1 (* ========================================================================= *)
2 (* Various convenient background stuff. *)
4 (* (c) Copyright, John Harrison 1998-2008 *)
5 (* ========================================================================= *)
9 (* ------------------------------------------------------------------------- *)
10 (* A couple of extra tactics used in some proofs below. *)
11 (* ------------------------------------------------------------------------- *)
14 SUBGOAL_THEN tm STRIP_ASSUME_TAC;;
17 MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC tm THEN CONJ_TAC;;
19 (* ------------------------------------------------------------------------- *)
20 (* Miscellaneous lemmas. *)
21 (* ------------------------------------------------------------------------- *)
23 let EXISTS_DIFF = prove
24 (`(?s:A->bool. P(UNIV DIFF s)) <=> (?s. P s)`,
25 MESON_TAC[prove(`UNIV DIFF (UNIV DIFF s) = s`,SET_TAC[])]);;
29 REWRITE_TAC[GE; LE_REFL]);;
31 let FORALL_SUC = prove
32 (`(!n. ~(n = 0) ==> P n) <=> (!n. P(SUC n))`,
33 MESON_TAC[num_CASES; NOT_SUC]);;
35 let SEQ_MONO_LEMMA = prove
36 (`!d e. (!n. n >= m ==> d(n) < e(n)) /\ (!n. n >= m ==> e(n) <= e(m))
37 ==> !n:num. n >= m ==> d(n) < e(m)`,
38 MESON_TAC[GE; REAL_LTE_TRANS]);;
41 (`(!e. &0 < e / &2 <=> &0 < e) /\
42 (!e. e / &2 + e / &2 = e) /\
43 (!e. &2 * (e / &2) = e)`,
46 let UPPER_BOUND_FINITE_SET = prove
47 (`!f:(A->num) s. FINITE(s) ==> ?a. !x. x IN s ==> f(x) <= a`,
48 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
49 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
50 MESON_TAC[LE_CASES; LE_REFL; LE_TRANS]);;
52 let UPPER_BOUND_FINITE_SET_REAL = prove
53 (`!f:(A->real) s. FINITE(s) ==> ?a. !x. x IN s ==> f(x) <= a`,
54 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
55 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
56 MESON_TAC[REAL_LE_TOTAL; REAL_LE_REFL; REAL_LE_TRANS]);;
58 let LOWER_BOUND_FINITE_SET = prove
59 (`!f:(A->num) s. FINITE(s) ==> ?a. !x. x IN s ==> a <= f(x)`,
60 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
61 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
62 MESON_TAC[LE_CASES; LE_REFL; LE_TRANS]);;
64 let LOWER_BOUND_FINITE_SET_REAL = prove
65 (`!f:(A->real) s. FINITE(s) ==> ?a. !x. x IN s ==> a <= f(x)`,
66 GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
67 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN
68 MESON_TAC[REAL_LE_TOTAL; REAL_LE_REFL; REAL_LE_TRANS]);;
70 let REAL_CONVEX_BOUND2_LT = prove
71 (`!x y a u v. x < a /\ y < b /\ &0 <= u /\ &0 <= v /\ u + v = &1
72 ==> u * x + v * y < u * a + v * b`,
73 REPEAT GEN_TAC THEN ASM_CASES_TAC `u = &0` THENL
74 [ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_LID] THEN REPEAT STRIP_TAC;
75 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LTE_ADD2 THEN
76 ASM_SIMP_TAC[REAL_LE_LMUL; REAL_LT_IMP_LE]] THEN
77 MATCH_MP_TAC REAL_LT_LMUL THEN ASM_REAL_ARITH_TAC);;
79 let REAL_CONVEX_BOUND_LT = prove
80 (`!x y a u v. x < a /\ y < a /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
81 ==> u * x + v * y < a`,
82 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
83 EXISTS_TAC `u * a + v * a:real` THEN CONJ_TAC THENL
