1 (* ========================================================================= *)
2 (* Real vectors in Euclidean space, and elementary linear algebra. *)
4 (* (c) Copyright, John Harrison 1998-2008 *)
5 (* ========================================================================= *)
7 (* Feb 2, 2012, patch of a file not yet committed to SVN.
8 Sent by email by Harrison. Delete in a few weeks after Harrison commits changes *)
10 needs "Multivariate/misc.ml";;
12 (* ------------------------------------------------------------------------- *)
13 (* Some common special cases. *)
14 (* ------------------------------------------------------------------------- *)
17 (`(!i. 1 <= i /\ i <= 1 ==> P i) <=> P 1`,
18 MESON_TAC[LE_ANTISYM]);;
21 (`!P. (!i. 1 <= i /\ i <= 2 ==> P i) <=> P 1 /\ P 2`,
22 MESON_TAC[ARITH_RULE `1 <= i /\ i <= 2 <=> i = 1 \/ i = 2`]);;
25 (`!P. (!i. 1 <= i /\ i <= 3 ==> P i) <=> P 1 /\ P 2 /\ P 3`,
26 MESON_TAC[ARITH_RULE `1 <= i /\ i <= 3 <=> i = 1 \/ i = 2 \/ i = 3`]);;
29 (`sum(1..1) f = f(1)`,
30 REWRITE_TAC[SUM_SING_NUMSEG]);;
33 (`!t. sum(1..2) t = t(1) + t(2)`,
34 REWRITE_TAC[num_CONV `2`; SUM_CLAUSES_NUMSEG] THEN
35 REWRITE_TAC[SUM_SING_NUMSEG; ARITH; REAL_ADD_ASSOC]);;
38 (`!t. sum(1..3) t = t(1) + t(2) + t(3)`,
39 REWRITE_TAC[num_CONV `3`; num_CONV `2`; SUM_CLAUSES_NUMSEG] THEN
40 REWRITE_TAC[SUM_SING_NUMSEG; ARITH; REAL_ADD_ASSOC]);;
42 (* ------------------------------------------------------------------------- *)
43 (* Basic componentwise operations on vectors. *)
44 (* ------------------------------------------------------------------------- *)
46 let vector_add = new_definition
47 `(vector_add:real^N->real^N->real^N) x y = lambda i. x$i + y$i`;;
49 let vector_sub = new_definition
50 `(vector_sub:real^N->real^N->real^N) x y = lambda i. x$i - y$i`;;
52 let vector_neg = new_definition
53 `(vector_neg:real^N->real^N) x = lambda i. --(x$i)`;;
55 overload_interface ("+",`(vector_add):real^N->real^N->real^N`);;
56 overload_interface ("-",`(vector_sub):real^N->real^N->real^N`);;
57 overload_interface ("--",`(vector_neg):real^N->real^N`);;
61 let prioritize_vector = let ty = `:real^N` in
62 fun () -> prioritize_overload ty;;
64 (* ------------------------------------------------------------------------- *)
65 (* Also the scalar-vector multiplication. *)
66 (* ------------------------------------------------------------------------- *)
68 parse_as_infix("%",(21,"right"));;
70 let vector_mul = new_definition
71 `((%):real->real^N->real^N) c x = lambda i. c * x$i`;;
73 (* ------------------------------------------------------------------------- *)
74 (* Vectors corresponding to small naturals. Perhaps should overload "&"? *)
75 (* ------------------------------------------------------------------------- *)
77 let vec = new_definition
78 `(vec:num->real^N) n = lambda i. &n`;;
80 (* ------------------------------------------------------------------------- *)
82 (* ------------------------------------------------------------------------- *)
84 parse_as_infix("dot",(20,"right"));;
86 let dot = new_definition
87 `(x:real^N) dot (y:real^N) = sum(1..dimindex(:N)) (\i. x$i * y$i)`;;
90 (`(x:real^1) dot (y:real^1) = x$1 * y$1`,
91 REWRITE_TAC[dot; DIMINDEX_1; SUM_1]);;
94 (`(x:real^2) dot (y:real^2) = x$1 * y$1 + x$2 * y$2`,
95 REWRITE_TAC[dot; DIMINDEX_2; SUM_2]);;
98 (`(x:real^3) dot (y:real^3) = x$1 * y$1 + x$2 * y$2 + x$3 * y$3`,
99 REWRITE_TAC[dot; DIMINDEX_3; SUM_3]);;
101 (* ------------------------------------------------------------------------- *)
102 (* A naive proof procedure to lift really trivial arithmetic stuff from R. *)
103 (* ------------------------------------------------------------------------- *)
105 let VECTOR_ARITH_TAC =
106 let RENAMED_LAMBDA_BETA th =
107 if fst(dest_fun_ty(type_of(funpow 3 rand (concl th)))) = aty
108 then INST_TYPE [aty,bty; bty,aty] LAMBDA_BETA else LAMBDA_BETA in
109 POP_ASSUM_LIST(K ALL_TAC) THEN
110 REPEAT(GEN_TAC ORELSE CONJ_TAC ORELSE DISCH_TAC ORELSE EQ_TAC) THEN
111 REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
112 REWRITE_TAC[dot; GSYM SUM_ADD_NUMSEG; GSYM SUM_SUB_NUMSEG;
113 GSYM SUM_LMUL; GSYM SUM_RMUL; GSYM SUM_NEG] THEN
114 (MATCH_MP_TAC SUM_EQ_NUMSEG ORELSE MATCH_MP_TAC SUM_EQ_0_NUMSEG ORELSE
115 GEN_REWRITE_TAC ONCE_DEPTH_CONV [CART_EQ]) THEN
116 REWRITE_TAC[AND_FORALL_THM] THEN TRY EQ_TAC THEN
117 TRY(MATCH_MP_TAC MONO_FORALL) THEN GEN_TAC THEN
118 REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> a ==> b /\ c`;
119 TAUT `(a ==> b) \/ (a ==> c) <=> a ==> b \/ c`] THEN
120 TRY(MATCH_MP_TAC(TAUT `(a ==> b ==> c) ==> (a ==> b) ==> (a ==> c)`)) THEN
121 REWRITE_TAC[vector_add; vector_sub; vector_neg; vector_mul; vec] THEN
122 DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP(RENAMED_LAMBDA_BETA th) th]) THEN
125 let VECTOR_ARITH tm = prove(tm,VECTOR_ARITH_TAC);;
127 (* ------------------------------------------------------------------------- *)
128 (* Obvious "component-pushing". *)
129 (* ------------------------------------------------------------------------- *)
131 let VEC_COMPONENT = prove
132 (`!k i. (vec k :real^N)$i = &k`,
134 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k`
136 [REWRITE_TAC[FINITE_INDEX_INRANGE];
137 ASM_SIMP_TAC[vec; CART_EQ; LAMBDA_BETA]]);;
139 let VECTOR_ADD_COMPONENT = prove
140 (`!x:real^N y i. (x + y)$i = x$i + y$i`,
142 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k`
144 [REWRITE_TAC[FINITE_INDEX_INRANGE];
145 ASM_SIMP_TAC[vector_add; CART_EQ; LAMBDA_BETA]]);;
147 let VECTOR_SUB_COMPONENT = prove
148 (`!x:real^N y i. (x - y)$i = x$i - y$i`,
150 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k`
152 [REWRITE_TAC[FINITE_INDEX_INRANGE];
153 ASM_SIMP_TAC[vector_sub; CART_EQ; LAMBDA_BETA]]);;
155 let VECTOR_NEG_COMPONENT = prove
156 (`!x:real^N i. (--x)$i = --(x$i)`,
158 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k`
160 [REWRITE_TAC[FINITE_INDEX_INRANGE];
161 ASM_SIMP_TAC[vector_neg; CART_EQ; LAMBDA_BETA]]);;
163 let VECTOR_MUL_COMPONENT = prove
164 (`!c x:real^N i. (c % x)$i = c * x$i`,
166 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\ !z:real^N. z$i = z$k`
168 [REWRITE_TAC[FINITE_INDEX_INRANGE];
169 ASM_SIMP_TAC[vector_mul; CART_EQ; LAMBDA_BETA]]);;
171 let COND_COMPONENT = prove
172 (`(if b then x else y)$i = if b then x$i else y$i`,
175 (* ------------------------------------------------------------------------- *)
176 (* Some frequently useful arithmetic lemmas over vectors. *)
177 (* ------------------------------------------------------------------------- *)
179 let VECTOR_ADD_SYM = VECTOR_ARITH `!x y:real^N. x + y = y + x`;;
181 let VECTOR_ADD_LID = VECTOR_ARITH `!x. vec 0 + x = x`;;
183 let VECTOR_ADD_RID = VECTOR_ARITH `!x. x + vec 0 = x`;;
185 let VECTOR_SUB_REFL = VECTOR_ARITH `!x. x - x = vec 0`;;
187 let VECTOR_ADD_LINV = VECTOR_ARITH `!x. --x + x = vec 0`;;
189 let VECTOR_ADD_RINV = VECTOR_ARITH `!x. x + --x = vec 0`;;
191 let VECTOR_SUB_RADD = VECTOR_ARITH `!x y. x - (x + y) = --y:real^N`;;
193 let VECTOR_NEG_SUB = VECTOR_ARITH `!x:real^N y. --(x - y) = y - x`;;
195 let VECTOR_SUB_EQ = VECTOR_ARITH `!x y. (x - y = vec 0) <=> (x = y)`;;
197 let VECTOR_MUL_ASSOC = VECTOR_ARITH `!a b x. a % (b % x) = (a * b) % x`;;
199 let VECTOR_MUL_LID = VECTOR_ARITH `!x. &1 % x = x`;;
201 let VECTOR_MUL_LZERO = VECTOR_ARITH `!x. &0 % x = vec 0`;;
203 let VECTOR_SUB_ADD = VECTOR_ARITH `(x - y) + y = x:real^N`;;
205 let VECTOR_SUB_ADD2 = VECTOR_ARITH `y + (x - y) = x:real^N`;;
207 let VECTOR_ADD_LDISTRIB = VECTOR_ARITH `c % (x + y) = c % x + c % y`;;
209 let VECTOR_SUB_LDISTRIB = VECTOR_ARITH `c % (x - y) = c % x - c % y`;;
211 let VECTOR_ADD_RDISTRIB = VECTOR_ARITH `(a + b) % x = a % x + b % x`;;
213 let VECTOR_SUB_RDISTRIB = VECTOR_ARITH `(a - b) % x = a % x - b % x`;;
215 let VECTOR_ADD_SUB = VECTOR_ARITH `(x + y:real^N) - x = y`;;
217 let VECTOR_EQ_ADDR = VECTOR_ARITH `(x + y = x) <=> (y = vec 0)`;;
219 let VECTOR_SUB = VECTOR_ARITH `x - y = x + --(y:real^N)`;;
221 let VECTOR_SUB_RZERO = VECTOR_ARITH `x - vec 0 = x`;;
223 let VECTOR_MUL_RZERO = VECTOR_ARITH `c % vec 0 = vec 0`;;
225 let VECTOR_NEG_MINUS1 = VECTOR_ARITH `--x = (--(&1)) % x`;;
227 let VECTOR_ADD_ASSOC = VECTOR_ARITH `(x:real^N) + y + z = (x + y) + z`;;
229 let VECTOR_SUB_LZERO = VECTOR_ARITH `vec 0 - x = --x`;;
231 let VECTOR_NEG_NEG = VECTOR_ARITH `--(--(x:real^N)) = x`;;
233 let VECTOR_MUL_LNEG = VECTOR_ARITH `--c % x = --(c % x)`;;
235 let VECTOR_MUL_RNEG = VECTOR_ARITH `c % --x = --(c % x)`;;
237 let VECTOR_NEG_0 = VECTOR_ARITH `--(vec 0) = vec 0`;;
239 let VECTOR_NEG_EQ_0 = VECTOR_ARITH `--x = vec 0 <=> x = vec 0`;;
241 let VECTOR_ADD_AC = VECTOR_ARITH
242 `(m + n = n + m:real^N) /\
243 ((m + n) + p = m + n + p) /\
244 (m + n + p = n + m + p)`;;
247 (`!m n. (vec m = vec n) <=> (m = n)`,
248 SIMP_TAC[CART_EQ; VEC_COMPONENT; REAL_OF_NUM_EQ] THEN
249 MESON_TAC[LE_REFL; DIMINDEX_GE_1]);;
251 (* ------------------------------------------------------------------------- *)
252 (* Infinitude of Euclidean space. *)
253 (* ------------------------------------------------------------------------- *)
255 let EUCLIDEAN_SPACE_INFINITE = prove
256 (`INFINITE(:real^N)`,
257 REWRITE_TAC[INFINITE] THEN DISCH_TAC THEN
258 FIRST_ASSUM(MP_TAC o ISPEC `vec:num->real^N` o
259 MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] FINITE_IMAGE_INJ)) THEN
260 REWRITE_TAC[VEC_EQ; SET_RULE `{x | f x IN UNIV} = UNIV`] THEN
261 REWRITE_TAC[GSYM INFINITE; num_INFINITE]);;
263 (* ------------------------------------------------------------------------- *)
264 (* Properties of the dot product. *)
265 (* ------------------------------------------------------------------------- *)
267 let DOT_SYM = VECTOR_ARITH `!x y. x dot y = y dot x`;;
269 let DOT_LADD = VECTOR_ARITH `!x y z. (x + y) dot z = (x dot z) + (y dot z)`;;
271 let DOT_RADD = VECTOR_ARITH `!x y z. x dot (y + z) = (x dot y) + (x dot z)`;;
273 let DOT_LSUB = VECTOR_ARITH `!x y z. (x - y) dot z = (x dot z) - (y dot z)`;;
275 let DOT_RSUB = VECTOR_ARITH `!x y z. x dot (y - z) = (x dot y) - (x dot z)`;;
277 let DOT_LMUL = VECTOR_ARITH `!c x y. (c % x) dot y = c * (x dot y)`;;
279 let DOT_RMUL = VECTOR_ARITH `!c x y. x dot (c % y) = c * (x dot y)`;;
281 let DOT_LNEG = VECTOR_ARITH `!x y. (--x) dot y = --(x dot y)`;;
283 let DOT_RNEG = VECTOR_ARITH `!x y. x dot (--y) = --(x dot y)`;;
285 let DOT_LZERO = VECTOR_ARITH `!x. (vec 0) dot x = &0`;;
287 let DOT_RZERO = VECTOR_ARITH `!x. x dot (vec 0) = &0`;;
289 let DOT_POS_LE = prove
290 (`!x. &0 <= x dot x`,
291 SIMP_TAC[dot; SUM_POS_LE_NUMSEG; REAL_LE_SQUARE]);;
294 (`!x:real^N. ((x dot x = &0) <=> (x = vec 0))`,
295 REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[DOT_LZERO]] THEN
296 SIMP_TAC[dot; CART_EQ; vec; LAMBDA_BETA] THEN DISCH_TAC THEN
297 ONCE_REWRITE_TAC[GSYM(REWRITE_CONV[REAL_ENTIRE] `x * x = &0`)] THEN
298 MATCH_MP_TAC SUM_POS_EQ_0_NUMSEG THEN ASM_REWRITE_TAC[REAL_LE_SQUARE]);;
300 let DOT_POS_LT = prove
301 (`!x. (&0 < x dot x) <=> ~(x = vec 0)`,
302 REWRITE_TAC[REAL_LT_LE; DOT_POS_LE] THEN MESON_TAC[DOT_EQ_0]);;
304 let FORALL_DOT_EQ_0 = prove
305 (`(!y. (!x. x dot y = &0) <=> y = vec 0) /\
306 (!x. (!y. x dot y = &0) <=> x = vec 0)`,
307 MESON_TAC[DOT_LZERO; DOT_RZERO; DOT_EQ_0]);;
309 (* ------------------------------------------------------------------------- *)
310 (* Introduce norms, but defer many properties till we get square roots. *)
311 (* ------------------------------------------------------------------------- *)
313 make_overloadable "norm" `:A->real`;;
314 overload_interface("norm",`vector_norm:real^N->real`);;
316 let vector_norm = new_definition
317 `norm x = sqrt(x dot x)`;;
319 (* ------------------------------------------------------------------------- *)
320 (* Useful for the special cases of 1 dimension. *)
321 (* ------------------------------------------------------------------------- *)
323 let FORALL_DIMINDEX_1 = prove
324 (`(!i. 1 <= i /\ i <= dimindex(:1) ==> P i) <=> P 1`,
325 MESON_TAC[DIMINDEX_1; LE_ANTISYM]);;
327 (* ------------------------------------------------------------------------- *)
328 (* The collapse of the general concepts to the real line R^1. *)
329 (* ------------------------------------------------------------------------- *)
331 let VECTOR_ONE = prove
332 (`!x:real^1. x = lambda i. x$1`,
333 SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN MESON_TAC[DIMINDEX_1; LE_ANTISYM]);;
335 let FORALL_REAL_ONE = prove
336 (`(!x:real^1. P x) <=> (!x. P(lambda i. x))`,
337 EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN GEN_TAC THEN
338 FIRST_X_ASSUM(MP_TAC o SPEC `(x:real^1)$1`) THEN
339 REWRITE_TAC[GSYM VECTOR_ONE]);;
341 let NORM_REAL = prove
342 (`!x:real^1. norm(x) = abs(x$1)`,
343 REWRITE_TAC[vector_norm; dot; DIMINDEX_1; SUM_SING_NUMSEG;
344 GSYM REAL_POW_2; POW_2_SQRT_ABS]);;
346 (* ------------------------------------------------------------------------- *)
347 (* Metric function. *)
348 (* ------------------------------------------------------------------------- *)
350 override_interface("dist",`distance:real^N#real^N->real`);;
352 let dist = new_definition
353 `dist(x,y) = norm(x - y)`;;
355 let DIST_REAL = prove
356 (`!x:real^1 y. dist(x,y) = abs(x$1 - y$1)`,
357 SIMP_TAC[dist; NORM_REAL; vector_sub; LAMBDA_BETA; LE_REFL; DIMINDEX_1]);;
359 (* ------------------------------------------------------------------------- *)
360 (* A connectedness or intermediate value lemma with several applications. *)
361 (* ------------------------------------------------------------------------- *)
363 let CONNECTED_REAL_LEMMA = prove
364 (`!f:real->real^N a b e1 e2.
365 a <= b /\ f(a) IN e1 /\ f(b) IN e2 /\
366 (!e x. a <= x /\ x <= b /\ &0 < e
368 !y. abs(y - x) < d ==> dist(f(y),f(x)) < e) /\
369 (!y. y IN e1 ==> ?e. &0 < e /\ !y'. dist(y',y) < e ==> y' IN e1) /\
370 (!y. y IN e2 ==> ?e. &0 < e /\ !y'. dist(y',y) < e ==> y' IN e2) /\
371 ~(?x. a <= x /\ x <= b /\ f(x) IN e1 /\ f(x) IN e2)
372 ==> ?x. a <= x /\ x <= b /\ ~(f(x) IN e1) /\ ~(f(x) IN e2)`,
373 let tac = ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_TOTAL; REAL_LE_ANTISYM] in
374 REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN
375 MP_TAC(SPEC `\c. !x. a <= x /\ x <= c ==> (f(x):real^N) IN e1`
377 REWRITE_TAC[] THEN ANTS_TAC THENL [tac; ALL_TAC] THEN
378 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real` THEN STRIP_TAC THEN
379 SUBGOAL_THEN `a <= x /\ x <= b` STRIP_ASSUME_TAC THENL [tac; ALL_TAC] THEN
380 ASM_REWRITE_TAC[] THEN
381 SUBGOAL_THEN `!z. a <= z /\ z < x ==> (f(z):real^N) IN e1` ASSUME_TAC THENL
382 [ASM_MESON_TAC[REAL_NOT_LT; REAL_LT_IMP_LE]; ALL_TAC] THEN
383 REPEAT STRIP_TAC THENL
385 `?d. &0 < d /\ !y. abs(y - x) < d ==> (f(y):real^N) IN e1`
386 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
387 ASM_MESON_TAC[REAL_ARITH `z <= x + e /\ e < d ==> z < x \/ abs(z - x) < d`;
388 REAL_ARITH `&0 < e ==> ~(x + e <= x)`; REAL_DOWN];
390 `?d. &0 < d /\ !y. abs(y - x) < d ==> (f(y):real^N) IN e2`
391 STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
392 MP_TAC(SPECL [`x - a`; `d:real`] REAL_DOWN2) THEN ANTS_TAC THENL
393 [ASM_MESON_TAC[REAL_LT_LE; REAL_SUB_LT]; ALL_TAC] THEN
394 ASM_MESON_TAC[REAL_ARITH `e < x - a ==> a <= x - e`;
395 REAL_ARITH `&0 < e /\ x <= b ==> x - e <= b`;
396 REAL_ARITH `&0 < e /\ e < d ==> x - e < x /\ abs((x - e) - x) < d`]]);;
398 (* ------------------------------------------------------------------------- *)
399 (* One immediately useful corollary is the existence of square roots! *)
400 (* ------------------------------------------------------------------------- *)
402 let SQUARE_BOUND_LEMMA = prove
403 (`!x. x < (&1 + x) * (&1 + x)`,
404 GEN_TAC THEN REWRITE_TAC[REAL_POW_2] THEN
405 MAP_EVERY (fun t -> MP_TAC(SPEC t REAL_LE_SQUARE)) [`x:real`; `&1 + x`] THEN
408 let SQUARE_CONTINUOUS = prove
410 ==> ?d. &0 < d /\ !y. abs(y - x) < d ==> abs(y * y - x * x) < e`,
411 REPEAT STRIP_TAC THEN ASM_CASES_TAC `x = &0` THENL
412 [ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_SUB_RZERO] THEN
413 EXISTS_TAC `inv(&1 + inv(e))` THEN
414 ASM_SIMP_TAC[REAL_LT_INV_EQ; REAL_LT_ADD; REAL_LT_01] THEN
415 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
416 EXISTS_TAC `inv(&1 + inv(e)) * inv(&1 + inv(e))` THEN
417 ASM_SIMP_TAC[REAL_ABS_MUL; REAL_LT_MUL2; REAL_ABS_POS] THEN
418 REWRITE_TAC[GSYM REAL_INV_MUL] THEN
419 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN
420 MATCH_MP_TAC REAL_LE_INV2 THEN
421 ASM_SIMP_TAC[REAL_LT_IMP_LE; SQUARE_BOUND_LEMMA; REAL_LT_INV_EQ];
422 MP_TAC(SPECL [`abs(x)`; `e / (&3 * abs(x))`] REAL_DOWN2)THEN
423 ASM_SIMP_TAC[GSYM REAL_ABS_NZ; REAL_LT_DIV; REAL_LT_MUL; REAL_OF_NUM_LT;
424 ARITH; REAL_LT_RDIV_EQ] THEN
425 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
426 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real` THEN
427 REWRITE_TAC[REAL_ARITH `x * x - y * y = (x - y) * (x + y)`] THEN
428 DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN
429 EXISTS_TAC `d * &3 * abs(x)` THEN ASM_REWRITE_TAC[REAL_ABS_MUL] THEN
430 MATCH_MP_TAC REAL_LE_MUL2 THEN
431 ASM_SIMP_TAC[REAL_ABS_POS; REAL_LT_IMP_LE] THEN
432 MAP_EVERY UNDISCH_TAC [`abs (y - x) < d`; `d < abs(x)`] THEN
435 let SQRT_WORKS = prove
436 (`!x. &0 <= x ==> &0 <= sqrt(x) /\ (sqrt(x) pow 2 = x)`,
437 GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN STRIP_TAC THENL
439 ASM_MESON_TAC[SQRT_0; REAL_POW_2; REAL_LE_REFL; REAL_MUL_LZERO]] THEN
440 REWRITE_TAC[sqrt] THEN CONV_TAC SELECT_CONV THEN
441 MP_TAC(ISPECL [`(\u. lambda i. u):real->real^1`; `&0`; `&1 + x`;
442 `{u:real^1 | u$1 * u$1 < x}`; `{u:real^1 | u$1 * u$1 > x}`]
443 CONNECTED_REAL_LEMMA) THEN
444 SIMP_TAC[LAMBDA_BETA; LE_REFL; DIMINDEX_1; DIST_REAL;
445 EXTENSION; IN_INTER; IN_ELIM_THM; NOT_IN_EMPTY;
446 REAL_MUL_LZERO; FORALL_REAL_ONE; real_gt] THEN
447 ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_POW_2; REAL_LT_TOTAL]] THEN
448 ASM_SIMP_TAC[REAL_LT_ANTISYM; REAL_ARITH `&0 < x ==> &0 <= &1 + x`] THEN
449 REWRITE_TAC[SQUARE_BOUND_LEMMA] THEN
450 MESON_TAC[SQUARE_CONTINUOUS; REAL_SUB_LT;
451 REAL_ARITH `abs(z2 - x2) < y - x2 ==> z2 < y`;
452 REAL_ARITH `abs(z2 - x2) < x2 - y ==> y < z2`]);;
454 let SQRT_POS_LE = prove
455 (`!x. &0 <= x ==> &0 <= sqrt(x)`,
456 MESON_TAC[SQRT_WORKS]);;
458 let SQRT_POW_2 = prove
459 (`!x. &0 <= x ==> (sqrt(x) pow 2 = x)`,
460 MESON_TAC[SQRT_WORKS]);;
463 (`!x y. &0 <= x /\ &0 <= y
464 ==> (sqrt(x * y) = sqrt x * sqrt y)`,
465 ASM_MESON_TAC[REAL_POW_2; SQRT_WORKS; REAL_LE_MUL; SQRT_UNIQUE;
466 REAL_ARITH `(x * y) * (x * y) = (x * x) * y * y`]);;
469 (`!x. &0 <= x ==> (sqrt (inv x) = inv(sqrt x))`,
470 MESON_TAC[SQRT_UNIQUE; SQRT_WORKS; REAL_POW_INV; REAL_LE_INV_EQ]);;
473 (`!x y. &0 <= x /\ &0 <= y ==> (sqrt (x / y) = sqrt x / sqrt y)`,
474 SIMP_TAC[real_div; SQRT_MUL; SQRT_INV; REAL_LE_INV_EQ]);;
476 let SQRT_POW2 = prove
477 (`!x. (sqrt(x) pow 2 = x) <=> &0 <= x`,
478 MESON_TAC[REAL_POW_2; REAL_LE_SQUARE; SQRT_POW_2]);;
480 let SQRT_MONO_LT = prove
481 (`!x y. &0 <= x /\ x < y ==> sqrt(x) < sqrt(y)`,
482 REWRITE_TAC[GSYM REAL_NOT_LE] THEN
483 MESON_TAC[REAL_LT_IMP_LE; REAL_NOT_LE; REAL_LE_TRANS;
484 REAL_POW_LE2; SQRT_WORKS]);;
486 let SQRT_MONO_LE = prove
487 (`!x y. &0 <= x /\ x <= y ==> sqrt(x) <= sqrt(y)`,
488 MESON_TAC[REAL_LE_LT; SQRT_MONO_LT]);;
490 let SQRT_MONO_LT_EQ = prove
491 (`!x y. &0 <= x /\ &0 <= y ==> (sqrt(x) < sqrt(y) <=> x < y)`,
492 MESON_TAC[REAL_NOT_LT; SQRT_MONO_LT; SQRT_MONO_LE]);;
494 let SQRT_MONO_LE_EQ = prove
495 (`!x y. &0 <= x /\ &0 <= y ==> (sqrt(x) <= sqrt(y) <=> x <= y)`,
496 MESON_TAC[REAL_NOT_LT; SQRT_MONO_LT; SQRT_MONO_LE]);;
499 (`!x y. &0 <= x /\ &0 <= y ==> ((sqrt(x) = sqrt(y)) <=> (x = y))`,
500 SIMP_TAC[GSYM REAL_LE_ANTISYM; SQRT_MONO_LE_EQ]);;
502 let SQRT_LT_0 = prove
503 (`!x. &0 <= x ==> (&0 < sqrt x <=> &0 < x)`,
504 MESON_TAC[SQRT_0; REAL_LE_REFL; SQRT_MONO_LT_EQ]);;
506 let SQRT_EQ_0 = prove
507 (`!x. &0 <= x ==> ((sqrt x = &0) <=> (x = &0))`,
508 MESON_TAC[SQRT_INJ; SQRT_0; REAL_LE_REFL]);;
510 let SQRT_POS_LT = prove
511 (`!x. &0 < x ==> &0 < sqrt(x)`,
512 MESON_TAC[REAL_LT_LE; SQRT_POS_LE; SQRT_EQ_0]);;
514 let REAL_LE_LSQRT = prove
515 (`!x y. &0 <= x /\ &0 <= y /\ x <= y pow 2 ==> sqrt(x) <= y`,
516 MESON_TAC[SQRT_MONO_LE; REAL_POW_LE; POW_2_SQRT]);;
518 let REAL_LE_RSQRT = prove
519 (`!x y. x pow 2 <= y ==> x <= sqrt(y)`,
520 MESON_TAC[REAL_LE_TOTAL; SQRT_MONO_LE; SQRT_POS_LE; REAL_POW_2;
521 REAL_LE_SQUARE; REAL_LE_TRANS; POW_2_SQRT]);;
523 let REAL_LT_RSQRT = prove
524 (`!x y. x pow 2 < y ==> x < sqrt(y)`,
525 REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `abs x < a ==> x < a`) THEN
526 REWRITE_TAC[GSYM POW_2_SQRT_ABS] THEN MATCH_MP_TAC SQRT_MONO_LT THEN
527 ASM_REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE]);;
529 let SQRT_EVEN_POW2 = prove
530 (`!n. EVEN n ==> (sqrt(&2 pow n) = &2 pow (n DIV 2))`,
531 SIMP_TAC[EVEN_EXISTS; LEFT_IMP_EXISTS_THM; DIV_MULT; ARITH_EQ] THEN
532 MESON_TAC[SQRT_UNIQUE; REAL_POW_POW; MULT_SYM; REAL_POW_LE; REAL_POS]);;
534 let REAL_DIV_SQRT = prove
535 (`!x. &0 <= x ==> (x / sqrt(x) = sqrt(x))`,
536 REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THENL
537 [ALL_TAC; ASM_MESON_TAC[SQRT_0; real_div; REAL_MUL_LZERO]] THEN
538 ASM_SIMP_TAC[REAL_EQ_LDIV_EQ; SQRT_POS_LT; GSYM REAL_POW_2] THEN
539 ASM_SIMP_TAC[SQRT_POW_2; REAL_LT_IMP_LE]);;
541 let REAL_RSQRT_LE = prove
542 (`!x y. &0 <= x /\ &0 <= y /\ x <= sqrt y ==> x pow 2 <= y`,
543 MESON_TAC[REAL_POW_LE2; SQRT_POW_2]);;
545 let REAL_LSQRT_LE = prove
546 (`!x y. &0 <= x /\ sqrt x <= y ==> x <= y pow 2`,
547 MESON_TAC[REAL_POW_LE2; SQRT_POS_LE; REAL_LE_TRANS; SQRT_POW_2]);;
549 (* ------------------------------------------------------------------------- *)
550 (* Hence derive more interesting properties of the norm. *)
551 (* ------------------------------------------------------------------------- *)
555 REWRITE_TAC[vector_norm; DOT_LZERO; SQRT_0]);;
557 let NORM_POS_LE = prove
559 GEN_TAC THEN SIMP_TAC[DOT_POS_LE; vector_norm; SQRT_POS_LE]);;
562 (`!x. norm(--x) = norm x`,
563 REWRITE_TAC[vector_norm; DOT_LNEG; DOT_RNEG; REAL_NEG_NEG]);;
566 (`!x y. norm(x - y) = norm(y - x)`,
567 MESON_TAC[NORM_NEG; VECTOR_NEG_SUB]);;
570 (`!a x. norm(a % x) = abs(a) * norm x`,
571 REWRITE_TAC[vector_norm; DOT_LMUL; DOT_RMUL; REAL_MUL_ASSOC] THEN
572 SIMP_TAC[SQRT_MUL; SQRT_POS_LE; DOT_POS_LE; REAL_LE_SQUARE] THEN
573 REWRITE_TAC[GSYM REAL_POW_2; POW_2_SQRT_ABS]);;
575 let NORM_EQ_0_DOT = prove
576 (`!x. (norm x = &0) <=> (x dot x = &0)`,
577 SIMP_TAC[vector_norm; SQRT_EQ_0; DOT_POS_LE]);;
579 let NORM_EQ_0 = prove
580 (`!x. (norm x = &0) <=> (x = vec 0)`,
581 SIMP_TAC[vector_norm; DOT_EQ_0; SQRT_EQ_0; DOT_POS_LE]);;
583 let NORM_POS_LT = prove
584 (`!x. &0 < norm x <=> ~(x = vec 0)`,
585 MESON_TAC[REAL_LT_LE; NORM_POS_LE; NORM_EQ_0]);;
587 let NORM_POW_2 = prove
588 (`!x. norm(x) pow 2 = x dot x`,
589 SIMP_TAC[vector_norm; SQRT_POW_2; DOT_POS_LE]);;
591 let NORM_EQ_0_IMP = prove
592 (`!x. (norm x = &0) ==> (x = vec 0)`,
593 MESON_TAC[NORM_EQ_0]);;
595 let NORM_LE_0 = prove
596 (`!x. norm x <= &0 <=> (x = vec 0)`,
597 MESON_TAC[REAL_LE_ANTISYM; NORM_EQ_0; NORM_POS_LE]);;
599 let VECTOR_MUL_EQ_0 = prove
600 (`!a x. (a % x = vec 0) <=> (a = &0) \/ (x = vec 0)`,
601 REWRITE_TAC[GSYM NORM_EQ_0; NORM_MUL; REAL_ABS_ZERO; REAL_ENTIRE]);;
603 let VECTOR_MUL_LCANCEL = prove
604 (`!a x y. (a % x = a % y) <=> (a = &0) \/ (x = y)`,
605 MESON_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_LDISTRIB; VECTOR_SUB_EQ]);;
607 let VECTOR_MUL_RCANCEL = prove
608 (`!a b x. (a % x = b % x) <=> (a = b) \/ (x = vec 0)`,
609 MESON_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_RDISTRIB; REAL_SUB_0; VECTOR_SUB_EQ]);;
611 let VECTOR_MUL_LCANCEL_IMP = prove
612 (`!a x y. ~(a = &0) /\ (a % x = a % y) ==> (x = y)`,
613 MESON_TAC[VECTOR_MUL_LCANCEL]);;
615 let VECTOR_MUL_RCANCEL_IMP = prove
616 (`!a b x. ~(x = vec 0) /\ (a % x = b % x) ==> (a = b)`,
617 MESON_TAC[VECTOR_MUL_RCANCEL]);;
619 let NORM_CAUCHY_SCHWARZ = prove
620 (`!(x:real^N) y. x dot y <= norm(x) * norm(y)`,
621 REPEAT STRIP_TAC THEN MAP_EVERY ASM_CASES_TAC
622 [`norm(x:real^N) = &0`; `norm(y:real^N) = &0`] THEN
623 ASM_SIMP_TAC[NORM_EQ_0_IMP; DOT_LZERO; DOT_RZERO;
624 REAL_MUL_LZERO; REAL_MUL_RZERO] THEN
625 MP_TAC(ISPEC `norm(y:real^N) % x - norm(x:real^N) % y` DOT_POS_LE) THEN
626 REWRITE_TAC[DOT_RSUB; DOT_LSUB; DOT_LMUL; DOT_RMUL; GSYM NORM_POW_2;
627 REAL_POW_2; REAL_LE_REFL] THEN
628 REWRITE_TAC[DOT_SYM; REAL_ARITH
629 `&0 <= y * (y * x * x - x * d) - x * (y * d - x * y * y) <=>
630 x * y * d <= x * y * x * y`] THEN
631 ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_LE; NORM_POS_LE]);;
633 let NORM_CAUCHY_SCHWARZ_ABS = prove
634 (`!x:real^N y. abs(x dot y) <= norm(x) * norm(y)`,
635 REPEAT GEN_TAC THEN MP_TAC(ISPEC `x:real^N` NORM_CAUCHY_SCHWARZ) THEN
636 DISCH_THEN(fun th -> MP_TAC(SPEC `y:real^N` th) THEN
637 MP_TAC(SPEC `--(y:real^N)` th)) THEN
638 REWRITE_TAC[DOT_RNEG; NORM_NEG] THEN REAL_ARITH_TAC);;
640 let REAL_ABS_NORM = prove
641 (`!x. abs(norm x) = norm x`,
642 REWRITE_TAC[NORM_POS_LE; REAL_ABS_REFL]);;
644 let NORM_CAUCHY_SCHWARZ_DIV = prove
645 (`!x:real^N y. abs((x dot y) / (norm x * norm y)) <= &1`,
647 MAP_EVERY ASM_CASES_TAC [`x:real^N = vec 0`; `y:real^N = vec 0`] THEN
648 ASM_REWRITE_TAC[NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO; real_div;
649 REAL_INV_1; DOT_LZERO; DOT_RZERO; REAL_ABS_NUM; REAL_POS] THEN
650 ASM_SIMP_TAC[GSYM real_div; REAL_ABS_DIV; REAL_LE_LDIV_EQ; REAL_LT_MUL;
651 REAL_ABS_INV; NORM_POS_LT; REAL_ABS_MUL; REAL_ABS_NORM] THEN
652 REWRITE_TAC[REAL_MUL_LID; NORM_CAUCHY_SCHWARZ_ABS]);;
654 let NORM_TRIANGLE = prove
655 (`!x y. norm(x + y) <= norm(x) + norm(y)`,
656 REPEAT GEN_TAC THEN REWRITE_TAC[vector_norm] THEN
657 MATCH_MP_TAC REAL_LE_LSQRT THEN
658 SIMP_TAC[GSYM vector_norm; DOT_POS_LE; NORM_POS_LE; REAL_LE_ADD] THEN
659 REWRITE_TAC[DOT_LADD; DOT_RADD; REAL_POW_2; GSYM NORM_POW_2] THEN
660 SIMP_TAC[NORM_CAUCHY_SCHWARZ; DOT_SYM; REAL_ARITH
661 `d <= x * y ==> (x * x + d) + (d + y * y) <= (x + y) * (x + y)`]);;
663 let NORM_TRIANGLE_SUB = prove
664 (`!x y:real^N. norm(x) <= norm(y) + norm(x - y)`,
665 MESON_TAC[NORM_TRIANGLE; VECTOR_SUB_ADD2]);;
667 let NORM_TRIANGLE_LE = prove
668 (`!x y. norm(x) + norm(y) <= e ==> norm(x + y) <= e`,
669 MESON_TAC[REAL_LE_TRANS; NORM_TRIANGLE]);;
671 let NORM_TRIANGLE_LT = prove
672 (`!x y. norm(x) + norm(y) < e ==> norm(x + y) < e`,
673 MESON_TAC[REAL_LET_TRANS; NORM_TRIANGLE]);;
675 let COMPONENT_LE_NORM = prove
676 (`!x:real^N i. 1 <= i /\ i <= dimindex(:N)
677 ==> abs(x$i) <= norm x`,
678 REPEAT STRIP_TAC THEN REWRITE_TAC[vector_norm] THEN
679 MATCH_MP_TAC REAL_LE_RSQRT THEN REWRITE_TAC[GSYM REAL_ABS_POW] THEN
680 REWRITE_TAC[real_abs; REAL_POW_2; REAL_LE_SQUARE] THEN
682 `x$i * (x:real^N)$i =
683 sum(1..dimindex(:N)) (\k. if k = i then x$i * x$i else &0)`
685 [REWRITE_TAC[SUM_DELTA] THEN ASM_REWRITE_TAC[IN_NUMSEG]; ALL_TAC] THEN
686 REWRITE_TAC[dot] THEN MATCH_MP_TAC SUM_LE THEN
687 REWRITE_TAC[FINITE_NUMSEG] THEN
688 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
689 ASM_REWRITE_TAC[REAL_LE_REFL; REAL_LE_SQUARE]);;
691 let NORM_BOUND_COMPONENT_LE = prove
692 (`!x:real^N e. norm(x) <= e
693 ==> !i. 1 <= i /\ i <= dimindex(:N) ==> abs(x$i) <= e`,
694 MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]);;
696 let NORM_BOUND_COMPONENT_LT = prove
697 (`!x:real^N e. norm(x) < e
698 ==> !i. 1 <= i /\ i <= dimindex(:N) ==> abs(x$i) < e`,
699 MESON_TAC[COMPONENT_LE_NORM; REAL_LET_TRANS]);;
701 let NORM_LE_L1 = prove
702 (`!x:real^N. norm x <= sum(1..dimindex(:N)) (\i. abs(x$i))`,
703 REPEAT GEN_TAC THEN REWRITE_TAC[vector_norm; dot] THEN
704 MATCH_MP_TAC REAL_LE_LSQRT THEN REWRITE_TAC[REAL_POW_2] THEN
705 SIMP_TAC[SUM_POS_LE; FINITE_NUMSEG; REAL_LE_SQUARE; REAL_ABS_POS] THEN
706 SPEC_TAC(`dimindex(:N)`,`n:num`) THEN INDUCT_TAC THEN
707 REWRITE_TAC[SUM_CLAUSES_NUMSEG; ARITH_EQ; ARITH_RULE `1 <= SUC n`] THEN
708 SIMP_TAC[REAL_MUL_LZERO; REAL_LE_REFL] THEN
709 MATCH_MP_TAC(REAL_ARITH
710 `a2 <= a * a /\ &0 <= a * b /\ b2 <= b * b
711 ==> a2 + b2 <= (a + b) * (a + b)`) THEN
712 ASM_SIMP_TAC[SUM_POS_LE; REAL_LE_MUL; REAL_ABS_POS; FINITE_NUMSEG] THEN
713 REWRITE_TAC[GSYM REAL_ABS_MUL] THEN REAL_ARITH_TAC);;
715 let REAL_ABS_SUB_NORM = prove
716 (`abs(norm(x) - norm(y)) <= norm(x - y)`,
717 REWRITE_TAC[REAL_ARITH `abs(x - y) <= a <=> x <= y + a /\ y <= x + a`] THEN
718 MESON_TAC[NORM_TRIANGLE_SUB; NORM_SUB]);;
721 (`!x y. norm(x) <= norm(y) <=> x dot x <= y dot y`,
722 REWRITE_TAC[vector_norm] THEN MESON_TAC[SQRT_MONO_LE_EQ; DOT_POS_LE]);;
725 (`!x y. norm(x) < norm(y) <=> x dot x < y dot y`,
726 REWRITE_TAC[vector_norm] THEN MESON_TAC[SQRT_MONO_LT_EQ; DOT_POS_LE]);;
729 (`!x y. (norm x = norm y) <=> (x dot x = y dot y)`,
730 REWRITE_TAC[GSYM REAL_LE_ANTISYM; NORM_LE]);;
732 let NORM_EQ_1 = prove
733 (`!x. norm(x) = &1 <=> x dot x = &1`,
734 GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM SQRT_1] THEN
735 SIMP_TAC[vector_norm; SQRT_INJ; DOT_POS_LE; REAL_POS]);;
737 let NORM_LE_COMPONENTWISE = prove
738 (`!x:real^N y:real^N.
739 (!i. 1 <= i /\ i <= dimindex(:N) ==> abs(x$i) <= abs(y$i))
740 ==> norm(x) <= norm(y)`,
741 REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_LE; dot] THEN
742 MATCH_MP_TAC SUM_LE_NUMSEG THEN
743 ASM_SIMP_TAC[GSYM REAL_POW_2; GSYM REAL_LE_SQUARE_ABS]);;
745 (* ------------------------------------------------------------------------- *)
746 (* Squaring equations and inequalities involving norms. *)
747 (* ------------------------------------------------------------------------- *)
749 let DOT_SQUARE_NORM = prove
750 (`!x. x dot x = norm(x) pow 2`,
751 SIMP_TAC[vector_norm; SQRT_POW_2; DOT_POS_LE]);;
753 let NORM_EQ_SQUARE = prove
754 (`!x:real^N. norm(x) = a <=> &0 <= a /\ x dot x = a pow 2`,
755 REWRITE_TAC[DOT_SQUARE_NORM] THEN
756 ONCE_REWRITE_TAC[REAL_RING `x pow 2 = a pow 2 <=> x = a \/ x + a = &0`] THEN
757 GEN_TAC THEN MP_TAC(ISPEC `x:real^N` NORM_POS_LE) THEN REAL_ARITH_TAC);;
759 let NORM_LE_SQUARE = prove
760 (`!x:real^N. norm(x) <= a <=> &0 <= a /\ x dot x <= a pow 2`,
761 REWRITE_TAC[DOT_SQUARE_NORM; GSYM REAL_LE_SQUARE_ABS] THEN
762 GEN_TAC THEN MP_TAC(ISPEC `x:real^N` NORM_POS_LE) THEN REAL_ARITH_TAC);;
764 let NORM_GE_SQUARE = prove
765 (`!x:real^N. norm(x) >= a <=> a <= &0 \/ x dot x >= a pow 2`,
766 REWRITE_TAC[real_ge; DOT_SQUARE_NORM; GSYM REAL_LE_SQUARE_ABS] THEN
767 GEN_TAC THEN MP_TAC(ISPEC `x:real^N` NORM_POS_LE) THEN REAL_ARITH_TAC);;
769 let NORM_LT_SQUARE = prove
770 (`!x:real^N. norm(x) < a <=> &0 < a /\ x dot x < a pow 2`,
771 REWRITE_TAC[REAL_ARITH `x < a <=> ~(x >= a)`; NORM_GE_SQUARE] THEN
774 let NORM_GT_SQUARE = prove
775 (`!x:real^N. norm(x) > a <=> a < &0 \/ x dot x > a pow 2`,
776 REWRITE_TAC[REAL_ARITH `x > a <=> ~(x <= a)`; NORM_LE_SQUARE] THEN
779 let NORM_LT_SQUARE_ALT = prove
780 (`!x:real^N. norm(x) < a <=> &0 <= a /\ x dot x < a pow 2`,
781 REWRITE_TAC[REAL_ARITH `x < a <=> ~(x >= a)`; NORM_GE_SQUARE] THEN
782 REPEAT GEN_TAC THEN ASM_CASES_TAC `a = &0` THENL
783 [ASM_REWRITE_TAC[real_ge] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
784 REWRITE_TAC[DOT_POS_LE];
785 ASM_REAL_ARITH_TAC]);;
787 (* ------------------------------------------------------------------------- *)
788 (* General linear decision procedure for normed spaces. *)
789 (* ------------------------------------------------------------------------- *)
792 let find_normedterms =
793 let augment_norm b tm acc =
795 Comb(Const("vector_norm",_),v) -> insert (b,v) acc
797 let rec find_normedterms tm acc =
799 Comb(Comb(Const("real_add",_),l),r) ->
800 find_normedterms l (find_normedterms r acc)
801 | Comb(Comb(Const("real_mul",_),c),n) ->
802 if not (is_ratconst c) then acc else
803 augment_norm (rat_of_term c >=/ Int 0) n acc
804 | _ -> augment_norm true tm acc in
806 let lincomb_neg t = mapf minus_num t in
807 let lincomb_cmul c t = if c =/ Int 0 then undefined else mapf (( */ ) c) t in
808 let lincomb_add l r = combine (+/) (fun x -> x =/ Int 0) l r in
809 let lincomb_sub l r = lincomb_add l (lincomb_neg r) in
810 let lincomb_eq l r = lincomb_sub l r = undefined in
811 let rec vector_lincomb tm =
813 Comb(Comb(Const("vector_add",_),l),r) ->
814 lincomb_add (vector_lincomb l) (vector_lincomb r)
815 | Comb(Comb(Const("vector_sub",_),l),r) ->
816 lincomb_sub (vector_lincomb l) (vector_lincomb r)
817 | Comb(Comb(Const("%",_),l),r) ->
818 lincomb_cmul (rat_of_term l) (vector_lincomb r)
819 | Comb(Const("vector_neg",_),t) ->
820 lincomb_neg (vector_lincomb t)
821 | Comb(Const("vec",_),n) when is_numeral n & dest_numeral n =/ Int 0 ->
823 | _ -> (tm |=> Int 1) in
824 let vector_lincombs tms =
826 if can (assoc t) fns then fns else
827 let f = vector_lincomb t in
828 try let _,f' = find (fun (_,f') -> lincomb_eq f f') fns in
830 with Failure _ -> (t,f)::fns) tms [] in
831 let rec replacenegnorms fn tm =
833 Comb(Comb(Const("real_add",_),l),r) ->
834 BINOP_CONV (replacenegnorms fn) tm
835 | Comb(Comb(Const("real_mul",_),c),n) when rat_of_term c </ Int 0 ->
839 if defined eq v then (v |-> minus_num(apply eq v)) eq else eq in
840 let rec allsubsets s =
843 | (a::t) -> let res = allsubsets t in
844 map (fun b -> a::b) res @ res in
845 let evaluate env lin =
846 foldr (fun x c s -> s +/ c */ apply env x) lin (Int 0) in
847 let rec solve (vs,eqs) =
849 [],[] -> (0 |=> Int 1)
851 let v = hd(intersect vs (dom eq)) in
852 let c = apply eq v in
853 let vdef = lincomb_cmul (Int(-1) // c) eq in
855 if not(defined eqn v) then eqn else
856 lincomb_add (lincomb_cmul (apply eqn v) vdef) eqn in
857 let soln = solve (subtract vs [v],map eliminate oeqs) in
858 (v |-> evaluate soln (undefine v vdef)) soln in
859 let rec combinations k l =
860 if k = 0 then [[]] else
863 | h::t -> map (fun c -> h::c) (combinations (k - 1) t) @
865 let vertices vs eqs =
867 let soln = solve(vs,cmb) in
868 map (fun v -> tryapplyd soln v (Int 0)) vs in
869 let rawvs = mapfilter vertex (combinations (length vs) eqs) in
870 let unset = filter (forall (fun c -> c >=/ Int 0)) rawvs in
871 itlist (insert' (forall2 (=/))) unset [] in
872 let subsumes l m = forall2 (fun x y -> abs_num x <=/ abs_num y) l m in
873 let rec subsume todo dun =
876 | v::ovs -> let dun' = if exists (fun w -> subsumes w v) dun then dun
877 else v::(filter (fun w -> not(subsumes v w)) dun) in
880 let MATCH_pth = (MATCH_MP o prove)
881 (`!b x. b >= norm(x) ==> !c. abs(c) * b >= norm(c % x)`,
882 SIMP_TAC[NORM_MUL; real_ge; REAL_LE_LMUL; REAL_ABS_POS]) in
883 fun c th -> ISPEC(term_of_rat c) (MATCH_pth th) in
885 let MATCH_pth = (MATCH_MP o prove)
886 (`!b1 b2 x1 x2. b1 >= norm(x1) /\ b2 >= norm(x2)
887 ==> b1 + b2 >= norm(x1 + x2)`,
888 REWRITE_TAC[real_ge] THEN REPEAT STRIP_TAC THEN
889 MATCH_MP_TAC NORM_TRIANGLE_LE THEN ASM_SIMP_TAC[REAL_LE_ADD2]) in
890 fun th1 th2 -> MATCH_pth (CONJ th1 th2) in
891 let INEQUALITY_CANON_RULE =
892 CONV_RULE(LAND_CONV REAL_POLY_CONV) o
893 CONV_RULE(LAND_CONV REAL_RAT_REDUCE_CONV) o
894 GEN_REWRITE_RULE I [REAL_ARITH `s >= t <=> s - t >= &0`] in
895 let NORM_CANON_CONV =
896 let APPLY_pth1 = GEN_REWRITE_CONV I
897 [VECTOR_ARITH `x:real^N = &1 % x`]
898 and APPLY_pth2 = GEN_REWRITE_CONV I
899 [VECTOR_ARITH `x - y:real^N = x + --y`]
900 and APPLY_pth3 = GEN_REWRITE_CONV I
901 [VECTOR_ARITH `--x:real^N = -- &1 % x`]
902 and APPLY_pth4 = GEN_REWRITE_CONV I
903 [VECTOR_ARITH `&0 % x:real^N = vec 0`;
904 VECTOR_ARITH `c % vec 0:real^N = vec 0`]
905 and APPLY_pth5 = GEN_REWRITE_CONV I
906 [VECTOR_ARITH `c % (d % x) = (c * d) % x`]
907 and APPLY_pth6 = GEN_REWRITE_CONV I
908 [VECTOR_ARITH `c % (x + y) = c % x + c % y`]
909 and APPLY_pth7 = GEN_REWRITE_CONV I
910 [VECTOR_ARITH `vec 0 + x = x`;
911 VECTOR_ARITH `x + vec 0 = x`]
913 GEN_REWRITE_CONV I [VECTOR_ARITH `c % x + d % x = (c + d) % x`] THENC
914 LAND_CONV REAL_RAT_ADD_CONV THENC
915 GEN_REWRITE_CONV TRY_CONV [VECTOR_ARITH `&0 % x = vec 0`]
918 [VECTOR_ARITH `(c % x + z) + d % x = (c + d) % x + z`;
919 VECTOR_ARITH `c % x + (d % x + z) = (c + d) % x + z`;
920 VECTOR_ARITH `(c % x + w) + (d % x + z) = (c + d) % x + (w + z)`] THENC
921 LAND_CONV(LAND_CONV REAL_RAT_ADD_CONV)
923 GEN_REWRITE_CONV I [VECTOR_ARITH `&0 % x + y = y`]
926 [VECTOR_ARITH `c % x + d % y = c % x + d % y`;
927 VECTOR_ARITH `(c % x + z) + d % y = c % x + (z + d % y)`;
928 VECTOR_ARITH `c % x + (d % y + z) = c % x + (d % y + z)`;
929 VECTOR_ARITH `(c % x + w) + (d % y + z) = c % x + (w + (d % y + z))`]
932 [VECTOR_ARITH `c % x + d % y = d % y + c % x`;
933 VECTOR_ARITH `(c % x + z) + d % y = d % y + (c % x + z)`;
934 VECTOR_ARITH `c % x + (d % y + z) = d % y + (c % x + z)`;
935 VECTOR_ARITH `(c % x + w) + (d % y + z) = d % y + ((c % x + w) + z)`]
937 GEN_REWRITE_CONV TRY_CONV
938 [VECTOR_ARITH `x + vec 0 = x`] in
941 Comb(Comb(Const("vector_add",_),Comb(Comb(Const("%",_),l),v)),r) -> v
942 | Comb(Comb(Const("%",_),l),v) -> v
943 | _ -> failwith "headvector: non-canonical term" in
944 let rec VECTOR_CMUL_CONV tm =
945 ((APPLY_pth5 THENC LAND_CONV REAL_RAT_MUL_CONV) ORELSEC
946 (APPLY_pth6 THENC BINOP_CONV VECTOR_CMUL_CONV)) tm
947 and VECTOR_ADD_CONV tm =
948 try APPLY_pth7 tm with Failure _ ->
949 try APPLY_pth8 tm with Failure _ ->
951 Comb(Comb(Const("vector_add",_),lt),rt) ->
952 let l = headvector lt and r = headvector rt in
953 if l < r then (APPLY_pthb THENC
954 RAND_CONV VECTOR_ADD_CONV THENC
956 else if r < l then (APPLY_pthc THENC
957 RAND_CONV VECTOR_ADD_CONV THENC
960 ((APPLY_ptha THENC VECTOR_ADD_CONV) ORELSEC
961 RAND_CONV VECTOR_ADD_CONV THENC
964 let rec VECTOR_CANON_CONV tm =
966 Comb(Comb(Const("vector_add",_),l),r) ->
967 let lth = VECTOR_CANON_CONV l and rth = VECTOR_CANON_CONV r in
968 let th = MK_COMB(AP_TERM (rator(rator tm)) lth,rth) in
969 CONV_RULE (RAND_CONV VECTOR_ADD_CONV) th
970 | Comb(Comb(Const("%",_),l),r) ->
971 let rth = AP_TERM (rator tm) (VECTOR_CANON_CONV r) in
972 CONV_RULE (RAND_CONV(APPLY_pth4 ORELSEC VECTOR_CMUL_CONV)) rth
973 | Comb(Comb(Const("vector_sub",_),l),r) ->
974 (APPLY_pth2 THENC VECTOR_CANON_CONV) tm
975 | Comb(Const("vector_neg",_),t) ->
976 (APPLY_pth3 THENC VECTOR_CANON_CONV) tm
977 | Comb(Const("vec",_),n) when is_numeral n & dest_numeral n =/ Int 0 ->
979 | _ -> APPLY_pth1 tm in
982 Comb(Const("vector_norm",_),e) -> RAND_CONV VECTOR_CANON_CONV tm
983 | _ -> failwith "NORM_CANON_CONV" in
984 let REAL_VECTOR_COMBO_PROVER =
985 let pth_zero = prove(`norm(vec 0:real^N) = &0`,REWRITE_TAC[NORM_0])
986 and tv_n = mk_vartype "N" in
987 fun translator (nubs,ges,gts) ->
988 let sources = map (rand o rand o concl) nubs
989 and rawdests = itlist (find_normedterms o lhand o concl) (ges @ gts) [] in
990 if not (forall fst rawdests) then failwith "Sanity check" else
991 let dests = setify (map snd rawdests) in
992 let srcfuns = map vector_lincomb sources
993 and destfuns = map vector_lincomb dests in
994 let vvs = itlist (union o dom) (srcfuns @ destfuns) [] in
995 let n = length srcfuns in
997 let srccombs = zip srcfuns nvs in
1000 let inp = if defined d x then 0 |=> minus_num(apply d x)
1002 itlist (fun (f,v) g -> if defined f x then (v |-> apply f x) g else g)
1004 let equations = map coefficients vvs
1005 and inequalities = map (fun n -> (n |=> Int 1)) nvs in
1006 let plausiblevertices f =
1007 let flippedequations = map (itlist flip f) equations in
1008 let constraints = flippedequations @ inequalities in
1009 let rawverts = vertices nvs constraints in
1010 let check_solution v =
1011 let f = itlist2 (|->) nvs v (0 |=> Int 1) in
1012 forall (fun e -> evaluate f e =/ Int 0) flippedequations in
1013 let goodverts = filter check_solution rawverts in
1014 let signfixups = map (fun n -> if mem n f then -1 else 1) nvs in
1015 map (map2 (fun s c -> Int s */ c) signfixups) goodverts in
1016 let allverts = itlist (@) (map plausiblevertices (allsubsets nvs)) [] in
1017 subsume allverts [] in
1018 let compute_ineq v =
1019 let ths = mapfilter (fun (v,t) -> if v =/ Int 0 then fail()
1020 else NORM_CMUL_RULE v t)
1022 INEQUALITY_CANON_RULE (end_itlist NORM_ADD_RULE ths) in
1023 let ges' = mapfilter compute_ineq (itlist ((@) o consider) destfuns []) @
1024 map INEQUALITY_CANON_RULE nubs @ ges in
1025 let zerodests = filter
1026 (fun t -> dom(vector_lincomb t) = []) (map snd rawdests) in
1027 REAL_LINEAR_PROVER translator
1028 (map (fun t -> INST_TYPE [last(snd(dest_type(type_of t))),tv_n] pth_zero)
1030 map (CONV_RULE(ONCE_DEPTH_CONV NORM_CANON_CONV THENC
1031 LAND_CONV REAL_POLY_CONV)) ges',
1032 map (CONV_RULE(ONCE_DEPTH_CONV NORM_CANON_CONV THENC
1033 LAND_CONV REAL_POLY_CONV)) gts) in
1034 let REAL_VECTOR_INEQ_PROVER =
1036 (`norm(x) = n ==> norm(x) >= &0 /\ n >= norm(x)`,
1037 DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
1038 REWRITE_TAC[real_ge; NORM_POS_LE] THEN REAL_ARITH_TAC) in
1039 let NORM_MP = MATCH_MP pth in
1040 fun translator (ges,gts) ->
1041 let ntms = itlist find_normedterms (map (lhand o concl) (ges @ gts)) [] in
1042 let lctab = vector_lincombs (map snd (filter (not o fst) ntms)) in
1043 let asl = map (fun (t,_) ->
1044 ASSUME(mk_eq(mk_icomb(mk_const("vector_norm",[]),t),
1045 genvar `:real`))) lctab in
1046 let replace_conv = GEN_REWRITE_CONV TRY_CONV asl in
1047 let replace_rule = CONV_RULE (LAND_CONV (replacenegnorms replace_conv)) in
1049 itlist (fun th ths -> CONJUNCT1(NORM_MP th)::ths)
1050 asl (map replace_rule ges)
1051 and gts' = map replace_rule gts
1052 and nubs = map (CONJUNCT2 o NORM_MP) asl in
1053 let th1 = REAL_VECTOR_COMBO_PROVER translator (nubs,ges',gts') in
1055 (map (fun th -> let l,r = dest_eq(concl th) in (l,r)) asl) th1 in
1056 itlist PROVE_HYP (map (REFL o lhand o concl) asl) th2 in
1057 let REAL_VECTOR_PROVER =
1059 GEN_REWRITE_RULE I [REAL_ARITH `x = &0 <=> x >= &0 /\ --x >= &0`] in
1060 let splitequation th acc =
1061 let th1,th2 = CONJ_PAIR(rawrule th) in
1062 th1::CONV_RULE(LAND_CONV REAL_POLY_NEG_CONV) th2::acc in
1063 fun translator (eqs,ges,gts) ->
1064 REAL_VECTOR_INEQ_PROVER translator
1065 (itlist splitequation eqs ges,gts) in
1067 (`(!x y:real^N. x = y <=> norm(x - y) <= &0) /\
1068 (!x y:real^N. ~(x = y) <=> ~(norm(x - y) <= &0))`,
1069 REWRITE_TAC[NORM_LE_0; VECTOR_SUB_EQ]) in
1070 let conv1 = GEN_REWRITE_CONV TRY_CONV [pth] in
1071 let conv2 tm = (conv1 tm,conv1(mk_neg tm)) in
1072 let init = GEN_REWRITE_CONV ONCE_DEPTH_CONV [DECIMAL] THENC
1073 REAL_RAT_REDUCE_CONV THENC
1074 GEN_REWRITE_CONV ONCE_DEPTH_CONV [dist] THENC
1075 GEN_NNF_CONV true (conv1,conv2)
1076 and pure = GEN_REAL_ARITH REAL_VECTOR_PROVER in
1077 fun tm -> let th = init tm in EQ_MP (SYM th) (pure(rand(concl th)));;
1079 let NORM_ARITH_TAC = CONV_TAC NORM_ARITH;;
1081 let ASM_NORM_ARITH_TAC =
1082 REPEAT(FIRST_X_ASSUM(MP_TAC o check (not o is_forall o concl))) THEN
1085 (* ------------------------------------------------------------------------- *)
1086 (* Dot product in terms of the norm rather than conversely. *)
1087 (* ------------------------------------------------------------------------- *)
1089 let DOT_NORM = prove
1090 (`!x y. x dot y = (norm(x + y) pow 2 - norm(x) pow 2 - norm(y) pow 2) / &2`,
1091 REWRITE_TAC[NORM_POW_2; DOT_LADD; DOT_RADD; DOT_SYM] THEN REAL_ARITH_TAC);;
1093 let DOT_NORM_NEG = prove
1094 (`!x y. x dot y = ((norm(x) pow 2 + norm(y) pow 2) - norm(x - y) pow 2) / &2`,
1095 REWRITE_TAC[NORM_POW_2; DOT_LADD; DOT_RADD; DOT_LSUB; DOT_RSUB; DOT_SYM] THEN
1098 (* ------------------------------------------------------------------------- *)
1099 (* Equality of vectors in terms of dot products. *)
1100 (* ------------------------------------------------------------------------- *)
1102 let VECTOR_EQ = prove
1103 (`!x y. (x = y) <=> (x dot x = x dot y) /\ (y dot y = x dot x)`,
1104 REPEAT GEN_TAC THEN EQ_TAC THENL [SIMP_TAC[]; ALL_TAC] THEN
1105 ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
1106 REWRITE_TAC[GSYM DOT_EQ_0] THEN
1107 SIMP_TAC[DOT_LSUB; DOT_RSUB; DOT_SYM] THEN REAL_ARITH_TAC);;
1109 (* ------------------------------------------------------------------------- *)
1110 (* Hence more metric properties. *)
1111 (* ------------------------------------------------------------------------- *)
1113 let DIST_REFL = prove
1114 (`!x. dist(x,x) = &0`,
1117 let DIST_SYM = prove
1118 (`!x y. dist(x,y) = dist(y,x)`,
1121 let DIST_POS_LE = prove
1122 (`!x y. &0 <= dist(x,y)`,
1125 let DIST_TRIANGLE = prove
1126 (`!x:real^N y z. dist(x,z) <= dist(x,y) + dist(y,z)`,
1129 let DIST_TRIANGLE_ALT = prove
1130 (`!x y z. dist(y,z) <= dist(x,y) + dist(x,z)`,
1133 let DIST_EQ_0 = prove
1134 (`!x y. (dist(x,y) = &0) <=> (x = y)`,
1137 let DIST_POS_LT = prove
1138 (`!x y. ~(x = y) ==> &0 < dist(x,y)`,
1142 (`!x y. ~(x = y) <=> &0 < dist(x,y)`,
1145 let DIST_TRIANGLE_LE = prove
1146 (`!x y z e. dist(x,z) + dist(y,z) <= e ==> dist(x,y) <= e`,
1149 let DIST_TRIANGLE_LT = prove
1150 (`!x y z e. dist(x,z) + dist(y,z) < e ==> dist(x,y) < e`,
1153 let DIST_TRIANGLE_HALF_L = prove
1154 (`!x1 x2 y. dist(x1,y) < e / &2 /\ dist(x2,y) < e / &2 ==> dist(x1,x2) < e`,
1157 let DIST_TRIANGLE_HALF_R = prove
1158 (`!x1 x2 y. dist(y,x1) < e / &2 /\ dist(y,x2) < e / &2 ==> dist(x1,x2) < e`,
1161 let DIST_TRIANGLE_ADD = prove
1162 (`!x x' y y'. dist(x + y,x' + y') <= dist(x,x') + dist(y,y')`,
1165 let DIST_MUL = prove
1166 (`!x y c. dist(c % x,c % y) = abs(c) * dist(x,y)`,
1167 REWRITE_TAC[dist; GSYM VECTOR_SUB_LDISTRIB; NORM_MUL]);;
1169 let DIST_TRIANGLE_ADD_HALF = prove
1170 (`!x x' y y':real^N.
