1 (* ========================================================================= *)
2 (* The real and complex gamma functions and Euler-Mascheroni constant. *)
3 (* ========================================================================= *)
5 needs "Multivariate/cauchy.ml";;
7 (* ------------------------------------------------------------------------- *)
8 (* Euler-Macheroni constant. *)
9 (* ------------------------------------------------------------------------- *)
11 let euler_mascheroni = new_definition
13 reallim sequentially (\n. sum(1..n) (\k. inv(&k)) - log(&n))`;;
15 let EULER_MASCHERONI = prove
16 (`((\n. sum(1..n) (\k. inv(&k)) - log(&n)) ---> euler_mascheroni)
18 REWRITE_TAC[euler_mascheroni; reallim] THEN CONV_TAC SELECT_CONV THEN
20 `real_summable (from 1) (\k. inv(&k) + (log(&k) - log(&(k + 1))))`
22 [MATCH_MP_TAC REAL_SUMMABLE_COMPARISON THEN
23 EXISTS_TAC `\k. &2 / &k pow 2` THEN CONJ_TAC THENL
24 [REWRITE_TAC[real_div] THEN MATCH_MP_TAC REAL_SUMMABLE_LMUL THEN
25 MATCH_MP_TAC REAL_SUMMABLE_ZETA_INTEGER THEN REWRITE_TAC[LE_REFL];
26 EXISTS_TAC `2` THEN REWRITE_TAC[GE; IN_FROM] THEN
27 X_GEN_TAC `n:num` THEN STRIP_TAC THEN
28 ONCE_REWRITE_TAC[REAL_ARITH `a + (b - c):real = a - (c - b)`] THEN
29 ASM_SIMP_TAC[GSYM LOG_DIV; REAL_OF_NUM_LT; LE_1;
30 ARITH_RULE `0 < n + 1`] THEN
31 ASM_SIMP_TAC[GSYM REAL_OF_NUM_ADD; REAL_OF_NUM_LT; LE_1; REAL_FIELD
32 `&0 < n ==> (n + &1) / n = &1 + inv(n)`] THEN
33 MP_TAC(ISPECL [`1`; `Cx(inv(&n))`] TAYLOR_CLOG)];
34 REWRITE_TAC[real_summable; real_sums; FROM_INTER_NUMSEG] THEN
35 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real` THEN
36 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REALLIM_TRANSFORM) THEN
37 REWRITE_TAC[SUM_ADD_NUMSEG; REAL_ARITH
38 `(x + y) - (x - z):real = y + z`] THEN
39 REWRITE_TAC[SUM_DIFFS; LOG_1; COND_RAND; COND_RATOR] THEN
40 REWRITE_TAC[REAL_ARITH `&0 - x + y = --(x - y)`] THEN
41 MATCH_MP_TAC REALLIM_NULL_COMPARISON THEN
42 EXISTS_TAC `\n. &2 / &n` THEN CONJ_TAC THENL
43 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `2` THEN
44 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
45 COND_CASES_TAC THENL [ALL_TAC; ASM_ARITH_TAC] THEN
46 ASM_SIMP_TAC[REAL_ABS_NEG; GSYM LOG_DIV; REAL_OF_NUM_LT; LE_1;
47 ARITH_RULE `0 < n + 1`] THEN
48 ASM_SIMP_TAC[GSYM REAL_OF_NUM_ADD; REAL_OF_NUM_LT; LE_1; REAL_FIELD
49 `&0 < n ==> (n + &1) / n = &1 + inv(n)`] THEN
50 MP_TAC(ISPECL [`0`; `Cx(inv(&n))`] TAYLOR_CLOG);
51 REWRITE_TAC[real_div] THEN MATCH_MP_TAC REALLIM_NULL_LMUL THEN
52 REWRITE_TAC[REALLIM_1_OVER_N]]] THEN
53 REWRITE_TAC[GSYM CX_ADD; VSUM_SING_NUMSEG; COMPLEX_NORM_CX] THEN
54 REWRITE_TAC[VSUM_CLAUSES_NUMSEG] THEN CONV_TAC NUM_REDUCE_CONV THEN
55 REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NUM; COMPLEX_DIV_1] THEN
56 ASM_SIMP_TAC[COMPLEX_POW_1; REAL_INV_LT_1; REAL_OF_NUM_LT;
57 ARITH_RULE `1 < n <=> 2 <= n`] THEN
58 REWRITE_TAC[COMPLEX_POW_2; COMPLEX_MUL_LNEG; COMPLEX_MUL_LID] THEN
59 ASM_SIMP_TAC[COMPLEX_NEG_NEG; GSYM CX_LOG; REAL_LT_ADD; REAL_OF_NUM_LT;
60 LE_1; ARITH; REAL_LT_INV_EQ; GSYM CX_SUB] THEN
61 REWRITE_TAC[REAL_POW_1; VECTOR_SUB_RZERO] THEN
62 REWRITE_TAC[COMPLEX_NORM_CX] THEN REWRITE_TAC[REAL_ABS_SUB] THEN
63 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
64 REWRITE_TAC[real_div; REAL_POW_INV] THEN
65 GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN
66 MATCH_MP_TAC REAL_LE_LMUL THEN
67 SIMP_TAC[REAL_LE_INV_EQ; REAL_POW_LE; REAL_POS] THEN
68 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN
69 MATCH_MP_TAC REAL_LE_INV2 THEN
70 CONV_TAC REAL_RAT_REDUCE_CONV THEN
71 REWRITE_TAC[REAL_ARITH `&1 / &2 <= &1 - &1 / n <=> inv(n) <= inv(&2)`] THEN
72 MATCH_MP_TAC REAL_LE_INV2 THEN
73 REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC);;
75 (* ------------------------------------------------------------------------- *)
76 (* Simple-minded estimation of gamma using Euler-Maclaurin summation. *)
77 (* ------------------------------------------------------------------------- *)
79 let LOG2_APPROX_40 = prove
80 (`abs(log(&2) - &381061692393 / &549755813888) <= inv(&2 pow 40)`,
81 MP_TAC(SPECL [`41`; `Cx(--inv(&2))`] TAYLOR_CLOG) THEN
82 SIMP_TAC[GSYM CX_DIV; GSYM CX_POW; GSYM CX_NEG; GSYM CX_ADD; GSYM CX_MUL;
83 VSUM_CX; COMPLEX_NORM_CX; GSYM CX_LOG; GSYM CX_SUB;
84 REAL_ARITH `&0 < &1 + --inv(&2)`] THEN
85 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
86 REWRITE_TAC[REAL_ARITH `a * b / c:real = a / c * b`] THEN
87 CONV_TAC(ONCE_DEPTH_CONV EXPAND_SUM_CONV) THEN
88 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
89 REWRITE_TAC[real_div; REAL_MUL_LID] THEN
90 SIMP_TAC[LOG_INV; REAL_ARITH `&0 < &2`] THEN REAL_ARITH_TAC);;
92 let EULER_MASCHERONI_APPROX_32 = prove
93 (`abs(euler_mascheroni - &2479122403 / &4294967296) <= inv(&2 pow 32)`,
95 (`!m n. 1 <= m /\ m <= n
96 ==> abs((sum (1..n) (\k. inv(&k)) - log(&n)) -
97 ((sum (1..m - 1) (\k. inv(&k)) - log(&m)) +
98 (inv(&m) + inv(&n)) / &2 +
99 &1 / &12 * (inv(&m pow 2) - inv(&n pow 2)) +
100 -- &1 / &120 * (inv(&m pow 4) - inv(&n pow 4))))
101 <= inv(&60 * &m pow 5)`,
102 REPEAT STRIP_TAC THEN
103 MP_TAC(SPECL [`\k. inv(&k)`; `1:num`; `m - 1`; `n:num`] SUM_COMBINE_R) THEN
104 ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN
105 ASM_SIMP_TAC[ARITH_RULE `1 <= m ==> m - 1 + 1 = m`] THEN
107 [`\n x. --(&1) pow n * &(FACT n) / x pow (n + 1)`;
108 `m:num`; `n:num`; `2`] REAL_EULER_MACLAURIN) THEN
109 ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN ANTS_TAC THENL
110 [REWRITE_TAC[IN_REAL_INTERVAL] THEN REPEAT
111 STRIP_TAC THEN REAL_DIFF_TAC THEN
112 REWRITE_TAC[REAL_MUL_LZERO; ADD_SUB; REAL_MUL_RID; REAL_SUB_LZERO] THEN
113 REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC; REAL_POW_EQ_0;
114 REAL_MUL_LNEG; REAL_MUL_RNEG] THEN
115 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN
116 ASM_CASES_TAC `x = &0` THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
117 REWRITE_TAC[GSYM real_div; REAL_POW_POW] THEN
118 ASM_SIMP_TAC[REAL_DIV_POW2_ALT; ARITH_RULE `~((k + 1) * 2 < k)`] THEN
119 REWRITE_TAC[ARITH_RULE `(k + 1) * 2 - k = SUC(SUC k)`;
120 ARITH_RULE `(k + 1) + 1 = SUC(SUC k)`] THEN
121 REWRITE_TAC[REAL_POW_ADD; REAL_POW_1] THEN
122 REWRITE_TAC[REAL_MUL_RNEG; REAL_MUL_LNEG; REAL_MUL_RID] THEN
123 AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[real_div] THEN
124 REWRITE_TAC[REAL_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN
125 REWRITE_TAC[GSYM ADD1; FACT; REAL_OF_NUM_MUL; MULT_AC];
126 DISCH_THEN(MP_TAC o CONJUNCT2) THEN
127 REWRITE_TAC[real_div; REAL_MUL_LID; real_pow] THEN
128 CONV_TAC(ONCE_DEPTH_CONV EXPAND_SUM_CONV) THEN
129 CONV_TAC NUM_REDUCE_CONV THEN
130 CONV_TAC(ONCE_DEPTH_CONV BERNOULLI_CONV) THEN
131 REWRITE_TAC[GSYM(BERNOULLI_CONV `bernoulli 5 x`)] THEN
132 CONV_TAC REAL_RAT_REDUCE_CONV THEN
133 REWRITE_TAC[REAL_MUL_LNEG; REAL_MUL_LID; REAL_POW_1] THEN
134 REWRITE_TAC[real_div; REAL_MUL_LID] THEN DISCH_THEN SUBST1_TAC] THEN
135 MP_TAC(ISPECL [`\x. log x`; `\x:real. inv x`; `&m`; `&n`]
136 REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
137 ASM_REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE] THEN
139 [REWRITE_TAC[IN_REAL_INTERVAL; GSYM REAL_OF_NUM_ADD] THEN
