = &0 <=> s = {} \/ t = {} \/ ~(s INTER t = {}))`,
15947 ONCE_REWRITE_TAC[SETDIST_SYM] THEN
15948 SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED] THEN SET_TAC[]);;
15950 let SETDIST_EQ_0_BOUNDED = prove
15951 (`!s t:real^N->bool.
15952 (bounded s \/ bounded t)
15953 ==> (setdist(s,t) = &0 <=>
15954 s = {} \/ t = {} \/ ~(closure(s) INTER closure(t) = {}))`,
15955 REPEAT GEN_TAC THEN
15956 MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
15957 ASM_REWRITE_TAC[SETDIST_EMPTY] THEN STRIP_TAC THEN
15958 ONCE_REWRITE_TAC[MESON[SETDIST_CLOSURE]
15959 `setdist(s,t) = setdist(closure s,closure t)`] THEN
15960 ASM_SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; SETDIST_EQ_0_CLOSED_COMPACT;
15961 COMPACT_CLOSURE; CLOSED_CLOSURE; CLOSURE_EQ_EMPTY]);;
15964 let SETDIST_TRANSLATION = prove
15966 setdist(IMAGE (\x. a + x) s,IMAGE (\x. a + x) t) = setdist(s,t)`,
15967 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[SETDIST_DIFFERENCES] THEN
15968 AP_TERM_TAC THEN AP_TERM_TAC THEN
15969 REWRITE_TAC[SET_RULE
15970 `{f x y | x IN IMAGE g s /\ y IN IMAGE g t} =
15971 {f (g x) (g y) | x IN s /\ y IN t}`] THEN
15972 REWRITE_TAC[VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`]);;
15974 add_translation_invariants [SETDIST_TRANSLATION];;
15976 let SETDIST_LINEAR_IMAGE = prove
15977 (`!f:real^M->real^N s t.
15978 linear f /\ (!x. norm(f x) = norm x)
15979 ==> setdist(IMAGE f s,IMAGE f t) = setdist(s,t)`,
15980 REPEAT STRIP_TAC THEN REWRITE_TAC[setdist; IMAGE_EQ_EMPTY] THEN
15981 COND_CASES_TAC THEN ASM_REWRITE_TAC[dist] THEN AP_TERM_TAC THEN
15982 REWRITE_TAC[SET_RULE
15983 `{f x y | x IN IMAGE g s /\ y IN IMAGE g t} =
15984 {f (g x) (g y) | x IN s /\ y IN t}`] THEN
15985 FIRST_X_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_SUB th)]) THEN
15986 ASM_REWRITE_TAC[]);;
15988 add_linear_invariants [SETDIST_LINEAR_IMAGE];;
15990 let SETDIST_UNIQUE = prove
15991 (`!s t a b:real^N d.
15992 a IN s /\ b IN t /\ dist(a,b) = d /\
15993 (!x y. x IN s /\ y IN t ==> dist(a,b) <= dist(x,y))
15994 ==> setdist(s,t) = d`,
15995 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL
15996 [ASM_MESON_TAC[SETDIST_LE_DIST];
15997 MATCH_MP_TAC REAL_LE_SETDIST THEN ASM SET_TAC[]]);;
15999 let SETDIST_CLOSEST_POINT = prove
16001 closed s /\ ~(s = {}) ==> setdist({a},s) = dist(a,closest_point s a)`,
16002 REPEAT STRIP_TAC THEN MATCH_MP_TAC SETDIST_UNIQUE THEN
16003 REWRITE_TAC[RIGHT_EXISTS_AND_THM; IN_SING; UNWIND_THM2] THEN
16004 EXISTS_TAC `closest_point s (a:real^N)` THEN
16005 ASM_MESON_TAC[CLOSEST_POINT_EXISTS; DIST_SYM]);;
16007 let SETDIST_EQ_0_SING = prove
16008 (`(!s x:real^N. setdist({x},s) = &0 <=> s = {} \/ x IN closure s) /\
16009 (!s x:real^N. setdist(s,{x}) = &0 <=> s = {} \/ x IN closure s)`,
16010 SIMP_TAC[SETDIST_EQ_0_BOUNDED; BOUNDED_SING; CLOSURE_SING] THEN SET_TAC[]);;
16012 let SETDIST_EQ_0_CLOSED = prove
16013 (`!s x. closed s ==> (setdist({x},s) = &0 <=> s = {} \/ x IN s)`,
16014 SIMP_TAC[SETDIST_EQ_0_COMPACT_CLOSED; COMPACT_SING] THEN SET_TAC[]);;
16016 let SETDIST_EQ_0_CLOSED_IN = prove
16017 (`!u s x. closed_in (subtopology euclidean u) s /\ x IN u
16018 ==> (setdist({x},s) = &0 <=> s = {} \/ x IN s)`,
16019 REWRITE_TAC[SETDIST_EQ_0_SING; CLOSED_IN_INTER_CLOSURE] THEN SET_TAC[]);;
16021 let SETDIST_SING_IN_SET = prove
16022 (`!x s. x IN s ==> setdist({x},s) = &0`,
16023 SIMP_TAC[SETDIST_EQ_0_SING; REWRITE_RULE[SUBSET] CLOSURE_SUBSET]);;
16025 let SETDIST_SING_TRIANGLE = prove
16026 (`!s x y:real^N. abs(setdist({x},s) - setdist({y},s)) <= dist(x,y)`,
16027 REPEAT GEN_TAC THEN ASM_CASES_TAC `s:real^N->bool = {}` THEN
16028 ASM_REWRITE_TAC[SETDIST_EMPTY; REAL_SUB_REFL; REAL_ABS_NUM; DIST_POS_LE] THEN
16029 REWRITE_TAC[GSYM REAL_BOUNDS_LE; REAL_NEG_SUB] THEN REPEAT STRIP_TAC THEN
16030 ONCE_REWRITE_TAC[REAL_ARITH `a - b <= c <=> a - c <= b`;
16031 REAL_ARITH `--a <= b - c <=> c - a <= b`] THEN
16032 MATCH_MP_TAC REAL_LE_SETDIST THEN ASM_REWRITE_TAC[NOT_INSERT_EMPTY] THEN
16033 SIMP_TAC[IN_SING; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_UNWIND_THM2] THEN
16034 X_GEN_TAC `z:real^N` THEN DISCH_TAC THENL
16035 [MATCH_MP_TAC(NORM_ARITH
16036 `a <= dist(y:real^N,z) ==> a - dist(x,y) <= dist(x,z)`);
16037 MATCH_MP_TAC(NORM_ARITH
16038 `a <= dist(x:real^N,z) ==> a - dist(x,y) <= dist(y,z)`)] THEN
16039 MATCH_MP_TAC SETDIST_LE_DIST THEN ASM_REWRITE_TAC[IN_SING]);;
16041 let SETDIST_LE_SING = prove
16042 (`!s t x:real^N. x IN s ==> setdist(s,t) <= setdist({x},t)`,
16043 REPEAT STRIP_TAC THEN MATCH_MP_TAC SETDIST_SUBSET_LEFT THEN ASM SET_TAC[]);;
16045 let SETDIST_BALLS = prove
16046 (`(!a b:real^N r s.
16047 setdist(ball(a,r),ball(b,s)) =
16048 if r <= &0 \/ s <= &0 then &0 else max (&0) (dist(a,b) - (r + s))) /\
16050 setdist(ball(a,r),cball(b,s)) =
16051 if r <= &0 \/ s < &0 then &0 else max (&0) (dist(a,b) - (r + s))) /\
16053 setdist(cball(a,r),ball(b,s)) =
16054 if r < &0 \/ s <= &0 then &0 else max (&0) (dist(a,b) - (r + s))) /\
16056 setdist(cball(a,r),cball(b,s)) =
16057 if r < &0 \/ s < &0 then &0 else max (&0) (dist(a,b) - (r + s)))`,
16058 REWRITE_TAC[MESON[]
16059 `(x = if p then y else z) <=> (p ==> x = y) /\ (~p ==> x = z)`] THEN
16060 SIMP_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
16061 SIMP_TAC[BALL_EMPTY; CBALL_EMPTY; SETDIST_EMPTY; DE_MORGAN_THM] THEN
16062 ONCE_REWRITE_TAC[MESON[SETDIST_CLOSURE]
16063 `setdist(s,t) = setdist(closure s,closure t)`] THEN
16064 SIMP_TAC[REAL_NOT_LE; REAL_NOT_LT; CLOSURE_BALL] THEN
16065 REWRITE_TAC[SETDIST_CLOSURE] THEN
16066 MATCH_MP_TAC(TAUT `(s ==> p /\ q /\ r) /\ s ==> p /\ q /\ r /\ s`) THEN
16067 CONJ_TAC THENL [MESON_TAC[REAL_LT_IMP_LE]; REPEAT GEN_TAC] THEN
16068 REWRITE_TAC[real_max; REAL_SUB_LE] THEN COND_CASES_TAC THEN
16069 SIMP_TAC[SETDIST_EQ_0_BOUNDED; BOUNDED_CBALL; CLOSED_CBALL; CLOSURE_CLOSED;
16070 CBALL_EQ_EMPTY; INTER_BALLS_EQ_EMPTY]
16071 THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
16072 ASM_CASES_TAC `b:real^N = a` THENL
16073 [FIRST_X_ASSUM SUBST_ALL_TAC THEN
16074 RULE_ASSUM_TAC(REWRITE_RULE[DIST_REFL]) THEN
16075 ASM_CASES_TAC `r = &0 /\ s = &0` THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
16076 ASM_SIMP_TAC[CBALL_SING; SETDIST_SINGS] THEN REAL_ARITH_TAC;
16078 REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL
16080 MATCH_MP_TAC REAL_LE_SETDIST THEN
16081 ASM_REWRITE_TAC[CBALL_EQ_EMPTY; REAL_NOT_LT; IN_CBALL] THEN
16082 CONV_TAC NORM_ARITH] THEN
16083 MATCH_MP_TAC REAL_LE_TRANS THEN
16084 EXISTS_TAC `dist(a + r / dist(a,b) % (b - a):real^N,
16085 b - s / dist(a,b) % (b - a))` THEN
16087 [MATCH_MP_TAC SETDIST_LE_DIST THEN
16088 REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(a:real^N,a + x) = norm x`;
16089 NORM_ARITH `dist(a:real^N,a - x) = norm x`] THEN
16090 ONCE_REWRITE_TAC[DIST_SYM] THEN
16091 REWRITE_TAC[dist; NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
16092 ASM_SIMP_TAC[REAL_DIV_RMUL; VECTOR_SUB_EQ; NORM_EQ_0] THEN
16093 ASM_REAL_ARITH_TAC;
16094 REWRITE_TAC[dist; VECTOR_ARITH
16095 `(a + d % (b - a)) - (b - e % (b - a)):real^N =
16096 (&1 - d - e) % (a - b)`] THEN
16097 REWRITE_TAC[NORM_MUL; REAL_ARITH
16098 `&1 - r / y - s / y = &1 - (r + s) / y`] THEN
16099 ONCE_REWRITE_TAC[GSYM REAL_ABS_NORM] THEN
16100 REWRITE_TAC[GSYM REAL_ABS_MUL] THEN REWRITE_TAC[REAL_ABS_NORM] THEN
16101 ASM_SIMP_TAC[VECTOR_SUB_EQ; NORM_EQ_0; REAL_FIELD
16102 `~(n = &0) ==> (&1 - x / n) * n = n - x`] THEN
16103 REWRITE_TAC[GSYM dist] THEN ASM_REAL_ARITH_TAC]);;
16105 (* ------------------------------------------------------------------------- *)
16106 (* Use set distance for an easy proof of separation properties etc. *)
16107 (* ------------------------------------------------------------------------- *)
16109 let SEPARATION_CLOSURES = prove
16110 (`!s t:real^N->bool.
16111 s INTER closure(t) = {} /\ t INTER closure(s) = {}
16112 ==> ?u v. DISJOINT u v /\ open u /\ open v /\
16113 s SUBSET u /\ t SUBSET v`,
16114 REPEAT STRIP_TAC THEN
16115 ASM_CASES_TAC `s:real^N->bool = {}` THENL
16116 [MAP_EVERY EXISTS_TAC [`{}:real^N->bool`; `(:real^N)`] THEN
16117 ASM_REWRITE_TAC[OPEN_EMPTY; OPEN_UNIV] THEN ASM SET_TAC[];
16119 ASM_CASES_TAC `t:real^N->bool = {}` THENL
16120 [MAP_EVERY EXISTS_TAC [`(:real^N)`; `{}:real^N->bool`] THEN
16121 ASM_REWRITE_TAC[OPEN_EMPTY; OPEN_UNIV] THEN ASM SET_TAC[];
16123 EXISTS_TAC `{x | x IN (:real^N) /\
16124 lift(setdist({x},t) - setdist({x},s)) IN
16125 {x | &0 < x$1}}` THEN
16126 EXISTS_TAC `{x | x IN (:real^N) /\
16127 lift(setdist({x},t) - setdist({x},s)) IN
16128 {x | x$1 < &0}}` THEN
16129 REPEAT CONJ_TAC THENL
16130 [REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. x IN s /\ x IN t ==> F`] THEN
16131 REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN REAL_ARITH_TAC;
16132 MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE THEN
16133 SIMP_TAC[REWRITE_RULE[real_gt] OPEN_HALFSPACE_COMPONENT_GT; OPEN_UNIV] THEN
16134 SIMP_TAC[LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST];
16135 MATCH_MP_TAC CONTINUOUS_OPEN_PREIMAGE THEN
16136 SIMP_TAC[OPEN_HALFSPACE_COMPONENT_LT; OPEN_UNIV] THEN
16137 SIMP_TAC[LIFT_SUB; CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST];
16138 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV; GSYM drop; LIFT_DROP] THEN
16139 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH
16140 `&0 <= x /\ y = &0 /\ ~(x = &0) ==> &0 < x - y`);
16141 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV; GSYM drop; LIFT_DROP] THEN
16142 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(REAL_ARITH
16143 `&0 <= y /\ x = &0 /\ ~(y = &0) ==> x - y < &0`)] THEN
16144 ASM_SIMP_TAC[SETDIST_POS_LE; SETDIST_EQ_0_BOUNDED; BOUNDED_SING] THEN
16145 ASM_SIMP_TAC[CLOSED_SING; CLOSURE_CLOSED; NOT_INSERT_EMPTY;
16146 REWRITE_RULE[SUBSET] CLOSURE_SUBSET;
16147 SET_RULE `{a} INTER s = {} <=> ~(a IN s)`] THEN
16150 let SEPARATION_NORMAL = prove
16151 (`!s t:real^N->bool.
16152 closed s /\ closed t /\ s INTER t = {}
16153 ==> ?u v. open u /\ open v /\
16154 s SUBSET u /\ t SUBSET v /\ u INTER v = {}`,
16155 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM DISJOINT] THEN
16156 ONCE_REWRITE_TAC[TAUT
16157 `a /\ b /\ c /\ d /\ e <=> e /\ a /\ b /\ c /\ d`] THEN
16158 MATCH_MP_TAC SEPARATION_CLOSURES THEN
16159 ASM_SIMP_TAC[CLOSURE_CLOSED] THEN ASM SET_TAC[]);;
16161 let SEPARATION_NORMAL_LOCAL = prove
16162 (`!s t u:real^N->bool.
16163 closed_in (subtopology euclidean u) s /\
16164 closed_in (subtopology euclidean u) t /\
16166 ==> ?s' t'. open_in (subtopology euclidean u) s' /\
16167 open_in (subtopology euclidean u) t' /\
16168 s SUBSET s' /\ t SUBSET t' /\ s' INTER t' = {}`,
16169 REPEAT STRIP_TAC THEN
16170 ASM_CASES_TAC `s:real^N->bool = {}` THENL
16171 [MAP_EVERY EXISTS_TAC [`{}:real^N->bool`; `u:real^N->bool`] THEN
16172 ASM_SIMP_TAC[OPEN_IN_REFL; OPEN_IN_EMPTY; INTER_EMPTY; EMPTY_SUBSET] THEN
16173 ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET];
16175 ASM_CASES_TAC `t:real^N->bool = {}` THENL
16176 [MAP_EVERY EXISTS_TAC [`u:real^N->bool`; `{}:real^N->bool`] THEN
16177 ASM_SIMP_TAC[OPEN_IN_REFL; OPEN_IN_EMPTY; INTER_EMPTY; EMPTY_SUBSET] THEN
16178 ASM_MESON_TAC[CLOSED_IN_IMP_SUBSET];
16180 EXISTS_TAC `{x:real^N | x IN u /\ setdist({x},s) < setdist({x},t)}` THEN
16181 EXISTS_TAC `{x:real^N | x IN u /\ setdist({x},t) < setdist({x},s)}` THEN
16182 SIMP_TAC[EXTENSION; SUBSET; IN_ELIM_THM; SETDIST_SING_IN_SET; IN_INTER;
16183 NOT_IN_EMPTY; SETDIST_POS_LE; CONJ_ASSOC;
16184 REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN
16185 CONJ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_ANTISYM]] THEN
16186 ONCE_REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL
16188 ASM_MESON_TAC[SETDIST_EQ_0_CLOSED_IN; CLOSED_IN_IMP_SUBSET; SUBSET;
16189 MEMBER_NOT_EMPTY; IN_INTER]] THEN
16190 ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN
16191 ONCE_REWRITE_TAC[MESON[LIFT_DROP] `&0 < x <=> &0 < drop(lift x)`] THEN
16192 REWRITE_TAC[SET_RULE
16193 `{x | x IN u /\ &0 < drop(f x)} =
16194 {x | x IN u /\ f x IN {x | &0 < drop x}}`] THEN
16195 REWRITE_TAC[drop] THEN CONJ_TAC THEN
16196 MATCH_MP_TAC CONTINUOUS_OPEN_IN_PREIMAGE THEN
16197 REWRITE_TAC[OPEN_HALFSPACE_COMPONENT_LT; LIFT_SUB;
16198 REWRITE_RULE[real_gt] OPEN_HALFSPACE_COMPONENT_GT; OPEN_UNIV] THEN
16199 SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_LIFT_SETDIST]);;
16201 let SEPARATION_NORMAL_COMPACT = prove
16202 (`!s t:real^N->bool.