84 [ASM_SIMP_TAC[REAL_CONVEX_BOUND2_LT];
85 MATCH_MP_TAC REAL_EQ_IMP_LE THEN
86 UNDISCH_TAC `u + v = &1` THEN CONV_TAC REAL_RING]);;
88 let REAL_CONVEX_BOUND_LE = prove
89 (`!x y a u v. x <= a /\ y <= a /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
90 ==> u * x + v * y <= a`,
92 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(u + v) * a` THEN
93 CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[REAL_LE_REFL; REAL_MUL_LID]] THEN
94 ASM_SIMP_TAC[REAL_ADD_RDISTRIB; REAL_LE_ADD2; REAL_LE_LMUL]);;
96 let INFINITE_ENUMERATE = prove
99 ==> ?r:num->num. (!m n. m < n ==> r(m) < r(n)) /\ (!n. r n IN s)`,
100 GEN_TAC THEN REWRITE_TAC[INFINITE; num_FINITE; NOT_EXISTS_THM] THEN
101 REWRITE_TAC[NOT_FORALL_THM; NOT_IMP; NOT_LE; SKOLEM_THM] THEN
102 DISCH_THEN(X_CHOOSE_TAC `next:num->num`) THEN
103 (MP_TAC o prove_recursive_functions_exist num_RECURSION)
104 `(f(0) = next 0) /\ (!n. f(SUC n) = next(f n))` THEN
105 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN CONJ_TAC THENL
106 [GEN_TAC; ALL_TAC] THEN
107 INDUCT_TAC THEN ASM_REWRITE_TAC[LT] THEN ASM_MESON_TAC[LT_TRANS]);;
109 let APPROACHABLE_LT_LE = prove
110 (`!P f. (?d. &0 < d /\ !x. f(x) < d ==> P x) =
111 (?d. &0 < d /\ !x. f(x) <= d ==> P x)`,
113 (`&0 < d ==> x <= d / &2 ==> x < d`,
114 SIMP_TAC[REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN REAL_ARITH_TAC) in
115 MESON_TAC[REAL_LT_IMP_LE; lemma; REAL_HALF]);;
117 let REAL_LE_BETWEEN = prove
118 (`!a b. a <= b <=> ?x. a <= x /\ x <= b`,
119 MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL]);;
121 let REAL_LT_BETWEEN = prove
122 (`!a b. a < b <=> ?x. a < x /\ x < b`,
123 REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_TRANS]] THEN
124 DISCH_TAC THEN EXISTS_TAC `(a + b) / &2` THEN
125 SIMP_TAC[REAL_LT_RDIV_EQ; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
126 POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
128 let TRIANGLE_LEMMA = prove
129 (`!x y z. &0 <= x /\ &0 <= y /\ &0 <= z /\ x pow 2 <= y pow 2 + z pow 2
132 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
133 ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
134 REWRITE_TAC[REAL_NOT_LE] THEN DISCH_TAC THEN
135 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `(y + z) pow 2` THEN
136 ASM_SIMP_TAC[REAL_POW_LT2; REAL_LE_ADD; ARITH_EQ] THEN
137 ASM_SIMP_TAC[REAL_LE_MUL; REAL_POW_2; REAL_ARITH
138 `x * x + y * y <= (x + y) * (x + y) <=> &0 <= x * y`]);;
140 let LAMBDA_SKOLEM = prove
141 (`(!i. 1 <= i /\ i <= dimindex(:N) ==> ?x. P i x) =
142 (?x:A^N. !i. 1 <= i /\ i <= dimindex(:N) ==> P i (x$i))`,
143 REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN EQ_TAC THENL
144 [DISCH_THEN(X_CHOOSE_TAC `x:num->A`) THEN
145 EXISTS_TAC `(lambda i. x i):A^N` THEN ASM_SIMP_TAC[LAMBDA_BETA];
146 DISCH_THEN(X_CHOOSE_TAC `x:A^N`) THEN
147 EXISTS_TAC `\i. (x:A^N)$i` THEN ASM_REWRITE_TAC[]]);;
149 let LAMBDA_PAIR = prove
150 (`(\(x,y). P x y) = (\p. P (FST p) (SND p))`,
151 REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM] THEN
152 CONV_TAC(ONCE_DEPTH_CONV GEN_BETA_CONV) THEN REWRITE_TAC[]);;
154 let EPSILON_DELTA_MINIMAL = prove
155 (`!P:real->A->bool Q.