140 REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN
141 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN
143 DISCH_THEN(SUBST1_TAC o MATCH_MP REAL_INTEGRAL_UNIQUE)] THEN
144 MATCH_MP_TAC REAL_LE_TRANS THEN
147 abs(real_integral (real_interval[&m,&n])
148 (\x. bernoulli 5 (frac x) *
149 --(&120 * inv(x pow 6))))` THEN
150 CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
151 REWRITE_TAC[REAL_ARITH `b * --(n * inv x):real = --n * b / x`] THEN
153 `(\x. bernoulli 5 (frac x) / x pow 6)
154 real_measurable_on real_interval[&m,&n]`
157 REAL_CONTINUOUS_AE_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN
158 EXISTS_TAC `integer` THEN
159 REWRITE_TAC[REAL_LEBESGUE_MEASURABLE_INTERVAL] THEN
160 SIMP_TAC[REAL_NEGLIGIBLE_COUNTABLE; COUNTABLE_INTEGER] THEN
161 REWRITE_TAC[REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
162 REWRITE_TAC[IN_DIFF; IN_REAL_INTERVAL] THEN REWRITE_TAC[IN] THEN
163 X_GEN_TAC `x:real` THEN STRIP_TAC THEN
164 MATCH_MP_TAC REAL_CONTINUOUS_ATREAL_WITHINREAL THEN
165 MATCH_MP_TAC REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL THEN
166 REAL_DIFFERENTIABLE_TAC THEN ASM_REWRITE_TAC[REAL_POW_EQ_0] THEN
167 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN
171 [`\x. inv(-- &60 * x pow 5)`; `\x. inv(&12 * x pow 6)`;
172 `&m`; `&n`] REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
173 ASM_REWRITE_TAC[REAL_OF_NUM_LE] THEN ANTS_TAC THENL
174 [REWRITE_TAC[IN_REAL_INTERVAL] THEN REPEAT STRIP_TAC THEN
175 REAL_DIFF_TAC THEN CONV_TAC NUM_REDUCE_CONV THEN
176 SUBGOAL_THEN `~(x = &0)` MP_TAC THENL
177 [RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN
179 CONV_TAC REAL_FIELD];
180 REWRITE_TAC[REAL_INV_MUL; REAL_ARITH
181 `inv(-- &60) * x - inv(-- &60) * y = (y - x) / &60`] THEN
182 REWRITE_TAC[GSYM REAL_INV_MUL] THEN DISCH_TAC] THEN
184 `!x. x IN real_interval[&m,&n]
185 ==> abs(bernoulli 5 (frac x) / x pow 6) <= inv(&12 * x pow 6)`
187 [REWRITE_TAC[IN_REAL_INTERVAL] THEN REPEAT STRIP_TAC THEN
188 REWRITE_TAC[REAL_INV_MUL; real_div; REAL_ABS_MUL] THEN
189 MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[REAL_ABS_POS] THEN
191 [MP_TAC(ISPECL [`5`; `frac x`] BERNOULLI_BOUND) THEN
192 SIMP_TAC[IN_REAL_INTERVAL; FLOOR_FRAC; REAL_LT_IMP_LE] THEN
193 CONV_TAC NUM_REDUCE_CONV THEN
194 REWRITE_TAC[BERNOULLI_CONV `bernoulli 4 (&0)`] THEN
196 REWRITE_TAC[REAL_ARITH `abs x <= x <=> &0 <= x`] THEN
197 REWRITE_TAC[REAL_LE_INV_EQ] THEN MATCH_MP_TAC REAL_POW_LE THEN
198 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN
202 `(\x. bernoulli 5 (frac x) / x pow 6)
203 real_integrable_on real_interval[&m,&n]`
205 [MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE THEN
206 EXISTS_TAC `\x:real. inv(&12 * x pow 6)` THEN
207 ASM_REWRITE_TAC[REAL_INTEGRABLE_CONST] THEN
208 ASM_MESON_TAC[real_integrable_on];
209 ASM_SIMP_TAC[REAL_INTEGRAL_LMUL; REAL_ABS_MUL; REAL_MUL_ASSOC]] THEN
210 CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_LID] THEN
211 TRANS_TAC REAL_LE_TRANS
212 `real_integral (real_interval [&m,&n]) (\x. inv(&12 * x pow 6))` THEN
214 [MATCH_MP_TAC REAL_INTEGRAL_ABS_BOUND_INTEGRAL THEN
215 ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[real_integrable_on];
216 FIRST_ASSUM(SUBST1_TAC o MATCH_MP REAL_INTEGRAL_UNIQUE) THEN
217 REWRITE_TAC[REAL_INV_MUL] THEN
218 REWRITE_TAC[REAL_LE_INV_EQ; REAL_ARITH
219 `(x - y) / &60 <= inv(&60) * x <=> &0 <= y`] THEN
220 SIMP_TAC[REAL_POW_LE; REAL_POS]])
223 (f ---> l) sequentially
224 ==> (g ---> m) sequentially /\
225 eventually (\n. abs(f n - g n) <= d) sequentially /\
227 ==> abs(l - k) <= e`,
228 REPEAT GEN_TAC THEN DISCH_TAC THEN
229 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
230 MATCH_MP_TAC(REAL_ARITH
231 `abs(l - m) <= d ==> abs (m - k) <= e - d ==> abs (l - k) <= e`) THEN
232 REWRITE_TAC[REAL_ABS_BOUNDS] THEN CONJ_TAC THENL
233 [MATCH_MP_TAC(ISPEC `sequentially` REALLIM_LBOUND);
234 MATCH_MP_TAC(ISPEC `sequentially` REALLIM_UBOUND)] THEN
235 EXISTS_TAC `(\n. f n - g n):num->real` THEN
236 ASM_SIMP_TAC[REALLIM_SUB; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
237 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT]
238 EVENTUALLY_MONO)) THEN
240 MATCH_MP_TAC(MATCH_MP lemma2 EULER_MASCHERONI) THEN
242 `(?a b c m:num. P (a m) (b m) (c m)) ==> ?a b c. P a b c`) THEN
244 `\m n. (sum(1..m - 1) (\k. inv(&k)) - log(&m)) +
245 (inv(&m) + inv(&n)) / &2 +
246 &1 / &12 * (inv(&m pow 2) - inv(&n pow 2)) +
247 -- &1 / &120 * (inv(&m pow 4) - inv(&n pow 4))` THEN
249 `\m. (sum(1..m - 1) (\k. inv(&k)) - log(&m)) +
251 &1 / &12 * inv(&m pow 2) +
252 -- &1 / &120 * inv(&m pow 4)` THEN
253 EXISTS_TAC `\m. inv (&60 * &m pow 5)` THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN
254 MATCH_MP_TAC(MESON[] `(!n. P n) /\ (?n. Q n) ==> (?n. P n /\ Q n)`) THEN
255 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
257 REPEAT(MATCH_MP_TAC REALLIM_ADD THEN CONJ_TAC) THEN
258 REWRITE_TAC[REALLIM_CONST] THEN
259 REWRITE_TAC[REAL_ARITH `x / &2 = inv(&2) * x`] THEN
260 MATCH_MP_TAC REALLIM_LMUL THEN REWRITE_TAC[real_sub] THEN
261 GEN_REWRITE_TAC LAND_CONV [GSYM REAL_ADD_RID] THEN
262 MATCH_MP_TAC REALLIM_ADD THEN REWRITE_TAC[REALLIM_CONST] THEN
263 REWRITE_TAC[REALLIM_NULL_NEG] THEN
264 REWRITE_TAC[REALLIM_1_OVER_N] THEN
265 MATCH_MP_TAC REALLIM_1_OVER_POW THEN CONV_TAC NUM_REDUCE_CONV;
267 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
268 MATCH_MP_TAC(MESON[] `(?a. P a a /\ Q a) ==> ?a. (?b. P a b) /\ Q a`) THEN
269 EXISTS_TAC `64` THEN CONJ_TAC THENL
270 [REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma1 THEN ASM_ARITH_TAC; ALL_TAC] THEN
271 SUBGOAL_THEN `log(&64) = &6 * log(&2)` SUBST1_TAC THENL
272 [SIMP_TAC[GSYM LOG_POW; REAL_ARITH `&0 < &2`] THEN
273 AP_TERM_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV;
274 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
275 CONV_TAC(ONCE_DEPTH_CONV EXPAND_SUM_CONV) THEN
276 CONV_TAC REAL_RAT_REDUCE_CONV THEN
277 MP_TAC LOG2_APPROX_40 THEN REAL_ARITH_TAC]);;
279 (* ------------------------------------------------------------------------- *)
280 (* Start with the log-gamma function. It's otherwise quite tedious to repeat *)
281 (* essentially the same argument when we want the logarithm of the gamma *)
282 (* function, since we can't just take the usual principal value of log. *)
283 (* ------------------------------------------------------------------------- *)
285 let lgamma = new_definition
286 `lgamma z = lim sequentially
287 (\n. z * clog(Cx(&n)) - clog z -
288 vsum(1..n) (\m. clog((Cx(&m) + z) / Cx(&m))))`;;
290 let LGAMMA,COMPLEX_DIFFERENTIABLE_AT_LGAMMA = (CONJ_PAIR o prove)
291 (`(!z. (!n. ~(z + Cx(&n) = Cx(&0)))
292 ==> ((\n. z * clog(Cx(&n)) - clog z -
293 vsum(1..n) (\m. clog((Cx(&m) + z) / Cx(&m))))
294 --> lgamma(z)) sequentially) /\
295 (!z. (Im z = &0 ==> &0 < Re z) ==> lgamma complex_differentiable at z)`,
296 SUBGOAL_THEN `open {z | !n. ~(z + Cx(&n) = Cx(&0))}` ASSUME_TAC THENL
297 [REWRITE_TAC[SET_RULE `{z | !n. P n z} = UNIV DIFF {z | ?n. ~P n z}`] THEN
298 REWRITE_TAC[GSYM closed] THEN MATCH_MP_TAC DISCRETE_IMP_CLOSED THEN
299 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; IMP_CONJ] THEN
300 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