16203 compact s /\ closed t /\ s INTER t = {}
16204 ==> ?u v. open u /\ compact(closure u) /\ open v /\
16205 s SUBSET u /\ t SUBSET v /\ u INTER v = {}`,
16206 REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED; CLOSED_CLOSURE] THEN
16207 REPEAT STRIP_TAC THEN FIRST_ASSUM
16208 (MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN
16209 DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
16210 MP_TAC(ISPECL [`s:real^N->bool`; `t UNION ((:real^N) DIFF ball(vec 0,r))`]
16211 SEPARATION_NORMAL) THEN
16212 ASM_SIMP_TAC[CLOSED_UNION; GSYM OPEN_CLOSED; OPEN_BALL] THEN
16213 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
16214 REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN
16215 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
16216 CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_CLOSURE; ASM SET_TAC[]] THEN
16217 MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `ball(vec 0:real^N,r)` THEN
16218 REWRITE_TAC[BOUNDED_BALL] THEN ASM SET_TAC[]);;
16220 let SEPARATION_HAUSDORFF = prove
16223 ==> ?u v. open u /\ open v /\ x IN u /\ y IN v /\ (u INTER v = {})`,
16224 REPEAT STRIP_TAC THEN
16225 MP_TAC(SPECL [`{x:real^N}`; `{y:real^N}`] SEPARATION_NORMAL) THEN
16226 REWRITE_TAC[SING_SUBSET; CLOSED_SING] THEN
16227 DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[]);;
16229 let SEPARATION_T2 = prove
16231 ~(x = y) <=> ?u v. open u /\ open v /\ x IN u /\ y IN v /\
16233 REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[SEPARATION_HAUSDORFF] THEN
16234 REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY] THEN MESON_TAC[]);;
16236 let SEPARATION_T1 = prove
16238 ~(x = y) <=> ?u v. open u /\ open v /\ x IN u /\ ~(y IN u) /\
16239 ~(x IN v) /\ y IN v`,
16240 REPEAT STRIP_TAC THEN EQ_TAC THENL
16241 [ASM_SIMP_TAC[SEPARATION_T2; EXTENSION; NOT_IN_EMPTY; IN_INTER];
16242 ALL_TAC] THEN MESON_TAC[]);;
16244 let SEPARATION_T0 = prove
16245 (`!x:real^N y. ~(x = y) <=> ?u. open u /\ ~(x IN u <=> y IN u)`,
16246 MESON_TAC[SEPARATION_T1]);;
16248 (* ------------------------------------------------------------------------- *)
16249 (* Hausdorff distance between sets. *)
16250 (* ------------------------------------------------------------------------- *)
16252 let hausdist = new_definition
16253 `hausdist(s:real^N->bool,t:real^N->bool) =
16254 let ds = {setdist({x},t) | x IN s} UNION {setdist({y},s) | y IN t} in
16255 if ~(ds = {}) /\ (?b. !d. d IN ds ==> d <= b) then sup ds
16258 let HAUSDIST_POS_LE = prove
16259 (`!s t:real^N->bool. &0 <= hausdist(s,t)`,
16260 REPEAT GEN_TAC THEN REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF] THEN
16261 REWRITE_TAC[FORALL_IN_GSPEC; FORALL_IN_UNION] THEN
16262 COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL] THEN
16263 MATCH_MP_TAC REAL_LE_SUP THEN
16264 ASM_REWRITE_TAC[FORALL_IN_GSPEC; FORALL_IN_UNION; SETDIST_POS_LE] THEN
16265 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
16266 ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
16267 MATCH_MP_TAC(SET_RULE
16268 `~(s = {}) /\ (!x. x IN s ==> P x) ==> ?y. y IN s /\ P y`) THEN
16269 ASM_REWRITE_TAC[FORALL_IN_GSPEC; FORALL_IN_UNION; SETDIST_POS_LE]);;
16271 let HAUSDIST_REFL = prove
16272 (`!s:real^N->bool. hausdist(s,s) = &0`,
16273 GEN_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM; HAUSDIST_POS_LE] THEN
16274 REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF] THEN
16275 COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL] THEN
16276 MATCH_MP_TAC REAL_SUP_LE THEN
16277 REWRITE_TAC[FORALL_IN_GSPEC; FORALL_IN_UNION] THEN
16278 ASM_SIMP_TAC[SETDIST_SING_IN_SET; REAL_LE_REFL]);;
16280 let HAUSDIST_SYM = prove
16281 (`!s t:real^N->bool. hausdist(s,t) = hausdist(t,s)`,
16282 REPEAT GEN_TAC THEN REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF] THEN
16283 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [UNION_COMM] THEN
16286 let HAUSDIST_EMPTY = prove
16287 (`(!t:real^N->bool. hausdist ({},t) = &0) /\
16288 (!s:real^N->bool. hausdist (s,{}) = &0)`,
16289 REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF; SETDIST_EMPTY] THEN
16290 REWRITE_TAC[SET_RULE `{f x | x IN {}} = {}`; UNION_EMPTY] THEN
16291 REWRITE_TAC[SET_RULE `{c |x| x IN s} = {} <=> s = {}`] THEN
16292 X_GEN_TAC `s:real^N->bool` THEN
16293 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THEN
16294 ASM_SIMP_TAC[SET_RULE `~(s = {}) ==> {c |x| x IN s} = {c}`] THEN
16295 REWRITE_TAC[SUP_SING; COND_ID]);;
16297 let HAUSDIST_SINGS = prove
16298 (`!x y:real^N. hausdist({x},{y}) = dist(x,y)`,
16299 REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF; SETDIST_SINGS] THEN
16300 REWRITE_TAC[SET_RULE `{f x | x IN {a}} = {f a}`] THEN
16301 REWRITE_TAC[DIST_SYM; UNION_IDEMPOT; SUP_SING; NOT_INSERT_EMPTY] THEN
16302 REWRITE_TAC[IN_SING; FORALL_UNWIND_THM2] THEN
16303 MESON_TAC[REAL_LE_REFL]);;
16305 let HAUSDIST_EQ = prove
16306 (`!s t:real^M->bool s' t':real^N->bool.
16307 (!b. (!x. x IN s ==> setdist({x},t) <= b) /\
16308 (!y. y IN t ==> setdist({y},s) <= b) <=>
16309 (!x. x IN s' ==> setdist({x},t') <= b) /\
16310 (!y. y IN t' ==> setdist({y},s') <= b))
16311 ==> hausdist(s,t) = hausdist(s',t')`,
16312 REPEAT STRIP_TAC THEN REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF] THEN
16313 MATCH_MP_TAC(MESON[]
16314 `(p <=> p') /\ s = s'
16315 ==> (if p then s else &0) = (if p' then s' else &0)`) THEN
16318 [PURE_REWRITE_TAC[SET_RULE `s = {} <=> !x. x IN s ==> F`];
16319 AP_TERM_TAC THEN ABS_TAC];
16320 MATCH_MP_TAC SUP_EQ] THEN
16321 PURE_REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_GSPEC] THEN
16322 ASM_REWRITE_TAC[] THEN
16323 REWRITE_TAC[DE_MORGAN_THM; NOT_FORALL_THM; MEMBER_NOT_EMPTY] THEN
16324 REWRITE_TAC[GSYM DE_MORGAN_THM] THEN AP_TERM_TAC THEN EQ_TAC THEN
16325 DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN ASSUME_TAC th) THEN
16326 ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
16327 DISCH_THEN(MP_TAC o SPEC `--(&1):real`) THEN
16328 SIMP_TAC[SETDIST_POS_LE; REAL_ARITH `&0 <= x ==> ~(x <= --(&1))`] THEN
16331 let HAUSDIST_TRANSLATION = prove
16332 (`!a s t:real^N->bool.
16333 hausdist(IMAGE (\x. a + x) s,IMAGE (\x. a + x) t) = hausdist(s,t)`,
16334 REPEAT GEN_TAC THEN REWRITE_TAC[hausdist] THEN
16335 REWRITE_TAC[SET_RULE `{f x | x IN IMAGE g s} = {f(g x) | x IN s}`] THEN
16336 REWRITE_TAC[SET_RULE `{a + x:real^N} = IMAGE (\x. a + x) {x}`] THEN
16337 REWRITE_TAC[SETDIST_TRANSLATION]);;
16339 add_translation_invariants [HAUSDIST_TRANSLATION];;
16341 let HAUSDIST_LINEAR_IMAGE = prove
16342 (`!f:real^M->real^N s t.
16343 linear f /\ (!x. norm(f x) = norm x)
16344 ==> hausdist(IMAGE f s,IMAGE f t) = hausdist(s,t)`,
16345 REPEAT STRIP_TAC THEN
16346 REPEAT GEN_TAC THEN REWRITE_TAC[hausdist] THEN
16347 REWRITE_TAC[SET_RULE `{f x | x IN IMAGE g s} = {f(g x) | x IN s}`] THEN
16348 ONCE_REWRITE_TAC[SET_RULE `{(f:real^M->real^N) x} = IMAGE f {x}`] THEN
16349 ASM_SIMP_TAC[SETDIST_LINEAR_IMAGE]);;
16351 add_linear_invariants [HAUSDIST_LINEAR_IMAGE];;
16353 let HAUSDIST_CLOSURE = prove
16354 (`(!s t:real^N->bool. hausdist(closure s,t) = hausdist(s,t)) /\
16355 (!s t:real^N->bool. hausdist(s,closure t) = hausdist(s,t))`,
16356 REPEAT STRIP_TAC THEN MATCH_MP_TAC HAUSDIST_EQ THEN
16357 GEN_TAC THEN BINOP_TAC THEN REWRITE_TAC[SETDIST_CLOSURE] THEN
16358 PURE_ONCE_REWRITE_TAC[SET_RULE
16359 `(!x. P x ==> Q x) <=> (!x. P x ==> x IN {x | Q x})`] THEN
16360 MATCH_MP_TAC FORALL_IN_CLOSURE_EQ THEN
16361 REWRITE_TAC[EMPTY_GSPEC; CONTINUOUS_ON_ID; CLOSED_EMPTY] THEN
16362 ONCE_REWRITE_TAC[MESON[LIFT_DROP] `x <= b <=> drop(lift x) <= b`] THEN
16363 REWRITE_TAC[SET_RULE
16364 `{x | drop(lift(f x)) <= b} =
16365 {x | x IN UNIV /\ lift(f x) IN {x | drop x <= b}}`] THEN
16366 MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN
16367 REWRITE_TAC[CLOSED_UNIV; CONTINUOUS_ON_LIFT_SETDIST] THEN
16368 REWRITE_TAC[drop; CLOSED_HALFSPACE_COMPONENT_LE]);;
16370 let REAL_HAUSDIST_LE = prove
16371 (`!s t:real^N->bool b.
16372 ~(s = {}) /\ ~(t = {}) /\
16373 (!x. x IN s ==> setdist({x},t) <= b) /\
16374 (!y. y IN t ==> setdist({y},s) <= b)
16375 ==> hausdist(s,t) <= b`,
16376 REPEAT STRIP_TAC THEN
16377 REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF; SETDIST_SINGS] THEN
16378 ASM_REWRITE_TAC[EMPTY_UNION; SET_RULE `{f x | x IN s} = {} <=> s = {}`] THEN
16379 REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_GSPEC] THEN
16380 COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
16381 MATCH_MP_TAC REAL_SUP_LE THEN
16382 ASM_REWRITE_TAC[EMPTY_UNION; SET_RULE `{f x | x IN s} = {} <=> s = {}`] THEN
16383 ASM_REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_GSPEC]);;
16385 let REAL_HAUSDIST_LE_SUMS = prove
16386 (`!s t:real^N->bool b.
16387 ~(s = {}) /\ ~(t = {}) /\
16388 s SUBSET {y + z | y IN t /\ z IN cball(vec 0,b)} /\
16389 t SUBSET {y + z | y IN s /\ z IN cball(vec 0,b)}
16390 ==> hausdist(s,t) <= b`,
16391 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_CBALL_0] THEN
16392 REWRITE_TAC[VECTOR_ARITH `a:real^N = b + x <=> a - b = x`;
16393 ONCE_REWRITE_RULE[CONJ_SYM] UNWIND_THM1] THEN
16394 REWRITE_TAC[GSYM dist] THEN REPEAT STRIP_TAC THEN
16395 MATCH_MP_TAC REAL_HAUSDIST_LE THEN
16396 ASM_MESON_TAC[SETDIST_LE_DIST; REAL_LE_TRANS; IN_SING]);;
16398 let REAL_LE_HAUSDIST = prove
16399 (`!s t:real^N->bool a b c z.
16400 ~(s = {}) /\ ~(t = {}) /\
16401 (!x. x IN s ==> setdist({x},t) <= b) /\
16402 (!y. y IN t ==> setdist({y},s) <= c) /\
16403 (z IN s /\ a <= setdist({z},t) \/ z IN t /\ a <= setdist({z},s))
16404 ==> a <= hausdist(s,t)`,
16405 REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
16406 REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF; SETDIST_SINGS] THEN
16407 ASM_REWRITE_TAC[EMPTY_UNION; SET_RULE `{f x | x IN s} = {} <=> s = {}`] THEN
16408 REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_GSPEC] THEN COND_CASES_TAC THENL
16409 [MATCH_MP_TAC REAL_LE_SUP THEN
16410 ASM_SIMP_TAC[EMPTY_UNION; SET_RULE `{f x | x IN s} = {} <=> s = {}`] THEN
16411 REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_GSPEC];
16412 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
16413 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
16414 REWRITE_TAC[NOT_FORALL_THM]] THEN
16415 EXISTS_TAC `max b c:real` THEN
16416 ASM_SIMP_TAC[REAL_LE_MAX] THEN ASM SET_TAC[]);;
16418 let SETDIST_LE_HAUSDIST = prove
16419 (`!s t:real^N->bool.
16420 bounded s /\ bounded t ==> setdist(s,t) <= hausdist(s,t)`,
16421 REPEAT STRIP_TAC THEN
16422 ASM_CASES_TAC `s:real^N->bool = {}` THEN
16423 ASM_REWRITE_TAC[SETDIST_EMPTY; HAUSDIST_EMPTY; REAL_LE_REFL] THEN
16424 ASM_CASES_TAC `t:real^N->bool = {}` THEN
16425 ASM_REWRITE_TAC[SETDIST_EMPTY; HAUSDIST_EMPTY; REAL_LE_REFL] THEN
16426 MATCH_MP_TAC REAL_LE_HAUSDIST THEN REWRITE_TAC[CONJ_ASSOC] THEN
16427 ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM; LEFT_EXISTS_AND_THM] THEN
16429 [ALL_TAC; ASM_MESON_TAC[SETDIST_LE_SING; MEMBER_NOT_EMPTY]] THEN
16430 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
16431 ASM_REWRITE_TAC[] THEN REWRITE_TAC[bounded; FORALL_IN_GSPEC; GSYM dist] THEN
16432 DISCH_THEN(X_CHOOSE_TAC `b:real`) THEN
16433 CONJ_TAC THEN EXISTS_TAC `b:real` THEN REPEAT STRIP_TAC THEN
16434 ASM_MESON_TAC[REAL_LE_TRANS; SETDIST_LE_DIST; MEMBER_NOT_EMPTY; IN_SING;
16437 let SETDIST_SING_LE_HAUSDIST = prove
16439 bounded s /\ bounded t /\ x IN s ==> setdist({x},t) <= hausdist(s,t)`,
16440 REPEAT GEN_TAC THEN
16441 ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
16442 ASM_CASES_TAC `t:real^N->bool = {}` THEN
16443 ASM_REWRITE_TAC[SETDIST_EMPTY; HAUSDIST_EMPTY; REAL_LE_REFL] THEN
16444 STRIP_TAC THEN MATCH_MP_TAC REAL_LE_HAUSDIST THEN
16445 ASM_REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
16446 REWRITE_TAC[LEFT_EXISTS_AND_THM; EXISTS_OR_THM; CONJ_ASSOC] THEN
16447 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LE_REFL]] THEN CONJ_TAC THEN
16448 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
16449 ASM_REWRITE_TAC[] THEN REWRITE_TAC[bounded; FORALL_IN_GSPEC] THEN
16450 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[GSYM dist] THEN GEN_TAC THENL
16451 [ALL_TAC; ONCE_REWRITE_TAC[SWAP_FORALL_THM]] THEN
16452 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^N` THEN
16453 REPEAT STRIP_TAC THENL
16454 [UNDISCH_TAC `~(t:real^N->bool = {})`;
16455 UNDISCH_TAC `~(s:real^N->bool = {})`] THEN
16456 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
16457 DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
16458 FIRST_X_ASSUM(MP_TAC o SPEC `z:real^N`) THEN ASM_REWRITE_TAC[] THEN
16459 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THENL
16460 [ALL_TAC; ONCE_REWRITE_TAC[DIST_SYM]] THEN
16461 MATCH_MP_TAC SETDIST_LE_DIST THEN ASM_REWRITE_TAC[IN_SING]);;
16463 let UPPER_LOWER_HEMICONTINUOUS = prove
16464 (`!f:real^M->real^N->bool t s.
16465 (!x. x IN s ==> f(x) SUBSET t) /\
16466 (!u. open_in (subtopology euclidean t) u
16467 ==> open_in (subtopology euclidean s)
16468 {x | x IN s /\ f(x) SUBSET u}) /\
16469 (!u. closed_in (subtopology euclidean t) u
16470 ==> closed_in (subtopology euclidean s)
16471 {x | x IN s /\ f(x) SUBSET u})
16472 ==> !x e. x IN s /\ &0 < e /\ bounded(f x)
16474 !x'. x' IN s /\ dist(x,x') < d
16475 ==> hausdist(f x,f x') < e`,
16476 REPEAT GEN_TAC THEN DISCH_TAC THEN REPEAT STRIP_TAC THEN
16477 ASM_CASES_TAC `(f:real^M->real^N->bool) x = {}` THENL
16478 [ASM_REWRITE_TAC[HAUSDIST_EMPTY] THEN MESON_TAC[REAL_LT_01]; ALL_TAC] THEN
16479 FIRST_ASSUM(MP_TAC o SPECL [`x:real^M`; `e / &2`] o MATCH_MP
16480 UPPER_LOWER_HEMICONTINUOUS_EXPLICIT) THEN
16481 ASM_REWRITE_TAC[REAL_HALF] THEN
16482 DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
16483 FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP BOUNDED_SUBSET_BALL) THEN
16484 DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
16485 FIRST_ASSUM(MP_TAC o SPEC `t INTER ball(vec 0:real^N,r)` o
16486 CONJUNCT1 o CONJUNCT2) THEN
16487 SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN REWRITE_TAC[open_in] THEN
16488 DISCH_THEN(MP_TAC o SPEC `x:real^M` o CONJUNCT2) THEN
16489 ASM_SIMP_TAC[SUBSET_INTER; IN_ELIM_THM] THEN
16490 DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
16491 EXISTS_TAC `min d1 d2:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
16492 X_GEN_TAC `x':real^M` THEN STRIP_TAC THEN
16493 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x':real^M`)) THEN
16494 ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REWRITE_TAC[] THEN
16495 STRIP_TAC THEN STRIP_TAC THEN
16496 ASM_CASES_TAC `(f:real^M->real^N->bool) x' = {}` THEN
16497 ASM_REWRITE_TAC[HAUSDIST_EMPTY] THEN
16498 MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN
16499 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_HAUSDIST_LE THEN
16500 ASM_MESON_TAC[SETDIST_LE_DIST; DIST_SYM; REAL_LE_TRANS;
16501 IN_SING; REAL_LT_IMP_LE]);;
16503 let HAUSDIST_NONTRIVIAL = prove
16504 (`!s t:real^N->bool.
16505 bounded s /\ bounded t /\ ~(s = {}) /\ ~(t = {})
16506 ==> hausdist(s,t) =
16507 sup({setdist ({x},t) | x IN s} UNION {setdist ({y},s) | y IN t})`,
16508 REPEAT STRIP_TAC THEN REWRITE_TAC[hausdist; LET_DEF; LET_END_DEF] THEN
16509 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
16510 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DE_MORGAN_THM]) THEN
16511 ASM_SIMP_TAC[EMPTY_UNION; SIMPLE_IMAGE; IMAGE_EQ_EMPTY] THEN
16512 MATCH_MP_TAC(TAUT `p ==> ~p ==> q`) THEN
16513 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
16514 ASM_REWRITE_TAC[bounded; FORALL_IN_UNION; FORALL_IN_IMAGE; GSYM dist] THEN
16515 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[FORALL_IN_GSPEC] THEN
16516 ASM_MESON_TAC[SETDIST_LE_DIST; dist; DIST_SYM; REAL_LE_TRANS;
16517 MEMBER_NOT_EMPTY; IN_SING]);;
16519 let HAUSDIST_NONTRIVIAL_ALT = prove
16520 (`!s t:real^N->bool.
16521 bounded s /\ bounded t /\ ~(s = {}) /\ ~(t = {})
16522 ==> hausdist(s,t) = max (sup {setdist ({x},t) | x IN s})
16523 (sup {setdist ({y},s) | y IN t})`,
16524 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[HAUSDIST_NONTRIVIAL] THEN
16525 MATCH_MP_TAC SUP_UNION THEN
16526 ASM_REWRITE_TAC[SIMPLE_IMAGE; FORALL_IN_IMAGE; IMAGE_EQ_EMPTY] THEN
16528 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
16529 ASM_REWRITE_TAC[bounded; FORALL_IN_UNION; FORALL_IN_IMAGE; GSYM dist] THEN
16530 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[FORALL_IN_GSPEC; GSYM dist] THEN
16531 ASM_MESON_TAC[SETDIST_LE_DIST; dist; DIST_SYM; REAL_LE_TRANS;
16532 MEMBER_NOT_EMPTY; IN_SING]);;
16534 let REAL_HAUSDIST_LE_EQ = prove
16535 (`!s t:real^N->bool b.