157 (!d e x. Q x /\ &0 < e /\ e < d ==> P d x ==> P e x) /\
158 (!x. Q x ==> ?d. &0 < d /\ P d x)
159 ==> ?d. &0 < d /\ !x. Q x ==> P d x`,
160 REWRITE_TAC[IMP_IMP] THEN REPEAT STRIP_TAC THEN
161 ASM_CASES_TAC `{x:A | Q x} = {}` THENL
162 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
163 REWRITE_TAC[NOT_IN_EMPTY; IN_ELIM_THM] THEN
164 DISCH_TAC THEN EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[REAL_LT_01];
165 FIRST_X_ASSUM(MP_TAC o
166 GEN_REWRITE_RULE BINDER_CONV [RIGHT_IMP_EXISTS_THM]) THEN
167 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
168 X_GEN_TAC `d:A->real` THEN DISCH_TAC THEN
169 EXISTS_TAC `inf(IMAGE d {x:A | Q x})` THEN
170 ASM_SIMP_TAC[REAL_LT_INF_FINITE; FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
171 ASM_SIMP_TAC[FORALL_IN_IMAGE; FORALL_IN_GSPEC] THEN
172 X_GEN_TAC `a:A` THEN DISCH_TAC THEN
174 `&0 < inf(IMAGE d {x:A | Q x}) /\ inf(IMAGE d {x | Q x}) <= d a`
176 [ASM_SIMP_TAC[REAL_LT_INF_FINITE; REAL_INF_LE_FINITE;
177 FINITE_IMAGE; IMAGE_EQ_EMPTY] THEN
178 REWRITE_TAC[EXISTS_IN_IMAGE; FORALL_IN_IMAGE; IN_ELIM_THM] THEN
179 ASM_MESON_TAC[REAL_LE_REFL];
180 REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN
181 FIRST_X_ASSUM MATCH_MP_TAC THEN
182 EXISTS_TAC `(d:A->real) a` THEN ASM_SIMP_TAC[]]]);;
184 (* ------------------------------------------------------------------------- *)
185 (* A generic notion of "hull" (convex, affine, conic hull and closure). *)
186 (* ------------------------------------------------------------------------- *)
188 parse_as_infix("hull",(21,"left"));;
190 let hull = new_definition
191 `P hull s = INTERS {t | P t /\ s SUBSET t}`;;
194 (`!P s. P s ==> (P hull s = s)`,
195 REWRITE_TAC[hull; EXTENSION; IN_INTERS; IN_ELIM_THM] THEN
199 (`!P s. (!f. (!s. s IN f ==> P s) ==> P(INTERS f)) ==> P(P hull s)`,
200 REWRITE_TAC[hull] THEN SIMP_TAC[IN_ELIM_THM]);;
203 (`!P s. (!f. (!s. s IN f ==> P s) ==> P(INTERS f))
204 ==> ((P hull s = s) <=> P s)`,
205 MESON_TAC[P_HULL; HULL_P]);;
207 let HULL_HULL = prove
208 (`!P s. P hull (P hull s) = P hull s`,
209 REWRITE_TAC[hull; EXTENSION; IN_INTERS; IN_ELIM_THM; SUBSET] THEN
212 let HULL_SUBSET = prove
213 (`!P s. s SUBSET (P hull s)`,
214 REWRITE_TAC[hull; SUBSET; IN_INTERS; IN_ELIM_THM] THEN MESON_TAC[]);;
216 let HULL_MONO = prove
217 (`!P s t. s SUBSET t ==> (P hull s) SUBSET (P hull t)`,
218 REWRITE_TAC[hull; SUBSET; IN_INTERS; IN_ELIM_THM] THEN MESON_TAC[]);;
220 let HULL_ANTIMONO = prove
221 (`!P Q s. P SUBSET Q ==> (Q hull s) SUBSET (P hull s)`,
222 REWRITE_TAC[SUBSET; hull; IN_INTERS; IN_ELIM_THM] THEN MESON_TAC[IN]);;
224 let HULL_MINIMAL = prove
225 (`!P s t. s SUBSET t /\ P t ==> (P hull s) SUBSET t`,
226 REWRITE_TAC[hull; SUBSET; IN_INTERS; IN_ELIM_THM] THEN MESON_TAC[]);;
228 let SUBSET_HULL = prove
229 (`!P s t. P t ==> ((P hull s) SUBSET t <=> s SUBSET t)`,
230 REWRITE_TAC[hull; SUBSET; IN_INTERS; IN_ELIM_THM] THEN MESON_TAC[]);;
232 let HULL_UNIQUE = prove
233 (`!P s t. s SUBSET t /\ P t /\ (!t'. s SUBSET t' /\ P t' ==> t SUBSET t')