301 SIMP_TAC[COMPLEX_RING `x + y = Cx(&0) <=> x = --y`] THEN
302 REWRITE_TAC[COMPLEX_RING `--x - --y:complex = y - x`] THEN
303 REWRITE_TAC[COMPLEX_EQ_NEG2; CX_INJ; GSYM CX_SUB; COMPLEX_NORM_CX] THEN
304 SIMP_TAC[GSYM REAL_EQ_INTEGERS; INTEGER_CLOSED];
307 `!y. (!n. ~(y + Cx(&n) = Cx(&0)))
310 ==> ?N. !n z. N <= n /\ z IN cball(y,d)
311 ==> dist(z * clog(Cx(&n)) -
313 (\m. clog((Cx(&m) + z) / Cx(&m))),
316 [REPEAT STRIP_TAC THEN
317 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
318 REWRITE_TAC[OPEN_CONTAINS_CBALL] THEN
319 DISCH_THEN(MP_TAC o SPEC `y:complex`) THEN
320 ASM_REWRITE_TAC[IN_ELIM_THM] THEN
321 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
322 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
323 REWRITE_TAC[UNIFORMLY_CONVERGENT_EQ_CAUCHY_ALT] THEN
324 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
327 (\m. Cx(&2 * ((norm(y:complex) + d) +
328 (norm(y) + d) pow 2)) / Cx(&m) pow 2)`
330 [REWRITE_TAC[complex_div; COMPLEX_MUL_ASSOC] THEN
331 MATCH_MP_TAC SUMMABLE_COMPLEX_LMUL THEN
332 MATCH_MP_TAC SUMMABLE_ZETA_INTEGER THEN REWRITE_TAC[LE_REFL];
334 REWRITE_TAC[summable; SERIES_CAUCHY; GE] THEN
335 DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
336 DISCH_THEN(X_CHOOSE_THEN `M:num` (LABEL_TAC "M")) THEN
337 MP_TAC(SPEC `&2 * (norm(y:complex) + d) + &1` REAL_ARCH_SIMPLE) THEN
338 DISCH_THEN(X_CHOOSE_THEN `N:num` (LABEL_TAC "N")) THEN
339 EXISTS_TAC `MAX (MAX 1 2) (MAX M N)` THEN
340 MAP_EVERY X_GEN_TAC [`m:num`; `n:num`; `z:complex`] THEN
341 REWRITE_TAC[GE; ARITH_RULE `MAX a b <= c <=> a <= c /\ b <= c`] THEN
342 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`m + 1`; `n:num`]) THEN
343 ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
344 REWRITE_TAC[FROM_INTER_NUMSEG_MAX; ARITH_RULE `MAX 1 (m + 1) = m + 1`] THEN
345 REWRITE_TAC[dist] THEN
348 ==> z * clog(Cx(&n)) - vsum(1..n) (\m. clog((Cx(&m) + z) / Cx(&m))) =
349 vsum(1..n) (\m. z * (clog(Cx(&(m + 1) - &1)) -
351 clog((Cx(&m) + z) / Cx(&m))) +
353 (fun th -> ASM_SIMP_TAC[th])
355 [REWRITE_TAC[VSUM_SUB_NUMSEG] THEN
356 ASM_SIMP_TAC[VSUM_COMPLEX_LMUL; FINITE_NUMSEG; VSUM_DIFFS_ALT] THEN
357 REWRITE_TAC[GSYM REAL_OF_NUM_ADD; REAL_ARITH `(x + &1) - &1 = x`;
359 REPEAT STRIP_TAC THEN CONV_TAC COMPLEX_RING;
361 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN
362 SUBGOAL_THEN `1 <= m + 1 /\ m <= n` MP_TAC THENL
364 DISCH_THEN(fun th -> REWRITE_TAC[GSYM(MATCH_MP VSUM_COMBINE_R th)])] THEN
365 REWRITE_TAC[COMPLEX_RING `(x + a) - ((x + y) + a):complex = --y`] THEN
366 REWRITE_TAC[NORM_NEG] THEN MATCH_MP_TAC COMPLEX_NORM_VSUM_BOUND THEN
367 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `k:num` THEN
368 STRIP_TAC THEN REWRITE_TAC[GSYM CX_POW; GSYM CX_DIV; REAL_CX; RE_CX] THEN
369 REWRITE_TAC[COMPLEX_RING `z * (a - b) - c:complex = --(z * (b - a) + c)`;
370 NORM_NEG; GSYM REAL_OF_NUM_ADD; REAL_ARITH `(x + &1) - &1 = x`] THEN
371 SUBGOAL_THEN `1 <= k /\ 1 < k /\ 2 <= k` STRIP_ASSUME_TAC THENL
372 [ASM_ARITH_TAC; ALL_TAC] THEN
374 `clog(Cx(&k - &1)) - clog(Cx(&k)) = clog(Cx(&1) - inv(Cx(&k)))`
376 [ASM_SIMP_TAC[GSYM CX_LOG; REAL_OF_NUM_LE; REAL_SUB_LT; REAL_INV_LT_1;
377 REAL_ARITH `&2 <= x ==> &0 < x /\ &1 < x /\ &0 < x - &1`;
378 GSYM CX_SUB; GSYM CX_INV; GSYM LOG_DIV] THEN
379 AP_TERM_TAC THEN AP_TERM_TAC THEN UNDISCH_TAC `2 <= k` THEN
380 REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN CONV_TAC REAL_FIELD;
382 MP_TAC(ISPECL [`1`; `z / Cx(&k)`] TAYLOR_CLOG) THEN
383 MP_TAC(ISPECL [`1`; `--inv(Cx(&k))`] TAYLOR_CLOG) THEN
384 REWRITE_TAC[VSUM_SING_NUMSEG] THEN CONV_TAC NUM_REDUCE_CONV THEN
385 REWRITE_TAC[GSYM VECTOR_SUB; NORM_NEG] THEN
386 REWRITE_TAC[COMPLEX_NORM_INV; COMPLEX_NORM_DIV; COMPLEX_NORM_CX] THEN
387 REWRITE_TAC[COMPLEX_POW_NEG; ARITH; REAL_ABS_NUM; COMPLEX_POW_ONE] THEN
388 ASM_SIMP_TAC[REAL_INV_LT_1; REAL_OF_NUM_LT; COMPLEX_DIV_1] THEN
389 ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; LE_1; COMPLEX_POW_1] THEN
390 REWRITE_TAC[REAL_MUL_LID; COMPLEX_MUL_LID] THEN
391 DISCH_THEN(MP_TAC o SPEC `norm(z:complex)` o MATCH_MP
392 (REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_LMUL)) THEN
393 REWRITE_TAC[NORM_POS_LE; GSYM COMPLEX_NORM_MUL] THEN
394 SUBGOAL_THEN `norm(z:complex) < &k` ASSUME_TAC THENL
395 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_CBALL]) THEN
396 FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl)) THEN
397 REPEAT(POP_ASSUM MP_TAC) THEN
398 REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD] THEN
400 ASM_REWRITE_TAC[COMPLEX_SUB_LDISTRIB]] THEN
401 ASM_SIMP_TAC[CX_INJ; REAL_OF_NUM_EQ; LE_1; COMPLEX_FIELD
402 `~(k = Cx(&0)) ==> (k + z) / k = Cx(&1) + z / k`] THEN
403 MATCH_MP_TAC(NORM_ARITH
404 `x' = --y' /\ d + e <= f
405 ==> norm(x - x') <= d ==> norm(y - y') <= e
406 ==> norm(x + y) <= f`) THEN
407 REWRITE_TAC[complex_div; COMPLEX_MUL_LNEG; COMPLEX_MUL_RNEG] THEN
408 REWRITE_TAC[REAL_POW_DIV; REAL_POW_INV; REAL_ARITH
409 `n * inv k / d + n pow 2 / k / e <= (&2 * (x + x pow 2)) / k <=>
410 (n * (&1 / d + n / e)) / k <= (x * (&2 + &2 * x)) / k`] THEN
411 ASM_SIMP_TAC[REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; LE_1; REAL_POW_LT] THEN
412 MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[NORM_POS_LE] THEN
413 SUBGOAL_THEN `norm(z:complex) <= norm(y:complex) + d` ASSUME_TAC THENL
414 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_CBALL]) THEN
417 REPEAT CONJ_TAC THENL
418 [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_CBALL]) THEN
420 MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC THEN
421 MATCH_MP_TAC REAL_LE_DIV THEN
422 REWRITE_TAC[REAL_SUB_LE; NORM_POS_LE; REAL_POS] THEN
423 REWRITE_TAC[REAL_ARITH `inv x = &1 / x`] THEN
424 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1];
425 REWRITE_TAC[REAL_ARITH `&2 + &2 * x = &1 * &2 + x * &2`] THEN
426 ONCE_REWRITE_TAC[real_div] THEN MATCH_MP_TAC REAL_LE_ADD2 THEN
427 CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
428 ASM_REWRITE_TAC[NORM_POS_LE; REAL_POS; REAL_LE_REFL; REAL_LE_INV_EQ] THEN
429 REWRITE_TAC[REAL_SUB_LE] THEN
430 REWRITE_TAC[REAL_ARITH `inv x = &1 / x`] THEN
431 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN
432 ASM_REWRITE_TAC[REAL_MUL_LID; REAL_OF_NUM_LE] THEN
434 [RULE_ASSUM_TAC(REWRITE_RULE
435 [GSYM REAL_OF_NUM_LT; GSYM REAL_OF_NUM_ADD]) THEN
436 ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
437 REWRITE_TAC[real_div; REAL_MUL_LID] THEN
438 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN
439 MATCH_MP_TAC REAL_LE_INV2 THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
440 REWRITE_TAC[REAL_ARITH `&1 / &2 <= &1 - x <=> x <= &1 / &2`] THEN
441 REWRITE_TAC[real_div; REAL_MUL_LID] THEN REWRITE_TAC[GSYM real_div] THEN
442 REWRITE_TAC[REAL_ARITH `inv x = &1 / x`] THEN
443 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1]] THEN
444 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_CBALL]) THEN
445 FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl)) THEN
446 REPEAT(POP_ASSUM MP_TAC) THEN
447 REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD] THEN
449 GEN_REWRITE_TAC (LAND_CONV o REDEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