1171 dist(x,x') < e / &2 /\ dist(y,y') < e / &2 ==> dist(x + y,x' + y') < e`,
1174 let DIST_LE_0 = prove
1175 (`!x y. dist(x,y) <= &0 <=> x = y`,
1179 (`!w x y z. dist(w,x) = dist(y,z) <=> dist(w,x) pow 2 = dist(y,z) pow 2`,
1180 REWRITE_TAC[dist; NORM_POW_2; NORM_EQ]);;
1183 (`!x. dist(x,vec 0) = norm(x) /\ dist(vec 0,x) = norm(x)`,
1186 (* ------------------------------------------------------------------------- *)
1187 (* Sums of vectors. *)
1188 (* ------------------------------------------------------------------------- *)
1190 let NEUTRAL_VECTOR_ADD = prove
1191 (`neutral(+) = vec 0:real^N`,
1192 REWRITE_TAC[neutral] THEN MATCH_MP_TAC SELECT_UNIQUE THEN
1193 REWRITE_TAC[VECTOR_ARITH `x + y = y <=> x = vec 0`;
1194 VECTOR_ARITH `x + y = x <=> y = vec 0`]);;
1196 let MONOIDAL_VECTOR_ADD = prove
1197 (`monoidal((+):real^N->real^N->real^N)`,
1198 REWRITE_TAC[monoidal; NEUTRAL_VECTOR_ADD] THEN
1199 REPEAT CONJ_TAC THEN VECTOR_ARITH_TAC);;
1201 let vsum = new_definition
1202 `(vsum:(A->bool)->(A->real^N)->real^N) s f = lambda i. sum s (\x. f(x)$i)`;;
1204 let VSUM_CLAUSES = prove
1205 (`(!f. vsum {} f = vec 0) /\
1207 ==> (vsum (x INSERT s) f =
1208 if x IN s then vsum s f else f(x) + vsum s f))`,
1209 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; SUM_CLAUSES] THEN
1210 SIMP_TAC[VEC_COMPONENT] THEN REPEAT STRIP_TAC THEN
1211 COND_CASES_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_COMPONENT]);;
1214 (`!f s. FINITE s ==> vsum s f = iterate (+) s f`,
1215 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1216 ASM_SIMP_TAC[VSUM_CLAUSES; ITERATE_CLAUSES; MONOIDAL_VECTOR_ADD] THEN
1217 REWRITE_TAC[NEUTRAL_VECTOR_ADD]);;
1219 let VSUM_EQ_0 = prove
1220 (`!f s. (!x:A. x IN s ==> (f(x) = vec 0)) ==> (vsum s f = vec 0)`,
1221 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; vec; SUM_EQ_0]);;
1224 (`vsum s (\x. vec 0) = vec 0`,
1225 SIMP_TAC[VSUM_EQ_0]);;
1227 let VSUM_LMUL = prove
1228 (`!f c s. vsum s (\x. c % f(x)) = c % vsum s f`,
1229 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_MUL_COMPONENT; SUM_LMUL]);;
1231 let VSUM_RMUL = prove
1232 (`!c s v. vsum s (\x. c x % v) = (sum s c) % v`,
1233 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_MUL_COMPONENT; SUM_RMUL]);;
1235 let VSUM_ADD = prove
1236 (`!f g s. FINITE s ==> (vsum s (\x. f x + g x) = vsum s f + vsum s g)`,
1237 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; SUM_ADD]);;
1239 let VSUM_SUB = prove
1240 (`!f g s. FINITE s ==> (vsum s (\x. f x - g x) = vsum s f - vsum s g)`,
1241 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_SUB_COMPONENT; SUM_SUB]);;
1243 let VSUM_CONST = prove
1244 (`!c s. FINITE s ==> (vsum s (\n. c) = &(CARD s) % c)`,
1245 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_CONST; VECTOR_MUL_COMPONENT]);;
1247 let VSUM_COMPONENT = prove
1248 (`!s f i. 1 <= i /\ i <= dimindex(:N)
1249 ==> ((vsum s (f:A->real^N))$i = sum s (\x. f(x)$i))`,
1250 SIMP_TAC[vsum; LAMBDA_BETA]);;
1252 let VSUM_IMAGE = prove
1253 (`!f g s. FINITE s /\ (!x y. x IN s /\ y IN s /\ (f x = f y) ==> (x = y))
1254 ==> (vsum (IMAGE f s) g = vsum s (g o f))`,
1255 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN
1256 W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE o lhs o snd) THEN
1257 ASM_REWRITE_TAC[o_DEF]);;
1259 let VSUM_UNION = prove
1260 (`!f s t. FINITE s /\ FINITE t /\ DISJOINT s t
1261 ==> (vsum (s UNION t) f = vsum s f + vsum t f)`,
1262 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_UNION; VECTOR_ADD_COMPONENT]);;
1264 let VSUM_DIFF = prove
1265 (`!f s t. FINITE s /\ t SUBSET s
1266 ==> (vsum (s DIFF t) f = vsum s f - vsum t f)`,
1267 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_DIFF; VECTOR_SUB_COMPONENT]);;
1269 let VSUM_DELETE = prove
1270 (`!f s a. FINITE s /\ a IN s
1271 ==> vsum (s DELETE a) f = vsum s f - f a`,
1272 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_DELETE; VECTOR_SUB_COMPONENT]);;
1274 let VSUM_INCL_EXCL = prove
1275 (`!s t (f:A->real^N).
1276 FINITE s /\ FINITE t
1277 ==> vsum s f + vsum t f = vsum (s UNION t) f + vsum (s INTER t) f`,
1278 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN
1279 SIMP_TAC[SUM_INCL_EXCL]);;
1281 let VSUM_NEG = prove
1282 (`!f s. vsum s (\x. --f x) = --vsum s f`,
1283 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_NEG; VECTOR_NEG_COMPONENT]);;
1286 (`!f g s. (!x. x IN s ==> (f x = g x)) ==> (vsum s f = vsum s g)`,
1287 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA] THEN
1288 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ THEN ASM_SIMP_TAC[]);;
1290 let VSUM_SUPERSET = prove
1292 u SUBSET v /\ (!x. x IN v /\ ~(x IN u) ==> (f(x) = vec 0))
1293 ==> (vsum v f = vsum u f)`,
1294 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VEC_COMPONENT; SUM_SUPERSET]);;
1296 let VSUM_EQ_SUPERSET = prove
1298 FINITE t /\ t SUBSET s /\
1299 (!x. x IN t ==> (f x = g x)) /\
1300 (!x. x IN s /\ ~(x IN t) ==> f(x) = vec 0)
1301 ==> vsum s f = vsum t g`,
1302 MESON_TAC[VSUM_SUPERSET; VSUM_EQ]);;
1304 let VSUM_UNION_RZERO = prove
1306 FINITE u /\ (!x. x IN v /\ ~(x IN u) ==> (f(x) = vec 0))
1307 ==> (vsum (u UNION v) f = vsum u f)`,
1308 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VEC_COMPONENT; SUM_UNION_RZERO]);;
1310 let VSUM_UNION_LZERO = prove
1312 FINITE v /\ (!x. x IN u /\ ~(x IN v) ==> (f(x) = vec 0))
1313 ==> (vsum (u UNION v) f = vsum v f)`,
1314 MESON_TAC[VSUM_UNION_RZERO; UNION_COMM]);;
1316 let VSUM_RESTRICT = prove
1318 ==> (vsum s (\x. if x IN s then f(x) else vec 0) = vsum s f)`,
1319 REPEAT STRIP_TAC THEN MATCH_MP_TAC VSUM_EQ THEN ASM_SIMP_TAC[]);;
1321 let VSUM_RESTRICT_SET = prove
1322 (`!P s f. vsum {x | x IN s /\ P x} f =
1323 vsum s (\x. if P x then f x else vec 0)`,
1324 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VEC_COMPONENT; SUM_RESTRICT_SET;
1327 let VSUM_CASES = prove
1328 (`!s P f g. FINITE s
1329 ==> vsum s (\x:A. if P x then (f x):real^N else g x) =
1330 vsum {x | x IN s /\ P x} f + vsum {x | x IN s /\ ~P x} g`,
1331 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT; SUM_CASES;
1334 let VSUM_SING = prove
1335 (`!f x. vsum {x} f = f(x)`,
1336 SIMP_TAC[VSUM_CLAUSES; FINITE_RULES; NOT_IN_EMPTY; VECTOR_ADD_RID]);;
1338 let VSUM_NORM = prove
1339 (`!f s. FINITE s ==> norm(vsum s f) <= sum s (\x. norm(f x))`,
1340 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1341 SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; NORM_0; REAL_LE_REFL] THEN
1344 let VSUM_NORM_LE = prove
1346 FINITE s /\ (!x. x IN s ==> norm(f x) <= g(x))
1347 ==> norm(vsum s f) <= sum s g`,
1348 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
1349 EXISTS_TAC `sum s (\x:A. norm(f x :real^N))` THEN
1350 ASM_SIMP_TAC[VSUM_NORM; SUM_LE]);;
1352 let VSUM_NORM_TRIANGLE = prove
1353 (`!s f b. FINITE s /\ sum s (\a. norm(f a)) <= b
1354 ==> norm(vsum s f) <= b`,
1355 MESON_TAC[VSUM_NORM; REAL_LE_TRANS]);;
1357 let VSUM_NORM_BOUND = prove
1358 (`!s f b. FINITE s /\ (!x:A. x IN s ==> norm(f(x)) <= b)
1359 ==> norm(vsum s f) <= &(CARD s) * b`,
1360 SIMP_TAC[GSYM SUM_CONST; VSUM_NORM_LE]);;
1362 let VSUM_CLAUSES_NUMSEG = prove
1363 (`(!m. vsum(m..0) f = if m = 0 then f(0) else vec 0) /\
1364 (!m n. vsum(m..SUC n) f = if m <= SUC n then vsum(m..n) f + f(SUC n)
1365 else vsum(m..n) f)`,
1366 REWRITE_TAC[NUMSEG_CLAUSES] THEN REPEAT STRIP_TAC THEN
1368 ASM_SIMP_TAC[VSUM_SING; VSUM_CLAUSES; FINITE_NUMSEG; IN_NUMSEG] THEN
1369 REWRITE_TAC[ARITH_RULE `~(SUC n <= n)`; VECTOR_ADD_AC]);;
1371 let VSUM_CLAUSES_RIGHT = prove
1372 (`!f m n. 0 < n /\ m <= n ==> vsum(m..n) f = vsum(m..n-1) f + (f n):real^N`,
1373 GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN
1374 SIMP_TAC[LT_REFL; VSUM_CLAUSES_NUMSEG; SUC_SUB1]);;
1376 let VSUM_CMUL_NUMSEG = prove
1377 (`!f c m n. vsum (m..n) (\x. c % f x) = c % vsum (m..n) f`,
1378 SIMP_TAC[VSUM_LMUL; FINITE_NUMSEG]);;
1380 let VSUM_EQ_NUMSEG = prove
1382 (!x. m <= x /\ x <= n ==> (f x = g x))
1383 ==> (vsum(m .. n) f = vsum(m .. n) g)`,
1384 REPEAT STRIP_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
1385 ASM_SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG]);;
1387 let VSUM_IMAGE_GEN = prove
1391 vsum (IMAGE f s) (\y. vsum {x | x IN s /\ (f(x) = y)} g))`,
1392 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_IMAGE_GEN]);;
1394 let VSUM_GROUP = prove
1396 FINITE s /\ IMAGE f s SUBSET t
1397 ==> vsum t (\y. vsum {x | x IN s /\ f(x) = y} g) = vsum s g`,
1398 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; SUM_GROUP]);;
1400 let VSUM_VMUL = prove
1401 (`!f v s. FINITE s ==> ((sum s f) % v = vsum s (\x. f(x) % v))`,
1402 GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1403 ASM_SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES] THEN
1404 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[VECTOR_ADD_RDISTRIB] THEN
1407 let VSUM_DELTA = prove
1408 (`!s a. vsum s (\x. if x = a then b else vec 0) =
1409 if a IN s then b else vec 0`,
1410 SIMP_TAC[CART_EQ; LAMBDA_BETA; VSUM_COMPONENT; COND_COMPONENT] THEN
1411 SIMP_TAC[VEC_COMPONENT; SUM_DELTA]);;
1413 let VSUM_ADD_NUMSEG = prove
1414 (`!f g m n. vsum(m..n) (\i. f i + g i) = vsum(m..n) f + vsum(m..n) g`,
1415 SIMP_TAC[VSUM_ADD; FINITE_NUMSEG]);;
1417 let VSUM_SUB_NUMSEG = prove
1418 (`!f g m n. vsum(m..n) (\i. f i - g i) = vsum(m..n) f - vsum(m..n) g`,
1419 SIMP_TAC[VSUM_SUB; FINITE_NUMSEG]);;
1421 let VSUM_ADD_SPLIT = prove
1423 m <= n + 1 ==> vsum(m..n + p) f = vsum(m..n) f + vsum(n + 1..n + p) f`,
1424 SIMP_TAC[CART_EQ; LAMBDA_BETA; VSUM_COMPONENT; VECTOR_ADD_COMPONENT;
1427 let VSUM_VSUM_PRODUCT = prove
1428 (`!s:A->bool t:A->B->bool x.
1429 FINITE s /\ (!i. i IN s ==> FINITE(t i))
1430 ==> vsum s (\i. vsum (t i) (x i)) =
1431 vsum {i,j | i IN s /\ j IN t i} (\(i,j). x i j)`,
1432 SIMP_TAC[CART_EQ; LAMBDA_BETA; VSUM_COMPONENT; COND_COMPONENT] THEN
1433 SIMP_TAC[SUM_SUM_PRODUCT] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN
1434 REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM]);;
1436 let VSUM_IMAGE_NONZERO = prove
1437 (`!d:B->real^N i:A->B s.
1439 (!x y. x IN s /\ y IN s /\ ~(x = y) /\ i x = i y ==> d(i x) = vec 0)
1440 ==> vsum (IMAGE i s) d = vsum s (d o i)`,
1441 GEN_TAC THEN GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN
1442 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1443 SIMP_TAC[IMAGE_CLAUSES; VSUM_CLAUSES; FINITE_IMAGE] THEN
1444 MAP_EVERY X_GEN_TAC [`a:A`; `s:A->bool`] THEN
1445 REWRITE_TAC[IN_INSERT] THEN REPEAT STRIP_TAC THEN
1446 SUBGOAL_THEN `vsum s ((d:B->real^N) o (i:A->B)) = vsum (IMAGE i s) d`
1447 SUBST1_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
1448 COND_CASES_TAC THEN ASM_REWRITE_TAC[o_THM] THEN
1449 REWRITE_TAC[VECTOR_ARITH `a = x + a <=> x = vec 0`] THEN
1450 ASM_MESON_TAC[IN_IMAGE]);;
1452 let VSUM_UNION_NONZERO = prove
1453 (`!f s t. FINITE s /\ FINITE t /\ (!x. x IN s INTER t ==> f(x) = vec 0)
1454 ==> vsum (s UNION t) f = vsum s f + vsum t f`,
1455 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN
1456 SIMP_TAC[VEC_COMPONENT; SUM_UNION_NONZERO]);;
1458 let VSUM_UNIONS_NONZERO = prove
1459 (`!f s. FINITE s /\ (!t:A->bool. t IN s ==> FINITE t) /\
1460 (!t1 t2 x. t1 IN s /\ t2 IN s /\ ~(t1 = t2) /\ x IN t1 /\ x IN t2
1462 ==> vsum (UNIONS s) f = vsum s (\t. vsum t f)`,
1463 GEN_TAC THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN
1464 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1465 REWRITE_TAC[UNIONS_0; UNIONS_INSERT; VSUM_CLAUSES; IN_INSERT] THEN
1466 MAP_EVERY X_GEN_TAC [`t:A->bool`; `s:(A->bool)->bool`] THEN
1467 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
1468 ONCE_REWRITE_TAC[IMP_CONJ] THEN ASM_SIMP_TAC[VSUM_CLAUSES] THEN
1469 ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN(SUBST_ALL_TAC o SYM)] THEN
1470 STRIP_TAC THEN MATCH_MP_TAC VSUM_UNION_NONZERO THEN
1471 ASM_SIMP_TAC[FINITE_UNIONS; IN_INTER; IN_UNIONS] THEN ASM_MESON_TAC[]);;
1473 let VSUM_CLAUSES_LEFT = prove
1474 (`!f m n. m <= n ==> vsum(m..n) f = f m + vsum(m + 1..n) f`,
1475 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN
1476 SIMP_TAC[VEC_COMPONENT; SUM_CLAUSES_LEFT]);;
1478 let VSUM_DIFFS = prove
1479 (`!m n. vsum(m..n) (\k. f(k) - f(k + 1)) =
1480 if m <= n then f(m) - f(n + 1) else vec 0`,
1481 GEN_TAC THEN INDUCT_TAC THEN ASM_SIMP_TAC[VSUM_CLAUSES_NUMSEG; LE] THEN
1482 ASM_CASES_TAC `m = SUC n` THEN
1483 ASM_REWRITE_TAC[ARITH_RULE `~(SUC n <= n)`; VECTOR_ADD_LID] THEN
1484 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
1485 REWRITE_TAC[GSYM ADD1] THEN VECTOR_ARITH_TAC);;
1487 let VSUM_DIFFS_ALT = prove
1488 (`!m n. vsum(m..n) (\k. f(k + 1) - f(k)) =
1489 if m <= n then f(n + 1) - f(m) else vec 0`,
1490 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_NEG_SUB] THEN
1491 SIMP_TAC[VSUM_NEG; VSUM_DIFFS] THEN
1492 COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_NEG_SUB; VECTOR_NEG_0]);;
1494 let VSUM_DELETE_CASES = prove
1497 ==> vsum(s DELETE x) f = if x IN s then vsum s f - f x else vsum s f`,
1498 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
1499 ASM_SIMP_TAC[SET_RULE `~(x IN s) ==> s DELETE x = s`] THEN
1500 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
1501 [MATCH_MP (SET_RULE `x IN s ==> s = x INSERT (s DELETE x)`) th]) THEN
1502 ASM_SIMP_TAC[VSUM_CLAUSES; FINITE_DELETE; IN_DELETE] THEN VECTOR_ARITH_TAC);;
1504 let VSUM_EQ_GENERAL = prove
1505 (`!s:A->bool t:B->bool (f:A->real^N) g h.
1506 (!y. y IN t ==> ?!x. x IN s /\ h x = y) /\
1507 (!x. x IN s ==> h x IN t /\ g(h x) = f x)
1508 ==> vsum s f = vsum t g`,
1509 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA] THEN
1510 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ_GENERAL THEN
1511 EXISTS_TAC `h:A->B` THEN ASM_MESON_TAC[]);;
1513 let VSUM_EQ_GENERAL_INVERSES = prove
1514 (`!s t (f:A->real^N) (g:B->real^N) h k.
1515 (!y. y IN t ==> k y IN s /\ h (k y) = y) /\
1516 (!x. x IN s ==> h x IN t /\ k (h x) = x /\ g (h x) = f x)
1517 ==> vsum s f = vsum t g`,
1518 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA] THEN
1519 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ_GENERAL_INVERSES THEN
1520 MAP_EVERY EXISTS_TAC [`h:A->B`; `k:B->A`] THEN ASM_MESON_TAC[]);;
1522 let VSUM_NORM_ALLSUBSETS_BOUND = prove
1525 (!q. q SUBSET p ==> norm(vsum q f) <= e)
1526 ==> sum p (\x. norm(f x)) <= &2 * &(dimindex(:N)) * e`,
1527 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
1529 `sum p (\x:A. sum (1..dimindex(:N)) (\i. abs((f x:real^N)$i)))` THEN
1531 [MATCH_MP_TAC SUM_LE THEN ASM_SIMP_TAC[NORM_LE_L1]; ALL_TAC] THEN
1532 W(MP_TAC o PART_MATCH (lhand o rand) SUM_SWAP o lhand o snd) THEN
1533 ASM_REWRITE_TAC[FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN
1534 ONCE_REWRITE_TAC[REAL_ARITH `&2 * &n * e = &n * &2 * e`] THEN
1535 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o RAND_CONV)
1536 [GSYM CARD_NUMSEG_1] THEN
1537 MATCH_MP_TAC SUM_BOUND THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
1538 X_GEN_TAC `k:num` THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
1539 EXISTS_TAC `sum {x:A | x IN p /\ &0 <= (f x:real^N)$k} (\x. abs((f x)$k)) +
1540 sum {x | x IN p /\ (f x)$k < &0} (\x. abs((f x)$k))` THEN
1542 [MATCH_MP_TAC(REAL_ARITH `a = b ==> b <= a`) THEN
1543 MATCH_MP_TAC SUM_UNION_EQ THEN
1544 ASM_SIMP_TAC[EXTENSION; NOT_IN_EMPTY; IN_INTER; IN_UNION; IN_ELIM_THM] THEN
1545 CONJ_TAC THEN X_GEN_TAC `x:A` THEN ASM_CASES_TAC `(x:A) IN p` THEN
1546 ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
1548 MATCH_MP_TAC(REAL_ARITH `x <= e /\ y <= e ==> x + y <= &2 * e`) THEN
1549 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM REAL_ABS_NEG] THEN
1550 CONJ_TAC THEN MATCH_MP_TAC(REAL_ARITH
1551 `!g. sum s g = sum s f /\ sum s g <= e ==> sum s f <= e`)
1553 [EXISTS_TAC `\x. ((f:A->real^N) x)$k`;
1554 EXISTS_TAC `\x. --(((f:A->real^N) x)$k)`] THEN
1556 [MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[IN_ELIM_THM] THEN REAL_ARITH_TAC;
1558 ASM_SIMP_TAC[GSYM VSUM_COMPONENT; SUM_NEG; FINITE_RESTRICT] THEN
1559 MATCH_MP_TAC(REAL_ARITH `abs(x) <= e ==> x <= e`) THEN
1560 REWRITE_TAC[REAL_ABS_NEG] THEN
1561 MATCH_MP_TAC(REAL_ARITH
1562 `abs((vsum q f)$k) <= norm(vsum q f) /\
1564 ==> abs((vsum q f)$k) <= e`) THEN
1565 ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN
1566 FIRST_X_ASSUM MATCH_MP_TAC THEN SET_TAC[]);;
1568 let DOT_LSUM = prove
1569 (`!s f y. FINITE s ==> (vsum s f) dot y = sum s (\x. f(x) dot y)`,
1570 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
1571 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1572 ASM_SIMP_TAC[VSUM_CLAUSES; SUM_CLAUSES; DOT_LZERO; DOT_LADD]);;
1574 let DOT_RSUM = prove
1575 (`!s f x. FINITE s ==> x dot (vsum s f) = sum s (\y. x dot f(y))`,
1576 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
1577 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
1578 ASM_SIMP_TAC[VSUM_CLAUSES; SUM_CLAUSES; DOT_RZERO; DOT_RADD]);;
1580 let VSUM_OFFSET = prove
1581 (`!f m p. vsum(m + p..n + p) f = vsum(m..n) (\i. f (i + p))`,
1582 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VEC_COMPONENT; SUM_OFFSET]);;
1584 let VSUM_OFFSET_0 = prove
1585 (`!f m n. m <= n ==> vsum(m..n) f = vsum(0..n - m) (\i. f (i + m))`,
1586 SIMP_TAC[vsum; CART_EQ; LAMBDA_BETA; VEC_COMPONENT; SUM_OFFSET_0]);;
1588 let VSUM_TRIV_NUMSEG = prove
1589 (`!f m n. n < m ==> vsum(m..n) f = vec 0`,
1590 SIMP_TAC[GSYM NUMSEG_EMPTY; VSUM_CLAUSES]);;
1592 let VSUM_CONST_NUMSEG = prove
1593 (`!c m n. vsum(m..n) (\n. c) = &((n + 1) - m) % c`,
1594 SIMP_TAC[VSUM_CONST; FINITE_NUMSEG; CARD_NUMSEG]);;
1596 let VSUM_SUC = prove
1597 (`!f m n. vsum (SUC n..SUC m) f = vsum (n..m) (f o SUC)`,
1599 SUBGOAL_THEN `SUC n..SUC m = IMAGE SUC (n..m)` SUBST1_TAC THENL
1600 [REWRITE_TAC [ADD1; NUMSEG_OFFSET_IMAGE] THEN
1601 REWRITE_TAC [ONE; ADD_SUC; ADD_0; ETA_AX];
1602 SIMP_TAC [VSUM_IMAGE; FINITE_NUMSEG; SUC_INJ]]);;
1604 let VSUM_BIJECTION = prove
1605 (`!f:A->real^N p s:A->bool.
1606 (!x. x IN s ==> p(x) IN s) /\
1607 (!y. y IN s ==> ?!x. x IN s /\ p(x) = y)
1608 ==> vsum s f = vsum s (f o p)`,
1609 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
1610 MATCH_MP_TAC VSUM_EQ_GENERAL THEN EXISTS_TAC `p:A->A` THEN
1611 ASM_REWRITE_TAC[o_THM]);;
1613 let VSUM_PARTIAL_SUC = prove
1614 (`!f g:num->real^N m n.
1615 vsum (m..n) (\k. f(k) % (g(k + 1) - g(k))) =
1616 if m <= n then f(n + 1) % g(n + 1) - f(m) % g(m) -
1617 vsum (m..n) (\k. (f(k + 1) - f(k)) % g(k + 1))
1619 GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN
1620 COND_CASES_TAC THEN ASM_SIMP_TAC[VSUM_TRIV_NUMSEG; GSYM NOT_LE] THEN
1621 ASM_REWRITE_TAC[VSUM_CLAUSES_NUMSEG] THENL
1622 [COND_CASES_TAC THEN ASM_SIMP_TAC[ARITH] THENL
1623 [VECTOR_ARITH_TAC; ASM_ARITH_TAC];
1625 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LE]) THEN
1626 DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
1627 ASM_SIMP_TAC[GSYM NOT_LT; VSUM_TRIV_NUMSEG; ARITH_RULE `n < SUC n`] THEN
1628 ASM_SIMP_TAC[GSYM ADD1; ADD_CLAUSES] THEN VECTOR_ARITH_TAC);;
1630 let VSUM_PARTIAL_PRE = prove
1631 (`!f g:num->real^N m n.
1632 vsum (m..n) (\k. f(k) % (g(k) - g(k - 1))) =
1633 if m <= n then f(n + 1) % g(n) - f(m) % g(m - 1) -
1634 vsum (m..n) (\k. (f(k + 1) - f(k)) % g(k))
1637 MP_TAC(ISPECL [`f:num->real`; `\k. (g:num->real^N)(k - 1)`;
1638 `m:num`; `n:num`] VSUM_PARTIAL_SUC) THEN
1639 REWRITE_TAC[ADD_SUB] THEN DISCH_THEN SUBST1_TAC THEN
1640 COND_CASES_TAC THEN REWRITE_TAC[]);;
1642 let VSUM_COMBINE_L = prove
1644 0 < n /\ m <= n /\ n <= p + 1
1645 ==> vsum(m..n - 1) f + vsum(n..p) f = vsum(m..p) f`,
1646 SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VSUM_COMPONENT; SUM_COMBINE_L]);;
1648 let VSUM_COMBINE_R = prove
1650 m <= n + 1 /\ n <= p
1651 ==> vsum(m..n) f + vsum(n + 1..p) f = vsum(m..p) f`,
1652 SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VSUM_COMPONENT; SUM_COMBINE_R]);;
1654 let VSUM_INJECTION = prove
1657 (!x. x IN s ==> p x IN s) /\
1658 (!x y. x IN s /\ y IN s /\ p x = p y ==> x = y)
1659 ==> vsum s (f o p) = vsum s f`,
1660 REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP SUM_INJECTION) THEN
1661 SIMP_TAC[CART_EQ; VSUM_COMPONENT; o_DEF]);;
1663 let VSUM_SWAP = prove
1665 FINITE s /\ FINITE t
1666 ==> vsum s (\i. vsum t (f i)) = vsum t (\j. vsum s (\i. f i j))`,
1667 SIMP_TAC[CART_EQ; VSUM_COMPONENT] THEN REPEAT STRIP_TAC THEN
1668 W(MP_TAC o PART_MATCH (lhs o rand) SUM_SWAP o lhs o snd) THEN
1669 ASM_REWRITE_TAC[]);;
1671 let VSUM_SWAP_NUMSEG = prove
1673 vsum (a..b) (\i. vsum (c..d) (f i)) =
1674 vsum (c..d) (\j. vsum (a..b) (\i. f i j))`,
1675 REPEAT GEN_TAC THEN MATCH_MP_TAC VSUM_SWAP THEN REWRITE_TAC[FINITE_NUMSEG]);;
1677 let VSUM_ADD_GEN = prove
1679 FINITE {x | x IN s /\ ~(f x = vec 0)} /\
1680 FINITE {x | x IN s /\ ~(g x = vec 0)}
1681 ==> vsum s (\x. f x + g x) = vsum s f + vsum s g`,
1682 REPEAT GEN_TAC THEN DISCH_TAC THEN
1683 SIMP_TAC[CART_EQ; vsum; LAMBDA_BETA; VECTOR_ADD_COMPONENT] THEN
1684 REPEAT GEN_TAC THEN DISCH_THEN(K ALL_TAC) THEN
1685 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_ADD_GEN THEN
1686 POP_ASSUM MP_TAC THEN MATCH_MP_TAC MONO_AND THEN
1687 CONJ_TAC THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] FINITE_SUBSET) THEN
1688 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN GEN_TAC THEN
1689 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[DE_MORGAN_THM] THEN
1690 STRIP_TAC THEN ASM_REWRITE_TAC[VEC_COMPONENT]);;
1692 let VSUM_CASES_1 = prove
1693 (`!s a. FINITE s /\ a IN s
1694 ==> vsum s (\x. if x = a then y else f(x)) = vsum s f + (y - f a)`,
1695 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[VSUM_CASES] THEN
1696 ASM_SIMP_TAC[GSYM DELETE; VSUM_DELETE] THEN
1697 ASM_SIMP_TAC[SET_RULE `a IN s ==> {x | x IN s /\ x = a} = {a}`] THEN
1698 REWRITE_TAC[VSUM_SING] THEN VECTOR_ARITH_TAC);;
1700 let VSUM_SING_NUMSEG = prove
1701 (`vsum(n..n) f = f n`,
1702 REWRITE_TAC[NUMSEG_SING; VSUM_SING]);;
1705 (`vsum(1..1) f = f(1)`,
1706 REWRITE_TAC[VSUM_SING_NUMSEG]);;
1709 (`!t. vsum(1..2) t = t(1) + t(2)`,
1710 REWRITE_TAC[num_CONV `2`; VSUM_CLAUSES_NUMSEG] THEN
1711 REWRITE_TAC[VSUM_SING_NUMSEG; ARITH; REAL_ADD_ASSOC]);;
1714 (`!t. vsum(1..3) t = t(1) + t(2) + t(3)`,
1715 REWRITE_TAC[num_CONV `3`; num_CONV `2`; VSUM_CLAUSES_NUMSEG] THEN
1716 REWRITE_TAC[VSUM_SING_NUMSEG; ARITH; VECTOR_ADD_ASSOC]);;
1718 let VSUM_PAIR = prove
1719 (`!f:num->real^N m n.
1720 vsum(2*m..2*n+1) f = vsum(m..n) (\i. f(2*i) + f(2*i+1))`,
1721 SIMP_TAC[CART_EQ; VSUM_COMPONENT; VECTOR_ADD_COMPONENT; SUM_PAIR]);;
1723 let VSUM_PAIR_0 = prove
1724 (`!f:num->real^N n. vsum(0..2*n+1) f = vsum(0..n) (\i. f(2*i) + f(2*i+1))`,
1726 MP_TAC(ISPECL [`f:num->real^N`; `0`; `n:num`] VSUM_PAIR) THEN
1727 ASM_REWRITE_TAC[ARITH]);;
1729 (* ------------------------------------------------------------------------- *)
1730 (* Add useful congruences to the simplifier. *)
1731 (* ------------------------------------------------------------------------- *)
1734 (`(!f g s. (!x. x IN s ==> f(x) = g(x))
1735 ==> vsum s (\i. f(i)) = vsum s g) /\
1736 (!f g a b. (!i. a <= i /\ i <= b ==> f(i) = g(i))
1737 ==> vsum(a..b) (\i. f(i)) = vsum(a..b) g) /\
1738 (!f g p. (!x. p x ==> f x = g x)
1739 ==> vsum {y | p y} (\i. f(i)) = vsum {y | p y} g)`,
1740 REPEAT STRIP_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
1741 ASM_SIMP_TAC[IN_ELIM_THM; IN_NUMSEG]) in
1742 extend_basic_congs (map SPEC_ALL (CONJUNCTS th));;
1744 (* ------------------------------------------------------------------------- *)
1745 (* A conversion for evaluation of `vsum(m..n) f` for numerals m and n. *)
1746 (* ------------------------------------------------------------------------- *)
1748 let EXPAND_VSUM_CONV =
1749 let pth_0,pth_1 = (CONJ_PAIR o prove)
1750 (`vsum(0..0) (f:num->real^N) = f(0) /\
1751 vsum(0..SUC n) f = vsum(0..n) f + f(SUC n)`,
1752 REWRITE_TAC[VSUM_CLAUSES_NUMSEG; LE_0; VECTOR_ADD_AC]) in
1753 let conv_0 = REWR_CONV pth_0 and conv_1 = REWR_CONV pth_1 in
1755 try (LAND_CONV(RAND_CONV num_CONV) THENC conv_1 THENC
1756 NUM_REDUCE_CONV THENC LAND_CONV conv) tm
1757 with Failure _ -> conv_0 tm in
1759 (REDEPTH_CONV BETA_CONV) THENC
1760 GEN_REWRITE_CONV TOP_DEPTH_CONV [GSYM VECTOR_ADD_ASSOC];;
1762 (* ------------------------------------------------------------------------- *)
1763 (* Basis vectors in coordinate directions. *)
1764 (* ------------------------------------------------------------------------- *)
1766 let basis = new_definition
1767 `basis k = lambda i. if i = k then &1 else &0`;;
1769 let NORM_BASIS = prove
1770 (`!k. 1 <= k /\ k <= dimindex(:N)
1771 ==> (norm(basis k :real^N) = &1)`,
1772 REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[basis; dot; vector_norm] THEN
1773 GEN_REWRITE_TAC RAND_CONV [GSYM SQRT_1] THEN AP_TERM_TAC THEN
1774 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC
1775 `sum (1..dimindex(:N)) (\i. if i = k then &1 else &0)` THEN
1777 [MATCH_MP_TAC SUM_EQ_NUMSEG THEN
1778 ASM_SIMP_TAC[LAMBDA_BETA; IN_NUMSEG; EQ_SYM_EQ] THEN
1779 REPEAT STRIP_TAC THEN COND_CASES_TAC THEN REAL_ARITH_TAC;
1780 ASM_REWRITE_TAC[SUM_DELTA; IN_NUMSEG]]);;
1782 let NORM_BASIS_1 = prove
1783 (`norm(basis 1) = &1`,
1784 SIMP_TAC[NORM_BASIS; ARITH_EQ; ARITH_RULE `1 <= k <=> ~(k = 0)`;
1785 DIMINDEX_NONZERO]);;
1787 let VECTOR_CHOOSE_SIZE = prove
1788 (`!c. &0 <= c ==> ?x:real^N. norm(x) = c`,
1789 REPEAT STRIP_TAC THEN EXISTS_TAC `c % basis 1 :real^N` THEN
1790 ASM_REWRITE_TAC[NORM_MUL; real_abs; NORM_BASIS_1; REAL_MUL_RID]);;
1792 let VECTOR_CHOOSE_DIST = prove
1793 (`!x e. &0 <= e ==> ?y:real^N. dist(x,y) = e`,
1794 REPEAT STRIP_TAC THEN
1795 SUBGOAL_THEN `?c:real^N. norm(c) = e` CHOOSE_TAC THENL
1796 [ASM_SIMP_TAC[VECTOR_CHOOSE_SIZE]; ALL_TAC] THEN
1797 EXISTS_TAC `x - c:real^N` THEN REWRITE_TAC[dist] THEN
1798 ASM_REWRITE_TAC[VECTOR_ARITH `x - (x - c) = c:real^N`]);;
1800 let BASIS_INJ = prove
1801 (`!i j. 1 <= i /\ i <= dimindex(:N) /\
1802 1 <= j /\ j <= dimindex(:N) /\
1803 (basis i :real^N = basis j)
1805 SIMP_TAC[basis; CART_EQ; LAMBDA_BETA] THEN REPEAT GEN_TAC THEN
1806 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
1807 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
1808 ASM_SIMP_TAC[REAL_OF_NUM_EQ; ARITH_EQ]);;
1810 let BASIS_NE = prove
1811 (`!i j. 1 <= i /\ i <= dimindex(:N) /\
1812 1 <= j /\ j <= dimindex(:N) /\
1814 ==> ~(basis i :real^N = basis j)`,
1815 MESON_TAC[BASIS_INJ]);;
1817 let BASIS_COMPONENT = prove
1818 (`!k i. 1 <= i /\ i <= dimindex(:N)
1819 ==> ((basis k :real^N)$i = if i = k then &1 else &0)`,
1820 SIMP_TAC[basis; LAMBDA_BETA] THEN MESON_TAC[]);;
1822 let BASIS_EXPANSION = prove
1823 (`!x:real^N. vsum(1..dimindex(:N)) (\i. x$i % basis i) = x`,
1824 SIMP_TAC[CART_EQ; VSUM_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN
1825 ONCE_REWRITE_TAC[COND_RAND] THEN REWRITE_TAC[REAL_MUL_RZERO] THEN
1826 REPEAT STRIP_TAC THEN
1827 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN
1828 ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG; REAL_MUL_RID]);;
1830 let BASIS_EXPANSION_UNIQUE = prove
1831 (`!f x:real^N. (vsum(1..dimindex(:N)) (\i. f(i) % basis i) = x) <=>
1832 (!i. 1 <= i /\ i <= dimindex(:N) ==> f(i) = x$i)`,
1833 SIMP_TAC[CART_EQ; VSUM_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT] THEN
1834 REPEAT GEN_TAC THEN REWRITE_TAC[COND_RAND; REAL_MUL_RZERO; REAL_MUL_RID] THEN
1835 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV o RAND_CONV o LAND_CONV o
1836 ONCE_DEPTH_CONV) [EQ_SYM_EQ] THEN
1837 SIMP_TAC[SUM_DELTA; IN_NUMSEG]);;
1839 let DOT_BASIS = prove
1841 1 <= i /\ i <= dimindex(:N)
1842 ==> ((basis i) dot x = x$i) /\ (x dot (basis i) = x$i)`,
1843 SIMP_TAC[dot; basis; LAMBDA_BETA] THEN
1844 REWRITE_TAC[COND_RATOR; COND_RAND] THEN
1845 REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO] THEN
1846 SIMP_TAC[SUM_DELTA; IN_NUMSEG; REAL_MUL_LID; REAL_MUL_RID]);;
1848 let DOT_BASIS_BASIS = prove
1849 (`!i j. 1 <= i /\ i <= dimindex(:N) /\
1850 1 <= j /\ j <= dimindex(:N)
1851 ==> (basis i:real^N) dot (basis j) = if i = j then &1 else &0`,
1852 SIMP_TAC[DOT_BASIS; BASIS_COMPONENT]);;
1854 let DOT_BASIS_BASIS_UNEQUAL = prove
1855 (`!i j. ~(i = j) ==> (basis i) dot (basis j) = &0`,
1856 SIMP_TAC[basis; dot; LAMBDA_BETA] THEN ONCE_REWRITE_TAC[COND_RAND] THEN
1857 SIMP_TAC[SUM_0; REAL_MUL_RZERO; REAL_MUL_LZERO; COND_ID]);;
1859 let BASIS_EQ_0 = prove
1860 (`!i. (basis i :real^N = vec 0) <=> ~(i IN 1..dimindex(:N))`,
1861 SIMP_TAC[CART_EQ; BASIS_COMPONENT; VEC_COMPONENT; IN_NUMSEG] THEN
1862 MESON_TAC[REAL_ARITH `~(&1 = &0)`]);;
1864 let BASIS_NONZERO = prove
1865 (`!k. 1 <= k /\ k <= dimindex(:N)
1866 ==> ~(basis k :real^N = vec 0)`,
1867 REWRITE_TAC[BASIS_EQ_0; IN_NUMSEG]);;
1869 let VECTOR_EQ_LDOT = prove
1870 (`!y z. (!x. x dot y = x dot z) <=> y = z`,
1871 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[] THEN
1872 REWRITE_TAC[CART_EQ] THEN MESON_TAC[DOT_BASIS]);;
1874 let VECTOR_EQ_RDOT = prove
1875 (`!x y. (!z. x dot z = y dot z) <=> x = y`,
1876 REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[] THEN
1877 REWRITE_TAC[CART_EQ] THEN MESON_TAC[DOT_BASIS]);;
1879 (* ------------------------------------------------------------------------- *)
1880 (* Orthogonality. *)
1881 (* ------------------------------------------------------------------------- *)
1883 let orthogonal = new_definition
1884 `orthogonal x y <=> (x dot y = &0)`;;
1886 let ORTHOGONAL_0 = prove
1887 (`!x. orthogonal (vec 0) x /\ orthogonal x (vec 0)`,
1888 REWRITE_TAC[orthogonal; DOT_LZERO; DOT_RZERO]);;
1890 let ORTHOGONAL_REFL = prove
1891 (`!x. orthogonal x x <=> x = vec 0`,
1892 REWRITE_TAC[orthogonal; DOT_EQ_0]);;
1894 let ORTHOGONAL_SYM = prove
1895 (`!x y. orthogonal x y <=> orthogonal y x`,
1896 REWRITE_TAC[orthogonal; DOT_SYM]);;
1898 let ORTHOGONAL_LNEG = prove
1899 (`!x y. orthogonal (--x) y <=> orthogonal x y`,
1900 REWRITE_TAC[orthogonal; DOT_LNEG; REAL_NEG_EQ_0]);;
1902 let ORTHOGONAL_RNEG = prove
1903 (`!x y. orthogonal x (--y) <=> orthogonal x y`,
1904 REWRITE_TAC[orthogonal; DOT_RNEG; REAL_NEG_EQ_0]);;
1906 let ORTHOGONAL_BASIS = prove
1907 (`!x:real^N i. 1 <= i /\ i <= dimindex(:N)
1908 ==> (orthogonal (basis i) x <=> (x$i = &0))`,
1909 REPEAT STRIP_TAC THEN SIMP_TAC[orthogonal; dot; basis; LAMBDA_BETA] THEN
1910 REWRITE_TAC[COND_RAND; COND_RATOR; REAL_MUL_LZERO] THEN
1911 ASM_SIMP_TAC[SUM_DELTA; IN_NUMSEG; REAL_MUL_LID]);;
1913 let ORTHOGONAL_BASIS_BASIS = prove
1914 (`!i j. 1 <= i /\ i <= dimindex(:N) /\
1915 1 <= j /\ j <= dimindex(:N)
1916 ==> (orthogonal (basis i :real^N) (basis j) <=> ~(i = j))`,
1917 ASM_SIMP_TAC[ORTHOGONAL_BASIS] THEN ASM_SIMP_TAC[BASIS_COMPONENT] THEN
1918 MESON_TAC[REAL_ARITH `~(&1 = &0)`]);;
1920 let ORTHOGONAL_CLAUSES = prove
1921 (`(!a. orthogonal a (vec 0)) /\
1922 (!a x c. orthogonal a x ==> orthogonal a (c % x)) /\
1923 (!a x. orthogonal a x ==> orthogonal a (--x)) /\
1924 (!a x y. orthogonal a x /\ orthogonal a y ==> orthogonal a (x + y)) /\
1925 (!a x y. orthogonal a x /\ orthogonal a y ==> orthogonal a (x - y)) /\
1926 (!a. orthogonal (vec 0) a) /\
1927 (!a x c. orthogonal x a ==> orthogonal (c % x) a) /\
1928 (!a x. orthogonal x a ==> orthogonal (--x) a) /\
1929 (!a x y. orthogonal x a /\ orthogonal y a ==> orthogonal (x + y) a) /\
1930 (!a x y. orthogonal x a /\ orthogonal y a ==> orthogonal (x - y) a)`,
1931 REWRITE_TAC[orthogonal; DOT_RNEG; DOT_RMUL; DOT_RADD; DOT_RSUB;
1932 DOT_LZERO; DOT_RZERO; DOT_LNEG; DOT_LMUL; DOT_LADD; DOT_LSUB] THEN
1933 SIMP_TAC[] THEN REAL_ARITH_TAC);;
1935 (* ------------------------------------------------------------------------- *)
1936 (* Explicit vector construction from lists. *)
1937 (* ------------------------------------------------------------------------- *)
1939 let VECTOR_1 = prove
1940 (`(vector[x]:A^1)$1 = x`,
1941 SIMP_TAC[vector; LAMBDA_BETA; DIMINDEX_1; ARITH; LENGTH; EL; HD; TL]);;
1943 let VECTOR_2 = prove
1944 (`(vector[x;y]:A^2)$1 = x /\
1945 (vector[x;y]:A^2)$2 = y`,
1946 SIMP_TAC[vector; LAMBDA_BETA; DIMINDEX_2; ARITH; LENGTH; EL] THEN
1947 REWRITE_TAC[num_CONV `1`; HD; TL; EL]);;
1949 let VECTOR_3 = prove
1950 (`(vector[x;y;z]:A^3)$1 = x /\
1951 (vector[x;y;z]:A^3)$2 = y /\
1952 (vector[x;y;z]:A^3)$3 = z`,
1953 SIMP_TAC[vector; LAMBDA_BETA; DIMINDEX_3; ARITH; LENGTH; EL] THEN
1954 REWRITE_TAC[num_CONV `2`; num_CONV `1`; HD; TL; EL]);;
1956 let FORALL_VECTOR_1 = prove
1957 (`(!v:A^1. P v) <=> !x. P(vector[x])`,
1958 EQ_TAC THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN
1959 FIRST_X_ASSUM(MP_TAC o SPEC `(v:A^1)$1`) THEN
1960 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1961 REWRITE_TAC[CART_EQ; FORALL_1; VECTOR_1; DIMINDEX_1]);;
1963 let FORALL_VECTOR_2 = prove
1964 (`(!v:A^2. P v) <=> !x y. P(vector[x;y])`,
1965 EQ_TAC THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN
1966 FIRST_X_ASSUM(MP_TAC o SPECL [`(v:A^2)$1`; `(v:A^2)$2`]) THEN
1967 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1968 REWRITE_TAC[CART_EQ; FORALL_2; VECTOR_2; DIMINDEX_2]);;
1970 let FORALL_VECTOR_3 = prove
1971 (`(!v:A^3. P v) <=> !x y z. P(vector[x;y;z])`,
1972 EQ_TAC THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN
1973 FIRST_X_ASSUM(MP_TAC o SPECL
1974 [`(v:A^3)$1`; `(v:A^3)$2`; `(v:A^3)$3`]) THEN
1975 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
1976 REWRITE_TAC[CART_EQ; FORALL_3; VECTOR_3; DIMINDEX_3]);;
1978 let EXISTS_VECTOR_1 = prove
1979 (`(?v:A^1. P v) <=> ?x. P(vector[x])`,
1980 REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN
1981 REWRITE_TAC[FORALL_VECTOR_1]);;
1983 let EXISTS_VECTOR_2 = prove
1984 (`(?v:A^2. P v) <=> ?x y. P(vector[x;y])`,
1985 REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN
1986 REWRITE_TAC[FORALL_VECTOR_2]);;
1988 let EXISTS_VECTOR_3 = prove
1989 (`(?v:A^3. P v) <=> ?x y z. P(vector[x;y;z])`,
1990 REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN
1991 REWRITE_TAC[FORALL_VECTOR_3]);;
1993 (* ------------------------------------------------------------------------- *)
1994 (* Linear functions. *)
1995 (* ------------------------------------------------------------------------- *)
1997 let linear = new_definition
1998 `linear (f:real^M->real^N) <=>
1999 (!x y. f(x + y) = f(x) + f(y)) /\
2000 (!c x. f(c % x) = c % f(x))`;;
2002 let LINEAR_COMPOSE_CMUL = prove
2003 (`!f c. linear f ==> linear (\x. c % f(x))`,
2004 SIMP_TAC[linear] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
2006 let LINEAR_COMPOSE_NEG = prove
2007 (`!f. linear f ==> linear (\x. --(f(x)))`,
2008 SIMP_TAC[linear] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
2010 let LINEAR_COMPOSE_ADD = prove
2011 (`!f g. linear f /\ linear g ==> linear (\x. f(x) + g(x))`,
2012 SIMP_TAC[linear] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
2014 let LINEAR_COMPOSE_SUB = prove
2015 (`!f g. linear f /\ linear g ==> linear (\x. f(x) - g(x))`,
2016 SIMP_TAC[linear] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
2018 let LINEAR_COMPOSE = prove
2019 (`!f g. linear f /\ linear g ==> linear (g o f)`,
2020 SIMP_TAC[linear; o_THM]);;
2022 let LINEAR_ID = prove
2024 REWRITE_TAC[linear]);;
2026 let LINEAR_I = prove
2028 REWRITE_TAC[I_DEF; LINEAR_ID]);;
2030 let LINEAR_ZERO = prove
2031 (`linear (\x. vec 0)`,
2032 REWRITE_TAC[linear] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);;
2034 let LINEAR_NEGATION = prove
2036 REWRITE_TAC[linear] THEN VECTOR_ARITH_TAC);;
2038 let LINEAR_COMPOSE_VSUM = prove
2039 (`!f s. FINITE s /\ (!a. a IN s ==> linear(f a))
2040 ==> linear(\x. vsum s (\a. f a x))`,
2041 GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
2042 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
2043 SIMP_TAC[VSUM_CLAUSES; LINEAR_ZERO] THEN
2044 ASM_SIMP_TAC[ETA_AX; IN_INSERT; LINEAR_COMPOSE_ADD]);;
2046 let LINEAR_VMUL_COMPONENT = prove
2047 (`!f:real^M->real^N v k.