235 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
236 REWRITE_TAC[hull; SUBSET; IN_INTERS; IN_ELIM_THM] THEN
237 ASM_MESON_TAC[SUBSET_HULL; SUBSET]);;
239 let HULL_UNION_SUBSET = prove
240 (`!P s t. (P hull s) UNION (P hull t) SUBSET (P hull (s UNION t))`,
241 SIMP_TAC[UNION_SUBSET; HULL_MONO; SUBSET_UNION]);;
243 let HULL_UNION = prove
244 (`!P s t. P hull (s UNION t) = P hull (P hull s UNION P hull t)`,
245 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[hull] THEN
246 AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; UNION_SUBSET] THEN
247 MESON_TAC[SUBSET_HULL]);;
249 let HULL_UNION_LEFT = prove
251 P hull (s UNION t) = P hull (P hull s UNION t)`,
252 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[hull] THEN
253 AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; UNION_SUBSET] THEN
254 MESON_TAC[SUBSET_HULL]);;
256 let HULL_UNION_RIGHT = prove
258 P hull (s UNION t) = P hull (s UNION P hull t)`,
259 REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[hull] THEN
260 AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM; UNION_SUBSET] THEN
261 MESON_TAC[SUBSET_HULL]);;
263 let HULL_REDUNDANT_EQ = prove
264 (`!P a s. a IN (P hull s) <=> (P hull (a INSERT s) = P hull s)`,
265 REWRITE_TAC[hull] THEN SET_TAC[]);;
267 let HULL_REDUNDANT = prove
268 (`!P a s. a IN (P hull s) ==> (P hull (a INSERT s) = P hull s)`,
269 REWRITE_TAC[HULL_REDUNDANT_EQ]);;
271 let HULL_INDUCT = prove
272 (`!P p s. (!x:A. x IN s ==> p x) /\ P {x | p x}
273 ==> !x. x IN P hull s ==> p x`,
275 MP_TAC(ISPECL [`P:(A->bool)->bool`; `s:A->bool`; `{x:A | p x}`]
277 REWRITE_TAC[SUBSET; IN_ELIM_THM]);;
280 (`!P s x. x IN s ==> x IN P hull s`,
281 MESON_TAC[REWRITE_RULE[SUBSET] HULL_SUBSET]);;
283 let HULL_IMAGE_SUBSET = prove
284 (`!P f s. P(P hull s) /\ (!s. P s ==> P(IMAGE f s))
285 ==> P hull (IMAGE f s) SUBSET (IMAGE f (P hull s))`,
286 REPEAT STRIP_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN
287 ASM_SIMP_TAC[IMAGE_SUBSET; HULL_SUBSET]);;
289 let HULL_IMAGE_GALOIS = prove
290 (`!P f g s. (!s. P(P hull s)) /\
291 (!s. P s ==> P(IMAGE f s)) /\ (!s. P s ==> P(IMAGE g s)) /\
292 (!s t. s SUBSET IMAGE g t <=> IMAGE f s SUBSET t)
293 ==> P hull (IMAGE f s) = IMAGE f (P hull s)`,
294 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
295 ASM_SIMP_TAC[HULL_IMAGE_SUBSET] THEN
296 FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [GSYM th]) THEN
297 MATCH_MP_TAC HULL_MINIMAL THEN
298 ASM_SIMP_TAC[HULL_SUBSET]);;
300 let HULL_IMAGE = prove
301 (`!P f s. (!s. P(P hull s)) /\ (!s. P(IMAGE f s) <=> P s) /\
302 (!x y:A. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
303 ==> P hull (IMAGE f s) = IMAGE f (P hull s)`,
305 REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
306 REWRITE_TAC[BIJECTIVE_LEFT_RIGHT_INVERSE] THEN
307 DISCH_THEN(X_CHOOSE_THEN `g:A->A` STRIP_ASSUME_TAC) THEN
308 MATCH_MP_TAC HULL_IMAGE_GALOIS THEN EXISTS_TAC `g:A->A` THEN
309 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
310 X_GEN_TAC `s:A->bool` THEN