450 REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
451 [`dd:complex->real`; `ll:complex->complex->complex`] THEN
452 DISCH_THEN(LABEL_TAC "*")] THEN
454 `!z. (!n. ~(z + Cx(&n) = Cx(&0)))
455 ==> ((\n. z * clog(Cx(&n)) -
456 vsum (1..n) (\m. clog((Cx(&m) + z) / Cx(&m)))) --> ll z z)
459 [X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
460 FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN
461 ASM_REWRITE_TAC[LIM_SEQUENTIALLY; GSYM SKOLEM_THM] THEN
462 MESON_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE];
464 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
465 [REPEAT STRIP_TAC THEN
466 REWRITE_TAC[lgamma; lim] THEN CONV_TAC SELECT_CONV THEN
467 EXISTS_TAC `(ll:complex->complex->complex) z z - clog z` THEN
468 ONCE_REWRITE_TAC[COMPLEX_RING `w - z - v:complex = (w - v) - z`] THEN
469 MATCH_MP_TAC LIM_SUB THEN REWRITE_TAC[LIM_CONST] THEN ASM_MESON_TAC[];
471 X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
472 SUBGOAL_THEN `!n. ~(z + Cx(&n) = Cx(&0))` ASSUME_TAC THENL
474 REWRITE_TAC[COMPLEX_RING `z + x = Cx(&0) <=> z = --x`] THEN
475 DISCH_THEN SUBST_ALL_TAC THEN
476 FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
477 REWRITE_TAC[IM_NEG; RE_NEG; IM_CX; RE_CX] THEN REAL_ARITH_TAC;
479 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
480 EXISTS_TAC `\z. (ll:complex->complex->complex) z z - clog z` THEN
481 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
482 REWRITE_TAC[OPEN_CONTAINS_BALL] THEN
483 DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN
484 ASM_REWRITE_TAC[IN_ELIM_THM; SUBSET; IN_BALL] THEN
485 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN
486 ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
487 [X_GEN_TAC `w:complex` THEN ONCE_REWRITE_TAC[DIST_SYM] THEN DISCH_TAC THEN
488 MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
489 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
490 EXISTS_TAC `\n. w * clog(Cx(&n)) - clog w -
491 vsum(1..n) (\m. clog((Cx(&m) + w) / Cx(&m)))` THEN
493 ONCE_REWRITE_TAC[COMPLEX_RING `w - z - v:complex = (w - v) - z`] THEN
494 MATCH_MP_TAC LIM_SUB THEN REWRITE_TAC[LIM_CONST] THEN ASM_MESON_TAC[];
496 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_SUB THEN
497 ASM_SIMP_TAC[COMPLEX_DIFFERENTIABLE_AT_CLOG] THEN
498 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
499 EXISTS_TAC `(ll:complex->complex->complex) z` THEN
500 EXISTS_TAC `min e (dd(z:complex))` THEN
501 ASM_SIMP_TAC[REAL_LT_MIN] THEN CONJ_TAC THENL
502 [X_GEN_TAC `w:complex` THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
504 MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
505 EXISTS_TAC `\n. w * clog(Cx(&n)) -
506 vsum(1..n) (\m. clog((Cx(&m) + w) / Cx(&m)))` THEN
507 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIM_SEQUENTIALLY] THEN
508 CONJ_TAC THEN X_GEN_TAC `r:real` THEN STRIP_TAC THENL
509 [REMOVE_THEN "*" (MP_TAC o SPEC `z:complex`);
510 REMOVE_THEN "*" (MP_TAC o SPEC `w:complex`)] THEN
512 REWRITE_TAC[GSYM SKOLEM_THM; RIGHT_EXISTS_IMP_THM] THEN
513 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `r:real`)) THEN
514 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
515 REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
516 ASM_SIMP_TAC[CENTRE_IN_CBALL; REAL_LT_IMP_LE] THEN
517 ASM_SIMP_TAC[IN_CBALL; REAL_LT_IMP_LE];
519 SUBGOAL_THEN `open {z | Im z = &0 ==> &0 < Re z}` MP_TAC THENL
521 `{z | Im z = &0 ==> &0 < Re z} =
522 (:complex) DIFF ({z | Im z = &0} INTER {z | Re z <= &0})`
523 (fun th -> SIMP_TAC[th; CLOSED_HALFSPACE_RE_LE; GSYM closed;
524 CLOSED_HALFSPACE_IM_EQ; CLOSED_INTER]) THEN
525 REWRITE_TAC[IN_ELIM_THM; IN_UNIV; IN_DIFF; IN_INTER; EXTENSION] THEN
527 REWRITE_TAC[OPEN_CONTAINS_CBALL; IN_ELIM_THM; SUBSET] THEN
528 DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN
529 DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC)] THEN
531 `(ll:complex->complex->complex) z continuous_on cball(z,min r (dd z)) /\
532 ll z holomorphic_on ball(z,min r (dd z))`
534 [MATCH_MP_TAC(ISPEC `sequentially` HOLOMORPHIC_UNIFORM_LIMIT) THEN
535 EXISTS_TAC `\n z. z * clog(Cx(&n)) -
536 vsum(1..n) (\m. clog((Cx(&m) + z) / Cx(&m)))` THEN
537 ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN
538 REWRITE_TAC[CBALL_MIN_INTER; IN_INTER] THEN
539 CONJ_TAC THENL [ALL_TAC; SIMP_TAC[GSYM dist] THEN ASM_MESON_TAC[]] THEN
540 EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
541 MATCH_MP_TAC(MESON[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;
542 HOLOMORPHIC_ON_SUBSET]
543 `t SUBSET s /\ f holomorphic_on s
544 ==> f continuous_on s /\ f holomorphic_on t`) THEN
545 REWRITE_TAC[BALL_SUBSET_CBALL; GSYM CBALL_MIN_INTER] THEN
546 MATCH_MP_TAC HOLOMORPHIC_ON_SUB THEN
547 SIMP_TAC[HOLOMORPHIC_ON_MUL; HOLOMORPHIC_ON_ID;
548 HOLOMORPHIC_ON_CONST] THEN
549 MATCH_MP_TAC HOLOMORPHIC_ON_VSUM THEN
550 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
551 X_GEN_TAC `m:num` THEN STRIP_TAC THEN
552 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
553 MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE THEN
554 SIMP_TAC[HOLOMORPHIC_ON_ADD; HOLOMORPHIC_ON_ID;
555 HOLOMORPHIC_ON_CONST; complex_div; HOLOMORPHIC_ON_MUL] THEN
556 MATCH_MP_TAC HOLOMORPHIC_ON_CLOG THEN
557 REWRITE_TAC[FORALL_IN_IMAGE; GSYM complex_div; IMP_CONJ] THEN
558 ASM_SIMP_TAC[RE_DIV_CX; IM_DIV_CX; REAL_DIV_EQ_0; RE_ADD; IM_ADD] THEN
559 X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
560 ASM_SIMP_TAC[REAL_LT_IMP_NZ; REAL_OF_NUM_LT; LE_1] THEN
561 REWRITE_TAC[IM_CX; RE_CX; REAL_ADD_LID] THEN DISCH_TAC THEN
562 MATCH_MP_TAC REAL_LT_DIV THEN
563 ASM_SIMP_TAC[REAL_LT_IMP_NZ; REAL_OF_NUM_LT; LE_1] THEN
564 MATCH_MP_TAC(REAL_ARITH `&0 < x ==> &0 < &m + x`) THEN
565 RULE_ASSUM_TAC(REWRITE_RULE[CBALL_MIN_INTER; IN_INTER]) THEN
567 SIMP_TAC[HOLOMORPHIC_ON_OPEN; OPEN_BALL; complex_differentiable] THEN
568 DISCH_THEN(MATCH_MP_TAC o CONJUNCT2) THEN
569 ASM_SIMP_TAC[CENTRE_IN_BALL; REAL_LT_MIN]]);;
571 let LGAMMA_ALT = prove
572 (`!z. (!n. ~(z + Cx(&n) = Cx(&0)))
573 ==> ((\n. (z * clog(Cx(&n)) + clog(Cx(&(FACT n)))) -
574 vsum(0..n) (\m. clog(Cx(&m) + z)))
575 --> lgamma(z)) sequentially`,
576 REPEAT STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP LGAMMA) THEN
577 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM) THEN
578 MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
579 EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
580 SIMP_TAC[VECTOR_SUB_EQ; VSUM_CLAUSES_LEFT; LE_0; COMPLEX_ADD_LID] THEN
581 MATCH_MP_TAC(COMPLEX_RING
582 `a:complex = d - c ==> x - y - a = (x + c) - (y + d)`) THEN
583 REWRITE_TAC[GSYM NPRODUCT_FACT; ADD_CLAUSES] THEN
584 SIMP_TAC[REAL_OF_NUM_NPRODUCT; FINITE_NUMSEG; GSYM CX_LOG; LOG_PRODUCT;
585 PRODUCT_POS_LT; IN_NUMSEG; REAL_OF_NUM_LT; LE_1; GSYM VSUM_CX] THEN
586 REWRITE_TAC[GSYM VSUM_SUB_NUMSEG] THEN MATCH_MP_TAC VSUM_EQ_NUMSEG THEN
587 REWRITE_TAC[complex_div] THEN ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN
588 ASM_SIMP_TAC[CLOG_MUL_CX; REAL_LT_INV_EQ; REAL_OF_NUM_LT; LE_1;
589 GSYM CX_INV; LOG_INV; CX_NEG; GSYM complex_sub]);;
591 let LGAMMA_ALT2 = prove
592 (`!z. (!n. ~(z + Cx(&n) = Cx(&0)))
593 ==> ((\n. vsum(1..n) (\m. z * clog(Cx(&1) + Cx(&1) / Cx(&m)) -
594 clog(Cx(&1) + z / Cx(&m))) -