2048 linear f /\ 1 <= k /\ k <= dimindex(:N)
2049 ==> linear (\x. f(x)$k % v)`,
2050 SIMP_TAC[linear; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
2051 REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
2053 let LINEAR_0 = prove
2054 (`!f. linear f ==> (f(vec 0) = vec 0)`,
2055 MESON_TAC[VECTOR_MUL_LZERO; linear]);;
2057 let LINEAR_CMUL = prove
2058 (`!f c x. linear f ==> (f(c % x) = c % f(x))`,
2061 let LINEAR_NEG = prove
2062 (`!f x. linear f ==> (f(--x) = --(f x))`,
2063 ONCE_REWRITE_TAC[VECTOR_NEG_MINUS1] THEN SIMP_TAC[LINEAR_CMUL]);;
2065 let LINEAR_ADD = prove
2066 (`!f x y. linear f ==> (f(x + y) = f(x) + f(y))`,
2069 let LINEAR_SUB = prove
2070 (`!f x y. linear f ==> (f(x - y) = f(x) - f(y))`,
2071 SIMP_TAC[VECTOR_SUB; LINEAR_ADD; LINEAR_NEG]);;
2073 let LINEAR_VSUM = prove
2074 (`!f g s. linear f /\ FINITE s ==> (f(vsum s g) = vsum s (f o g))`,
2075 GEN_TAC THEN GEN_TAC THEN SIMP_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
2076 DISCH_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
2077 SIMP_TAC[VSUM_CLAUSES] THEN FIRST_ASSUM(fun th ->
2078 SIMP_TAC[MATCH_MP LINEAR_0 th; MATCH_MP LINEAR_ADD th; o_THM]));;
2080 let LINEAR_VSUM_MUL = prove
2082 linear f /\ FINITE s
2083 ==> f(vsum s (\i. c i % v i)) = vsum s (\i. c(i) % f(v i))`,
2084 SIMP_TAC[LINEAR_VSUM; o_DEF; LINEAR_CMUL]);;
2086 let LINEAR_INJECTIVE_0 = prove
2088 ==> ((!x y. (f(x) = f(y)) ==> (x = y)) <=>
2089 (!x. (f(x) = vec 0) ==> (x = vec 0)))`,
2090 REPEAT STRIP_TAC THEN
2091 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM VECTOR_SUB_EQ] THEN
2092 ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN MESON_TAC[VECTOR_SUB_RZERO]);;
2094 let LINEAR_BOUNDED = prove
2095 (`!f:real^M->real^N. linear f ==> ?B. !x. norm(f x) <= B * norm(x)`,
2096 REPEAT STRIP_TAC THEN EXISTS_TAC
2097 `sum(1..dimindex(:M)) (\i. norm((f:real^M->real^N)(basis i)))` THEN
2099 GEN_REWRITE_TAC (LAND_CONV o funpow 2 RAND_CONV) [GSYM BASIS_EXPANSION] THEN
2100 ASM_SIMP_TAC[LINEAR_VSUM; FINITE_NUMSEG] THEN
2101 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM SUM_LMUL] THEN
2102 MATCH_MP_TAC VSUM_NORM_LE THEN
2103 SIMP_TAC[FINITE_CROSS; FINITE_NUMSEG; IN_NUMSEG] THEN
2104 ASM_SIMP_TAC[o_DEF; NORM_MUL; LINEAR_CMUL] THEN
2105 ASM_SIMP_TAC[REAL_LE_RMUL; NORM_POS_LE; COMPONENT_LE_NORM]);;
2107 let LINEAR_BOUNDED_POS = prove
2108 (`!f:real^M->real^N. linear f ==> ?B. &0 < B /\ !x. norm(f x) <= B * norm(x)`,
2109 REPEAT STRIP_TAC THEN
2110 FIRST_ASSUM(X_CHOOSE_TAC `B:real` o MATCH_MP LINEAR_BOUNDED) THEN
2111 EXISTS_TAC `abs(B) + &1` THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
2112 POP_ASSUM MP_TAC THEN MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
2113 MATCH_MP_TAC(REAL_ARITH `a <= b ==> x <= a ==> x <= b`) THEN
2114 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
2117 let SYMMETRIC_LINEAR_IMAGE = prove
2118 (`!f s. (!x. x IN s ==> --x IN s) /\ linear f
2119 ==> !x. x IN (IMAGE f s) ==> --x IN (IMAGE f s)`,
2120 REWRITE_TAC[FORALL_IN_IMAGE] THEN
2121 SIMP_TAC[GSYM LINEAR_NEG] THEN SET_TAC[]);;
2123 (* ------------------------------------------------------------------------- *)
2124 (* Bilinear functions. *)
2125 (* ------------------------------------------------------------------------- *)
2127 let bilinear = new_definition
2128 `bilinear f <=> (!x. linear(\y. f x y)) /\ (!y. linear(\x. f x y))`;;
2130 let BILINEAR_LADD = prove
2131 (`!h x y z. bilinear h ==> h (x + y) z = (h x z) + (h y z)`,
2132 SIMP_TAC[bilinear; linear]);;
2134 let BILINEAR_RADD = prove
2135 (`!h x y z. bilinear h ==> h x (y + z) = (h x y) + (h x z)`,
2136 SIMP_TAC[bilinear; linear]);;
2138 let BILINEAR_LMUL = prove
2139 (`!h c x y. bilinear h ==> h (c % x) y = c % (h x y)`,
2140 SIMP_TAC[bilinear; linear]);;
2142 let BILINEAR_RMUL = prove
2143 (`!h c x y. bilinear h ==> h x (c % y) = c % (h x y)`,
2144 SIMP_TAC[bilinear; linear]);;
2146 let BILINEAR_LNEG = prove
2147 (`!h x y. bilinear h ==> h (--x) y = --(h x y)`,
2148 ONCE_REWRITE_TAC[VECTOR_NEG_MINUS1] THEN SIMP_TAC[BILINEAR_LMUL]);;
2150 let BILINEAR_RNEG = prove
2151 (`!h x y. bilinear h ==> h x (--y) = --(h x y)`,
2152 ONCE_REWRITE_TAC[VECTOR_NEG_MINUS1] THEN SIMP_TAC[BILINEAR_RMUL]);;
2154 let BILINEAR_LZERO = prove
2155 (`!h x. bilinear h ==> h (vec 0) x = vec 0`,
2156 ONCE_REWRITE_TAC[VECTOR_ARITH `x = vec 0 <=> x + x = x`] THEN
2157 SIMP_TAC[GSYM BILINEAR_LADD; VECTOR_ADD_LID]);;
2159 let BILINEAR_RZERO = prove
2160 (`!h x. bilinear h ==> h x (vec 0) = vec 0`,
2161 ONCE_REWRITE_TAC[VECTOR_ARITH `x = vec 0 <=> x + x = x`] THEN
2162 SIMP_TAC[GSYM BILINEAR_RADD; VECTOR_ADD_LID]);;
2164 let BILINEAR_LSUB = prove
2165 (`!h x y z. bilinear h ==> h (x - y) z = (h x z) - (h y z)`,
2166 SIMP_TAC[VECTOR_SUB; BILINEAR_LNEG; BILINEAR_LADD]);;
2168 let BILINEAR_RSUB = prove
2169 (`!h x y z. bilinear h ==> h x (y - z) = (h x y) - (h x z)`,
2170 SIMP_TAC[VECTOR_SUB; BILINEAR_RNEG; BILINEAR_RADD]);;
2172 let BILINEAR_VSUM = prove
2173 (`!h:real^M->real^N->real^P.
2174 bilinear h /\ FINITE s /\ FINITE t
2175 ==> h (vsum s f) (vsum t g) = vsum (s CROSS t) (\(i,j). h (f i) (g j))`,
2176 REPEAT GEN_TAC THEN SIMP_TAC[bilinear; ETA_AX] THEN
2177 ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c /\ d <=> (a /\ d) /\ (b /\ c)`] THEN
2178 DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
2179 ONCE_REWRITE_TAC[LEFT_AND_FORALL_THM] THEN DISCH_TAC THEN
2180 FIRST_ASSUM(MP_TAC o GEN_ALL o MATCH_MP LINEAR_VSUM o SPEC_ALL) THEN
2181 SIMP_TAC[] THEN ASM_SIMP_TAC[LINEAR_VSUM; o_DEF; VSUM_VSUM_PRODUCT] THEN
2182 REWRITE_TAC[GSYM CROSS]);;
2184 let BILINEAR_BOUNDED = prove
2185 (`!h:real^M->real^N->real^P.
2186 bilinear h ==> ?B. !x y. norm(h x y) <= B * norm(x) * norm(y)`,
2187 REPEAT STRIP_TAC THEN
2188 EXISTS_TAC `sum ((1..dimindex(:M)) CROSS (1..dimindex(:N)))
2189 (\(i,j). norm((h:real^M->real^N->real^P)
2190 (basis i) (basis j)))` THEN
2191 REPEAT GEN_TAC THEN GEN_REWRITE_TAC
2192 (LAND_CONV o RAND_CONV o BINOP_CONV) [GSYM BASIS_EXPANSION] THEN
2193 ASM_SIMP_TAC[BILINEAR_VSUM; FINITE_NUMSEG] THEN
2194 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM SUM_LMUL] THEN
2195 MATCH_MP_TAC VSUM_NORM_LE THEN
2196 SIMP_TAC[FINITE_CROSS; FINITE_NUMSEG; FORALL_PAIR_THM; IN_CROSS] THEN
2197 REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
2198 ASM_SIMP_TAC[BILINEAR_LMUL; NORM_MUL] THEN
2199 ASM_SIMP_TAC[BILINEAR_RMUL; NORM_MUL; REAL_MUL_ASSOC] THEN
2200 MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
2201 ASM_SIMP_TAC[COMPONENT_LE_NORM; REAL_ABS_POS; REAL_LE_MUL2]);;
2203 let BILINEAR_BOUNDED_POS = prove
2205 ==> ?B. &0 < B /\ !x y. norm(h x y) <= B * norm(x) * norm(y)`,
2206 REPEAT STRIP_TAC THEN
2207 FIRST_ASSUM(X_CHOOSE_TAC `B:real` o MATCH_MP BILINEAR_BOUNDED) THEN
2208 EXISTS_TAC `abs(B) + &1` THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
2209 POP_ASSUM MP_TAC THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
2210 MATCH_MP_TAC(REAL_ARITH `a <= b ==> x <= a ==> x <= b`) THEN
2211 REPEAT(MATCH_MP_TAC REAL_LE_RMUL THEN
2212 SIMP_TAC[NORM_POS_LE; REAL_LE_MUL]) THEN
2215 let BILINEAR_VSUM_PARTIAL_SUC = prove
2216 (`!f g h:real^M->real^N->real^P m n.
2218 ==> vsum (m..n) (\k. h (f k) (g(k + 1) - g(k))) =
2219 if m <= n then h (f(n + 1)) (g(n + 1)) - h (f m) (g m) -
2220 vsum (m..n) (\k. h (f(k + 1) - f(k)) (g(k + 1)))
2222 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN
2223 GEN_TAC THEN INDUCT_TAC THEN
2224 COND_CASES_TAC THEN ASM_SIMP_TAC[VSUM_TRIV_NUMSEG; GSYM NOT_LE] THEN
2225 ASM_REWRITE_TAC[VSUM_CLAUSES_NUMSEG] THENL
2226 [COND_CASES_TAC THEN ASM_SIMP_TAC[ARITH] THENL
2227 [ASM_SIMP_TAC[BILINEAR_RSUB; BILINEAR_LSUB] THEN VECTOR_ARITH_TAC;
2230 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LE]) THEN
2231 DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC ASSUME_TAC) THEN
2232 ASM_SIMP_TAC[GSYM NOT_LT; VSUM_TRIV_NUMSEG; ARITH_RULE `n < SUC n`] THEN
2233 ASM_SIMP_TAC[GSYM ADD1; ADD_CLAUSES] THEN
2234 ASM_SIMP_TAC[BILINEAR_RSUB; BILINEAR_LSUB] THEN VECTOR_ARITH_TAC);;
2236 let BILINEAR_VSUM_PARTIAL_PRE = prove
2237 (`!f g h:real^M->real^N->real^P m n.
2239 ==> vsum (m..n) (\k. h (f k) (g(k) - g(k - 1))) =
2240 if m <= n then h (f(n + 1)) (g(n)) - h (f m) (g(m - 1)) -
2241 vsum (m..n) (\k. h (f(k + 1) - f(k)) (g(k)))
2243 REPEAT STRIP_TAC THEN
2244 FIRST_ASSUM(MP_TAC o ISPECL [`f:num->real^M`; `\k. (g:num->real^N)(k - 1)`;
2245 `m:num`; `n:num`] o MATCH_MP BILINEAR_VSUM_PARTIAL_SUC) THEN
2246 REWRITE_TAC[ADD_SUB] THEN DISCH_THEN SUBST1_TAC THEN
2247 COND_CASES_TAC THEN REWRITE_TAC[]);;
2249 (* ------------------------------------------------------------------------- *)
2251 (* ------------------------------------------------------------------------- *)
2253 let adjoint = new_definition
2254 `adjoint(f:real^M->real^N) = @f'. !x y. f(x) dot y = x dot f'(y)`;;
2256 let ADJOINT_WORKS = prove
2257 (`!f:real^M->real^N. linear f ==> !x y. f(x) dot y = x dot (adjoint f)(y)`,
2258 GEN_TAC THEN DISCH_TAC THEN SIMP_TAC[adjoint] THEN CONV_TAC SELECT_CONV THEN
2259 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN ONCE_REWRITE_TAC[GSYM SKOLEM_THM] THEN
2260 X_GEN_TAC `y:real^N` THEN
2261 EXISTS_TAC `(lambda i. (f:real^M->real^N) (basis i) dot y):real^M` THEN
2262 X_GEN_TAC `x:real^M` THEN
2263 GEN_REWRITE_TAC (funpow 2 LAND_CONV o RAND_CONV) [GSYM BASIS_EXPANSION] THEN
2264 ASM_SIMP_TAC[LINEAR_VSUM; FINITE_NUMSEG] THEN
2265 SIMP_TAC[dot; LAMBDA_BETA; VSUM_COMPONENT; GSYM SUM_LMUL; GSYM SUM_RMUL] THEN
2266 GEN_REWRITE_TAC RAND_CONV [SUM_SWAP_NUMSEG] THEN
2267 ASM_SIMP_TAC[o_THM; VECTOR_MUL_COMPONENT; LINEAR_CMUL; REAL_MUL_ASSOC]);;
2269 let ADJOINT_LINEAR = prove
2270 (`!f:real^M->real^N. linear f ==> linear(adjoint f)`,
2271 REPEAT STRIP_TAC THEN REWRITE_TAC[linear; GSYM VECTOR_EQ_LDOT] THEN
2272 ASM_SIMP_TAC[DOT_RMUL; DOT_RADD; GSYM ADJOINT_WORKS]);;
2274 let ADJOINT_CLAUSES = prove
2275 (`!f:real^M->real^N.
2276 linear f ==> (!x y. x dot (adjoint f)(y) = f(x) dot y) /\
2277 (!x y. (adjoint f)(y) dot x = y dot f(x))`,
2278 MESON_TAC[ADJOINT_WORKS; DOT_SYM]);;
2280 let ADJOINT_ADJOINT = prove
2281 (`!f:real^M->real^N. linear f ==> adjoint(adjoint f) = f`,
2282 SIMP_TAC[FUN_EQ_THM; GSYM VECTOR_EQ_LDOT; ADJOINT_CLAUSES; ADJOINT_LINEAR]);;
2284 let ADJOINT_UNIQUE = prove
2285 (`!f f'. linear f /\ (!x y. f'(x) dot y = x dot f(y))
2286 ==> f' = adjoint f`,
2287 SIMP_TAC[FUN_EQ_THM; GSYM VECTOR_EQ_RDOT; ADJOINT_CLAUSES]);;
2289 (* ------------------------------------------------------------------------- *)
2290 (* Matrix notation. NB: an MxN matrix is of type real^N^M, not real^M^N. *)
2291 (* We could define a special type if we're going to use them a lot. *)
2292 (* ------------------------------------------------------------------------- *)
2294 overload_interface ("--",`(matrix_neg):real^N^M->real^N^M`);;
2295 overload_interface ("+",`(matrix_add):real^N^M->real^N^M->real^N^M`);;
2296 overload_interface ("-",`(matrix_sub):real^N^M->real^N^M->real^N^M`);;
2298 make_overloadable "**" `:A->B->C`;;
2300 overload_interface ("**",`(matrix_mul):real^N^M->real^P^N->real^P^M`);;
2301 overload_interface ("**",`(matrix_vector_mul):real^N^M->real^N->real^M`);;
2302 overload_interface ("**",`(vector_matrix_mul):real^M->real^N^M->real^N`);;
2304 parse_as_infix("%%",(21,"right"));;
2308 let matrix_cmul = new_definition
2309 `((%%):real->real^N^M->real^N^M) c A = lambda i j. c * A$i$j`;;
2311 let matrix_neg = new_definition
2312 `!A:real^N^M. --A = lambda i j. --(A$i$j)`;;
2314 let matrix_add = new_definition
2315 `!A:real^N^M B:real^N^M. A + B = lambda i j. A$i$j + B$i$j`;;
2317 let matrix_sub = new_definition
2318 `!A:real^N^M B:real^N^M. A - B = lambda i j. A$i$j - B$i$j`;;
2320 let matrix_mul = new_definition
2321 `!A:real^N^M B:real^P^N.
2323 lambda i j. sum(1..dimindex(:N)) (\k. A$i$k * B$k$j)`;;
2325 let matrix_vector_mul = new_definition
2326 `!A:real^N^M x:real^N.
2327 A ** x = lambda i. sum(1..dimindex(:N)) (\j. A$i$j * x$j)`;;
2329 let vector_matrix_mul = new_definition
2330 `!A:real^N^M x:real^M.
2331 x ** A = lambda j. sum(1..dimindex(:M)) (\i. A$i$j * x$i)`;;
2333 let mat = new_definition
2334 `(mat:num->real^N^M) k = lambda i j. if i = j then &k else &0`;;
2336 let transp = new_definition
2337 `(transp:real^N^M->real^M^N) A = lambda i j. A$j$i`;;
2339 let row = new_definition
2340 `(row:num->real^N^M->real^N) i A = lambda j. A$i$j`;;
2342 let column = new_definition
2343 `(column:num->real^N^M->real^M) j A = lambda i. A$i$j`;;
2345 let rows = new_definition
2346 `rows(A:real^N^M) = { row i A | 1 <= i /\ i <= dimindex(:M)}`;;
2348 let columns = new_definition
2349 `columns(A:real^N^M) = { column i A | 1 <= i /\ i <= dimindex(:N)}`;;
2351 let MATRIX_CMUL_COMPONENT = prove
2352 (`!c A:real^N^M i. (c %% A)$i$j = c * A$i$j`,
2354 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:M) /\ !A:real^N^M. A$i = A$k`
2355 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2356 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$j = z$l`
2357 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2358 ASM_SIMP_TAC[matrix_cmul; CART_EQ; LAMBDA_BETA]);;
2360 let MATRIX_ADD_COMPONENT = prove
2361 (`!A B:real^N^M i j. (A + B)$i$j = A$i$j + B$i$j`,
2363 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:M) /\ !A:real^N^M. A$i = A$k`
2364 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2365 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$j = z$l`
2366 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2367 ASM_SIMP_TAC[matrix_add; LAMBDA_BETA]);;
2369 let MATRIX_SUB_COMPONENT = prove
2370 (`!A B:real^N^M i j. (A - B)$i$j = A$i$j - B$i$j`,
2372 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:M) /\ !A:real^N^M. A$i = A$k`
2373 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2374 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$j = z$l`
2375 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2376 ASM_SIMP_TAC[matrix_sub; LAMBDA_BETA]);;
2378 let MATRIX_NEG_COMPONENT = prove
2379 (`!A:real^N^M i j. (--A)$i$j = --(A$i$j)`,
2381 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:M) /\ !A:real^N^M. A$i = A$k`
2382 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2383 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:N) /\ !z:real^N. z$j = z$l`
2384 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE]; ALL_TAC] THEN
2385 ASM_SIMP_TAC[matrix_neg; LAMBDA_BETA]);;
2387 let TRANSP_COMPONENT = prove
2388 (`!A:real^N^M i j. (transp A)$i$j = A$j$i`,
2390 SUBGOAL_THEN `?k. 1 <= k /\ k <= dimindex(:N) /\
2391 (!A:real^M^N. A$i = A$k) /\ (!z:real^N. z$i = z$k)`
2392 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE_2]; ALL_TAC] THEN
2393 SUBGOAL_THEN `?l. 1 <= l /\ l <= dimindex(:M) /\
2394 (!A:real^N^M. A$j = A$l) /\ (!z:real^M. z$j = z$l)`
2395 CHOOSE_TAC THENL [REWRITE_TAC[FINITE_INDEX_INRANGE_2]; ALL_TAC] THEN
2396 ASM_SIMP_TAC[transp; LAMBDA_BETA]);;
2398 let MAT_COMPONENT = prove
2400 1 <= i /\ i <= dimindex(:M) /\
2401 1 <= j /\ j <= dimindex(:N)
2402 ==> (mat n:real^N^M)$i$j = if i = j then &n else &0`,
2403 SIMP_TAC[mat; LAMBDA_BETA]);;
2405 let MATRIX_CMUL_ASSOC = prove
2406 (`!a b X:real^M^N. a %% (b %% X) = (a * b) %% X`,
2407 SIMP_TAC[CART_EQ; matrix_cmul; LAMBDA_BETA; REAL_MUL_ASSOC]);;
2409 let MATRIX_CMUL_LID = prove
2410 (`!X:real^M^N. &1 %% X = X`,
2411 SIMP_TAC[CART_EQ; matrix_cmul; LAMBDA_BETA; REAL_MUL_LID]);;
2413 let MATRIX_ADD_SYM = prove
2414 (`!A:real^N^M B. A + B = B + A`,
2415 SIMP_TAC[matrix_add; CART_EQ; LAMBDA_BETA; REAL_ADD_AC]);;
2417 let MATRIX_ADD_ASSOC = prove
2418 (`!A:real^N^M B C. A + (B + C) = (A + B) + C`,
2419 SIMP_TAC[matrix_add; CART_EQ; LAMBDA_BETA; REAL_ADD_AC]);;
2421 let MATRIX_ADD_LID = prove
2422 (`!A. mat 0 + A = A`,
2423 SIMP_TAC[matrix_add; mat; COND_ID; CART_EQ; LAMBDA_BETA; REAL_ADD_LID]);;
2425 let MATRIX_ADD_RID = prove
2426 (`!A. A + mat 0 = A`,
2427 SIMP_TAC[matrix_add; mat; COND_ID; CART_EQ; LAMBDA_BETA; REAL_ADD_RID]);;
2429 let MATRIX_ADD_LNEG = prove
2430 (`!A. --A + A = mat 0`,
2431 SIMP_TAC[matrix_neg; matrix_add; mat; COND_ID;
2432 CART_EQ; LAMBDA_BETA; REAL_ADD_LINV]);;
2434 let MATRIX_ADD_RNEG = prove
2435 (`!A. A + --A = mat 0`,
2436 SIMP_TAC[matrix_neg; matrix_add; mat; COND_ID;
2437 CART_EQ; LAMBDA_BETA; REAL_ADD_RINV]);;
2439 let MATRIX_SUB = prove
2440 (`!A:real^N^M B. A - B = A + --B`,
2441 SIMP_TAC[matrix_neg; matrix_add; matrix_sub; CART_EQ; LAMBDA_BETA;
2444 let MATRIX_SUB_REFL = prove
2445 (`!A. A - A = mat 0`,
2446 REWRITE_TAC[MATRIX_SUB; MATRIX_ADD_RNEG]);;
2448 let MATRIX_ADD_LDISTRIB = prove
2449 (`!A:real^N^M B:real^P^N C. A ** (B + C) = A ** B + A ** C`,
2450 SIMP_TAC[matrix_mul; matrix_add; CART_EQ; LAMBDA_BETA;
2451 GSYM SUM_ADD_NUMSEG] THEN
2452 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN
2453 ASM_SIMP_TAC[LAMBDA_BETA; REAL_ADD_LDISTRIB]);;
2455 let MATRIX_MUL_LID = prove
2456 (`!A:real^N^M. mat 1 ** A = A`,
2457 REWRITE_TAC[matrix_mul;
2458 GEN_REWRITE_RULE (RAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ]
2459 (SPEC_ALL mat)] THEN
2460 SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REWRITE_TAC[COND_RATOR; COND_RAND] THEN
2461 SIMP_TAC[SUM_DELTA; REAL_MUL_LZERO; IN_NUMSEG; REAL_MUL_LID]);;
2463 let MATRIX_MUL_RID = prove
2464 (`!A:real^N^M. A ** mat 1 = A`,
2465 REWRITE_TAC[matrix_mul; mat] THEN
2466 SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REWRITE_TAC[COND_RATOR; COND_RAND] THEN
2467 SIMP_TAC[SUM_DELTA; REAL_MUL_RZERO; IN_NUMSEG; REAL_MUL_RID]);;
2469 let MATRIX_MUL_ASSOC = prove
2470 (`!A:real^N^M B:real^P^N C:real^Q^P. A ** B ** C = (A ** B) ** C`,
2472 SIMP_TAC[matrix_mul; CART_EQ; LAMBDA_BETA; GSYM SUM_LMUL; GSYM SUM_RMUL] THEN
2473 REWRITE_TAC[REAL_MUL_ASSOC] THEN REPEAT STRIP_TAC THEN
2474 GEN_REWRITE_TAC RAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[]);;
2476 let MATRIX_MUL_LZERO = prove
2477 (`!A. (mat 0:real^N^M) ** (A:real^P^N) = mat 0`,
2478 SIMP_TAC[matrix_mul; mat; CART_EQ; LAMBDA_BETA; COND_ID; REAL_MUL_LZERO] THEN
2479 REWRITE_TAC[SUM_0]);;
2481 let MATRIX_MUL_RZERO = prove
2482 (`!A. (A:real^N^M) ** (mat 0:real^P^N) = mat 0`,
2483 SIMP_TAC[matrix_mul; mat; CART_EQ; LAMBDA_BETA; COND_ID; REAL_MUL_RZERO] THEN
2484 REWRITE_TAC[SUM_0]);;
2486 let MATRIX_ADD_RDISTRIB = prove
2487 (`!A:real^N^M B C:real^P^N. (A + B) ** C = A ** C + B ** C`,
2488 SIMP_TAC[matrix_mul; matrix_add; CART_EQ; LAMBDA_BETA] THEN
2489 REWRITE_TAC[REAL_ADD_RDISTRIB; SUM_ADD_NUMSEG]);;
2491 let MATRIX_SUB_LDISTRIB = prove
2492 (`!A:real^N^M B C:real^P^N. A ** (B - C) = A ** B - A ** C`,
2493 SIMP_TAC[matrix_mul; matrix_sub; CART_EQ; LAMBDA_BETA] THEN
2494 REWRITE_TAC[REAL_SUB_LDISTRIB; SUM_SUB_NUMSEG]);;
2496 let MATRIX_SUB_RDISTRIB = prove
2497 (`!A:real^N^M B C:real^P^N. (A - B) ** C = A ** C - B ** C`,
2498 SIMP_TAC[matrix_mul; matrix_sub; CART_EQ; LAMBDA_BETA] THEN
2499 REWRITE_TAC[REAL_SUB_RDISTRIB; SUM_SUB_NUMSEG]);;
2501 let MATRIX_MUL_LMUL = prove
2502 (`!A:real^N^M B:real^P^N c. (c %% A) ** B = c %% (A ** B)`,
2503 SIMP_TAC[matrix_mul; matrix_cmul; CART_EQ; LAMBDA_BETA] THEN
2504 REWRITE_TAC[GSYM REAL_MUL_ASSOC; SUM_LMUL]);;
2506 let MATRIX_MUL_RMUL = prove
2507 (`!A:real^N^M B:real^P^N c. A ** (c %% B) = c %% (A ** B)`,
2508 SIMP_TAC[matrix_mul; matrix_cmul; CART_EQ; LAMBDA_BETA] THEN
2509 ONCE_REWRITE_TAC[REAL_ARITH `A * c * B:real = c * A * B`] THEN
2510 REWRITE_TAC[SUM_LMUL]);;
2512 let MATRIX_CMUL_ADD_LDISTRIB = prove
2513 (`!A:real^N^M B c. c %% (A + B) = c %% A + c %% B`,
2514 SIMP_TAC[matrix_cmul; matrix_add; CART_EQ; LAMBDA_BETA] THEN
2515 REWRITE_TAC[REAL_ADD_LDISTRIB]);;
2517 let MATRIX_CMUL_SUB_LDISTRIB = prove
2518 (`!A:real^N^M B c. c %% (A - B) = c %% A - c %% B`,
2519 SIMP_TAC[matrix_cmul; matrix_sub; CART_EQ; LAMBDA_BETA] THEN
2520 REWRITE_TAC[REAL_SUB_LDISTRIB]);;
2522 let MATRIX_CMUL_ADD_RDISTRIB = prove
2523 (`!A:real^N^M b c. (b + c) %% A = b %% A + c %% A`,
2524 SIMP_TAC[matrix_cmul; matrix_add; CART_EQ; LAMBDA_BETA] THEN
2525 REWRITE_TAC[REAL_ADD_RDISTRIB]);;
2527 let MATRIX_CMUL_SUB_RDISTRIB = prove
2528 (`!A:real^N^M b c. (b - c) %% A = b %% A - c %% A`,
2529 SIMP_TAC[matrix_cmul; matrix_sub; CART_EQ; LAMBDA_BETA] THEN
2530 REWRITE_TAC[REAL_SUB_RDISTRIB]);;
2532 let MATRIX_CMUL_RZERO = prove
2533 (`!c. c %% mat 0 = mat 0`,
2534 SIMP_TAC[matrix_cmul; mat; CART_EQ; LAMBDA_BETA; COND_ID; REAL_MUL_RZERO]);;
2536 let MATRIX_CMUL_LZERO = prove
2537 (`!A. &0 %% A = mat 0`,
2538 SIMP_TAC[matrix_cmul; mat; CART_EQ; LAMBDA_BETA; COND_ID; REAL_MUL_LZERO]);;
2540 let MATRIX_NEG_MINUS1 = prove
2541 (`!A:real^N^M. --A = --(&1) %% A`,
2542 REWRITE_TAC[matrix_cmul; matrix_neg; CART_EQ; LAMBDA_BETA] THEN
2543 REWRITE_TAC[GSYM REAL_NEG_MINUS1]);;
2545 let MATRIX_ADD_AC = prove
2546 (`(A:real^N^M) + B = B + A /\
2547 (A + B) + C = A + (B + C) /\
2548 A + (B + C) = B + (A + C)`,
2549 MESON_TAC[MATRIX_ADD_ASSOC; MATRIX_ADD_SYM]);;
2551 let MATRIX_NEG_ADD = prove
2552 (`!A B:real^N^M. --(A + B) = --A + --B`,
2553 SIMP_TAC[matrix_neg; matrix_add; CART_EQ; LAMBDA_BETA; REAL_NEG_ADD]);;
2555 let MATRIX_NEG_SUB = prove
2556 (`!A B:real^N^M. --(A - B) = B - A`,
2557 SIMP_TAC[matrix_neg; matrix_sub; CART_EQ; LAMBDA_BETA; REAL_NEG_SUB]);;
2559 let MATRIX_NEG_0 = prove
2560 (`--(mat 0) = mat 0`,
2561 SIMP_TAC[CART_EQ; mat; matrix_neg; LAMBDA_BETA; REAL_NEG_0; COND_ID]);;
2563 let MATRIX_SUB_RZERO = prove
2564 (`!A:real^N^M. A - mat 0 = A`,
2565 SIMP_TAC[CART_EQ; mat; matrix_sub; LAMBDA_BETA; REAL_SUB_RZERO; COND_ID]);;
2567 let MATRIX_SUB_LZERO = prove
2568 (`!A:real^N^M. mat 0 - A = --A`,
2569 SIMP_TAC[CART_EQ; mat; matrix_sub; matrix_neg;
2570 LAMBDA_BETA; REAL_SUB_LZERO; COND_ID]);;
2572 let MATRIX_NEG_EQ_0 = prove
2573 (`!A:real^N^M. --A = mat 0 <=> A = mat 0`,
2574 SIMP_TAC[CART_EQ; matrix_neg; mat; LAMBDA_BETA; REAL_NEG_EQ_0; COND_ID]);;
2576 let MATRIX_VECTOR_MUL_ASSOC = prove
2577 (`!A:real^N^M B:real^P^N x:real^P. A ** B ** x = (A ** B) ** x`,
2579 SIMP_TAC[matrix_mul; matrix_vector_mul;
2580 CART_EQ; LAMBDA_BETA; GSYM SUM_LMUL; GSYM SUM_RMUL] THEN
2581 REWRITE_TAC[REAL_MUL_ASSOC] THEN REPEAT STRIP_TAC THEN
2582 GEN_REWRITE_TAC RAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[]);;
2584 let MATRIX_VECTOR_MUL_LID = prove
2585 (`!x:real^N. mat 1 ** x = x`,
2586 REWRITE_TAC[matrix_vector_mul;
2587 GEN_REWRITE_RULE (RAND_CONV o ONCE_DEPTH_CONV) [EQ_SYM_EQ]
2588 (SPEC_ALL mat)] THEN
2589 SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN REWRITE_TAC[COND_RATOR; COND_RAND] THEN
2590 SIMP_TAC[SUM_DELTA; REAL_MUL_LZERO; IN_NUMSEG; REAL_MUL_LID]);;
2592 let MATRIX_VECTOR_MUL_LZERO = prove
2593 (`!x:real^N. mat 0 ** x = vec 0`,
2594 SIMP_TAC[mat; matrix_vector_mul; CART_EQ; VEC_COMPONENT; LAMBDA_BETA;
2595 COND_ID; REAL_MUL_LZERO; SUM_0]);;
2597 let MATRIX_VECTOR_MUL_RZERO = prove
2598 (`!A:real^M^N. A ** vec 0 = vec 0`,
2599 SIMP_TAC[mat; matrix_vector_mul; CART_EQ; VEC_COMPONENT; LAMBDA_BETA;
2600 COND_ID; REAL_MUL_RZERO; SUM_0]);;
2602 let MATRIX_VECTOR_MUL_ADD_LDISTRIB = prove
2603 (`!A:real^M^N x:real^M y. A ** (x + y) = A ** x + A ** y`,
2604 SIMP_TAC[CART_EQ; matrix_vector_mul; VECTOR_ADD_COMPONENT; LAMBDA_BETA;
2605 SUM_ADD_NUMSEG; REAL_ADD_LDISTRIB]);;
2607 let MATRIX_VECTOR_MUL_SUB_LDISTRIB = prove
2608 (`!A:real^M^N x:real^M y. A ** (x - y) = A ** x - A ** y`,
2609 SIMP_TAC[CART_EQ; matrix_vector_mul; VECTOR_SUB_COMPONENT; LAMBDA_BETA;
2610 SUM_SUB_NUMSEG; REAL_SUB_LDISTRIB]);;
2612 let MATRIX_VECTOR_MUL_ADD_RDISTRIB = prove
2613 (`!A:real^M^N B x. (A + B) ** x = (A ** x) + (B ** x)`,
2614 SIMP_TAC[CART_EQ; matrix_vector_mul; matrix_add; LAMBDA_BETA;
2615 VECTOR_ADD_COMPONENT; REAL_ADD_RDISTRIB; SUM_ADD_NUMSEG]);;
2617 let MATRIX_VECTOR_MUL_SUB_RDISTRIB = prove
2618 (`!A:real^M^N B x. (A - B) ** x = (A ** x) - (B ** x)`,
2619 SIMP_TAC[CART_EQ; matrix_vector_mul; matrix_sub; LAMBDA_BETA;
2620 VECTOR_SUB_COMPONENT; REAL_SUB_RDISTRIB; SUM_SUB_NUMSEG]);;
2622 let MATRIX_VECTOR_MUL_RMUL = prove
2623 (`!A:real^M^N x:real^M c. A ** (c % x) = c % (A ** x)`,
2624 SIMP_TAC[CART_EQ; VECTOR_MUL_COMPONENT; matrix_vector_mul; LAMBDA_BETA] THEN
2625 REWRITE_TAC[GSYM SUM_LMUL] THEN REWRITE_TAC[REAL_MUL_AC]);;
2627 let MATRIX_TRANSP_MUL = prove
2628 (`!A B. transp(A ** B) = transp(B) ** transp(A)`,
2629 SIMP_TAC[matrix_mul; transp; CART_EQ; LAMBDA_BETA] THEN
2630 REWRITE_TAC[REAL_MUL_AC]);;
2632 let MATRIX_EQ = prove
2633 (`!A:real^N^M B. (A = B) = !x:real^N. A ** x = B ** x`,
2634 REPEAT GEN_TAC THEN EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
2635 DISCH_THEN(MP_TAC o GEN `i:num` o SPEC `(basis i):real^N`) THEN
2636 SIMP_TAC[CART_EQ; matrix_vector_mul; LAMBDA_BETA; basis] THEN
2637 SIMP_TAC[SUM_DELTA; COND_RAND; REAL_MUL_RZERO] THEN
2638 REWRITE_TAC[TAUT `(if p then b else T) <=> p ==> b`] THEN
2639 SIMP_TAC[REAL_MUL_RID; IN_NUMSEG]);;
2641 let MATRIX_VECTOR_MUL_COMPONENT = prove
2643 1 <= k /\ k <= dimindex(:M) ==> ((A ** x)$k = (A$k) dot x)`,
2644 SIMP_TAC[matrix_vector_mul; LAMBDA_BETA; dot]);;
2646 let DOT_LMUL_MATRIX = prove
2647 (`!A:real^N^M x:real^M y:real^N. (x ** A) dot y = x dot (A ** y)`,
2648 SIMP_TAC[dot; matrix_vector_mul; vector_matrix_mul; dot; LAMBDA_BETA] THEN
2649 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM SUM_LMUL] THEN
2650 REWRITE_TAC[GSYM SUM_RMUL] THEN
2651 GEN_REWRITE_TAC RAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[REAL_MUL_AC]);;
2653 let TRANSP_MATRIX_CMUL = prove
2654 (`!A:real^M^N c. transp(c %% A) = c %% transp A`,
2655 SIMP_TAC[CART_EQ; transp; MATRIX_CMUL_COMPONENT; LAMBDA_BETA]);;
2657 let TRANSP_MATRIX_ADD = prove
2658 (`!A B:real^N^M. transp(A + B) = transp A + transp B`,
2659 SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; matrix_add]);;
2661 let TRANSP_MATRIX_SUB = prove
2662 (`!A B:real^N^M. transp(A - B) = transp A - transp B`,
2663 SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; matrix_sub]);;
2665 let TRANSP_MATRIX_NEG = prove
2666 (`!A:real^N^M. transp(--A) = --(transp A)`,
2667 SIMP_TAC[CART_EQ; transp; LAMBDA_BETA; matrix_neg]);;
2669 let TRANSP_MAT = prove
2670 (`!n. transp(mat n) = mat n`,
2671 SIMP_TAC[transp; mat; LAMBDA_BETA; CART_EQ; EQ_SYM_EQ]);;
2673 let TRANSP_TRANSP = prove
2674 (`!A:real^N^M. transp(transp A) = A`,
2675 SIMP_TAC[CART_EQ; transp; LAMBDA_BETA]);;
2677 let TRANSP_EQ = prove
2678 (`!A B:real^M^N. transp A = transp B <=> A = B`,
2679 MESON_TAC[TRANSP_TRANSP]);;
2681 let ROW_TRANSP = prove
2683 1 <= i /\ i <= dimindex(:N) ==> row i (transp A) = column i A`,
2684 SIMP_TAC[row; column; transp; CART_EQ; LAMBDA_BETA]);;
2686 let COLUMN_TRANSP = prove
2688 1 <= i /\ i <= dimindex(:M) ==> column i (transp A) = row i A`,
2689 SIMP_TAC[row; column; transp; CART_EQ; LAMBDA_BETA]);;
2691 let ROWS_TRANSP = prove
2692 (`!A:real^N^M. rows(transp A) = columns A`,
2693 REWRITE_TAC[rows; columns; EXTENSION; IN_ELIM_THM] THEN
2694 MESON_TAC[ROW_TRANSP]);;
2696 let COLUMNS_TRANSP = prove
2697 (`!A:real^N^M. columns(transp A) = rows A`,
2698 MESON_TAC[TRANSP_TRANSP; ROWS_TRANSP]);;
2700 let VECTOR_MATRIX_MUL_TRANSP = prove
2701 (`!A:real^M^N x:real^N. x ** A = transp A ** x`,
2702 REWRITE_TAC[matrix_vector_mul; vector_matrix_mul; transp] THEN
2703 SIMP_TAC[LAMBDA_BETA; CART_EQ]);;
2705 let MATRIX_VECTOR_MUL_TRANSP = prove
2706 (`!A:real^M^N x:real^M. A ** x = x ** transp A`,
2707 REWRITE_TAC[VECTOR_MATRIX_MUL_TRANSP; TRANSP_TRANSP]);;
2709 (* ------------------------------------------------------------------------- *)
2710 (* Two sometimes fruitful ways of looking at matrix-vector multiplication. *)
2711 (* ------------------------------------------------------------------------- *)
2713 let MATRIX_MUL_DOT = prove
2714 (`!A:real^N^M x. A ** x = lambda i. A$i dot x`,
2715 REWRITE_TAC[matrix_vector_mul; dot] THEN SIMP_TAC[CART_EQ; LAMBDA_BETA]);;
2717 let MATRIX_MUL_VSUM = prove
2718 (`!A:real^N^M x. A ** x = vsum(1..dimindex(:N)) (\i. x$i % column i A)`,
2719 SIMP_TAC[matrix_vector_mul; CART_EQ; VSUM_COMPONENT; LAMBDA_BETA;
2720 VECTOR_MUL_COMPONENT; column; REAL_MUL_AC]);;
2722 (* ------------------------------------------------------------------------- *)
2723 (* Slightly gruesome lemmas: better to define sums over vectors really... *)
2724 (* ------------------------------------------------------------------------- *)
2726 let VECTOR_COMPONENTWISE = prove
2728 x = lambda j. sum(1..dimindex(:N))
2729 (\i. x$i * (basis i :real^N)$j)`,
2730 SIMP_TAC[CART_EQ; LAMBDA_BETA; basis] THEN
2731 ONCE_REWRITE_TAC[ARITH_RULE `(m:num = n) <=> (n = m)`] THEN
2732 SIMP_TAC[COND_RAND; REAL_MUL_RZERO; SUM_DELTA; IN_NUMSEG] THEN
2733 REWRITE_TAC[REAL_MUL_RID; COND_ID]);;
2735 let LINEAR_COMPONENTWISE = prove
2736 (`!f:real^M->real^N.
2738 ==> !x j. 1 <= j /\ j <= dimindex(:N)
2740 sum(1..dimindex(:M)) (\i. x$i * f(basis i)$j))`,
2741 REWRITE_TAC[linear] THEN REPEAT STRIP_TAC THEN
2742 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV)
2743 [VECTOR_COMPONENTWISE] THEN
2744 SPEC_TAC(`dimindex(:M)`,`n:num`) THEN
2745 INDUCT_TAC THEN REWRITE_TAC[SUM_CLAUSES_NUMSEG; ARITH] THENL
2746 [REWRITE_TAC[GSYM vec] THEN
2747 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV)
2748 [GSYM VECTOR_MUL_LZERO] THEN
2749 ASM_REWRITE_TAC[] THEN REWRITE_TAC[VECTOR_MUL_LZERO] THEN
2750 ASM_SIMP_TAC[vec; LAMBDA_BETA];
2751 REWRITE_TAC[ARITH_RULE `1 <= SUC n`] THEN
2752 ASSUM_LIST(fun thl -> REWRITE_TAC(map GSYM thl)) THEN
2753 SIMP_TAC[GSYM VECTOR_MUL_COMPONENT;
2754 ASSUME `1 <= j`; ASSUME `j <= dimindex(:N)`] THEN
2755 ASSUM_LIST(fun thl -> REWRITE_TAC(map GSYM thl)) THEN
2756 SIMP_TAC[GSYM VECTOR_ADD_COMPONENT;
2757 ASSUME `1 <= j`; ASSUME `j <= dimindex(:N)`] THEN
2758 ASSUM_LIST(fun thl -> REWRITE_TAC(map GSYM thl)) THEN
2759 AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
2760 ASM_SIMP_TAC[CART_EQ; VECTOR_ADD_COMPONENT; LAMBDA_BETA] THEN
2761 SIMP_TAC[VECTOR_MUL_COMPONENT]]);;
2763 (* ------------------------------------------------------------------------- *)
2764 (* Inverse matrices (not necessarily square, but it's vacuous otherwise). *)
2765 (* ------------------------------------------------------------------------- *)
2767 let invertible = new_definition
2768 `invertible(A:real^N^M) <=>
2769 ?A':real^M^N. (A ** A' = mat 1) /\ (A' ** A = mat 1)`;;
2771 let matrix_inv = new_definition
2772 `matrix_inv(A:real^N^M) =
2773 @A':real^M^N. (A ** A' = mat 1) /\ (A' ** A = mat 1)`;;
2775 let MATRIX_INV = prove
2777 invertible A ==> A ** matrix_inv A = mat 1 /\ matrix_inv A ** A = mat 1`,
2778 GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[matrix_inv; invertible] THEN
2779 CONV_TAC SELECT_CONV THEN ASM_REWRITE_TAC[GSYM invertible]);;
2781 (* ------------------------------------------------------------------------- *)
2782 (* Correspondence between matrices and linear operators. *)
2783 (* ------------------------------------------------------------------------- *)
2785 let matrix = new_definition
2786 `(matrix:(real^M->real^N)->real^M^N) f = lambda i j. f(basis j)$i`;;
2788 let MATRIX_VECTOR_MUL_LINEAR = prove
2789 (`!A:real^N^M. linear(\x. A ** x)`,
2790 REWRITE_TAC[linear; matrix_vector_mul] THEN
2791 SIMP_TAC[CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
2792 VECTOR_MUL_COMPONENT] THEN
2793 REWRITE_TAC[GSYM SUM_ADD_NUMSEG; GSYM SUM_LMUL; REAL_ADD_LDISTRIB] THEN
2794 REWRITE_TAC[REAL_ADD_AC; REAL_MUL_AC]);;
2796 let MATRIX_WORKS = prove
2797 (`!f:real^M->real^N. linear f ==> !x. matrix f ** x = f(x)`,
2798 REWRITE_TAC[matrix; matrix_vector_mul] THEN
2799 SIMP_TAC[CART_EQ; LAMBDA_BETA] THEN GEN_TAC THEN DISCH_TAC THEN
2800 REPEAT GEN_TAC THEN DISCH_TAC THEN
2801 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
2802 ASM_SIMP_TAC[GSYM LINEAR_COMPONENTWISE]);;
2804 let MATRIX_VECTOR_MUL = prove
2805 (`!f:real^M->real^N. linear f ==> f = \x. matrix f ** x`,
2806 SIMP_TAC[FUN_EQ_THM; MATRIX_WORKS]);;
2808 let MATRIX_OF_MATRIX_VECTOR_MUL = prove
2809 (`!A:real^N^M. matrix(\x. A ** x) = A`,
2810 SIMP_TAC[MATRIX_EQ; MATRIX_VECTOR_MUL_LINEAR; MATRIX_WORKS]);;
2812 let MATRIX_COMPOSE = prove
2813 (`!f g. linear f /\ linear g ==> (matrix(g o f) = matrix g ** matrix f)`,
2814 SIMP_TAC[MATRIX_EQ; MATRIX_WORKS; LINEAR_COMPOSE;
2815 GSYM MATRIX_VECTOR_MUL_ASSOC; o_THM]);;
2817 let MATRIX_VECTOR_COLUMN = prove
2819 A ** x = vsum(1..dimindex(:N)) (\i. x$i % (transp A)$i)`,
2820 REWRITE_TAC[matrix_vector_mul; transp] THEN
2821 SIMP_TAC[CART_EQ; LAMBDA_BETA; VSUM_COMPONENT; VECTOR_MUL_COMPONENT] THEN
2822 REWRITE_TAC[REAL_MUL_AC]);;
2824 let MATRIX_MUL_COMPONENT = prove
2825 (`!i. 1 <= i /\ i <= dimindex(:N)
2826 ==> ((A:real^N^N) ** (B:real^N^N))$i = transp B ** A$i`,
2827 SIMP_TAC[matrix_mul; LAMBDA_BETA; matrix_vector_mul; vector_matrix_mul;
2828 transp; CART_EQ] THEN
2829 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUM_EQ_NUMSEG THEN
2830 REWRITE_TAC[REAL_MUL_AC]);;
2832 let ADJOINT_MATRIX = prove
2833 (`!A:real^N^M. adjoint(\x. A ** x) = (\x. transp A ** x)`,
2834 GEN_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC ADJOINT_UNIQUE THEN
2835 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN REPEAT GEN_TAC THEN
2836 SIMP_TAC[transp; dot; LAMBDA_BETA; matrix_vector_mul;
2837 GSYM SUM_LMUL; GSYM SUM_RMUL] THEN
2838 GEN_REWRITE_TAC LAND_CONV [SUM_SWAP_NUMSEG] THEN REWRITE_TAC[REAL_MUL_AC]);;
2840 let MATRIX_ADJOINT = prove
2841 (`!f. linear f ==> matrix(adjoint f) = transp(matrix f)`,
2842 GEN_TAC THEN DISCH_THEN
2843 (fun th -> GEN_REWRITE_TAC (LAND_CONV o funpow 2 RAND_CONV)
2844 [MATCH_MP MATRIX_VECTOR_MUL th]) THEN
2845 REWRITE_TAC[ADJOINT_MATRIX; MATRIX_OF_MATRIX_VECTOR_MUL]);;
2847 let MATRIX_ID = prove
2848 (`matrix(\x. x) = mat 1`,
2849 SIMP_TAC[MATRIX_EQ; LINEAR_ID; MATRIX_WORKS; MATRIX_VECTOR_MUL_LID]);;
2851 let MATRIX_I = prove
2852 (`matrix I = mat 1`,
2853 REWRITE_TAC[I_DEF; MATRIX_ID]);;
2855 let LINEAR_EQ_MATRIX = prove
2856 (`!f g. linear f /\ linear g /\ matrix f = matrix g ==> f = g`,
2857 REPEAT STRIP_TAC THEN
2858 REPEAT(FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP MATRIX_VECTOR_MUL)) THEN
2859 ASM_REWRITE_TAC[]);;
2861 (* ------------------------------------------------------------------------- *)
2862 (* Operator norm. *)
2863 (* ------------------------------------------------------------------------- *)
2865 let onorm = new_definition
2866 `onorm (f:real^M->real^N) = sup { norm(f x) | norm(x) = &1 }`;;
2868 let NORM_BOUND_GENERALIZE = prove
2869 (`!f:real^M->real^N b.
2871 ==> ((!x. (norm(x) = &1) ==> norm(f x) <= b) <=>
2872 (!x. norm(f x) <= b * norm(x)))`,
2873 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
2874 [ALL_TAC; ASM_MESON_TAC[REAL_MUL_RID]] THEN
2875 X_GEN_TAC `x:real^M` THEN ASM_CASES_TAC `x:real^M = vec 0` THENL
2876 [ASM_REWRITE_TAC[NORM_0; REAL_MUL_RZERO] THEN
2877 ASM_MESON_TAC[LINEAR_0; NORM_0; REAL_LE_REFL];
2879 ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; NORM_POS_LT; real_div] THEN
2880 MATCH_MP_TAC(REAL_ARITH `abs(a * b) <= c ==> b * a <= c`) THEN
2881 REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM; GSYM NORM_MUL] THEN
2882 FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN
2883 ASM_SIMP_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM; REAL_MUL_LINV;
2887 (`!f:real^M->real^N.
2889 ==> (!x. norm(f x) <= onorm f * norm(x)) /\
2890 (!b. (!x. norm(f x) <= b * norm(x)) ==> onorm f <= b)`,
2891 GEN_TAC THEN DISCH_TAC THEN
2892 MP_TAC(SPEC `{ norm((f:real^M->real^N) x) | norm(x) = &1 }` SUP) THEN
2893 SIMP_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
2894 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
2895 REWRITE_TAC[LEFT_FORALL_IMP_THM; RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN
2896 ASM_SIMP_TAC[NORM_BOUND_GENERALIZE; GSYM onorm; GSYM MEMBER_NOT_EMPTY] THEN
2897 DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
2898 ASM_MESON_TAC[VECTOR_CHOOSE_SIZE; LINEAR_BOUNDED; REAL_POS]);;
2900 let ONORM_POS_LE = prove
2901 (`!f. linear f ==> &0 <= onorm f`,
2902 MESON_TAC[ONORM; VECTOR_CHOOSE_SIZE; REAL_POS; REAL_MUL_RID; NORM_POS_LE;
2905 let ONORM_EQ_0 = prove
2906 (`!f:real^M->real^N. linear f ==> ((onorm f = &0) <=> (!x. f x = vec 0))`,
2907 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THEN
2908 MP_TAC(SPEC `f:real^M->real^N` ONORM) THEN
2909 ASM_SIMP_TAC[GSYM REAL_LE_ANTISYM; ONORM_POS_LE; NORM_0; REAL_MUL_LZERO;
2910 NORM_LE_0; REAL_LE_REFL]);;
2912 let ONORM_CONST = prove
2913 (`!y:real^N. onorm(\x:real^M. y) = norm(y)`,
2914 GEN_TAC THEN REWRITE_TAC[onorm] THEN
2915 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sup {norm(y:real^N)}` THEN
2917 [AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE
2918 `(?x. P x) ==> {f y | x | P x} = {f y}`) THEN
2919 EXISTS_TAC `basis 1 :real^M` THEN
2920 SIMP_TAC[NORM_BASIS; DIMINDEX_GE_1; LE_REFL];
2921 MATCH_MP_TAC REAL_SUP_UNIQUE THEN SET_TAC[REAL_LE_REFL]]);;
2923 let ONORM_POS_LT = prove
2924 (`!f. linear f ==> (&0 < onorm f <=> ~(!x. f x = vec 0))`,
2925 SIMP_TAC[GSYM ONORM_EQ_0; ONORM_POS_LE;
2926 REAL_ARITH `(&0 < x <=> ~(x = &0)) <=> &0 <= x`]);;
2928 let ONORM_COMPOSE = prove
2929 (`!f g. linear f /\ linear g ==> onorm(f o g) <= onorm f * onorm g`,
2930 MESON_TAC[ONORM; LINEAR_COMPOSE; o_THM; REAL_MUL_ASSOC; REAL_LE_TRANS; ONORM;
2931 REAL_LE_LMUL; ONORM_POS_LE]);;
2933 let ONORM_NEG_LEMMA = prove
2934 (`!f. linear f ==> onorm(\x. --(f x)) <= onorm f`,
2935 REPEAT STRIP_TAC THEN
2936 FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP ONORM o
2937 MATCH_MP LINEAR_COMPOSE_NEG) THEN
2938 FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[NORM_NEG; ONORM]);;
2940 let ONORM_NEG = prove
2941 (`!f:real^M->real^N. linear f ==> (onorm(\x. --(f x)) = onorm f)`,
2942 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN
2943 ASM_SIMP_TAC[ONORM_NEG_LEMMA] THEN
2944 SUBGOAL_THEN `f:real^M->real^N = \x. --(--(f x))`
2945 (fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [th]) THEN
2946 ASM_SIMP_TAC[ONORM_NEG_LEMMA; LINEAR_COMPOSE_NEG] THEN
2947 REWRITE_TAC[VECTOR_NEG_NEG; ETA_AX]);;
2949 let ONORM_TRIANGLE = prove
2950 (`!f:real^M->real^N g.