311 FIRST_X_ASSUM(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
312 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN ASM SET_TAC[]);;
315 (`!P s. (!f. (!s. s IN f ==> P s) ==> P(INTERS f))
316 ==> (P s <=> ?t. s = P hull t)`,
317 MESON_TAC[HULL_P; P_HULL]);;
321 (!f. (!s. s IN f ==> P s) ==> P (INTERS f)) /\
322 s SUBSET P hull t /\ t SUBSET P hull s
323 ==> P hull s = P hull t`,
324 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_ANTISYM THEN
325 CONJ_TAC THEN MATCH_MP_TAC HULL_MINIMAL THEN
326 ASM_SIMP_TAC[P_HULL]);;
328 let HULL_P_AND_Q = prove
329 (`!P Q. (!f. (!s. s IN f ==> P s) ==> P(INTERS f)) /\
330 (!f. (!s. s IN f ==> Q s) ==> Q(INTERS f)) /\
331 (!s. Q s ==> Q(P hull s))
332 ==> (\x. P x /\ Q x) hull s = P hull (Q hull s)`,
333 REPEAT STRIP_TAC THEN
334 MATCH_MP_TAC HULL_UNIQUE THEN ASM_SIMP_TAC[HULL_INC; SUBSET_HULL] THEN
335 ASM_MESON_TAC[P_HULL; HULL_SUBSET; SUBSET_TRANS]);;
337 (* ------------------------------------------------------------------------- *)
338 (* More variants of the Archimedian property and useful consequences. *)
339 (* ------------------------------------------------------------------------- *)
341 let REAL_ARCH_INV = prove
342 (`!e. &0 < e <=> ?n. ~(n = 0) /\ &0 < inv(&n) /\ inv(&n) < e`,
343 GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_TRANS]] THEN
344 DISCH_TAC THEN MP_TAC(SPEC `inv(e)` REAL_ARCH_LT) THEN
345 MATCH_MP_TAC MONO_EXISTS THEN
346 ASM_MESON_TAC[REAL_LT_INV2; REAL_INV_INV; REAL_LT_INV_EQ; REAL_LT_TRANS;
349 let REAL_POW_LBOUND = prove
350 (`!x n. &0 <= x ==> &1 + &n * x <= (&1 + x) pow n`,
351 GEN_TAC THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN
353 REWRITE_TAC[real_pow; REAL_MUL_LZERO; REAL_ADD_RID; REAL_LE_REFL] THEN
354 REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN
355 MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `(&1 + x) * (&1 + &n * x)` THEN
356 ASM_SIMP_TAC[REAL_LE_LMUL; REAL_ARITH `&0 <= x ==> &0 <= &1 + x`] THEN
357 ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS; REAL_ARITH
358 `&1 + (n + &1) * x <= (&1 + x) * (&1 + n * x) <=> &0 <= n * x * x`]);;
360 let REAL_ARCH_POW = prove
361 (`!x y. &1 < x ==> ?n. y < x pow n`,
362 REPEAT STRIP_TAC THEN
363 MP_TAC(SPEC `x - &1` REAL_ARCH) THEN ASM_REWRITE_TAC[REAL_SUB_LT] THEN
364 DISCH_THEN(MP_TAC o SPEC `y:real`) THEN MATCH_MP_TAC MONO_EXISTS THEN
365 X_GEN_TAC `n:num` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LTE_TRANS THEN
366 EXISTS_TAC `&1 + &n * (x - &1)` THEN
367 ASM_SIMP_TAC[REAL_ARITH `x < y ==> x < &1 + y`] THEN
368 ASM_MESON_TAC[REAL_POW_LBOUND; REAL_SUB_ADD2; REAL_ARITH
369 `&1 < x ==> &0 <= x - &1`]);;
371 let REAL_ARCH_POW2 = prove
372 (`!x. ?n. x < &2 pow n`,
373 SIMP_TAC[REAL_ARCH_POW; REAL_OF_NUM_LT; ARITH]);;
375 let REAL_ARCH_POW_INV = prove
376 (`!x y. &0 < y /\ x < &1 ==> ?n. x pow n < y`,
377 REPEAT STRIP_TAC THEN ASM_CASES_TAC `&0 < x` THENL
378 [ALL_TAC; ASM_MESON_TAC[REAL_POW_1; REAL_LET_TRANS; REAL_NOT_LT]] THEN
379 SUBGOAL_THEN `inv(&1) < inv(x)` MP_TAC THENL