16536 ~(s = {}) /\ ~(t = {}) /\ bounded s /\ bounded t
16537 ==> (hausdist(s,t) <= b <=>
16538 (!x. x IN s ==> setdist({x},t) <= b) /\
16539 (!y. y IN t ==> setdist({y},s) <= b))`,
16540 REPEAT STRIP_TAC THEN
16541 ASM_SIMP_TAC[HAUSDIST_NONTRIVIAL_ALT; REAL_MAX_LE] THEN
16543 ONCE_REWRITE_TAC[SET_RULE `(!x. x IN s ==> f x <= b) <=>
16544 (!y. y IN {f x | x IN s} ==> y <= b)`] THEN
16545 MATCH_MP_TAC REAL_SUP_LE_EQ THEN
16546 ASM_REWRITE_TAC[SIMPLE_IMAGE; IMAGE_EQ_EMPTY; FORALL_IN_IMAGE] THEN
16547 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
16548 ASM_REWRITE_TAC[bounded; FORALL_IN_UNION; FORALL_IN_IMAGE; GSYM dist] THEN
16549 MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[FORALL_IN_GSPEC; GSYM dist] THEN
16550 ASM_MESON_TAC[SETDIST_LE_DIST; dist; DIST_SYM; REAL_LE_TRANS;
16551 MEMBER_NOT_EMPTY; IN_SING]);;
16553 let HAUSDIST_COMPACT_EXISTS = prove
16554 (`!s t:real^N->bool.
16555 bounded s /\ compact t /\ ~(t = {})
16556 ==> !x. x IN s ==> ?y. y IN t /\ dist(x,y) <= hausdist(s,t)`,
16557 REPEAT STRIP_TAC THEN
16558 ASM_CASES_TAC `s:real^N->bool = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN
16559 MP_TAC(ISPECL [`{x:real^N}`; `t:real^N->bool`]
16560 SETDIST_COMPACT_CLOSED) THEN
16561 ASM_SIMP_TAC[COMPACT_SING; COMPACT_IMP_CLOSED; NOT_INSERT_EMPTY] THEN
16562 REWRITE_TAC[IN_SING; UNWIND_THM2; RIGHT_EXISTS_AND_THM; UNWIND_THM1] THEN
16563 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN
16564 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
16565 MATCH_MP_TAC REAL_LE_HAUSDIST THEN
16566 ASM_REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM] THEN
16567 REWRITE_TAC[CONJ_ASSOC] THEN
16568 CONJ_TAC THENL [CONJ_TAC; ASM_MESON_TAC[REAL_LE_REFL]] THEN
16569 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`] BOUNDED_DIFFS) THEN
16570 ASM_SIMP_TAC[COMPACT_IMP_BOUNDED] THEN
16571 REWRITE_TAC[bounded; FORALL_IN_GSPEC; GSYM dist] THEN
16572 MATCH_MP_TAC MONO_EXISTS THEN
16573 ASM_MESON_TAC[SETDIST_LE_DIST; dist; DIST_SYM; REAL_LE_TRANS;
16574 MEMBER_NOT_EMPTY; IN_SING]);;
16576 let HAUSDIST_COMPACT_SUMS = prove
16577 (`!s t:real^N->bool.
16578 bounded s /\ compact t /\ ~(t = {})
16579 ==> s SUBSET {y + z | y IN t /\ z IN cball(vec 0,hausdist(s,t))}`,
16580 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_CBALL_0] THEN
16581 REWRITE_TAC[VECTOR_ARITH `a:real^N = b + x <=> a - b = x`;
16582 ONCE_REWRITE_RULE[CONJ_SYM] UNWIND_THM1] THEN
16583 REWRITE_TAC[GSYM dist; HAUSDIST_COMPACT_EXISTS]);;
16585 let HAUSDIST_TRANS = prove
16586 (`!s t u:real^N->bool.
16587 bounded s /\ bounded t /\ bounded u /\ ~(t = {})
16588 ==> hausdist(s,u) <= hausdist(s,t) + hausdist(t,u)`,
16590 (`!s t u:real^N->bool.
16591 bounded s /\ bounded t /\ bounded u /\
16592 ~(s = {}) /\ ~(t = {}) /\ ~(u = {})
16593 ==> !x. x IN s ==> setdist({x},u) <= hausdist(s,t) + hausdist(t,u)`,
16594 REPEAT STRIP_TAC THEN
16595 MP_TAC(ISPECL [`closure s:real^N->bool`; `closure t:real^N->bool`]
16596 HAUSDIST_COMPACT_EXISTS) THEN
16597 ASM_SIMP_TAC[COMPACT_CLOSURE; BOUNDED_CLOSURE; CLOSURE_EQ_EMPTY] THEN
16598 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
16599 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; HAUSDIST_CLOSURE] THEN
16600 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
16601 MP_TAC(ISPECL [`closure t:real^N->bool`; `closure u:real^N->bool`]
16602 HAUSDIST_COMPACT_EXISTS) THEN
16603 ASM_SIMP_TAC[COMPACT_CLOSURE; BOUNDED_CLOSURE; CLOSURE_EQ_EMPTY] THEN
16604 DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN
16605 ASM_SIMP_TAC[REWRITE_RULE[SUBSET] CLOSURE_SUBSET; HAUSDIST_CLOSURE] THEN
16606 DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
16607 TRANS_TAC REAL_LE_TRANS `dist(x:real^N,z)` THEN CONJ_TAC THENL
16608 [ASM_MESON_TAC[SETDIST_CLOSURE; SETDIST_LE_DIST; IN_SING]; ALL_TAC] THEN
16609 TRANS_TAC REAL_LE_TRANS `dist(x:real^N,y) + dist(y,z)` THEN
16610 REWRITE_TAC[DIST_TRIANGLE] THEN ASM_REAL_ARITH_TAC) in
16611 REPEAT STRIP_TAC THEN
16612 ASM_CASES_TAC `s:real^N->bool = {}` THEN
16613 ASM_REWRITE_TAC[HAUSDIST_EMPTY; REAL_ADD_LID; HAUSDIST_POS_LE] THEN
16614 ASM_CASES_TAC `u:real^N->bool = {}` THEN
16615 ASM_REWRITE_TAC[HAUSDIST_EMPTY; REAL_ADD_RID; HAUSDIST_POS_LE] THEN
16616 ASM_SIMP_TAC[REAL_HAUSDIST_LE_EQ] THEN
16617 ASM_MESON_TAC[lemma; HAUSDIST_SYM; SETDIST_SYM; REAL_ADD_SYM]);;
16619 let HAUSDIST_EQ_0 = prove
16620 (`!s t:real^N->bool.
16621 bounded s /\ bounded t
16622 ==> (hausdist(s,t) = &0 <=> s = {} \/ t = {} \/ closure s = closure t)`,
16623 REPEAT STRIP_TAC THEN
16624 MAP_EVERY ASM_CASES_TAC [`s:real^N->bool = {}`; `t:real^N->bool = {}`] THEN
16625 ASM_REWRITE_TAC[HAUSDIST_EMPTY] THEN
16626 ASM_SIMP_TAC[GSYM REAL_LE_ANTISYM; HAUSDIST_POS_LE; REAL_HAUSDIST_LE_EQ] THEN
16627 SIMP_TAC[SETDIST_POS_LE; REAL_ARITH `&0 <= x ==> (x <= &0 <=> x = &0)`] THEN
16628 ASM_REWRITE_TAC[SETDIST_EQ_0_SING; GSYM SUBSET_ANTISYM_EQ; SUBSET] THEN
16629 SIMP_TAC[FORALL_IN_CLOSURE_EQ; CLOSED_CLOSURE; CONTINUOUS_ON_ID]);;
16631 let HAUSDIST_COMPACT_NONTRIVIAL = prove
16632 (`!s t:real^N->bool.
16633 compact s /\ compact t /\ ~(s = {}) /\ ~(t = {})
16634 ==> hausdist(s,t) =
16635 inf {e | &0 <= e /\
16636 s SUBSET {x + y | x IN t /\ norm y <= e} /\
16637 t SUBSET {x + y | x IN s /\ norm y <= e}}`,
16638 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
16639 MATCH_MP_TAC REAL_INF_UNIQUE THEN
16640 REWRITE_TAC[FORALL_IN_GSPEC; EXISTS_IN_GSPEC] THEN
16641 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
16642 REWRITE_TAC[VECTOR_ARITH `a:real^N = b + x <=> a - b = x`;
16643 ONCE_REWRITE_RULE[CONJ_SYM] UNWIND_THM1] THEN
16644 REWRITE_TAC[GSYM dist] THEN CONJ_TAC THENL
16645 [REPEAT STRIP_TAC THEN
16646 MATCH_MP_TAC REAL_HAUSDIST_LE THEN
16647 ASM_MESON_TAC[SETDIST_LE_DIST; DIST_SYM; REAL_LE_TRANS;
16648 IN_SING; REAL_LT_IMP_LE];
16649 REPEAT STRIP_TAC THEN EXISTS_TAC `hausdist(s:real^N->bool,t)` THEN
16650 ASM_REWRITE_TAC[HAUSDIST_POS_LE] THEN
16651 ASM_MESON_TAC[DIST_SYM; HAUSDIST_SYM;
16652 HAUSDIST_COMPACT_EXISTS; COMPACT_IMP_BOUNDED]]);;
16654 let HAUSDIST_BALLS = prove
16655 (`(!a b:real^N r s.
16656 hausdist(ball(a,r),ball(b,s)) =
16657 if r <= &0 \/ s <= &0 then &0 else dist(a,b) + abs(r - s)) /\
16659 hausdist(ball(a,r),cball(b,s)) =
16660 if r <= &0 \/ s < &0 then &0 else dist(a,b) + abs(r - s)) /\
16662 hausdist(cball(a,r),ball(b,s)) =
16663 if r < &0 \/ s <= &0 then &0 else dist(a,b) + abs(r - s)) /\
16665 hausdist(cball(a,r),cball(b,s)) =
16666 if r < &0 \/ s < &0 then &0 else dist(a,b) + abs(r - s))`,
16667 REWRITE_TAC[MESON[]
16668 `(x = if p then y else z) <=> (p ==> x = y) /\ (~p ==> x = z)`] THEN
16669 SIMP_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
16670 SIMP_TAC[BALL_EMPTY; CBALL_EMPTY; HAUSDIST_EMPTY; DE_MORGAN_THM] THEN
16671 ONCE_REWRITE_TAC[MESON[HAUSDIST_CLOSURE]
16672 `hausdist(s,t) = hausdist(closure s,closure t)`] THEN
16673 SIMP_TAC[REAL_NOT_LE; REAL_NOT_LT; CLOSURE_BALL] THEN
16674 REWRITE_TAC[HAUSDIST_CLOSURE] THEN
16675 MATCH_MP_TAC(TAUT `(s ==> p /\ q /\ r) /\ s ==> p /\ q /\ r /\ s`) THEN
16676 CONJ_TAC THENL [MESON_TAC[REAL_LT_IMP_LE]; REPEAT STRIP_TAC] THEN
16677 ASM_SIMP_TAC[HAUSDIST_NONTRIVIAL; BOUNDED_CBALL; CBALL_EQ_EMPTY;
16679 MATCH_MP_TAC SUP_UNIQUE THEN
16680 REWRITE_TAC[FORALL_IN_GSPEC; FORALL_IN_UNION] THEN
16681 REWRITE_TAC[MESON[CBALL_SING] `{a} = cball(a:real^N,&0)`] THEN
16682 ASM_REWRITE_TAC[SETDIST_BALLS; REAL_LT_REFL] THEN
16683 X_GEN_TAC `c:real` THEN REWRITE_TAC[IN_CBALL] THEN
16684 EQ_TAC THENL [ALL_TAC; NORM_ARITH_TAC] THEN
16685 ASM_CASES_TAC `b:real^N = a` THENL
16686 [ASM_REWRITE_TAC[DIST_SYM; DIST_REFL; REAL_MAX_LE] THEN
16687 DISCH_THEN(CONJUNCTS_THEN2
16688 (MP_TAC o SPEC `a + r % basis 1:real^N`)
16689 (MP_TAC o SPEC `a + s % basis 1:real^N`)) THEN
16690 REWRITE_TAC[NORM_ARITH `dist(a:real^N,a + x) = norm x`] THEN
16691 SIMP_TAC[NORM_MUL; NORM_BASIS; LE_REFL; DIMINDEX_GE_1] THEN
16692 ASM_REAL_ARITH_TAC;
16693 DISCH_THEN(CONJUNCTS_THEN2
16694 (MP_TAC o SPEC `a - r / dist(a,b) % (b - a):real^N`)
16695 (MP_TAC o SPEC `b - s / dist(a,b) % (a - b):real^N`)) THEN
16696 REWRITE_TAC[NORM_ARITH `dist(a:real^N,a - x) = norm x`] THEN
16697 REWRITE_TAC[dist; NORM_MUL; VECTOR_ARITH
16698 `b - e % (a - b) - a:real^N = (&1 + e) % (b - a)`] THEN
16699 ONCE_REWRITE_TAC[GSYM REAL_ABS_NORM] THEN
16700 REWRITE_TAC[GSYM REAL_ABS_MUL] THEN REWRITE_TAC[REAL_ABS_NORM] THEN
16701 REWRITE_TAC[NORM_SUB; REAL_ADD_RDISTRIB; REAL_MUL_LID] THEN
16702 ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
16703 ASM_REAL_ARITH_TAC]);;
16705 let HAUSDIST_ALT = prove
16706 (`!s t:real^N->bool.
16707 bounded s /\ bounded t /\ ~(s = {}) /\ ~(t = {})
16708 ==> hausdist(s,t) =
16709 sup {abs(setdist({x},s) - setdist({x},t)) | x IN (:real^N)}`,
16710 REPEAT GEN_TAC THEN
16711 ONCE_REWRITE_TAC[GSYM COMPACT_CLOSURE; GSYM(CONJUNCT2 SETDIST_CLOSURE);
16712 GSYM CLOSURE_EQ_EMPTY; MESON[HAUSDIST_CLOSURE]
16713 `hausdist(s:real^N->bool,t) = hausdist(closure s,closure t)`] THEN
16714 SPEC_TAC(`closure t:real^N->bool`,`t:real^N->bool`) THEN
16715 SPEC_TAC(`closure s:real^N->bool`,`s:real^N->bool`) THEN
16716 REPEAT STRIP_TAC THEN
16717 ASM_SIMP_TAC[HAUSDIST_NONTRIVIAL; COMPACT_IMP_BOUNDED] THEN
16718 MATCH_MP_TAC SUP_EQ THEN
16719 REWRITE_TAC[FORALL_IN_UNION; FORALL_IN_GSPEC; IN_UNIV] THEN
16720 REWRITE_TAC[REAL_ARITH `abs(y - x) <= b <=> x <= y + b /\ y <= x + b`] THEN
16721 GEN_TAC THEN REWRITE_TAC[FORALL_AND_THM] THEN BINOP_TAC THEN
16722 (EQ_TAC THENL [ALL_TAC; MESON_TAC[SETDIST_SING_IN_SET; REAL_ADD_LID]]) THEN
16723 DISCH_TAC THEN X_GEN_TAC `z:real^N` THENL
16724 [MP_TAC(ISPECL[`{z:real^N}`; `s:real^N->bool`] SETDIST_CLOSED_COMPACT);
16725 MP_TAC(ISPECL[`{z:real^N}`; `t:real^N->bool`] SETDIST_CLOSED_COMPACT)] THEN
16726 ASM_REWRITE_TAC[CLOSED_SING; NOT_INSERT_EMPTY] THEN
16727 REWRITE_TAC[IN_SING; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
16728 DISCH_THEN(X_CHOOSE_THEN `y:real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
16729 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THENL
16730 [MP_TAC(ISPECL[`{y:real^N}`; `t:real^N->bool`] SETDIST_CLOSED_COMPACT);
16731 MP_TAC(ISPECL[`{y:real^N}`; `s:real^N->bool`] SETDIST_CLOSED_COMPACT)] THEN
16732 ASM_REWRITE_TAC[CLOSED_SING; NOT_INSERT_EMPTY] THEN
16733 REWRITE_TAC[IN_SING; RIGHT_EXISTS_AND_THM; UNWIND_THM2] THEN
16734 DISCH_THEN(X_CHOOSE_THEN `x:real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
16735 ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
16736 TRANS_TAC REAL_LE_TRANS `dist(z:real^N,x)` THEN
16737 ASM_SIMP_TAC[SETDIST_LE_DIST; IN_SING] THEN
16738 UNDISCH_TAC `dist(y:real^N,x) <= b` THEN CONV_TAC NORM_ARITH);;
16740 let CONTINUOUS_DIAMETER = prove
16741 (`!s:real^N->bool e.
16742 bounded s /\ ~(s = {}) /\ &0 < e
16744 !t. bounded t /\ ~(t = {}) /\ hausdist(s,t) < d
16745 ==> abs(diameter s - diameter t) < e`,
16746 REPEAT STRIP_TAC THEN EXISTS_TAC `e / &2` THEN
16747 ASM_REWRITE_TAC[REAL_HALF] THEN REPEAT STRIP_TAC THEN
16748 SUBGOAL_THEN `diameter(s:real^N->bool) - diameter(t:real^N->bool) =
16749 diameter(closure s) - diameter(closure t)`
16750 SUBST1_TAC THENL [ASM_MESON_TAC[DIAMETER_CLOSURE]; ALL_TAC] THEN
16751 MATCH_MP_TAC REAL_LET_TRANS THEN
16752 EXISTS_TAC `&2 * hausdist(s:real^N->bool,t)` THEN
16753 CONJ_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
16754 MP_TAC(ISPECL [`vec 0:real^N`; `hausdist(s:real^N->bool,t)`]
16755 DIAMETER_CBALL) THEN
16756 ASM_SIMP_TAC[HAUSDIST_POS_LE; GSYM REAL_NOT_LE] THEN
16757 DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC(REAL_ARITH
16758 `x <= y + e /\ y <= x + e ==> abs(x - y) <= e`) THEN
16760 W(MP_TAC o PART_MATCH (rand o rand) DIAMETER_SUMS o rand o snd) THEN
16761 ASM_SIMP_TAC[BOUNDED_CBALL; BOUNDED_CLOSURE] THEN
16762 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LE_TRANS) THEN
16763 MATCH_MP_TAC DIAMETER_SUBSET THEN
16764 ASM_SIMP_TAC[BOUNDED_SUMS; BOUNDED_CBALL; BOUNDED_CLOSURE] THEN
16765 ONCE_REWRITE_TAC[MESON[HAUSDIST_CLOSURE]
16766 `hausdist(s:real^N->bool,t) = hausdist(closure s,closure t)`]
16767 THENL [ALL_TAC; ONCE_REWRITE_TAC[HAUSDIST_SYM]] THEN
16768 MATCH_MP_TAC HAUSDIST_COMPACT_SUMS THEN
16769 ASM_SIMP_TAC[COMPACT_CLOSURE; BOUNDED_CLOSURE; CLOSURE_EQ_EMPTY]);;
16771 (* ------------------------------------------------------------------------- *)
16772 (* Isometries are embeddings, and even surjective in the compact case. *)
16773 (* ------------------------------------------------------------------------- *)
16775 let ISOMETRY_IMP_OPEN_MAP = prove
16776 (`!f:real^M->real^N s t u.