2951 linear f /\ linear g ==> onorm(\x. f x + g x) <= onorm f + onorm g`,
2952 REPEAT GEN_TAC THEN DISCH_TAC THEN
2953 FIRST_ASSUM(MATCH_MP_TAC o CONJUNCT2 o MATCH_MP ONORM o MATCH_MP
2954 LINEAR_COMPOSE_ADD) THEN
2955 REWRITE_TAC[REAL_ADD_RDISTRIB] THEN
2956 ASM_MESON_TAC[REAL_LE_ADD2; REAL_LE_TRANS; NORM_TRIANGLE; ONORM]);;
2958 let ONORM_TRIANGLE_LE = prove
2959 (`!f g. linear f /\ linear g /\ onorm(f) + onorm(g) <= e
2960 ==> onorm(\x. f x + g x) <= e`,
2961 MESON_TAC[REAL_LE_TRANS; ONORM_TRIANGLE]);;
2963 let ONORM_TRIANGLE_LT = prove
2964 (`!f g. linear f /\ linear g /\ onorm(f) + onorm(g) < e
2965 ==> onorm(\x. f x + g x) < e`,
2966 MESON_TAC[REAL_LET_TRANS; ONORM_TRIANGLE]);;
2968 (* ------------------------------------------------------------------------- *)
2969 (* It's handy to "lift" from R to R^1 and "drop" from R^1 to R. *)
2970 (* ------------------------------------------------------------------------- *)
2972 let lift = new_definition
2973 `(lift:real->real^1) x = lambda i. x`;;
2975 let drop = new_definition
2976 `(drop:real^1->real) x = x$1`;;
2978 let LIFT_COMPONENT = prove
2979 (`!x. (lift x)$1 = x`,
2980 SIMP_TAC[lift; LAMBDA_BETA; DIMINDEX_1; LE_ANTISYM]);;
2982 let LIFT_DROP = prove
2983 (`(!x. lift(drop x) = x) /\ (!x. drop(lift x) = x)`,
2984 SIMP_TAC[lift; drop; CART_EQ; LAMBDA_BETA; DIMINDEX_1; LE_ANTISYM]);;
2986 let IMAGE_LIFT_DROP = prove
2987 (`(!s. IMAGE (lift o drop) s = s) /\ (!s. IMAGE (drop o lift) s = s)`,
2988 REWRITE_TAC[o_DEF; LIFT_DROP] THEN SET_TAC[]);;
2990 let IN_IMAGE_LIFT_DROP = prove
2991 (`(!x s. x IN IMAGE lift s <=> drop x IN s) /\
2992 (!x s. x IN IMAGE drop s <=> lift x IN s)`,
2993 REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[LIFT_DROP]);;
2995 let FORALL_LIFT = prove
2996 (`(!x. P x) = (!x. P(lift x))`,
2997 MESON_TAC[LIFT_DROP]);;
2999 let EXISTS_LIFT = prove
3000 (`(?x. P x) = (?x. P(lift x))`,
3001 MESON_TAC[LIFT_DROP]);;
3003 let FORALL_DROP = prove
3004 (`(!x. P x) = (!x. P(drop x))`,
3005 MESON_TAC[LIFT_DROP]);;
3007 let EXISTS_DROP = prove
3008 (`(?x. P x) = (?x. P(drop x))`,
3009 MESON_TAC[LIFT_DROP]);;
3011 let FORALL_LIFT_FUN = prove
3012 (`!P:(A->real^1)->bool. (!f. P f) <=> (!f. P(lift o f))`,
3013 GEN_TAC THEN EQ_TAC THEN SIMP_TAC[] THEN DISCH_TAC THEN
3014 X_GEN_TAC `f:A->real^1` THEN
3015 FIRST_X_ASSUM(MP_TAC o SPEC `drop o (f:A->real^1)`) THEN
3016 REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]);;
3018 let FORALL_DROP_FUN = prove
3019 (`!P:(A->real)->bool. (!f. P f) <=> (!f. P(drop o f))`,
3020 REWRITE_TAC[FORALL_LIFT_FUN; o_DEF; LIFT_DROP; ETA_AX]);;
3022 let EXISTS_LIFT_FUN = prove
3023 (`!P:(A->real^1)->bool. (?f. P f) <=> (?f. P(lift o f))`,
3024 ONCE_REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN
3025 REWRITE_TAC[FORALL_LIFT_FUN]);;
3027 let EXISTS_DROP_FUN = prove
3028 (`!P:(A->real)->bool. (?f. P f) <=> (?f. P(drop o f))`,
3029 ONCE_REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN
3030 REWRITE_TAC[FORALL_DROP_FUN]);;
3033 (`!x y. (lift x = lift y) <=> (x = y)`,
3034 MESON_TAC[LIFT_DROP]);;
3037 (`!x y. (drop x = drop y) <=> (x = y)`,
3038 MESON_TAC[LIFT_DROP]);;
3040 let LIFT_IN_IMAGE_LIFT = prove
3041 (`!x s. (lift x) IN (IMAGE lift s) <=> x IN s`,
3042 REWRITE_TAC[IN_IMAGE] THEN MESON_TAC[LIFT_DROP]);;
3044 let LIFT_NUM = prove
3045 (`!n. lift(&n) = vec n`,
3046 SIMP_TAC[CART_EQ; lift; vec; LAMBDA_BETA]);;
3048 let LIFT_ADD = prove
3049 (`!x y. lift(x + y) = lift x + lift y`,
3050 SIMP_TAC[CART_EQ; lift; LAMBDA_BETA; VECTOR_ADD_COMPONENT]);;
3052 let LIFT_SUB = prove
3053 (`!x y. lift(x - y) = lift x - lift y`,
3054 SIMP_TAC[CART_EQ; lift; LAMBDA_BETA; VECTOR_SUB_COMPONENT]);;
3056 let LIFT_CMUL = prove
3057 (`!x c. lift(c * x) = c % lift(x)`,
3058 SIMP_TAC[CART_EQ; lift; LAMBDA_BETA; VECTOR_MUL_COMPONENT]);;
3060 let LIFT_NEG = prove
3061 (`!x. lift(--x) = --(lift x)`,
3062 SIMP_TAC[CART_EQ; lift; LAMBDA_BETA; VECTOR_NEG_COMPONENT]);;
3064 let LIFT_EQ_CMUL = prove
3065 (`!x. lift x = x % vec 1`,
3066 REWRITE_TAC[GSYM LIFT_NUM; GSYM LIFT_CMUL; REAL_MUL_RID]);;
3068 let LIFT_SUM = prove
3069 (`!k x. FINITE k ==> (lift(sum k x) = vsum k (lift o x))`,
3070 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
3071 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3072 SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; o_THM; LIFT_ADD; LIFT_NUM]);;
3074 let DROP_LAMBDA = prove
3075 (`!x. drop(lambda i. x i) = x 1`,
3076 SIMP_TAC[drop; LAMBDA_BETA; DIMINDEX_1; LE_REFL]);;
3078 let DROP_VEC = prove
3079 (`!n. drop(vec n) = &n`,
3080 MESON_TAC[LIFT_DROP; LIFT_NUM]);;
3082 let DROP_ADD = prove
3083 (`!x y. drop(x + y) = drop x + drop y`,
3084 MESON_TAC[LIFT_DROP; LIFT_ADD]);;
3086 let DROP_SUB = prove
3087 (`!x y. drop(x - y) = drop x - drop y`,
3088 MESON_TAC[LIFT_DROP; LIFT_SUB]);;
3090 let DROP_CMUL = prove
3091 (`!x c. drop(c % x) = c * drop(x)`,
3092 MESON_TAC[LIFT_DROP; LIFT_CMUL]);;
3094 let DROP_NEG = prove
3095 (`!x. drop(--x) = --(drop x)`,
3096 MESON_TAC[LIFT_DROP; LIFT_NEG]);;
3098 let DROP_VSUM = prove
3099 (`!k x. FINITE k ==> (drop(vsum k x) = sum k (drop o x))`,
3100 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
3101 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3102 SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; o_THM; DROP_ADD; DROP_VEC]);;
3104 let NORM_LIFT = prove
3105 (`!x. norm(lift x) = abs(x)`,
3106 SIMP_TAC[lift; NORM_REAL; LIFT_COMPONENT]);;
3108 let DIST_LIFT = prove
3109 (`!x y. dist(lift x,lift y) = abs(x - y)`,
3110 REWRITE_TAC[DIST_REAL; LIFT_COMPONENT]);;
3112 let ABS_DROP = prove
3113 (`!x. norm x = abs(drop x)`,
3114 REWRITE_TAC[FORALL_LIFT; LIFT_DROP; NORM_LIFT]);;
3116 let LINEAR_VMUL_DROP = prove
3117 (`!f v. linear f ==> linear (\x. drop(f x) % v)`,
3118 SIMP_TAC[drop; LINEAR_VMUL_COMPONENT; DIMINDEX_1; LE_REFL]);;
3120 let LINEAR_FROM_REALS = prove
3121 (`!f:real^1->real^N. linear f ==> f = \x. drop x % column 1 (matrix f)`,
3122 GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
3123 DISCH_THEN(fun th -> REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN
3124 SIMP_TAC[CART_EQ; matrix_vector_mul; vector_mul; LAMBDA_BETA;
3125 DIMINDEX_1; SUM_SING_NUMSEG; drop; column] THEN
3126 REWRITE_TAC[REAL_MUL_AC]);;
3128 let LINEAR_TO_REALS = prove
3129 (`!f:real^N->real^1. linear f ==> f = \x. lift(row 1 (matrix f) dot x)`,
3130 GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN
3131 DISCH_THEN(fun th -> REWRITE_TAC[GSYM(MATCH_MP MATRIX_WORKS th)]) THEN
3132 SIMP_TAC[CART_EQ; matrix_vector_mul; dot; LAMBDA_BETA;
3133 DIMINDEX_1; SUM_SING_NUMSEG; lift; row; LE_ANTISYM]);;
3135 let DROP_EQ_0 = prove
3136 (`!x. drop x = &0 <=> x = vec 0`,
3137 REWRITE_TAC[GSYM DROP_EQ; DROP_VEC]);;
3139 let VSUM_REAL = prove
3140 (`!f s. FINITE s ==> vsum s f = lift(sum s (drop o f))`,
3141 SIMP_TAC[LIFT_SUM; o_DEF; LIFT_DROP; ETA_AX]);;
3143 let DROP_WLOG_LE = prove
3144 (`(!x y. P x y <=> P y x) /\ (!x y. drop x <= drop y ==> P x y)
3146 MESON_TAC[REAL_LE_TOTAL]);;
3148 let IMAGE_LIFT_UNIV = prove
3149 (`IMAGE lift (:real) = (:real^1)`,
3150 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_UNIV] THEN MESON_TAC[LIFT_DROP]);;
3152 let IMAGE_DROP_UNIV = prove
3153 (`IMAGE drop (:real^1) = (:real)`,
3154 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_UNIV] THEN MESON_TAC[LIFT_DROP]);;
3156 let SUM_VSUM = prove
3157 (`!f s. FINITE s ==> sum s f = drop(vsum s (lift o f))`,
3158 SIMP_TAC[VSUM_REAL; o_DEF; LIFT_DROP; ETA_AX]);;
3160 let LINEAR_LIFT_DOT = prove
3161 (`!a. linear(\x. lift(a dot x))`,
3162 REWRITE_TAC[linear; DOT_RMUL; DOT_RADD; LIFT_ADD; LIFT_CMUL]);;
3164 let LINEAR_LIFT_COMPONENT = prove
3165 (`!k. linear(\x:real^N. lift(x$k))`,
3167 SUBGOAL_THEN `?j. 1 <= j /\ j <= dimindex(:N) /\ !z:real^N. z$k = z$j`
3169 [REWRITE_TAC[FINITE_INDEX_INRANGE];
3170 MP_TAC(ISPEC `basis j:real^N` LINEAR_LIFT_DOT) THEN
3171 ASM_SIMP_TAC[DOT_BASIS]]);;
3173 (* ------------------------------------------------------------------------- *)
3174 (* Pasting vectors. *)
3175 (* ------------------------------------------------------------------------- *)
3177 let LINEAR_FSTCART = prove
3179 SIMP_TAC[linear; fstcart; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
3180 VECTOR_MUL_COMPONENT; DIMINDEX_FINITE_SUM;
3181 ARITH_RULE `x <= a ==> x <= a + b:num`]);;
3183 let LINEAR_SNDCART = prove
3185 SIMP_TAC[linear; sndcart; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
3186 VECTOR_MUL_COMPONENT; DIMINDEX_FINITE_SUM;
3187 ARITH_RULE `x <= a ==> x <= a + b:num`;
3188 ARITH_RULE `x <= b ==> x + a <= a + b:num`]);;
3190 let FSTCART_VEC = prove
3191 (`!n. fstcart(vec n) = vec n`,
3192 SIMP_TAC[vec; fstcart; LAMBDA_BETA; CART_EQ; DIMINDEX_FINITE_SUM;
3193 ARITH_RULE `m <= n:num ==> m <= n + p`]);;
3195 let FSTCART_ADD = prove
3196 (`!x:real^(M,N)finite_sum y. fstcart(x + y) = fstcart(x) + fstcart(y)`,
3197 REWRITE_TAC[REWRITE_RULE[linear] LINEAR_FSTCART]);;
3199 let FSTCART_CMUL = prove
3200 (`!x:real^(M,N)finite_sum c. fstcart(c % x) = c % fstcart(x)`,
3201 REWRITE_TAC[REWRITE_RULE[linear] LINEAR_FSTCART]);;
3203 let FSTCART_NEG = prove
3204 (`!x:real^(M,N)finite_sum. --(fstcart x) = fstcart(--x)`,
3205 ONCE_REWRITE_TAC[VECTOR_ARITH `--x = --(&1) % x`] THEN
3206 REWRITE_TAC[FSTCART_CMUL]);;
3208 let FSTCART_SUB = prove
3209 (`!x:real^(M,N)finite_sum y. fstcart(x - y) = fstcart(x) - fstcart(y)`,
3210 REWRITE_TAC[VECTOR_SUB; FSTCART_NEG; FSTCART_ADD]);;
3212 let FSTCART_VSUM = prove
3213 (`!k x. FINITE k ==> (fstcart(vsum k x) = vsum k (\i. fstcart(x i)))`,
3214 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
3215 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3216 SIMP_TAC[VSUM_CLAUSES; FINITE_RULES; FSTCART_ADD; FSTCART_VEC]);;
3218 let SNDCART_VEC = prove
3219 (`!n. sndcart(vec n) = vec n`,
3220 SIMP_TAC[vec; sndcart; LAMBDA_BETA; CART_EQ; DIMINDEX_FINITE_SUM;
3221 ARITH_RULE `x <= a ==> x <= a + b:num`;
3222 ARITH_RULE `x <= b ==> x + a <= a + b:num`]);;
3224 let SNDCART_ADD = prove
3225 (`!x:real^(M,N)finite_sum y. sndcart(x + y) = sndcart(x) + sndcart(y)`,
3226 REWRITE_TAC[REWRITE_RULE[linear] LINEAR_SNDCART]);;
3228 let SNDCART_CMUL = prove
3229 (`!x:real^(M,N)finite_sum c. sndcart(c % x) = c % sndcart(x)`,
3230 REWRITE_TAC[REWRITE_RULE[linear] LINEAR_SNDCART]);;
3232 let SNDCART_NEG = prove
3233 (`!x:real^(M,N)finite_sum. --(sndcart x) = sndcart(--x)`,
3234 ONCE_REWRITE_TAC[VECTOR_ARITH `--x = --(&1) % x`] THEN
3235 REWRITE_TAC[SNDCART_CMUL]);;
3237 let SNDCART_SUB = prove
3238 (`!x:real^(M,N)finite_sum y. sndcart(x - y) = sndcart(x) - sndcart(y)`,
3239 REWRITE_TAC[VECTOR_SUB; SNDCART_NEG; SNDCART_ADD]);;
3241 let SNDCART_VSUM = prove
3242 (`!k x. FINITE k ==> (sndcart(vsum k x) = vsum k (\i. sndcart(x i)))`,
3243 REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
3244 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3245 SIMP_TAC[VSUM_CLAUSES; FINITE_RULES; SNDCART_ADD; SNDCART_VEC]);;
3247 let PASTECART_VEC = prove
3248 (`!n. pastecart (vec n) (vec n) = vec n`,
3249 REWRITE_TAC[PASTECART_EQ; FSTCART_VEC; SNDCART_VEC;
3250 FSTCART_PASTECART; SNDCART_PASTECART]);;
3252 let PASTECART_ADD = prove
3253 (`!x1 y1 x2:real^M y2:real^N.
3254 pastecart x1 y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)`,
3255 REWRITE_TAC[PASTECART_EQ; FSTCART_ADD; SNDCART_ADD;
3256 FSTCART_PASTECART; SNDCART_PASTECART]);;
3258 let PASTECART_CMUL = prove
3259 (`!x1 y1 c. pastecart (c % x1) (c % y1) = c % pastecart x1 y1`,
3260 REWRITE_TAC[PASTECART_EQ; FSTCART_CMUL; SNDCART_CMUL;
3261 FSTCART_PASTECART; SNDCART_PASTECART]);;
3263 let PASTECART_NEG = prove
3264 (`!x:real^M y:real^N. pastecart (--x) (--y) = --(pastecart x y)`,
3265 ONCE_REWRITE_TAC[VECTOR_ARITH `--x = --(&1) % x`] THEN
3266 REWRITE_TAC[PASTECART_CMUL]);;
3268 let PASTECART_SUB = prove
3269 (`!x1 y1 x2:real^M y2:real^N.
3270 pastecart x1 y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)`,
3271 REWRITE_TAC[VECTOR_SUB; GSYM PASTECART_NEG; PASTECART_ADD]);;
3273 let PASTECART_VSUM = prove
3274 (`!k x y. FINITE k ==> (pastecart (vsum k x) (vsum k y) =
3275 vsum k (\i. pastecart (x i) (y i)))`,
3276 SIMP_TAC[PASTECART_EQ; FSTCART_VSUM; SNDCART_VSUM;
3277 FSTCART_PASTECART; SNDCART_PASTECART; ETA_AX]);;
3279 let PASTECART_EQ_VEC = prove
3280 (`!x y n. pastecart x y = vec n <=> x = vec n /\ y = vec n`,
3281 REWRITE_TAC[PASTECART_EQ; FSTCART_VEC; SNDCART_VEC;
3282 FSTCART_PASTECART; SNDCART_PASTECART]);;
3284 let NORM_FSTCART = prove
3285 (`!x. norm(fstcart x) <= norm x`,
3287 GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM PASTECART_FST_SND] THEN
3288 SIMP_TAC[SQRT_MONO_LE_EQ; DOT_POS_LE; vector_norm] THEN
3289 SIMP_TAC[pastecart; dot; DIMINDEX_FINITE_SUM; LAMBDA_BETA; DIMINDEX_NONZERO;
3290 SUM_ADD_SPLIT; REAL_LE_ADDR; SUM_POS_LE; FINITE_NUMSEG;
3291 REAL_LE_SQUARE; ARITH_RULE `x <= a ==> x <= a + b:num`;
3292 ARITH_RULE `~(d = 0) ==> 1 <= d + 1`]);;
3294 let DIST_FSTCART = prove
3295 (`!x y. dist(fstcart x,fstcart y) <= dist(x,y)`,
3296 REWRITE_TAC[dist; GSYM FSTCART_SUB; NORM_FSTCART]);;
3298 let NORM_SNDCART = prove
3299 (`!x. norm(sndcart x) <= norm x`,
3301 GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM PASTECART_FST_SND] THEN
3302 SIMP_TAC[SQRT_MONO_LE_EQ; DOT_POS_LE; vector_norm] THEN
3303 SIMP_TAC[pastecart; dot; DIMINDEX_FINITE_SUM; LAMBDA_BETA; DIMINDEX_NONZERO;
3304 SUM_ADD_SPLIT; ARITH_RULE `x <= a ==> x <= a + b:num`;
3305 ARITH_RULE `~(d = 0) ==> 1 <= d + 1`] THEN
3306 ONCE_REWRITE_TAC[ADD_SYM] THEN REWRITE_TAC[NUMSEG_OFFSET_IMAGE] THEN
3307 SIMP_TAC[SUM_IMAGE; FINITE_NUMSEG; EQ_ADD_RCANCEL; o_DEF; ADD_SUB] THEN
3308 SIMP_TAC[ARITH_RULE `1 <= x ==> ~(x + a <= a)`; SUM_POS_LE; FINITE_NUMSEG;
3309 REAL_LE_ADDL; REAL_LE_SQUARE]);;
3311 let DIST_SNDCART = prove
3312 (`!x y. dist(sndcart x,sndcart y) <= dist(x,y)`,
3313 REWRITE_TAC[dist; GSYM SNDCART_SUB; NORM_SNDCART]);;
3315 let DOT_PASTECART = prove
3316 (`!x1 x2 y1 y2. (pastecart x1 x2) dot (pastecart y1 y2) =
3317 x1 dot y1 + x2 dot y2`,
3318 SIMP_TAC[pastecart; dot; LAMBDA_BETA; DIMINDEX_FINITE_SUM] THEN
3319 SIMP_TAC[SUM_ADD_SPLIT; ARITH_RULE `~(d = 0) ==> 1 <= d + 1`;
3320 DIMINDEX_NONZERO; REAL_LE_LADD] THEN
3321 ONCE_REWRITE_TAC[ADD_SYM] THEN REWRITE_TAC[NUMSEG_OFFSET_IMAGE] THEN
3322 SIMP_TAC[SUM_IMAGE; FINITE_NUMSEG; EQ_ADD_RCANCEL; o_DEF; ADD_SUB] THEN
3323 SIMP_TAC[ARITH_RULE `1 <= x ==> ~(x + a <= a)`; REAL_LE_REFL]);;
3325 let NORM_PASTECART = prove
3326 (`!x y. norm(pastecart x y) = sqrt(norm(x) pow 2 + norm(y) pow 2)`,
3327 REWRITE_TAC[NORM_EQ_SQUARE] THEN
3328 SIMP_TAC[SQRT_POS_LE; SQRT_POW_2; REAL_LE_ADD; REAL_LE_POW_2] THEN
3329 REWRITE_TAC[DOT_PASTECART; NORM_POW_2]);;
3331 let NORM_PASTECART_LE = prove
3332 (`!x y. norm(pastecart x y) <= norm(x) + norm(y)`,
3333 REPEAT GEN_TAC THEN MATCH_MP_TAC TRIANGLE_LEMMA THEN
3334 REWRITE_TAC[NORM_POS_LE; NORM_POW_2; DOT_PASTECART; REAL_LE_REFL]);;
3336 let NORM_LE_PASTECART = prove
3337 (`!x:real^M y:real^M.
3338 norm(x) <= norm(pastecart x y) /\
3339 norm(y) <= norm(pastecart x y)`,
3340 REPEAT GEN_TAC THEN REWRITE_TAC[NORM_PASTECART] THEN CONJ_TAC THEN
3341 MATCH_MP_TAC REAL_LE_RSQRT THEN
3342 REWRITE_TAC[REAL_LE_ADDL; REAL_LE_ADDR; REAL_LE_POW_2]);;
3344 (* ------------------------------------------------------------------------- *)
3345 (* A bit of linear algebra. *)
3346 (* ------------------------------------------------------------------------- *)
3348 let subspace = new_definition
3351 (!x y. x IN s /\ y IN s ==> (x + y) IN s) /\
3352 (!c x. x IN s ==> (c % x) IN s)`;;
3354 let span = new_definition
3355 `span s = subspace hull s`;;
3357 let dependent = new_definition
3358 `dependent s <=> ?a. a IN s /\ a IN span(s DELETE a)`;;
3360 let independent = new_definition
3361 `independent s <=> ~(dependent s)`;;
3363 (* ------------------------------------------------------------------------- *)
3364 (* Closure properties of subspaces. *)
3365 (* ------------------------------------------------------------------------- *)
3367 let SUBSPACE_UNIV = prove
3368 (`subspace(UNIV:real^N->bool)`,
3369 REWRITE_TAC[subspace; IN_UNIV]);;
3371 let SUBSPACE_IMP_NONEMPTY = prove
3372 (`!s. subspace s ==> ~(s = {})`,
3373 REWRITE_TAC[subspace] THEN SET_TAC[]);;
3375 let SUBSPACE_0 = prove
3376 (`subspace s ==> vec(0) IN s`,
3377 SIMP_TAC[subspace]);;
3379 let SUBSPACE_ADD = prove
3380 (`!x y s. subspace s /\ x IN s /\ y IN s ==> (x + y) IN s`,
3381 SIMP_TAC[subspace]);;
3383 let SUBSPACE_MUL = prove
3384 (`!x c s. subspace s /\ x IN s ==> (c % x) IN s`,
3385 SIMP_TAC[subspace]);;
3387 let SUBSPACE_NEG = prove
3388 (`!x s. subspace s /\ x IN s ==> (--x) IN s`,
3389 SIMP_TAC[VECTOR_ARITH `--x = --(&1) % x`; SUBSPACE_MUL]);;
3391 let SUBSPACE_SUB = prove
3392 (`!x y s. subspace s /\ x IN s /\ y IN s ==> (x - y) IN s`,
3393 SIMP_TAC[VECTOR_SUB; SUBSPACE_ADD; SUBSPACE_NEG]);;
3395 let SUBSPACE_VSUM = prove
3396 (`!s f t. subspace s /\ FINITE t /\ (!x. x IN t ==> f(x) IN s)
3397 ==> (vsum t f) IN s`,
3398 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3399 GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
3400 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
3401 ASM_SIMP_TAC[VSUM_CLAUSES; SUBSPACE_0; IN_INSERT; SUBSPACE_ADD]);;
3403 let SUBSPACE_LINEAR_IMAGE = prove
3404 (`!f s. linear f /\ subspace s ==> subspace(IMAGE f s)`,
3405 REWRITE_TAC[subspace; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3406 REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE] THEN
3407 MESON_TAC[linear; LINEAR_0]);;
3409 let SUBSPACE_LINEAR_PREIMAGE = prove
3410 (`!f s. linear f /\ subspace s ==> subspace {x | f(x) IN s}`,
3411 REWRITE_TAC[subspace; IN_ELIM_THM] THEN
3412 MESON_TAC[linear; LINEAR_0]);;
3414 let SUBSPACE_TRIVIAL = prove
3415 (`subspace {vec 0}`,
3416 SIMP_TAC[subspace; IN_SING] THEN CONJ_TAC THEN VECTOR_ARITH_TAC);;
3418 let SUBSPACE_INTER = prove
3419 (`!s t. subspace s /\ subspace t ==> subspace (s INTER t)`,
3420 REWRITE_TAC[subspace; IN_INTER] THEN MESON_TAC[]);;
3422 let SUBSPACE_INTERS = prove
3423 (`!f. (!s. s IN f ==> subspace s) ==> subspace(INTERS f)`,
3424 SIMP_TAC[subspace; IMP_CONJ; RIGHT_FORALL_IMP_THM; IN_INTERS]);;
3426 let LINEAR_INJECTIVE_0_SUBSPACE = prove
3427 (`!f:real^M->real^N s.
3428 linear f /\ subspace s
3429 ==> ((!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) <=>
3430 (!x. x IN s /\ f x = vec 0 ==> x = vec 0))`,
3431 REPEAT STRIP_TAC THEN
3432 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM VECTOR_SUB_EQ] THEN
3433 ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN
3434 ASM_MESON_TAC[VECTOR_SUB_RZERO; SUBSPACE_SUB; SUBSPACE_0]);;
3436 let SUBSPACE_UNION_CHAIN = prove
3437 (`!s t:real^N->bool.
3438 subspace s /\ subspace t /\ subspace(s UNION t)
3439 ==> s SUBSET t \/ t SUBSET s`,
3440 REPEAT STRIP_TAC THEN REWRITE_TAC[SET_RULE
3441 `s SUBSET t \/ t SUBSET s <=>
3442 ~(?x y. x IN s /\ ~(x IN t) /\ y IN t /\ ~(y IN s))`] THEN
3443 STRIP_TAC THEN SUBGOAL_THEN `(x + y:real^N) IN s UNION t` MP_TAC THENL
3444 [MATCH_MP_TAC SUBSPACE_ADD THEN ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
3445 REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN
3446 ASM_MESON_TAC[SUBSPACE_SUB; VECTOR_ARITH
3447 `(x + y) - x:real^N = y /\ (x + y) - y = x`]]);;
3449 (* ------------------------------------------------------------------------- *)
3451 (* ------------------------------------------------------------------------- *)
3453 let SPAN_SPAN = prove
3454 (`!s. span(span s) = span s`,
3455 REWRITE_TAC[span; HULL_HULL]);;
3457 let SPAN_MONO = prove
3458 (`!s t. s SUBSET t ==> span s SUBSET span t`,
3459 REWRITE_TAC[span; HULL_MONO]);;
3461 let SUBSPACE_SPAN = prove
3462 (`!s. subspace(span s)`,
3463 GEN_TAC THEN REWRITE_TAC[span] THEN MATCH_MP_TAC P_HULL THEN
3464 SIMP_TAC[subspace; IN_INTERS]);;
3466 let SPAN_CLAUSES = prove
3467 (`(!a s. a IN s ==> a IN span s) /\
3468 (vec(0) IN span s) /\
3469 (!x y s. x IN span s /\ y IN span s ==> (x + y) IN span s) /\
3470 (!x c s. x IN span s ==> (c % x) IN span s)`,
3471 MESON_TAC[span; HULL_SUBSET; SUBSET; SUBSPACE_SPAN; subspace]);;
3473 let SPAN_INDUCT = prove
3474 (`!s h. (!x. x IN s ==> x IN h) /\ subspace h ==> !x. x IN span(s) ==> h(x)`,
3475 REWRITE_TAC[span] THEN MESON_TAC[SUBSET; HULL_MINIMAL; IN]);;
3477 let SPAN_EMPTY = prove
3478 (`span {} = {vec 0}`,
3479 REWRITE_TAC[span] THEN MATCH_MP_TAC HULL_UNIQUE THEN
3480 SIMP_TAC[subspace; SUBSET; IN_SING; NOT_IN_EMPTY] THEN
3481 REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
3483 let INDEPENDENT_EMPTY = prove
3485 REWRITE_TAC[independent; dependent; NOT_IN_EMPTY]);;
3487 let INDEPENDENT_NONZERO = prove
3488 (`!s. independent s ==> ~(vec 0 IN s)`,
3489 REWRITE_TAC[independent; dependent] THEN MESON_TAC[SPAN_CLAUSES]);;
3491 let INDEPENDENT_MONO = prove
3492 (`!s t. independent t /\ s SUBSET t ==> independent s`,
3493 REWRITE_TAC[independent; dependent] THEN
3494 ASM_MESON_TAC[SPAN_MONO; SUBSET; IN_DELETE]);;
3496 let DEPENDENT_MONO = prove
3497 (`!s t:real^N->bool. dependent s /\ s SUBSET t ==> dependent t`,
3498 ONCE_REWRITE_TAC[TAUT `p /\ q ==> r <=> ~r /\ q ==> ~p`] THEN
3499 REWRITE_TAC[GSYM independent; INDEPENDENT_MONO]);;
3501 let SPAN_SUBSPACE = prove
3502 (`!b s. b SUBSET s /\ s SUBSET (span b) /\ subspace s ==> (span b = s)`,
3503 MESON_TAC[SUBSET_ANTISYM; span; HULL_MINIMAL]);;
3505 let SPAN_INDUCT_ALT = prove
3507 (!c x y. x IN s /\ h(y) ==> h(c % x + y))
3508 ==> !x:real^N. x IN span(s) ==> h(x)`,
3509 REPEAT GEN_TAC THEN DISCH_TAC THEN
3510 FIRST_ASSUM(MP_TAC o prove_inductive_relations_exist o concl) THEN
3511 DISCH_THEN(X_CHOOSE_THEN `g:real^N->bool` STRIP_ASSUME_TAC) THEN
3512 SUBGOAL_THEN `!x:real^N. x IN span(s) ==> g(x)`
3513 (fun th -> ASM_MESON_TAC[th]) THEN
3514 MATCH_MP_TAC SPAN_INDUCT THEN REWRITE_TAC[subspace; IN_ELIM_THM] THEN
3515 REWRITE_TAC[IN; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3516 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN
3517 REPEAT CONJ_TAC THEN TRY(FIRST_X_ASSUM MATCH_MP_TAC) THEN
3518 REWRITE_TAC[VECTOR_ADD_LDISTRIB; VECTOR_MUL_ASSOC] THEN
3519 ASM_MESON_TAC[IN; VECTOR_ADD_LID; VECTOR_ADD_ASSOC; VECTOR_ADD_SYM;
3520 VECTOR_MUL_LID; VECTOR_MUL_RZERO]);;
3522 (* ------------------------------------------------------------------------- *)
3523 (* Individual closure properties. *)
3524 (* ------------------------------------------------------------------------- *)
3526 let SPAN_SUPERSET = prove
3527 (`!x. x IN s ==> x IN span s`,
3528 MESON_TAC[SPAN_CLAUSES]);;
3530 let SPAN_INC = prove
3531 (`!s. s SUBSET span s`,
3532 REWRITE_TAC[SUBSET; SPAN_SUPERSET]);;
3534 let SPAN_UNION_SUBSET = prove
3535 (`!s t. span s UNION span t SUBSET span(s UNION t)`,
3536 REWRITE_TAC[span; HULL_UNION_SUBSET]);;
3538 let SPAN_UNIV = prove
3539 (`span(:real^N) = (:real^N)`,
3540 SIMP_TAC[SPAN_INC; SET_RULE `UNIV SUBSET s ==> s = UNIV`]);;
3543 (`vec(0) IN span s`,
3544 MESON_TAC[SUBSPACE_SPAN; SUBSPACE_0]);;
3546 let SPAN_ADD = prove
3547 (`!x y s. x IN span s /\ y IN span s ==> (x + y) IN span s`,
3548 MESON_TAC[SUBSPACE_SPAN; SUBSPACE_ADD]);;
3550 let SPAN_MUL = prove
3551 (`!x c s. x IN span s ==> (c % x) IN span s`,
3552 MESON_TAC[SUBSPACE_SPAN; SUBSPACE_MUL]);;
3554 let SPAN_MUL_EQ = prove
3555 (`!x:real^N c s. ~(c = &0) ==> ((c % x) IN span s <=> x IN span s)`,
3556 REPEAT(STRIP_TAC ORELSE EQ_TAC) THEN ASM_SIMP_TAC[SPAN_MUL] THEN
3557 SUBGOAL_THEN `(inv(c) % c % x:real^N) IN span s` MP_TAC THENL
3558 [ASM_SIMP_TAC[SPAN_MUL];
3559 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID]]);;
3561 let SPAN_NEG = prove
3562 (`!x s. x IN span s ==> (--x) IN span s`,
3563 MESON_TAC[SUBSPACE_SPAN; SUBSPACE_NEG]);;
3565 let SPAN_NEG_EQ = prove
3566 (`!x s. --x IN span s <=> x IN span s`,
3567 MESON_TAC[SPAN_NEG; VECTOR_NEG_NEG]);;
3569 let SPAN_SUB = prove
3570 (`!x y s. x IN span s /\ y IN span s ==> (x - y) IN span s`,
3571 MESON_TAC[SUBSPACE_SPAN; SUBSPACE_SUB]);;
3573 let SPAN_VSUM = prove
3574 (`!s f t. FINITE t /\ (!x. x IN t ==> f(x) IN span(s))
3575 ==> (vsum t f) IN span(s)`,
3576 MESON_TAC[SUBSPACE_SPAN; SUBSPACE_VSUM]);;
3578 let SPAN_ADD_EQ = prove
3579 (`!s x y. x IN span s ==> ((x + y) IN span s <=> y IN span s)`,
3580 MESON_TAC[SPAN_ADD; SPAN_SUB; VECTOR_ARITH `(x + y) - x:real^N = y`]);;
3582 let SPAN_EQ_SELF = prove
3583 (`!s. span s = s <=> subspace s`,
3584 GEN_TAC THEN EQ_TAC THENL [MESON_TAC[SUBSPACE_SPAN]; ALL_TAC] THEN
3585 DISCH_TAC THEN MATCH_MP_TAC SPAN_SUBSPACE THEN
3586 ASM_REWRITE_TAC[SUBSET_REFL; SPAN_INC]);;
3588 let SPAN_SUBSET_SUBSPACE = prove
3589 (`!s t:real^N->bool. s SUBSET t /\ subspace t ==> span s SUBSET t`,
3590 MESON_TAC[SPAN_MONO; SPAN_EQ_SELF]);;
3592 let SUBSPACE_TRANSLATION_SELF = prove
3593 (`!s a. subspace s /\ a IN s ==> IMAGE (\x. a + x) s = s`,
3594 REPEAT STRIP_TAC THEN MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
3595 FIRST_ASSUM(SUBST1_TAC o SYM o GEN_REWRITE_RULE I [GSYM SPAN_EQ_SELF]) THEN
3596 ASM_SIMP_TAC[SPAN_ADD_EQ; SPAN_CLAUSES] THEN
3597 REWRITE_TAC[VECTOR_ARITH `a + x:real^N = y <=> x = y - a`; EXISTS_REFL]);;
3599 let SUBSPACE_TRANSLATION_SELF_EQ = prove
3600 (`!s a:real^N. subspace s ==> (IMAGE (\x. a + x) s = s <=> a IN s)`,
3601 REPEAT STRIP_TAC THEN EQ_TAC THEN
3602 ASM_SIMP_TAC[SUBSPACE_TRANSLATION_SELF] THEN
3603 DISCH_THEN(MP_TAC o AP_TERM `\s. (a:real^N) IN s`) THEN
3604 REWRITE_TAC[] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
3605 REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^N` THEN
3606 ASM_MESON_TAC[subspace; VECTOR_ADD_RID]);;
3608 let SUBSPACE_SUMS = prove
3609 (`!s t. subspace s /\ subspace t
3610 ==> subspace {x + y | x IN s /\ y IN t}`,
3611 REWRITE_TAC[subspace; FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3612 REWRITE_TAC[IN_ELIM_THM] THEN REPEAT STRIP_TAC THENL
3613 [ASM_MESON_TAC[VECTOR_ADD_LID];
3614 ONCE_REWRITE_TAC[VECTOR_ARITH
3615 `(x + y) + (x' + y'):real^N = (x + x') + (y + y')`] THEN
3617 REWRITE_TAC[VECTOR_ADD_LDISTRIB] THEN ASM_MESON_TAC[]]);;
3619 let SPAN_UNION = prove
3620 (`!s t. span(s UNION t) = {x + y:real^N | x IN span s /\ y IN span t}`,
3621 REPEAT GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3622 [MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN
3623 SIMP_TAC[SUBSPACE_SUMS; SUBSPACE_SPAN] THEN
3624 REWRITE_TAC[SUBSET; IN_UNION; IN_ELIM_THM] THEN
3625 X_GEN_TAC `x:real^N` THEN STRIP_TAC THENL
3626 [MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
3627 ASM_SIMP_TAC[SPAN_SUPERSET; SPAN_0; VECTOR_ADD_RID];
3628 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN
3629 ASM_SIMP_TAC[SPAN_SUPERSET; SPAN_0; VECTOR_ADD_LID]];
3630 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN
3631 REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_ADD THEN
3632 ASM_MESON_TAC[SPAN_MONO; SUBSET_UNION; SUBSET]]);;
3634 (* ------------------------------------------------------------------------- *)
3635 (* Mapping under linear image. *)
3636 (* ------------------------------------------------------------------------- *)
3638 let SPAN_LINEAR_IMAGE = prove
3639 (`!f:real^M->real^N s. linear f ==> (span(IMAGE f s) = IMAGE f (span s))`,
3640 REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN
3641 X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
3642 [SPEC_TAC(`x:real^N`,`x:real^N`) THEN MATCH_MP_TAC SPAN_INDUCT THEN
3643 REWRITE_TAC[SET_RULE `(\x. x IN s) = s`] THEN
3644 ASM_SIMP_TAC[SUBSPACE_SPAN; SUBSPACE_LINEAR_IMAGE] THEN
3645 REWRITE_TAC[FORALL_IN_IMAGE] THEN REWRITE_TAC[IN_IMAGE] THEN
3646 MESON_TAC[SPAN_SUPERSET; SUBSET];
3647 SPEC_TAC(`x:real^N`,`x:real^N`) THEN REWRITE_TAC[FORALL_IN_IMAGE] THEN
3648 MATCH_MP_TAC SPAN_INDUCT THEN
3649 REWRITE_TAC[SET_RULE `(\x. f x IN span(s)) = {x | f(x) IN span s}`] THEN
3650 ASM_SIMP_TAC[SUBSPACE_LINEAR_PREIMAGE; SUBSPACE_SPAN] THEN
3651 REWRITE_TAC[IN_ELIM_THM] THEN
3652 MESON_TAC[SPAN_SUPERSET; SUBSET; IN_IMAGE]]);;
3654 let DEPENDENT_LINEAR_IMAGE_EQ = prove
3655 (`!f:real^M->real^N s.
3656 linear f /\ (!x y. f x = f y ==> x = y)
3657 ==> (dependent(IMAGE f s) <=> dependent s)`,
3658 REPEAT STRIP_TAC THEN REWRITE_TAC[dependent; EXISTS_IN_IMAGE] THEN
3659 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `a:real^M` THEN
3660 ASM_CASES_TAC `(a:real^M) IN s` THEN ASM_REWRITE_TAC[] THEN
3661 MATCH_MP_TAC EQ_TRANS THEN
3662 EXISTS_TAC `(f:real^M->real^N) a IN span(IMAGE f (s DELETE a))` THEN
3664 [AP_TERM_TAC THEN AP_TERM_TAC THEN ASM SET_TAC[];
3665 ASM_SIMP_TAC[SPAN_LINEAR_IMAGE] THEN ASM SET_TAC[]]);;
3667 let DEPENDENT_LINEAR_IMAGE = prove
3668 (`!f:real^M->real^N s.
3669 linear f /\ (!x y. x IN s /\ y IN s /\ f x = f y ==> x = y) /\
3671 ==> dependent(IMAGE f s)`,
3673 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
3674 REWRITE_TAC[dependent; EXISTS_IN_IMAGE] THEN
3675 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^M` THEN
3676 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
3677 SUBGOAL_THEN `IMAGE (f:real^M->real^N) s DELETE f a = IMAGE f (s DELETE a)`
3678 (fun th -> ASM_SIMP_TAC[FUN_IN_IMAGE; SPAN_LINEAR_IMAGE; th]) THEN
3681 let INDEPENDENT_LINEAR_IMAGE_EQ = prove
3682 (`!f:real^M->real^N s.
3683 linear f /\ (!x y. f x = f y ==> x = y)
3684 ==> (independent(IMAGE f s) <=> independent s)`,
3685 REWRITE_TAC[independent; TAUT `(~p <=> ~q) <=> (p <=> q)`] THEN
3686 REWRITE_TAC[DEPENDENT_LINEAR_IMAGE_EQ]);;
3688 (* ------------------------------------------------------------------------- *)
3689 (* The key breakdown property. *)
3690 (* ------------------------------------------------------------------------- *)
3692 let SPAN_BREAKDOWN = prove
3694 b IN s /\ a IN span s ==> ?k. (a - k % b) IN span(s DELETE b)`,
3695 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
3696 REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SPAN_INDUCT THEN
3697 REWRITE_TAC[subspace; IN_ELIM_THM] THEN CONJ_TAC THENL
3698 [GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `a:real^N = b`; ALL_TAC] THEN
3699 ASM_MESON_TAC[SPAN_CLAUSES; IN_DELETE; VECTOR_ARITH
3700 `(a - &1 % a = vec 0) /\ (a - &0 % b = a) /\
3701 ((x + y) - (k1 + k2) % b = (x - k1 % b) + (y - k2 % b)) /\
3702 (c % x - (c * k) % y = c % (x - k % y))`]);;
3704 let SPAN_BREAKDOWN_EQ = prove
3705 (`!a:real^N s. (x IN span(a INSERT s) <=> (?k. (x - k % a) IN span s))`,
3706 REPEAT STRIP_TAC THEN EQ_TAC THENL
3707 [DISCH_THEN(MP_TAC o CONJ(SET_RULE `(a:real^N) IN (a INSERT s)`)) THEN
3708 DISCH_THEN(MP_TAC o MATCH_MP SPAN_BREAKDOWN) THEN
3709 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real` THEN
3710 SPEC_TAC(`x - k % a:real^N`,`y:real^N`) THEN
3711 REWRITE_TAC[GSYM SUBSET] THEN MATCH_MP_TAC SPAN_MONO THEN SET_TAC[];
3712 DISCH_THEN(X_CHOOSE_TAC `k:real`) THEN
3713 SUBST1_TAC(VECTOR_ARITH `x = (x - k % a) + k % a:real^N`) THEN
3714 MATCH_MP_TAC SPAN_ADD THEN
3715 ASM_MESON_TAC[SPAN_MONO; SUBSET; IN_INSERT; SPAN_CLAUSES]]);;
3717 let SPAN_INSERT_0 = prove
3718 (`!s. span(vec 0 INSERT s) = span s`,
3719 SIMP_TAC[EXTENSION; SPAN_BREAKDOWN_EQ; VECTOR_MUL_RZERO; VECTOR_SUB_RZERO]);;
3721 let SPAN_SING = prove
3722 (`!a. span {a} = {u % a | u IN (:real)}`,
3723 REWRITE_TAC[EXTENSION; IN_ELIM_THM; SPAN_BREAKDOWN_EQ; SPAN_EMPTY] THEN
3724 REWRITE_TAC[IN_UNIV; IN_SING; VECTOR_SUB_EQ]);;
3727 (`!a b. span {a,b} = {u % a + v % b | u IN (:real) /\ v IN (:real)}`,
3728 REWRITE_TAC[EXTENSION; IN_ELIM_THM; SPAN_BREAKDOWN_EQ; SPAN_EMPTY] THEN
3729 REWRITE_TAC[IN_UNIV; IN_SING; VECTOR_SUB_EQ] THEN
3730 REWRITE_TAC[VECTOR_ARITH `x - y:real^N = z <=> x = y + z`]);;
3733 (`!a b c. span {a,b,c} =
3734 {u % a + v % b + w % c | u IN (:real) /\ v IN (:real) /\ w IN (:real)}`,
3735 REWRITE_TAC[EXTENSION; IN_ELIM_THM; SPAN_BREAKDOWN_EQ; SPAN_EMPTY] THEN
3736 REWRITE_TAC[IN_UNIV; IN_SING; VECTOR_SUB_EQ] THEN
3737 REWRITE_TAC[VECTOR_ARITH `x - y:real^N = z <=> x = y + z`]);;
3739 (* ------------------------------------------------------------------------- *)
3740 (* Hence some "reversal" results. *)
3741 (* ------------------------------------------------------------------------- *)
3743 let IN_SPAN_INSERT = prove
3745 a IN span(b INSERT s) /\ ~(a IN span s) ==> b IN span(a INSERT s)`,
3746 REPEAT STRIP_TAC THEN
3747 MP_TAC(ISPECL [`b:real^N`; `(b:real^N) INSERT s`; `a:real^N`]
3748 SPAN_BREAKDOWN) THEN ASM_REWRITE_TAC[IN_INSERT] THEN
3749 DISCH_THEN(X_CHOOSE_THEN `k:real` MP_TAC) THEN ASM_CASES_TAC `k = &0` THEN
3750 ASM_REWRITE_TAC[VECTOR_ARITH `a - &0 % b = a`; DELETE_INSERT] THENL
3751 [ASM_MESON_TAC[SPAN_MONO; SUBSET; DELETE_SUBSET]; ALL_TAC] THEN
3752 DISCH_THEN(MP_TAC o SPEC `inv(k)` o MATCH_MP SPAN_MUL) THEN
3753 ASM_SIMP_TAC[VECTOR_SUB_LDISTRIB; VECTOR_MUL_ASSOC; REAL_MUL_LINV] THEN
3754 DISCH_TAC THEN SUBST1_TAC(VECTOR_ARITH
3755 `b:real^N = inv(k) % a - (inv(k) % a - &1 % b)`) THEN
3756 MATCH_MP_TAC SPAN_SUB THEN
3757 ASM_MESON_TAC[SPAN_CLAUSES; IN_INSERT; SUBSET; IN_DELETE; SPAN_MONO]);;
3759 let IN_SPAN_DELETE = prove
3761 a IN span s /\ ~(a IN span (s DELETE b))
3762 ==> b IN span (a INSERT (s DELETE b))`,
3763 ASM_MESON_TAC[IN_SPAN_INSERT; SPAN_MONO; SUBSET; IN_INSERT; IN_DELETE]);;
3765 let EQ_SPAN_INSERT_EQ = prove
3766 (`!s x y:real^N. (x - y) IN span s ==> span(x INSERT s) = span(y INSERT s)`,
3767 REPEAT STRIP_TAC THEN REWRITE_TAC[SPAN_BREAKDOWN_EQ; EXTENSION] THEN
3768 ASM_MESON_TAC[SPAN_ADD; SPAN_SUB; SPAN_MUL;
3769 VECTOR_ARITH `(z - k % y) - k % (x - y) = z - k % x`;
3770 VECTOR_ARITH `(z - k % x) + k % (x - y) = z - k % y`]);;
3772 (* ------------------------------------------------------------------------- *)
3773 (* Transitivity property. *)
3774 (* ------------------------------------------------------------------------- *)
3776 let SPAN_TRANS = prove
3777 (`!x y:real^N s. x IN span(s) /\ y IN span(x INSERT s) ==> y IN span(s)`,
3778 REPEAT STRIP_TAC THEN
3779 MP_TAC(SPECL [`x:real^N`; `(x:real^N) INSERT s`; `y:real^N`]
3780 SPAN_BREAKDOWN) THEN
3781 ASM_REWRITE_TAC[IN_INSERT] THEN
3782 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
3783 SUBST1_TAC(VECTOR_ARITH `y:real^N = (y - k % x) + k % x`) THEN
3784 MATCH_MP_TAC SPAN_ADD THEN ASM_SIMP_TAC[SPAN_MUL] THEN
3785 ASM_MESON_TAC[SPAN_MONO; SUBSET; IN_INSERT; IN_DELETE]);;
3787 (* ------------------------------------------------------------------------- *)
3788 (* An explicit expansion is sometimes needed. *)
3789 (* ------------------------------------------------------------------------- *)
3791 let SPAN_EXPLICIT = prove
3792 (`!(p:real^N -> bool).
3794 {y | ?s u. FINITE s /\ s SUBSET p /\
3795 vsum s (\v. u v % v) = y}`,
3796 GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
3798 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
3799 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
3800 MATCH_MP_TAC SPAN_VSUM THEN ASM_REWRITE_TAC[] THEN
3801 ASM_MESON_TAC[SPAN_SUPERSET; SPAN_MUL]] THEN
3802 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
3803 MATCH_MP_TAC SPAN_INDUCT_ALT THEN CONJ_TAC THENL
3804 [EXISTS_TAC `{}:real^N->bool` THEN
3805 REWRITE_TAC[FINITE_RULES; VSUM_CLAUSES; EMPTY_SUBSET; NOT_IN_EMPTY];
3807 MAP_EVERY X_GEN_TAC [`c:real`; `x:real^N`; `y:real^N`] THEN
3808 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
3809 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
3810 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `u:real^N->real`] THEN
3811 STRIP_TAC THEN EXISTS_TAC `(x:real^N) INSERT s` THEN
3812 EXISTS_TAC `\y. if y = x then (if x IN s then (u:real^N->real) y + c else c)
3814 ASM_SIMP_TAC[FINITE_INSERT; IN_INSERT; VSUM_CLAUSES] THEN
3815 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
3816 FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
3817 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
3818 [FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
3819 `x IN s ==> s = x INSERT (s DELETE x)`)) THEN
3820 ASM_SIMP_TAC[VSUM_CLAUSES; FINITE_INSERT; FINITE_DELETE; IN_DELETE] THEN
3821 MATCH_MP_TAC(VECTOR_ARITH
3822 `y = z ==> (c + d) % x + y = d % x + c % x + z`);
3824 MATCH_MP_TAC VSUM_EQ THEN ASM_MESON_TAC[IN_DELETE]);;
3826 let DEPENDENT_EXPLICIT = prove
3827 (`!p. dependent (p:real^N -> bool) <=>
3828 ?s u. FINITE s /\ s SUBSET p /\
3829 (?v. v IN s /\ ~(u v = &0)) /\
3830 vsum s (\v. u v % v) = vec 0`,
3831 GEN_TAC THEN REWRITE_TAC[dependent; SPAN_EXPLICIT; IN_ELIM_THM] THEN
3832 REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN
3833 EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
3834 [MAP_EVERY X_GEN_TAC [`a:real^N`; `s:real^N->bool`; `u:real^N->real`] THEN
3835 STRIP_TAC THEN MAP_EVERY EXISTS_TAC
3836 [`(a:real^N) INSERT s`;
3837 `\y. if y = a then -- &1 else (u:real^N->real) y`;
3839 ASM_REWRITE_TAC[IN_INSERT; INSERT_SUBSET; FINITE_INSERT] THEN
3840 CONJ_TAC THENL [ASM SET_TAC[]; CONV_TAC REAL_RAT_REDUCE_CONV] THEN
3841 ASM_SIMP_TAC[VSUM_CLAUSES] THEN
3842 COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3843 REWRITE_TAC[VECTOR_ARITH `-- &1 % a + s = vec 0 <=> a = s`] THEN
3844 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC LAND_CONV [SYM th]) THEN
3845 MATCH_MP_TAC VSUM_EQ THEN ASM SET_TAC[];
3846 MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `u:real^N->real`; `a:real^N`] THEN
3847 STRIP_TAC THEN MAP_EVERY EXISTS_TAC
3848 [`a:real^N`; `s DELETE (a:real^N)`;
3849 `\i. --((u:real^N->real) i) / (u a)`] THEN
3850 ASM_SIMP_TAC[VSUM_DELETE; FINITE_DELETE] THEN
3851 REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
3852 REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
3853 ASM_REWRITE_TAC[VECTOR_MUL_LNEG; GSYM VECTOR_MUL_ASSOC; VSUM_LMUL;
3854 VSUM_NEG; VECTOR_MUL_RNEG; VECTOR_MUL_RZERO] THEN
3855 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV] THEN VECTOR_ARITH_TAC]);;
3857 let DEPENDENT_FINITE = prove
3860 ==> (dependent s <=> ?u. (?v. v IN s /\ ~(u v = &0)) /\
3861 vsum s (\v. u(v) % v) = vec 0)`,
3862 REPEAT STRIP_TAC THEN REWRITE_TAC[DEPENDENT_EXPLICIT] THEN EQ_TAC THEN
3863 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
3864 [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
3865 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
3866 EXISTS_TAC `\v:real^N. if v IN t then u(v) else &0` THEN
3867 REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
3868 ONCE_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
3869 ASM_SIMP_TAC[VECTOR_MUL_LZERO; GSYM VSUM_RESTRICT_SET] THEN
3870 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`];
3871 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
3872 MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `u:real^N->real`] THEN
3873 ASM_REWRITE_TAC[SUBSET_REFL]]);;
3875 let SPAN_FINITE = prove
3877 FINITE s ==> span s = {y | ?u. vsum s (\v. u v % v) = y}`,
3878 REPEAT STRIP_TAC THEN REWRITE_TAC[SPAN_EXPLICIT; EXTENSION; IN_ELIM_THM] THEN
3879 X_GEN_TAC `y:real^N` THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL
3880 [MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
3881 STRIP_TAC THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
3882 EXISTS_TAC `\x:real^N. if x IN t then u(x) else &0` THEN
3883 REWRITE_TAC[COND_RAND; COND_RATOR; VECTOR_MUL_LZERO] THEN
3884 ASM_SIMP_TAC[GSYM VSUM_RESTRICT_SET] THEN
3885 ASM_SIMP_TAC[SET_RULE `t SUBSET s ==> {x | x IN s /\ x IN t} = t`];
3886 X_GEN_TAC `u:real^N->real` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
3887 MAP_EVERY EXISTS_TAC [`s:real^N->bool`; `u:real^N->real`] THEN
3888 ASM_REWRITE_TAC[SUBSET_REFL]]);;
3890 (* ------------------------------------------------------------------------- *)
3891 (* Standard bases are a spanning set, and obviously finite. *)
3892 (* ------------------------------------------------------------------------- *)
3894 let SPAN_STDBASIS = prove
3895 (`span {basis i :real^N | 1 <= i /\ i <= dimindex(:N)} = UNIV`,
3896 REWRITE_TAC[EXTENSION; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN
3897 GEN_REWRITE_TAC LAND_CONV [GSYM BASIS_EXPANSION] THEN
3898 MATCH_MP_TAC SPAN_VSUM THEN SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
3899 REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
3900 MATCH_MP_TAC SPAN_SUPERSET THEN REWRITE_TAC[IN_ELIM_THM] THEN
3903 let HAS_SIZE_STDBASIS = prove
3904 (`{basis i :real^N | 1 <= i /\ i <= dimindex(:N)} HAS_SIZE
3906 ONCE_REWRITE_TAC[SET_RULE `{f x | P x} = IMAGE f {x | P x}`] THEN
3907 MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN
3908 REWRITE_TAC[GSYM numseg; HAS_SIZE_NUMSEG_1; IN_NUMSEG] THEN
3909 MESON_TAC[BASIS_INJ]);;
3911 let FINITE_STDBASIS = prove
3912 (`FINITE {basis i :real^N | 1 <= i /\ i <= dimindex(:N)}`,
3913 MESON_TAC[HAS_SIZE_STDBASIS; HAS_SIZE]);;
3915 let CARD_STDBASIS = prove
3916 (`CARD {basis i :real^N | 1 <= i /\ i <= dimindex(:N)} =
3918 MESON_TAC[HAS_SIZE_STDBASIS; HAS_SIZE]);;
3920 let IN_SPAN_IMAGE_BASIS = prove
3922 x IN span(IMAGE basis s) <=>
3923 !i. 1 <= i /\ i <= dimindex(:N) /\ ~(i IN s) ==> x$i = &0`,
3924 REPEAT GEN_TAC THEN EQ_TAC THENL
3925 [SPEC_TAC(`x:real^N`,`x:real^N`) THEN MATCH_MP_TAC SPAN_INDUCT THEN
3926 SIMP_TAC[subspace; IN_ELIM_THM; VEC_COMPONENT; VECTOR_ADD_COMPONENT;
3927 VECTOR_MUL_COMPONENT; REAL_MUL_RZERO; REAL_ADD_RID] THEN
3928 SIMP_TAC[FORALL_IN_IMAGE; BASIS_COMPONENT] THEN MESON_TAC[];
3929 DISCH_TAC THEN REWRITE_TAC[SPAN_EXPLICIT; IN_ELIM_THM] THEN
3930 EXISTS_TAC `(IMAGE basis ((1..dimindex(:N)) INTER s)):real^N->bool` THEN
3931 SIMP_TAC[FINITE_IMAGE; FINITE_INTER; FINITE_NUMSEG] THEN
3932 REWRITE_TAC[RIGHT_EXISTS_AND_THM] THEN
3933 CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
3934 EXISTS_TAC `\v:real^N. x dot v` THEN
3935 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhand o snd) THEN
3937 [SIMP_TAC[FINITE_IMAGE; FINITE_INTER; FINITE_NUMSEG] THEN
3938 REWRITE_TAC[IN_INTER; IN_NUMSEG] THEN MESON_TAC[BASIS_INJ];
3939 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[]] THEN
3940 REWRITE_TAC[o_DEF] THEN
3941 SIMP_TAC[CART_EQ; VSUM_COMPONENT; VECTOR_MUL_COMPONENT;
3942 BASIS_COMPONENT] THEN
3943 ONCE_REWRITE_TAC[COND_RAND] THEN
3944 ONCE_REWRITE_TAC[MESON[]
3945 `(if x = y then p else q) = (if y = x then p else q)`] THEN
3946 SIMP_TAC[SUM_DELTA; REAL_MUL_RZERO; IN_INTER; IN_NUMSEG; DOT_BASIS] THEN
3947 ASM_MESON_TAC[REAL_MUL_RID]]);;
3949 let INDEPENDENT_STDBASIS = prove
3950 (`independent {basis i :real^N | 1 <= i /\ i <= dimindex(:N)}`,
3951 REWRITE_TAC[independent; dependent] THEN
3952 ONCE_REWRITE_TAC[SET_RULE `{f x | P x} = IMAGE f {x | P x}`] THEN
3953 REWRITE_TAC[EXISTS_IN_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM] THEN
3954 DISCH_THEN(X_CHOOSE_THEN `k:num` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
3956 `IMAGE basis {i | 1 <= i /\ i <= dimindex(:N)} DELETE
3958 IMAGE basis ({i | 1 <= i /\ i <= dimindex(:N)} DELETE k)`
3960 [REWRITE_TAC[EXTENSION; IN_IMAGE; IN_DELETE; IN_ELIM_THM] THEN
3961 GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
3962 ASM_MESON_TAC[BASIS_INJ];
3964 REWRITE_TAC[IN_SPAN_IMAGE_BASIS] THEN DISCH_THEN(MP_TAC o SPEC `k:num`) THEN
3965 ASM_SIMP_TAC[IN_DELETE; BASIS_COMPONENT; REAL_OF_NUM_EQ; ARITH]);;
3967 (* ------------------------------------------------------------------------- *)
3968 (* This is useful for building a basis step-by-step. *)
3969 (* ------------------------------------------------------------------------- *)
3971 let INDEPENDENT_INSERT = prove
3972 (`!a:real^N s. independent(a INSERT s) <=>
3973 if a IN s then independent s
3974 else independent s /\ ~(a IN span s)`,
3975 REPEAT GEN_TAC THEN ASM_CASES_TAC `(a:real^N) IN s` THEN
3976 ASM_SIMP_TAC[SET_RULE `x IN s ==> (x INSERT s = s)`] THEN
3978 [DISCH_TAC THEN CONJ_TAC THENL
3979 [ASM_MESON_TAC[INDEPENDENT_MONO; SUBSET; IN_INSERT];
3980 POP_ASSUM MP_TAC THEN REWRITE_TAC[independent; dependent] THEN
3981 ASM_MESON_TAC[IN_INSERT; SET_RULE
3982 `~(a IN s) ==> ((a INSERT s) DELETE a = s)`]];
3984 REWRITE_TAC[independent; dependent; NOT_EXISTS_THM] THEN
3985 STRIP_TAC THEN X_GEN_TAC `b:real^N` THEN
3986 REWRITE_TAC[IN_INSERT] THEN ASM_CASES_TAC `b:real^N = a` THEN
3987 ASM_SIMP_TAC[SET_RULE `~(a IN s) ==> ((a INSERT s) DELETE a = s)`] THEN
3988 ASM_SIMP_TAC[SET_RULE
3989 `~(a IN s) /\ ~(b = a)
3990 ==> ((a INSERT s) DELETE b = a INSERT (s DELETE b))`] THEN
3991 ASM_MESON_TAC[IN_SPAN_INSERT; SET_RULE
3992 `b IN s ==> (b INSERT (s DELETE b) = s)`]);;
3994 (* ------------------------------------------------------------------------- *)
3995 (* The degenerate case of the Exchange Lemma. *)
3996 (* ------------------------------------------------------------------------- *)
3998 let SPANNING_SUBSET_INDEPENDENT = prove
3999 (`!s t:real^N->bool.