380 [ASM_SIMP_TAC[REAL_LT_INV2]; REWRITE_TAC[REAL_INV_1]] THEN
381 DISCH_THEN(MP_TAC o SPEC `inv(y)` o MATCH_MP REAL_ARCH_POW) THEN
382 MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN
383 GEN_REWRITE_TAC BINOP_CONV [GSYM REAL_INV_INV] THEN
384 ASM_SIMP_TAC[GSYM REAL_POW_INV; REAL_LT_INV; REAL_LT_INV2]);;
386 let FORALL_POS_MONO = prove
387 (`!P. (!d e. d < e /\ P d ==> P e) /\ (!n. ~(n = 0) ==> P(inv(&n)))
388 ==> !e. &0 < e ==> P e`,
389 MESON_TAC[REAL_ARCH_INV; REAL_LT_TRANS]);;
391 let FORALL_POS_MONO_1 = prove
392 (`!P. (!d e. d < e /\ P d ==> P e) /\ (!n. P(inv(&n + &1)))
393 ==> !e. &0 < e ==> P e`,
394 REWRITE_TAC[REAL_OF_NUM_SUC; GSYM FORALL_SUC; FORALL_POS_MONO]);;
396 let REAL_ARCH_RDIV_EQ_0 = prove
397 (`!x c. &0 <= x /\ &0 <= c /\ (!m. 0 < m ==> &m * x <= c) ==> x = &0`,
398 SIMP_TAC [GSYM REAL_LE_ANTISYM; GSYM REAL_NOT_LT] THEN REPEAT STRIP_TAC THEN
399 POP_ASSUM (STRIP_ASSUME_TAC o SPEC `c:real` o MATCH_MP REAL_ARCH) THEN
400 ASM_CASES_TAC `n=0` THENL
401 [POP_ASSUM SUBST_ALL_TAC THEN
402 RULE_ASSUM_TAC (REWRITE_RULE [REAL_MUL_LZERO]) THEN
403 ASM_MESON_TAC [REAL_LET_ANTISYM];
404 ASM_MESON_TAC [REAL_LET_ANTISYM; REAL_MUL_SYM; LT_NZ]]);;
406 (* ------------------------------------------------------------------------- *)
407 (* Relate max and min to sup and inf. *)
408 (* ------------------------------------------------------------------------- *)
410 let REAL_MAX_SUP = prove
411 (`!x y. max x y = sup {x,y}`,
412 SIMP_TAC[GSYM REAL_LE_ANTISYM; REAL_SUP_LE_FINITE; REAL_LE_SUP_FINITE;
413 FINITE_RULES; NOT_INSERT_EMPTY; REAL_MAX_LE; REAL_LE_MAX] THEN
414 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN MESON_TAC[REAL_LE_TOTAL]);;
416 let REAL_MIN_INF = prove
417 (`!x y. min x y = inf {x,y}`,
418 SIMP_TAC[GSYM REAL_LE_ANTISYM; REAL_INF_LE_FINITE; REAL_LE_INF_FINITE;
419 FINITE_RULES; NOT_INSERT_EMPTY; REAL_MIN_LE; REAL_LE_MIN] THEN
420 REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN MESON_TAC[REAL_LE_TOTAL]);;
422 (* ------------------------------------------------------------------------- *)
423 (* Define square root here to decouple it from the existing analysis theory. *)
424 (* ------------------------------------------------------------------------- *)
426 let sqrt = new_definition
427 `sqrt(x) = @y. &0 <= y /\ (y pow 2 = x)`;;
429 let SQRT_UNIQUE = prove
430 (`!x y. &0 <= y /\ (y pow 2 = x) ==> (sqrt(x) = y)`,
431 REPEAT STRIP_TAC THEN REWRITE_TAC[sqrt] THEN MATCH_MP_TAC SELECT_UNIQUE THEN
432 FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[REAL_POW_2] THEN
433 REWRITE_TAC[REAL_ARITH `(x * x = y * y) <=> ((x + y) * (x - y) = &0)`] THEN
434 REWRITE_TAC[REAL_ENTIRE] THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
436 let POW_2_SQRT = prove
437 (`!x. &0 <= x ==> (sqrt(x pow 2) = x)`,
438 MESON_TAC[SQRT_UNIQUE]);;
442 MESON_TAC[SQRT_UNIQUE; REAL_POW_2; REAL_MUL_LZERO; REAL_POS]);;
446 MESON_TAC[SQRT_UNIQUE; REAL_POW_2; REAL_MUL_LID; REAL_POS]);;
448 let POW_2_SQRT_ABS = prove
449 (`!x. sqrt(x pow 2) = abs(x)`,
450 GEN_TAC THEN MATCH_MP_TAC SQRT_UNIQUE THEN