16778 (!x y. x IN s /\ y IN s ==> dist(f x,f y) = dist(x,y)) /\
16779 open_in (subtopology euclidean s) u
16780 ==> open_in (subtopology euclidean t) (IMAGE f u)`,
16781 REWRITE_TAC[open_in; FORALL_IN_IMAGE] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
16782 CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `x:real^M` THEN DISCH_TAC] THEN
16783 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN
16784 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
16785 STRIP_TAC THEN ASM_REWRITE_TAC[IMP_CONJ] THEN
16786 EXPAND_TAC "t" THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
16787 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN
16788 ASM_SIMP_TAC[IN_IMAGE] THEN ASM_MESON_TAC[]);;
16790 let ISOMETRY_IMP_EMBEDDING = prove
16791 (`!f:real^M->real^N s t.
16792 IMAGE f s = t /\ (!x y. x IN s /\ y IN s ==> dist(f x,f y) = dist(x,y))
16793 ==> ?g. homeomorphism (s,t) (f,g)`,
16794 REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMEOMORPHISM_INJECTIVE_OPEN_MAP THEN
16795 ASM_SIMP_TAC[ISOMETRY_ON_IMP_CONTINUOUS_ON] THEN
16796 CONJ_TAC THENL [ASM_MESON_TAC[DIST_EQ_0]; REPEAT STRIP_TAC] THEN
16797 MATCH_MP_TAC ISOMETRY_IMP_OPEN_MAP THEN ASM_MESON_TAC[]);;
16799 let ISOMETRY_IMP_HOMEOMORPHISM_COMPACT = prove
16800 (`!f s:real^N->bool.
16801 compact s /\ IMAGE f s SUBSET s /\
16802 (!x y. x IN s /\ y IN s ==> dist(f x,f y) = dist(x,y))
16803 ==> ?g. homeomorphism (s,s) (f,g)`,
16804 REPEAT STRIP_TAC THEN
16805 SUBGOAL_THEN `IMAGE (f:real^N->real^N) s = s`
16806 (fun th -> ASM_MESON_TAC[th; ISOMETRY_IMP_EMBEDDING]) THEN
16807 FIRST_ASSUM(ASSUME_TAC o MATCH_MP ISOMETRY_ON_IMP_CONTINUOUS_ON) THEN
16808 ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[SUBSET] THEN
16809 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
16810 SUBGOAL_THEN `setdist({x},IMAGE (f:real^N->real^N) s) = &0` MP_TAC THENL
16811 [MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ ~(&0 < x) ==> x = &0`) THEN
16812 REWRITE_TAC[SETDIST_POS_LE] THEN DISCH_TAC THEN
16813 (X_CHOOSE_THEN `z:num->real^N` STRIP_ASSUME_TAC o
16814 prove_recursive_functions_exist num_RECURSION)
16815 `z 0 = (x:real^N) /\ !n. z(SUC n) = f(z n)` THEN
16816 SUBGOAL_THEN `!n. (z:num->real^N) n IN s` ASSUME_TAC THENL
16817 [INDUCT_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN
16818 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [compact]) THEN
16819 DISCH_THEN(MP_TAC o SPEC `z:num->real^N`) THEN
16820 ASM_REWRITE_TAC[NOT_EXISTS_THM] THEN
16821 MAP_EVERY X_GEN_TAC [`l:real^N`; `r:num->num`] THEN STRIP_TAC THEN
16822 FIRST_ASSUM(MP_TAC o MATCH_MP CONVERGENT_IMP_CAUCHY) THEN
16823 REWRITE_TAC[cauchy] THEN
16824 DISCH_THEN(MP_TAC o SPEC `setdist({x},IMAGE (f:real^N->real^N) s)`) THEN
16825 ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `N:num`
16826 (MP_TAC o SPECL [`N:num`; `N + 1`])) THEN
16827 ANTS_TAC THENL [ARITH_TAC; REWRITE_TAC[REAL_NOT_LT; o_THM]] THEN
16828 SUBGOAL_THEN `(r:num->num) N < r (N + 1)` MP_TAC THENL
16829 [FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC;
16830 REWRITE_TAC[LT_EXISTS; LEFT_IMP_EXISTS_THM]] THEN
16831 X_GEN_TAC `d:num` THEN DISCH_THEN SUBST1_TAC THEN
16832 TRANS_TAC REAL_LE_TRANS `dist(x:real^N,z(SUC d))` THEN CONJ_TAC THENL
16833 [MATCH_MP_TAC SETDIST_LE_DIST THEN ASM SET_TAC[]; ALL_TAC] THEN
16834 MATCH_MP_TAC REAL_EQ_IMP_LE THEN
16835 SPEC_TAC(`(r:num->num) N`,`m:num`) THEN
16836 INDUCT_TAC THEN ASM_MESON_TAC[ADD_CLAUSES];
16837 REWRITE_TAC[SETDIST_EQ_0_SING; IMAGE_EQ_EMPTY] THEN
16838 ASM_MESON_TAC[COMPACT_IMP_CLOSED; NOT_IN_EMPTY;
16839 COMPACT_CONTINUOUS_IMAGE; CLOSURE_CLOSED]]);;
16841 (* ------------------------------------------------------------------------- *)
16842 (* Urysohn's lemma (for real^N, where the proof is easy using distances). *)
16843 (* ------------------------------------------------------------------------- *)
16845 let URYSOHN_LOCAL_STRONG = prove
16847 closed_in (subtopology euclidean u) s /\
16848 closed_in (subtopology euclidean u) t /\
16849 s INTER t = {} /\ ~(a = b)
16850 ==> ?f:real^N->real^M.
16851 f continuous_on u /\
16852 (!x. x IN u ==> f(x) IN segment[a,b]) /\
16853 (!x. x IN u ==> (f x = a <=> x IN s)) /\
16854 (!x. x IN u ==> (f x = b <=> x IN t))`,
16857 closed_in (subtopology euclidean u) s /\
16858 closed_in (subtopology euclidean u) t /\
16859 s INTER t = {} /\ ~(s = {}) /\ ~(t = {}) /\ ~(a = b)
16860 ==> ?f:real^N->real^M.
16861 f continuous_on u /\
16862 (!x. x IN u ==> f(x) IN segment[a,b]) /\
16863 (!x. x IN u ==> (f x = a <=> x IN s)) /\
16864 (!x. x IN u ==> (f x = b <=> x IN t))`,
16865 REPEAT STRIP_TAC THEN EXISTS_TAC
16866 `\x:real^N. a + setdist({x},s) / (setdist({x},s) + setdist({x},t)) %
16867 (b - a:real^M)` THEN REWRITE_TAC[] THEN
16869 `(!x:real^N. x IN u ==> (setdist({x},s) = &0 <=> x IN s)) /\
16870 (!x:real^N. x IN u ==> (setdist({x},t) = &0 <=> x IN t))`
16871 STRIP_ASSUME_TAC THENL
16872 [ASM_REWRITE_TAC[SETDIST_EQ_0_SING] THEN CONJ_TAC THENL
16873 [MP_TAC(ISPEC `s:real^N->bool` CLOSED_IN_CLOSED);
16874 MP_TAC(ISPEC `t:real^N->bool` CLOSED_IN_CLOSED)] THEN
16875 DISCH_THEN(MP_TAC o SPEC `u:real^N->bool`) THEN
16876 ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool`
16877 (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
16878 ASM_MESON_TAC[CLOSURE_CLOSED; INTER_SUBSET; SUBSET_CLOSURE; SUBSET;
16879 IN_INTER; CLOSURE_SUBSET];
16881 SUBGOAL_THEN `!x:real^N. x IN u ==> &0 < setdist({x},s) + setdist({x},t)`
16883 [REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH
16884 `&0 <= x /\ &0 <= y /\ ~(x = &0 /\ y = &0) ==> &0 < x + y`) THEN
16885 REWRITE_TAC[SETDIST_POS_LE] THEN ASM SET_TAC[];
16887 REPEAT CONJ_TAC THENL
16888 [MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
16889 REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THEN
16890 REPEAT(MATCH_MP_TAC CONTINUOUS_ON_MUL THEN CONJ_TAC) THEN
16891 REWRITE_TAC[CONTINUOUS_ON_CONST; o_DEF] THEN
16892 REWRITE_TAC[CONTINUOUS_ON_LIFT_SETDIST] THEN
16893 MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
16894 ASM_SIMP_TAC[REAL_LT_IMP_NZ] THEN
16895 REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN
16896 REWRITE_TAC[CONTINUOUS_ON_LIFT_SETDIST];
16897 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
16898 REWRITE_TAC[segment; IN_ELIM_THM] THEN
16899 REWRITE_TAC[VECTOR_MUL_EQ_0; LEFT_OR_DISTRIB; VECTOR_ARITH
16900 `a + x % (b - a):real^N = (&1 - u) % a + u % b <=>
16901 (x - u) % (b - a) = vec 0`;
16902 EXISTS_OR_THM] THEN
16903 DISJ1_TAC THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
16904 REWRITE_TAC[REAL_SUB_0; UNWIND_THM1] THEN
16905 ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_ADD; SETDIST_POS_LE; REAL_LE_LDIV_EQ;
16906 REAL_ARITH `a <= &1 * (a + b) <=> &0 <= b`];
16907 REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a <=> x = vec 0`];
16908 REWRITE_TAC[VECTOR_ARITH `a + x % (b - a):real^N = b <=>
16909 (x - &1) % (b - a) = vec 0`]] THEN
16910 ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
16911 ASM_SIMP_TAC[REAL_SUB_0; REAL_EQ_LDIV_EQ;
16912 REAL_MUL_LZERO; REAL_MUL_LID] THEN
16913 REWRITE_TAC[REAL_ARITH `x:real = x + y <=> y = &0`] THEN
16914 ASM_REWRITE_TAC[]) in
16915 MATCH_MP_TAC(MESON[]
16916 `(!s t. P s t <=> P t s) /\
16917 (!s t. ~(s = {}) /\ ~(t = {}) ==> P s t) /\
16918 P {} {} /\ (!t. ~(t = {}) ==> P {} t)
16919 ==> !s t. P s t`) THEN
16920 REPEAT CONJ_TAC THENL
16921 [REPEAT GEN_TAC THEN
16922 GEN_REWRITE_TAC (RAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
16923 REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
16924 REWRITE_TAC[SEGMENT_SYM; INTER_COMM; CONJ_ACI; EQ_SYM_EQ];
16926 REPEAT STRIP_TAC THEN EXISTS_TAC `(\x. midpoint(a,b)):real^N->real^M` THEN
16927 ASM_SIMP_TAC[NOT_IN_EMPTY; CONTINUOUS_ON_CONST; MIDPOINT_IN_SEGMENT] THEN
16928 REWRITE_TAC[midpoint] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN
16929 UNDISCH_TAC `~(a:real^M = b)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN
16931 REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = u` THENL
16932 [EXISTS_TAC `(\x. b):real^N->real^M` THEN
16933 ASM_REWRITE_TAC[NOT_IN_EMPTY; ENDS_IN_SEGMENT; IN_UNIV;
16934 CONTINUOUS_ON_CONST];
16935 SUBGOAL_THEN `?c:real^N. c IN u /\ ~(c IN t)` STRIP_ASSUME_TAC THENL
16936 [REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET)) THEN
16937 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM SET_TAC[];
16939 MP_TAC(ISPECL [`{c:real^N}`; `t:real^N->bool`; `u:real^N->bool`;
16940 `midpoint(a,b):real^M`; `b:real^M`] lemma) THEN
16941 ASM_REWRITE_TAC[CLOSED_IN_SING; MIDPOINT_EQ_ENDPOINT] THEN
16942 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
16943 MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[NOT_IN_EMPTY] THEN
16944 X_GEN_TAC `f:real^N->real^M` THEN STRIP_TAC THEN CONJ_TAC THENL
16946 `segment[midpoint(a,b):real^M,b] SUBSET segment[a,b]` MP_TAC
16948 [REWRITE_TAC[SUBSET; IN_SEGMENT; midpoint] THEN GEN_TAC THEN
16949 DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
16950 EXISTS_TAC `(&1 + u) / &2` THEN ASM_REWRITE_TAC[] THEN
16951 REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
16954 SUBGOAL_THEN `~(a IN segment[midpoint(a,b):real^M,b])` MP_TAC THENL
16955 [ALL_TAC; ASM_MESON_TAC[]] THEN
16956 DISCH_THEN(MP_TAC o CONJUNCT2 o MATCH_MP DIST_IN_CLOSED_SEGMENT) THEN
16957 REWRITE_TAC[DIST_MIDPOINT] THEN
16958 UNDISCH_TAC `~(a:real^M = b)` THEN NORM_ARITH_TAC]]]);;
16960 let URYSOHN_LOCAL = prove
16962 closed_in (subtopology euclidean u) s /\
16963 closed_in (subtopology euclidean u) t /\
16965 ==> ?f:real^N->real^M.
16966 f continuous_on u /\
16967 (!x. x IN u ==> f(x) IN segment[a,b]) /\
16968 (!x. x IN s ==> f x = a) /\
16969 (!x. x IN t ==> f x = b)`,
16970 REPEAT STRIP_TAC THEN ASM_CASES_TAC `a:real^M = b` THENL
16971 [EXISTS_TAC `(\x. b):real^N->real^M` THEN
16972 ASM_REWRITE_TAC[ENDS_IN_SEGMENT; CONTINUOUS_ON_CONST];
16973 MP_TAC(ISPECL [`s:real^N->bool`; `t:real^N->bool`; `u:real^N->bool`;
16974 `a:real^M`; `b:real^M`] URYSOHN_LOCAL_STRONG) THEN
16975 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN
16976 REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET)) THEN
16977 REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN SET_TAC[]]);;
16979 let URYSOHN_STRONG = prove
16981 closed s /\ closed t /\ s INTER t = {} /\ ~(a = b)
16982 ==> ?f:real^N->real^M.
16983 f continuous_on (:real^N) /\ (!x. f(x) IN segment[a,b]) /\
16984 (!x. f x = a <=> x IN s) /\ (!x. f x = b <=> x IN t)`,
16985 REPEAT GEN_TAC THEN REWRITE_TAC[CLOSED_IN] THEN
16986 ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN
16987 DISCH_THEN(MP_TAC o MATCH_MP URYSOHN_LOCAL_STRONG) THEN
16988 REWRITE_TAC[IN_UNIV]);;
16990 let URYSOHN = prove
16992 closed s /\ closed t /\ s INTER t = {}
16993 ==> ?f:real^N->real^M.
16994 f continuous_on (:real^N) /\ (!x. f(x) IN segment[a,b]) /\
16995 (!x. x IN s ==> f x = a) /\ (!x. x IN t ==> f x = b)`,
16996 REPEAT GEN_TAC THEN REWRITE_TAC[CLOSED_IN] THEN
16997 ONCE_REWRITE_TAC[GSYM SUBTOPOLOGY_UNIV] THEN DISCH_THEN
16998 (MP_TAC o ISPECL [`a:real^M`; `b:real^M`] o MATCH_MP URYSOHN_LOCAL) THEN
16999 REWRITE_TAC[IN_UNIV]);;
17001 (* ------------------------------------------------------------------------- *)
17002 (* Countability of some relevant sets. *)
17003 (* ------------------------------------------------------------------------- *)
17005 let COUNTABLE_INTEGER = prove
17006 (`COUNTABLE integer`,
17007 MATCH_MP_TAC COUNTABLE_SUBSET THEN EXISTS_TAC
17008 `IMAGE (\n. (&n:real)) (:num) UNION IMAGE (\n. --(&n)) (:num)` THEN
17009 SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_UNION; NUM_COUNTABLE] THEN
17010 REWRITE_TAC[SUBSET; IN_UNION; IN_IMAGE; IN_UNIV] THEN
17011 REWRITE_TAC[IN; INTEGER_CASES]);;
17013 let CARD_EQ_INTEGER = prove
17014 (`integer =_c (:num)`,
17015 REWRITE_TAC[GSYM CARD_LE_ANTISYM; GSYM COUNTABLE_ALT; COUNTABLE_INTEGER] THEN
17016 REWRITE_TAC[le_c] THEN EXISTS_TAC `real_of_num` THEN
17017 REWRITE_TAC[IN_UNIV; REAL_OF_NUM_EQ] THEN
17018 REWRITE_TAC[IN; INTEGER_CLOSED]);;
17020 let COUNTABLE_RATIONAL = prove
17021 (`COUNTABLE rational`,
17022 MATCH_MP_TAC COUNTABLE_SUBSET THEN
17023 EXISTS_TAC `IMAGE (\(x,y). x / y) (integer CROSS integer)` THEN
17024 SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_CROSS; COUNTABLE_INTEGER] THEN
17025 REWRITE_TAC[SUBSET; IN_IMAGE; EXISTS_PAIR_THM; IN_CROSS] THEN
17026 REWRITE_TAC[rational; IN] THEN MESON_TAC[]);;
17028 let CARD_EQ_RATIONAL = prove
17029 (`rational =_c (:num)`,
17030 REWRITE_TAC[GSYM CARD_LE_ANTISYM; GSYM COUNTABLE_ALT; COUNTABLE_RATIONAL] THEN
17031 REWRITE_TAC[le_c] THEN EXISTS_TAC `real_of_num` THEN
17032 REWRITE_TAC[IN_UNIV; REAL_OF_NUM_EQ] THEN
17033 REWRITE_TAC[IN; RATIONAL_CLOSED]);;
17035 let COUNTABLE_INTEGER_COORDINATES = prove
17036 (`COUNTABLE { x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }`,
17037 MATCH_MP_TAC COUNTABLE_CART THEN
17038 REWRITE_TAC[SET_RULE `{x | P x} = P`; COUNTABLE_INTEGER]);;
17040 let COUNTABLE_RATIONAL_COORDINATES = prove
17041 (`COUNTABLE { x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) }`,
17042 MATCH_MP_TAC COUNTABLE_CART THEN
17043 REWRITE_TAC[SET_RULE `{x | P x} = P`; COUNTABLE_RATIONAL]);;
17045 (* ------------------------------------------------------------------------- *)
17046 (* Density of points with rational, or just dyadic rational, coordinates. *)
17047 (* ------------------------------------------------------------------------- *)
17049 let CLOSURE_DYADIC_RATIONALS = prove
17050 (`closure { inv(&2 pow n) % x |n,x|
17051 !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) } = (:real^N)`,
17052 REWRITE_TAC[EXTENSION; CLOSURE_APPROACHABLE; IN_UNIV; EXISTS_IN_GSPEC] THEN
17053 MAP_EVERY X_GEN_TAC [`x:real^N`; `e:real`] THEN DISCH_TAC THEN
17054 MP_TAC(SPECL [`inv(&2)`; `e / &(dimindex(:N))`] REAL_ARCH_POW_INV) THEN
17055 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1;
17056 REAL_POW_INV; REAL_LT_RDIV_EQ] THEN
17057 CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC MONO_EXISTS THEN
17058 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
17059 EXISTS_TAC `(lambda i. floor(&2 pow n * (x:real^N)$i)):real^N` THEN
17060 ASM_SIMP_TAC[LAMBDA_BETA; FLOOR; dist; NORM_MUL] THEN
17061 MATCH_MP_TAC(MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS)
17062 (SPEC_ALL NORM_LE_L1)) THEN
17063 SIMP_TAC[LAMBDA_BETA; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT] THEN
17064 MATCH_MP_TAC REAL_LET_TRANS THEN
17065 EXISTS_TAC `&(dimindex(:N)) * inv(&2 pow n)` THEN ASM_REWRITE_TAC[] THEN
17066 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV) [GSYM CARD_NUMSEG_1] THEN
17067 MATCH_MP_TAC SUM_BOUND THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
17068 X_GEN_TAC `k:num` THEN STRIP_TAC THEN
17069 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN
17070 SIMP_TAC[REAL_ABS_MUL; REAL_POW_EQ_0; REAL_OF_NUM_EQ; ARITH;
17071 REAL_FIELD `~(a = &0) ==> inv a * b - x = inv a * (b - a * x)`] THEN
17072 MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[REAL_ABS_POS] THEN
17073 REWRITE_TAC[REAL_LE_REFL; REAL_ABS_POW; REAL_ABS_INV; REAL_ABS_NUM] THEN
17074 MP_TAC(SPEC `&2 pow n * (x:real^N)$k` FLOOR) THEN REAL_ARITH_TAC);;
17076 let CLOSURE_RATIONAL_COORDINATES = prove
17077 (`closure { x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) } =
17079 MATCH_MP_TAC(SET_RULE `!s. s SUBSET t /\ s = UNIV ==> t = UNIV`) THEN
17081 `closure { inv(&2 pow n) % x:real^N |n,x|
17082 !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }` THEN
17084 CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[CLOSURE_DYADIC_RATIONALS]] THEN
17085 MATCH_MP_TAC SUBSET_CLOSURE THEN
17086 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_ELIM_THM; VECTOR_MUL_COMPONENT] THEN
17087 ASM_SIMP_TAC[RATIONAL_CLOSED]);;
17089 let CLOSURE_DYADIC_RATIONALS_IN_OPEN_SET = prove
17092 ==> closure(s INTER
17093 { inv(&2 pow n) % x | n,x |
17094 !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }) =
17096 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_OPEN_INTER_SUPERSET THEN
17097 ASM_REWRITE_TAC[CLOSURE_DYADIC_RATIONALS; SUBSET_UNIV]);;
17099 let CLOSURE_RATIONALS_IN_OPEN_SET = prove
17102 ==> closure(s INTER
17103 { inv(&2 pow n) % x | n,x |
17104 !i. 1 <= i /\ i <= dimindex(:N) ==> integer(x$i) }) =
17106 REPEAT STRIP_TAC THEN MATCH_MP_TAC CLOSURE_OPEN_INTER_SUPERSET THEN
17107 ASM_REWRITE_TAC[CLOSURE_DYADIC_RATIONALS; SUBSET_UNIV]);;
17109 (* ------------------------------------------------------------------------- *)
17110 (* Various separability-type properties. *)
17111 (* ------------------------------------------------------------------------- *)
17113 let UNIV_SECOND_COUNTABLE = prove
17114 (`?b. COUNTABLE b /\ (!c. c IN b ==> open c) /\
17115 !s:real^N->bool. open s ==> ?u. u SUBSET b /\ s = UNIONS u`,
17117 `IMAGE (\(v:real^N,q). ball(v,q))
17118 ({v | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(v$i)} CROSS
17120 REPEAT CONJ_TAC THENL
17121 [MATCH_MP_TAC COUNTABLE_IMAGE THEN MATCH_MP_TAC COUNTABLE_CROSS THEN
17122 REWRITE_TAC[COUNTABLE_RATIONAL] THEN MATCH_MP_TAC COUNTABLE_CART THEN
17123 REWRITE_TAC[COUNTABLE_RATIONAL; SET_RULE `{x | P x} = P`];
17124 REWRITE_TAC[FORALL_IN_IMAGE; CROSS; FORALL_IN_GSPEC; OPEN_BALL];
17125 REPEAT STRIP_TAC THEN
17126 ASM_CASES_TAC `s:real^N->bool = {}` THENL
17127 [EXISTS_TAC `{}:(real^N->bool)->bool` THEN
17128 ASM_REWRITE_TAC[UNIONS_0; EMPTY_SUBSET];
17130 EXISTS_TAC `{c | c IN IMAGE (\(v:real^N,q). ball(v,q))
17131 ({v | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(v$i)} CROSS
17132 rational) /\ c SUBSET s}` THEN
17133 CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
17134 MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN
17135 REWRITE_TAC[SUBSET; IN_UNIONS; IN_ELIM_THM] THEN
17136 REWRITE_TAC[GSYM CONJ_ASSOC; EXISTS_IN_IMAGE] THEN
17137 REWRITE_TAC[CROSS; EXISTS_PAIR_THM; EXISTS_IN_GSPEC] THEN
17138 REWRITE_TAC[IN_ELIM_PAIR_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
17139 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
17140 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
17141 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM; SUBSET; IN_BALL] THEN
17142 X_GEN_TAC `e:real` THEN STRIP_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
17143 MP_TAC(REWRITE_RULE[EXTENSION; IN_UNIV] CLOSURE_RATIONAL_COORDINATES) THEN
17144 REWRITE_TAC[CLOSURE_APPROACHABLE] THEN
17145 DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `e / &4`]) THEN
17146 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_ELIM_THM]] THEN
17147 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
17148 SUBGOAL_THEN `?x. rational x /\ e / &3 < x /\ x < e / &2`
17149 (X_CHOOSE_THEN `q:real` STRIP_ASSUME_TAC)
17151 [MP_TAC(ISPECL [`&5 / &12 * e`; `e / &12`] RATIONAL_APPROXIMATION) THEN
17152 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN
17153 SIMP_TAC[] THEN REAL_ARITH_TAC;
17154 EXISTS_TAC `q:real` THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
17155 [ASM_REWRITE_TAC[IN];
17156 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
17157 REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC;
17158 ASM_REAL_ARITH_TAC]]]);;
17160 let UNIV_SECOND_COUNTABLE_SEQUENCE = prove
17161 (`?b:num->real^N->bool.