4000 t SUBSET s /\ independent s /\ s SUBSET span(t) ==> (s = t)`,
4001 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
4002 ASM_REWRITE_TAC[] THEN REWRITE_TAC[SUBSET] THEN
4003 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
4004 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [independent]) THEN
4005 REWRITE_TAC[dependent; NOT_EXISTS_THM] THEN
4006 DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN
4007 ASM_MESON_TAC[SPAN_MONO; SUBSET; IN_DELETE]);;
4009 (* ------------------------------------------------------------------------- *)
4010 (* The general case of the Exchange Lemma, the key to what follows. *)
4011 (* ------------------------------------------------------------------------- *)
4013 let EXCHANGE_LEMMA = prove
4014 (`!s t:real^N->bool.
4015 FINITE t /\ independent s /\ s SUBSET span t
4016 ==> ?t'. t' HAS_SIZE (CARD t) /\
4017 s SUBSET t' /\ t' SUBSET (s UNION t) /\ s SUBSET (span t')`,
4019 WF_INDUCT_TAC `CARD(t DIFF s :real^N->bool)` THEN
4020 ASM_CASES_TAC `(s:real^N->bool) SUBSET t` THENL
4021 [ASM_MESON_TAC[HAS_SIZE; SUBSET_UNION]; ALL_TAC] THEN
4022 ASM_CASES_TAC `t SUBSET (s:real^N->bool)` THENL
4023 [ASM_MESON_TAC[SPANNING_SUBSET_INDEPENDENT; HAS_SIZE]; ALL_TAC] THEN
4025 FIRST_X_ASSUM(MP_TAC o REWRITE_RULE[SUBSET] o check(is_neg o concl)) THEN
4026 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
4027 DISCH_THEN(X_CHOOSE_THEN `b:real^N` STRIP_ASSUME_TAC) THEN
4028 ASM_CASES_TAC `s SUBSET span(t DELETE (b:real^N))` THENL
4029 [FIRST_X_ASSUM(MP_TAC o
4030 SPECL [`t DELETE (b:real^N)`; `s:real^N->bool`]) THEN
4031 ASM_REWRITE_TAC[SET_RULE `s DELETE a DIFF t = (s DIFF t) DELETE a`] THEN
4032 ASM_SIMP_TAC[CARD_DELETE; FINITE_DIFF; IN_DIFF; FINITE_DELETE;
4033 CARD_EQ_0; ARITH_RULE `n - 1 < n <=> ~(n = 0)`] THEN
4035 [UNDISCH_TAC `~((s:real^N->bool) SUBSET t)` THEN ASM SET_TAC[];
4037 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
4038 EXISTS_TAC `(b:real^N) INSERT u` THEN
4039 ASM_SIMP_TAC[SUBSET_INSERT; INSERT_SUBSET; IN_UNION] THEN CONJ_TAC THENL
4040 [UNDISCH_TAC `(u:real^N->bool) HAS_SIZE CARD(t:real^N->bool) - 1` THEN
4041 SIMP_TAC[HAS_SIZE; FINITE_RULES; CARD_CLAUSES] THEN STRIP_TAC THEN
4042 COND_CASES_TAC THENL
4043 [ASM_MESON_TAC[SUBSET; IN_UNION; IN_DELETE]; ALL_TAC] THEN
4044 ASM_MESON_TAC[ARITH_RULE `~(n = 0) ==> (SUC(n - 1) = n)`;
4045 CARD_EQ_0; MEMBER_NOT_EMPTY];
4048 [UNDISCH_TAC `u SUBSET s UNION t DELETE (b:real^N)` THEN SET_TAC[];
4049 ASM_MESON_TAC[SUBSET; SPAN_MONO; IN_INSERT]];
4051 UNDISCH_TAC `~(s SUBSET span (t DELETE (b:real^N)))` THEN
4052 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SUBSET] THEN
4053 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
4054 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
4055 SUBGOAL_THEN `~(a:real^N = b)` ASSUME_TAC THENL
4056 [ASM_MESON_TAC[]; ALL_TAC] THEN
4057 SUBGOAL_THEN `~((a:real^N) IN t)` ASSUME_TAC THENL
4058 [ASM_MESON_TAC[IN_DELETE; SPAN_CLAUSES]; ALL_TAC] THEN
4059 FIRST_X_ASSUM(MP_TAC o SPECL
4060 [`(a:real^N) INSERT (t DELETE b)`; `s:real^N->bool`]) THEN
4062 [ASM_SIMP_TAC[SET_RULE
4063 `a IN s ==> ((a INSERT (t DELETE b) DIFF s) = (t DIFF s) DELETE b)`] THEN
4064 ASM_SIMP_TAC[CARD_DELETE; FINITE_DELETE; FINITE_DIFF; IN_DIFF] THEN
4065 ASM_SIMP_TAC[ARITH_RULE `n - 1 < n <=> ~(n = 0)`; CARD_EQ_0;
4067 UNDISCH_TAC `~((s:real^N->bool) SUBSET t)` THEN ASM SET_TAC[];
4070 [ASM_SIMP_TAC[FINITE_RULES; FINITE_DELETE] THEN
4071 REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:real^N` THEN
4072 DISCH_TAC THEN MATCH_MP_TAC SPAN_TRANS THEN EXISTS_TAC `b:real^N` THEN
4073 ASM_MESON_TAC[IN_SPAN_DELETE; SUBSET; SPAN_MONO;
4074 SET_RULE `t SUBSET (b INSERT (a INSERT (t DELETE b)))`];
4076 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `u:real^N->bool` THEN
4077 ASM_SIMP_TAC[HAS_SIZE; CARD_CLAUSES; CARD_DELETE; FINITE_DELETE; IN_DELETE;
4078 ARITH_RULE `(SUC(n - 1) = n) <=> ~(n = 0)`;
4080 UNDISCH_TAC `(b:real^N) IN t` THEN ASM SET_TAC[]);;
4082 (* ------------------------------------------------------------------------- *)
4083 (* This implies corresponding size bounds. *)
4084 (* ------------------------------------------------------------------------- *)
4086 let INDEPENDENT_SPAN_BOUND = prove
4087 (`!s t. FINITE t /\ independent s /\ s SUBSET span(t)
4088 ==> FINITE s /\ CARD(s) <= CARD(t)`,
4089 REPEAT GEN_TAC THEN DISCH_TAC THEN
4090 FIRST_ASSUM(MP_TAC o MATCH_MP EXCHANGE_LEMMA) THEN
4091 ASM_MESON_TAC[HAS_SIZE; CARD_SUBSET; FINITE_SUBSET]);;
4093 let INDEPENDENT_BOUND = prove
4095 independent s ==> FINITE s /\ CARD(s) <= dimindex(:N)`,
4096 REPEAT GEN_TAC THEN DISCH_TAC THEN
4097 ONCE_REWRITE_TAC[GSYM CARD_STDBASIS] THEN
4098 MATCH_MP_TAC INDEPENDENT_SPAN_BOUND THEN
4099 ASM_REWRITE_TAC[FINITE_STDBASIS; SPAN_STDBASIS; SUBSET_UNIV]);;
4101 let DEPENDENT_BIGGERSET = prove
4102 (`!s:real^N->bool. (FINITE s ==> CARD(s) > dimindex(:N)) ==> dependent s`,
4103 MP_TAC INDEPENDENT_BOUND THEN MATCH_MP_TAC MONO_FORALL THEN
4104 REWRITE_TAC[GT; GSYM NOT_LE; independent] THEN MESON_TAC[]);;
4106 let INDEPENDENT_IMP_FINITE = prove
4107 (`!s:real^N->bool. independent s ==> FINITE s`,
4108 SIMP_TAC[INDEPENDENT_BOUND]);;
4110 (* ------------------------------------------------------------------------- *)
4111 (* Explicit formulation of independence. *)
4112 (* ------------------------------------------------------------------------- *)
4114 let INDEPENDENT_EXPLICIT = prove
4118 !c. vsum b (\v. c(v) % v) = vec 0 ==> !v. v IN b ==> c(v) = &0`,
4120 ASM_CASES_TAC `FINITE(b:real^N->bool)` THENL
4121 [ALL_TAC; ASM_MESON_TAC[INDEPENDENT_BOUND]] THEN
4122 ASM_SIMP_TAC[independent; DEPENDENT_FINITE] THEN MESON_TAC[]);;
4124 let INDEPENDENT_2 = prove
4126 independent{a,b} /\ ~(a = b)
4127 ==> (x % a + y % b = vec 0 <=> x = &0 /\ y = &0)`,
4128 REWRITE_TAC[INDEPENDENT_EXPLICIT] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN
4129 SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN DISCH_TAC THEN
4130 FIRST_X_ASSUM(MP_TAC o SPEC `\c:real^N. if c = a then x else y:real`) THEN
4131 SIMP_TAC[VSUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
4132 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; VECTOR_ADD_RID] THEN
4133 DISCH_THEN(fun th ->
4134 MP_TAC(SPEC `a:real^N` th) THEN MP_TAC(SPEC `b:real^N` th)) THEN
4137 let INDEPENDENT_3 = prove
4138 (`!a b c:real^N x y z.
4139 independent{a,b,c} /\ ~(a = b) /\ ~(a = c) /\ ~(b = c)
4140 ==> (x % a + y % b + z % c = vec 0 <=> x = &0 /\ y = &0 /\ z = &0)`,
4141 REWRITE_TAC[INDEPENDENT_EXPLICIT] THEN REPEAT STRIP_TAC THEN EQ_TAC THEN
4142 SIMP_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN DISCH_TAC THEN
4143 FIRST_X_ASSUM(MP_TAC o SPEC
4144 `\v:real^N. if v = a then x else if v = b then y else z:real`) THEN
4145 SIMP_TAC[VSUM_CLAUSES; FINITE_INSERT; FINITE_EMPTY] THEN
4146 ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; VECTOR_ADD_RID] THEN
4147 DISCH_THEN(fun th ->
4148 MP_TAC(SPEC `a:real^N` th) THEN MP_TAC(SPEC `b:real^N` th) THEN
4149 MP_TAC(SPEC `c:real^N` th)) THEN
4152 (* ------------------------------------------------------------------------- *)
4153 (* Hence we can create a maximal independent subset. *)
4154 (* ------------------------------------------------------------------------- *)
4156 let MAXIMAL_INDEPENDENT_SUBSET_EXTEND = prove
4157 (`!s v:real^N->bool.
4158 s SUBSET v /\ independent s
4159 ==> ?b. s SUBSET b /\ b SUBSET v /\ independent b /\
4162 WF_INDUCT_TAC `dimindex(:N) - CARD(s:real^N->bool)` THEN
4163 REPEAT STRIP_TAC THEN
4164 ASM_CASES_TAC `v SUBSET (span(s:real^N->bool))` THENL
4165 [ASM_MESON_TAC[SUBSET_REFL]; ALL_TAC] THEN
4166 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [SUBSET]) THEN
4167 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP] THEN
4168 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
4169 FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N) INSERT s`) THEN
4170 REWRITE_TAC[IMP_IMP] THEN ANTS_TAC THENL
4171 [ALL_TAC; MESON_TAC[INSERT_SUBSET]] THEN
4172 SUBGOAL_THEN `independent ((a:real^N) INSERT s)` ASSUME_TAC THENL
4173 [ASM_REWRITE_TAC[INDEPENDENT_INSERT; COND_ID]; ALL_TAC] THEN
4174 ASM_REWRITE_TAC[INSERT_SUBSET] THEN
4175 MATCH_MP_TAC(ARITH_RULE `(b = a + 1) /\ b <= n ==> n - b < n - a`) THEN
4176 ASM_SIMP_TAC[CARD_CLAUSES; INDEPENDENT_BOUND] THEN
4177 ASM_MESON_TAC[SPAN_SUPERSET; ADD1]);;
4179 let MAXIMAL_INDEPENDENT_SUBSET = prove
4180 (`!v:real^N->bool. ?b. b SUBSET v /\ independent b /\ v SUBSET (span b)`,
4181 MP_TAC(SPEC `EMPTY:real^N->bool` MAXIMAL_INDEPENDENT_SUBSET_EXTEND) THEN
4182 REWRITE_TAC[EMPTY_SUBSET; INDEPENDENT_EMPTY]);;
4184 (* ------------------------------------------------------------------------- *)
4185 (* Notion of dimension. *)
4186 (* ------------------------------------------------------------------------- *)
4188 let dim = new_definition
4189 `dim v = @n. ?b. b SUBSET v /\ independent b /\ v SUBSET (span b) /\
4192 let BASIS_EXISTS = prove
4193 (`!v. ?b. b SUBSET v /\ independent b /\ v SUBSET (span b) /\
4194 b HAS_SIZE (dim v)`,
4195 GEN_TAC THEN REWRITE_TAC[dim] THEN CONV_TAC SELECT_CONV THEN
4196 MESON_TAC[MAXIMAL_INDEPENDENT_SUBSET; HAS_SIZE; INDEPENDENT_BOUND]);;
4198 let BASIS_EXISTS_FINITE = prove
4199 (`!v. ?b. FINITE b /\
4202 v SUBSET (span b) /\
4203 b HAS_SIZE (dim v)`,
4204 MESON_TAC[BASIS_EXISTS; INDEPENDENT_IMP_FINITE]);;
4206 let BASIS_SUBSPACE_EXISTS = prove
4214 REPEAT STRIP_TAC THEN
4215 MP_TAC(ISPEC `s:real^N->bool` BASIS_EXISTS) THEN
4216 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN
4217 ASM_REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
4218 ASM_MESON_TAC[SPAN_EQ_SELF; SPAN_MONO; INDEPENDENT_IMP_FINITE]);;
4220 (* ------------------------------------------------------------------------- *)
4221 (* Consequences of independence or spanning for cardinality. *)
4222 (* ------------------------------------------------------------------------- *)
4224 let INDEPENDENT_CARD_LE_DIM = prove
4225 (`!v b:real^N->bool.
4226 b SUBSET v /\ independent b ==> FINITE b /\ CARD(b) <= dim v`,
4227 MESON_TAC[BASIS_EXISTS; INDEPENDENT_SPAN_BOUND; HAS_SIZE;SUBSET_TRANS]);;
4229 let SPAN_CARD_GE_DIM = prove
4230 (`!v b:real^N->bool.
4231 v SUBSET (span b) /\ FINITE b ==> dim(v) <= CARD(b)`,
4232 MESON_TAC[BASIS_EXISTS; INDEPENDENT_SPAN_BOUND; HAS_SIZE;SUBSET_TRANS]);;
4234 let BASIS_CARD_EQ_DIM = prove
4235 (`!v b. b SUBSET v /\ v SUBSET (span b) /\ independent b
4236 ==> FINITE b /\ (CARD b = dim v)`,
4237 MESON_TAC[LE_ANTISYM; INDEPENDENT_CARD_LE_DIM; SPAN_CARD_GE_DIM]);;
4239 let BASIS_HAS_SIZE_DIM = prove
4240 (`!v b. independent b /\ span b = v ==> b HAS_SIZE (dim v)`,
4241 REPEAT STRIP_TAC THEN REWRITE_TAC[HAS_SIZE] THEN
4242 MATCH_MP_TAC BASIS_CARD_EQ_DIM THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN
4243 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[SPAN_INC]);;
4245 let DIM_UNIQUE = prove
4246 (`!v b. b SUBSET v /\ v SUBSET (span b) /\ independent b /\ b HAS_SIZE n
4248 MESON_TAC[BASIS_CARD_EQ_DIM; HAS_SIZE]);;
4250 let DIM_LE_CARD = prove
4251 (`!s. FINITE s ==> dim s <= CARD s`,
4252 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SPAN_CARD_GE_DIM THEN
4253 ASM_REWRITE_TAC[SPAN_INC; SUBSET_REFL]);;
4255 (* ------------------------------------------------------------------------- *)
4256 (* More lemmas about dimension. *)
4257 (* ------------------------------------------------------------------------- *)
4259 let DIM_UNIV = prove
4260 (`dim(:real^N) = dimindex(:N)`,
4261 MATCH_MP_TAC DIM_UNIQUE THEN
4262 EXISTS_TAC `{basis i :real^N | 1 <= i /\ i <= dimindex(:N)}` THEN
4263 REWRITE_TAC[SUBSET_UNIV; SPAN_STDBASIS; HAS_SIZE_STDBASIS;
4264 INDEPENDENT_STDBASIS]);;
4266 let DIM_SUBSET = prove
4267 (`!s t:real^N->bool. s SUBSET t ==> dim(s) <= dim(t)`,
4268 MESON_TAC[BASIS_EXISTS; INDEPENDENT_SPAN_BOUND; SUBSET; HAS_SIZE]);;
4270 let DIM_SUBSET_UNIV = prove
4271 (`!s:real^N->bool. dim(s) <= dimindex(:N)`,
4272 GEN_TAC THEN REWRITE_TAC[GSYM DIM_UNIV] THEN
4273 MATCH_MP_TAC DIM_SUBSET THEN REWRITE_TAC[SUBSET_UNIV]);;
4275 let BASIS_HAS_SIZE_UNIV = prove
4276 (`!b. independent b /\ span b = (:real^N) ==> b HAS_SIZE (dimindex(:N))`,
4277 REWRITE_TAC[GSYM DIM_UNIV; BASIS_HAS_SIZE_DIM]);;
4279 (* ------------------------------------------------------------------------- *)
4280 (* Converses to those. *)
4281 (* ------------------------------------------------------------------------- *)
4283 let CARD_GE_DIM_INDEPENDENT = prove
4284 (`!v b:real^N->bool.
4285 b SUBSET v /\ independent b /\ dim v <= CARD(b)
4286 ==> v SUBSET (span b)`,
4287 REPEAT STRIP_TAC THEN
4288 SUBGOAL_THEN `!a:real^N. ~(a IN v /\ ~(a IN span b))` MP_TAC THENL
4289 [ALL_TAC; SET_TAC[]] THEN
4290 X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN
4291 SUBGOAL_THEN `independent((a:real^N) INSERT b)` ASSUME_TAC THENL
4292 [ASM_MESON_TAC[INDEPENDENT_INSERT]; ALL_TAC] THEN
4293 MP_TAC(ISPECL [`v:real^N->bool`; `(a:real^N) INSERT b`]
4294 INDEPENDENT_CARD_LE_DIM) THEN
4295 ASM_SIMP_TAC[INSERT_SUBSET; CARD_CLAUSES; INDEPENDENT_BOUND] THEN
4296 ASM_MESON_TAC[SPAN_SUPERSET; SUBSET; ARITH_RULE
4297 `x <= y ==> ~(SUC y <= x)`]);;
4299 let CARD_LE_DIM_SPANNING = prove
4300 (`!v b:real^N->bool.
4301 v SUBSET (span b) /\ FINITE b /\ CARD(b) <= dim v
4303 REPEAT STRIP_TAC THEN REWRITE_TAC[independent; dependent] THEN
4304 DISCH_THEN(X_CHOOSE_THEN `a:real^N` STRIP_ASSUME_TAC) THEN
4305 SUBGOAL_THEN `dim(v:real^N->bool) <= CARD(b DELETE (a:real^N))` MP_TAC THENL
4307 ASM_SIMP_TAC[CARD_DELETE] THEN MATCH_MP_TAC
4308 (ARITH_RULE `b <= n /\ ~(b = 0) ==> ~(n <= b - 1)`) THEN
4309 ASM_SIMP_TAC[CARD_EQ_0] THEN ASM_MESON_TAC[MEMBER_NOT_EMPTY]] THEN
4310 MATCH_MP_TAC SPAN_CARD_GE_DIM THEN ASM_SIMP_TAC[FINITE_DELETE] THEN
4311 REWRITE_TAC[SUBSET] THEN REPEAT STRIP_TAC THEN
4312 MATCH_MP_TAC SPAN_TRANS THEN EXISTS_TAC `a:real^N` THEN
4313 ASM_SIMP_TAC[SET_RULE `a IN b ==> (a INSERT (b DELETE a) = b)`] THEN
4314 ASM_MESON_TAC[SUBSET]);;
4316 let CARD_EQ_DIM = prove
4317 (`!v b. b SUBSET v /\ b HAS_SIZE (dim v)
4318 ==> (independent b <=> v SUBSET (span b))`,
4319 REWRITE_TAC[HAS_SIZE; GSYM LE_ANTISYM] THEN
4320 MESON_TAC[CARD_LE_DIM_SPANNING; CARD_GE_DIM_INDEPENDENT]);;
4322 (* ------------------------------------------------------------------------- *)
4323 (* More general size bound lemmas. *)
4324 (* ------------------------------------------------------------------------- *)
4326 let INDEPENDENT_BOUND_GENERAL = prove
4327 (`!s:real^N->bool. independent s ==> FINITE s /\ CARD(s) <= dim(s)`,
4328 MESON_TAC[INDEPENDENT_CARD_LE_DIM; INDEPENDENT_BOUND; SUBSET_REFL]);;
4330 let DEPENDENT_BIGGERSET_GENERAL = prove
4331 (`!s:real^N->bool. (FINITE s ==> CARD(s) > dim(s)) ==> dependent s`,
4332 MP_TAC INDEPENDENT_BOUND_GENERAL THEN MATCH_MP_TAC MONO_FORALL THEN
4333 REWRITE_TAC[GT; GSYM NOT_LE; independent] THEN MESON_TAC[]);;
4335 let DIM_SPAN = prove
4336 (`!s:real^N->bool. dim(span s) = dim s`,
4337 GEN_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN CONJ_TAC THENL
4339 MATCH_MP_TAC DIM_SUBSET THEN MESON_TAC[SUBSET; SPAN_SUPERSET]] THEN
4340 MP_TAC(ISPEC `s:real^N->bool` BASIS_EXISTS) THEN
4341 REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC THEN
4342 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
4343 MATCH_MP_TAC SPAN_CARD_GE_DIM THEN ASM_REWRITE_TAC[] THEN
4344 GEN_REWRITE_TAC RAND_CONV [GSYM SPAN_SPAN] THEN
4345 MATCH_MP_TAC SPAN_MONO THEN ASM_REWRITE_TAC[]);;
4347 let DIM_INSERT_0 = prove
4348 (`!s:real^N->bool. dim(vec 0 INSERT s) = dim s`,
4349 ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
4350 REWRITE_TAC[SPAN_INSERT_0]);;
4352 let DIM_EQ_CARD = prove
4353 (`!s:real^N->bool. independent s ==> dim s = CARD s`,
4354 REPEAT STRIP_TAC THEN MP_TAC
4355 (ISPECL [`span s:real^N->bool`; `s:real^N->bool`] BASIS_CARD_EQ_DIM) THEN
4356 ASM_SIMP_TAC[SUBSET_REFL; SPAN_INC; DIM_SPAN]);;
4358 let SUBSET_LE_DIM = prove
4359 (`!s t:real^N->bool. s SUBSET (span t) ==> dim s <= dim t`,
4360 MESON_TAC[DIM_SPAN; DIM_SUBSET]);;
4362 let SPAN_EQ_DIM = prove
4363 (`!s t. span s = span t ==> dim s = dim t`,
4364 MESON_TAC[DIM_SPAN]);;
4366 let SPANS_IMAGE = prove
4367 (`!f b v. linear f /\ v SUBSET (span b)
4368 ==> (IMAGE f v) SUBSET span(IMAGE f b)`,
4369 SIMP_TAC[SPAN_LINEAR_IMAGE; IMAGE_SUBSET]);;
4371 let DIM_LINEAR_IMAGE_LE = prove
4372 (`!f:real^M->real^N s. linear f ==> dim(IMAGE f s) <= dim s`,
4373 REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^M->bool` BASIS_EXISTS) THEN
4374 REWRITE_TAC[HAS_SIZE] THEN STRIP_TAC THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
4375 MATCH_MP_TAC LE_TRANS THEN EXISTS_TAC `CARD(IMAGE (f:real^M->real^N) b)` THEN
4376 ASM_SIMP_TAC[CARD_IMAGE_LE] THEN MATCH_MP_TAC SPAN_CARD_GE_DIM THEN
4377 ASM_MESON_TAC[SPAN_LINEAR_IMAGE; SPANS_IMAGE; SUBSET_IMAGE; FINITE_IMAGE]);;
4379 (* ------------------------------------------------------------------------- *)
4380 (* Some stepping theorems. *)
4381 (* ------------------------------------------------------------------------- *)
4383 let DIM_EMPTY = prove
4384 (`dim({}:real^N->bool) = 0`,
4385 MATCH_MP_TAC DIM_UNIQUE THEN EXISTS_TAC `{}:real^N->bool` THEN
4386 REWRITE_TAC[SUBSET_REFL; SPAN_EMPTY; INDEPENDENT_EMPTY; HAS_SIZE_0;
4389 let DIM_INSERT = prove
4390 (`!x:real^N s. dim(x INSERT s) = if x IN span s then dim s else dim s + 1`,
4391 REPEAT GEN_TAC THEN COND_CASES_TAC THENL
4392 [MATCH_MP_TAC SPAN_EQ_DIM THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
4393 ASM_MESON_TAC[SPAN_TRANS; SUBSET; SPAN_MONO; IN_INSERT];
4395 X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC
4396 (ISPEC `span s:real^N->bool` BASIS_EXISTS) THEN
4397 ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
4398 MATCH_MP_TAC DIM_UNIQUE THEN
4399 EXISTS_TAC `(x:real^N) INSERT b` THEN REPEAT CONJ_TAC THENL
4400 [REWRITE_TAC[INSERT_SUBSET] THEN
4401 ASM_MESON_TAC[SUBSET; SPAN_MONO; IN_INSERT; SPAN_SUPERSET];
4402 REWRITE_TAC[SUBSET; SPAN_BREAKDOWN_EQ] THEN
4403 ASM_MESON_TAC[SUBSET];
4404 REWRITE_TAC[INDEPENDENT_INSERT] THEN
4405 ASM_MESON_TAC[SUBSET; SPAN_SUPERSET; SPAN_MONO; SPAN_SPAN];
4406 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
4407 ASM_SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_INSERT; ADD1] THEN
4408 ASM_MESON_TAC[SUBSET; SPAN_SUPERSET; SPAN_MONO; SPAN_SPAN]]);;
4410 let DIM_SING = prove
4411 (`!x. dim{x} = if x = vec 0 then 0 else 1`,
4412 REWRITE_TAC[DIM_INSERT; DIM_EMPTY; SPAN_EMPTY; IN_SING; ARITH]);;
4414 let DIM_EQ_0 = prove
4415 (`!s:real^N->bool. dim s = 0 <=> s SUBSET {vec 0}`,
4416 REPEAT STRIP_TAC THEN EQ_TAC THEN DISCH_TAC THENL
4417 [MATCH_MP_TAC(SET_RULE
4418 `~(?b. ~(b = a) /\ {b} SUBSET s) ==> s SUBSET {a}`) THEN
4419 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP DIM_SUBSET);
4420 MATCH_MP_TAC(ARITH_RULE `!m. m = 0 /\ n <= m ==> n = 0`) THEN
4421 EXISTS_TAC `dim{vec 0:real^N}` THEN ASM_SIMP_TAC[DIM_SUBSET]] THEN
4422 ASM_REWRITE_TAC[DIM_SING; ARITH]);;
4424 (* ------------------------------------------------------------------------- *)
4425 (* Relation between bases and injectivity/surjectivity of map. *)
4426 (* ------------------------------------------------------------------------- *)
4428 let SPANNING_SURJECTIVE_IMAGE = prove
4429 (`!f:real^M->real^N s.
4430 UNIV SUBSET (span s) /\ linear f /\ (!y. ?x. f(x) = y)
4431 ==> UNIV SUBSET span(IMAGE f s)`,
4432 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
4433 EXISTS_TAC `IMAGE (f:real^M->real^N) UNIV` THEN
4434 ASM_SIMP_TAC[SPANS_IMAGE] THEN
4435 REWRITE_TAC[SUBSET; IN_UNIV; IN_IMAGE] THEN ASM_MESON_TAC[]);;
4437 let INDEPENDENT_INJECTIVE_IMAGE_GEN = prove
4438 (`!f:real^M->real^N s.
4439 independent s /\ linear f /\
4440 (!x y. x IN span s /\ y IN span s /\ f(x) = f(y) ==> x = y)
4441 ==> independent (IMAGE f s)`,
4443 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
4444 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
4445 REWRITE_TAC[independent; DEPENDENT_EXPLICIT] THEN
4446 REWRITE_TAC[CONJ_ASSOC; FINITE_SUBSET_IMAGE] THEN
4448 `(?s u. ((?t. p t /\ s = f t) /\ q s u) /\ r s u) <=>
4449 (?t u. p t /\ q (f t) u /\ r (f t) u)`] THEN
4450 REWRITE_TAC[EXISTS_IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
4451 MAP_EVERY X_GEN_TAC [`t:real^M->bool`; `u:real^N->real`] THEN
4452 DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
4453 MAP_EVERY EXISTS_TAC
4454 [`t:real^M->bool`; `(u:real^N->real) o (f:real^M->real^N)`] THEN
4455 ASM_REWRITE_TAC[o_THM] THEN
4456 FIRST_ASSUM MATCH_MP_TAC THEN REPEAT CONJ_TAC THENL
4457 [MATCH_MP_TAC SPAN_VSUM THEN ASM_REWRITE_TAC[] THEN
4458 REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
4459 MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[];
4460 REWRITE_TAC[SPAN_0];
4461 ASM_SIMP_TAC[LINEAR_VSUM] THEN
4462 FIRST_ASSUM(SUBST1_TAC o MATCH_MP LINEAR_0) THEN
4463 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN CONV_TAC SYM_CONV THEN
4464 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhand o snd) THEN
4465 ASM_SIMP_TAC[o_DEF; LINEAR_CMUL] THEN DISCH_THEN MATCH_MP_TAC THEN
4466 ASM_MESON_TAC[SPAN_SUPERSET; SUBSET]]);;
4468 let INDEPENDENT_INJECTIVE_IMAGE = prove
4469 (`!f:real^M->real^N s.
4470 independent s /\ linear f /\ (!x y. (f(x) = f(y)) ==> (x = y))
4471 ==> independent (IMAGE f s)`,
4472 REPEAT STRIP_TAC THEN MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE_GEN THEN
4475 (* ------------------------------------------------------------------------- *)
4476 (* Picking an orthogonal replacement for a spanning set. *)
4477 (* ------------------------------------------------------------------------- *)
4479 let VECTOR_SUB_PROJECT_ORTHOGONAL = prove
4480 (`!b:real^N x. b dot (x - ((b dot x) / (b dot b)) % b) = &0`,
4481 REPEAT GEN_TAC THEN ASM_CASES_TAC `b = vec 0 :real^N` THENL
4482 [ASM_REWRITE_TAC[DOT_LZERO]; ALL_TAC] THEN
4483 ASM_SIMP_TAC[DOT_RSUB; DOT_RMUL] THEN
4484 ASM_SIMP_TAC[REAL_SUB_REFL; REAL_DIV_RMUL; DOT_EQ_0]);;
4486 let BASIS_ORTHOGONAL = prove
4489 ==> ?c. FINITE c /\ CARD c <= CARD b /\
4490 span c = span b /\ pairwise orthogonal c`,
4491 REWRITE_TAC[pairwise; orthogonal] THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
4493 [EXISTS_TAC `{}:real^N->bool` THEN
4494 REWRITE_TAC[FINITE_RULES; NOT_IN_EMPTY; LE_REFL];
4496 MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N->bool`] THEN
4497 DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC)
4498 STRIP_ASSUME_TAC) THEN
4499 EXISTS_TAC `(a - vsum c (\x. ((x dot a) / (x dot x)) % x):real^N)
4501 ASM_SIMP_TAC[FINITE_RULES; CARD_CLAUSES] THEN REPEAT CONJ_TAC THENL
4503 REWRITE_TAC[EXTENSION; SPAN_BREAKDOWN_EQ] THEN
4504 FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN GEN_TAC THEN
4505 AP_TERM_TAC THEN ABS_TAC THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB] THEN
4506 REWRITE_TAC[VECTOR_ARITH `a - (x - y):real^N = y + (a - x)`] THEN
4507 MATCH_MP_TAC SPAN_ADD_EQ THEN MATCH_MP_TAC SPAN_MUL THEN
4508 MATCH_MP_TAC SPAN_VSUM THEN ASM_REWRITE_TAC[] THEN
4509 REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
4510 ASM_SIMP_TAC[SPAN_SUPERSET];
4511 REWRITE_TAC[IN_INSERT] THEN REPEAT STRIP_TAC THENL
4513 FIRST_X_ASSUM SUBST_ALL_TAC;
4514 FIRST_X_ASSUM SUBST_ALL_TAC;
4515 ASM_MESON_TAC[]] THEN
4516 REWRITE_TAC[DOT_LSUB; DOT_RSUB; REAL_SUB_0] THEN
4517 FIRST_ASSUM(SUBST1_TAC o MATCH_MP (SET_RULE
4518 `x IN s ==> s = x INSERT (s DELETE x)`)) THEN
4519 ASM_SIMP_TAC[VSUM_CLAUSES; FINITE_DELETE; IN_DELETE] THEN
4520 REWRITE_TAC[DOT_LADD; DOT_RADD; DOT_LMUL; DOT_RMUL] THEN
4521 MATCH_MP_TAC(REAL_ARITH `s = &0 /\ a = b ==> b = a + s`) THEN
4522 ASM_SIMP_TAC[DOT_LSUM; DOT_RSUM; FINITE_DELETE] THEN
4524 [MATCH_MP_TAC SUM_EQ_0 THEN
4525 ASM_SIMP_TAC[DOT_LMUL; DOT_RMUL; IN_DELETE;
4526 REAL_MUL_RZERO; REAL_MUL_LZERO];
4527 W(MP_TAC o PART_MATCH (lhand o rand) REAL_DIV_RMUL o lhand o snd) THEN
4528 REWRITE_TAC[DOT_SYM] THEN
4529 MATCH_MP_TAC(TAUT `(p ==> q) ==> (~p ==> q) ==> q`) THEN
4530 SIMP_TAC[] THEN SIMP_TAC[DOT_EQ_0; DOT_RZERO; DOT_LZERO] THEN
4531 REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO]])]);;
4533 let ORTHOGONAL_BASIS_EXISTS = prove
4535 ?b. independent b /\
4539 pairwise orthogonal b`,
4540 GEN_TAC THEN MP_TAC(ISPEC `v:real^N->bool` BASIS_EXISTS) THEN
4541 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4542 MP_TAC(SPEC `b:real^N->bool` BASIS_ORTHOGONAL) THEN
4543 ANTS_TAC THENL [ASM_MESON_TAC[HAS_SIZE]; ALL_TAC] THEN
4544 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN
4545 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
4546 [MATCH_MP_TAC CARD_LE_DIM_SPANNING THEN ASM_REWRITE_TAC[] THEN
4547 EXISTS_TAC `span(v):real^N->bool` THEN CONJ_TAC THENL
4548 [ASM_MESON_TAC[SPAN_SPAN; SPAN_MONO];
4549 ASM_MESON_TAC[LE_TRANS; HAS_SIZE; DIM_SPAN]];
4550 ASM_MESON_TAC[SUBSET_TRANS; SPAN_INC; SPAN_SPAN; SPAN_MONO];
4551 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
4552 ASM_REWRITE_TAC[HAS_SIZE; GSYM LE_ANTISYM] THEN
4553 CONJ_TAC THENL [ASM_MESON_TAC[LE_TRANS]; ALL_TAC] THEN
4554 ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN MATCH_MP_TAC SPAN_CARD_GE_DIM THEN
4555 ASM_REWRITE_TAC[] THEN
4556 ASM_MESON_TAC[SPAN_SPAN; SPAN_MONO; SUBSET_TRANS; SPAN_INC]]);;
4559 (`!s t. span s = span t <=> s SUBSET span t /\ t SUBSET span s`,
4560 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ] THEN
4561 MESON_TAC[SUBSET_TRANS; SPAN_SPAN; SPAN_MONO; SPAN_INC]);;
4563 (* ------------------------------------------------------------------------- *)
4564 (* We can extend a linear basis-basis injection to the whole set. *)
4565 (* ------------------------------------------------------------------------- *)
4567 let LINEAR_INDEP_IMAGE_LEMMA = prove
4568 (`!f b. linear(f:real^M->real^N) /\
4570 independent (IMAGE f b) /\
4571 (!x y. x IN b /\ y IN b /\ (f x = f y) ==> (x = y))
4572 ==> !x. x IN span b ==> (f(x) = vec 0) ==> (x = vec 0)`,
4573 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
4574 GEN_TAC THEN DISCH_TAC THEN
4575 GEN_REWRITE_TAC (BINDER_CONV o RAND_CONV) [IMP_IMP] THEN
4576 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
4577 CONJ_TAC THENL [SIMP_TAC[IN_SING; SPAN_EMPTY]; ALL_TAC] THEN
4578 MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M->bool`] THEN STRIP_TAC THEN
4579 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
4581 [ASM_MESON_TAC[INDEPENDENT_MONO; IMAGE_CLAUSES; SUBSET; IN_INSERT];
4583 DISCH_TAC THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
4584 MP_TAC(ISPECL [`a:real^M`; `(a:real^M) INSERT b`; `x:real^M`]
4585 SPAN_BREAKDOWN) THEN
4586 ASM_REWRITE_TAC[IN_INSERT] THEN
4587 SIMP_TAC[ASSUME `~((a:real^M) IN b)`; SET_RULE
4588 `~(a IN b) ==> ((a INSERT b) DELETE a = b)`] THEN
4589 DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN DISCH_TAC THEN
4590 SUBGOAL_THEN `(f:real^M->real^N)(x - k % a) IN span(IMAGE f b)` MP_TAC THENL
4591 [ASM_MESON_TAC[SPAN_LINEAR_IMAGE; IN_IMAGE]; ALL_TAC] THEN
4592 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_SUB th]) THEN
4593 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN
4594 ASM_REWRITE_TAC[VECTOR_ARITH `vec 0 - k % x = (--k) % x`] THEN
4595 ASM_CASES_TAC `k = &0` THENL
4596 [ASM_MESON_TAC[VECTOR_ARITH `x - &0 % y = x`]; ALL_TAC] THEN
4597 DISCH_THEN(MP_TAC o SPEC `--inv(k)` o MATCH_MP SPAN_MUL) THEN
4598 REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LNEG; REAL_MUL_RNEG] THEN
4599 SIMP_TAC[REAL_NEGNEG; REAL_MUL_LINV; ASSUME `~(k = &0)`] THEN
4600 REWRITE_TAC[VECTOR_MUL_LID] THEN
4601 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [independent]) THEN
4602 REWRITE_TAC[dependent; NOT_EXISTS_THM] THEN
4603 DISCH_THEN(MP_TAC o SPEC `(f:real^M->real^N) a`) THEN
4605 `IMAGE (f:real^M->real^N) (a INSERT b) DELETE f a =
4606 IMAGE f ((a INSERT b) DELETE a)`
4608 [REWRITE_TAC[EXTENSION; IN_IMAGE; IN_DELETE; IN_INSERT] THEN
4609 ASM_MESON_TAC[IN_INSERT];
4611 ASM_REWRITE_TAC[DELETE_INSERT] THEN
4612 SIMP_TAC[SET_RULE `~(a IN b) ==> (b DELETE a = b)`;
4613 ASSUME `~(a:real^M IN b)`] THEN
4614 SIMP_TAC[IMAGE_CLAUSES; IN_INSERT]);;
4616 (* ------------------------------------------------------------------------- *)
4617 (* We can extend a linear mapping from basis. *)
4618 (* ------------------------------------------------------------------------- *)
4620 let LINEAR_INDEPENDENT_EXTEND_LEMMA = prove
4623 ==> ?g:real^M->real^N.
4624 (!x y. x IN span b /\ y IN span b
4625 ==> (g(x + y) = g(x) + g(y))) /\
4626 (!x c. x IN span b ==> (g(c % x) = c % g(x))) /\
4627 (!x. x IN b ==> (g x = f x))`,
4628 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
4629 REWRITE_TAC[NOT_IN_EMPTY; INDEPENDENT_INSERT] THEN CONJ_TAC THENL
4630 [REPEAT STRIP_TAC THEN EXISTS_TAC `(\x. vec 0):real^M->real^N` THEN
4631 SIMP_TAC[SPAN_EMPTY] THEN REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC;
4633 SIMP_TAC[] THEN MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M->bool`] THEN
4634 DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
4635 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
4636 DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN
4637 ABBREV_TAC `h = \z:real^M. @k. (z - k % a) IN span b` THEN
4638 SUBGOAL_THEN `!z:real^M. z IN span(a INSERT b)
4639 ==> (z - h(z) % a) IN span(b) /\
4640 !k. (z - k % a) IN span(b) ==> (k = h(z))`
4642 [GEN_TAC THEN DISCH_TAC THEN
4643 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
4644 [EXPAND_TAC "h" THEN CONV_TAC SELECT_CONV THEN
4645 ASM_MESON_TAC[SPAN_BREAKDOWN_EQ];
4647 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN GEN_TAC THEN
4648 DISCH_THEN(MP_TAC o MATCH_MP SPAN_SUB) THEN
4649 REWRITE_TAC[VECTOR_ARITH `(z - a % v) - (z - b % v) = (b - a) % v`] THEN
4650 ASM_CASES_TAC `k = (h:real^M->real) z` THEN ASM_REWRITE_TAC[] THEN
4651 DISCH_THEN(MP_TAC o SPEC `inv(k - (h:real^M->real) z)` o
4652 MATCH_MP SPAN_MUL) THEN
4653 ASM_SIMP_TAC[REAL_MUL_LINV; VECTOR_MUL_ASSOC; REAL_SUB_0] THEN
4654 ASM_REWRITE_TAC[VECTOR_MUL_LID];
4656 REWRITE_TAC[TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN
4657 REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN
4658 GEN_REWRITE_TAC LAND_CONV [FORALL_AND_THM] THEN STRIP_TAC THEN
4659 EXISTS_TAC `\z:real^M. h(z) % (f:real^M->real^N)(a) + g(z - h(z) % a)` THEN
4660 REPEAT CONJ_TAC THENL
4661 [MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN
4662 SUBGOAL_THEN `(h:real^M->real)(x + y) = h(x) + h(y)` ASSUME_TAC THENL
4663 [CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
4664 REWRITE_TAC[VECTOR_ARITH
4665 `(x + y) - (k + l) % a = (x - k % a) + (y - l % a)`] THEN
4666 CONJ_TAC THEN MATCH_MP_TAC SPAN_ADD THEN ASM_REWRITE_TAC[] THEN
4669 ASM_REWRITE_TAC[VECTOR_ARITH
4670 `(x + y) - (k + l) % a = (x - k % a) + (y - l % a)`] THEN
4671 ASM_SIMP_TAC[] THEN VECTOR_ARITH_TAC;
4672 MAP_EVERY X_GEN_TAC [`x:real^M`; `c:real`] THEN STRIP_TAC THEN
4673 SUBGOAL_THEN `(h:real^M->real)(c % x) = c * h(x)` ASSUME_TAC THENL
4674 [CONV_TAC SYM_CONV THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
4675 REWRITE_TAC[VECTOR_ARITH
4676 `c % x - (c * k) % a = c % (x - k % a)`] THEN
4677 CONJ_TAC THEN MATCH_MP_TAC SPAN_MUL THEN ASM_REWRITE_TAC[] THEN
4680 ASM_REWRITE_TAC[VECTOR_ARITH
4681 `c % x - (c * k) % a = c % (x - k % a)`] THEN
4682 ASM_SIMP_TAC[] THEN VECTOR_ARITH_TAC;
4684 X_GEN_TAC `x:real^M` THEN REWRITE_TAC[IN_INSERT] THEN
4685 DISCH_THEN(DISJ_CASES_THEN2 SUBST_ALL_TAC ASSUME_TAC) THENL
4686 [SUBGOAL_THEN `&1 = h(a:real^M)` (SUBST1_TAC o SYM) THENL
4687 [FIRST_X_ASSUM MATCH_MP_TAC; ALL_TAC] THEN
4688 REWRITE_TAC[VECTOR_ARITH `a - &1 % a = vec 0`; SPAN_0] THENL
4689 [ASM_MESON_TAC[SPAN_SUPERSET; SUBSET; IN_INSERT]; ALL_TAC] THEN
4690 FIRST_X_ASSUM(MP_TAC o SPECL [`vec 0:real^M`; `vec 0:real^M`]) THEN
4691 REWRITE_TAC[SPAN_0; VECTOR_ADD_LID] THEN
4692 REWRITE_TAC[VECTOR_ARITH `(a = a + a) <=> (a = vec 0)`] THEN
4693 DISCH_THEN SUBST1_TAC THEN VECTOR_ARITH_TAC;
4695 SUBGOAL_THEN `&0 = h(x:real^M)` (SUBST1_TAC o SYM) THENL
4696 [FIRST_X_ASSUM MATCH_MP_TAC; ALL_TAC] THEN
4697 REWRITE_TAC[VECTOR_ADD_LID; VECTOR_MUL_LZERO; VECTOR_SUB_RZERO] THEN
4698 ASM_MESON_TAC[SUBSET; IN_INSERT; SPAN_SUPERSET]);;
4700 let LINEAR_INDEPENDENT_EXTEND = prove
4701 (`!f b. independent b
4702 ==> ?g:real^M->real^N. linear g /\ (!x. x IN b ==> (g x = f x))`,
4703 REPEAT STRIP_TAC THEN
4704 MP_TAC(ISPECL [`b:real^M->bool`; `(:real^M)`]
4705 MAXIMAL_INDEPENDENT_SUBSET_EXTEND) THEN
4706 ASM_REWRITE_TAC[SUBSET_UNIV; UNIV_SUBSET] THEN
4707 REWRITE_TAC[EXTENSION; IN_UNIV] THEN
4708 DISCH_THEN(X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC) THEN
4709 MP_TAC(ISPECL [`f:real^M->real^N`; `c:real^M->bool`]
4710 LINEAR_INDEPENDENT_EXTEND_LEMMA) THEN
4711 ASM_SIMP_TAC[INDEPENDENT_BOUND; linear] THEN
4712 ASM_MESON_TAC[SUBSET]);;
4714 (* ------------------------------------------------------------------------- *)
4715 (* Linear functions are equal on a subspace if they are on a spanning set. *)
4716 (* ------------------------------------------------------------------------- *)
4718 let SUBSPACE_KERNEL = prove
4719 (`!f. linear f ==> subspace {x | f(x) = vec 0}`,
4720 REWRITE_TAC[subspace; IN_ELIM_THM] THEN
4721 SIMP_TAC[LINEAR_ADD; LINEAR_CMUL; VECTOR_ADD_LID; VECTOR_MUL_RZERO] THEN
4722 MESON_TAC[LINEAR_0]);;
4724 let LINEAR_EQ_0_SPAN = prove
4725 (`!f:real^M->real^N b.
4726 linear f /\ (!x. x IN b ==> f(x) = vec 0)
4727 ==> !x. x IN span(b) ==> f(x) = vec 0`,
4728 REPEAT GEN_TAC THEN STRIP_TAC THEN
4729 RULE_ASSUM_TAC(REWRITE_RULE[IN]) THEN
4730 MATCH_MP_TAC SPAN_INDUCT THEN ASM_REWRITE_TAC[IN] THEN
4731 MP_TAC(ISPEC `f:real^M->real^N` SUBSPACE_KERNEL) THEN
4732 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN
4733 AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM]);;
4735 let LINEAR_EQ_0 = prove
4736 (`!f b s. linear f /\ s SUBSET (span b) /\ (!x. x IN b ==> f(x) = vec 0)
4737 ==> !x. x IN s ==> f(x) = vec 0`,
4738 MESON_TAC[LINEAR_EQ_0_SPAN; SUBSET]);;
4740 let LINEAR_EQ = prove
4741 (`!f g b s. linear f /\ linear g /\ s SUBSET (span b) /\
4742 (!x. x IN b ==> f(x) = g(x))
4743 ==> !x. x IN s ==> f(x) = g(x)`,
4744 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
4745 STRIP_TAC THEN MATCH_MP_TAC LINEAR_EQ_0 THEN
4746 ASM_MESON_TAC[LINEAR_COMPOSE_SUB]);;
4748 let LINEAR_EQ_STDBASIS = prove
4749 (`!f:real^M->real^N g.
4750 linear f /\ linear g /\
4751 (!i. 1 <= i /\ i <= dimindex(:M)
4752 ==> f(basis i) = g(basis i))
4754 REPEAT STRIP_TAC THEN
4755 SUBGOAL_THEN `!x. x IN UNIV ==> (f:real^M->real^N) x = g x`
4756 (fun th -> MP_TAC th THEN REWRITE_TAC[FUN_EQ_THM; IN_UNIV]) THEN
4757 MATCH_MP_TAC LINEAR_EQ THEN
4758 EXISTS_TAC `{basis i :real^M | 1 <= i /\ i <= dimindex(:M)}` THEN
4759 ASM_REWRITE_TAC[SPAN_STDBASIS; SUBSET_REFL; IN_ELIM_THM] THEN
4762 (* ------------------------------------------------------------------------- *)
4763 (* Similar results for bilinear functions. *)
4764 (* ------------------------------------------------------------------------- *)
4766 let BILINEAR_EQ = prove
4767 (`!f:real^M->real^N->real^P g b c s.
4768 bilinear f /\ bilinear g /\
4769 s SUBSET (span b) /\ t SUBSET (span c) /\
4770 (!x y. x IN b /\ y IN c ==> f x y = g x y)
4771 ==> !x y. x IN s /\ y IN t ==> f x y = g x y`,
4772 REPEAT STRIP_TAC THEN SUBGOAL_THEN
4773 `!x:real^M. x IN span b
4774 ==> !y:real^N. y IN span c ==> (f x y :real^P = g x y)`
4775 (fun th -> ASM_MESON_TAC[th; SUBSET]) THEN
4776 MATCH_MP_TAC SPAN_INDUCT THEN REWRITE_TAC[subspace; IN_ELIM_THM] THEN
4778 [GEN_TAC THEN DISCH_TAC;
4779 ASM_SIMP_TAC[BILINEAR_LADD; BILINEAR_LMUL] THEN
4780 ASM_MESON_TAC[BILINEAR_LZERO]] THEN
4781 MATCH_MP_TAC SPAN_INDUCT THEN REWRITE_TAC[subspace; IN_ELIM_THM] THEN
4782 ASM_SIMP_TAC[BILINEAR_RADD; BILINEAR_RMUL] THEN
4783 ASM_MESON_TAC[BILINEAR_RZERO]);;
4785 let BILINEAR_EQ_STDBASIS = prove
4786 (`!f:real^M->real^N->real^P g.
4787 bilinear f /\ bilinear g /\
4788 (!i j. 1 <= i /\ i <= dimindex(:M) /\ 1 <= j /\ j <= dimindex(:N)
4789 ==> f (basis i) (basis j) = g (basis i) (basis j))
4791 REPEAT STRIP_TAC THEN SUBGOAL_THEN
4792 `!x y. x IN UNIV /\ y IN UNIV ==> (f:real^M->real^N->real^P) x y = g x y`
4793 (fun th -> MP_TAC th THEN REWRITE_TAC[FUN_EQ_THM; IN_UNIV]) THEN
4794 MATCH_MP_TAC BILINEAR_EQ THEN
4795 EXISTS_TAC `{basis i :real^M | 1 <= i /\ i <= dimindex(:M)}` THEN
4796 EXISTS_TAC `{basis i :real^N | 1 <= i /\ i <= dimindex(:N)}` THEN
4797 ASM_REWRITE_TAC[SPAN_STDBASIS; SUBSET_REFL; IN_ELIM_THM] THEN
4800 (* ------------------------------------------------------------------------- *)
4801 (* Detailed theorems about left and right invertibility in general case. *)
4802 (* ------------------------------------------------------------------------- *)
4804 let LEFT_INVERTIBLE_TRANSP = prove
4806 (?B:real^N^M. B ** transp A = mat 1) <=> (?B:real^M^N. A ** B = mat 1)`,
4807 MESON_TAC[MATRIX_TRANSP_MUL; TRANSP_MAT; TRANSP_TRANSP]);;
4809 let RIGHT_INVERTIBLE_TRANSP = prove
4811 (?B:real^N^M. transp A ** B = mat 1) <=> (?B:real^M^N. B ** A = mat 1)`,
4812 MESON_TAC[MATRIX_TRANSP_MUL; TRANSP_MAT; TRANSP_TRANSP]);;
4814 let LINEAR_INJECTIVE_LEFT_INVERSE = prove
4815 (`!f:real^M->real^N.
4816 linear f /\ (!x y. f x = f y ==> x = y)
4817 ==> ?g. linear g /\ g o f = I`,
4818 REWRITE_TAC[INJECTIVE_LEFT_INVERSE] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN
4819 `?h. linear(h:real^N->real^M) /\
4820 !x. x IN IMAGE (f:real^M->real^N)
4821 {basis i | 1 <= i /\ i <= dimindex(:M)} ==> h x = g x`
4823 [MATCH_MP_TAC LINEAR_INDEPENDENT_EXTEND THEN
4824 MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE THEN
4825 ASM_MESON_TAC[INJECTIVE_LEFT_INVERSE; INDEPENDENT_STDBASIS];
4826 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^N->real^M` THEN
4827 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_ELIM_THM] THEN STRIP_TAC THEN
4828 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LINEAR_EQ_STDBASIS THEN
4829 ASM_SIMP_TAC[I_DEF; LINEAR_COMPOSE; LINEAR_ID; o_THM] THEN
4832 let LINEAR_SURJECTIVE_RIGHT_INVERSE = prove
4833 (`!f:real^M->real^N.