596 --> lgamma(z)) sequentially`,
597 REPEAT STRIP_TAC THEN
598 SIMP_TAC[CX_INJ; REAL_OF_NUM_EQ; LE_1;
599 COMPLEX_FIELD `~(m = Cx(&0)) ==> Cx(&1) + z / m = (z + m) / m`] THEN
600 REWRITE_TAC[GSYM CX_ADD; GSYM CX_DIV] THEN
601 SIMP_TAC[GSYM CX_LOG; LOG_DIV; REAL_LT_DIV; REAL_ARITH `&0 < &1 + &m`;
602 REAL_OF_NUM_LT; LE_1] THEN
603 SIMP_TAC[VSUM_SUB_NUMSEG; VSUM_COMPLEX_LMUL; FINITE_NUMSEG] THEN
604 REWRITE_TAC[CX_SUB; REAL_OF_NUM_ADD; ARITH_RULE `1 + m = m + 1`] THEN
605 REWRITE_TAC[VSUM_DIFFS_ALT; LOG_1; COMPLEX_SUB_RZERO] THEN
606 FIRST_ASSUM(MP_TAC o MATCH_MP LGAMMA) THEN
607 DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
608 LIM_SUBSEQUENCE)) THEN
609 DISCH_THEN(MP_TAC o SPEC `\r. r + 1`) THEN
610 REWRITE_TAC[] THEN ANTS_TAC THENL [ARITH_TAC; ALL_TAC] THEN
611 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM) THEN
612 REWRITE_TAC[o_DEF] THEN
613 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
614 EXISTS_TAC `\n. --clog((Cx(&(n + 1)) + z) / Cx(&(n + 1)))` THEN
616 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
617 X_GEN_TAC `n:num` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
618 REWRITE_TAC[GSYM ADD1; VSUM_CLAUSES_NUMSEG; ARITH_RULE `1 <= SUC n`] THEN
619 REWRITE_TAC[COMPLEX_RING `Cx(&n) + z = z + Cx(&n)`] THEN
620 SIMP_TAC[CX_LOG; REAL_OF_NUM_LT; LT_0] THEN
621 CONV_TAC COMPLEX_RING;
622 REWRITE_TAC[COMPLEX_VEC_0] THEN MATCH_MP_TAC LIM_NULL_COMPLEX_NEG THEN
623 SIMP_TAC[REAL_OF_NUM_EQ; CX_INJ; ARITH_EQ; ADD_EQ_0;
624 COMPLEX_FIELD `~(y = Cx(&0)) ==> (y + z) / y = Cx(&1) + z / y`] THEN
625 SUBGOAL_THEN `Cx(&0) = clog (Cx (&1) + Cx(&0))` SUBST1_TAC THENL
626 [REWRITE_TAC[COMPLEX_ADD_RID; CLOG_1]; ALL_TAC] THEN
627 MP_TAC(ISPECL [`\z. clog(Cx(&1) + z)`; `sequentially`]
628 LIM_CONTINUOUS_FUNCTION) THEN
629 REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL
630 [MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT THEN
631 COMPLEX_DIFFERENTIABLE_TAC THEN REWRITE_TAC[RE_ADD; RE_CX] THEN
632 REWRITE_TAC[REAL_ADD_RID; REAL_LT_01];
633 REWRITE_TAC[complex_div] THEN MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
634 MATCH_MP_TAC(REWRITE_RULE[o_DEF]
635 (ISPECL [`f:num->complex`; `\n. n + 1`] LIM_SUBSEQUENCE)) THEN
636 CONJ_TAC THENL [ARITH_TAC; ALL_TAC] THEN
637 REWRITE_TAC[GSYM CX_INV; REWRITE_RULE[o_DEF] (GSYM REALLIM_COMPLEX)] THEN
638 REWRITE_TAC[REALLIM_1_OVER_N]]]);;
640 (* ------------------------------------------------------------------------- *)
641 (* The complex gamma function (defined using the Gauss/Euler product). *)
642 (* Note that this totalizes it with the value zero at the poles. *)
643 (* ------------------------------------------------------------------------- *)
645 let cgamma = new_definition
646 `cgamma(z) = lim sequentially (\n. (Cx(&n) cpow z * Cx(&(FACT n))) /
647 cproduct(0..n) (\m. z + Cx(&m)))`;;
649 let [CGAMMA;CGAMMA_EQ_0;CGAMMA_LGAMMA] = (CONJUNCTS o prove)
650 (`(!z. ((\n. (Cx(&n) cpow z * Cx(&(FACT n))) / cproduct(0..n) (\m. z + Cx(&m)))
651 --> cgamma(z)) sequentially) /\
652 (!z. cgamma(z) = Cx(&0) <=> ?n. z + Cx(&n) = Cx(&0)) /\
653 (!z. cgamma(z) = if ?n. z + Cx(&n) = Cx(&0) then Cx(&0)
654 else cexp(lgamma z))`,
655 REWRITE_TAC[AND_FORALL_THM] THEN X_GEN_TAC `y:complex` THEN
656 ASM_CASES_TAC `?n. y + Cx(&n) = Cx(&0)` THENL
657 [ASM_REWRITE_TAC[GSYM NOT_EXISTS_THM] THEN
658 FIRST_X_ASSUM(X_CHOOSE_TAC `N:num`) THEN
659 REWRITE_TAC[cgamma; lim] THEN MATCH_MP_TAC(MESON[LIM_UNIQUE]
660 `~trivial_limit net /\ (f --> a) net
661 ==> (f --> @a. (f --> a) net) net /\ (@a. (f --> a) net) = a`) THEN
662 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
663 MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
664 EXISTS_TAC `N:num` THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[GE] THEN
666 SIMP_TAC[COMPLEX_ENTIRE; COMPLEX_DIV_EQ_0; CPRODUCT_EQ_0; FINITE_NUMSEG;
667 IN_NUMSEG; LE_0] THEN
668 ASM_MESON_TAC[LE_REFL];
669 ASM_REWRITE_TAC[] THEN RULE_ASSUM_TAC(REWRITE_RULE[NOT_EXISTS_THM])] THEN
671 `((\n. (Cx(&n) cpow y * Cx(&(FACT n))) / cproduct(0..n) (\m. y + Cx(&m)))
672 --> cexp(lgamma y)) sequentially`
674 [MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
675 EXISTS_TAC `\n. cexp(y * clog(Cx(&n)) - clog y -
676 vsum(1..n) (\m. clog((Cx(&m) + y) / Cx(&m))))` THEN
678 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
679 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
680 ASM_SIMP_TAC[CEXP_SUB; cpow; CX_INJ; REAL_OF_NUM_EQ; LE_1] THEN
681 REWRITE_TAC[complex_div; GSYM COMPLEX_MUL_ASSOC] THEN AP_TERM_TAC THEN
682 SIMP_TAC[GSYM NPRODUCT_FACT; REAL_OF_NUM_NPRODUCT; FINITE_NUMSEG] THEN
683 SIMP_TAC[CX_PRODUCT; FINITE_NUMSEG; GSYM CPRODUCT_INV] THEN
684 SIMP_TAC[CPRODUCT_CLAUSES_LEFT; LE_0] THEN
685 GEN_REWRITE_TAC RAND_CONV [COMPLEX_RING
686 `a * b * c:complex = b * a * c`] THEN
687 REWRITE_TAC[COMPLEX_ADD_RID] THEN BINOP_TAC THENL
688 [ASM_MESON_TAC[COMPLEX_ADD_RID; CEXP_CLOG]; ALL_TAC] THEN
689 SIMP_TAC[ADD_CLAUSES; GSYM CPRODUCT_MUL; FINITE_NUMSEG] THEN
690 REWRITE_TAC[GSYM CEXP_NEG; GSYM VSUM_NEG] THEN
691 SIMP_TAC[CEXP_VSUM; FINITE_NUMSEG] THEN MATCH_MP_TAC CPRODUCT_EQ THEN
692 REWRITE_TAC[IN_NUMSEG] THEN X_GEN_TAC `m:num` THEN STRIP_TAC THEN
693 REWRITE_TAC[CEXP_NEG] THEN
694 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [GSYM COMPLEX_INV_INV] THEN
695 REWRITE_TAC[GSYM COMPLEX_INV_MUL] THEN AP_TERM_TAC THEN
696 GEN_REWRITE_TAC RAND_CONV [COMPLEX_MUL_SYM] THEN
697 GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [COMPLEX_ADD_SYM] THEN
698 MATCH_MP_TAC CEXP_CLOG THEN
699 ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN
700 ASM_REWRITE_TAC[COMPLEX_ENTIRE; COMPLEX_INV_EQ_0] THEN
701 REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ] THEN ASM_ARITH_TAC;
702 MATCH_MP_TAC(ISPEC `cexp` LIM_CONTINUOUS_FUNCTION) THEN
703 ASM_SIMP_TAC[LGAMMA; CONTINUOUS_AT_CEXP]];
704 MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
705 [REWRITE_TAC[cgamma; lim] THEN CONV_TAC SELECT_CONV THEN ASM_MESON_TAC[];
707 MATCH_MP_TAC(TAUT `q /\ (q ==> p) ==> p /\ q`) THEN CONJ_TAC THENL
708 [MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
709 ASM_MESON_TAC[TRIVIAL_LIMIT_SEQUENTIALLY];
710 SIMP_TAC[CEXP_NZ]]]);;
712 let CGAMMA_RECURRENCE_ALT = prove
713 (`!z. cgamma(z) = cgamma(z + Cx(&1)) / z`,
714 GEN_TAC THEN ASM_CASES_TAC `?n. z + Cx(&n) = Cx(&0)` THENL
715 [FIRST_ASSUM(fun th -> REWRITE_TAC[REWRITE_RULE[GSYM CGAMMA_EQ_0] th]) THEN
716 CONV_TAC SYM_CONV THEN REWRITE_TAC[COMPLEX_DIV_EQ_0] THEN
717 ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN
718 FIRST_X_ASSUM(MP_TAC o check (is_exists o concl)) THEN
719 REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN INDUCT_TAC THEN
720 ASM_REWRITE_TAC[COMPLEX_ADD_RID; CGAMMA_EQ_0] THEN
721 REWRITE_TAC[GSYM COMPLEX_ADD_ASSOC; GSYM CX_ADD; REAL_OF_NUM_ADD] THEN
722 MESON_TAC[ADD1; ADD_SYM];
723 RULE_ASSUM_TAC(REWRITE_RULE[NOT_EXISTS_THM])] THEN
724 SUBGOAL_THEN `!n. ~((z + Cx(&1)) + Cx(&n) = Cx(&0))` ASSUME_TAC THENL
725 [REWRITE_TAC[GSYM COMPLEX_ADD_ASSOC; GSYM CX_ADD; REAL_OF_NUM_ADD] THEN
728 SUBGOAL_THEN `~(z = Cx(&0))` ASSUME_TAC THENL
729 [ASM_MESON_TAC[COMPLEX_ADD_LID]; ALL_TAC] THEN
730 MATCH_MP_TAC(COMPLEX_FIELD