17162 (!m n. b m = b n <=> m = n) /\
17164 (!s. open s ==> ?k. s = UNIONS {b n | n IN k})`,
17165 X_CHOOSE_THEN `bb:(real^N->bool)->bool` STRIP_ASSUME_TAC
17166 UNIV_SECOND_COUNTABLE THEN
17167 MP_TAC(ISPEC `bb:(real^N->bool)->bool` COUNTABLE_AS_INJECTIVE_IMAGE) THEN
17169 [ASM_REWRITE_TAC[INFINITE] THEN DISCH_TAC THEN
17171 `INFINITE {ball(vec 0:real^N,inv(&n + &1)) | n IN (:num)}`
17173 [REWRITE_TAC[SIMPLE_IMAGE] THEN MATCH_MP_TAC(REWRITE_RULE
17174 [RIGHT_IMP_FORALL_THM; IMP_IMP] INFINITE_IMAGE_INJ) THEN
17175 REWRITE_TAC[num_INFINITE] THEN MATCH_MP_TAC WLOG_LT THEN SIMP_TAC[] THEN
17176 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
17177 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN
17178 REWRITE_TAC[EXTENSION] THEN
17179 DISCH_THEN(MP_TAC o SPEC `inv(&n + &1) % basis 1:real^N`) THEN
17180 REWRITE_TAC[IN_BALL; DIST_0; NORM_MUL; REAL_ABS_INV] THEN
17181 SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL; REAL_MUL_RID] THEN
17182 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN
17183 REWRITE_TAC[REAL_ARITH `abs(&n + &1) = &n + &1`; REAL_LT_REFL] THEN
17184 MATCH_MP_TAC REAL_LT_INV2 THEN
17185 REWRITE_TAC[REAL_OF_NUM_LT; REAL_OF_NUM_ADD] THEN ASM_ARITH_TAC;
17186 REWRITE_TAC[INFINITE; SIMPLE_IMAGE] THEN
17187 MATCH_MP_TAC FINITE_SUBSET THEN
17188 EXISTS_TAC `IMAGE UNIONS {u | u SUBSET bb} :(real^N->bool)->bool` THEN
17189 ASM_SIMP_TAC[FINITE_IMAGE; FINITE_POWERSET] THEN
17190 GEN_REWRITE_TAC I [SUBSET] THEN SIMP_TAC[FORALL_IN_IMAGE; IN_UNIV] THEN
17191 X_GEN_TAC `n:num` THEN REWRITE_TAC[IN_IMAGE; IN_ELIM_THM] THEN
17192 ASM_MESON_TAC[OPEN_BALL]];
17193 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:num->real^N->bool` THEN
17194 DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
17195 RULE_ASSUM_TAC(REWRITE_RULE[FORALL_IN_IMAGE; IN_UNIV]) THEN
17196 REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC]) THEN
17197 X_GEN_TAC `s:real^N->bool` THEN DISCH_TAC THEN
17198 FIRST_X_ASSUM(MP_TAC o SPEC `s:real^N->bool`) THEN
17199 ASM_REWRITE_TAC[SUBSET_IMAGE; LEFT_AND_EXISTS_THM; SUBSET_UNIV] THEN
17200 ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
17201 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[SIMPLE_IMAGE]]);;
17203 let SUBSET_SECOND_COUNTABLE = prove
17206 (!c. c IN b ==> ~(c = {}) /\ open_in(subtopology euclidean s) c) /\
17207 !t. open_in(subtopology euclidean s) t
17208 ==> ?u. u SUBSET b /\ t = UNIONS u`,
17211 `?b. COUNTABLE b /\
17212 (!c:real^N->bool. c IN b ==> open_in(subtopology euclidean s) c) /\
17213 !t. open_in(subtopology euclidean s) t
17214 ==> ?u. u SUBSET b /\ t = UNIONS u`
17215 STRIP_ASSUME_TAC THENL
17216 [X_CHOOSE_THEN `B:(real^N->bool)->bool` STRIP_ASSUME_TAC
17217 UNIV_SECOND_COUNTABLE THEN
17218 EXISTS_TAC `{s INTER c :real^N->bool | c IN B}` THEN
17219 ASM_SIMP_TAC[SIMPLE_IMAGE; COUNTABLE_IMAGE] THEN
17220 ASM_SIMP_TAC[FORALL_IN_IMAGE; EXISTS_SUBSET_IMAGE; OPEN_IN_OPEN_INTER] THEN
17221 REWRITE_TAC[OPEN_IN_OPEN] THEN
17222 X_GEN_TAC `t:real^N->bool` THEN
17223 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
17224 FIRST_X_ASSUM SUBST_ALL_TAC THEN
17225 SUBGOAL_THEN `?b. b SUBSET B /\ u:real^N->bool = UNIONS b`
17226 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
17227 FIRST_X_ASSUM SUBST_ALL_TAC THEN
17228 EXISTS_TAC `b:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[] THEN
17229 REWRITE_TAC[INTER_UNIONS] THEN AP_TERM_TAC THEN SET_TAC[];
17230 EXISTS_TAC `b DELETE ({}:real^N->bool)` THEN
17231 ASM_SIMP_TAC[COUNTABLE_DELETE; IN_DELETE; SUBSET_DELETE] THEN
17232 X_GEN_TAC `t:real^N->bool` THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
17233 DISCH_THEN(X_CHOOSE_THEN `u:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
17234 EXISTS_TAC `u DELETE ({}:real^N->bool)` THEN
17235 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
17236 FIRST_X_ASSUM SUBST_ALL_TAC THEN
17237 REWRITE_TAC[EXTENSION; IN_UNIONS] THEN
17238 GEN_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
17239 REWRITE_TAC[IN_DELETE] THEN SET_TAC[]]);;
17241 let SEPARABLE = prove
17243 ?t. COUNTABLE t /\ t SUBSET s /\ s SUBSET closure t`,
17244 MP_TAC SUBSET_SECOND_COUNTABLE THEN
17245 MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `s:real^N->bool` THEN
17246 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; LEFT_AND_EXISTS_THM] THEN
17247 DISCH_THEN(X_CHOOSE_THEN `B:(real^N->bool)->bool`
17248 (CONJUNCTS_THEN2 ASSUME_TAC (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC))) THEN
17249 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
17250 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
17251 X_GEN_TAC `f:(real^N->bool)->real^N` THEN DISCH_TAC THEN
17252 EXISTS_TAC `IMAGE (f:(real^N->bool)->real^N) B` THEN
17253 ASM_SIMP_TAC[COUNTABLE_IMAGE] THEN CONJ_TAC THENL
17254 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
17255 X_GEN_TAC `c:real^N->bool` THEN DISCH_TAC THEN
17256 FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN
17257 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
17258 FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_SUBSET) THEN
17259 REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN] THEN ASM SET_TAC[];
17260 REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE; EXISTS_IN_IMAGE] THEN
17261 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
17262 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
17265 open_in (subtopology euclidean s) t
17266 ==> (?u. u SUBSET B /\ t = UNIONS u)`
17267 (MP_TAC o SPEC `s INTER ball(x:real^N,e)`) THEN
17268 SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; LEFT_IMP_EXISTS_THM] THEN
17269 X_GEN_TAC `b:(real^N->bool)->bool` THEN
17270 ASM_CASES_TAC `b:(real^N->bool)->bool = {}` THENL
17271 [MATCH_MP_TAC(TAUT `~b ==> a /\ b ==> c`) THEN
17272 ASM_REWRITE_TAC[EXTENSION; IN_INTER; NOT_IN_EMPTY; UNIONS_0] THEN
17273 ASM_MESON_TAC[CENTRE_IN_BALL];
17275 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
17276 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN
17277 DISCH_TAC THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
17278 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
17279 DISCH_THEN(MP_TAC o SPEC `(f:(real^N->bool)->real^N) c`) THEN
17280 ONCE_REWRITE_TAC[DIST_SYM] THEN REWRITE_TAC[IN_INTER; IN_BALL] THEN
17281 MATCH_MP_TAC(TAUT `a /\ c ==> (a /\ b <=> c) ==> b`) THEN
17282 CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
17283 FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN
17284 ANTS_TAC THENL [ASM SET_TAC[]; STRIP_TAC] THEN
17285 FIRST_X_ASSUM(MP_TAC o MATCH_MP OPEN_IN_SUBSET) THEN
17286 REWRITE_TAC[TOPSPACE_SUBTOPOLOGY; TOPSPACE_EUCLIDEAN] THEN
17289 let OPEN_SET_RATIONAL_COORDINATES = prove
17290 (`!s. open s /\ ~(s = {})
17291 ==> ?x:real^N. x IN s /\
17292 !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)`,
17293 REPEAT STRIP_TAC THEN
17295 `~(closure { x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) } INTER
17296 (s:real^N->bool) = {})`
17298 [ASM_REWRITE_TAC[CLOSURE_RATIONAL_COORDINATES; INTER_UNIV]; ALL_TAC] THEN
17299 REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; CLOSURE_APPROACHABLE; IN_INTER;
17301 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
17302 FIRST_X_ASSUM(MP_TAC o SPEC `a:real^N` o REWRITE_RULE[open_def]) THEN
17303 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);;
17305 let OPEN_COUNTABLE_UNION_OPEN_INTERVALS,
17306 OPEN_COUNTABLE_UNION_CLOSED_INTERVALS = (CONJ_PAIR o prove)
17307 (`(!s:real^N->bool.
17309 ==> ?D. COUNTABLE D /\
17310 (!i. i IN D ==> i SUBSET s /\ ?a b. i = interval(a,b)) /\
17314 ==> ?D. COUNTABLE D /\
17315 (!i. i IN D ==> i SUBSET s /\ ?a b. i = interval[a,b]) /\
17317 REPEAT STRIP_TAC THENL
17319 `{i | i IN IMAGE (\(a:real^N,b). interval(a,b))
17320 ({x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)} CROSS
17321 {x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)}) /\
17324 `{i | i IN IMAGE (\(a:real^N,b). interval[a,b])
17325 ({x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)} CROSS
17326 {x | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i)}) /\
17328 (SIMP_TAC[COUNTABLE_RESTRICT; COUNTABLE_IMAGE; COUNTABLE_CROSS;
17329 COUNTABLE_RATIONAL_COORDINATES] THEN
17330 REWRITE_TAC[IN_ELIM_THM; UNIONS_GSPEC; IMP_CONJ; GSYM CONJ_ASSOC] THEN
17331 REWRITE_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN
17332 REWRITE_TAC[FORALL_PAIR_THM; EXISTS_PAIR_THM; IN_CROSS; IN_ELIM_THM] THEN
17333 CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
17334 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
17335 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL [SET_TAC[]; DISCH_TAC] THEN
17336 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o REWRITE_RULE[open_def]) THEN
17337 ASM_REWRITE_TAC[] THEN
17338 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
17340 `!i. 1 <= i /\ i <= dimindex(:N)
17341 ==> ?a b. rational a /\ rational b /\
17342 a < (x:real^N)$i /\ (x:real^N)$i < b /\
17343 abs(b - a) < e / &(dimindex(:N))`
17345 [REPEAT STRIP_TAC THEN MATCH_MP_TAC RATIONAL_APPROXIMATION_STRADDLE THEN
17346 ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1];
17347 REWRITE_TAC[LAMBDA_SKOLEM]] THEN
17348 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
17349 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N` THEN
17350 DISCH_TAC THEN ASM_SIMP_TAC[SUBSET; IN_INTERVAL; REAL_LT_IMP_LE] THEN
17351 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
17352 REWRITE_TAC[dist] THEN MP_TAC(ISPEC `y - x:real^N` NORM_LE_L1) THEN
17353 MATCH_MP_TAC(REAL_ARITH `s < e ==> n <= s ==> n < e`) THEN
17354 MATCH_MP_TAC SUM_BOUND_LT_GEN THEN
17355 REWRITE_TAC[FINITE_NUMSEG; NUMSEG_EMPTY; NOT_LT; CARD_NUMSEG_1] THEN
17356 REWRITE_TAC[DIMINDEX_GE_1; IN_NUMSEG; VECTOR_SUB_COMPONENT] THEN
17357 X_GEN_TAC `k:num` THEN STRIP_TAC THEN
17358 REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `k:num`)) THEN ASM_REWRITE_TAC[] THEN
17359 ASM_REAL_ARITH_TAC));;
17361 let LINDELOF = prove
17362 (`!f:(real^N->bool)->bool.
17363 (!s. s IN f ==> open s)
17364 ==> ?f'. f' SUBSET f /\ COUNTABLE f' /\ UNIONS f' = UNIONS f`,
17365 REPEAT STRIP_TAC THEN
17367 `?b. COUNTABLE b /\
17368 (!c:real^N->bool. c IN b ==> open c) /\
17369 (!s. open s ==> ?u. u SUBSET b /\ s = UNIONS u)`
17370 STRIP_ASSUME_TAC THENL [ASM_REWRITE_TAC[UNIV_SECOND_COUNTABLE]; ALL_TAC] THEN
17372 `d = {s:real^N->bool | s IN b /\ ?u. u IN f /\ s SUBSET u}` THEN
17374 `COUNTABLE d /\ UNIONS f :real^N->bool = UNIONS d`
17375 STRIP_ASSUME_TAC THENL
17376 [EXPAND_TAC "d" THEN ASM_SIMP_TAC[COUNTABLE_RESTRICT] THEN ASM SET_TAC[];
17379 `!s:real^N->bool. ?u. s IN d ==> u IN f /\ s SUBSET u`
17380 MP_TAC THENL [EXPAND_TAC "d" THEN SET_TAC[]; ALL_TAC] THEN
17381 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
17382 X_GEN_TAC `g:(real^N->bool)->(real^N->bool)` THEN STRIP_TAC THEN
17383 EXISTS_TAC `IMAGE (g:(real^N->bool)->(real^N->bool)) d` THEN
17384 ASM_SIMP_TAC[COUNTABLE_IMAGE; UNIONS_IMAGE] THEN
17385 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN ASM SET_TAC[]);;
17387 let LINDELOF_OPEN_IN = prove
17388 (`!f u:real^N->bool.