4834 linear f /\ (!y. ?x. f x = y) ==> ?g. linear g /\ f o g = I`,
4835 REWRITE_TAC[SURJECTIVE_RIGHT_INVERSE] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN
4836 `?h. linear(h:real^N->real^M) /\
4837 !x. x IN {basis i | 1 <= i /\ i <= dimindex(:N)} ==> h x = g x`
4839 [MATCH_MP_TAC LINEAR_INDEPENDENT_EXTEND THEN
4840 REWRITE_TAC[INDEPENDENT_STDBASIS];
4841 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `h:real^N->real^M` THEN
4842 ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_ELIM_THM] THEN STRIP_TAC THEN
4843 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LINEAR_EQ_STDBASIS THEN
4844 ASM_SIMP_TAC[I_DEF; LINEAR_COMPOSE; LINEAR_ID; o_THM] THEN
4847 let MATRIX_LEFT_INVERTIBLE_INJECTIVE = prove
4849 (?B:real^M^N. B ** A = mat 1) <=>
4850 !x y:real^N. A ** x = A ** y ==> x = y`,
4851 GEN_TAC THEN EQ_TAC THENL
4852 [STRIP_TAC THEN REPEAT GEN_TAC THEN
4853 DISCH_THEN(MP_TAC o AP_TERM `\x:real^M. (B:real^M^N) ** x`) THEN
4854 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_VECTOR_MUL_LID];
4855 DISCH_TAC THEN MP_TAC(ISPEC
4856 `\x:real^N. (A:real^N^M) ** x` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
4857 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR; FUN_EQ_THM; I_THM; o_THM] THEN
4858 DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN
4859 EXISTS_TAC `matrix(g):real^M^N` THEN
4860 REWRITE_TAC[MATRIX_EQ; MATRIX_VECTOR_MUL_LID] THEN
4861 ASM_MESON_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_WORKS]]);;
4863 let MATRIX_LEFT_INVERTIBLE_KER = prove
4865 (?B:real^M^N. B ** A = mat 1) <=> !x. A ** x = vec 0 ==> x = vec 0`,
4866 GEN_TAC THEN REWRITE_TAC[MATRIX_LEFT_INVERTIBLE_INJECTIVE] THEN
4867 MATCH_MP_TAC LINEAR_INJECTIVE_0 THEN REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR]);;
4869 let MATRIX_RIGHT_INVERTIBLE_SURJECTIVE = prove
4871 (?B:real^M^N. A ** B = mat 1) <=> !y. ?x. A ** x = y`,
4872 GEN_TAC THEN EQ_TAC THENL
4873 [STRIP_TAC THEN X_GEN_TAC `y:real^M` THEN
4874 EXISTS_TAC `(B:real^M^N) ** (y:real^M)` THEN
4875 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_VECTOR_MUL_LID];
4876 DISCH_TAC THEN MP_TAC(ISPEC
4877 `\x:real^N. (A:real^N^M) ** x` LINEAR_SURJECTIVE_RIGHT_INVERSE) THEN
4878 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR; FUN_EQ_THM; I_THM; o_THM] THEN
4879 DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^N` STRIP_ASSUME_TAC) THEN
4880 EXISTS_TAC `matrix(g):real^M^N` THEN
4881 REWRITE_TAC[MATRIX_EQ; MATRIX_VECTOR_MUL_LID] THEN
4882 ASM_MESON_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_WORKS]]);;
4884 let MATRIX_LEFT_INVERTIBLE_INDEPENDENT_COLUMNS = prove
4885 (`!A:real^N^M. (?B:real^M^N. B ** A = mat 1) <=>
4886 !c. vsum(1..dimindex(:N)) (\i. c(i) % column i A) = vec 0 ==>
4887 !i. 1 <= i /\ i <= dimindex(:N) ==> c(i) = &0`,
4888 GEN_TAC THEN REWRITE_TAC[MATRIX_LEFT_INVERTIBLE_KER; MATRIX_MUL_VSUM] THEN
4889 EQ_TAC THEN DISCH_TAC THENL
4890 [X_GEN_TAC `c:num->real` THEN DISCH_TAC THEN
4891 FIRST_X_ASSUM(MP_TAC o SPEC `(lambda i. c(i)):real^N`);
4892 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
4893 FIRST_X_ASSUM(MP_TAC o SPEC `\i. (x:real^N)$i`)] THEN
4894 ASM_SIMP_TAC[LAMBDA_BETA; CART_EQ; VEC_COMPONENT]);;
4896 let MATRIX_RIGHT_INVERTIBLE_INDEPENDENT_ROWS = prove
4897 (`!A:real^N^M. (?B:real^M^N. A ** B = mat 1) <=>
4898 !c. vsum(1..dimindex(:M)) (\i. c(i) % row i A) = vec 0 ==>
4899 !i. 1 <= i /\ i <= dimindex(:M) ==> c(i) = &0`,
4900 ONCE_REWRITE_TAC[GSYM LEFT_INVERTIBLE_TRANSP] THEN
4901 REWRITE_TAC[MATRIX_LEFT_INVERTIBLE_INDEPENDENT_COLUMNS] THEN
4902 SIMP_TAC[COLUMN_TRANSP]);;
4904 let MATRIX_RIGHT_INVERTIBLE_SPAN_COLUMNS = prove
4905 (`!A:real^N^M. (?B:real^M^N. A ** B = mat 1) <=> span(columns A) = (:real^M)`,
4906 GEN_TAC THEN REWRITE_TAC[MATRIX_RIGHT_INVERTIBLE_SURJECTIVE] THEN
4907 REWRITE_TAC[MATRIX_MUL_VSUM; EXTENSION; IN_UNIV] THEN
4908 AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `y:real^M` THEN
4910 [DISCH_THEN(X_CHOOSE_THEN `x:real^N` (SUBST1_TAC o SYM)) THEN
4911 MATCH_MP_TAC SPAN_VSUM THEN REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
4912 X_GEN_TAC `i:num` THEN STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
4913 MATCH_MP_TAC(CONJUNCT1 SPAN_CLAUSES) THEN
4914 REWRITE_TAC[columns; IN_ELIM_THM] THEN ASM_MESON_TAC[];
4916 SPEC_TAC(`y:real^M`,`y:real^M`) THEN MATCH_MP_TAC SPAN_INDUCT_ALT THEN
4918 [EXISTS_TAC `vec 0 :real^N` THEN
4919 SIMP_TAC[VEC_COMPONENT; VECTOR_MUL_LZERO; VSUM_0];
4921 MAP_EVERY X_GEN_TAC [`c:real`; `y1:real^M`; `y2:real^M`] THEN
4922 REWRITE_TAC[columns; IN_ELIM_THM] THEN DISCH_THEN(CONJUNCTS_THEN2
4923 (X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC)
4924 (X_CHOOSE_THEN `x:real^N` (SUBST1_TAC o SYM))) THEN
4925 EXISTS_TAC `(lambda j. if j = i then c + (x:real^N)$i else x$j):real^N` THEN
4926 SUBGOAL_THEN `1..dimindex(:N) = i INSERT ((1..dimindex(:N)) DELETE i)`
4927 SUBST1_TAC THENL [ASM_MESON_TAC[INSERT_DELETE; IN_NUMSEG]; ALL_TAC] THEN
4928 SIMP_TAC[VSUM_CLAUSES; FINITE_DELETE; FINITE_NUMSEG; IN_DELETE] THEN
4929 ASM_SIMP_TAC[LAMBDA_BETA; VECTOR_ADD_RDISTRIB; VECTOR_ADD_ASSOC] THEN
4930 AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
4931 SIMP_TAC[FINITE_DELETE; IN_DELETE; FINITE_NUMSEG; LAMBDA_BETA; IN_NUMSEG]);;
4933 let MATRIX_LEFT_INVERTIBLE_SPAN_ROWS = prove
4934 (`!A:real^N^M. (?B:real^M^N. B ** A = mat 1) <=> span(rows A) = (:real^N)`,
4935 MESON_TAC[RIGHT_INVERTIBLE_TRANSP; COLUMNS_TRANSP;
4936 MATRIX_RIGHT_INVERTIBLE_SPAN_COLUMNS]);;
4938 (* ------------------------------------------------------------------------- *)
4939 (* An injective map real^N->real^N is also surjective. *)
4940 (* ------------------------------------------------------------------------- *)
4942 let LINEAR_INJECTIVE_IMP_SURJECTIVE = prove
4943 (`!f:real^N->real^N.
4944 linear f /\ (!x y. (f(x) = f(y)) ==> (x = y))
4945 ==> !y. ?x. f(x) = y`,
4946 REPEAT STRIP_TAC THEN
4947 MP_TAC(ISPEC `(:real^N)` BASIS_EXISTS) THEN
4948 REWRITE_TAC[SUBSET_UNIV; HAS_SIZE] THEN
4949 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4950 SUBGOAL_THEN `UNIV SUBSET span(IMAGE (f:real^N->real^N) b)` MP_TAC THENL
4951 [MATCH_MP_TAC CARD_GE_DIM_INDEPENDENT THEN
4952 ASM_MESON_TAC[INDEPENDENT_INJECTIVE_IMAGE; LE_REFL;
4953 SUBSET_UNIV; CARD_IMAGE_INJ];
4954 ASM_SIMP_TAC[SPAN_LINEAR_IMAGE] THEN
4955 ASM_MESON_TAC[SUBSET; IN_IMAGE; IN_UNIV]]);;
4957 (* ------------------------------------------------------------------------- *)
4958 (* And vice versa. *)
4959 (* ------------------------------------------------------------------------- *)
4961 let LINEAR_SURJECTIVE_IMP_INJECTIVE = prove
4962 (`!f:real^N->real^N.
4963 linear f /\ (!y. ?x. f(x) = y)
4964 ==> !x y. (f(x) = f(y)) ==> (x = y)`,
4965 REPEAT GEN_TAC THEN STRIP_TAC THEN
4966 MP_TAC(ISPEC `(:real^N)` BASIS_EXISTS) THEN
4967 REWRITE_TAC[SUBSET_UNIV; HAS_SIZE] THEN
4968 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
4970 `!x. x IN span b ==> (f:real^N->real^N) x = vec 0 ==> x = vec 0`
4971 (fun th -> ASM_MESON_TAC[th; LINEAR_INJECTIVE_0; SUBSET; IN_UNIV]) THEN
4972 MATCH_MP_TAC LINEAR_INDEP_IMAGE_LEMMA THEN
4973 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
4974 [MATCH_MP_TAC CARD_LE_DIM_SPANNING THEN
4975 EXISTS_TAC `(:real^N)` THEN
4976 ASM_SIMP_TAC[SUBSET_UNIV; FINITE_IMAGE; SPAN_LINEAR_IMAGE] THEN
4977 REWRITE_TAC[SUBSET; IN_UNIV; IN_IMAGE] THEN
4978 ASM_MESON_TAC[CARD_IMAGE_LE; SUBSET; IN_UNIV];
4980 SUBGOAL_THEN `dim(:real^N) <= CARD(IMAGE (f:real^N->real^N) b)`
4982 [MATCH_MP_TAC SPAN_CARD_GE_DIM THEN
4983 ASM_SIMP_TAC[SUBSET_UNIV; FINITE_IMAGE] THEN
4984 ASM_SIMP_TAC[SPAN_LINEAR_IMAGE] THEN MATCH_MP_TAC SUBSET_TRANS THEN
4985 EXISTS_TAC `IMAGE (f:real^N->real^N) UNIV` THEN
4986 ASM_SIMP_TAC[IMAGE_SUBSET] THEN
4987 ASM_REWRITE_TAC[SUBSET; IN_IMAGE; IN_UNIV] THEN ASM_MESON_TAC[];
4989 FIRST_X_ASSUM(MP_TAC o ISPEC `f:real^N->real^N` o
4990 MATCH_MP CARD_IMAGE_LE) THEN
4991 ASM_REWRITE_TAC[IMP_IMP; LE_ANTISYM] THEN DISCH_TAC THEN
4993 [`b:real^N->bool`; `IMAGE (f:real^N->real^N) b`; `f:real^N->real^N`]
4994 SURJECTIVE_IFF_INJECTIVE_GEN) THEN
4995 ASM_SIMP_TAC[FINITE_IMAGE; INDEPENDENT_BOUND; SUBSET_REFL] THEN
4996 REWRITE_TAC[FORALL_IN_IMAGE] THEN MESON_TAC[]);;
4998 let LINEAR_SURJECTIVE_IFF_INJECTIVE = prove
4999 (`!f:real^N->real^N.
5000 linear f ==> ((!y. ?x. f x = y) <=> (!x y. f x = f y ==> x = y))`,
5001 MESON_TAC[LINEAR_INJECTIVE_IMP_SURJECTIVE;
5002 LINEAR_SURJECTIVE_IMP_INJECTIVE]);;
5004 (* ------------------------------------------------------------------------- *)
5005 (* Hence either is enough for isomorphism. *)
5006 (* ------------------------------------------------------------------------- *)
5008 let LEFT_RIGHT_INVERSE_EQ = prove
5009 (`!f:A->A g h. f o g = I /\ g o h = I ==> f = h`,
5010 MESON_TAC[o_ASSOC; I_O_ID]);;
5012 let ISOMORPHISM_EXPAND = prove
5013 (`!f g. f o g = I /\ g o f = I <=> (!x. f(g x) = x) /\ (!x. g(f x) = x)`,
5014 REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM]);;
5016 let LINEAR_INJECTIVE_ISOMORPHISM = prove
5017 (`!f:real^N->real^N.
5018 linear f /\ (!x y. f x = f y ==> x = y)
5019 ==> ?f'. linear f' /\ (!x. f'(f x) = x) /\ (!x. f(f' x) = x)`,
5020 REPEAT STRIP_TAC THEN
5021 REWRITE_TAC[GSYM ISOMORPHISM_EXPAND] THEN
5022 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_SURJECTIVE_RIGHT_INVERSE) THEN
5023 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
5024 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_INJECTIVE_IMP_SURJECTIVE) THEN
5025 ASM_REWRITE_TAC[] THEN SIMP_TAC[] THEN MESON_TAC[LEFT_RIGHT_INVERSE_EQ]);;
5027 let LINEAR_SURJECTIVE_ISOMORPHISM = prove
5028 (`!f:real^N->real^N.
5029 linear f /\ (!y. ?x. f x = y)
5030 ==> ?f'. linear f' /\ (!x. f'(f x) = x) /\ (!x. f(f' x) = x)`,
5031 REPEAT STRIP_TAC THEN
5032 REWRITE_TAC[GSYM ISOMORPHISM_EXPAND] THEN
5033 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_SURJECTIVE_RIGHT_INVERSE) THEN
5034 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
5035 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_SURJECTIVE_IMP_INJECTIVE) THEN
5036 ASM_REWRITE_TAC[] THEN SIMP_TAC[] THEN MESON_TAC[LEFT_RIGHT_INVERSE_EQ]);;
5038 (* ------------------------------------------------------------------------- *)
5039 (* Left and right inverses are the same for R^N->R^N. *)
5040 (* ------------------------------------------------------------------------- *)
5042 let LINEAR_INVERSE_LEFT = prove
5043 (`!f:real^N->real^N f'.
5044 linear f /\ linear f' ==> ((f o f' = I) <=> (f' o f = I))`,
5046 `!f:real^N->real^N f'.
5047 linear f /\ linear f' /\ (f o f' = I) ==> (f' o f = I)`
5048 (fun th -> MESON_TAC[th]) THEN
5049 REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REPEAT STRIP_TAC THEN
5050 MP_TAC(ISPEC `f:real^N->real^N` LINEAR_SURJECTIVE_ISOMORPHISM) THEN
5053 (* ------------------------------------------------------------------------- *)
5054 (* Moreover, a one-sided inverse is automatically linear. *)
5055 (* ------------------------------------------------------------------------- *)
5057 let LEFT_INVERSE_LINEAR = prove
5058 (`!f g:real^N->real^N. linear f /\ (g o f = I) ==> linear g`,
5059 REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
5060 STRIP_TAC THEN SUBGOAL_THEN
5061 `?h:real^N->real^N. linear h /\ (!x. h(f x) = x) /\ (!x. f(h x) = x)`
5063 [MATCH_MP_TAC LINEAR_INJECTIVE_ISOMORPHISM THEN ASM_MESON_TAC[];
5064 SUBGOAL_THEN `g:real^N->real^N = h` (fun th -> ASM_REWRITE_TAC[th]) THEN
5065 REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[]]);;
5067 let RIGHT_INVERSE_LINEAR = prove
5068 (`!f g:real^N->real^N. linear f /\ (f o g = I) ==> linear g`,
5069 REPEAT GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
5070 STRIP_TAC THEN SUBGOAL_THEN
5071 `?h:real^N->real^N. linear h /\ (!x. h(f x) = x) /\ (!x. f(h x) = x)`
5072 CHOOSE_TAC THENL [ASM_MESON_TAC[LINEAR_SURJECTIVE_ISOMORPHISM]; ALL_TAC] THEN
5073 SUBGOAL_THEN `g:real^N->real^N = h` (fun th -> ASM_REWRITE_TAC[th]) THEN
5074 REWRITE_TAC[FUN_EQ_THM] THEN ASM_MESON_TAC[]);;
5076 (* ------------------------------------------------------------------------- *)
5077 (* Without (ostensible) constraints on types, though dimensions must match. *)
5078 (* ------------------------------------------------------------------------- *)
5080 let LEFT_RIGHT_INVERSE_LINEAR = prove
5081 (`!f g:real^M->real^N.
5082 linear f /\ g o f = I /\ f o g = I ==> linear g`,
5083 REWRITE_TAC[linear; FUN_EQ_THM; o_THM; I_THM] THEN MESON_TAC[]);;
5085 let LINEAR_BIJECTIVE_LEFT_RIGHT_INVERSE = prove
5086 (`!f:real^M->real^N.
5087 linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
5088 ==> ?g. linear g /\ (!x. g(f x) = x) /\ (!y. f(g y) = y)`,
5089 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
5090 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [BIJECTIVE_LEFT_RIGHT_INVERSE]) THEN
5091 MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5092 MATCH_MP_TAC LEFT_RIGHT_INVERSE_LINEAR THEN
5093 EXISTS_TAC `f:real^M->real^N` THEN
5094 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM]);;
5096 (* ------------------------------------------------------------------------- *)
5097 (* The same result in terms of square matrices. *)
5098 (* ------------------------------------------------------------------------- *)
5100 let MATRIX_LEFT_RIGHT_INVERSE = prove
5101 (`!A:real^N^N A':real^N^N. (A ** A' = mat 1) <=> (A' ** A = mat 1)`,
5103 `!A:real^N^N A':real^N^N. (A ** A' = mat 1) ==> (A' ** A = mat 1)`
5104 (fun th -> MESON_TAC[th]) THEN
5105 REPEAT STRIP_TAC THEN
5106 MP_TAC(ISPEC `\x:real^N. A:(real^N^N) ** x`
5107 LINEAR_SURJECTIVE_ISOMORPHISM) THEN
5108 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN ANTS_TAC THENL
5109 [X_GEN_TAC `x:real^N` THEN EXISTS_TAC `(A':real^N^N) ** (x:real^N)` THEN
5110 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_ASSOC; MATRIX_VECTOR_MUL_LID];
5112 DISCH_THEN(X_CHOOSE_THEN `f':real^N->real^N` STRIP_ASSUME_TAC) THEN
5113 SUBGOAL_THEN `matrix (f':real^N->real^N) ** (A:real^N^N) = mat 1`
5115 [ASM_SIMP_TAC[MATRIX_EQ; MATRIX_WORKS; GSYM MATRIX_VECTOR_MUL_ASSOC;
5116 MATRIX_VECTOR_MUL_LID];
5118 DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
5119 DISCH_THEN(MP_TAC o AP_TERM `(\m:real^N^N. m ** (A':real^N^N))`) THEN
5120 REWRITE_TAC[GSYM MATRIX_MUL_ASSOC] THEN
5121 ASM_REWRITE_TAC[MATRIX_MUL_RID; MATRIX_MUL_LID] THEN ASM_MESON_TAC[]);;
5123 (* ------------------------------------------------------------------------- *)
5124 (* Invertibility of matrices and corresponding linear functions. *)
5125 (* ------------------------------------------------------------------------- *)
5127 let MATRIX_LEFT_INVERTIBLE = prove
5128 (`!f:real^M->real^N.
5129 linear f ==> ((?B:real^N^M. B ** matrix f = mat 1) <=>
5130 (?g. linear g /\ g o f = I))`,
5131 GEN_TAC THEN DISCH_TAC THEN EQ_TAC THEN STRIP_TAC THENL
5132 [EXISTS_TAC `\y:real^N. (B:real^N^M) ** y` THEN
5133 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN
5134 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV)
5135 [MATCH_MP MATRIX_VECTOR_MUL th]) THEN
5136 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM; MATRIX_VECTOR_MUL_ASSOC;
5137 MATRIX_VECTOR_MUL_LID];
5138 EXISTS_TAC `matrix(g:real^N->real^M)` THEN
5139 ASM_SIMP_TAC[GSYM MATRIX_COMPOSE; MATRIX_I]]);;
5141 let MATRIX_RIGHT_INVERTIBLE = prove
5142 (`!f:real^M->real^N.
5143 linear f ==> ((?B:real^N^M. matrix f ** B = mat 1) <=>
5144 (?g. linear g /\ f o g = I))`,
5145 GEN_TAC THEN DISCH_TAC THEN EQ_TAC THEN STRIP_TAC THENL
5146 [EXISTS_TAC `\y:real^N. (B:real^N^M) ** y` THEN
5147 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN
5148 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV)
5149 [MATCH_MP MATRIX_VECTOR_MUL th]) THEN
5150 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM; MATRIX_VECTOR_MUL_ASSOC;
5151 MATRIX_VECTOR_MUL_LID];
5152 EXISTS_TAC `matrix(g:real^N->real^M)` THEN
5153 ASM_SIMP_TAC[GSYM MATRIX_COMPOSE; MATRIX_I]]);;
5155 let INVERTIBLE_LEFT_INVERSE = prove
5156 (`!A:real^N^N. invertible(A) <=> ?B:real^N^N. B ** A = mat 1`,
5157 MESON_TAC[invertible; MATRIX_LEFT_RIGHT_INVERSE]);;
5159 let INVERTIBLE_RIGHT_INVERSE = prove
5160 (`!A:real^N^N. invertible(A) <=> ?B:real^N^N. A ** B = mat 1`,
5161 MESON_TAC[invertible; MATRIX_LEFT_RIGHT_INVERSE]);;
5163 let MATRIX_INVERTIBLE = prove
5164 (`!f:real^N->real^N.
5166 ==> (invertible(matrix f) <=>
5167 ?g. linear g /\ f o g = I /\ g o f = I)`,
5168 SIMP_TAC[INVERTIBLE_LEFT_INVERSE; MATRIX_LEFT_INVERTIBLE] THEN
5169 MESON_TAC[LINEAR_INVERSE_LEFT]);;
5171 (* ------------------------------------------------------------------------- *)
5172 (* Left-invertible linear transformation has a lower bound. *)
5173 (* ------------------------------------------------------------------------- *)
5175 let LINEAR_INVERTIBLE_BOUNDED_BELOW_POS = prove
5176 (`!f:real^M->real^N g.
5177 linear f /\ linear g /\ (g o f = I)
5178 ==> ?B. &0 < B /\ !x. B * norm(x) <= norm(f x)`,
5179 REPEAT STRIP_TAC THEN
5180 MP_TAC(ISPEC `g:real^N->real^M` LINEAR_BOUNDED_POS) THEN
5181 ASM_REWRITE_TAC[] THEN
5182 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
5183 EXISTS_TAC `inv B:real` THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN
5184 X_GEN_TAC `x:real^M` THEN MATCH_MP_TAC REAL_LE_TRANS THEN
5185 EXISTS_TAC `inv(B) * norm(((g:real^N->real^M) o (f:real^M->real^N)) x)` THEN
5186 CONJ_TAC THENL [ASM_SIMP_TAC[I_THM; REAL_LE_REFL]; ALL_TAC] THEN
5187 REWRITE_TAC[REAL_ARITH `inv B * x = x / B`] THEN
5188 ASM_SIMP_TAC[o_THM; REAL_LE_LDIV_EQ] THEN
5189 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_REWRITE_TAC[]);;
5191 let LINEAR_INVERTIBLE_BOUNDED_BELOW = prove
5192 (`!f:real^M->real^N g.
5193 linear f /\ linear g /\ (g o f = I)
5194 ==> ?B. !x. B * norm(x) <= norm(f x)`,
5195 MESON_TAC[LINEAR_INVERTIBLE_BOUNDED_BELOW_POS]);;
5197 let LINEAR_INJECTIVE_BOUNDED_BELOW_POS = prove
5198 (`!f:real^M->real^N.
5199 linear f /\ (!x y. f x = f y ==> x = y)
5200 ==> ?B. &0 < B /\ !x. norm(x) * B <= norm(f x)`,
5201 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
5202 MATCH_MP_TAC LINEAR_INVERTIBLE_BOUNDED_BELOW_POS THEN
5203 ASM_MESON_TAC[LINEAR_INJECTIVE_LEFT_INVERSE]);;
5205 (* ------------------------------------------------------------------------- *)
5206 (* Preservation of dimension by injective map. *)
5207 (* ------------------------------------------------------------------------- *)
5209 let DIM_INJECTIVE_LINEAR_IMAGE = prove
5210 (`!f:real^M->real^N s.
5211 linear f /\ (!x y. f x = f y ==> x = y) ==> dim(IMAGE f s) = dim s`,
5212 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM LE_ANTISYM] THEN
5213 CONJ_TAC THENL [ASM_MESON_TAC[DIM_LINEAR_IMAGE_LE]; ALL_TAC] THEN
5214 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
5215 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
5216 DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
5217 MATCH_MP_TAC LE_TRANS THEN
5218 EXISTS_TAC `dim(IMAGE (g:real^N->real^M) (IMAGE (f:real^M->real^N) s))` THEN
5220 [ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID; LE_REFL];
5221 MATCH_MP_TAC DIM_LINEAR_IMAGE_LE THEN ASM_REWRITE_TAC[]]);;
5223 (* ------------------------------------------------------------------------- *)
5224 (* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *)
5225 (* ------------------------------------------------------------------------- *)
5227 let rowvector = new_definition
5228 `(rowvector:real^N->real^N^1) v = lambda i j. v$j`;;
5230 let columnvector = new_definition
5231 `(columnvector:real^N->real^1^N) v = lambda i j. v$i`;;
5233 let TRANSP_COLUMNVECTOR = prove
5234 (`!v. transp(columnvector v) = rowvector v`,
5235 SIMP_TAC[transp; columnvector; rowvector; CART_EQ; LAMBDA_BETA]);;
5237 let TRANSP_ROWVECTOR = prove
5238 (`!v. transp(rowvector v) = columnvector v`,
5239 SIMP_TAC[transp; columnvector; rowvector; CART_EQ; LAMBDA_BETA]);;
5241 let DOT_ROWVECTOR_COLUMNVECTOR = prove
5242 (`!A:real^N^M v:real^N. columnvector(A ** v) = A ** columnvector v`,
5243 REWRITE_TAC[rowvector; columnvector; matrix_mul; matrix_vector_mul] THEN
5244 SIMP_TAC[CART_EQ; LAMBDA_BETA]);;
5246 let DOT_MATRIX_PRODUCT = prove
5247 (`!x y:real^N. x dot y = (rowvector x ** columnvector y)$1$1`,
5248 REWRITE_TAC[matrix_mul; columnvector; rowvector; dot] THEN
5249 SIMP_TAC[LAMBDA_BETA; DIMINDEX_1; LE_REFL]);;
5251 let DOT_MATRIX_VECTOR_MUL = prove
5252 (`!A:real^N^N B:real^N^N x:real^N y:real^N.
5253 (A ** x) dot (B ** y) =
5254 ((rowvector x) ** (transp(A) ** B) ** (columnvector y))$1$1`,
5255 REWRITE_TAC[DOT_MATRIX_PRODUCT] THEN
5256 ONCE_REWRITE_TAC[GSYM TRANSP_COLUMNVECTOR] THEN
5257 REWRITE_TAC[DOT_ROWVECTOR_COLUMNVECTOR; MATRIX_TRANSP_MUL] THEN
5258 REWRITE_TAC[MATRIX_MUL_ASSOC]);;
5260 (* ------------------------------------------------------------------------- *)
5261 (* Rank of a matrix. Equivalence of row and column rank is taken from *)
5262 (* George Mackiw's paper, Mathematics Magazine 1995, p. 285. *)
5263 (* ------------------------------------------------------------------------- *)
5265 let MATRIX_VECTOR_MUL_IN_COLUMNSPACE = prove
5266 (`!A:real^M^N x:real^M. (A ** x) IN span(columns A)`,
5267 REPEAT GEN_TAC THEN REWRITE_TAC[MATRIX_VECTOR_COLUMN; columns] THEN
5268 MATCH_MP_TAC SPAN_VSUM THEN
5269 SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; transp; LAMBDA_BETA] THEN
5270 X_GEN_TAC `k:num` THEN STRIP_TAC THEN MATCH_MP_TAC SPAN_MUL THEN
5271 MATCH_MP_TAC SPAN_SUPERSET THEN
5272 REWRITE_TAC[IN_ELIM_THM; column] THEN EXISTS_TAC `k:num` THEN
5273 ASM_REWRITE_TAC[]);;
5275 let SUBSPACE_ORTHOGONAL_TO_VECTOR = prove
5276 (`!x. subspace {y | orthogonal x y}`,
5277 SIMP_TAC[subspace; IN_ELIM_THM; ORTHOGONAL_CLAUSES]);;
5279 let SUBSPACE_ORTHOGONAL_TO_VECTORS = prove
5280 (`!s. subspace {y | (!x. x IN s ==> orthogonal x y)}`,
5281 SIMP_TAC[subspace; IN_ELIM_THM; ORTHOGONAL_CLAUSES]);;
5283 let ORTHOGONAL_TO_SPAN = prove
5284 (`!s x. (!y. y IN s ==> orthogonal x y)
5285 ==> !y. y IN span(s) ==> orthogonal x y`,
5286 REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SPAN_INDUCT THEN
5287 REWRITE_TAC[SET_RULE `(\y. orthogonal x y) = {y | orthogonal x y}`] THEN
5288 ASM_SIMP_TAC[SUBSPACE_ORTHOGONAL_TO_VECTOR; IN_ELIM_THM]);;
5290 let ORTHOGONAL_TO_SPAN_EQ = prove
5291 (`!s x. (!y. y IN span(s) ==> orthogonal x y) <=>
5292 (!y. y IN s ==> orthogonal x y)`,
5293 MESON_TAC[SPAN_SUPERSET; ORTHOGONAL_TO_SPAN]);;
5295 let ORTHOGONAL_TO_SPANS_EQ = prove
5296 (`!s t. (!x y. x IN span(s) /\ y IN span(t) ==> orthogonal x y) <=>
5297 (!x y. x IN s /\ y IN t ==> orthogonal x y)`,
5298 MESON_TAC[ORTHOGONAL_TO_SPAN_EQ; ORTHOGONAL_SYM]);;
5300 let ORTHOGONAL_NULLSPACE_ROWSPACE = prove
5301 (`!A:real^M^N x y:real^M.
5302 A ** x = vec 0 /\ y IN span(rows A) ==> orthogonal x y`,
5303 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5304 REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC SPAN_INDUCT THEN
5305 REWRITE_TAC[SET_RULE `(\y. orthogonal x y) = {y | orthogonal x y}`] THEN
5306 REWRITE_TAC[SUBSPACE_ORTHOGONAL_TO_VECTOR; rows; FORALL_IN_GSPEC] THEN
5307 X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
5308 FIRST_X_ASSUM(MP_TAC o AP_TERM `\y:real^N. y$k`) THEN
5309 ASM_SIMP_TAC[MATRIX_VECTOR_MUL_COMPONENT; VEC_COMPONENT; row; dot;
5310 orthogonal; LAMBDA_BETA] THEN
5311 REWRITE_TAC[REAL_MUL_SYM]);;
5313 let NULLSPACE_INTER_ROWSPACE = prove
5314 (`!A:real^M^N x:real^M. A ** x = vec 0 /\ x IN span(rows A) <=> x = vec 0`,
5315 REPEAT GEN_TAC THEN EQ_TAC THENL
5316 [MESON_TAC[ORTHOGONAL_NULLSPACE_ROWSPACE; ORTHOGONAL_REFL];
5317 SIMP_TAC[MATRIX_VECTOR_MUL_RZERO; SPAN_0]]);;
5319 let MATRIX_VECTOR_MUL_INJECTIVE_ON_ROWSPACE = prove
5320 (`!A:real^M^N x y:real^M.
5321 x IN span(rows A) /\ y IN span(rows A) /\ A ** x = A ** y ==> x = y`,
5322 ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
5323 REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_SUB_LDISTRIB] THEN
5324 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM NULLSPACE_INTER_ROWSPACE] THEN
5325 ASM_SIMP_TAC[SPAN_SUB]);;
5327 let DIM_ROWS_LE_DIM_COLUMNS = prove
5328 (`!A:real^M^N. dim(rows A) <= dim(columns A)`,
5329 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
5330 X_CHOOSE_THEN `b:real^M->bool` STRIP_ASSUME_TAC
5331 (ISPEC `span(rows(A:real^M^N))` BASIS_EXISTS) THEN
5332 SUBGOAL_THEN `FINITE(IMAGE (\x:real^M. (A:real^M^N) ** x) b) /\
5333 CARD (IMAGE (\x:real^M. (A:real^M^N) ** x) b) <=
5334 dim(span(columns A))`
5336 [MATCH_MP_TAC INDEPENDENT_CARD_LE_DIM THEN
5337 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; MATRIX_VECTOR_MUL_IN_COLUMNSPACE] THEN
5338 MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE_GEN THEN
5339 ASM_REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR] THEN
5340 SUBGOAL_THEN `span(b) = span(rows(A:real^M^N))` SUBST1_TAC THENL
5341 [ALL_TAC; ASM_MESON_TAC[MATRIX_VECTOR_MUL_INJECTIVE_ON_ROWSPACE]] THEN
5342 MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[] THEN
5343 GEN_REWRITE_TAC RAND_CONV [GSYM SPAN_SPAN] THEN
5344 ASM_SIMP_TAC[SPAN_MONO];
5345 DISCH_THEN(MP_TAC o CONJUNCT2) THEN MATCH_MP_TAC EQ_IMP THEN
5346 AP_THM_TAC THEN AP_TERM_TAC THEN
5347 FIRST_ASSUM(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM) o
5348 GEN_REWRITE_RULE I [HAS_SIZE]) THEN
5349 MATCH_MP_TAC CARD_IMAGE_INJ THEN ASM_REWRITE_TAC[] THEN
5350 REPEAT STRIP_TAC THEN MATCH_MP_TAC
5351 (ISPEC `A:real^M^N` MATRIX_VECTOR_MUL_INJECTIVE_ON_ROWSPACE) THEN
5354 let rank = new_definition
5355 `rank(A:real^M^N) = dim(columns A)`;;
5357 let RANK_ROW = prove
5358 (`!A:real^M^N. rank(A) = dim(rows A)`,
5359 GEN_TAC THEN REWRITE_TAC[rank] THEN
5360 MP_TAC(ISPEC `A:real^M^N` DIM_ROWS_LE_DIM_COLUMNS) THEN
5361 MP_TAC(ISPEC `transp(A:real^M^N)` DIM_ROWS_LE_DIM_COLUMNS) THEN
5362 REWRITE_TAC[ROWS_TRANSP; COLUMNS_TRANSP] THEN ARITH_TAC);;
5364 let RANK_TRANSP = prove
5365 (`!A:real^M^N. rank(transp A) = rank A`,
5366 GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [RANK_ROW] THEN
5367 REWRITE_TAC[rank; COLUMNS_TRANSP]);;
5369 let MATRIX_VECTOR_MUL_BASIS = prove
5370 (`!A:real^M^N k. 1 <= k /\ k <= dimindex(:M)
5371 ==> A ** (basis k) = column k A`,
5372 SIMP_TAC[CART_EQ; column; MATRIX_VECTOR_MUL_COMPONENT; DOT_BASIS;
5375 let COLUMNS_IMAGE_BASIS = prove
5377 columns A = IMAGE (\x. A ** x) {basis i | 1 <= i /\ i <= dimindex(:M)}`,
5378 GEN_TAC THEN REWRITE_TAC[columns] THEN
5379 ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
5380 REWRITE_TAC[GSYM IMAGE_o; o_DEF] THEN
5381 MATCH_MP_TAC(SET_RULE
5382 `(!x. x IN s ==> f x = g x) ==> IMAGE f s = IMAGE g s`) THEN
5383 SIMP_TAC[IN_ELIM_THM; MATRIX_VECTOR_MUL_BASIS]);;
5385 let RANK_DIM_IM = prove
5386 (`!A:real^M^N. rank A = dim(IMAGE (\x. A ** x) (:real^M))`,
5387 GEN_TAC THEN REWRITE_TAC[rank] THEN
5388 MATCH_MP_TAC SPAN_EQ_DIM THEN REWRITE_TAC[COLUMNS_IMAGE_BASIS] THEN
5389 SIMP_TAC[SPAN_LINEAR_IMAGE; MATRIX_VECTOR_MUL_LINEAR] THEN
5390 AP_TERM_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM SPAN_SPAN] THEN
5391 REWRITE_TAC[SPAN_STDBASIS]);;
5393 let DIM_EQ_SPAN = prove
5394 (`!s t:real^N->bool. s SUBSET t /\ dim t <= dim s ==> span s = span t`,
5395 REPEAT STRIP_TAC THEN
5396 X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC
5397 (ISPEC `span s:real^N->bool` BASIS_EXISTS) THEN
5398 MP_TAC(ISPECL [`span t:real^N->bool`; `b:real^N->bool`]
5399 CARD_GE_DIM_INDEPENDENT) THEN
5400 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
5401 ASM_REWRITE_TAC[DIM_SPAN] THEN
5402 ASM_MESON_TAC[SPAN_MONO; SPAN_SPAN; SUBSET_TRANS; SUBSET_ANTISYM]);;
5404 let DIM_EQ_FULL = prove
5405 (`!s:real^N->bool. dim s = dimindex(:N) <=> span s = (:real^N)`,
5406 GEN_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN EQ_TAC THEN
5407 SIMP_TAC[DIM_UNIV] THEN DISCH_TAC THEN
5408 GEN_REWRITE_TAC RAND_CONV [GSYM SPAN_UNIV] THEN MATCH_MP_TAC DIM_EQ_SPAN THEN
5409 ASM_REWRITE_TAC[SUBSET_UNIV; DIM_UNIV] THEN
5410 ASM_MESON_TAC[LE_REFL; DIM_SPAN]);;
5412 let DIM_PSUBSET = prove
5413 (`!s t. (span s) PSUBSET (span t) ==> dim s < dim t`,
5414 ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
5415 SIMP_TAC[PSUBSET; DIM_SUBSET; LT_LE] THEN
5416 MESON_TAC[EQ_IMP_LE; DIM_EQ_SPAN; SPAN_SPAN]);;
5418 let RANK_BOUND = prove
5419 (`!A:real^M^N. rank(A) <= MIN (dimindex(:M)) (dimindex(:N))`,
5420 GEN_TAC THEN REWRITE_TAC[ARITH_RULE `x <= MIN a b <=> x <= a /\ x <= b`] THEN
5422 [REWRITE_TAC[DIM_SUBSET_UNIV; RANK_ROW];
5423 REWRITE_TAC[DIM_SUBSET_UNIV; rank]]);;
5425 let FULL_RANK_INJECTIVE = prove
5427 rank A = dimindex(:M) <=>
5428 (!x y:real^M. A ** x = A ** y ==> x = y)`,
5429 REWRITE_TAC[GSYM MATRIX_LEFT_INVERTIBLE_INJECTIVE] THEN
5430 REWRITE_TAC[MATRIX_LEFT_INVERTIBLE_SPAN_ROWS] THEN
5431 REWRITE_TAC[RANK_ROW; DIM_EQ_FULL]);;
5433 let FULL_RANK_SURJECTIVE = prove
5435 rank A = dimindex(:N) <=> (!y:real^N. ?x:real^M. A ** x = y)`,
5436 REWRITE_TAC[GSYM MATRIX_RIGHT_INVERTIBLE_SURJECTIVE] THEN
5437 REWRITE_TAC[GSYM LEFT_INVERTIBLE_TRANSP] THEN
5438 REWRITE_TAC[MATRIX_LEFT_INVERTIBLE_INJECTIVE] THEN
5439 REWRITE_TAC[GSYM FULL_RANK_INJECTIVE; RANK_TRANSP]);;
5441 let MATRIX_FULL_LINEAR_EQUATIONS = prove
5442 (`!A:real^M^N b:real^N.
5443 rank A = dimindex(:N) ==> ?x. A ** x = b`,
5444 SIMP_TAC[FULL_RANK_SURJECTIVE]);;
5446 let MATRIX_NONFULL_LINEAR_EQUATIONS_EQ = prove
5448 (?x. ~(x = vec 0) /\ A ** x = vec 0) <=> ~(rank A = dimindex(:M))`,
5449 REPEAT GEN_TAC THEN REWRITE_TAC[FULL_RANK_INJECTIVE] THEN
5450 SIMP_TAC[LINEAR_INJECTIVE_0; MATRIX_VECTOR_MUL_LINEAR] THEN
5453 let MATRIX_NONFULL_LINEAR_EQUATIONS = prove
5455 ~(rank A = dimindex(:M)) ==> ?x. ~(x = vec 0) /\ A ** x = vec 0`,
5456 REWRITE_TAC[MATRIX_NONFULL_LINEAR_EQUATIONS_EQ]);;
5458 let MATRIX_TRIVIAL_LINEAR_EQUATIONS = prove
5460 dimindex(:N) < dimindex(:M)
5461 ==> ?x. ~(x = vec 0) /\ A ** x = vec 0`,
5462 REPEAT STRIP_TAC THEN MATCH_MP_TAC MATRIX_NONFULL_LINEAR_EQUATIONS THEN
5463 MATCH_MP_TAC(ARITH_RULE
5464 `!a. x <= MIN b a /\ a < b ==> ~(x = b)`) THEN
5465 EXISTS_TAC `dimindex(:N)` THEN ASM_REWRITE_TAC[RANK_BOUND]);;
5467 let RANK_EQ_0 = prove
5468 (`!A:real^M^N. rank A = 0 <=> A = mat 0`,
5469 REWRITE_TAC[RANK_DIM_IM; DIM_EQ_0; SUBSET; FORALL_IN_IMAGE; IN_SING;
5471 GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [CART_EQ] THEN
5472 SIMP_TAC[CART_EQ; MATRIX_MUL_DOT; VEC_COMPONENT; LAMBDA_BETA; mat] THEN
5473 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
5474 REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_DOT_EQ_0; COND_ID] THEN
5475 REWRITE_TAC[CART_EQ; VEC_COMPONENT]);;
5479 REWRITE_TAC[RANK_EQ_0]);;
5481 let RANK_MUL_LE_RIGHT = prove
5482 (`!A:real^N^M B:real^P^N. rank(A ** B) <= rank(B)`,
5483 REPEAT GEN_TAC THEN MATCH_MP_TAC LE_TRANS THEN
5484 EXISTS_TAC `dim(IMAGE (\y. (A:real^N^M) ** y)
5485 (IMAGE (\x. (B:real^P^N) ** x) (:real^P)))` THEN
5486 REWRITE_TAC[RANK_DIM_IM] THEN CONJ_TAC THENL
5487 [REWRITE_TAC[GSYM IMAGE_o; o_DEF; MATRIX_VECTOR_MUL_ASSOC; LE_REFL];
5488 MATCH_MP_TAC DIM_LINEAR_IMAGE_LE THEN
5489 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR]]);;
5491 let RANK_MUL_LE_LEFT = prove
5492 (`!A:real^N^M B:real^P^N. rank(A ** B) <= rank(A)`,
5493 ONCE_REWRITE_TAC[GSYM RANK_TRANSP] THEN
5494 REWRITE_TAC[MATRIX_TRANSP_MUL] THEN
5495 REWRITE_TAC[RANK_MUL_LE_RIGHT]);;
5497 (* ------------------------------------------------------------------------- *)
5498 (* A non-injective linear function maps into a hyperplane. *)
5499 (* ------------------------------------------------------------------------- *)
5501 let ADJOINT_INJECTIVE = prove
5502 (`!f:real^M->real^N.
5504 ==> ((!x y. adjoint f x = adjoint f y ==> x = y) <=>
5505 (!y. ?x. f x = y))`,
5506 REPEAT STRIP_TAC THEN
5507 FIRST_ASSUM(ASSUME_TAC o GSYM o MATCH_MP MATRIX_WORKS o MATCH_MP
5508 ADJOINT_LINEAR) THEN
5509 FIRST_ASSUM(ASSUME_TAC o GSYM o MATCH_MP MATRIX_WORKS) THEN
5510 ASM_REWRITE_TAC[GSYM FULL_RANK_INJECTIVE; GSYM FULL_RANK_SURJECTIVE] THEN
5511 ASM_SIMP_TAC[MATRIX_ADJOINT; RANK_TRANSP]);;
5513 let ADJOINT_SURJECTIVE = prove
5514 (`!f:real^M->real^N.
5516 ==> ((!y. ?x. adjoint f x = y) <=> (!x y. f x = f y ==> x = y))`,
5517 REPEAT STRIP_TAC THEN
5518 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
5519 [GSYM(MATCH_MP ADJOINT_ADJOINT th)]) THEN
5520 ASM_SIMP_TAC[ADJOINT_INJECTIVE; ADJOINT_LINEAR]);;
5522 let ADJOINT_INJECTIVE_INJECTIVE = prove
5523 (`!f:real^N->real^N.
5525 ==> ((!x y. adjoint f x = adjoint f y ==> x = y) <=>
5526 (!x y. f x = f y ==> x = y))`,
5527 SIMP_TAC[ADJOINT_INJECTIVE] THEN
5528 MESON_TAC[LINEAR_INJECTIVE_IMP_SURJECTIVE;
5529 LINEAR_SURJECTIVE_IMP_INJECTIVE]);;
5531 let ADJOINT_INJECTIVE_INJECTIVE_0 = prove
5532 (`!f:real^N->real^N.
5534 ==> ((!x. adjoint f x = vec 0 ==> x = vec 0) <=>
5535 (!x. f x = vec 0 ==> x = vec 0))`,
5536 REPEAT STRIP_TAC THEN
5537 FIRST_ASSUM(MP_TAC o MATCH_MP ADJOINT_INJECTIVE_INJECTIVE) THEN
5538 FIRST_ASSUM(ASSUME_TAC o MATCH_MP ADJOINT_LINEAR) THEN
5539 ASM_MESON_TAC[LINEAR_INJECTIVE_0]);;
5541 let LINEAR_SINGULAR_INTO_HYPERPLANE = prove
5542 (`!f:real^N->real^N.
5544 ==> (~(!x y. f(x) = f(y) ==> x = y) <=>
5545 ?a. ~(a = vec 0) /\ !x. a dot f(x) = &0)`,
5546 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[DOT_SYM] THEN
5547 ASM_SIMP_TAC[ADJOINT_WORKS; FORALL_DOT_EQ_0] THEN
5548 REWRITE_TAC[MESON[] `(?a. ~p a /\ q a) <=> ~(!a. q a ==> p a)`] THEN
5549 ASM_SIMP_TAC[ADJOINT_INJECTIVE_INJECTIVE_0; LINEAR_INJECTIVE_0]);;
5551 let LINEAR_SINGULAR_IMAGE_HYPERPLANE = prove
5552 (`!f:real^N->real^N.
5553 linear f /\ ~(!x y. f(x) = f(y) ==> x = y)
5554 ==> ?a. ~(a = vec 0) /\ !s. IMAGE f s SUBSET {x | a dot x = &0}`,
5555 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
5556 ASM_SIMP_TAC[LINEAR_SINGULAR_INTO_HYPERPLANE] THEN
5557 SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN MESON_TAC[]);;
5559 let LOWDIM_EXPAND_DIMENSION = prove
5560 (`!s:real^N->bool n.
5561 dim s <= n /\ n <= dimindex(:N)
5562 ==> ?t. dim(t) = n /\ span s SUBSET span t`,
5564 GEN_REWRITE_TAC (BINDER_CONV o LAND_CONV o LAND_CONV) [LE_EXISTS] THEN
5565 SIMP_TAC[LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN
5566 ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
5567 REWRITE_TAC[RIGHT_FORALL_IMP_THM; LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
5568 INDUCT_TAC THENL [MESON_TAC[ADD_CLAUSES; SUBSET_REFL]; ALL_TAC] THEN
5569 REWRITE_TAC[ARITH_RULE `s + SUC d <= n <=> s + d < n`] THEN
5570 DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
5571 ASM_SIMP_TAC[LT_IMP_LE; LEFT_IMP_EXISTS_THM] THEN
5572 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
5573 REWRITE_TAC[ADD_CLAUSES] THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
5574 SUBGOAL_THEN `~(span t = (:real^N))` MP_TAC THENL
5575 [REWRITE_TAC[GSYM DIM_EQ_FULL] THEN ASM_ARITH_TAC; ALL_TAC] THEN
5576 REWRITE_TAC[EXTENSION; IN_UNIV; NOT_FORALL_THM; LEFT_IMP_EXISTS_THM] THEN
5577 X_GEN_TAC `a:real^N` THEN DISCH_TAC THEN
5578 EXISTS_TAC `(a:real^N) INSERT t` THEN ASM_REWRITE_TAC[DIM_INSERT; ADD1] THEN
5579 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `span(t:real^N->bool)` THEN
5580 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SPAN_MONO THEN SET_TAC[]);;
5582 let LOWDIM_EXPAND_BASIS = prove
5583 (`!s:real^N->bool n.
5584 dim s <= n /\ n <= dimindex(:N)
5585 ==> ?b. b HAS_SIZE n /\ independent b /\ span s SUBSET span b`,
5586 REPEAT GEN_TAC THEN DISCH_TAC THEN
5587 FIRST_ASSUM(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC o
5588 MATCH_MP LOWDIM_EXPAND_DIMENSION) THEN
5589 MP_TAC(ISPEC `t:real^N->bool` BASIS_EXISTS) THEN
5590 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `b:real^N->bool` THEN
5591 ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5592 ASM_MESON_TAC[SPAN_SPAN; SUBSET_TRANS; SPAN_MONO]);;
5594 (* ------------------------------------------------------------------------- *)
5595 (* Orthogonal bases, Gram-Schmidt process, and related theorems. *)
5596 (* ------------------------------------------------------------------------- *)
5598 let SPAN_DELETE_0 = prove
5599 (`!s:real^N->bool. span(s DELETE vec 0) = span s`,
5600 GEN_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
5601 SIMP_TAC[DELETE_SUBSET; SPAN_MONO] THEN
5602 MATCH_MP_TAC SUBSET_TRANS THEN
5603 EXISTS_TAC `span((vec 0:real^N) INSERT (s DELETE vec 0))` THEN CONJ_TAC THENL
5604 [MATCH_MP_TAC SPAN_MONO THEN SET_TAC[];
5605 SIMP_TAC[SUBSET; SPAN_BREAKDOWN_EQ; VECTOR_MUL_RZERO; VECTOR_SUB_RZERO]]);;
5607 let SPAN_IMAGE_SCALE = prove
5608 (`!c s. FINITE s /\ (!x. x IN s ==> ~(c x = &0))
5609 ==> span (IMAGE (\x:real^N. c(x) % x) s) = span s`,
5610 GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
5611 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
5612 SIMP_TAC[IMAGE_CLAUSES; SPAN_BREAKDOWN_EQ; EXTENSION; FORALL_IN_INSERT] THEN
5613 MAP_EVERY X_GEN_TAC [`x:real^N`; `t:real^N->bool`] THEN
5614 STRIP_TAC THEN STRIP_TAC THEN X_GEN_TAC `y:real^N` THEN
5615 REWRITE_TAC[VECTOR_MUL_ASSOC] THEN EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
5616 DISCH_THEN(X_CHOOSE_TAC `k:real`) THEN
5617 EXISTS_TAC `k / (c:real^N->real) x` THEN
5618 ASM_SIMP_TAC[REAL_DIV_RMUL]);;
5620 let PAIRWISE_ORTHOGONAL_INDEPENDENT = prove
5622 pairwise orthogonal s /\ ~(vec 0 IN s) ==> independent s`,
5623 REWRITE_TAC[pairwise; orthogonal] THEN REPEAT STRIP_TAC THEN
5624 REWRITE_TAC[independent; dependent] THEN
5625 DISCH_THEN(X_CHOOSE_THEN `a:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
5626 REWRITE_TAC[SPAN_EXPLICIT; IN_ELIM_THM; NOT_EXISTS_THM] THEN
5627 MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:real^N->real`] THEN
5628 REWRITE_TAC[SUBSET; IN_DELETE] THEN STRIP_TAC THEN
5629 FIRST_X_ASSUM(MP_TAC o AP_TERM `\x:real^N. a dot x`) THEN
5630 ASM_SIMP_TAC[DOT_RSUM; DOT_RMUL; REAL_MUL_RZERO; SUM_0] THEN
5631 ASM_MESON_TAC[DOT_EQ_0]);;
5633 let PAIRWISE_ORTHOGONAL_IMP_FINITE = prove
5634 (`!s:real^N->bool. pairwise orthogonal s ==> FINITE s`,
5635 REPEAT STRIP_TAC THEN
5636 SUBGOAL_THEN `independent (s DELETE (vec 0:real^N))` MP_TAC THENL
5637 [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN
5638 REWRITE_TAC[IN_DELETE] THEN MATCH_MP_TAC PAIRWISE_MONO THEN
5639 EXISTS_TAC `s:real^N->bool` THEN
5640 ASM_SIMP_TAC[SUBSET; IN_DELETE];
5641 DISCH_THEN(MP_TAC o MATCH_MP INDEPENDENT_IMP_FINITE) THEN
5642 REWRITE_TAC[FINITE_DELETE]]);;
5644 let GRAM_SCHMIDT_STEP = prove
5646 pairwise orthogonal s /\ x IN span s
5647 ==> orthogonal x (a - vsum s (\b:real^N. (b dot a) / (b dot b) % b))`,
5648 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5649 REWRITE_TAC[ONCE_REWRITE_RULE[ORTHOGONAL_SYM] ORTHOGONAL_TO_SPAN_EQ] THEN
5650 X_GEN_TAC `s:real^N->bool` THEN STRIP_TAC THEN
5651 MAP_EVERY X_GEN_TAC [`a:real^N`; `x:real^N`] THEN DISCH_TAC THEN
5652 FIRST_ASSUM(ASSUME_TAC o MATCH_MP PAIRWISE_ORTHOGONAL_IMP_FINITE) THEN
5653 REWRITE_TAC[orthogonal; DOT_RSUB] THEN ASM_SIMP_TAC[DOT_RSUM] THEN
5654 REWRITE_TAC[REAL_SUB_0; DOT_RMUL] THEN MATCH_MP_TAC EQ_TRANS THEN
5655 EXISTS_TAC `sum s (\y:real^N. if y = x then y dot a else &0)` THEN
5656 CONJ_TAC THENL [ASM_SIMP_TAC[SUM_DELTA; DOT_SYM]; ALL_TAC] THEN
5657 MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
5658 RULE_ASSUM_TAC(REWRITE_RULE[pairwise; orthogonal]) THEN
5659 ASM_CASES_TAC `x:real^N = y` THEN ASM_SIMP_TAC[DOT_LMUL; REAL_MUL_RZERO] THEN
5660 ASM_CASES_TAC `y:real^N = vec 0` THEN
5661 ASM_SIMP_TAC[REAL_DIV_RMUL; DOT_EQ_0; DOT_LZERO; REAL_MUL_RZERO]);;
5663 let ORTHOGONAL_EXTENSION = prove
5664 (`!t s:real^N->bool.
5665 FINITE t /\ FINITE s /\ pairwise orthogonal s
5666 ==> ?u. pairwise orthogonal (s UNION u) /\
5667 span (s UNION u) = span (s UNION t)`,
5668 REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5669 MATCH_MP_TAC FINITE_INDUCT_STRONG THEN CONJ_TAC THENL
5670 [REPEAT STRIP_TAC THEN EXISTS_TAC `{}:real^N->bool` THEN
5671 ASM_REWRITE_TAC[UNION_EMPTY];
5673 MAP_EVERY X_GEN_TAC [`a:real^N`; `t:real^N->bool`] THEN
5674 REWRITE_TAC[pairwise; orthogonal] THEN REPEAT STRIP_TAC THEN
5675 ABBREV_TAC `a' = a - vsum s (\b:real^N. (b dot a) / (b dot b) % b)` THEN
5676 FIRST_X_ASSUM(MP_TAC o SPEC `(a':real^N) INSERT s`) THEN
5677 ASM_REWRITE_TAC[FINITE_INSERT] THEN ANTS_TAC THENL
5678 [SUBGOAL_THEN `!x:real^N. x IN s ==> a' dot x = &0`
5679 (fun th -> REWRITE_TAC[IN_INSERT] THEN ASM_MESON_TAC[DOT_SYM; th]) THEN
5680 REPEAT STRIP_TAC THEN EXPAND_TAC "a'" THEN
5681 REWRITE_TAC[GSYM orthogonal] THEN ONCE_REWRITE_TAC[ORTHOGONAL_SYM] THEN
5682 MATCH_MP_TAC GRAM_SCHMIDT_STEP THEN
5683 ASM_SIMP_TAC[pairwise; orthogonal; SPAN_CLAUSES];
5684 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
5685 EXISTS_TAC `(a':real^N) INSERT u` THEN
5686 ASM_REWRITE_TAC[SET_RULE `s UNION a INSERT u = a INSERT s UNION u`] THEN
5687 REWRITE_TAC[SET_RULE `(x INSERT s) UNION t = x INSERT (s UNION t)`] THEN
5688 MATCH_MP_TAC EQ_SPAN_INSERT_EQ THEN EXPAND_TAC "a'" THEN
5689 REWRITE_TAC[VECTOR_ARITH `a - x - a:real^N = --x`] THEN
5690 MATCH_MP_TAC SPAN_NEG THEN MATCH_MP_TAC SPAN_VSUM THEN
5691 ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
5692 MATCH_MP_TAC SPAN_MUL THEN ASM_SIMP_TAC[SPAN_SUPERSET; IN_UNION]]);;
5694 let VECTOR_IN_ORTHOGONAL_SPANNINGSET = prove
5695 (`!a. ?s. a IN s /\ pairwise orthogonal s /\ span s = (:real^N)`,
5697 MP_TAC(ISPECL [`(IMAGE basis (1..dimindex(:N))):real^N->bool`;
5698 `{a:real^N}`] ORTHOGONAL_EXTENSION) THEN
5699 SIMP_TAC[FINITE_SING; PAIRWISE_SING; FINITE_IMAGE; FINITE_NUMSEG] THEN
5700 DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
5701 EXISTS_TAC `{a:real^N} UNION u` THEN ASM_REWRITE_TAC[IN_UNION; IN_SING] THEN
5702 MATCH_MP_TAC(SET_RULE `!s. s = UNIV /\ s SUBSET t ==> t = UNIV`) THEN
5703 EXISTS_TAC `span {basis i:real^N | 1 <= i /\ i <= dimindex (:N)}` THEN
5704 CONJ_TAC THENL [REWRITE_TAC[SPAN_STDBASIS]; MATCH_MP_TAC SPAN_MONO] THEN
5705 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; GSYM IN_NUMSEG] THEN SET_TAC[]);;
5707 let VECTOR_IN_ORTHOGONAL_BASIS = prove
5709 ==> ?s. a IN s /\ ~(vec 0 IN s) /\
5710 pairwise orthogonal s /\
5712 s HAS_SIZE (dimindex(:N)) /\
5713 span s = (:real^N)`,
5714 REPEAT STRIP_TAC THEN
5715 MP_TAC(ISPEC `a:real^N` VECTOR_IN_ORTHOGONAL_SPANNINGSET) THEN
5716 DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
5717 EXISTS_TAC `s DELETE (vec 0:real^N)` THEN ASM_REWRITE_TAC[IN_DELETE] THEN
5718 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5719 [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN
5720 ASM_SIMP_TAC[pairwise; IN_DELETE];
5722 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5723 [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_SIMP_TAC[IN_DELETE];
5725 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
5726 [ASM_MESON_TAC[SPAN_DELETE_0];
5727 DISCH_TAC THEN ASM_SIMP_TAC[BASIS_HAS_SIZE_UNIV]]);;
5729 let VECTOR_IN_ORTHONORMAL_BASIS = prove
5732 pairwise orthogonal s /\
5733 (!x. x IN s ==> norm x = &1) /\
5735 s HAS_SIZE (dimindex(:N)) /\
5736 span s = (:real^N)`,
5737 GEN_TAC THEN ASM_CASES_TAC `a:real^N = vec 0` THEN
5738 ASM_REWRITE_TAC[NORM_0; REAL_OF_NUM_EQ; ARITH_EQ] THEN DISCH_TAC THEN
5739 FIRST_ASSUM(MP_TAC o MATCH_MP VECTOR_IN_ORTHOGONAL_BASIS) THEN
5740 DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
5741 EXISTS_TAC `IMAGE (\x:real^N. inv(norm x) % x) s` THEN
5743 [REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `a:real^N` THEN
5744 ASM_REWRITE_TAC[REAL_INV_1; VECTOR_MUL_LID];
5746 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5747 [REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
5748 RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN
5749 ASM_MESON_TAC[ORTHOGONAL_CLAUSES];
5751 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5752 [REWRITE_TAC[FORALL_IN_IMAGE; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN
5753 ASM_MESON_TAC[REAL_MUL_LINV; NORM_EQ_0];
5755 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5756 [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_REWRITE_TAC[] THEN
5757 REWRITE_TAC[IN_IMAGE] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
5758 SIMP_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0] THEN ASM_MESON_TAC[];
5760 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
5761 [ALL_TAC; ASM_SIMP_TAC[BASIS_HAS_SIZE_UNIV]] THEN
5762 UNDISCH_THEN `span s = (:real^N)` (SUBST1_TAC o SYM) THEN
5763 MATCH_MP_TAC SPAN_IMAGE_SCALE THEN
5764 REWRITE_TAC[REAL_INV_EQ_0; NORM_EQ_0] THEN
5765 ASM_MESON_TAC[HAS_SIZE]);;
5767 (* ------------------------------------------------------------------------- *)
5768 (* Analogous theorems for existence of orthonormal basis for a subspace. *)
5769 (* ------------------------------------------------------------------------- *)
5771 let ORTHOGONAL_SPANNINGSET_SUBSPACE = prove
5774 ==> ?b. b SUBSET s /\ pairwise orthogonal b /\ span b = s`,
5775 REPEAT STRIP_TAC THEN MP_TAC(ISPEC `s:real^N->bool` BASIS_EXISTS) THEN
5776 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
5777 MP_TAC(ISPECL[`b:real^N->bool`; `{}:real^N->bool`] ORTHOGONAL_EXTENSION) THEN
5778 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
5779 ASM_REWRITE_TAC[FINITE_EMPTY; PAIRWISE_EMPTY; UNION_EMPTY] THEN
5780 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N->bool` THEN
5781 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
5782 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
5783 [MATCH_MP_TAC SPAN_SUBSPACE THEN ASM_REWRITE_TAC[];
5784 DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_MESON_TAC[SPAN_INC]]);;
5786 let ORTHOGONAL_BASIS_SUBSPACE = prove
5789 ==> ?b. ~(vec 0 IN b) /\
5791 pairwise orthogonal b /\
5793 b HAS_SIZE (dim s) /\
5795 REPEAT STRIP_TAC THEN
5796 FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_SPANNINGSET_SUBSPACE) THEN
5797 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
5798 EXISTS_TAC `b DELETE (vec 0:real^N)` THEN ASM_REWRITE_TAC[IN_DELETE] THEN
5799 CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
5800 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5801 [RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN
5802 ASM_SIMP_TAC[pairwise; IN_DELETE];
5804 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5805 [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_SIMP_TAC[IN_DELETE];
5807 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
5808 [ASM_MESON_TAC[SPAN_DELETE_0];
5809 DISCH_TAC THEN ASM_SIMP_TAC[BASIS_HAS_SIZE_DIM]]);;
5811 let ORTHONORMAL_BASIS_SUBSPACE = prove
5814 ==> ?b. b SUBSET span s /\
5815 pairwise orthogonal b /\
5816 (!x. x IN b ==> norm x = &1) /\
5818 b HAS_SIZE (dim s) /\
5820 REPEAT STRIP_TAC THEN
5821 FIRST_ASSUM(MP_TAC o MATCH_MP ORTHOGONAL_BASIS_SUBSPACE) THEN
5822 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
5823 EXISTS_TAC `IMAGE (\x:real^N. inv(norm x) % x) b` THEN
5825 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
5826 ASM_MESON_TAC[SPAN_MUL; SPAN_INC; SUBSET];
5828 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5829 [REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
5830 RULE_ASSUM_TAC(REWRITE_RULE[pairwise]) THEN
5831 ASM_MESON_TAC[ORTHOGONAL_CLAUSES];
5833 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5834 [REWRITE_TAC[FORALL_IN_IMAGE; NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN
5835 ASM_MESON_TAC[REAL_MUL_LINV; NORM_EQ_0];
5837 MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
5838 [MATCH_MP_TAC PAIRWISE_ORTHOGONAL_INDEPENDENT THEN ASM_REWRITE_TAC[] THEN
5839 REWRITE_TAC[IN_IMAGE] THEN ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN
5840 SIMP_TAC[VECTOR_MUL_EQ_0; REAL_INV_EQ_0; NORM_EQ_0] THEN ASM_MESON_TAC[];
5842 MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
5843 [ALL_TAC; ASM_SIMP_TAC[BASIS_HAS_SIZE_DIM]] THEN
5844 UNDISCH_THEN `span b = (s:real^N->bool)` (SUBST1_TAC o SYM) THEN
5845 MATCH_MP_TAC SPAN_IMAGE_SCALE THEN
5846 REWRITE_TAC[REAL_INV_EQ_0; NORM_EQ_0] THEN
5847 ASM_MESON_TAC[HAS_SIZE]);;
5849 let ORTHOGONAL_TO_SUBSPACE_EXISTS_GEN = prove
5850 (`!s t:real^N->bool.