731 `(a * b) / c = Cx(&1) /\ ~(a = Cx(&0)) /\ ~(c = Cx(&0)) ==> b = c / a`) THEN
732 ASM_REWRITE_TAC[CGAMMA_EQ_0] THEN
733 MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
735 `\n. (z * (Cx(&(n + 1)) cpow z * Cx(&(FACT(n + 1)))) /
736 cproduct(0..n+1) (\m. z + Cx(&m))) /
737 ((Cx(&n) cpow (z + Cx(&1)) * Cx(&(FACT n))) /
738 cproduct(0..n) (\m. (z + Cx(&1)) + Cx(&m)))` THEN
739 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL
740 [MATCH_MP_TAC LIM_COMPLEX_DIV THEN
741 ASM_REWRITE_TAC[CGAMMA; CGAMMA_EQ_0] THEN
742 MATCH_MP_TAC LIM_COMPLEX_LMUL THEN
743 MP_TAC(SPEC `z:complex` CGAMMA) THEN
744 DISCH_THEN(MP_TAC o SPEC `1` o MATCH_MP SEQ_OFFSET) THEN
747 REWRITE_TAC[GSYM COMPLEX_ADD_ASSOC; GSYM CX_ADD; REAL_OF_NUM_ADD] THEN
748 REWRITE_TAC[ARITH_RULE `1 + n = n + 1`] THEN
749 SIMP_TAC[SYM(ISPECL [`f:num->complex`; `m:num`; `1`] CPRODUCT_OFFSET)] THEN
750 SIMP_TAC[CPRODUCT_CLAUSES_LEFT; LE_0; ADD_CLAUSES] THEN
751 REWRITE_TAC[GSYM ADD1; FACT; CX_MUL; COMPLEX_MUL_ASSOC; GSYM REAL_OF_NUM_MUL;
752 ONCE_REWRITE_RULE[COMPLEX_MUL_SYM] (GSYM CPOW_SUC)] THEN
753 REWRITE_TAC[complex_div; COMPLEX_INV_MUL; COMPLEX_INV_INV] THEN
754 REWRITE_TAC[COMPLEX_ADD_RID; GSYM COMPLEX_MUL_ASSOC] THEN
755 REWRITE_TAC[COMPLEX_RING
756 `a * b * c * d * e * f * g * h:complex =
757 (a * d) * (c * g) * (h * e) * (b * f)`] THEN
758 ASM_SIMP_TAC[GSYM complex_div; COMPLEX_DIV_REFL; CX_INJ;
759 REAL_OF_NUM_EQ; FACT_NZ; COMPLEX_MUL_LID] THEN
760 GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_MUL_LID] THEN
761 MATCH_MP_TAC LIM_COMPLEX_MUL THEN CONJ_TAC THENL
762 [MATCH_MP_TAC LIM_EVENTUALLY THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN
763 X_GEN_TAC `n:num` THEN REWRITE_TAC[] THEN
764 MATCH_MP_TAC COMPLEX_DIV_REFL THEN
765 SIMP_TAC[CPRODUCT_EQ_0; FINITE_NUMSEG] THEN ASM_MESON_TAC[];
767 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
768 EXISTS_TAC `\n. cexp((z + Cx(&1)) * clog(Cx(&1) + inv(Cx(&n))))` THEN
770 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
771 SIMP_TAC[cpow; CX_INJ; REAL_OF_NUM_EQ; NOT_SUC; LE_1] THEN
772 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM CEXP_SUB] THEN AP_TERM_TAC THEN
773 REWRITE_TAC[GSYM COMPLEX_SUB_LDISTRIB] THEN AP_TERM_TAC THEN
774 ASM_SIMP_TAC[CX_INJ; REAL_OF_NUM_EQ; LE_1; COMPLEX_FIELD
775 `~(n = Cx(&0)) ==> Cx(&1) + inv n = (n + Cx(&1)) / n`] THEN
776 REWRITE_TAC[GSYM REAL_OF_NUM_SUC; GSYM CX_ADD] THEN
777 ASM_SIMP_TAC[REAL_OF_NUM_ADD; GSYM CX_DIV; GSYM CX_LOG; LE_1; LOG_DIV;
778 REAL_OF_NUM_LT; REAL_LT_DIV; ARITH_RULE `0 < n + 1`] THEN
781 GEN_REWRITE_TAC LAND_CONV [GSYM CEXP_0] THEN
782 MATCH_MP_TAC(ISPEC `cexp` LIM_CONTINUOUS_FUNCTION) THEN
783 REWRITE_TAC[CONTINUOUS_AT_CEXP] THEN
784 MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
785 MATCH_MP_TAC LIM_NULL_COMPARISON_COMPLEX THEN
786 EXISTS_TAC `\n. Cx(&2) * inv(Cx(&n))` THEN
787 SIMP_TAC[LIM_INV_N; LIM_NULL_COMPLEX_LMUL] THEN
788 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `2` THEN
789 REWRITE_TAC[GE; IN_FROM] THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN
790 MP_TAC(ISPECL [`0`; `Cx(inv(&n))`] TAYLOR_CLOG) THEN
791 SIMP_TAC[VSUM_CLAUSES_NUMSEG; ARITH; VECTOR_SUB_RZERO] THEN
792 REWRITE_TAC[REAL_POW_1; GSYM CX_ADD; COMPLEX_NORM_CX] THEN
793 REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NUM] THEN ANTS_TAC THENL
794 [MATCH_MP_TAC REAL_INV_LT_1 THEN REWRITE_TAC[REAL_OF_NUM_LT] THEN
796 REWRITE_TAC[CX_ADD; CX_INV]] THEN
797 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
798 ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN
799 REWRITE_TAC[real_div; COMPLEX_NORM_MUL] THEN
800 REWRITE_TAC[COMPLEX_NORM_INV; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
801 MATCH_MP_TAC REAL_LE_LMUL THEN
802 REWRITE_TAC[REAL_LE_INV_EQ; REAL_POS] THEN
803 GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN
804 MATCH_MP_TAC REAL_LE_INV2 THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
805 REWRITE_TAC[REAL_ARITH `&1 / &2 <= &1 - x <=> x <= inv(&2)`] THEN
806 REWRITE_TAC[real_div; REAL_MUL_LID] THEN
807 MATCH_MP_TAC REAL_LE_INV2 THEN
808 REWRITE_TAC[REAL_OF_NUM_LT; REAL_OF_NUM_LE] THEN ASM_ARITH_TAC);;
811 (`cgamma(Cx(&1)) = Cx(&1)`,
812 MP_TAC(SPEC `Cx(&1)` CGAMMA) THEN
813 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT; TRIVIAL_LIMIT_SEQUENTIALLY]
814 (ISPEC `sequentially` LIM_UNIQUE)) THEN
815 REWRITE_TAC[GSYM CX_ADD; REAL_OF_NUM_ADD; ARITH_RULE `1 + n = n + 1`] THEN
816 SIMP_TAC[SYM(ISPECL [`f:num->complex`; `m:num`; `1`] CPRODUCT_OFFSET)] THEN
817 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
818 EXISTS_TAC `\n. Cx(&n / (&n + &1))` THEN CONJ_TAC THENL
819 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
820 X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[CX_DIV] THEN
821 ASM_SIMP_TAC[CPOW_N; CX_INJ; REAL_OF_NUM_EQ; LE_1; COMPLEX_POW_1] THEN
822 REWRITE_TAC[complex_div; GSYM COMPLEX_MUL_ASSOC] THEN AP_TERM_TAC THEN
823 ONCE_REWRITE_TAC [GSYM COMPLEX_INV_INV] THEN
824 AP_TERM_TAC THEN REWRITE_TAC[COMPLEX_INV_INV; COMPLEX_INV_MUL] THEN
825 MATCH_MP_TAC(COMPLEX_FIELD
826 `a * b = c /\ ~(b = Cx(&0)) ==> a = inv b * c`) THEN
827 REWRITE_TAC[GSYM CX_MUL; CX_INJ; REAL_OF_NUM_EQ; FACT_NZ] THEN
828 REWRITE_TAC[REAL_OF_NUM_SUC; REAL_OF_NUM_MUL; GSYM(CONJUNCT2 FACT)] THEN
829 REWRITE_TAC[ADD_CLAUSES; ADD1] THEN SPEC_TAC(`n + 1`,`m:num`) THEN
830 INDUCT_TAC THEN REWRITE_TAC[FACT; CPRODUCT_CLAUSES_NUMSEG; ARITH] THEN
831 ASM_REWRITE_TAC[CX_MUL; GSYM REAL_OF_NUM_MUL] THEN
832 REWRITE_TAC[ARITH_RULE `1 <= SUC n`; COMPLEX_MUL_SYM];
834 SIMP_TAC[REAL_FIELD `&n / (&n + &1) = &1 - inv(&n + &1)`] THEN
835 SUBST1_TAC(COMPLEX_RING `Cx(&1) = Cx(&1) - Cx(&0)`) THEN
836 REWRITE_TAC[CX_SUB] THEN MATCH_MP_TAC LIM_SUB THEN
837 REWRITE_TAC[LIM_CONST; REAL_OF_NUM_ADD] THEN
838 MATCH_MP_TAC(ISPECL [`f:num->complex`; `l:complex`; `1`] SEQ_OFFSET) THEN
839 REWRITE_TAC[CX_INV; LIM_INV_N]);;
841 let CGAMMA_RECURRENCE = prove
842 (`!z. cgamma(z + Cx(&1)) = if z = Cx(&0) then Cx(&1) else z * cgamma(z)`,
843 GEN_TAC THEN COND_CASES_TAC THEN
844 ASM_REWRITE_TAC[COMPLEX_ADD_LID; CGAMMA_1] THEN
845 MATCH_MP_TAC(COMPLEX_FIELD `a = b / c /\ ~(c = Cx(&0)) ==> b = c * a`) THEN
846 ASM_MESON_TAC[CGAMMA_RECURRENCE_ALT]);;
848 let CGAMMA_FACT = prove
849 (`!n. cgamma(Cx(&(n + 1))) = Cx(&(FACT n))`,
850 INDUCT_TAC THEN REWRITE_TAC[FACT; ADD_CLAUSES; CGAMMA_1] THEN
851 REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN REWRITE_TAC[CX_ADD] THEN
852 REWRITE_TAC[CGAMMA_RECURRENCE; CX_INJ; REAL_OF_NUM_EQ; ADD_EQ_0; ARITH] THEN
853 ASM_REWRITE_TAC[ADD1; GSYM REAL_OF_NUM_MUL; CX_MUL]);;
855 let CGAMMA_POLES = prove
856 (`!n. cgamma(--(Cx(&n))) = Cx(&0)`,
857 REWRITE_TAC[CGAMMA_EQ_0] THEN MESON_TAC[COMPLEX_ADD_LINV]);;
859 let COMPLEX_DIFFERENTIABLE_AT_CGAMMA = prove
860 (`!z. (!n. ~(z + Cx(&n) = Cx(&0))) ==> cgamma complex_differentiable at z`,
862 (`!z. (!n. ~(z + Cx(&n) = Cx(&0))) /\
863 cgamma complex_differentiable at (z + Cx(&1))
864 ==> cgamma complex_differentiable at z`,
865 REPEAT STRIP_TAC THEN
866 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
867 MAP_EVERY EXISTS_TAC [`\z. cgamma(z + Cx(&1)) / z`; `&1`] THEN
868 REWRITE_TAC[REAL_LT_01] THEN
869 CONJ_TAC THENL [REWRITE_TAC[GSYM CGAMMA_RECURRENCE_ALT]; ALL_TAC] THEN