17389 (!s. s IN f ==> open_in (subtopology euclidean u) s)
17390 ==> ?f'. f' SUBSET f /\ COUNTABLE f' /\ UNIONS f' = UNIONS f`,
17391 REPEAT GEN_TAC THEN REWRITE_TAC[OPEN_IN_OPEN] THEN
17392 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
17393 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
17394 X_GEN_TAC `v:(real^N->bool)->real^N->bool` THEN DISCH_TAC THEN
17395 MP_TAC(ISPEC `IMAGE (v:(real^N->bool)->real^N->bool) f` LINDELOF) THEN
17396 ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN
17397 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
17398 REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN
17399 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f':(real^N->bool)->bool` THEN
17400 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
17402 `!f'. f' SUBSET f ==> UNIONS f' = (u:real^N->bool) INTER UNIONS (IMAGE v f')`
17403 MP_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[SUBSET_REFL]]);;
17405 let COUNTABLE_DISJOINT_OPEN_SUBSETS = prove
17406 (`!f. (!s:real^N->bool. s IN f ==> open s) /\ pairwise DISJOINT f
17408 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LINDELOF) THEN
17409 DISCH_THEN(X_CHOOSE_THEN `g:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
17410 MATCH_MP_TAC COUNTABLE_SUBSET THEN
17411 EXISTS_TAC `({}:real^N->bool) INSERT g` THEN
17412 ASM_REWRITE_TAC[COUNTABLE_INSERT] THEN
17413 REWRITE_TAC[SUBSET; IN_INSERT] THEN
17414 REPEAT(POP_ASSUM MP_TAC) THEN
17415 REWRITE_TAC[EXTENSION; SUBSET] THEN
17416 REWRITE_TAC[IN_UNIONS; pairwise] THEN
17417 REWRITE_TAC[SET_RULE `DISJOINT s t <=> !x. ~(x IN s /\ x IN t)`] THEN
17418 REWRITE_TAC[NOT_IN_EMPTY] THEN MESON_TAC[]);;
17420 let CARD_EQ_OPEN_SETS = prove
17421 (`{s:real^N->bool | open s} =_c (:real)`,
17422 REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL
17423 [X_CHOOSE_THEN `b:(real^N->bool)->bool` STRIP_ASSUME_TAC
17424 UNIV_SECOND_COUNTABLE THEN
17425 TRANS_TAC CARD_LE_TRANS `{s:(real^N->bool)->bool | s SUBSET b}` THEN
17427 [REWRITE_TAC[LE_C] THEN
17428 EXISTS_TAC `UNIONS:((real^N->bool)->bool)->real^N->bool` THEN
17429 REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[];
17430 TRANS_TAC CARD_LE_TRANS `{s | s SUBSET (:num)}` THEN CONJ_TAC THENL
17431 [MATCH_MP_TAC CARD_LE_POWERSET THEN ASM_REWRITE_TAC[GSYM COUNTABLE_ALT];
17432 REWRITE_TAC[SUBSET_UNIV; UNIV_GSPEC] THEN
17433 MESON_TAC[CARD_EQ_IMP_LE; CARD_EQ_SYM; CARD_EQ_REAL]]];
17434 REWRITE_TAC[le_c; IN_UNIV; IN_ELIM_THM] THEN
17435 EXISTS_TAC `\x. ball(x % basis 1:real^N,&1)` THEN
17436 REWRITE_TAC[OPEN_BALL; GSYM SUBSET_ANTISYM_EQ; SUBSET_BALLS] THEN
17437 CONV_TAC REAL_RAT_REDUCE_CONV THEN
17438 REWRITE_TAC[NORM_ARITH `dist(p:real^N,q) + &1 <= &1 <=> p = q`] THEN
17439 REWRITE_TAC[VECTOR_MUL_RCANCEL; EQ_SYM_EQ] THEN
17440 SIMP_TAC[BASIS_NONZERO; DIMINDEX_GE_1; ARITH]]);;
17442 let CARD_EQ_CLOSED_SETS = prove
17443 (`{s:real^N->bool | closed s} =_c (:real)`,
17445 `{s:real^N->bool | closed s} =
17446 IMAGE (\s. (:real^N) DIFF s) {s | open s}`
17448 [CONV_TAC SYM_CONV THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
17449 REWRITE_TAC[IN_ELIM_THM; GSYM OPEN_CLOSED] THEN
17450 MESON_TAC[SET_RULE `UNIV DIFF (UNIV DIFF s) = s`];
17451 TRANS_TAC CARD_EQ_TRANS `{s:real^N->bool | open s}` THEN
17452 REWRITE_TAC[CARD_EQ_OPEN_SETS] THEN
17453 MATCH_MP_TAC CARD_EQ_IMAGE THEN SET_TAC[]]);;
17455 let CARD_EQ_COMPACT_SETS = prove
17456 (`{s:real^N->bool | compact s} =_c (:real)`,
17457 REWRITE_TAC[GSYM CARD_LE_ANTISYM] THEN CONJ_TAC THENL
17458 [TRANS_TAC CARD_LE_TRANS `{s:real^N->bool | closed s}` THEN
17459 SIMP_TAC[CARD_EQ_IMP_LE; CARD_EQ_CLOSED_SETS] THEN
17460 MATCH_MP_TAC CARD_LE_SUBSET THEN
17461 SIMP_TAC[SUBSET; IN_ELIM_THM; COMPACT_IMP_CLOSED];
17462 REWRITE_TAC[le_c; IN_UNIV; IN_ELIM_THM] THEN
17463 EXISTS_TAC `\x. {x % basis 1:real^N}` THEN
17464 REWRITE_TAC[COMPACT_SING; SET_RULE `{x} = {y} <=> x = y`] THEN
17465 SIMP_TAC[VECTOR_MUL_RCANCEL; BASIS_NONZERO; DIMINDEX_GE_1; ARITH]]);;
17467 let COUNTABLE_NON_CONDENSATION_POINTS = prove
17468 (`!s:real^N->bool. COUNTABLE(s DIFF {x | x condensation_point_of s})`,
17469 REPEAT STRIP_TAC THEN REWRITE_TAC[condensation_point_of] THEN
17470 MATCH_MP_TAC COUNTABLE_SUBSET THEN
17471 X_CHOOSE_THEN `b:(real^N->bool)->bool` STRIP_ASSUME_TAC
17472 UNIV_SECOND_COUNTABLE THEN
17474 `s INTER UNIONS { u:real^N->bool | u IN b /\ COUNTABLE(s INTER u)}` THEN
17475 REWRITE_TAC[INTER_UNIONS; IN_ELIM_THM] THEN CONJ_TAC THENL
17476 [MATCH_MP_TAC COUNTABLE_UNIONS THEN SIMP_TAC[FORALL_IN_GSPEC] THEN
17477 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
17478 ASM_SIMP_TAC[COUNTABLE_IMAGE; COUNTABLE_RESTRICT];
17479 SIMP_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM; IN_INTER; IN_DIFF] THEN
17480 X_GEN_TAC `x:real^N` THEN
17481 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
17482 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; LEFT_IMP_EXISTS_THM] THEN
17483 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
17484 SUBGOAL_THEN `?u:real^N->bool. x IN u /\ u IN b /\ u SUBSET t` MP_TAC THENL
17485 [ASM SET_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN
17486 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
17487 MATCH_MP_TAC COUNTABLE_SUBSET THEN
17488 EXISTS_TAC `s INTER t:real^N->bool` THEN ASM SET_TAC[]]);;
17490 let CARD_EQ_CONDENSATION_POINTS_IN_SET = prove
17492 ~(COUNTABLE s) ==> {x | x IN s /\ x condensation_point_of s} =_c s`,
17493 REPEAT STRIP_TAC THEN
17494 TRANS_TAC CARD_EQ_TRANS
17495 `(s DIFF {x | x condensation_point_of s}) +_c
17496 {x:real^N | x IN s /\ x condensation_point_of s}` THEN
17498 [ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN MATCH_MP_TAC CARD_ADD_ABSORB THEN
17499 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
17500 [POP_ASSUM MP_TAC THEN REWRITE_TAC[INFINITE; CONTRAPOS_THM] THEN
17501 DISCH_THEN(MP_TAC o CONJ (SPEC `s:real^N->bool`
17502 COUNTABLE_NON_CONDENSATION_POINTS) o MATCH_MP FINITE_IMP_COUNTABLE) THEN
17503 REWRITE_TAC[GSYM COUNTABLE_UNION] THEN MATCH_MP_TAC EQ_IMP THEN
17504 AP_TERM_TAC THEN SET_TAC[];
17505 REWRITE_TAC[INFINITE_CARD_LE] THEN
17506 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] CARD_LE_TRANS) THEN
17507 REWRITE_TAC[GSYM COUNTABLE_ALT; COUNTABLE_NON_CONDENSATION_POINTS]];
17508 ONCE_REWRITE_TAC[CARD_EQ_SYM] THEN
17509 W(MP_TAC o PART_MATCH (rand o rand) CARD_DISJOINT_UNION o rand o snd) THEN
17510 ANTS_TAC THENL [SET_TAC[]; ALL_TAC] THEN
17511 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN SET_TAC[]]);;
17513 let LIMPT_OF_CONDENSATION_POINTS,CONDENSATION_POINT_OF_CONDENSATION_POINTS =
17514 (CONJ_PAIR o prove)
17516 x limit_point_of {y | y condensation_point_of s} <=>
17517 x condensation_point_of s) /\
17519 x condensation_point_of {y | y condensation_point_of s} <=>
17520 x condensation_point_of s)`,
17521 REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT
17522 `(r ==> q) /\ (q ==> p) /\ (p ==> r)
17523 ==> (q <=> p) /\ (r <=> p)`) THEN
17524 REWRITE_TAC[CONDENSATION_POINT_IMP_LIMPT] THEN CONJ_TAC THENL
17525 [REWRITE_TAC[LIMPT_APPROACHABLE; CONDENSATION_POINT_INFINITE_BALL] THEN
17526 REPEAT GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN
17527 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
17528 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
17529 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
17530 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
17531 ASM_REWRITE_TAC[REAL_HALF; CONTRAPOS_THM] THEN
17532 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COUNTABLE_SUBSET) THEN
17533 SIMP_TAC[SUBSET; IN_INTER; IN_BALL] THEN
17534 REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC;
17535 ONCE_REWRITE_TAC[CONDENSATION_POINT_INFINITE_BALL] THEN DISCH_TAC THEN
17536 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
17537 FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
17538 ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(MP_TAC o MATCH_MP
17539 (MESON[CARD_EQ_CONDENSATION_POINTS_IN_SET; CARD_COUNTABLE_CONG]
17541 ==> ~COUNTABLE {x | x IN s /\ x condensation_point_of s}`)) THEN
17542 REWRITE_TAC[UNCOUNTABLE_REAL; CONTRAPOS_THM] THEN
17543 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] COUNTABLE_SUBSET) THEN
17544 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INTER] THEN X_GEN_TAC `y:real^N` THEN
17545 REPEAT STRIP_TAC THENL
17546 [ASM_MESON_TAC[CONDENSATION_POINT_OF_SUBSET; INTER_SUBSET]; ALL_TAC] THEN
17547 MATCH_MP_TAC(SET_RULE `!s. x IN s /\ s SUBSET t ==> x IN t`) THEN
17548 EXISTS_TAC `closure(s INTER ball(x:real^N,e / &2))` THEN CONJ_TAC THENL
17549 [REWRITE_TAC[closure; IN_UNION; IN_ELIM_THM] THEN DISJ2_TAC THEN
17550 ASM_SIMP_TAC[CONDENSATION_POINT_IMP_LIMPT];
17551 TRANS_TAC SUBSET_TRANS `closure(ball(x:real^N,e / &2))` THEN
17552 SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET] THEN
17553 ASM_SIMP_TAC[CLOSURE_BALL; REAL_HALF; SUBSET_BALLS; DIST_REFL] THEN
17554 ASM_REAL_ARITH_TAC]]);;
17556 let CLOSED_CONDENSATION_POINTS = prove
17557 (`!s:real^N->bool. closed {x | x condensation_point_of s}`,
17558 SIMP_TAC[CLOSED_LIMPT; LIMPT_OF_CONDENSATION_POINTS; IN_ELIM_THM]);;
17560 let CANTOR_BENDIXSON = prove
17563 ==> ?t u. closed t /\ (!x. x IN t ==> x limit_point_of t) /\
17564 COUNTABLE u /\ s = t UNION u`,
17565 REPEAT STRIP_TAC THEN MAP_EVERY EXISTS_TAC
17566 [`{x:real^N | x condensation_point_of s}`;
17567 `s DIFF {x:real^N | x condensation_point_of s}`] THEN
17568 REWRITE_TAC[COUNTABLE_NON_CONDENSATION_POINTS; CLOSED_CONDENSATION_POINTS;
17569 IN_ELIM_THM; LIMPT_OF_CONDENSATION_POINTS] THEN
17570 REWRITE_TAC[SET_RULE `s = t UNION (s DIFF t) <=> t SUBSET s`] THEN
17571 RULE_ASSUM_TAC(REWRITE_RULE[CLOSED_LIMPT]) THEN
17572 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
17573 ASM_MESON_TAC[CONDENSATION_POINT_IMP_LIMPT]);;
17575 (* ------------------------------------------------------------------------- *)
17576 (* A discrete set is countable, and an uncountable set has a limit point. *)
17577 (* ------------------------------------------------------------------------- *)
17579 let DISCRETE_IMP_COUNTABLE = prove
17581 (!x. x IN s ==> ?e. &0 < e /\
17582 !y. y IN s /\ ~(y = x) ==> e <= norm(y - x))
17584 REPEAT STRIP_TAC THEN
17587 ==> ?q. (!i. 1 <= i /\ i <= dimindex(:N) ==> rational(q$i)) /\
17588 !y:real^N. y IN s /\ ~(y = x) ==> norm(x - q) < norm(y - q)`
17590 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
17591 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
17592 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
17593 MP_TAC(SET_RULE `x IN (:real^N)`) THEN
17594 REWRITE_TAC[GSYM CLOSURE_RATIONAL_COORDINATES] THEN
17595 REWRITE_TAC[CLOSURE_APPROACHABLE; IN_ELIM_THM] THEN
17596 DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
17597 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `q:real^N` THEN
17598 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
17599 X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
17600 FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN
17601 REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC;
17602 POP_ASSUM(K ALL_TAC) THEN
17603 REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
17604 X_GEN_TAC `q:real^N->real^N` THEN DISCH_TAC THEN
17607 `{ x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) }`;
17608 `(:num)`] CARD_LE_TRANS) THEN
17609 REWRITE_TAC[COUNTABLE; ge_c] THEN DISCH_THEN MATCH_MP_TAC THEN
17610 SIMP_TAC[REWRITE_RULE[COUNTABLE; ge_c] COUNTABLE_RATIONAL_COORDINATES] THEN
17611 REWRITE_TAC[le_c] THEN EXISTS_TAC `q:real^N->real^N` THEN
17612 ASM_SIMP_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LT_ANTISYM]]);;
17614 let UNCOUNTABLE_CONTAINS_LIMIT_POINT = prove
17615 (`!s. ~(COUNTABLE s) ==> ?x. x IN s /\ x limit_point_of s`,
17616 GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP
17617 (ONCE_REWRITE_RULE[GSYM CONTRAPOS_THM] DISCRETE_IMP_COUNTABLE)) THEN
17618 REWRITE_TAC[LIMPT_APPROACHABLE; GSYM REAL_NOT_LT; dist] THEN
17621 (* ------------------------------------------------------------------------- *)
17622 (* The Brouwer reduction theorem. *)
17623 (* ------------------------------------------------------------------------- *)
17625 let BROUWER_REDUCTION_THEOREM_GEN = prove
17626 (`!P s:real^N->bool.
17627 (!f. (!n. closed(f n) /\ P(f n)) /\ (!n. f(SUC n) SUBSET f(n))
17628 ==> P(INTERS {f n | n IN (:num)})) /\
17630 ==> ?t. t SUBSET s /\ closed t /\ P t /\
17631 (!u. u SUBSET s /\ closed u /\ P u ==> ~(u PSUBSET t))`,
17632 REPEAT STRIP_TAC THEN
17634 `?b:num->real^N->bool.
17635 (!m n. b m = b n <=> m = n) /\
17636 (!n. open (b n)) /\
17637 (!s. open s ==> (?k. s = UNIONS {b n | n IN k}))`
17638 STRIP_ASSUME_TAC THENL
17639 [REWRITE_TAC[UNIV_SECOND_COUNTABLE_SEQUENCE]; ALL_TAC] THEN
17640 X_CHOOSE_THEN `a:num->real^N->bool` MP_TAC
17641 (prove_recursive_functions_exist num_RECURSION
17642 `a 0 = (s:real^N->bool) /\
17644 if ?u. u SUBSET a(n) /\ closed u /\ P u /\ u INTER (b n) = {}
17645 then @u. u SUBSET a(n) /\ closed u /\ P u /\ u INTER (b n) = {}
17647 DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "base") (LABEL_TAC "step")) THEN
17648 EXISTS_TAC `INTERS {a n :real^N->bool | n IN (:num)}` THEN
17649 SUBGOAL_THEN `!n. (a:num->real^N->bool)(SUC n) SUBSET a(n)` ASSUME_TAC THENL
17650 [GEN_TAC THEN ASM_REWRITE_TAC[] THEN
17651 COND_CASES_TAC THEN REWRITE_TAC[SUBSET_REFL] THEN
17652 FIRST_X_ASSUM(MP_TAC o SELECT_RULE) THEN MESON_TAC[];
17654 SUBGOAL_THEN `!n. (a:num->real^N->bool) n SUBSET s` ASSUME_TAC THENL
17655 [INDUCT_TAC THEN ASM_MESON_TAC[SUBSET_REFL; SUBSET_TRANS]; ALL_TAC] THEN
17656 SUBGOAL_THEN `!n. closed((a:num->real^N->bool) n) /\ P(a n)` ASSUME_TAC THENL
17657 [INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN
17658 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
17659 FIRST_X_ASSUM(MP_TAC o SELECT_RULE) THEN MESON_TAC[];
17661 REPEAT CONJ_TAC THENL
17663 MATCH_MP_TAC CLOSED_INTERS THEN
17664 ASM_REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN SET_TAC[];
17665 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
17666 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
17667 REWRITE_TAC[PSUBSET_ALT] THEN
17668 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
17669 REWRITE_TAC[INTERS_GSPEC; EXISTS_IN_GSPEC; IN_UNIV] THEN
17670 DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
17672 `?n. x IN (b:num->real^N->bool)(n) /\ t INTER b n = {}`
17673 STRIP_ASSUME_TAC THENL
17674 [MP_TAC(ISPEC `(:real^N) DIFF t` OPEN_CONTAINS_BALL) THEN
17675 ASM_REWRITE_TAC[GSYM closed] THEN
17676 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
17677 ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; LEFT_IMP_EXISTS_THM] THEN
17678 REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> t INTER s = {}`] THEN
17679 X_GEN_TAC `e:real` THEN
17680 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
17681 MP_TAC(ISPECL [`x:real^N`; `e:real`] CENTRE_IN_BALL) THEN
17682 FIRST_X_ASSUM(MP_TAC o SPEC `ball(x:real^N,e)`) THEN
17683 ASM_REWRITE_TAC[OPEN_BALL; LEFT_IMP_EXISTS_THM] THEN
17684 X_GEN_TAC `k:num->bool` THEN DISCH_THEN SUBST1_TAC THEN
17685 REWRITE_TAC[IN_UNIONS; INTER_UNIONS; EMPTY_UNIONS; FORALL_IN_GSPEC] THEN
17687 REMOVE_THEN "step" (MP_TAC o SPEC `n:num`) THEN
17688 COND_CASES_TAC THENL
17689 [DISCH_THEN(ASSUME_TAC o SYM) THEN
17690 FIRST_X_ASSUM(MP_TAC o SELECT_RULE) THEN ASM_REWRITE_TAC[] THEN
17692 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
17693 DISCH_THEN(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN
17694 ASM SET_TAC[]]]]);;
17696 let BROUWER_REDUCTION_THEOREM = prove
17697 (`!P s:real^N->bool.