5851 span s PSUBSET span t
5852 ==> ?x. ~(x = vec 0) /\ x IN span t /\
5853 (!y. y IN span s ==> orthogonal x y)`,
5854 REPEAT STRIP_TAC THEN
5855 MP_TAC(ISPEC `span s:real^N->bool` ORTHOGONAL_BASIS_SUBSPACE) THEN
5856 REWRITE_TAC[SUBSPACE_SPAN] THEN
5857 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
5858 FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN
5859 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [PSUBSET_ALT]) THEN
5860 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
5861 (X_CHOOSE_THEN `u:real^N` STRIP_ASSUME_TAC)) THEN
5862 MP_TAC(ISPECL [`{u:real^N}`; `b:real^N->bool`] ORTHOGONAL_EXTENSION) THEN
5863 ANTS_TAC THENL [ASM_MESON_TAC[FINITE_SING; HAS_SIZE]; ALL_TAC] THEN
5864 DISCH_THEN(X_CHOOSE_THEN `ns:real^N->bool` MP_TAC) THEN
5865 ASM_CASES_TAC `ns SUBSET (vec 0:real^N) INSERT b` THENL
5866 [DISCH_THEN(MP_TAC o AP_TERM `(IN) (u:real^N)` o CONJUNCT2) THEN
5867 SIMP_TAC[SPAN_SUPERSET; IN_UNION; IN_SING] THEN
5868 MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
5869 SUBGOAL_THEN `~(u IN span (b UNION {vec 0:real^N}))` MP_TAC THENL
5870 [ASM_REWRITE_TAC[SET_RULE `s UNION {a} = a INSERT s`; SPAN_INSERT_0];
5871 MATCH_MP_TAC(SET_RULE `s SUBSET t ==> ~(x IN t) ==> ~(x IN s)`) THEN
5872 MATCH_MP_TAC SPAN_MONO THEN ASM SET_TAC[]];
5874 FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
5875 `~(s SUBSET t) ==> ?z. z IN s /\ ~(z IN t)`)) THEN
5876 REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_INSERT; DE_MORGAN_THM] THEN
5877 X_GEN_TAC `n:real^N` THEN STRIP_TAC THEN
5878 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
5879 REWRITE_TAC[pairwise; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
5880 DISCH_THEN(MP_TAC o SPEC `n:real^N`) THEN ASM_REWRITE_TAC[IN_UNION] THEN
5881 REWRITE_TAC[IMP_IMP] THEN DISCH_TAC THEN EXISTS_TAC `n:real^N` THEN
5882 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
5883 [SUBGOAL_THEN `(n:real^N) IN span (b UNION ns)` MP_TAC THENL
5884 [MATCH_MP_TAC SPAN_SUPERSET THEN ASM SET_TAC[];
5885 ASM_REWRITE_TAC[] THEN SPEC_TAC(`n:real^N`,`n:real^N`) THEN
5886 REWRITE_TAC[GSYM SUBSET] THEN
5887 MATCH_MP_TAC SPAN_SUBSET_SUBSPACE THEN REWRITE_TAC[SUBSPACE_SPAN] THEN
5888 ASM_REWRITE_TAC[SET_RULE
5889 `s UNION {a} SUBSET t <=> s SUBSET t /\ a IN t`] THEN
5890 ASM_MESON_TAC[SPAN_INC; SUBSET_TRANS]];
5891 MATCH_MP_TAC SPAN_INDUCT THEN
5892 REWRITE_TAC[SET_RULE `(\y. orthogonal n y) = {y | orthogonal n y}`] THEN
5893 REWRITE_TAC[SUBSPACE_ORTHOGONAL_TO_VECTOR] THEN ASM SET_TAC[]]);;
5895 let ORTHOGONAL_TO_SUBSPACE_EXISTS = prove
5896 (`!s:real^N->bool. dim s < dimindex(:N)
5897 ==> ?x. ~(x = vec 0) /\ !y. y IN s ==> orthogonal x y`,
5898 REPEAT STRIP_TAC THEN
5899 MP_TAC(ISPECL [`s:real^N->bool`; `(:real^N)`]
5900 ORTHOGONAL_TO_SUBSPACE_EXISTS_GEN) THEN
5901 ANTS_TAC THENL [REWRITE_TAC[PSUBSET]; MESON_TAC[SPAN_SUPERSET]] THEN
5902 REWRITE_TAC[SPAN_UNIV; SUBSET_UNIV] THEN
5903 ASM_MESON_TAC[DIM_SPAN; DIM_UNIV; LT_REFL]);;
5905 let ORTHOGONAL_TO_VECTOR_EXISTS = prove
5906 (`!x:real^N. 2 <= dimindex(:N) ==> ?y. ~(y = vec 0) /\ orthogonal x y`,
5907 REPEAT STRIP_TAC THEN
5908 MP_TAC(ISPEC `{x:real^N}` ORTHOGONAL_TO_SUBSPACE_EXISTS) THEN
5909 SIMP_TAC[DIM_SING; IN_SING; LEFT_FORALL_IMP_THM; EXISTS_REFL] THEN
5910 ANTS_TAC THENL [ASM_ARITH_TAC; MESON_TAC[ORTHOGONAL_SYM]]);;
5912 let SPAN_NOT_UNIV_ORTHOGONAL = prove
5913 (`!s. ~(span s = (:real^N))
5914 ==> ?a. ~(a = vec 0) /\ !x. x IN span s ==> a dot x = &0`,
5915 REWRITE_TAC[GSYM DIM_EQ_FULL; GSYM LE_ANTISYM; DIM_SUBSET_UNIV;
5917 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM orthogonal] THEN
5918 MATCH_MP_TAC ORTHOGONAL_TO_SUBSPACE_EXISTS THEN ASM_REWRITE_TAC[DIM_SPAN]);;
5920 let SPAN_NOT_UNIV_SUBSET_HYPERPLANE = prove
5921 (`!s. ~(span s = (:real^N))
5922 ==> ?a. ~(a = vec 0) /\ span s SUBSET {x | a dot x = &0}`,
5923 REWRITE_TAC[SUBSET; IN_ELIM_THM; SPAN_NOT_UNIV_ORTHOGONAL]);;
5925 let LOWDIM_SUBSET_HYPERPLANE = prove
5926 (`!s. dim s < dimindex(:N)
5927 ==> ?a:real^N. ~(a = vec 0) /\ span s SUBSET {x | a dot x = &0}`,
5928 REPEAT STRIP_TAC THEN MATCH_MP_TAC SPAN_NOT_UNIV_SUBSET_HYPERPLANE THEN
5929 REWRITE_TAC[GSYM SUBSET_ANTISYM_EQ; SUBSET_UNIV] THEN
5930 DISCH_THEN(MP_TAC o MATCH_MP DIM_SUBSET) THEN
5931 ASM_REWRITE_TAC[NOT_LE; DIM_SPAN; DIM_UNIV]);;
5933 (* ------------------------------------------------------------------------- *)
5934 (* Decomposing a vector into parts in orthogonal subspaces. *)
5935 (* ------------------------------------------------------------------------- *)
5937 let ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE = prove
5938 (`!s t x y x' y':real^N.
5939 (!a b. a IN s /\ b IN t ==> orthogonal a b) /\
5940 x IN span s /\ x' IN span s /\ y IN span t /\ y' IN span t /\
5942 ==> x = x' /\ y = y'`,
5943 REWRITE_TAC[VECTOR_ARITH `x + y:real^N = x' + y' <=> x - x' = y' - y`] THEN
5944 ONCE_REWRITE_TAC[GSYM ORTHOGONAL_TO_SPANS_EQ] THEN
5945 REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[VECTOR_ARITH
5946 `x:real^N = x' /\ y:real^N = y' <=> x - x' = vec 0 /\ y' - y = vec 0`] THEN
5947 STRIP_TAC THEN ASM_REWRITE_TAC[GSYM ORTHOGONAL_REFL] THEN
5948 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [SYM th]) THEN
5949 ASM_MESON_TAC[ORTHOGONAL_CLAUSES; ORTHOGONAL_SYM]);;
5951 let ORTHOGONAL_SUBSPACE_DECOMP_EXISTS = prove
5952 (`!s x:real^N. ?y z. y IN span s /\ (!w. w IN span s ==> orthogonal z w) /\
5954 REPEAT STRIP_TAC THEN
5955 MP_TAC(ISPEC `span s:real^N->bool` ORTHOGONAL_BASIS_SUBSPACE) THEN
5956 REWRITE_TAC[SUBSPACE_SPAN; LEFT_IMP_EXISTS_THM] THEN
5957 X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
5958 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
5959 EXISTS_TAC `vsum t (\b:real^N. (b dot x) / (b dot b) % b)` THEN
5960 EXISTS_TAC `x - vsum t (\b:real^N. (b dot x) / (b dot b) % b)` THEN
5961 REPEAT CONJ_TAC THENL
5962 [MATCH_MP_TAC SPAN_VSUM THEN
5963 ASM_SIMP_TAC[INDEPENDENT_IMP_FINITE; SPAN_CLAUSES];
5964 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[ORTHOGONAL_SYM] THEN
5965 MATCH_MP_TAC GRAM_SCHMIDT_STEP THEN ASM_SIMP_TAC[];
5966 VECTOR_ARITH_TAC]);;
5968 let ORTHOGONAL_SUBSPACE_DECOMP = prove
5969 (`!s x. ?!(y,z). y IN span s /\
5970 z IN {z:real^N | !x. x IN span s ==> orthogonal z x} /\
5972 REWRITE_TAC[EXISTS_UNIQUE_DEF; IN_ELIM_THM] THEN
5973 REWRITE_TAC[EXISTS_PAIRED_THM; FORALL_PAIRED_THM] THEN
5974 REWRITE_TAC[FORALL_PAIR_THM; ORTHOGONAL_SUBSPACE_DECOMP_EXISTS] THEN
5975 REPEAT STRIP_TAC THEN REWRITE_TAC[PAIR_EQ] THEN
5976 MATCH_MP_TAC ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE THEN
5977 MAP_EVERY EXISTS_TAC
5978 [`s:real^N->bool`; `{z:real^N | !x. x IN span s ==> orthogonal z x}`] THEN
5979 ASM_SIMP_TAC[SPAN_CLAUSES; IN_ELIM_THM] THEN
5980 ASM_MESON_TAC[SPAN_CLAUSES; ORTHOGONAL_SYM]);;
5982 (* ------------------------------------------------------------------------- *)
5983 (* Existence of isometry between subspaces of same dimension. *)
5984 (* ------------------------------------------------------------------------- *)
5986 let ISOMETRY_SUBSPACES = prove
5987 (`!s:real^M->bool t:real^N->bool.
5988 subspace s /\ subspace t /\ dim s = dim t
5989 ==> ?f:real^M->real^N. linear f /\ IMAGE f s = t /\
5990 (!x. x IN s ==> norm(f x) = norm(x))`,
5991 REPEAT STRIP_TAC THEN ABBREV_TAC `n = dim(t:real^N->bool)` THEN
5992 MP_TAC(ISPEC `t:real^N->bool` ORTHONORMAL_BASIS_SUBSPACE) THEN
5993 MP_TAC(ISPEC `s:real^M->bool` ORTHONORMAL_BASIS_SUBSPACE) THEN
5994 ASM_REWRITE_TAC[] THEN
5995 DISCH_THEN(X_CHOOSE_THEN `b:real^M->bool` STRIP_ASSUME_TAC) THEN
5996 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
5997 MP_TAC(ISPECL [`b:real^M->bool`; `c:real^N->bool`] CARD_EQ_BIJECTIONS) THEN
5998 RULE_ASSUM_TAC(REWRITE_RULE[HAS_SIZE]) THEN
5999 ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
6000 MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^N->real^M`] THEN
6002 MP_TAC(ISPECL [`f:real^M->real^N`; `b:real^M->bool`]
6003 LINEAR_INDEPENDENT_EXTEND) THEN
6004 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
6005 X_GEN_TAC `h:real^M->real^N` THEN
6006 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
6007 [REWRITE_TAC[SYM(ASSUME `span(b:real^M->bool) = s`);
6008 SYM(ASSUME `span(c:real^N->bool) = t`)] THEN
6009 MATCH_MP_TAC SUBSET_ANTISYM THEN
6010 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
6011 CONJ_TAC THEN MATCH_MP_TAC SPAN_INDUCT THEN REWRITE_TAC[IN_ELIM_THM] THENL
6012 [GEN_REWRITE_TAC (RAND_CONV o RAND_CONV)
6013 [SET_RULE `(\x. p x) = {x | p x}`] THEN
6014 ASM_SIMP_TAC[SUBSPACE_LINEAR_PREIMAGE] THEN
6015 ASM_MESON_TAC[SPAN_INC; SUBSET];
6016 ONCE_REWRITE_TAC[SET_RULE `(\x. x IN s) = s`] THEN
6017 ASM_SIMP_TAC[SUBSPACE_LINEAR_IMAGE; SUBSPACE_SPAN] THEN
6018 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_IMAGE] THEN
6019 ASM_MESON_TAC[SPAN_INC; SUBSET]];
6020 X_GEN_TAC `x:real^M` THEN
6021 REWRITE_TAC[SYM(ASSUME `span(b:real^M->bool) = s`)] THEN
6022 SIMP_TAC[SPAN_FINITE; ASSUME `FINITE(b:real^M->bool) /\ CARD b = n`] THEN
6023 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
6024 X_GEN_TAC `u:real^M->real` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
6025 ASM_SIMP_TAC[LINEAR_VSUM; NORM_EQ] THEN
6026 ASM_SIMP_TAC[o_DEF; LINEAR_CMUL] THEN
6027 ASM_SIMP_TAC[DOT_LSUM; DOT_RSUM; DOT_RMUL; DOT_LMUL] THEN
6028 MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
6029 REWRITE_TAC[] THEN MATCH_MP_TAC SUM_EQ THEN X_GEN_TAC `y:real^M` THEN
6030 DISCH_TAC THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
6031 RULE_ASSUM_TAC(REWRITE_RULE[pairwise; orthogonal]) THEN
6032 ASM_CASES_TAC `x:real^M = y` THEN ASM_REWRITE_TAC[GSYM NORM_EQ] THEN
6035 let ISOMETRY_UNIV_SUBSPACE = prove
6036 (`!s. subspace s /\ dimindex(:M) = dim s
6037 ==> ?f:real^M->real^N.
6038 linear f /\ IMAGE f (:real^M) = s /\
6039 (!x. norm(f x) = norm(x))`,
6040 REPEAT STRIP_TAC THEN
6041 MP_TAC(ISPECL [`(:real^M)`; `s:real^N->bool`] ISOMETRY_SUBSPACES) THEN
6042 ASM_REWRITE_TAC[SUBSPACE_UNIV; IN_UNIV; DIM_UNIV]);;
6044 let ISOMETRY_UNIV_SUPERSET_SUBSPACE = prove
6045 (`!s. subspace s /\ dim s <= dimindex(:M) /\ dimindex(:M) <= dimindex(:N)
6046 ==> ?f:real^M->real^N.
6047 linear f /\ s SUBSET (IMAGE f (:real^M)) /\
6048 (!x. norm(f x) = norm(x))`,
6049 GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
6050 FIRST_ASSUM(MP_TAC o MATCH_MP LOWDIM_EXPAND_DIMENSION) THEN
6051 DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
6052 MP_TAC(ISPECL [`(:real^M)`; `span t:real^N->bool`] ISOMETRY_SUBSPACES) THEN
6053 ASM_REWRITE_TAC[SUBSPACE_SPAN; SUBSPACE_UNIV; DIM_UNIV; DIM_SPAN] THEN
6054 MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[IN_UNIV] THEN
6055 ASM_MESON_TAC[SUBSET; SPAN_INC]);;
6057 let ISOMETRY_UNIV_UNIV = prove
6058 (`dimindex(:M) <= dimindex(:N)
6059 ==> ?f:real^M->real^N. linear f /\ (!x. norm(f x) = norm(x))`,
6061 MP_TAC(ISPEC `{vec 0:real^N}`ISOMETRY_UNIV_SUPERSET_SUBSPACE) THEN
6062 ASM_REWRITE_TAC[SUBSPACE_TRIVIAL] THEN
6063 ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
6064 MATCH_MP_TAC(ARITH_RULE `x = 0 /\ 1 <= y ==> x <= y`) THEN
6065 ASM_REWRITE_TAC[DIM_EQ_0; DIMINDEX_GE_1] THEN SET_TAC[]);;
6067 let SUBSPACE_ISOMORPHISM = prove
6068 (`!s t. subspace s /\ subspace t /\ dim(s) = dim(t)
6069 ==> ?f:real^M->real^N.
6070 linear f /\ (IMAGE f s = t) /\
6071 (!x y. x IN s /\ y IN s /\ f x = f y ==> (x = y))`,
6072 REPEAT GEN_TAC THEN DISCH_TAC THEN
6073 FIRST_ASSUM(MP_TAC o MATCH_MP ISOMETRY_SUBSPACES) THEN
6074 MATCH_MP_TAC MONO_EXISTS THEN
6075 ASM_SIMP_TAC[LINEAR_INJECTIVE_0_SUBSPACE] THEN MESON_TAC[NORM_EQ_0]);;
6077 let ISOMORPHISMS_UNIV_UNIV = prove
6078 (`dimindex(:M) = dimindex(:N)
6079 ==> ?f:real^M->real^N g.
6080 linear f /\ linear g /\
6081 (!x. norm(f x) = norm x) /\ (!y. norm(g y) = norm y) /\
6082 (!x. g(f x) = x) /\ (!y. f(g y) = y)`,
6083 REPEAT STRIP_TAC THEN
6084 EXISTS_TAC `(\x. lambda i. x$i):real^M->real^N` THEN
6085 EXISTS_TAC `(\x. lambda i. x$i):real^N->real^M` THEN
6086 SIMP_TAC[vector_norm; dot; LAMBDA_BETA] THEN
6087 SIMP_TAC[linear; CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
6089 FIRST_ASSUM SUBST1_TAC THEN SIMP_TAC[LAMBDA_BETA] THEN
6090 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN SIMP_TAC[LAMBDA_BETA]);;
6092 (* ------------------------------------------------------------------------- *)
6093 (* Properties of special hyperplanes. *)
6094 (* ------------------------------------------------------------------------- *)
6096 let SUBSPACE_HYPERPLANE = prove
6097 (`!a. subspace {x:real^N | a dot x = &0}`,
6098 SIMP_TAC[subspace; DOT_RADD; DOT_RMUL; IN_ELIM_THM; REAL_ADD_LID;
6099 REAL_MUL_RZERO; DOT_RZERO]);;
6101 let SUBSPACE_SPECIAL_HYPERPLANE = prove
6102 (`!k. subspace {x:real^N | x$k = &0}`,
6103 SIMP_TAC[subspace; IN_ELIM_THM; VEC_COMPONENT;
6104 VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN REAL_ARITH_TAC);;
6106 let SPECIAL_HYPERPLANE_SPAN = prove
6107 (`!k. 1 <= k /\ k <= dimindex(:N)
6108 ==> {x:real^N | x$k = &0} =
6109 span(IMAGE basis ((1..dimindex(:N)) DELETE k))`,
6110 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SPAN_SUBSPACE THEN
6111 ASM_SIMP_TAC[SUBSPACE_SPECIAL_HYPERPLANE] THEN CONJ_TAC THENL
6112 [REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN
6113 ASM_SIMP_TAC[BASIS_COMPONENT; IN_DELETE];
6114 REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN
6115 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
6116 GEN_REWRITE_TAC LAND_CONV [GSYM BASIS_EXPANSION] THEN
6117 SIMP_TAC[SPAN_FINITE; FINITE_IMAGE; FINITE_DELETE; FINITE_NUMSEG] THEN
6118 REWRITE_TAC[IN_ELIM_THM] THEN
6119 EXISTS_TAC `\v:real^N. x dot v` THEN
6120 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhs o snd) THEN
6122 [REWRITE_TAC[FINITE_DELETE; FINITE_NUMSEG; IN_NUMSEG; IN_DELETE] THEN
6123 MESON_TAC[BASIS_INJ];
6124 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF] THEN
6125 ASM_SIMP_TAC[VSUM_DELETE; FINITE_NUMSEG; IN_NUMSEG; DOT_BASIS] THEN
6126 REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_SUB_RZERO]]]);;
6128 let DIM_SPECIAL_HYPERPLANE = prove
6129 (`!k. 1 <= k /\ k <= dimindex(:N)
6130 ==> dim {x:real^N | x$k = &0} = dimindex(:N) - 1`,
6131 SIMP_TAC[SPECIAL_HYPERPLANE_SPAN] THEN REPEAT STRIP_TAC THEN
6132 MATCH_MP_TAC DIM_UNIQUE THEN
6133 EXISTS_TAC `IMAGE (basis:num->real^N) ((1..dimindex(:N)) DELETE k)` THEN
6134 REWRITE_TAC[SUBSET_REFL; SPAN_INC] THEN CONJ_TAC THENL
6135 [MATCH_MP_TAC INDEPENDENT_MONO THEN
6136 EXISTS_TAC `{basis i:real^N | 1 <= i /\ i <= dimindex(:N)}` THEN
6137 REWRITE_TAC[INDEPENDENT_STDBASIS; SUBSET; FORALL_IN_IMAGE] THEN
6138 REWRITE_TAC[IN_DELETE; IN_NUMSEG; IN_ELIM_THM] THEN MESON_TAC[];
6139 MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN CONJ_TAC THENL
6140 [REWRITE_TAC[FINITE_DELETE; FINITE_NUMSEG; IN_NUMSEG; IN_DELETE] THEN
6141 MESON_TAC[BASIS_INJ];
6142 ASM_SIMP_TAC[HAS_SIZE; FINITE_DELETE; FINITE_NUMSEG; CARD_DELETE;
6143 FINITE_IMAGE; IN_NUMSEG; CARD_NUMSEG_1]]]);;
6145 (* ------------------------------------------------------------------------- *)
6146 (* More theorems about dimensions of different subspaces. *)
6147 (* ------------------------------------------------------------------------- *)
6149 let DIM_IMAGE_KERNEL_GEN = prove
6150 (`!f:real^M->real^N s.
6151 linear f /\ subspace s
6152 ==> dim(IMAGE f s) + dim {x | x IN s /\ f x = vec 0} = dim(s)`,
6153 REPEAT STRIP_TAC THEN MP_TAC
6154 (ISPEC `{x | x IN s /\ (f:real^M->real^N) x = vec 0}` BASIS_EXISTS) THEN
6155 DISCH_THEN(X_CHOOSE_THEN `v:real^M->bool` STRIP_ASSUME_TAC) THEN
6156 MP_TAC(ISPECL [`v:real^M->bool`; `s:real^M->bool`]
6157 MAXIMAL_INDEPENDENT_SUBSET_EXTEND) THEN
6158 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
6159 DISCH_THEN(X_CHOOSE_THEN `w:real^M->bool` STRIP_ASSUME_TAC) THEN
6160 SUBGOAL_THEN `span(w:real^M->bool) = s`
6161 (fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [SYM th] THEN
6163 THENL [ASM_SIMP_TAC[SPAN_SUBSPACE]; ALL_TAC] THEN
6164 SUBGOAL_THEN `subspace {x | x IN s /\ (f:real^M->real^N) x = vec 0}`
6166 [REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} = s INTER {x | P x}`] THEN
6167 ASM_SIMP_TAC[SUBSPACE_INTER; SUBSPACE_KERNEL];
6169 SUBGOAL_THEN `{x | x IN s /\ (f:real^M->real^N) x = vec 0} = span v`
6171 [ASM_MESON_TAC[SUBSET_ANTISYM; SPAN_SUBSET_SUBSPACE; SUBSPACE_KERNEL];
6173 ASM_SIMP_TAC[DIM_SPAN; DIM_EQ_CARD] THEN
6175 `!x. x IN span(w DIFF v) /\ (f:real^M->real^N) x = vec 0 ==> x = vec 0`
6176 (LABEL_TAC "*") THENL
6177 [MATCH_MP_TAC(SET_RULE
6178 `!t. s SUBSET t /\ (!x. x IN s /\ x IN t /\ P x ==> Q x)
6179 ==> (!x. x IN s /\ P x ==> Q x)`) THEN
6180 EXISTS_TAC `s:real^M->bool` THEN CONJ_TAC THENL
6181 [ASM_MESON_TAC[SPAN_MONO; SUBSET_DIFF]; ALL_TAC] THEN
6182 ASM_SIMP_TAC[SPAN_FINITE; IN_ELIM_THM; IMP_CONJ; FINITE_DIFF;
6183 INDEPENDENT_IMP_FINITE; LEFT_IMP_EXISTS_THM] THEN
6184 GEN_TAC THEN X_GEN_TAC `u:real^M->real` THEN
6185 DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[IMP_IMP] THEN
6186 ONCE_REWRITE_TAC[SET_RULE
6187 `y IN s /\ f y = a <=> y IN {x | x IN s /\ f x = a}`] THEN
6188 ASM_REWRITE_TAC[] THEN
6189 ASM_SIMP_TAC[SPAN_FINITE; INDEPENDENT_IMP_FINITE; IN_ELIM_THM] THEN
6190 DISCH_THEN(X_CHOOSE_TAC `t:real^M->real`) THEN
6191 MP_TAC(ISPEC `w:real^M->bool` INDEPENDENT_EXPLICIT) THEN
6192 ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
6193 DISCH_THEN(MP_TAC o SPEC
6194 `(\x. if x IN w DIFF v then --u x else t x):real^M->real`) THEN
6195 ASM_REWRITE_TAC[COND_RAND] THEN ONCE_REWRITE_TAC[COND_RATOR] THEN
6196 ASM_SIMP_TAC[VSUM_CASES; INDEPENDENT_IMP_FINITE] THEN
6197 REWRITE_TAC[SET_RULE `{x | x IN w /\ x IN (w DIFF v)} = w DIFF v`] THEN
6198 SIMP_TAC[ASSUME `(v:real^M->bool) SUBSET w`; SET_RULE
6199 `v SUBSET w ==> {x | x IN w /\ ~(x IN (w DIFF v))} = v`] THEN
6200 ASM_REWRITE_TAC[VECTOR_MUL_LNEG; VSUM_NEG; VECTOR_ADD_LINV] THEN
6201 DISCH_THEN(fun th -> MATCH_MP_TAC VSUM_EQ_0 THEN MP_TAC th) THEN
6202 REWRITE_TAC[REAL_NEG_EQ_0; VECTOR_MUL_EQ_0; IN_DIFF] THEN MESON_TAC[];
6204 SUBGOAL_THEN `!x y. x IN (w DIFF v) /\ y IN (w DIFF v) /\
6205 (f:real^M->real^N) x = f y ==> x = y`
6207 [REMOVE_THEN "*" MP_TAC THEN
6208 ASM_SIMP_TAC[GSYM LINEAR_INJECTIVE_0_SUBSPACE; SUBSPACE_SPAN] THEN
6209 MP_TAC(ISPEC `w DIFF v:real^M->bool` SPAN_INC) THEN SET_TAC[];
6211 SUBGOAL_THEN `IMAGE (f:real^M->real^N) s = span(IMAGE f (w DIFF v))`
6213 [MATCH_MP_TAC SUBSET_ANTISYM THEN CONJ_TAC THENL
6215 ASM_MESON_TAC[SUBSPACE_LINEAR_IMAGE; SPAN_MONO; IMAGE_SUBSET;
6216 SUBSET_TRANS; SUBSET_DIFF; SPAN_EQ_SELF]] THEN
6217 SIMP_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `x:real^M` THEN
6218 DISCH_TAC THEN UNDISCH_TAC `span w:real^M->bool = s` THEN
6219 REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
6220 ASM_REWRITE_TAC[] THEN
6221 REMOVE_THEN "*" (MP_TAC o SPEC `x:real^M`) THEN
6222 (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 4)
6223 [IN_UNIV; SPAN_FINITE; INDEPENDENT_IMP_FINITE; IN_ELIM_THM;
6224 FINITE_IMAGE; FINITE_DIFF; ASSUME `independent(w:real^M->bool)`] THEN
6225 REWRITE_TAC[IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN DISCH_TAC THEN
6226 X_GEN_TAC `u:real^M->real` THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
6227 FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INJECTIVE_ON_LEFT_INVERSE]) THEN
6228 DISCH_THEN(X_CHOOSE_TAC `g:real^N->real^M`) THEN
6229 EXISTS_TAC `(u:real^M->real) o (g:real^N->real^M)` THEN
6230 W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhand o snd) THEN
6231 ASM_REWRITE_TAC[] THEN
6232 ASM_SIMP_TAC[FINITE_DIFF; INDEPENDENT_IMP_FINITE; LINEAR_VSUM] THEN
6233 DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[o_DEF] THEN
6234 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_EQ_SUPERSET THEN
6235 SIMP_TAC[SUBSET_DIFF; FINITE_DIFF; INDEPENDENT_IMP_FINITE;
6236 LINEAR_CMUL; IN_DIFF; TAUT `a /\ ~(a /\ ~b) <=> a /\ b`;
6237 ASSUME `independent(w:real^M->bool)`;
6238 ASSUME `linear(f:real^M->real^N)`] THEN
6239 REWRITE_TAC[VECTOR_MUL_EQ_0] THEN ASM SET_TAC[];
6240 SUBGOAL_THEN `independent(IMAGE (f:real^M->real^N) (w DIFF v))`
6242 [MATCH_MP_TAC INDEPENDENT_INJECTIVE_IMAGE_GEN THEN
6243 ASM_SIMP_TAC[LINEAR_INJECTIVE_0_SUBSPACE; SUBSPACE_SPAN] THEN
6244 ASM_MESON_TAC[INDEPENDENT_MONO; SUBSET_DIFF];
6245 ASM_SIMP_TAC[DIM_SPAN; DIM_EQ_CARD] THEN
6246 W(MP_TAC o PART_MATCH (lhs o rand) CARD_IMAGE_INJ o
6247 lhand o lhand o snd) THEN
6248 ASM_REWRITE_TAC[] THEN
6249 ASM_SIMP_TAC[FINITE_DIFF; CARD_DIFF; INDEPENDENT_IMP_FINITE] THEN
6250 DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC SUB_ADD THEN
6251 ASM_MESON_TAC[CARD_SUBSET; INDEPENDENT_IMP_FINITE]]]);;
6253 let DIM_IMAGE_KERNEL = prove
6254 (`!f:real^M->real^N.
6256 ==> dim(IMAGE f (:real^M)) + dim {x | f x = vec 0} = dimindex(:M)`,
6257 REPEAT STRIP_TAC THEN
6258 MP_TAC(ISPECL [`f:real^M->real^N`; `(:real^M)`] DIM_IMAGE_KERNEL_GEN) THEN
6259 ASM_REWRITE_TAC[SUBSPACE_UNIV; IN_UNIV; DIM_UNIV]);;
6261 let DIM_SUMS_INTER = prove
6262 (`!s t:real^N->bool.
6263 subspace s /\ subspace t
6264 ==> dim {x + y | x IN s /\ y IN t} + dim(s INTER t) = dim(s) + dim(t)`,
6265 REPEAT STRIP_TAC THEN
6266 MP_TAC(ISPEC `s INTER t:real^N->bool` BASIS_EXISTS) THEN
6267 DISCH_THEN(X_CHOOSE_THEN `b:real^N->bool` STRIP_ASSUME_TAC) THEN
6268 MP_TAC(ISPECL [`b:real^N->bool`; `s:real^N->bool`]
6269 MAXIMAL_INDEPENDENT_SUBSET_EXTEND) THEN
6270 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
6271 DISCH_THEN(X_CHOOSE_THEN `c:real^N->bool` STRIP_ASSUME_TAC) THEN
6272 MP_TAC(ISPECL [`b:real^N->bool`; `t:real^N->bool`]
6273 MAXIMAL_INDEPENDENT_SUBSET_EXTEND) THEN
6274 ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
6275 DISCH_THEN(X_CHOOSE_THEN `d:real^N->bool` STRIP_ASSUME_TAC) THEN
6276 SUBGOAL_THEN `(c:real^N->bool) INTER d = b` ASSUME_TAC THENL
6277 [MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN
6278 REWRITE_TAC[SUBSET; IN_INTER] THEN X_GEN_TAC `x:real^N` THEN
6279 STRIP_TAC THEN MP_TAC(ISPEC `c:real^N->bool` independent) THEN
6280 ASM_REWRITE_TAC[dependent; NOT_EXISTS_THM] THEN
6281 DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
6282 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN STRIP_TAC THEN
6284 SUBGOAL_THEN `(x:real^N) IN span b` MP_TAC THENL
6285 [ASM_MESON_TAC[SUBSET; IN_INTER; SPAN_INC];
6286 MP_TAC(ISPECL [`b:real^N->bool`; `c DELETE (x:real^N)`] SPAN_MONO) THEN
6290 `dim (s INTER t:real^N->bool) = CARD(b:real^N->bool) /\
6291 dim s = CARD c /\ dim t = CARD d /\
6292 dim {x + y:real^N | x IN s /\ y IN t} = CARD(c UNION d:real^N->bool)`
6293 (REPEAT_TCL CONJUNCTS_THEN SUBST1_TAC) THENL
6295 ASM_SIMP_TAC[CARD_UNION_GEN; INDEPENDENT_IMP_FINITE] THEN
6296 MATCH_MP_TAC(ARITH_RULE `b:num <= c ==> (c + d) - b + b = c + d`) THEN
6297 ASM_SIMP_TAC[CARD_SUBSET; INDEPENDENT_IMP_FINITE]] THEN
6298 REPEAT CONJ_TAC THEN MATCH_MP_TAC DIM_UNIQUE THENL
6299 [EXISTS_TAC `b:real^N->bool`;
6300 EXISTS_TAC `c:real^N->bool`;
6301 EXISTS_TAC `d:real^N->bool`;
6302 EXISTS_TAC `c UNION d:real^N->bool`] THEN
6303 ASM_SIMP_TAC[HAS_SIZE; INDEPENDENT_IMP_FINITE; FINITE_UNION] THEN
6304 REWRITE_TAC[UNION_SUBSET; GSYM CONJ_ASSOC] THEN
6305 REWRITE_TAC[SUBSET; IN_ELIM_THM; FORALL_IN_GSPEC] THEN REPEAT CONJ_TAC THENL
6306 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
6307 MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN
6308 ASM_SIMP_TAC[SUBSPACE_0; VECTOR_ADD_RID] THEN ASM SET_TAC[];
6309 X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
6310 MAP_EVERY EXISTS_TAC [`vec 0:real^N`; `x:real^N`] THEN
6311 ASM_SIMP_TAC[SUBSPACE_0; VECTOR_ADD_LID] THEN ASM SET_TAC[];
6312 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
6313 MATCH_MP_TAC SPAN_ADD THEN CONJ_TAC THENL
6314 [MP_TAC(ISPECL[`c:real^N->bool`; `c UNION d:real^N->bool`] SPAN_MONO);
6315 MP_TAC(ISPECL[`d:real^N->bool`; `c UNION d:real^N->bool`] SPAN_MONO)] THEN
6316 REWRITE_TAC[SUBSET_UNION] THEN REWRITE_TAC[SUBSET] THEN
6317 DISCH_THEN MATCH_MP_TAC THEN ASM SET_TAC[];
6319 ASM_SIMP_TAC[INDEPENDENT_EXPLICIT; FINITE_UNION; INDEPENDENT_IMP_FINITE] THEN
6320 X_GEN_TAC `a:real^N->real` THEN
6321 GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
6322 [SET_RULE `s UNION t = s UNION (t DIFF s)`] THEN
6323 ASM_SIMP_TAC[VSUM_UNION; SET_RULE `DISJOINT c (d DIFF c)`;
6324 INDEPENDENT_IMP_FINITE; FINITE_DIFF; FINITE_UNION] THEN
6327 `(vsum (d DIFF c) (\v:real^N. a v % v)) IN span b`
6329 [FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
6330 REWRITE_TAC[IN_INTER] THEN CONJ_TAC THENL
6331 [FIRST_X_ASSUM(SUBST1_TAC o MATCH_MP (VECTOR_ARITH
6332 `a + b = vec 0 ==> b = --a`)) THEN
6333 MATCH_MP_TAC SUBSPACE_NEG THEN ASM_REWRITE_TAC[];
6335 MATCH_MP_TAC SUBSPACE_VSUM THEN
6336 ASM_SIMP_TAC[FINITE_DIFF; INDEPENDENT_IMP_FINITE] THEN
6337 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSPACE_MUL THEN
6338 ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
6340 ASM_SIMP_TAC[SPAN_FINITE; INDEPENDENT_IMP_FINITE; IN_ELIM_THM] THEN
6341 DISCH_THEN(X_CHOOSE_TAC `e:real^N->real`) THEN
6342 MP_TAC(ISPEC `c:real^N->bool` INDEPENDENT_EXPLICIT) THEN
6343 ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
6344 (MP_TAC o SPEC `(\x. if x IN b then a x + e x else a x):real^N->real`)) THEN
6345 REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COND_RAND] THEN
6346 ONCE_REWRITE_TAC[COND_RATOR] THEN ASM_SIMP_TAC[VSUM_CASES] THEN
6347 REWRITE_TAC[VECTOR_ADD_RDISTRIB; GSYM DIFF] THEN
6348 ASM_SIMP_TAC[SET_RULE `b SUBSET c ==> {x | x IN c /\ x IN b} = b`] THEN
6349 ASM_SIMP_TAC[VSUM_ADD; INDEPENDENT_IMP_FINITE] THEN
6350 ONCE_REWRITE_TAC[VECTOR_ARITH `(a + b) + c:real^N = (a + c) + b`] THEN
6351 ASM_SIMP_TAC[GSYM VSUM_UNION; FINITE_DIFF; INDEPENDENT_IMP_FINITE;
6352 SET_RULE `DISJOINT b (c DIFF b)`] THEN
6353 ASM_SIMP_TAC[SET_RULE `b SUBSET c ==> b UNION (c DIFF b) = c`] THEN
6355 SUBGOAL_THEN `!v:real^N. v IN (c DIFF b) ==> a v = &0` ASSUME_TAC THENL
6356 [ASM SET_TAC[]; ALL_TAC] THEN
6357 MP_TAC(ISPEC `d:real^N->bool` INDEPENDENT_EXPLICIT) THEN
6358 ASM_REWRITE_TAC[] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
6359 (MP_TAC o SPEC `a:real^N->real`)) THEN
6360 SUBGOAL_THEN `d:real^N->bool = b UNION (d DIFF c)`
6361 (fun th -> GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [th])
6362 THENL [ASM SET_TAC[]; ALL_TAC] THEN
6363 ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
6364 ASM_SIMP_TAC[VSUM_UNION; FINITE_DIFF; INDEPENDENT_IMP_FINITE;
6365 SET_RULE `c INTER d = b ==> DISJOINT b (d DIFF c)`] THEN
6366 SUBGOAL_THEN `vsum b (\x:real^N. a x % x) = vsum c (\x. a x % x)`
6367 (fun th -> ASM_REWRITE_TAC[th]) THEN
6368 CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
6369 ASM_SIMP_TAC[VECTOR_MUL_EQ_0] THEN ASM_MESON_TAC[]);;
6371 let DIM_KERNEL_COMPOSE = prove
6372 (`!f:real^M->real^N g:real^N->real^P.
6373 linear f /\ linear g
6374 ==> dim {x | (g o f) x = vec 0} <=
6375 dim {x | f(x) = vec 0} +
6376 dim {y | g(y) = vec 0}`,
6377 REPEAT STRIP_TAC THEN
6378 MP_TAC(ISPEC `{x | (f:real^M->real^N) x = vec 0}` BASIS_EXISTS_FINITE) THEN
6379 DISCH_THEN(X_CHOOSE_THEN `b:real^M->bool` STRIP_ASSUME_TAC) THEN
6382 IMAGE f c SUBSET {y | g(y):real^P = vec 0} /\
6383 independent (IMAGE (f:real^M->real^N) c) /\
6384 IMAGE f (:real^M) INTER {y | g(y) = vec 0} SUBSET span(IMAGE f c) /\
6385 (!x y. x IN c /\ y IN c ==> (f x = f y <=> x = y)) /\
6386 (IMAGE f c) HAS_SIZE dim (IMAGE f (:real^M) INTER {y | g(y) = vec 0})`
6387 STRIP_ASSUME_TAC THENL
6388 [MP_TAC(ISPEC `IMAGE (f:real^M->real^N) (:real^M) INTER
6389 {x | (g:real^N->real^P) x = vec 0}` BASIS_EXISTS_FINITE) THEN
6390 REWRITE_TAC[SUBSET_INTER; GSYM CONJ_ASSOC; EXISTS_FINITE_SUBSET_IMAGE] THEN
6391 DISCH_THEN(X_CHOOSE_THEN `c:real^M->bool` STRIP_ASSUME_TAC) THEN
6392 MP_TAC(ISPECL [`f:real^M->real^N`; `c:real^M->bool`]
6393 IMAGE_INJECTIVE_IMAGE_OF_SUBSET) THEN
6394 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^M->bool` THEN
6395 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
6396 (CONJUNCTS_THEN2 SUBST_ALL_TAC ASSUME_TAC)) THEN
6397 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[FINITE_SUBSET];
6399 MATCH_MP_TAC LE_TRANS THEN
6400 EXISTS_TAC `dim(span(b UNION c:real^M->bool))` THEN CONJ_TAC THENL
6401 [MATCH_MP_TAC DIM_SUBSET THEN
6402 REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; o_THM] THEN
6403 X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
6404 SUBGOAL_THEN `(f:real^M->real^N) x IN span(IMAGE f c)` MP_TAC THENL
6405 [ASM SET_TAC[]; ALL_TAC] THEN
6406 ASM_SIMP_TAC[SPAN_LINEAR_IMAGE; IN_IMAGE; LEFT_IMP_EXISTS_THM] THEN
6407 X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
6408 SUBST1_TAC(VECTOR_ARITH `x:real^M = y + (x - y)`) THEN
6409 MATCH_MP_TAC SPAN_ADD THEN CONJ_TAC THENL
6410 [ASM_MESON_TAC[SUBSET_UNION; SPAN_MONO; SUBSET]; ALL_TAC] THEN
6411 MATCH_MP_TAC(SET_RULE
6412 `!t. x IN t /\ t SUBSET s ==> x IN s`) THEN
6413 EXISTS_TAC `{x | (f:real^M->real^N) x = vec 0}` THEN CONJ_TAC THENL
6414 [REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[LINEAR_SUB; VECTOR_SUB_EQ];
6415 ASM_MESON_TAC[SUBSET_TRANS; SUBSET_UNION; SPAN_MONO]];
6416 REWRITE_TAC[DIM_SPAN] THEN MATCH_MP_TAC LE_TRANS THEN
6417 EXISTS_TAC `CARD(b UNION c:real^M->bool)` THEN
6418 ASM_SIMP_TAC[DIM_LE_CARD; FINITE_UNION; INDEPENDENT_IMP_FINITE] THEN
6419 MATCH_MP_TAC LE_TRANS THEN
6420 EXISTS_TAC `CARD(b:real^M->bool) + CARD(c:real^M->bool)` THEN
6421 ASM_SIMP_TAC[CARD_UNION_LE] THEN MATCH_MP_TAC LE_ADD2 THEN CONJ_TAC THENL
6422 [ASM_SIMP_TAC[GSYM DIM_EQ_CARD; DIM_SUBSET]; ALL_TAC] THEN
6423 MATCH_MP_TAC LE_TRANS THEN
6424 EXISTS_TAC `dim(IMAGE (f:real^M->real^N) c)` THEN CONJ_TAC THENL
6425 [ASM_SIMP_TAC[DIM_EQ_CARD] THEN
6426 ASM_MESON_TAC[CARD_IMAGE_INJ; LE_REFL];
6427 ASM_SIMP_TAC[GSYM DIM_EQ_CARD; DIM_SUBSET]]]);;
6429 let DIM_ORTHOGONAL_SUM = prove
6430 (`!s t:real^N->bool.
6431 (!x y. x IN s /\ y IN t ==> x dot y = &0)
6432 ==> dim(s UNION t) = dim(s) + dim(t)`,
6433 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM DIM_SPAN] THEN
6434 REWRITE_TAC[SPAN_UNION] THEN
6435 SIMP_TAC[GSYM DIM_SUMS_INTER; SUBSPACE_SPAN] THEN
6436 REWRITE_TAC[ARITH_RULE `x = x + y <=> y = 0`] THEN
6437 REWRITE_TAC[DIM_EQ_0; SUBSET; IN_INTER] THEN
6439 `!x:real^N. x IN span s ==> !y:real^N. y IN span t ==> x dot y = &0`
6441 [MATCH_MP_TAC SPAN_INDUCT THEN CONJ_TAC THENL
6442 [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
6443 MATCH_MP_TAC SPAN_INDUCT THEN ASM_SIMP_TAC[IN_ELIM_THM] THEN
6444 SIMP_TAC[subspace; IN_ELIM_THM; DOT_RMUL; DOT_RADD; DOT_RZERO] THEN
6446 SIMP_TAC[subspace; IN_ELIM_THM; DOT_LMUL; DOT_LADD; DOT_LZERO] THEN
6448 REWRITE_TAC[IN_SING] THEN MESON_TAC[DOT_EQ_0]]);;
6450 (* ------------------------------------------------------------------------- *)
6451 (* More about rank from the rank/nullspace formula. *)
6452 (* ------------------------------------------------------------------------- *)
6454 let RANK_NULLSPACE = prove
6455 (`!A:real^M^N. rank A + dim {x | A ** x = vec 0} = dimindex(:M)`,
6456 GEN_TAC THEN REWRITE_TAC[RANK_DIM_IM] THEN
6457 MATCH_MP_TAC DIM_IMAGE_KERNEL THEN
6458 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR]);;
6460 let RANK_SYLVESTER = prove
6461 (`!A:real^N^M B:real^P^N.