870 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_DIV_AT THEN
871 REWRITE_TAC[COMPLEX_DIFFERENTIABLE_ID] THEN
872 CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[COMPLEX_ADD_RID]] THEN
873 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
874 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_COMPOSE_AT THEN
875 ASM_REWRITE_TAC[] THEN COMPLEX_DIFFERENTIABLE_TAC) in
876 REPEAT STRIP_TAC THEN MP_TAC(SPEC `abs(Re z) + &1` REAL_ARCH_SIMPLE) THEN
877 DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
879 `!n. n <= N ==> cgamma complex_differentiable (at (z + Cx(&N) - Cx(&n)))`
881 [ALL_TAC; MESON_TAC[LE_REFL; COMPLEX_SUB_REFL; COMPLEX_ADD_RID]] THEN
883 [DISCH_TAC THEN REWRITE_TAC[COMPLEX_SUB_RZERO] THEN
884 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
885 EXISTS_TAC `\z. cexp(lgamma z)` THEN
886 SUBGOAL_THEN `open {z | !n. ~(z + Cx(&n) = Cx(&0))}` MP_TAC THENL
887 [REWRITE_TAC[SET_RULE `{z | !n. P n z} = UNIV DIFF {z | ?n. ~P n z}`] THEN
888 REWRITE_TAC[GSYM closed] THEN MATCH_MP_TAC DISCRETE_IMP_CLOSED THEN
889 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01; IMP_CONJ] THEN
890 REWRITE_TAC[IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
891 SIMP_TAC[COMPLEX_RING `x + y = Cx(&0) <=> x = --y`] THEN
892 REWRITE_TAC[COMPLEX_RING `--x - --y:complex = y - x`] THEN
893 REWRITE_TAC[COMPLEX_EQ_NEG2; CX_INJ; GSYM CX_SUB; COMPLEX_NORM_CX] THEN
894 SIMP_TAC[GSYM REAL_EQ_INTEGERS; INTEGER_CLOSED];
895 REWRITE_TAC[open_def; IN_ELIM_THM] THEN
896 DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN
897 MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
898 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
899 [X_GEN_TAC `w:complex` THEN REWRITE_TAC[dist] THEN DISCH_TAC THEN
900 FIRST_X_ASSUM(MP_TAC o SPEC `w - Cx(&N)`) THEN
901 ASM_SIMP_TAC[dist; COMPLEX_RING `w - n - z:complex = w - (z + n)`] THEN
902 REWRITE_TAC[CGAMMA_LGAMMA] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN
903 FIRST_X_ASSUM(X_CHOOSE_TAC `n:num`) THEN
904 DISCH_THEN(MP_TAC o SPEC `n + N:num`) THEN
905 REWRITE_TAC[GSYM REAL_OF_NUM_ADD; CX_ADD] THEN
906 ASM_REWRITE_TAC[COMPLEX_RING `w - N + n + N:complex = w + n`];
907 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
908 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_COMPOSE_AT THEN
909 REWRITE_TAC[COMPLEX_DIFFERENTIABLE_AT_CEXP] THEN
910 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_AT_LGAMMA THEN
911 REWRITE_TAC[RE_ADD; RE_CX] THEN ASM_REAL_ARITH_TAC]];
912 DISCH_TAC THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC; CX_ADD] THEN
913 MATCH_MP_TAC lemma THEN
914 REWRITE_TAC[COMPLEX_RING `(z + N - (n + w)) + w:complex = z + N - n`] THEN
916 [ALL_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC] THEN
917 X_GEN_TAC `m:num` THEN
918 FIRST_X_ASSUM(MP_TAC o SPEC `(N + m) - (n + 1)`) THEN
919 SUBGOAL_THEN `n + 1 <= N + m`
920 (fun th -> SIMP_TAC[th; GSYM REAL_OF_NUM_SUB])
921 THENL [ASM_ARITH_TAC; ALL_TAC] THEN
922 REWRITE_TAC[GSYM REAL_OF_NUM_ADD; CX_ADD; CX_SUB] THEN
923 CONV_TAC COMPLEX_RING]);;
925 let COMPLEX_DIFFERENTIABLE_WITHIN_CGAMMA = prove
926 (`!z s. (!n. ~(z + Cx(&n) = Cx(&0)))
927 ==> cgamma complex_differentiable at z within s`,
928 SIMP_TAC[COMPLEX_DIFFERENTIABLE_AT_WITHIN;
929 COMPLEX_DIFFERENTIABLE_AT_CGAMMA]);;
931 let HOLOMORPHIC_ON_CGAMMA = prove
932 (`!s. s SUBSET {z | !n. ~(z + Cx(&n) = Cx(&0))}
933 ==> cgamma holomorphic_on s`,
934 REPEAT STRIP_TAC THEN REWRITE_TAC[HOLOMORPHIC_ON_DIFFERENTIABLE] THEN
935 REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_WITHIN_CGAMMA THEN
938 let CONTINUOUS_AT_CGAMMA = prove
939 (`!z. (!n. ~(z + Cx(&n) = Cx(&0))) ==> cgamma continuous at z`,
940 SIMP_TAC[COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT;
941 COMPLEX_DIFFERENTIABLE_AT_CGAMMA]);;
943 let CONTINUOUS_WITHIN_CGAMMA = prove
944 (`!z s. (!n. ~(z + Cx(&n) = Cx(&0)))
945 ==> cgamma continuous at z within s`,
946 SIMP_TAC[COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN;
947 COMPLEX_DIFFERENTIABLE_WITHIN_CGAMMA]);;
949 let CONTINUOUS_ON_CGAMMA = prove
950 (`!s. s SUBSET {z | !n. ~(z + Cx(&n) = Cx(&0))}
951 ==> cgamma continuous_on s`,
952 SIMP_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON; HOLOMORPHIC_ON_CGAMMA]);;
954 let CGAMMA_SIMPLE_POLES = prove
955 (`!n. ((\z. (z + Cx(&n)) * cgamma z) --> --Cx(&1) pow n / Cx(&(FACT n)))
958 [REWRITE_TAC[COMPLEX_ADD_RID; COMPLEX_NEG_0; FACT; complex_pow;
960 ONCE_REWRITE_TAC[CGAMMA_RECURRENCE_ALT] THEN
961 MATCH_MP_TAC LIM_TRANSFORM_AWAY_AT THEN
962 MAP_EVERY EXISTS_TAC [`\z. cgamma(z + Cx(&1))`; `Cx(&1)`] THEN
963 REWRITE_TAC[CONJ_ASSOC] THEN
964 CONJ_TAC THENL [CONV_TAC COMPLEX_FIELD; ALL_TAC] THEN
965 SUBGOAL_THEN `(\z. cgamma (z + Cx(&1))) continuous at (Cx(&0))`
967 [GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
968 MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN
969 CONJ_TAC THENL [CONTINUOUS_TAC; ALL_TAC] THEN
970 MATCH_MP_TAC CONTINUOUS_AT_CGAMMA THEN
971 REWRITE_TAC[GSYM CX_ADD; CX_INJ] THEN REAL_ARITH_TAC;
972 REWRITE_TAC[CONTINUOUS_AT; COMPLEX_ADD_LID; CGAMMA_1]];
973 REWRITE_TAC[FACT; CX_MUL; GSYM REAL_OF_NUM_MUL] THEN
974 ONCE_REWRITE_TAC[CGAMMA_RECURRENCE_ALT] THEN
975 REWRITE_TAC[complex_div; complex_pow; COMPLEX_INV_MUL] THEN
976 REWRITE_TAC[SIMPLE_COMPLEX_ARITH
977 `(--Cx(&1) * p) * is * i = (p * i) * --is`] THEN
978 REWRITE_TAC[COMPLEX_MUL_ASSOC] THEN MATCH_MP_TAC LIM_COMPLEX_MUL THEN
980 [FIRST_X_ASSUM(MP_TAC o
981 ISPECL [`at (--Cx(&(SUC n)))`; `\z. z + Cx(&1)`] o
982 MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT]
983 (REWRITE_RULE[CONJ_ASSOC] LIM_COMPOSE_AT))) THEN
984 REWRITE_TAC[o_DEF; GSYM REAL_OF_NUM_SUC; CX_ADD] THEN
985 REWRITE_TAC[COMPLEX_RING `(z + Cx(&1)) + w = z + (w + Cx(&1))`] THEN
986 REWRITE_TAC[GSYM complex_div] THEN DISCH_THEN MATCH_MP_TAC THEN
988 [LIM_TAC THEN CONV_TAC COMPLEX_RING;
989 REWRITE_TAC[EVENTUALLY_AT; GSYM DIST_NZ] THEN
990 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN
991 CONV_TAC COMPLEX_RING];
992 REWRITE_TAC[COMPLEX_NEG_INV; GSYM CONTINUOUS_AT] THEN
993 GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
994 MATCH_MP_TAC CONTINUOUS_COMPLEX_INV_AT THEN
995 REWRITE_TAC[GSYM CX_NEG; CX_INJ; GSYM REAL_OF_NUM_SUC] THEN
996 REWRITE_TAC[CONTINUOUS_AT_ID] THEN REAL_ARITH_TAC]]);;
998 let CNJ_CGAMMA = prove
999 (`!z. cnj(cgamma z) = cgamma(cnj z)`,
1000 GEN_TAC THEN MP_TAC(SPEC `cnj z` CGAMMA) THEN
1001 MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`]
1003 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
1004 GEN_REWRITE_TAC I [GSYM LIM_CNJ] THEN REWRITE_TAC[CNJ_CNJ] THEN
1005 MP_TAC(SPEC `z:complex` CGAMMA) THEN
1006 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
1007 SIMP_TAC[CNJ_DIV; CNJ_MUL; CNJ_CX; CNJ_CPRODUCT; FINITE_NUMSEG] THEN
1008 REWRITE_TAC[CNJ_ADD; CNJ_CNJ; CNJ_CX] THEN
1009 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
1010 X_GEN_TAC `n:num` THEN REWRITE_TAC[cpow; CNJ_EQ_0] THEN
1011 DISCH_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[CNJ_CX; CNJ_CEXP] THEN
1012 REWRITE_TAC[CNJ_MUL; CNJ_CNJ] THEN
1013 ASM_SIMP_TAC[CNJ_CLOG; RE_CX; REAL_OF_NUM_LT; LE_1; CNJ_CX]);;
1015 let REAL_GAMMA = prove