17698 (!f. (!n. compact(f n) /\ ~(f n = {}) /\ P(f n)) /\
17699 (!n. f(SUC n) SUBSET f(n))
17700 ==> P(INTERS {f n | n IN (:num)})) /\
17701 compact s /\ ~(s = {}) /\ P s
17702 ==> ?t. t SUBSET s /\ compact t /\ ~(t = {}) /\ P t /\
17703 (!u. u SUBSET s /\ closed u /\ ~(u = {}) /\ P u
17704 ==> ~(u PSUBSET t))`,
17705 REPEAT STRIP_TAC THEN
17706 MP_TAC(ISPECL [`\t:real^N->bool. ~(t = {}) /\ t SUBSET s /\ P t`;
17708 BROUWER_REDUCTION_THEOREM_GEN) THEN
17709 ASM_SIMP_TAC[COMPACT_IMP_CLOSED; SUBSET_REFL] THEN ANTS_TAC THENL
17710 [GEN_TAC THEN STRIP_TAC THEN
17711 SUBGOAL_THEN `!n. compact((f:num->real^N->bool) n)` ASSUME_TAC THENL
17712 [ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET]; ALL_TAC] THEN
17713 REPEAT CONJ_TAC THENL
17714 [MATCH_MP_TAC COMPACT_NEST THEN ASM_REWRITE_TAC[] THEN
17715 MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_SIMP_TAC[] THEN SET_TAC[];
17717 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]];
17718 MATCH_MP_TAC MONO_EXISTS THEN ASM_SIMP_TAC[] THEN
17719 ASM_MESON_TAC[COMPACT_EQ_BOUNDED_CLOSED; BOUNDED_SUBSET]]);;
17721 (* ------------------------------------------------------------------------- *)
17722 (* The Arzela-Ascoli theorem. *)
17723 (* ------------------------------------------------------------------------- *)
17725 let SUBSEQUENCE_DIAGONALIZATION_LEMMA = prove
17726 (`!P:num->(num->A)->bool.
17727 (!i r:num->A. ?k. (!m n. m < n ==> k m < k n) /\ P i (r o k)) /\
17728 (!i r:num->A k1 k2 N.
17729 P i (r o k1) /\ (!j. N <= j ==> ?j'. j <= j' /\ k2 j = k1 j')
17731 ==> !r:num->A. ?k. (!m n. m < n ==> k m < k n) /\ (!i. P i (r o k))`,
17732 REPEAT GEN_TAC THEN
17733 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
17734 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [SKOLEM_THM] THEN
17735 REWRITE_TAC[FORALL_AND_THM; TAUT
17736 `(p ==> q /\ r) <=> (p ==> q) /\ (p ==> r)`] THEN
17737 DISCH_THEN(X_CHOOSE_THEN
17738 `kk:num->(num->A)->num->num` STRIP_ASSUME_TAC) THEN
17739 X_GEN_TAC `r:num->A` THEN
17740 (STRIP_ASSUME_TAC o prove_recursive_functions_exist num_RECURSION)
17741 `(rr 0 = (kk:num->(num->A)->num->num) 0 r) /\
17742 (!n. rr(SUC n) = rr n o kk (SUC n) (r o rr n))` THEN
17743 EXISTS_TAC `\n. (rr:num->num->num) n n` THEN REWRITE_TAC[ETA_AX] THEN
17745 `(!i. (!m n. m < n ==> (rr:num->num->num) i m < rr i n)) /\
17746 (!i. (P:num->(num->A)->bool) i (r o rr i))`
17747 STRIP_ASSUME_TAC THENL
17748 [REWRITE_TAC[AND_FORALL_THM] THEN
17749 INDUCT_TAC THEN ASM_REWRITE_TAC[o_ASSOC] THEN
17750 REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[];
17752 SUBGOAL_THEN `!i j n. i <= j ==> (rr:num->num->num) i n <= rr j n`
17754 [REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [LE_EXISTS] THEN
17755 SIMP_TAC[LEFT_IMP_EXISTS_THM] THEN SPEC_TAC(`j:num`,`j:num`) THEN
17756 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN SIMP_TAC[FORALL_UNWIND_THM2] THEN
17757 INDUCT_TAC THEN REWRITE_TAC[ADD_CLAUSES; LE_REFL] THEN
17758 ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
17759 (REWRITE_RULE[IMP_CONJ] LE_TRANS)) THEN REWRITE_TAC[o_THM] THEN
17760 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
17763 (!m n. m < n ==> f m < f n) ==> (!m n. m <= n ==> f m <= f n)`) o
17764 SPEC `i + d:num`) THEN
17765 SPEC_TAC(`n:num`,`n:num`) THEN MATCH_MP_TAC MONOTONE_BIGGER THEN
17769 [MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN
17770 MATCH_MP_TAC LET_TRANS THEN
17771 EXISTS_TAC `(rr:num->num->num) n m` THEN
17772 ASM_MESON_TAC[LT_IMP_LE];
17775 `!m n i. n <= m ==> ?j. i <= j /\ (rr:num->num->num) m i = rr n j`
17778 X_GEN_TAC `i:num` THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
17779 EXISTS_TAC `(rr:num->num->num) i` THEN ASM_REWRITE_TAC[] THEN
17780 EXISTS_TAC `i:num` THEN ASM_MESON_TAC[]] THEN
17782 `!p d i. ?j. i <= j /\ (rr:num->num->num) (p + d) i = rr p j`
17783 (fun th -> MESON_TAC[LE_EXISTS; th]) THEN
17784 X_GEN_TAC `p:num` THEN MATCH_MP_TAC num_INDUCTION THEN
17785 ASM_REWRITE_TAC[ADD_CLAUSES] THEN CONJ_TAC THENL
17786 [MESON_TAC[LE_REFL]; ALL_TAC] THEN
17787 X_GEN_TAC `d:num` THEN DISCH_THEN(LABEL_TAC "+") THEN
17788 X_GEN_TAC `i:num` THEN ASM_REWRITE_TAC[o_THM] THEN
17789 REMOVE_THEN "+" (MP_TAC o SPEC
17790 `(kk:num->(num->A)->num->num) (SUC(p + d))
17791 ((r:num->A) o (rr:num->num->num) (p + d)) i`) THEN
17792 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `j:num` THEN
17793 MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
17794 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LE_TRANS) THEN
17795 SPEC_TAC(`i:num`,`i:num`) THEN MATCH_MP_TAC MONOTONE_BIGGER THEN
17796 ASM_REWRITE_TAC[o_THM] THEN ASM_MESON_TAC[]);;
17798 let FUNCTION_CONVERGENT_SUBSEQUENCE = prove
17799 (`!f:num->real^M->real^N s M.
17800 COUNTABLE s /\ (!n x. x IN s ==> norm(f n x) <= M)
17801 ==> ?k. (!m n:num. m < n ==> k m < k n) /\
17802 !x. x IN s ==> ?l. ((\n. f (k n) x) --> l) sequentially`,
17803 REPEAT STRIP_TAC THEN
17804 ASM_CASES_TAC `s:real^M->bool = {}` THENL
17805 [EXISTS_TAC `\n:num. n` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY];
17807 MP_TAC(ISPEC `s:real^M->bool` COUNTABLE_AS_IMAGE) THEN
17808 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
17809 X_GEN_TAC `X:num->real^M` THEN DISCH_THEN SUBST_ALL_TAC THEN
17811 `\i r. ?l. ((\n. ((f:num->real^M->real^N) o (r:num->num)) n
17812 ((X:num->real^M) i)) --> l) sequentially`
17813 SUBSEQUENCE_DIAGONALIZATION_LEMMA) THEN
17814 REWRITE_TAC[FORALL_IN_IMAGE; o_THM; IN_UNIV] THEN
17815 ANTS_TAC THENL [ALL_TAC; DISCH_THEN MATCH_ACCEPT_TAC] THEN CONJ_TAC THENL
17816 [RULE_ASSUM_TAC(REWRITE_RULE[FORALL_IN_IMAGE; IN_UNIV]) THEN
17817 MAP_EVERY X_GEN_TAC [`i:num`; `r:num->num`] THEN
17818 MP_TAC(ISPEC `cball(vec 0:real^N,M)` compact) THEN
17819 REWRITE_TAC[COMPACT_CBALL] THEN DISCH_THEN(MP_TAC o SPEC
17820 `\n. (f:num->real^M->real^N) ((r:num->num) n) (X(i:num))`) THEN
17821 ASM_REWRITE_TAC[IN_CBALL_0; o_DEF] THEN MESON_TAC[];
17822 REPEAT GEN_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY; GE] THEN
17823 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
17824 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
17825 MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
17826 MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
17827 ASM_MESON_TAC[LE_TRANS; ARITH_RULE `MAX a b <= c <=> a <= c /\ b <= c`]]);;
17829 let ARZELA_ASCOLI = prove
17830 (`!f:num->real^M->real^N s M.
17832 (!n x. x IN s ==> norm(f n x) <= M) /\
17833 (!x e. x IN s /\ &0 < e
17835 !n y. y IN s /\ norm(x - y) < d
17836 ==> norm(f n x - f n y) < e)
17837 ==> ?g. g continuous_on s /\
17838 ?r. (!m n:num. m < n ==> r m < r n) /\
17840 ==> ?N. !n x. n >= N /\ x IN s
17841 ==> norm(f(r n) x - g x) < e`,
17842 REPEAT STRIP_TAC THEN REWRITE_TAC[GE] THEN
17843 MATCH_MP_TAC(MESON[]
17844 `(!k g. V k g ==> N g) /\ (?k. M k /\ ?g. V k g)
17845 ==> ?g. N g /\ ?k. M k /\ V k g`) THEN
17847 [MAP_EVERY X_GEN_TAC [`k:num->num`; `g:real^M->real^N`] THEN
17848 STRIP_TAC THEN MATCH_MP_TAC(ISPEC `sequentially`
17849 CONTINUOUS_UNIFORM_LIMIT) THEN
17850 EXISTS_TAC `(f:num->real^M->real^N) o (k:num->num)` THEN
17851 ASM_SIMP_TAC[EVENTUALLY_SEQUENTIALLY; o_THM; TRIVIAL_LIMIT_SEQUENTIALLY;
17852 RIGHT_IMP_FORALL_THM; IMP_IMP] THEN
17853 EXISTS_TAC `0` THEN REWRITE_TAC[continuous_on; dist] THEN
17854 ASM_MESON_TAC[NORM_SUB];
17857 [`IMAGE (f:num->real^M->real^N) (:num)`;
17859 COMPACT_UNIFORMLY_EQUICONTINUOUS) THEN
17860 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_UNIV] THEN
17862 [REWRITE_TAC[dist] THEN ONCE_REWRITE_TAC[NORM_SUB] THEN ASM_MESON_TAC[];
17863 ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(K ALL_TAC o SPEC `x:real^M`)] THEN
17864 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN
17865 REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC; dist] THEN
17866 DISCH_THEN(ASSUME_TAC o ONCE_REWRITE_RULE[NORM_SUB]) THEN
17867 REWRITE_TAC[GSYM dist; UNIFORMLY_CONVERGENT_EQ_CAUCHY] THEN
17868 X_CHOOSE_THEN `r:real^M->bool` STRIP_ASSUME_TAC
17869 (ISPEC `s:real^M->bool` SEPARABLE) THEN
17870 MP_TAC(ISPECL [`f:num->real^M->real^N`; `r:real^M->bool`; `M:real`]
17871 FUNCTION_CONVERGENT_SUBSEQUENCE) THEN
17872 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
17873 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:num->num` THEN
17874 REWRITE_TAC[CONVERGENT_EQ_CAUCHY; cauchy] THEN
17875 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*")) THEN
17876 ASM_REWRITE_TAC[] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
17877 FIRST_X_ASSUM(MP_TAC o SPEC `e / &3`) THEN
17878 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
17879 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
17880 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [COMPACT_EQ_HEINE_BOREL]) THEN
17881 DISCH_THEN(MP_TAC o SPEC `IMAGE (\x:real^M. ball(x,d)) r`) THEN
17882 REWRITE_TAC[FORALL_IN_IMAGE; OPEN_BALL] THEN
17883 ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN
17884 REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN ANTS_TAC THENL
17885 [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `closure r:real^M->bool` THEN
17886 ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE] THEN
17887 X_GEN_TAC `x:real^M` THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN
17888 ASM_REWRITE_TAC[UNIONS_IMAGE; IN_ELIM_THM; IN_BALL];
17889 DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC)] THEN
17890 REMOVE_THEN "*" MP_TAC THEN REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN
17891 GEN_REWRITE_TAC LAND_CONV [SWAP_FORALL_THM] THEN
17892 DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
17893 ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN
17894 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
17895 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
17896 X_GEN_TAC `M:real^M->num` THEN DISCH_THEN(LABEL_TAC "*") THEN
17897 MP_TAC(ISPECL [`M:real^M->num`; `t:real^M->bool`]
17898 UPPER_BOUND_FINITE_SET) THEN
17899 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
17901 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`; `x:real^M`] THEN STRIP_TAC THEN
17902 UNDISCH_TAC `s SUBSET UNIONS (IMAGE (\x:real^M. ball (x,d)) t)` THEN
17903 REWRITE_TAC[SUBSET; UNIONS_IMAGE; IN_ELIM_THM] THEN
17904 DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
17905 ASM_REWRITE_TAC[IN_BALL; LEFT_IMP_EXISTS_THM; dist] THEN
17906 X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
17907 MATCH_MP_TAC(NORM_ARITH
17908 `norm(f (k(m:num)) y - f (k m) x) < e / &3 /\
17909 norm(f (k n) y - f (k n) x) < e / &3 /\
17910 norm(f (k m) y - f (k n) y) < e / &3
17911 ==> norm(f (k m) x - f (k n) x :real^M) < e`) THEN
17912 ASM_SIMP_TAC[] THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^M`) THEN
17913 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
17914 DISCH_THEN(MP_TAC o SPECL [`m:num`; `n:num`]) THEN
17915 ASM_REWRITE_TAC[dist; GE] THEN ASM_MESON_TAC[SUBSET; LE_TRANS]);;
17917 (* ------------------------------------------------------------------------- *)
17918 (* Two forms of the Baire propery of dense sets. *)
17919 (* ------------------------------------------------------------------------- *)
17922 (`!g s:real^N->bool.
17923 closed s /\ COUNTABLE g /\
17925 ==> open_in (subtopology euclidean s) t /\ s SUBSET closure t)
17926 ==> s SUBSET closure(INTERS g)`,
17927 REPEAT STRIP_TAC THEN ASM_CASES_TAC `g:(real^N->bool)->bool = {}` THEN
17928 ASM_REWRITE_TAC[INTERS_0; CLOSURE_UNIV; SUBSET_UNIV] THEN
17929 MP_TAC(ISPEC `g:(real^N->bool)->bool` COUNTABLE_AS_IMAGE) THEN
17930 ASM_REWRITE_TAC[] THEN
17931 MAP_EVERY (C UNDISCH_THEN (K ALL_TAC))
17932 [`COUNTABLE(g:(real^N->bool)->bool)`;
17933 `~(g:(real^N->bool)->bool = {})`] THEN
17934 DISCH_THEN(X_CHOOSE_THEN `g:num->real^N->bool` SUBST_ALL_TAC) THEN
17935 RULE_ASSUM_TAC(REWRITE_RULE[FORALL_IN_IMAGE; IN_UNIV]) THEN
17936 REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE] THEN
17937 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
17938 X_GEN_TAC `e:real` THEN DISCH_TAC THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
17939 REWRITE_TAC[GSYM IN_BALL; GSYM IN_INTER; MEMBER_NOT_EMPTY] THEN
17941 `?t:num->real^N->bool.
17942 (!n. open_in (subtopology euclidean s) (t n) /\ ~(t n = {}) /\
17943 s INTER closure(t n) SUBSET g n /\
17944 closure(t n) SUBSET ball(x,e)) /\
17945 (!n. t(SUC n) SUBSET t n)`
17946 STRIP_ASSUME_TAC THENL
17948 `!u n. open_in (subtopology euclidean s) u /\ ~(u = {}) /\
17949 closure u SUBSET ball(x,e)
17950 ==> ?y. open_in (subtopology euclidean s) y /\
17952 s INTER closure y SUBSET (g:num->real^N->bool) n /\
17953 closure y SUBSET ball(x,e) /\
17956 [MAP_EVERY X_GEN_TAC [`u:real^N->bool`; `n:num`] THEN STRIP_TAC THEN
17957 SUBGOAL_THEN `?y:real^N. y IN u /\ y IN g(n:num)` STRIP_ASSUME_TAC THENL
17958 [FIRST_X_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC o SPEC `n:num`) THEN
17959 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
17960 FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP OPEN_IN_IMP_SUBSET) THEN
17961 DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
17962 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [open_in]) THEN
17963 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `y:real^N`)) THEN
17964 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN
17965 STRIP_TAC THEN REWRITE_TAC[SUBSET; CLOSURE_APPROACHABLE] THEN
17969 `open_in (subtopology euclidean s) (u INTER g(n:num):real^N->bool)`
17970 MP_TAC THENL [ASM_SIMP_TAC[OPEN_IN_INTER]; ALL_TAC] THEN
17971 GEN_REWRITE_TAC LAND_CONV [OPEN_IN_CONTAINS_BALL] THEN
17972 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `y:real^N`)) THEN
17973 ASM_REWRITE_TAC[IN_INTER] THEN
17974 DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
17975 EXISTS_TAC `s INTER ball(y:real^N,d / &2)` THEN
17976 SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL] THEN REPEAT CONJ_TAC THENL
17977 [REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `y:real^N` THEN
17978 ASM_REWRITE_TAC[CENTRE_IN_BALL; REAL_HALF; IN_INTER] THEN
17980 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
17981 `b SUBSET u INTER g ==> !s. s SUBSET b ==> s SUBSET g`)) THEN
17982 MATCH_MP_TAC(SET_RULE
17983 `closure(s INTER b) SUBSET closure b /\ closure b SUBSET c
17984 ==> s INTER closure(s INTER b) SUBSET c INTER s`) THEN
17985 SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET] THEN
17986 ASM_SIMP_TAC[CLOSURE_BALL; SUBSET_BALLS; REAL_HALF; DIST_REFL] THEN
17987 ASM_REAL_ARITH_TAC;
17988 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
17989 SUBSET_TRANS)) THEN MATCH_MP_TAC SUBSET_CLOSURE;
17991 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
17992 `b INTER s SUBSET u INTER g ==> c SUBSET b
17993 ==> s INTER c SUBSET u`)) THEN
17994 REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN ASM_REAL_ARITH_TAC;
17995 MATCH_MP_TAC DEPENDENT_CHOICE THEN ASM_SIMP_TAC[GSYM CONJ_ASSOC] THEN
17996 FIRST_X_ASSUM(MP_TAC o SPECL [`s INTER ball(x:real^N,e / &2)`; `0`]) THEN
17997 ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; GSYM MEMBER_NOT_EMPTY] THEN
17998 ANTS_TAC THENL [REWRITE_TAC[LEFT_AND_EXISTS_THM]; MESON_TAC[]] THEN
17999 EXISTS_TAC `x:real^N` THEN
18000 ASM_REWRITE_TAC[CENTRE_IN_BALL; REAL_HALF; IN_INTER] THEN
18001 TRANS_TAC SUBSET_TRANS `closure(ball(x:real^N,e / &2))` THEN
18002 SIMP_TAC[SUBSET_CLOSURE; INTER_SUBSET] THEN
18003 ASM_SIMP_TAC[CLOSURE_BALL; SUBSET_BALLS; REAL_HALF; DIST_REFL] THEN
18004 ASM_REAL_ARITH_TAC];
18006 `(\n. s INTER closure(t n)):num->real^N->bool` COMPACT_NEST) THEN
18008 [REWRITE_TAC[FORALL_AND_THM] THEN REPEAT CONJ_TAC THENL
18009 [GEN_TAC THEN MATCH_MP_TAC CLOSED_INTER_COMPACT THEN
18010 ASM_MESON_TAC[BOUNDED_SUBSET; BOUNDED_BALL; COMPACT_EQ_BOUNDED_CLOSED;
18012 GEN_TAC THEN MATCH_MP_TAC(SET_RULE
18013 `~(t = {}) /\ t SUBSET s /\ t SUBSET closure t
18014 ==> ~(s INTER closure t = {})`) THEN
18015 ASM_MESON_TAC[CLOSURE_SUBSET; OPEN_IN_IMP_SUBSET];
18016 MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN
18017 ASM_SIMP_TAC[SUBSET_CLOSURE; SET_RULE
18018 `t SUBSET u ==> s INTER t SUBSET s INTER u`] THEN
18020 MATCH_MP_TAC(SET_RULE `s SUBSET t ==> ~(s = {}) ==> ~(t = {})`) THEN
18021 REWRITE_TAC[SUBSET_INTER] THEN
18022 REWRITE_TAC[SUBSET; IN_INTERS; FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN
18025 let BAIRE_ALT = prove
18026 (`!g s:real^N->bool.