6462 rank(A) + rank(B) <= rank(A ** B) + dimindex(:N)`,
6463 REPEAT GEN_TAC THEN MATCH_MP_TAC(ARITH_RULE
6469 ==> ra + rb <= rab + n`) THEN
6470 MAP_EVERY EXISTS_TAC
6471 [`dim {x | (A:real^N^M) ** x = vec 0}`;
6472 `dim {x | (B:real^P^N) ** x = vec 0}`;
6473 `dim {x | ((A:real^N^M) ** (B:real^P^N)) ** x = vec 0}`;
6474 `dimindex(:P)`] THEN
6475 REWRITE_TAC[RANK_NULLSPACE] THEN
6476 REWRITE_TAC[GSYM MATRIX_VECTOR_MUL_ASSOC] THEN
6477 ONCE_REWRITE_TAC[ADD_SYM] THEN
6478 MATCH_MP_TAC(REWRITE_RULE[o_DEF] DIM_KERNEL_COMPOSE) THEN
6479 CONJ_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM ETA_AX] THEN
6480 REWRITE_TAC[MATRIX_VECTOR_MUL_LINEAR]);;
6482 let RANK_GRAM = prove
6483 (`!A:real^M^N. rank(transp A ** A) = rank A`,
6484 GEN_TAC THEN MATCH_MP_TAC(ARITH_RULE
6485 `!n n' k. r + n:num = k /\ r' + n' = k /\ n = n' ==> r = r'`) THEN
6486 MAP_EVERY EXISTS_TAC
6487 [`dim {x | (transp A ** (A:real^M^N)) ** x = vec 0}`;
6488 `dim {x | (A:real^M^N) ** x = vec 0}`;
6489 `dimindex(:M)`] THEN
6490 REWRITE_TAC[RANK_NULLSPACE] THEN AP_TERM_TAC THEN
6491 MATCH_MP_TAC SUBSET_ANTISYM THEN
6492 SIMP_TAC[SUBSET; IN_ELIM_THM; GSYM MATRIX_VECTOR_MUL_ASSOC;
6493 MATRIX_VECTOR_MUL_RZERO] THEN
6494 X_GEN_TAC `x:real^M` THEN
6495 DISCH_THEN(MP_TAC o AP_TERM `(dot) (x:real^M)`) THEN
6496 ONCE_REWRITE_TAC[GSYM DOT_LMUL_MATRIX] THEN
6497 REWRITE_TAC[VECTOR_MATRIX_MUL_TRANSP; TRANSP_TRANSP; DOT_RZERO] THEN
6498 REWRITE_TAC[DOT_EQ_0]);;
6500 let RANK_TRIANGLE = prove
6501 (`!A B:real^M^N. rank(A + B) <= rank(A) + rank(B)`,
6502 REPEAT GEN_TAC THEN REWRITE_TAC[RANK_DIM_IM] THEN
6503 MP_TAC(ISPECL [`IMAGE (\x. (A:real^M^N) ** x) (:real^M)`;
6504 `IMAGE (\x. (B:real^M^N) ** x) (:real^M)`]
6505 DIM_SUMS_INTER) THEN
6506 ASM_SIMP_TAC[SUBSPACE_LINEAR_IMAGE; SUBSPACE_UNIV;
6507 MATRIX_VECTOR_MUL_LINEAR] THEN
6508 DISCH_THEN(SUBST1_TAC o SYM) THEN
6509 MATCH_MP_TAC(ARITH_RULE `x:num <= y ==> x <= y + z`) THEN
6510 MATCH_MP_TAC DIM_SUBSET THEN
6511 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV;
6512 MATRIX_VECTOR_MUL_ADD_RDISTRIB] THEN
6513 REWRITE_TAC[IN_ELIM_THM; IN_IMAGE; IN_UNIV] THEN MESON_TAC[]);;
6515 (* ------------------------------------------------------------------------- *)
6516 (* Infinity norm. *)
6517 (* ------------------------------------------------------------------------- *)
6519 let infnorm = define
6520 `infnorm (x:real^N) = sup { abs(x$i) | 1 <= i /\ i <= dimindex(:N) }`;;
6522 let NUMSEG_DIMINDEX_NONEMPTY = prove
6523 (`?i. i IN 1..dimindex(:N)`,
6524 REWRITE_TAC[MEMBER_NOT_EMPTY; NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1]);;
6526 let INFNORM_SET_IMAGE = prove
6527 (`{abs(x$i) | 1 <= i /\ i <= dimindex(:N)} =
6528 IMAGE (\i. abs(x$i)) (1..dimindex(:N))`,
6529 REWRITE_TAC[numseg] THEN SET_TAC[]);;
6531 let INFNORM_SET_LEMMA = prove
6532 (`FINITE {abs((x:real^N)$i) | 1 <= i /\ i <= dimindex(:N)} /\
6533 ~({abs(x$i) | 1 <= i /\ i <= dimindex(:N)} = {})`,
6534 SIMP_TAC[INFNORM_SET_IMAGE; FINITE_NUMSEG; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
6535 REWRITE_TAC[NUMSEG_EMPTY; NOT_LT; DIMINDEX_GE_1]);;
6537 let INFNORM_POS_LE = prove
6538 (`!x. &0 <= infnorm x`,
6539 REWRITE_TAC[infnorm] THEN
6540 SIMP_TAC[REAL_LE_SUP_FINITE; INFNORM_SET_LEMMA] THEN
6541 REWRITE_TAC[INFNORM_SET_IMAGE; NUMSEG_DIMINDEX_NONEMPTY;
6542 EXISTS_IN_IMAGE; REAL_ABS_POS]);;
6544 let INFNORM_TRIANGLE = prove
6545 (`!x y. infnorm(x + y) <= infnorm x + infnorm y`,
6546 REWRITE_TAC[infnorm] THEN
6547 SIMP_TAC[REAL_SUP_LE_FINITE; INFNORM_SET_LEMMA] THEN
6548 ONCE_REWRITE_TAC[GSYM REAL_LE_SUB_RADD] THEN
6549 SIMP_TAC[REAL_LE_SUP_FINITE; INFNORM_SET_LEMMA] THEN
6550 ONCE_REWRITE_TAC[REAL_ARITH `x - y <= z <=> x - z <= y`] THEN
6551 SIMP_TAC[REAL_LE_SUP_FINITE; INFNORM_SET_LEMMA] THEN
6552 REWRITE_TAC[INFNORM_SET_IMAGE; FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN
6553 SIMP_TAC[VECTOR_ADD_COMPONENT; GSYM IN_NUMSEG] THEN
6554 MESON_TAC[NUMSEG_DIMINDEX_NONEMPTY;
6555 REAL_ARITH `abs(x + y) - abs(x) <= abs(y)`]);;
6557 let INFNORM_EQ_0 = prove
6558 (`!x. infnorm x = &0 <=> x = vec 0`,
6559 REWRITE_TAC[GSYM REAL_LE_ANTISYM; INFNORM_POS_LE] THEN
6560 SIMP_TAC[infnorm; REAL_SUP_LE_FINITE; INFNORM_SET_LEMMA] THEN
6561 SIMP_TAC[FORALL_IN_IMAGE; INFNORM_SET_IMAGE; CART_EQ; VEC_COMPONENT] THEN
6562 REWRITE_TAC[IN_NUMSEG; REAL_ARITH `abs(x) <= &0 <=> x = &0`]);;
6564 let INFNORM_0 = prove
6565 (`infnorm(vec 0) = &0`,
6566 REWRITE_TAC[INFNORM_EQ_0]);;
6568 let INFNORM_NEG = prove
6569 (`!x. infnorm(--x) = infnorm x`,
6570 GEN_TAC THEN REWRITE_TAC[infnorm] THEN AP_TERM_TAC THEN
6571 REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
6572 MESON_TAC[REAL_ABS_NEG; VECTOR_NEG_COMPONENT]);;
6574 let INFNORM_SUB = prove
6575 (`!x y. infnorm(x - y) = infnorm(y - x)`,
6576 MESON_TAC[INFNORM_NEG; VECTOR_NEG_SUB]);;
6578 let REAL_ABS_SUB_INFNORM = prove
6579 (`abs(infnorm x - infnorm y) <= infnorm(x - y)`,
6580 MATCH_MP_TAC(REAL_ARITH
6581 `nx <= n + ny /\ ny <= n + nx ==> abs(nx - ny) <= n`) THEN
6582 MESON_TAC[INFNORM_SUB; VECTOR_SUB_ADD2; INFNORM_TRIANGLE; VECTOR_ADD_SYM]);;
6584 let REAL_ABS_INFNORM = prove
6585 (`!x. abs(infnorm x) = infnorm x`,
6586 REWRITE_TAC[real_abs; INFNORM_POS_LE]);;
6588 let COMPONENT_LE_INFNORM = prove
6589 (`!x:real^N i. 1 <= i /\ i <= dimindex (:N) ==> abs(x$i) <= infnorm x`,
6590 REPEAT GEN_TAC THEN REWRITE_TAC[infnorm] THEN
6591 MP_TAC(SPEC `{ abs((x:real^N)$i) | 1 <= i /\ i <= dimindex(:N) }`
6593 REWRITE_TAC[INFNORM_SET_LEMMA] THEN
6594 SIMP_TAC[INFNORM_SET_IMAGE; FORALL_IN_IMAGE; IN_NUMSEG]);;
6596 let INFNORM_MUL_LEMMA = prove
6597 (`!a x. infnorm(a % x) <= abs a * infnorm x`,
6598 REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [infnorm] THEN
6599 SIMP_TAC[REAL_SUP_LE_FINITE; INFNORM_SET_LEMMA] THEN
6600 REWRITE_TAC[FORALL_IN_IMAGE; INFNORM_SET_IMAGE] THEN
6601 SIMP_TAC[REAL_ABS_MUL; VECTOR_MUL_COMPONENT; IN_NUMSEG] THEN
6602 SIMP_TAC[COMPONENT_LE_INFNORM; REAL_LE_LMUL; REAL_ABS_POS]);;
6604 let INFNORM_MUL = prove
6605 (`!a x:real^N. infnorm(a % x) = abs a * infnorm x`,
6606 REPEAT GEN_TAC THEN ASM_CASES_TAC `a = &0` THEN
6607 ASM_REWRITE_TAC[VECTOR_MUL_LZERO; INFNORM_0; REAL_ABS_0; REAL_MUL_LZERO] THEN
6608 REWRITE_TAC[GSYM REAL_LE_ANTISYM; INFNORM_MUL_LEMMA] THEN
6609 GEN_REWRITE_TAC (LAND_CONV o funpow 2 RAND_CONV) [GSYM VECTOR_MUL_LID] THEN
6610 FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP REAL_MUL_LINV) THEN
6611 REWRITE_TAC[GSYM VECTOR_MUL_ASSOC] THEN
6612 MATCH_MP_TAC REAL_LE_TRANS THEN
6613 EXISTS_TAC `abs(a) * abs(inv a) * infnorm(a % x:real^N)` THEN
6614 ASM_SIMP_TAC[INFNORM_MUL_LEMMA; REAL_LE_LMUL; REAL_ABS_POS] THEN
6615 ASM_SIMP_TAC[REAL_MUL_ASSOC; GSYM REAL_ABS_MUL; REAL_MUL_RINV] THEN
6618 let INFNORM_POS_LT = prove
6619 (`!x. &0 < infnorm x <=> ~(x = vec 0)`,
6620 MESON_TAC[REAL_LT_LE; INFNORM_POS_LE; INFNORM_EQ_0]);;
6622 (* ------------------------------------------------------------------------- *)
6623 (* Prove that it differs only up to a bound from Euclidean norm. *)
6624 (* ------------------------------------------------------------------------- *)
6626 let INFNORM_LE_NORM = prove
6627 (`!x. infnorm(x) <= norm(x)`,
6628 SIMP_TAC[infnorm; REAL_SUP_LE_FINITE; INFNORM_SET_LEMMA] THEN
6629 REWRITE_TAC[IN_ELIM_THM] THEN MESON_TAC[COMPONENT_LE_NORM]);;
6631 let NORM_LE_INFNORM = prove
6632 (`!x:real^N. norm(x) <= sqrt(&(dimindex(:N))) * infnorm(x)`,
6633 GEN_TAC THEN GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o funpow 2 RAND_CONV)
6634 [GSYM CARD_NUMSEG_1] THEN
6635 REWRITE_TAC[vector_norm] THEN MATCH_MP_TAC REAL_LE_LSQRT THEN
6636 SIMP_TAC[DOT_POS_LE; SQRT_POS_LE; REAL_POS; REAL_LE_MUL; INFNORM_POS_LE;
6637 SQRT_POW_2; REAL_POW_MUL] THEN
6638 REWRITE_TAC[dot] THEN MATCH_MP_TAC SUM_BOUND THEN
6639 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
6640 REWRITE_TAC[GSYM REAL_POW_2] THEN ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN
6641 MATCH_MP_TAC REAL_POW_LE2 THEN REWRITE_TAC[REAL_ABS_POS] THEN
6642 MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= abs(y)`) THEN
6643 SIMP_TAC[infnorm; REAL_LE_SUP_FINITE; INFNORM_SET_LEMMA] THEN
6644 REWRITE_TAC[IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_LE_REFL]);;
6646 (* ------------------------------------------------------------------------- *)
6647 (* Equality in Cauchy-Schwarz and triangle inequalities. *)
6648 (* ------------------------------------------------------------------------- *)
6650 let NORM_CAUCHY_SCHWARZ_EQ = prove
6651 (`!x:real^N y. x dot y = norm(x) * norm(y) <=> norm(x) % y = norm(y) % x`,
6652 REPEAT STRIP_TAC THEN
6653 MAP_EVERY ASM_CASES_TAC [`x:real^N = vec 0`; `y:real^N = vec 0`] THEN
6654 ASM_REWRITE_TAC[NORM_0; REAL_MUL_LZERO; REAL_MUL_RZERO;
6655 DOT_LZERO; DOT_RZERO; VECTOR_MUL_LZERO; VECTOR_MUL_RZERO] THEN
6656 MP_TAC(ISPEC `norm(y:real^N) % x - norm(x:real^N) % y` DOT_EQ_0) THEN
6657 REWRITE_TAC[DOT_RSUB; DOT_LSUB; DOT_LMUL; DOT_RMUL; GSYM NORM_POW_2;
6658 REAL_POW_2; VECTOR_SUB_EQ] THEN
6659 REWRITE_TAC[DOT_SYM; REAL_ARITH
6660 `y * (y * x * x - x * d) - x * (y * d - x * y * y) =
6661 &2 * x * y * (x * y - d)`] THEN
6662 ASM_SIMP_TAC[REAL_ENTIRE; NORM_EQ_0; REAL_SUB_0; REAL_OF_NUM_EQ; ARITH] THEN
6663 REWRITE_TAC[EQ_SYM_EQ]);;
6665 let NORM_CAUCHY_SCHWARZ_ABS_EQ = prove
6666 (`!x:real^N y. abs(x dot y) = norm(x) * norm(y) <=>
6667 norm(x) % y = norm(y) % x \/ norm(x) % y = --norm(y) % x`,
6668 SIMP_TAC[REAL_ARITH `&0 <= a ==> (abs x = a <=> x = a \/ --x = a)`;
6669 REAL_LE_MUL; NORM_POS_LE; GSYM DOT_RNEG] THEN
6671 GEN_REWRITE_TAC (LAND_CONV o funpow 3 RAND_CONV) [GSYM NORM_NEG] THEN
6672 REWRITE_TAC[NORM_CAUCHY_SCHWARZ_EQ] THEN REWRITE_TAC[NORM_NEG] THEN
6673 BINOP_TAC THEN VECTOR_ARITH_TAC);;
6675 let NORM_TRIANGLE_EQ = prove
6676 (`!x y:real^N. norm(x + y) = norm(x) + norm(y) <=> norm(x) % y = norm(y) % x`,
6677 REPEAT GEN_TAC THEN REWRITE_TAC[GSYM NORM_CAUCHY_SCHWARZ_EQ] THEN
6678 MATCH_MP_TAC EQ_TRANS THEN
6679 EXISTS_TAC `norm(x + y:real^N) pow 2 = (norm(x) + norm(y)) pow 2` THEN
6681 [REWRITE_TAC[REAL_RING `x pow 2 = y pow 2 <=> x = y \/ x + y = &0`] THEN
6682 MAP_EVERY (MP_TAC o C ISPEC NORM_POS_LE)
6683 [`x + y:real^N`; `x:real^N`; `y:real^N`] THEN
6685 REWRITE_TAC[NORM_POW_2; DOT_LADD; DOT_RADD; REAL_ARITH
6686 `(x + y) pow 2 = x pow 2 + y pow 2 + &2 * x * y`] THEN
6687 REWRITE_TAC[DOT_SYM] THEN REAL_ARITH_TAC]);;
6689 let DIST_TRIANGLE_EQ = prove
6690 (`!x y z. dist(x,z) = dist(x,y) + dist(y,z) <=>
6691 norm (x - y) % (y - z) = norm (y - z) % (x - y)`,
6692 REWRITE_TAC[GSYM NORM_TRIANGLE_EQ] THEN NORM_ARITH_TAC);;
6694 let NORM_CROSS_MULTIPLY = prove
6696 a % x = b % y /\ &0 < a /\ &0 < b
6697 ==> norm y % x = norm x % y`,
6698 REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
6699 ASM_CASES_TAC `y:real^N = vec 0` THEN
6700 ASM_SIMP_TAC[VECTOR_MUL_EQ_0; REAL_LT_IMP_NZ; VECTOR_MUL_RZERO] THEN
6701 DISCH_THEN(MP_TAC o AP_TERM `\x:real^N. inv(a) % x`) THEN
6702 ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; REAL_LT_IMP_NZ; VECTOR_MUL_LID;
6703 NORM_MUL; REAL_ABS_MUL; REAL_ABS_INV] THEN
6704 ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; REAL_MUL_AC]);;
6706 (* ------------------------------------------------------------------------- *)
6708 (* ------------------------------------------------------------------------- *)
6710 let collinear = new_definition
6711 `collinear s <=> ?u. !x y. x IN s /\ y IN s ==> ?c. x - y = c % u`;;
6713 let COLLINEAR_SUBSET = prove
6714 (`!s t. collinear t /\ s SUBSET t ==> collinear s`,
6715 REWRITE_TAC[collinear] THEN SET_TAC[]);;
6717 let COLLINEAR_EMPTY = prove
6719 REWRITE_TAC[collinear; NOT_IN_EMPTY]);;
6721 let COLLINEAR_SING = prove
6722 (`!x. collinear {x}`,
6723 SIMP_TAC[collinear; IN_SING; VECTOR_SUB_REFL] THEN
6724 MESON_TAC[VECTOR_MUL_LZERO]);;
6726 let COLLINEAR_2 = prove
6727 (`!x y:real^N. collinear {x,y}`,
6728 REPEAT GEN_TAC THEN REWRITE_TAC[collinear; IN_INSERT; NOT_IN_EMPTY] THEN
6729 EXISTS_TAC `x - y:real^N` THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
6730 [EXISTS_TAC `&0`; EXISTS_TAC `&1`; EXISTS_TAC `-- &1`; EXISTS_TAC `&0`] THEN
6733 let COLLINEAR_SMALL = prove
6734 (`!s. FINITE s /\ CARD s <= 2 ==> collinear s`,
6735 REWRITE_TAC[ARITH_RULE `s <= 2 <=> s = 0 \/ s = 1 \/ s = 2`] THEN
6736 REWRITE_TAC[LEFT_OR_DISTRIB; GSYM HAS_SIZE] THEN
6737 CONV_TAC(ONCE_DEPTH_CONV HAS_SIZE_CONV) THEN
6738 REPEAT STRIP_TAC THEN
6739 ASM_REWRITE_TAC[COLLINEAR_EMPTY; COLLINEAR_SING; COLLINEAR_2]);;
6741 let COLLINEAR_3 = prove
6742 (`!x y z. collinear {x,y,z} <=> collinear {vec 0,x - y,z - y}`,
6744 REWRITE_TAC[collinear; FORALL_IN_INSERT; IMP_CONJ; RIGHT_FORALL_IMP_THM;
6746 AP_TERM_TAC THEN ABS_TAC THEN
6747 MESON_TAC[VECTOR_ARITH `x - y = (x - y) - vec 0`;
6748 VECTOR_ARITH `y - x = vec 0 - (x - y)`;
6749 VECTOR_ARITH `x - z:real^N = (x - y) - (z - y)`]);;
6751 let COLLINEAR_LEMMA = prove
6752 (`!x y:real^N. collinear {vec 0,x,y} <=>
6753 x = vec 0 \/ y = vec 0 \/ ?c. y = c % x`,
6755 MAP_EVERY ASM_CASES_TAC [`x:real^N = vec 0`; `y:real^N = vec 0`] THEN
6756 TRY(ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_SING; COLLINEAR_2] THEN NO_TAC) THEN
6757 ASM_REWRITE_TAC[collinear] THEN EQ_TAC THENL
6758 [DISCH_THEN(X_CHOOSE_THEN `u:real^N`
6759 (fun th -> MP_TAC(SPECL [`x:real^N`; `vec 0:real^N`] th) THEN
6760 MP_TAC(SPECL [`y:real^N`; `vec 0:real^N`] th))) THEN
6761 REWRITE_TAC[IN_INSERT; VECTOR_SUB_RZERO] THEN
6762 DISCH_THEN(X_CHOOSE_THEN `e:real` SUBST_ALL_TAC) THEN
6763 DISCH_THEN(X_CHOOSE_THEN `d:real` SUBST_ALL_TAC) THEN
6764 EXISTS_TAC `e / d` THEN REWRITE_TAC[VECTOR_MUL_ASSOC] THEN
6765 RULE_ASSUM_TAC(REWRITE_RULE[VECTOR_MUL_EQ_0; DE_MORGAN_THM]) THEN
6766 ASM_SIMP_TAC[REAL_DIV_RMUL];
6767 STRIP_TAC THEN EXISTS_TAC `x:real^N` THEN ASM_REWRITE_TAC[] THEN
6768 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN
6769 ASM_REWRITE_TAC[] THENL
6770 [EXISTS_TAC `&0`; EXISTS_TAC `-- &1`; EXISTS_TAC `--c`;
6771 EXISTS_TAC `&1`; EXISTS_TAC `&0`; EXISTS_TAC `&1 - c`;
6772 EXISTS_TAC `c:real`; EXISTS_TAC `c - &1`; EXISTS_TAC `&0`] THEN
6773 VECTOR_ARITH_TAC]);;
6775 let COLLINEAR_LEMMA_ALT = prove
6776 (`!x y. collinear {vec 0,x,y} <=> x = vec 0 \/ ?c. y = c % x`,
6777 REWRITE_TAC[COLLINEAR_LEMMA] THEN MESON_TAC[VECTOR_MUL_LZERO]);;
6779 let NORM_CAUCHY_SCHWARZ_EQUAL = prove
6780 (`!x y:real^N. abs(x dot y) = norm(x) * norm(y) <=> collinear {vec 0,x,y}`,
6781 REPEAT GEN_TAC THEN REWRITE_TAC[NORM_CAUCHY_SCHWARZ_ABS_EQ] THEN
6782 MAP_EVERY ASM_CASES_TAC [`x:real^N = vec 0`; `y:real^N = vec 0`] THEN
6783 TRY(ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_SING; COLLINEAR_2; NORM_0;
6784 VECTOR_MUL_LZERO; VECTOR_MUL_RZERO] THEN NO_TAC) THEN
6785 ASM_REWRITE_TAC[COLLINEAR_LEMMA] THEN EQ_TAC THENL
6787 [FIRST_X_ASSUM(MP_TAC o AP_TERM
6788 `(%) (inv(norm(x:real^N))):real^N->real^N`);
6789 FIRST_X_ASSUM(MP_TAC o AP_TERM
6790 `(%) (--inv(norm(x:real^N))):real^N->real^N`)] THEN
6791 ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LNEG] THEN
6792 ASM_SIMP_TAC[REAL_MUL_LINV; NORM_EQ_0; VECTOR_MUL_LNEG; VECTOR_MUL_LID;
6793 VECTOR_ARITH `--x = --y <=> x:real^N = y`] THEN
6795 STRIP_TAC THEN ASM_REWRITE_TAC[NORM_MUL; VECTOR_MUL_ASSOC] THEN
6796 MATCH_MP_TAC(MESON[]
6797 `t = a \/ t = b ==> t % x = a % x \/ t % x = b % x`) THEN
6798 REWRITE_TAC[GSYM REAL_MUL_LNEG;
6799 REAL_ARITH `x * c = d * x <=> x * (c - d) = &0`] THEN
6800 ASM_REWRITE_TAC[REAL_ENTIRE; NORM_EQ_0] THEN REAL_ARITH_TAC]);;
6802 let DOT_CAUCHY_SCHWARZ_EQUAL = prove
6804 (x dot y) pow 2 = (x dot x) * (y dot y) <=>
6805 collinear {vec 0,x,y}`,
6806 REWRITE_TAC[GSYM NORM_CAUCHY_SCHWARZ_EQUAL] THEN
6807 REPEAT GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH
6808 `&0 <= y /\ (u:real = v <=> x = abs y) ==> (u = v <=> x = y)`) THEN
6809 SIMP_TAC[NORM_POS_LE; REAL_LE_MUL] THEN
6810 REWRITE_TAC[REAL_EQ_SQUARE_ABS] THEN REWRITE_TAC[REAL_POW_MUL; NORM_POW_2]);;
6812 let COLLINEAR_3_EXPAND = prove
6813 (`!a b c:real^N. collinear{a,b,c} <=> a = c \/ ?u. b = u % a + (&1 - u) % c`,
6815 ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN
6816 ONCE_REWRITE_TAC[COLLINEAR_3] THEN
6817 REWRITE_TAC[COLLINEAR_LEMMA; VECTOR_SUB_EQ] THEN
6818 ASM_CASES_TAC `a:real^N = c` THEN ASM_REWRITE_TAC[] THEN
6819 ASM_CASES_TAC `b:real^N = c` THEN
6820 ASM_REWRITE_TAC[VECTOR_ARITH `u % c + (&1 - u) % c = c`] THENL
6821 [EXISTS_TAC `&0` THEN VECTOR_ARITH_TAC;
6822 AP_TERM_TAC THEN ABS_TAC THEN VECTOR_ARITH_TAC]);;
6824 let COLLINEAR_TRIPLES = prove
6827 ==> (collinear(a INSERT b INSERT s) <=>
6828 !x. x IN s ==> collinear{a,b,x})`,
6829 REPEAT STRIP_TAC THEN EQ_TAC THENL
6830 [REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
6831 (REWRITE_RULE[IMP_CONJ] COLLINEAR_SUBSET)) THEN
6833 ONCE_REWRITE_TAC[SET_RULE `{a,b,x} = {a,x,b}`] THEN
6834 ASM_REWRITE_TAC[COLLINEAR_3_EXPAND] THEN DISCH_TAC THEN
6836 `!x:real^N. x IN (a INSERT b INSERT s) ==> ?u. x = u % a + (&1 - u) % b`
6838 [ASM_REWRITE_TAC[FORALL_IN_INSERT] THEN CONJ_TAC THENL
6839 [EXISTS_TAC `&1` THEN VECTOR_ARITH_TAC;
6840 EXISTS_TAC `&0` THEN VECTOR_ARITH_TAC];
6841 POP_ASSUM_LIST(K ALL_TAC) THEN DISCH_TAC THEN
6842 REWRITE_TAC[collinear] THEN EXISTS_TAC `b - a:real^N` THEN
6843 MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
6844 FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `x:real^N` th) THEN MP_TAC(SPEC
6845 `y:real^N` th)) THEN
6846 ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
6847 ASM_REWRITE_TAC[VECTOR_ARITH
6848 `(u % a + (&1 - u) % b) - (v % a + (&1 - v) % b):real^N =
6849 (v - u) % (b - a)`] THEN
6852 let COLLINEAR_4_3 = prove
6855 ==> (collinear {a,b,c,d} <=> collinear{a,b,c} /\ collinear{a,b,d})`,
6856 REPEAT STRIP_TAC THEN
6857 MP_TAC(ISPECL [`{c:real^N,d}`; `a:real^N`; `b:real^N`]
6858 COLLINEAR_TRIPLES) THEN
6859 ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
6860 REWRITE_TAC[FORALL_IN_INSERT; NOT_IN_EMPTY]);;
6862 let COLLINEAR_3_TRANS = prove
6864 collinear{a,b,c} /\ collinear{b,c,d} /\ ~(b = c) ==> collinear{a,b,d}`,
6865 REPEAT STRIP_TAC THEN MATCH_MP_TAC COLLINEAR_SUBSET THEN
6866 EXISTS_TAC `{b:real^N,c,a,d}` THEN ASM_SIMP_TAC[COLLINEAR_4_3] THEN
6867 CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN
6868 REPEAT(POP_ASSUM MP_TAC) THEN SIMP_TAC[INSERT_AC]);;
6870 let ORTHOGONAL_TO_ORTHOGONAL_2D = prove
6872 ~(x = vec 0) /\ orthogonal x y /\ orthogonal x z
6873 ==> collinear {vec 0,y,z}`,
6874 REWRITE_TAC[orthogonal; GSYM DOT_CAUCHY_SCHWARZ_EQUAL; GSYM DOT_EQ_0] THEN
6875 REWRITE_TAC[DOT_2] THEN CONV_TAC REAL_RING);;
6877 let COLLINEAR_3_2D = prove
6878 (`!x y z:real^2. collinear{x,y,z} <=>
6879 (z$1 - x$1) * (y$2 - x$2) = (y$1 - x$1) * (z$2 - x$2)`,
6880 ONCE_REWRITE_TAC[COLLINEAR_3] THEN
6881 REWRITE_TAC[GSYM DOT_CAUCHY_SCHWARZ_EQUAL] THEN
6882 REWRITE_TAC[DOT_2; VECTOR_SUB_COMPONENT] THEN CONV_TAC REAL_RING);;
6884 (* ------------------------------------------------------------------------- *)
6886 (* ------------------------------------------------------------------------- *)
6888 let between = new_definition
6889 `between x (a,b) <=> dist(a,b) = dist(a,x) + dist(x,b)`;;
6891 let BETWEEN_REFL = prove
6892 (`!a b. between a (a,b) /\ between b (a,b) /\ between a (a,a)`,
6893 REWRITE_TAC[between] THEN NORM_ARITH_TAC);;
6895 let BETWEEN_REFL_EQ = prove
6896 (`!a x. between x (a,a) <=> x = a`,
6897 REWRITE_TAC[between] THEN NORM_ARITH_TAC);;
6899 let BETWEEN_SYM = prove
6900 (`!a b x. between x (a,b) <=> between x (b,a)`,
6901 REWRITE_TAC[between] THEN NORM_ARITH_TAC);;
6903 let BETWEEN_ANTISYM = prove
6904 (`!a b c. between a (b,c) /\ between b (a,c) ==> a = b`,
6905 REWRITE_TAC[between; DIST_SYM] THEN NORM_ARITH_TAC);;
6907 let BETWEEN_TRANS = prove
6908 (`!a b c d. between a (b,c) /\ between d (a,c) ==> between d (b,c)`,
6909 REWRITE_TAC[between; DIST_SYM] THEN NORM_ARITH_TAC);;
6911 let BETWEEN_TRANS_2 = prove
6912 (`!a b c d. between a (b,c) /\ between d (a,b) ==> between a (c,d)`,
6913 REWRITE_TAC[between; DIST_SYM] THEN NORM_ARITH_TAC);;
6915 let BETWEEN_NORM = prove
6917 between x (a,b) <=> norm(x - a) % (b - x) = norm(b - x) % (x - a)`,
6918 REPEAT GEN_TAC THEN REWRITE_TAC[between; DIST_TRIANGLE_EQ] THEN
6919 REWRITE_TAC[NORM_SUB] THEN VECTOR_ARITH_TAC);;
6921 let BETWEEN_DOT = prove
6923 between x (a,b) <=> (x - a) dot (b - x) = norm(x - a) * norm(b - x)`,
6924 REWRITE_TAC[BETWEEN_NORM; NORM_CAUCHY_SCHWARZ_EQ]);;
6926 let BETWEEN_IMP_COLLINEAR = prove
6927 (`!a b x:real^N. between x (a,b) ==> collinear {a,x,b}`,
6928 REPEAT GEN_TAC THEN MAP_EVERY
6929 (fun t -> ASM_CASES_TAC t THEN
6930 TRY(ASM_REWRITE_TAC[INSERT_AC; COLLINEAR_2] THEN NO_TAC))
6931 [`x:real^N = a`; `x:real^N = b`; `a:real^N = b`] THEN
6932 ONCE_REWRITE_TAC[COLLINEAR_3; BETWEEN_NORM] THEN
6933 DISCH_TAC THEN REWRITE_TAC[COLLINEAR_LEMMA] THEN
6934 REPEAT DISJ2_TAC THEN EXISTS_TAC `--(norm(b - x:real^N) / norm(x - a))` THEN
6935 MATCH_MP_TAC VECTOR_MUL_LCANCEL_IMP THEN EXISTS_TAC `norm(x - a:real^N)` THEN
6936 ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RNEG] THEN
6937 ASM_SIMP_TAC[REAL_DIV_LMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN
6940 let COLLINEAR_BETWEEN_CASES = prove
6942 collinear {a,b,c} <=>
6943 between a (b,c) \/ between b (c,a) \/ between c (a,b)`,
6944 REPEAT STRIP_TAC THEN EQ_TAC THENL
6945 [REWRITE_TAC[COLLINEAR_3_EXPAND] THEN
6946 ASM_CASES_TAC `c:real^N = a` THEN ASM_REWRITE_TAC[BETWEEN_REFL] THEN
6947 STRIP_TAC THEN ASM_REWRITE_TAC[between; dist] THEN
6948 REWRITE_TAC[VECTOR_ARITH `(u % a + (&1 - u) % c) - c = --u % (c - a)`;
6949 VECTOR_ARITH `(u % a + (&1 - u) % c) - a = (&1 - u) % (c - a)`;
6950 VECTOR_ARITH `c - (u % a + (&1 - u) % c) = u % (c - a)`;
6951 VECTOR_ARITH `a - (u % a + (&1 - u) % c) = (u - &1) % (c - a)`] THEN
6952 REWRITE_TAC[NORM_MUL] THEN
6953 SUBST1_TAC(NORM_ARITH `norm(a - c:real^N) = norm(c - a)`) THEN
6954 REWRITE_TAC[REAL_ARITH `a * c + c = (a + &1) * c`; GSYM REAL_ADD_RDISTRIB;
6955 REAL_ARITH `c + a * c = (a + &1) * c`] THEN
6956 ASM_REWRITE_TAC[REAL_EQ_MUL_RCANCEL;
6957 REAL_RING `n = x * n <=> n = &0 \/ x = &1`] THEN
6958 ASM_REWRITE_TAC[NORM_EQ_0; VECTOR_SUB_EQ] THEN REAL_ARITH_TAC;
6959 DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN (MP_TAC o MATCH_MP
6960 BETWEEN_IMP_COLLINEAR)) THEN
6961 REWRITE_TAC[INSERT_AC]]);;
6963 let COLLINEAR_DIST_BETWEEN = prove
6964 (`!a b x. collinear {x,a,b} /\
6965 dist(x,a) <= dist(a,b) /\ dist(x,b) <= dist(a,b)
6966 ==> between x (a,b)`,
6967 SIMP_TAC[COLLINEAR_BETWEEN_CASES; between; DIST_SYM] THEN NORM_ARITH_TAC);;
6969 let COLLINEAR_1 = prove
6970 (`!s:real^1->bool. collinear s`,
6971 GEN_TAC THEN MATCH_MP_TAC COLLINEAR_SUBSET THEN
6972 EXISTS_TAC `(vec 0:real^1) INSERT (vec 1) INSERT s` THEN
6973 CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN
6974 W(MP_TAC o PART_MATCH (lhs o rand) COLLINEAR_TRIPLES o snd) THEN
6975 REWRITE_TAC[VEC_EQ; ARITH_EQ] THEN DISCH_THEN SUBST1_TAC THEN
6976 REWRITE_TAC[COLLINEAR_BETWEEN_CASES] THEN
6977 REWRITE_TAC[between; DIST_REAL; GSYM drop; DROP_VEC; REAL_ABS_NUM] THEN
6980 (* ------------------------------------------------------------------------- *)
6981 (* Midpoint between two points. *)
6982 (* ------------------------------------------------------------------------- *)
6984 let midpoint = new_definition
6985 `midpoint(a,b) = inv(&2) % (a + b)`;;
6987 let MIDPOINT_REFL = prove
6988 (`!x. midpoint(x,x) = x`,
6989 REWRITE_TAC[midpoint] THEN VECTOR_ARITH_TAC);;
6991 let MIDPOINT_SYM = prove
6992 (`!a b. midpoint(a,b) = midpoint(b,a)`,
6993 REWRITE_TAC[midpoint; VECTOR_ADD_SYM]);;
6995 let DIST_MIDPOINT = prove
6996 (`!a b. dist(a,midpoint(a,b)) = dist(a,b) / &2 /\
6997 dist(b,midpoint(a,b)) = dist(a,b) / &2 /\
6998 dist(midpoint(a,b),a) = dist(a,b) / &2 /\
6999 dist(midpoint(a,b),b) = dist(a,b) / &2`,
7000 REWRITE_TAC[midpoint] THEN NORM_ARITH_TAC);;
7002 let MIDPOINT_EQ_ENDPOINT = prove
7003 (`!a b. (midpoint(a,b) = a <=> a = b) /\
7004 (midpoint(a,b) = b <=> a = b) /\
7005 (a = midpoint(a,b) <=> a = b) /\
7006 (b = midpoint(a,b) <=> a = b)`,
7007 REWRITE_TAC[midpoint] THEN NORM_ARITH_TAC);;
7009 let BETWEEN_MIDPOINT = prove
7010 (`!a b. between (midpoint(a,b)) (a,b) /\ between (midpoint(a,b)) (b,a)`,
7011 REWRITE_TAC[between; midpoint] THEN NORM_ARITH_TAC);;
7013 let MIDPOINT_LINEAR_IMAGE = prove
7014 (`!f a b. linear f ==> midpoint(f a,f b) = f(midpoint(a,b))`,
7015 SIMP_TAC[midpoint; LINEAR_ADD; LINEAR_CMUL]);;
7017 let COLLINEAR_MIDPOINT = prove
7018 (`!a b. collinear{a,midpoint(a,b),b}`,
7019 REPEAT GEN_TAC THEN REWRITE_TAC[COLLINEAR_3_EXPAND; midpoint] THEN
7020 DISJ2_TAC THEN EXISTS_TAC `&1 / &2` THEN VECTOR_ARITH_TAC);;
7022 let MIDPOINT_COLLINEAR = prove
7025 ==> (b = midpoint(a,c) <=> collinear{a,b,c} /\ dist(a,b) = dist(b,c))`,
7026 REPEAT STRIP_TAC THEN
7027 MATCH_MP_TAC(TAUT `(a ==> b) /\ (b ==> (a <=> c)) ==> (a <=> b /\ c)`) THEN
7028 SIMP_TAC[COLLINEAR_MIDPOINT] THEN ASM_REWRITE_TAC[COLLINEAR_3_EXPAND] THEN
7029 STRIP_TAC THEN ASM_REWRITE_TAC[midpoint; dist] THEN
7031 [VECTOR_ARITH `a - (u % a + (&1 - u) % c) = (&1 - u) % (a - c)`;
7032 VECTOR_ARITH `(u % a + (&1 - u) % c) - c = u % (a - c)`;
7033 VECTOR_ARITH `u % a + (&1 - u) % c = inv (&2) % (a + c) <=>
7034 (u - &1 / &2) % (a - c) = vec 0`] THEN
7035 ASM_SIMP_TAC[NORM_MUL; REAL_EQ_MUL_RCANCEL; NORM_EQ_0; VECTOR_SUB_EQ;
7036 VECTOR_MUL_EQ_0] THEN
7039 (* ------------------------------------------------------------------------- *)
7040 (* General "one way" lemma for properties preserved by injective map. *)
7041 (* ------------------------------------------------------------------------- *)
7043 let WLOG_LINEAR_INJECTIVE_IMAGE_2 = prove
7044 (`!P Q. (!f s. P s /\ linear f ==> Q(IMAGE f s)) /\
7045 (!g t. Q t /\ linear g ==> P(IMAGE g t))
7046 ==> !f:real^M->real^N.
7047 linear f /\ (!x y. f x = f y ==> x = y)
7048 ==> !s. Q(IMAGE f s) <=> P s`,
7049 REPEAT STRIP_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[] THEN DISCH_TAC THEN
7050 MP_TAC(ISPEC `f:real^M->real^N` LINEAR_INJECTIVE_LEFT_INVERSE) THEN
7051 ASM_REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN
7052 DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^M` STRIP_ASSUME_TAC) THEN
7053 FIRST_X_ASSUM(MP_TAC o SPECL
7054 [`g:real^N->real^M`; `IMAGE (f:real^M->real^N) s`]) THEN
7055 ASM_REWRITE_TAC[GSYM IMAGE_o; o_DEF; IMAGE_ID]);;
7057 let WLOG_LINEAR_INJECTIVE_IMAGE_2_ALT = prove
7058 (`!P Q f s. (!h u. P u /\ linear h ==> Q(IMAGE h u)) /\
7059 (!g t. Q t /\ linear g ==> P(IMAGE g t)) /\
7060 linear f /\ (!x y. f x = f y ==> x = y)
7061 ==> (Q(IMAGE f s) <=> P s)`,
7062 REPEAT GEN_TAC THEN STRIP_TAC THEN
7063 MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]
7064 WLOG_LINEAR_INJECTIVE_IMAGE_2) THEN
7065 ASM_REWRITE_TAC[]);;
7067 let WLOG_LINEAR_INJECTIVE_IMAGE = prove
7068 (`!P. (!f s. P s /\ linear f ==> P(IMAGE f s))
7069 ==> !f:real^N->real^N. linear f /\ (!x y. f x = f y ==> x = y)
7070 ==> !s. P(IMAGE f s) <=> P s`,
7071 GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC WLOG_LINEAR_INJECTIVE_IMAGE_2 THEN
7072 ASM_REWRITE_TAC[]);;
7074 let WLOG_LINEAR_INJECTIVE_IMAGE_ALT = prove
7075 (`!P f s. (!g t. P t /\ linear g ==> P(IMAGE g t)) /\
7076 linear f /\ (!x y. f x = f y ==> x = y)
7077 ==> (P(IMAGE f s) <=> P s)`,
7078 REPEAT GEN_TAC THEN STRIP_TAC THEN
7079 MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]
7080 WLOG_LINEAR_INJECTIVE_IMAGE) THEN
7081 ASM_REWRITE_TAC[]);;
7083 (* ------------------------------------------------------------------------- *)
7084 (* Inference rule to apply it conveniently. *)
7086 (* |- !f s. P s /\ linear f ==> P(IMAGE f s) [or /\ commuted] *)
7087 (* --------------------------------------------------------------- *)
7088 (* |- !f s. linear f /\ (!x y. f x = f y ==> x = y) *)
7089 (* ==> (Q(IMAGE f s) <=> P s) *)
7090 (* ------------------------------------------------------------------------- *)
7092 let LINEAR_INVARIANT_RULE th =
7093 let [f;s] = fst(strip_forall(concl th)) in
7094 let (rm,rn) = dest_fun_ty (type_of f) in
7095 let m = last(snd(dest_type rm)) and n = last(snd(dest_type rn)) in
7096 let th' = INST_TYPE [m,n; n,m] th in
7097 let th0 = CONJ th th' in
7098 let th1 = try MATCH_MP WLOG_LINEAR_INJECTIVE_IMAGE_2 th0
7100 MATCH_MP WLOG_LINEAR_INJECTIVE_IMAGE_2
7101 (GEN_REWRITE_RULE (BINOP_CONV o ONCE_DEPTH_CONV) [CONJ_SYM] th0) in
7102 GEN_REWRITE_RULE BINDER_CONV [RIGHT_IMP_FORALL_THM] th1;;
7104 (* ------------------------------------------------------------------------- *)
7105 (* Immediate application. *)
7106 (* ------------------------------------------------------------------------- *)
7108 let SUBSPACE_LINEAR_IMAGE_EQ = prove
7109 (`!f s. linear f /\ (!x y. f x = f y ==> x = y)
7110 ==> (subspace (IMAGE f s) <=> subspace s)`,
7111 MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE SUBSPACE_LINEAR_IMAGE));;
7113 (* ------------------------------------------------------------------------- *)
7114 (* Storage of useful "invariance under linear map / translation" theorems. *)
7115 (* ------------------------------------------------------------------------- *)
7117 let invariant_under_linear = ref([]:thm list);;
7119 let invariant_under_translation = ref([]:thm list);;
7121 let scaling_theorems = ref([]:thm list);;
7123 (* ------------------------------------------------------------------------- *)
7124 (* Scaling theorems and derivation from linear invariance. *)
7125 (* ------------------------------------------------------------------------- *)
7127 let LINEAR_SCALING = prove
7128 (`!c. linear(\x:real^N. c % x)`,
7129 REWRITE_TAC[linear] THEN VECTOR_ARITH_TAC);;
7131 let INJECTIVE_SCALING = prove
7132 (`!c. (!x y:real^N. c % x = c % y ==> x = y) <=> ~(c = &0)`,
7133 GEN_TAC THEN REWRITE_TAC[VECTOR_MUL_LCANCEL] THEN
7134 ASM_CASES_TAC `c:real = &0` THEN ASM_REWRITE_TAC[] THEN
7135 DISCH_THEN(MP_TAC o SPECL [`vec 0:real^N`; `vec 1:real^N`]) THEN
7136 REWRITE_TAC[VEC_EQ; ARITH]);;
7138 let SURJECTIVE_SCALING = prove
7139 (`!c. (!y:real^N. ?x. c % x = y) <=> ~(c = &0)`,
7140 ASM_SIMP_TAC[LINEAR_SURJECTIVE_IFF_INJECTIVE; LINEAR_SCALING] THEN
7141 REWRITE_TAC[INJECTIVE_SCALING]);;
7143 let SCALING_INVARIANT =
7144 let pths = (CONJUNCTS o UNDISCH o prove)
7146 ==> linear(\x:real^N. c % x) /\
7147 (!x y:real^N. c % x = c % y ==> x = y) /\
7148 (!y:real^N. ?x. c % x = y)`,
7149 SIMP_TAC[REAL_LT_IMP_NZ; LINEAR_SCALING;
7150 INJECTIVE_SCALING; SURJECTIVE_SCALING])
7151 and sc_tm = `\x:real^N. c % x`
7152 and sa_tm = `&0:real < c`
7153 and c_tm = `c:real` in
7155 let ith = BETA_RULE(ISPEC sc_tm th) in
7156 let avs,bod = strip_forall(concl ith) in
7157 let cjs = conjuncts(lhand bod) in
7158 let cths = map (fun t -> find(fun th -> aconv (concl th) t) pths) cjs in
7159 let oth = MP (SPECL avs ith) (end_itlist CONJ cths) in
7160 GEN c_tm (DISCH sa_tm (GENL avs oth));;
7162 let scaling_theorems = ref([]:thm list);;
7164 (* ------------------------------------------------------------------------- *)
7165 (* Augmentation of the lists. The "add_linear_invariants" also updates *)
7166 (* the scaling theorems automatically, so only a few of those will need *)
7167 (* to be added explicitly. *)
7168 (* ------------------------------------------------------------------------- *)
7170 let add_scaling_theorems thl =
7171 (scaling_theorems := (!scaling_theorems) @ thl);;
7173 let add_linear_invariants thl =
7174 ignore(mapfilter (fun th -> add_scaling_theorems[SCALING_INVARIANT th]) thl);
7175 (invariant_under_linear := (!invariant_under_linear) @ thl);;
7177 let add_translation_invariants thl =
7178 (invariant_under_translation := (!invariant_under_translation) @ thl);;
7180 (* ------------------------------------------------------------------------- *)
7181 (* Start with some basic set equivalences. *)
7182 (* We give them all an injectivity hypothesis even if it's not necessary. *)
7183 (* For just the intersection theorem we add surjectivity (more manageable *)
7184 (* than assuming that the set isn't empty). *)
7185 (* ------------------------------------------------------------------------- *)
7188 (`!f. (!x y. f x = f y ==> x = y)
7189 ==> (if p then f x else f y) = f(if p then x else y) /\
7190 (if p then IMAGE f s else IMAGE f t) =
7191 IMAGE f (if p then s else t) /\
7192 (f x) INSERT (IMAGE f s) = IMAGE f (x INSERT s) /\
7193 (IMAGE f s) DELETE (f x) = IMAGE f (s DELETE x) /\
7194 (IMAGE f s) INTER (IMAGE f t) = IMAGE f (s INTER t) /\
7195 (IMAGE f s) UNION (IMAGE f t) = IMAGE f (s UNION t) /\
7196 UNIONS(IMAGE (IMAGE f) u) = IMAGE f (UNIONS u) /\
7197 (IMAGE f s) DIFF (IMAGE f t) = IMAGE f (s DIFF t) /\
7198 ((f x) IN (IMAGE f s) <=> x IN s) /\
7199 ((f o xs) (n:num) = f(xs n)) /\
7200 ((f o pt) (tt:real^1) = f(pt tt)) /\
7201 (DISJOINT (IMAGE f s) (IMAGE f t) <=> DISJOINT s t) /\
7202 ((IMAGE f s) SUBSET (IMAGE f t) <=> s SUBSET t) /\
7203 ((IMAGE f s) PSUBSET (IMAGE f t) <=> s PSUBSET t) /\
7204 (IMAGE f s = IMAGE f t <=> s = t) /\
7205 ((IMAGE f s) HAS_SIZE n <=> s HAS_SIZE n) /\
7206 (FINITE(IMAGE f s) <=> FINITE s) /\
7207 (INFINITE(IMAGE f s) <=> INFINITE s)`,
7208 REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[IMAGE_UNIONS] THEN
7209 REWRITE_TAC[o_THM] THEN
7210 REPLICATE_TAC 2 (CONJ_TAC THENL [MESON_TAC[]; ALL_TAC]) THEN
7211 REWRITE_TAC[INFINITE; TAUT `(~p <=> ~q) <=> (p <=> q)`] THEN
7212 REPLICATE_TAC 10 (CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
7213 REWRITE_TAC[HAS_SIZE] THEN
7214 ASM_MESON_TAC[FINITE_IMAGE_INJ_EQ; CARD_IMAGE_INJ]) in
7215 let f = `f:real^M->real^N`
7216 and imf = `IMAGE (f:real^M->real^N)`
7218 and ima = `IMAGE (\x:real^N. a + x)`
7219 and vth = VECTOR_ARITH `!x y. a + x:real^N = a + y ==> x = y` in
7220 let th1 = UNDISCH(ISPEC f th_sets)
7222 (GEN_REWRITE_RULE LAND_CONV [INJECTIVE_IMAGE] (ISPEC imf th_sets))
7223 and th2 = MATCH_MP th_sets vth
7225 (BETA_RULE(GEN_REWRITE_RULE LAND_CONV [INJECTIVE_IMAGE] (ISPEC ima th_sets)))
7227 let fn a th = GENL (a::subtract (frees(concl th)) [a]) th in
7228 add_linear_invariants(map (fn f o DISCH_ALL) (CONJUNCTS th1 @ CONJUNCTS th1')),
7229 add_translation_invariants(map (fn a) (CONJUNCTS th2 @ CONJUNCTS th2'));;
7232 (`!f:A->B s. (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
7233 ==> INTERS (IMAGE (IMAGE f) s) = IMAGE f (INTERS s)`,
7234 REWRITE_TAC[INTERS_IMAGE] THEN SET_TAC[]) in
7237 INTERS (IMAGE (IMAGE (\x. a + x)) s) = IMAGE (\x. a + x) (INTERS s)`,
7238 REPEAT GEN_TAC THEN MATCH_MP_TAC th_set THEN
7239 REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
7240 REWRITE_TAC[VECTOR_ARITH `a + x:real^N = y <=> x = y - a`; EXISTS_REFL]) in
7241 add_linear_invariants [th_set],add_translation_invariants[th_vec];;
7243 (* ------------------------------------------------------------------------- *)
7244 (* Now add arithmetical equivalences. *)
7245 (* ------------------------------------------------------------------------- *)
7247 let PRESERVES_NORM_PRESERVES_DOT = prove
7248 (`!f:real^M->real^N x y.
7249 linear f /\ (!x. norm(f x) = norm x)
7250 ==> (f x) dot (f y) = x dot y`,
7251 REWRITE_TAC[NORM_EQ] THEN REPEAT STRIP_TAC THEN
7252 FIRST_ASSUM(MP_TAC o SPEC `x + y:real^M`) THEN
7253 FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_ADD th]) THEN
7254 ASM_REWRITE_TAC[DOT_LADD; DOT_RADD] THEN
7255 REWRITE_TAC[DOT_SYM] THEN REAL_ARITH_TAC);;
7257 let PRESERVES_NORM_INJECTIVE = prove
7258 (`!f:real^M->real^N.
7259 linear f /\ (!x. norm(f x) = norm x)
7260 ==> !x y. f x = f y ==> x = y`,
7261 SIMP_TAC[LINEAR_INJECTIVE_0; GSYM NORM_EQ_0]);;
7263 let ORTHOGONAL_LINEAR_IMAGE_EQ = prove
7264 (`!f:real^M->real^N x y.
7265 linear f /\ (!x. norm(f x) = norm x)
7266 ==> (orthogonal (f x) (f y) <=> orthogonal x y)`,
7267 SIMP_TAC[orthogonal; PRESERVES_NORM_PRESERVES_DOT]);;
7269 add_linear_invariants
7274 MIDPOINT_LINEAR_IMAGE;
7275 MESON[] `!f:real^M->real^N x.
7276 (!x. norm(f x) = norm x) ==> norm(f x) = norm x`;
7277 PRESERVES_NORM_PRESERVES_DOT;
7278 MESON[dist; LINEAR_SUB]
7279 `!f:real^M->real^N x y.
7280 linear f /\ (!x. norm(f x) = norm x)
7281 ==> dist(f x,f y) = dist(x,y)`;
7282 MESON[] `!f:real^M->real^N x y.
7283 (!x y. f x = f y ==> x = y) ==> (f x = f y <=> x = y)`;
7284 SUBSPACE_LINEAR_IMAGE_EQ;
7285 ORTHOGONAL_LINEAR_IMAGE_EQ;
7287 DEPENDENT_LINEAR_IMAGE_EQ;
7288 INDEPENDENT_LINEAR_IMAGE_EQ;
7289 DIM_INJECTIVE_LINEAR_IMAGE];;
7291 add_translation_invariants
7292 [VECTOR_ARITH `!a x y. a + x:real^N = a + y <=> x = y`;
7293 NORM_ARITH `!a x y. dist(a + x,a + y) = dist(x,y)`;
7294 VECTOR_ARITH `!a x y. &1 / &2 % ((a + x) + (a + y)) = a + &1 / &2 % (x + y)`;
7295 VECTOR_ARITH `!a x y. inv(&2) % ((a + x) + (a + y)) = a + inv(&2) % (x + y)`;
7296 VECTOR_ARITH `!a x y. (a + x) - (a + y):real^N = x - y`;
7297 (EQT_ELIM o (REWRITE_CONV[midpoint] THENC(EQT_INTRO o NORM_ARITH)))
7298 `!a x y. midpoint(a + x,a + y) = a + midpoint(x,y)`;
7299 (EQT_ELIM o (REWRITE_CONV[between] THENC(EQT_INTRO o NORM_ARITH)))
7300 `!a x y z. between (a + x) (a + y,a + z) <=> between x (y,z)`];;
7302 (* ------------------------------------------------------------------------- *)
7303 (* A few for lists. *)
7304 (* ------------------------------------------------------------------------- *)
7306 let MEM_TRANSLATION = prove
7307 (`!a:real^N x l. MEM (a + x) (MAP (\x. a + x) l) <=> MEM x l`,
7308 REWRITE_TAC[MEM_MAP; VECTOR_ARITH `a + x:real^N = a + y <=> x = y`] THEN
7311 add_translation_invariants [MEM_TRANSLATION];;
7313 let MEM_LINEAR_IMAGE = prove
7314 (`!f:real^M->real^N x l.
7315 linear f /\ (!x y. f x = f y ==> x = y)
7316 ==> (MEM (f x) (MAP f l) <=> MEM x l)`,
7317 REWRITE_TAC[MEM_MAP] THEN MESON_TAC[]);;
7319 add_linear_invariants [MEM_LINEAR_IMAGE];;
7321 let LENGTH_TRANSLATION = prove
7322 (`!a:real^N l. LENGTH(MAP (\x. a + x) l) = LENGTH l`,
7323 REWRITE_TAC[LENGTH_MAP]) in
7324 add_translation_invariants [LENGTH_TRANSLATION];;
7326 let LENGTH_LINEAR_IMAGE = prove
7327 (`!f:real^M->real^N l. linear f ==> LENGTH(MAP f l) = LENGTH l`,
7328 REWRITE_TAC[LENGTH_MAP]) in
7329 add_linear_invariants [LENGTH_LINEAR_IMAGE];;
7331 let CONS_TRANSLATION = prove
7333 CONS ((\x. a + x) h) (MAP (\x. a + x) t) = MAP (\x. a + x) (CONS h t)`,
7334 REWRITE_TAC[MAP]) in
7335 add_translation_invariants [CONS_TRANSLATION];;
7337 let CONS_LINEAR_IMAGE = prove
7338 (`!f:real^M->real^N h t.
7339 linear f ==> CONS (f h) (MAP f t) = MAP f (CONS h t)`,
7340 REWRITE_TAC[MAP]) in
7341 add_linear_invariants [CONS_LINEAR_IMAGE];;
7343 let APPEND_TRANSLATION = prove
7345 APPEND (MAP (\x. a + x) l1) (MAP (\x. a + x) l2) =
7346 MAP (\x. a + x) (APPEND l1 l2)`,
7347 REWRITE_TAC[MAP_APPEND]) in
7348 add_translation_invariants [APPEND_TRANSLATION];;
7350 let APPEND_LINEAR_IMAGE = prove
7351 (`!f:real^M->real^N l1 l2.
7352 linear f ==> APPEND (MAP f l1) (MAP f l2) = MAP f (APPEND l1 l2)`,
7353 REWRITE_TAC[MAP_APPEND]) in
7354 add_linear_invariants [APPEND_LINEAR_IMAGE];;
7356 let REVERSE_TRANSLATION = prove
7357 (`!a:real^N l. REVERSE(MAP (\x. a + x) l) = MAP (\x. a + x) (REVERSE l)`,
7358 REWRITE_TAC[MAP_REVERSE]) in
7359 add_translation_invariants [REVERSE_TRANSLATION];;
7361 let REVERSE_LINEAR_IMAGE = prove
7362 (`!f:real^M->real^N l. linear f ==> REVERSE(MAP f l) = MAP f (REVERSE l)`,
7363 REWRITE_TAC[MAP_REVERSE]) in
7364 add_linear_invariants [REVERSE_LINEAR_IMAGE];;
7366 (* ------------------------------------------------------------------------- *)
7367 (* A few scaling theorems that don't come from invariance theorems. Most are *)
7368 (* artificially weak with 0 < c hypotheses, so we don't bind them to names. *)
7369 (* ------------------------------------------------------------------------- *)
7371 let DOT_SCALING = prove
7372 (`!c. &0 < c ==> !x y. (c % x) dot (c % y) = c pow 2 * (x dot y)`,
7373 REWRITE_TAC[DOT_LMUL; DOT_RMUL] THEN REAL_ARITH_TAC) in
7374 add_scaling_theorems [DOT_SCALING];;
7376 let DIST_SCALING = prove
7377 (`!c. &0 < c ==> !x y. dist(c % x,c % y) = c * dist(x,y)`,
7378 SIMP_TAC[DIST_MUL; REAL_ARITH `&0 < c ==> abs c = c`]) in
7379 add_scaling_theorems [DIST_SCALING];;
7381 let ORTHOGONAL_SCALING = prove
7382 (`!c. &0 < c ==> !x y. orthogonal (c % x) (c % y) <=> orthogonal x y`,
7383 REWRITE_TAC[orthogonal; DOT_LMUL; DOT_RMUL] THEN CONV_TAC REAL_FIELD) in
7384 add_scaling_theorems [ORTHOGONAL_SCALING];;
7386 let NORM_SCALING = prove
7387 (`!c. &0 < c ==> !x. norm(c % x) = c * norm x`,
7388 SIMP_TAC[NORM_MUL; REAL_ARITH `&0 < c ==> abs c = c`]) in
7389 add_scaling_theorems [NORM_SCALING];;
7391 add_scaling_theorems
7392 [REAL_ARITH `!c. &0 < c ==> !a b. a * c * b = c * a * b`;
7393 REAL_ARITH `!c. &0 < c ==> !a b. c * a + c * b = c * (a + b)`;
7394 REAL_ARITH `!c. &0 < c ==> !a b. c * a - c * b = c * (a - b)`;
7395 REAL_FIELD `!c. &0 < c ==> !a b. c * a = c * b <=> a = b`;
7396 MESON[REAL_LT_LMUL_EQ] `!c. &0 < c ==> !a b. c * a < c * b <=> a < b`;
7397 MESON[REAL_LE_LMUL_EQ] `!c. &0 < c ==> !a b. c * a <= c * b <=> a <= b`;
7398 MESON[REAL_LT_LMUL_EQ; real_gt]
7399 `!c. &0 < c ==> !a b. c * a > c * b <=> a > b`;
7400 MESON[REAL_LE_LMUL_EQ; real_ge]
7401 `!c. &0 < c ==> !a b. c * a >= c * b <=> a >= b`;
7403 `!c. &0 < c ==> !a n. (c * a) pow n = c pow n * a pow n`;
7404 REAL_ARITH `!c. &0 < c ==> !a b n. a * c pow n * b = c pow n * a * b`;
7406 `!c. &0 < c ==> !a b n. c pow n * a + c pow n * b = c pow n * (a + b)`;
7408 `!c. &0 < c ==> !a b n. c pow n * a - c pow n * b = c pow n * (a - b)`;
7409 MESON[REAL_POW_LT; REAL_EQ_LCANCEL_IMP; REAL_LT_IMP_NZ]
7410 `!c. &0 < c ==> !a b n. c pow n * a = c pow n * b <=> a = b`;
7411 MESON[REAL_LT_LMUL_EQ; REAL_POW_LT]
7412 `!c. &0 < c ==> !a b n. c pow n * a < c pow n * b <=> a < b`;
7413 MESON[REAL_LE_LMUL_EQ; REAL_POW_LT]
7414 `!c. &0 < c ==> !a b n. c pow n * a <= c pow n * b <=> a <= b`;
7415 MESON[REAL_LT_LMUL_EQ; real_gt; REAL_POW_LT]
7416 `!c. &0 < c ==> !a b n. c pow n * a > c pow n * b <=> a > b`;
7417 MESON[REAL_LE_LMUL_EQ; real_ge; REAL_POW_LT]
7418 `!c. &0 < c ==> !a b n. c pow n * a >= c pow n * b <=> a >= b`];;
7420 (* ------------------------------------------------------------------------- *)
7421 (* Theorem deducing quantifier mappings from surjectivity. *)
7422 (* ------------------------------------------------------------------------- *)
7424 let QUANTIFY_SURJECTION_THM = prove
7427 ==> ((!P. (!x. P x) <=> (!x. P (f x))) /\
7428 (!P. (?x. P x) <=> (?x. P (f x))) /\
7429 (!Q. (!s. Q s) <=> (!s. Q(IMAGE f s))) /\
7430 (!Q. (?s. Q s) <=> (?s. Q(IMAGE f s)))) /\
7431 (!P. {x | P x} = IMAGE f {x | P(f x)})`,
7432 GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [SURJECTIVE_RIGHT_INVERSE] THEN
7433 DISCH_THEN(X_CHOOSE_TAC `g:B->A`) THEN
7434 SUBGOAL_THEN `!s. IMAGE (f:A->B) (IMAGE g s) = s` ASSUME_TAC THENL
7435 [ASM SET_TAC[]; CONJ_TAC THENL [ASM MESON_TAC[]; ASM SET_TAC[]]]);;
7437 let QUANTIFY_SURJECTION_HIGHER_THM = prove
7440 ==> ((!P. (!x. P x) <=> (!x. P (f x))) /\
7441 (!P. (?x. P x) <=> (?x. P (f x))) /\
7442 (!Q. (!s. Q s) <=> (!s. Q(IMAGE f s))) /\
7443 (!Q. (?s. Q s) <=> (?s. Q(IMAGE f s))) /\
7444 (!Q. (!s. Q s) <=> (!s. Q(IMAGE (IMAGE f) s))) /\
7445 (!Q. (?s. Q s) <=> (?s. Q(IMAGE (IMAGE f) s))) /\
7446 (!P. (!g:real^1->B. P g) <=> (!g. P(f o g))) /\
7447 (!P. (?g:real^1->B. P g) <=> (?g. P(f o g))) /\
7448 (!P. (!g:num->B. P g) <=> (!g. P(f o g))) /\
7449 (!P. (?g:num->B. P g) <=> (?g. P(f o g))) /\
7450 (!Q. (!l. Q l) <=> (!l. Q(MAP f l))) /\
7451 (!Q. (?l. Q l) <=> (?l. Q(MAP f l)))) /\
7452 ((!P. {x | P x} = IMAGE f {x | P(f x)}) /\
7453 (!Q. {s | Q s} = IMAGE (IMAGE f) {s | Q(IMAGE f s)}) /\
7454 (!R. {l | R l} = IMAGE (MAP f) {l | R(MAP f l)}))`,
7455 GEN_TAC THEN DISCH_TAC THEN CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
7456 ASM_REWRITE_TAC[GSYM SURJECTIVE_FORALL_THM; GSYM SURJECTIVE_EXISTS_THM;
7457 GSYM SURJECTIVE_IMAGE_THM; SURJECTIVE_IMAGE; SURJECTIVE_MAP] THEN
7458 REWRITE_TAC[FUN_EQ_THM; o_THM; GSYM SKOLEM_THM] THEN ASM_MESON_TAC[]);;
7460 (* ------------------------------------------------------------------------- *)
7461 (* Apply such quantifier and set expansions once per level at depth. *)
7462 (* In the PARTIAL version, avoid expanding named variables in list. *)
7463 (* ------------------------------------------------------------------------- *)
7465 let PARTIAL_EXPAND_QUANTS_CONV avoid th =
7466 let ath,sth = CONJ_PAIR th in
7467 let conv1 = GEN_REWRITE_CONV I [ath]
7468 and conv2 = GEN_REWRITE_CONV I [sth] in
7470 let th = conv1 tm in
7471 if mem (fst(dest_var(fst(dest_abs(rand tm))))) avoid
7472 then failwith "Not going to expand this variable" else th in
7474 ((conv1' THENC BINDER_CONV conv) ORELSEC
7476 RAND_CONV(RAND_CONV(ABS_CONV(BINDER_CONV(LAND_CONV conv))))) ORELSEC
7477 SUB_CONV conv) tm in
7480 let EXPAND_QUANTS_CONV = PARTIAL_EXPAND_QUANTS_CONV [];;