451 REWRITE_TAC[REAL_ABS_POS; REAL_POW_2; GSYM REAL_ABS_MUL] THEN
452 REWRITE_TAC[real_abs; REAL_LE_SQUARE]);;
454 (* ------------------------------------------------------------------------- *)
455 (* Geometric progression. *)
456 (* ------------------------------------------------------------------------- *)
458 let SUM_GP_BASIC = prove
459 (`!x n. (&1 - x) * sum(0..n) (\i. x pow i) = &1 - x pow (SUC n)`,
460 GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[SUM_CLAUSES_NUMSEG] THEN
461 REWRITE_TAC[real_pow; REAL_MUL_RID; LE_0] THEN
462 ASM_REWRITE_TAC[REAL_ADD_LDISTRIB; real_pow] THEN REAL_ARITH_TAC);;
464 let SUM_GP_MULTIPLIED = prove
466 ==> ((&1 - x) * sum(m..n) (\i. x pow i) = x pow m - x pow (SUC n))`,
467 REPEAT STRIP_TAC THEN ASM_SIMP_TAC
468 [SUM_OFFSET_0; REAL_POW_ADD; REAL_MUL_ASSOC; SUM_GP_BASIC; SUM_RMUL] THEN
469 REWRITE_TAC[REAL_SUB_RDISTRIB; GSYM REAL_POW_ADD; REAL_MUL_LID] THEN
470 ASM_SIMP_TAC[ARITH_RULE `m <= n ==> (SUC(n - m) + m = SUC n)`]);;
474 sum(m..n) (\i. x pow i) =
476 else if x = &1 then &((n + 1) - m)
477 else (x pow m - x pow (SUC n)) / (&1 - x)`,
479 DISJ_CASES_TAC(ARITH_RULE `n < m \/ ~(n < m) /\ m <= n:num`) THEN
480 ASM_SIMP_TAC[SUM_TRIV_NUMSEG] THEN COND_CASES_TAC THENL
481 [ASM_REWRITE_TAC[REAL_POW_ONE; SUM_CONST_NUMSEG; REAL_MUL_RID]; ALL_TAC] THEN
482 MATCH_MP_TAC REAL_EQ_LCANCEL_IMP THEN EXISTS_TAC `&1 - x` THEN
483 ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_SUB_0; SUM_GP_MULTIPLIED]);;
485 let SUM_GP_OFFSET = prove
486 (`!x m n. sum(m..m+n) (\i. x pow i) =
487 if x = &1 then &n + &1
488 else x pow m * (&1 - x pow (SUC n)) / (&1 - x)`,
489 REPEAT GEN_TAC THEN REWRITE_TAC[SUM_GP; ARITH_RULE `~(m + n < m:num)`] THEN
490 COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
491 [REWRITE_TAC[REAL_OF_NUM_ADD] THEN AP_TERM_TAC THEN ARITH_TAC;
492 REWRITE_TAC[real_div; real_pow; REAL_POW_ADD] THEN REAL_ARITH_TAC]);;
494 (* ------------------------------------------------------------------------- *)
495 (* Segment of natural numbers starting at a specific number. *)
496 (* ------------------------------------------------------------------------- *)
498 let from = new_definition
499 `from n = {m:num | n <= m}`;;
503 REWRITE_TAC[from; LE_0] THEN SET_TAC[]);;
505 let FROM_INTER_NUMSEG_GEN = prove
506 (`!k m n. (from k) INTER (m..n) = (if m < k then k..n else m..n)`,
507 REPEAT GEN_TAC THEN COND_CASES_TAC THEN POP_ASSUM MP_TAC THEN
508 REWRITE_TAC[from; IN_ELIM_THM; IN_INTER; IN_NUMSEG; EXTENSION] THEN
511 let FROM_INTER_NUMSEG = prove
512 (`!k n. (from k) INTER (0..n) = k..n`,
513 REWRITE_TAC[from; IN_ELIM_THM; IN_INTER; IN_NUMSEG; EXTENSION] THEN
517 (`!m n. m IN from n <=> n <= m`,
518 REWRITE_TAC[from; IN_ELIM_THM]);;
520 let INFINITE_FROM = prove
521 (`!n. INFINITE(from n)`,
523 SUBGOAL_THEN `from n = (:num) DIFF {i | i < n}`
524 (fun th -> SIMP_TAC[th; INFINITE_DIFF_FINITE; FINITE_NUMSEG_LT;
526 REWRITE_TAC[EXTENSION; from; IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN ARITH_TAC);;