18027 closed s /\ ~(s = {}) /\ COUNTABLE g /\ UNIONS g = s
18028 ==> ?t u. t IN g /\ open_in (subtopology euclidean s) u /\
18029 u SUBSET (closure t)`,
18030 REPEAT STRIP_TAC THEN MP_TAC(ISPECL
18031 [`IMAGE (\t:real^N->bool. s DIFF closure t) g`; `s:real^N->bool`] BAIRE) THEN
18032 ASM_SIMP_TAC[COUNTABLE_IMAGE; FORALL_IN_IMAGE] THEN
18033 MATCH_MP_TAC(TAUT `~q /\ (~r ==> p) ==> (p ==> q) ==> r`) THEN
18035 [MATCH_MP_TAC(SET_RULE
18036 `~(s = {}) /\ (t = {} ==> closure t = {}) /\ t = {}
18037 ==> ~(s SUBSET closure t)`) THEN
18038 ASM_SIMP_TAC[CLOSURE_EMPTY] THEN
18039 MATCH_MP_TAC(SET_RULE `i SUBSET s /\ s DIFF i = s ==> i = {}`) THEN
18040 CONJ_TAC THENL [REWRITE_TAC[INTERS_IMAGE] THEN ASM SET_TAC[]; ALL_TAC] THEN
18041 REWRITE_TAC[DIFF_INTERS] THEN
18042 REWRITE_TAC[SET_RULE `{f x | x IN IMAGE g s} = {f(g x) | x IN s}`] THEN
18043 REWRITE_TAC[SET_RULE `s DIFF (s DIFF t) = s INTER t`] THEN
18044 REWRITE_TAC[SET_RULE `{s INTER closure t | t IN g} =
18045 {s INTER t | t IN IMAGE closure g}`] THEN
18046 SIMP_TAC[GSYM INTER_UNIONS; SET_RULE `s INTER t = s <=> s SUBSET t`] THEN
18047 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
18048 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM IMAGE_ID] THEN
18049 MATCH_MP_TAC UNIONS_MONO_IMAGE THEN REWRITE_TAC[CLOSURE_SUBSET];
18050 REWRITE_TAC[NOT_EXISTS_THM] THEN STRIP_TAC THEN
18051 X_GEN_TAC `t:real^N->bool` THEN REPEAT STRIP_TAC THENL
18052 [ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s DIFF (s INTER t)`] THEN
18053 MATCH_MP_TAC OPEN_IN_DIFF THEN
18054 ASM_SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_CLOSURE; OPEN_IN_REFL];
18055 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18056 REWRITE_TAC[CLOSURE_APPROACHABLE] THEN
18057 X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
18058 [`t:real^N->bool`; `s INTER ball(x:real^N,e)`]) THEN
18059 ASM_SIMP_TAC[OPEN_IN_OPEN_INTER; OPEN_BALL; SUBSET; IN_INTER; IN_BALL;
18061 MESON_TAC[DIST_SYM]]]);;
18063 (* ------------------------------------------------------------------------- *)
18064 (* Several variants of paracompactness. *)
18065 (* ------------------------------------------------------------------------- *)
18067 let PARACOMPACT = prove
18068 (`!s c. (!t:real^N->bool. t IN c ==> open t) /\ s SUBSET UNIONS c
18069 ==> ?c'. s SUBSET UNIONS c' /\
18071 ==> open u /\ ?t. t IN c /\ u SUBSET t) /\
18073 ==> ?v. open v /\ x IN v /\
18074 FINITE {u | u IN c' /\ ~(u INTER v = {})})`,
18075 REPEAT STRIP_TAC THEN
18076 ASM_CASES_TAC `s:real^N->bool = {}` THENL
18077 [EXISTS_TAC `{}:(real^N->bool)->bool` THEN
18078 ASM_REWRITE_TAC[EMPTY_SUBSET; NOT_IN_EMPTY];
18082 ==> ?t u. x IN u /\ open u /\ closure u SUBSET t /\ t IN c`
18084 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18085 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [SUBSET]) THEN
18086 ASM_REWRITE_TAC[IN_UNIONS] THEN MATCH_MP_TAC MONO_EXISTS THEN
18087 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
18088 FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN
18089 ASM_REWRITE_TAC[] THEN
18090 GEN_REWRITE_TAC LAND_CONV [OPEN_CONTAINS_CBALL] THEN
18091 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
18092 DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
18093 EXISTS_TAC `ball(x:real^N,e)` THEN
18094 ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL; CLOSURE_BALL];
18095 GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
18096 REWRITE_TAC[LEFT_IMP_EXISTS_THM; SKOLEM_THM] THEN
18097 MAP_EVERY X_GEN_TAC
18098 [`f:real^N->real^N->bool`; `e:real^N->real^N->bool`] THEN
18100 MP_TAC(ISPEC `IMAGE (e:real^N->real^N->bool) s` LINDELOF) THEN
18101 ASM_SIMP_TAC[FORALL_IN_IMAGE] THEN
18102 ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
18103 REWRITE_TAC[EXISTS_COUNTABLE_SUBSET_IMAGE] THEN
18104 DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool`
18105 (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
18106 ASM_CASES_TAC `k:real^N->bool = {}` THENL
18107 [ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN
18108 MP_TAC(ISPEC `k:real^N->bool` COUNTABLE_AS_IMAGE) THEN
18109 ASM_REWRITE_TAC[] THEN
18110 DISCH_THEN(X_CHOOSE_THEN `a:num->real^N` SUBST_ALL_TAC) THEN
18111 STRIP_TAC THEN EXISTS_TAC
18112 `{ f(a n:real^N) DIFF UNIONS {closure(e(a m)):real^N->bool | m < n} |
18114 REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN REPEAT CONJ_TAC THENL
18115 [X_GEN_TAC `n:num` THEN CONJ_TAC THENL
18116 [MATCH_MP_TAC OPEN_DIFF THEN
18117 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
18118 MATCH_MP_TAC CLOSED_UNIONS THEN
18119 REWRITE_TAC[FORALL_IN_GSPEC; CLOSED_CLOSURE] THEN
18120 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
18121 SIMP_TAC[FINITE_IMAGE; FINITE_NUMSEG_LT];
18122 EXISTS_TAC `f((a:num->real^N) n):real^N->bool` THEN ASM SET_TAC[]];
18123 REWRITE_TAC[SUBSET; UNIONS_GSPEC; IN_ELIM_THM; IN_DIFF] THEN
18124 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18125 SUBGOAL_THEN `?n. x IN (f((a:num->real^N) n):real^N->bool)` MP_TAC THENL
18126 [RULE_ASSUM_TAC(REWRITE_RULE[UNIONS_IMAGE; EXISTS_IN_IMAGE]) THEN
18127 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
18128 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
18129 ASM_REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN
18130 DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN
18131 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
18132 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:num` THEN
18134 FIRST_X_ASSUM(MP_TAC o SPEC `(a:num->real^N) n`) THEN
18135 ANTS_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]];
18136 GEN_REWRITE_TAC LAND_CONV [num_WOP] THEN
18137 MATCH_MP_TAC MONO_EXISTS THEN ASM SET_TAC[]];
18138 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18139 RULE_ASSUM_TAC(REWRITE_RULE[UNIONS_IMAGE; EXISTS_IN_IMAGE]) THEN
18140 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
18141 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
18142 ASM_REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN
18143 DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN
18144 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
18145 DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN
18146 EXISTS_TAC `e((a:num->real^N) n):real^N->bool` THEN
18147 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
18148 REWRITE_TAC[SET_RULE
18149 `{u | (?n. u = f n) /\ P u} = IMAGE f {n |n| P(f n) /\ n IN (:num)}`] THEN
18150 MATCH_MP_TAC FINITE_IMAGE THEN MATCH_MP_TAC FINITE_SUBSET THEN
18151 EXISTS_TAC `{m:num | m <= n}` THEN REWRITE_TAC[FINITE_NUMSEG_LE] THEN
18152 REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_UNIV] THEN
18153 X_GEN_TAC `m:num` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
18154 REWRITE_TAC[NOT_LE] THEN DISCH_TAC THEN
18155 MATCH_MP_TAC(SET_RULE `u SUBSET t ==> (s DIFF t) INTER u = {}`) THEN
18156 REWRITE_TAC[SUBSET; IN_UNIONS; EXISTS_IN_GSPEC] THEN
18157 ASM_MESON_TAC[CLOSURE_SUBSET; SUBSET]]);;
18159 let PARACOMPACT_CLOSED_IN = prove
18160 (`!u:real^N->bool s c.
18161 closed_in (subtopology euclidean u) s /\
18162 (!t:real^N->bool. t IN c ==> open_in (subtopology euclidean u) t) /\
18164 ==> ?c'. s SUBSET UNIONS c' /\
18166 ==> open_in (subtopology euclidean u) v /\
18167 ?t. t IN c /\ v SUBSET t) /\
18169 ==> ?v. open_in (subtopology euclidean u) v /\ x IN v /\
18170 FINITE {n | n IN c' /\ ~(n INTER v = {})})`,
18171 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
18172 (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
18173 REWRITE_TAC[OPEN_IN_OPEN] THEN
18174 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
18175 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
18176 X_GEN_TAC `uu:(real^N->bool)->(real^N->bool)` THEN
18177 DISCH_THEN(ASSUME_TAC o GSYM) THEN
18178 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CLOSED_IN_CLOSED]) THEN
18179 DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool`
18180 (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
18183 `((:real^N) DIFF k) INSERT IMAGE (uu:(real^N->bool)->(real^N->bool)) c`]
18185 ASM_SIMP_TAC[FORALL_IN_IMAGE; UNIONS_IMAGE; UNIONS_INSERT; FORALL_IN_INSERT;
18186 EXISTS_IN_IMAGE; EXISTS_IN_INSERT; GSYM closed] THEN
18187 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
18188 DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
18189 EXISTS_TAC `{u INTER v:real^N->bool | v IN d /\ ~(v INTER k = {})}` THEN
18190 REPEAT CONJ_TAC THENL
18191 [REWRITE_TAC[UNIONS_GSPEC] THEN ASM SET_TAC[];
18192 REWRITE_TAC[FORALL_IN_GSPEC] THEN ASM SET_TAC[];
18193 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18194 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
18195 DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN
18196 EXISTS_TAC `u INTER v:real^N->bool` THEN ASM_REWRITE_TAC[IN_INTER] THEN
18197 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
18198 ONCE_REWRITE_TAC[SET_RULE
18199 `{y | y IN {f x | P x} /\ Q y} = IMAGE f {x | P x /\ Q(f x)}`] THEN
18200 MATCH_MP_TAC FINITE_IMAGE THEN
18201 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
18202 (REWRITE_RULE[IMP_CONJ] FINITE_SUBSET)) THEN SET_TAC[]]);;
18204 let PARACOMPACT_CLOSED = prove
18205 (`!s:real^N->bool c.
18206 closed s /\ (!t:real^N->bool. t IN c ==> open t) /\ s SUBSET UNIONS c
18207 ==> ?c'. s SUBSET UNIONS c' /\
18208 (!u. u IN c' ==> open u /\ ?t. t IN c /\ u SUBSET t) /\
18209 (!x. ?v. open v /\ x IN v /\
18210 FINITE {u | u IN c' /\ ~(u INTER v = {})})`,
18211 REPEAT STRIP_TAC THEN
18212 MP_TAC(ISPECL [`(:real^N)`; `s:real^N->bool`; `c:(real^N->bool)->bool`]
18213 PARACOMPACT_CLOSED_IN) THEN
18214 ASM_REWRITE_TAC[SUBTOPOLOGY_UNIV; GSYM OPEN_IN; GSYM CLOSED_IN; IN_UNIV]);;
18216 (* ------------------------------------------------------------------------- *)
18217 (* Partitions of unity subordinate to locally finite open coverings. *)
18218 (* ------------------------------------------------------------------------- *)
18220 let SUBORDINATE_PARTITION_OF_UNITY = prove
18221 (`!c s. s SUBSET UNIONS c /\ (!u. u IN c ==> open u) /\
18223 ==> ?v. open v /\ x IN v /\
18224 FINITE {u | u IN c /\ ~(u INTER v = {})})
18225 ==> ?f:(real^N->bool)->real^N->real.
18227 ==> (lift o f u) continuous_on s /\
18228 !x. x IN s ==> &0 <= f u x) /\
18229 (!x u. u IN c /\ x IN s /\ ~(x IN u) ==> f u x = &0) /\
18230 (!x. x IN s ==> sum c (\u. f u x) = &1) /\
18232 ==> ?n. open n /\ x IN n /\
18233 FINITE {u | u IN c /\
18234 ~(!x. x IN n ==> f u x = &0)})`,
18235 REPEAT STRIP_TAC THEN
18236 ASM_CASES_TAC `?u:real^N->bool. u IN c /\ s SUBSET u` THENL
18237 [FIRST_X_ASSUM(CHOOSE_THEN STRIP_ASSUME_TAC) THEN
18238 EXISTS_TAC `\v:real^N->bool x:real^N. if v = u then &1 else &0` THEN
18239 REWRITE_TAC[COND_RAND; COND_RATOR; o_DEF; REAL_POS;
18240 REAL_OF_NUM_EQ; ARITH_EQ;
18241 MESON[] `(if p then q else T) <=> p ==> q`] THEN
18242 ASM_SIMP_TAC[CONTINUOUS_ON_CONST; COND_ID; SUM_DELTA] THEN
18243 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
18244 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18245 EXISTS_TAC `ball(x:real^N,&1)` THEN
18246 REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN
18247 MATCH_MP_TAC FINITE_SUBSET THEN EXISTS_TAC `{u:real^N->bool}` THEN
18248 REWRITE_TAC[FINITE_SING; SUBSET; IN_ELIM_THM; IN_SING] THEN
18249 X_GEN_TAC `v:real^N->bool` THEN
18250 ASM_CASES_TAC `v:real^N->bool = u` THEN ASM_REWRITE_TAC[];
18252 EXISTS_TAC `\u:real^N->bool x:real^N.
18254 then setdist({x},s DIFF u) / sum c (\v. setdist({x},s DIFF v))
18256 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
18257 SIMP_TAC[SUM_POS_LE; SETDIST_POS_LE; REAL_LE_DIV] THEN
18258 SIMP_TAC[SETDIST_SING_IN_SET; IN_DIFF; real_div; REAL_MUL_LZERO] THEN
18259 REWRITE_TAC[SUM_RMUL] THEN REWRITE_TAC[GSYM real_div] THEN
18260 MATCH_MP_TAC(TAUT `r /\ p /\ q ==> p /\ q /\ r`) THEN CONJ_TAC THENL
18261 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18262 FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
18263 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:real^N->bool` THEN
18264 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
18265 ASM_REWRITE_TAC[] THEN
18266 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN
18267 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `u:real^N->bool` THEN
18268 ASM_CASES_TAC `(u:real^N->bool) IN c` THEN
18269 ASM_REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_TAC THEN
18270 X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
18271 REWRITE_TAC[real_div; REAL_ENTIRE] THEN
18272 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
18273 ASM_CASES_TAC `(y:real^N) IN u` THEN
18274 ASM_SIMP_TAC[SETDIST_SING_IN_SET; IN_DIFF; REAL_MUL_LZERO] THEN
18278 `!v x:real^N. v IN c /\ x IN s /\ x IN v ==> &0 < setdist({x},s DIFF v)`
18280 [REPEAT STRIP_TAC THEN
18281 SIMP_TAC[SETDIST_POS_LE; REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN
18282 MP_TAC(ISPECL [`s:real^N->bool`; `s DIFF v:real^N->bool`; `x:real^N`]
18283 SETDIST_EQ_0_CLOSED_IN) THEN
18284 ONCE_REWRITE_TAC[SET_RULE `s DIFF t = s INTER (UNIV DIFF t)`] THEN
18285 ASM_SIMP_TAC[CLOSED_IN_CLOSED_INTER; GSYM OPEN_CLOSED] THEN
18286 DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[] THEN
18287 ASM_REWRITE_TAC[IN_INTER; IN_DIFF; IN_UNION] THEN ASM SET_TAC[];
18290 `!x:real^N. x IN s ==> &0 < sum c (\v. setdist ({x},s DIFF v))`
18292 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18293 ONCE_REWRITE_TAC[GSYM SUM_SUPPORT] THEN
18294 REWRITE_TAC[support; NEUTRAL_REAL_ADD] THEN
18295 MATCH_MP_TAC SUM_POS_LT THEN REWRITE_TAC[SETDIST_POS_LE] THEN
18297 [FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
18298 DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
18299 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
18300 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN
18301 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `u:real^N->bool` THEN
18302 ASM_CASES_TAC `(x:real^N) IN u` THEN
18303 ASM_SIMP_TAC[SETDIST_SING_IN_SET; IN_DIFF] THEN ASM SET_TAC[];
18304 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
18305 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN REWRITE_TAC[IN_UNIONS] THEN
18306 ASM_REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
18307 ASM_MESON_TAC[REAL_LT_IMP_NZ]];
18309 ASM_SIMP_TAC[REAL_LT_IMP_NZ; REAL_DIV_REFL; o_DEF] THEN
18310 X_GEN_TAC `u:real^N->bool` THEN DISCH_TAC THEN
18311 MATCH_MP_TAC CONTINUOUS_ON_EQ THEN
18312 EXISTS_TAC `\x:real^N.
18313 lift(setdist({x},s DIFF u) / sum c (\v. setdist({x},s DIFF v)))` THEN
18314 SIMP_TAC[] THEN REWRITE_TAC[real_div; LIFT_CMUL] THEN
18315 MATCH_MP_TAC CONTINUOUS_ON_MUL THEN
18316 SIMP_TAC[CONTINUOUS_ON_LIFT_SETDIST; o_DEF] THEN
18317 MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
18318 ASM_SIMP_TAC[REAL_LT_IMP_NZ; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
18319 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
18320 FIRST_X_ASSUM(fun th ->
18321 MP_TAC(SPEC `x:real^N` th) THEN ASM_REWRITE_TAC[] THEN
18322 DISCH_THEN(X_CHOOSE_THEN `n:real^N->bool` STRIP_ASSUME_TAC)) THEN
18323 MATCH_MP_TAC CONTINUOUS_TRANSFORM_WITHIN_OPEN_IN THEN
18324 MAP_EVERY EXISTS_TAC
18325 [`\x:real^N. lift(sum {v | v IN c /\ ~(v INTER n = {})}
18326 (\v. setdist({x},s DIFF v)))`;
18327 `s INTER n:real^N->bool`] THEN
18328 ASM_SIMP_TAC[IN_INTER; OPEN_IN_OPEN_INTER] THEN CONJ_TAC THENL
18329 [X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN AP_TERM_TAC THEN
18330 CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_EQ_SUPERSET THEN
18331 ASM_REWRITE_TAC[SUBSET_RESTRICT] THEN X_GEN_TAC `v:real^N->bool` THEN
18332 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
18333 ASM_REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN
18334 MATCH_MP_TAC SETDIST_SING_IN_SET THEN ASM SET_TAC[];
18335 ASM_SIMP_TAC[LIFT_SUM; o_DEF] THEN MATCH_MP_TAC CONTINUOUS_VSUM THEN
18336 ASM_SIMP_TAC[CONTINUOUS_AT_LIFT_SETDIST; CONTINUOUS_AT_WITHIN]]);;