1016 (`!z. real z ==> real(cgamma z)`,
1017 SIMP_TAC[REAL_CNJ; CNJ_CGAMMA]);;
1019 let RE_POS_CGAMMA_REAL = prove
1020 (`!z. real z /\ &0 <= Re z ==> &0 <= Re(cgamma z)`,
1021 REWRITE_TAC[REAL_EXISTS; LEFT_IMP_EXISTS_THM; IMP_CONJ] THEN
1022 GEN_TAC THEN X_GEN_TAC `x:real` THEN DISCH_THEN SUBST1_TAC THEN
1023 REWRITE_TAC[RE_CX] THEN DISCH_TAC THEN MP_TAC(SPEC `Cx x` CGAMMA) THEN
1024 MATCH_MP_TAC(REWRITE_RULE[TAUT `p /\ q /\ r ==> s <=> p /\ r ==> q ==> s`]
1026 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN
1027 EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
1028 ASM_SIMP_TAC[cpow; CX_INJ; REAL_OF_NUM_EQ; LE_1; GSYM CX_LOG; GSYM CX_INV;
1029 REAL_OF_NUM_LT; GSYM CX_MUL; GSYM CX_EXP; RE_MUL_CX; RE_CX;
1030 complex_div; GSYM CX_ADD; GSYM CX_PRODUCT; FINITE_NUMSEG] THEN
1031 MATCH_MP_TAC REAL_LE_MUL THEN
1032 SIMP_TAC[REAL_LE_MUL; REAL_EXP_POS_LE; REAL_POS; REAL_LE_INV_EQ] THEN
1033 MATCH_MP_TAC PRODUCT_POS_LE_NUMSEG THEN ASM_REAL_ARITH_TAC);;
1035 let CGAMMA_LEGENDRE_ALT = prove
1036 (`!z. cgamma(z) * cgamma(z + Cx(&1) / Cx(&2)) =
1037 Cx(&2) cpow (Cx(&1) - Cx(&2) * z) *
1038 cgamma(Cx(&1) / Cx(&2)) * cgamma(Cx(&2) * z)`,
1039 REWRITE_TAC[GSYM CX_DIV] THEN
1041 `?f. !z. (!n. ~(Cx(&2) * z + Cx(&n) = Cx(&0)))
1042 ==> (f --> (cgamma(z) * cgamma(z + Cx(&1 / &2))) /
1043 (Cx(&2) cpow (Cx(&1) - Cx(&2) * z) * cgamma(Cx(&2) * z)))
1047 `\n. (Cx(&n) cpow Cx(&1 / &2) * inv (Cx(&2))) *
1048 (Cx(&(FACT n)) pow 2 * inv (Cx(&(FACT (2 * n))))) *
1049 Cx(&4) pow (n + 1) *
1050 inv(Cx(&(2 * n + 1)))` THEN
1051 REPEAT STRIP_TAC THEN MP_TAC(SPEC `Cx(&2) * z` CGAMMA) THEN
1052 DISCH_THEN(MP_TAC o SPEC `\n. 2 * n` o MATCH_MP
1053 (REWRITE_RULE[IMP_CONJ_ALT] LIM_SUBSEQUENCE)) THEN
1054 REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL [ARITH_TAC; ALL_TAC] THEN
1055 DISCH_THEN(MP_TAC o SPEC `Cx(&2) cpow (Cx(&1) - Cx(&2) * z)` o
1056 MATCH_MP LIM_COMPLEX_LMUL) THEN
1057 MP_TAC(CONJ (SPEC `z:complex` CGAMMA) (SPEC `z + Cx(&1 / &2)` CGAMMA)) THEN
1058 DISCH_THEN(MP_TAC o MATCH_MP LIM_COMPLEX_MUL) THEN
1059 REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP
1060 (ONCE_REWRITE_RULE[IMP_CONJ]
1061 (REWRITE_RULE[CONJ_ASSOC] LIM_COMPLEX_DIV))) THEN
1062 ASM_REWRITE_TAC[COMPLEX_ENTIRE; CPOW_EQ_0; CGAMMA_EQ_0] THEN
1063 REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ] THEN
1064 SUBGOAL_THEN `((\n. (Cx(&2) * z + Cx(&(2 * n + 1))) / Cx(&(2 * n + 1)))
1065 --> Cx(&1)) sequentially`
1067 [SIMP_TAC[complex_div; COMPLEX_MUL_RINV; CX_INJ; REAL_OF_NUM_EQ;
1068 COMPLEX_ADD_RDISTRIB; ARITH_RULE `~(2 * n + 1 = 0)`] THEN
1069 ONCE_REWRITE_TAC[LIM_NULL_COMPLEX] THEN
1070 REWRITE_TAC[COMPLEX_RING `(a + b) - b:complex = a`] THEN
1071 MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
1072 MP_TAC(SPEC `\n. 2 * n + 1`
1073 (MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] LIM_SUBSEQUENCE) LIM_INV_N)) THEN
1074 REWRITE_TAC[o_DEF] THEN DISCH_THEN MATCH_MP_TAC THEN ARITH_TAC;
1075 REWRITE_TAC[IMP_IMP] THEN
1076 DISCH_THEN(MP_TAC o MATCH_MP LIM_COMPLEX_MUL)] THEN
1077 REWRITE_TAC[COMPLEX_MUL_LID] THEN
1078 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
1079 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
1080 EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
1081 REWRITE_TAC[CPOW_ADD; complex_div; COMPLEX_INV_MUL; COMPLEX_INV_INV] THEN
1082 REWRITE_TAC[GSYM REAL_OF_NUM_MUL; CX_MUL] THEN
1083 SIMP_TAC[CPOW_MUL_REAL; REAL_CX; RE_CX; REAL_POS] THEN
1084 REWRITE_TAC[CPOW_SUB; CPOW_N; CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ] THEN
1085 SIMP_TAC[COMPLEX_POW_1; complex_div; COMPLEX_INV_INV; COMPLEX_INV_MUL] THEN
1086 REWRITE_TAC[GSYM COMPLEX_MUL_ASSOC] THEN REWRITE_TAC[COMPLEX_RING
1087 `x * y * a * b * c * d * e * f * g * h * i * j * k * l * m:complex =
1088 (x * y) * (e * h) * ((a * d) * k) *
1089 ((b * f) * l) * (i * j) * (m * c * g)`] THEN
1090 REWRITE_TAC[GSYM COMPLEX_POW_2; COMPLEX_MUL_2; CPOW_ADD] THEN
1091 ASM_SIMP_TAC[COMPLEX_MUL_RINV; COMPLEX_POW_EQ_0; CPOW_EQ_0;
1092 CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ; LE_1; COMPLEX_MUL_LID] THEN
1093 REWRITE_TAC[GSYM COMPLEX_MUL_2; GSYM COMPLEX_INV_MUL] THEN
1094 SIMP_TAC[GSYM CPRODUCT_MUL; FINITE_NUMSEG] THEN
1095 REWRITE_TAC[COMPLEX_RING
1096 `(z + m) * ((z + Cx(&1 / &2)) + m) =
1098 (Cx(&2) * z + Cx(&2) * m) * (Cx(&2) * z + Cx(&2) * m + Cx(&1))`] THEN
1099 SIMP_TAC[CPRODUCT_MUL; FINITE_NUMSEG; CPRODUCT_CONST_NUMSEG] THEN
1100 SIMP_TAC[GSYM CPRODUCT_MUL; SUB_0; FINITE_NUMSEG] THEN
1101 REWRITE_TAC[GSYM CX_ADD; GSYM CX_MUL;
1102 REAL_OF_NUM_ADD; REAL_OF_NUM_MUL] THEN
1103 REWRITE_TAC[GSYM CPRODUCT_PAIR] THEN
1104 SIMP_TAC[GSYM ADD1; CPRODUCT_CLAUSES_NUMSEG] THEN
1105 REWRITE_TAC[ADD1] THEN CONV_TAC NUM_REDUCE_CONV THEN
1106 SIMP_TAC[LE_0; COMPLEX_INV_MUL; COMPLEX_INV_INV; GSYM COMPLEX_POW_INV] THEN
1107 ONCE_REWRITE_TAC[COMPLEX_RING
1108 `x * a * b * c * d * e * f:complex = (c * e) * a * b * d * f * x`] THEN
1109 ASM_SIMP_TAC[COMPLEX_MUL_RINV; CPRODUCT_EQ_0; FINITE_NUMSEG] THEN
1110 REWRITE_TAC[COMPLEX_MUL_LID; COMPLEX_MUL_ASSOC] THEN
1111 AP_THM_TAC THEN AP_TERM_TAC THEN
1112 ASM_SIMP_TAC[GSYM COMPLEX_MUL_ASSOC; COMPLEX_MUL_LINV] THEN
1113 CONV_TAC COMPLEX_RING;
1115 X_GEN_TAC `z:complex` THEN
1116 ASM_CASES_TAC `!n. ~(Cx(&2) * z + Cx(&n) = Cx(&0))` THENL
1117 [FIRST_X_ASSUM(fun th ->
1118 MP_TAC(SPEC `z:complex` th) THEN MP_TAC(SPEC `Cx(&1 / &2)` th)) THEN
1119 ASM_REWRITE_TAC[GSYM CX_ADD; GSYM CX_MUL; GSYM CX_SUB; CX_INJ] THEN
1120 ANTS_TAC THENL [REAL_ARITH_TAC; CONV_TAC REAL_RAT_REDUCE_CONV] THEN
1121 REWRITE_TAC[CGAMMA_1; CPOW_N; CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ] THEN
1122 REWRITE_TAC[complex_pow; COMPLEX_MUL_RID; COMPLEX_DIV_1; IMP_IMP] THEN
1123 DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[IMP_CONJ_ALT]
1125 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN MATCH_MP_TAC(COMPLEX_FIELD
1126 `~(p = Cx(&0)) /\ ~(z2 = Cx(&0))
1127 ==> a = (z * z') / (p * z2) ==> z * z' = p * a * z2`) THEN
1128 ASM_REWRITE_TAC[CGAMMA_EQ_0; CPOW_EQ_0; CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ];
1129 MATCH_MP_TAC(COMPLEX_RING
1130 `z = Cx(&0) /\ ((w = Cx(&0)) \/ (y = Cx(&0)))
1131 ==> w * y = p * q * z`) THEN
1132 REWRITE_TAC[CGAMMA_EQ_0] THEN
1133 CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
1134 REWRITE_TAC[OR_EXISTS_THM; GSYM COMPLEX_ADD_ASSOC] THEN
1135 ONCE_REWRITE_TAC[COMPLEX_RING
1136 `z + n = Cx(&0) <=> Cx(&2) * z + Cx(&2) * n = Cx(&0)`] THEN
1137 REWRITE_TAC[GSYM CX_ADD; GSYM CX_MUL] THEN
1138 REWRITE_TAC[REAL_ARITH `&2 * (&1 / &2 + n) = &2 * n + &1`] THEN
1139 REWRITE_TAC[REAL_OF_NUM_SUC; REAL_OF_NUM_MUL] THEN
1140 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
1141 DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN
1142 MP_TAC(SPEC `n:num` EVEN_OR_ODD) THEN
1143 MESON_TAC[ODD_EXISTS; EVEN_EXISTS]]);;
1145 let CGAMMA_REFLECTION = prove
1146 (`!z. cgamma(z) * cgamma(Cx(&1) - z) = Cx pi / csin(Cx pi * z)`,
1148 (`!w z. (?n. integer n /\ w = Cx n) /\ (?n. integer n /\ z = Cx n) /\
1152 REPEAT(DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC)) THEN
1153 ASM_REWRITE_TAC[DIST_CX] THEN ASM_MESON_TAC[REAL_EQ_INTEGERS_IMP]) in
1155 `g = \z. if ?n. integer n /\ z = Cx n then Cx pi
1156 else cgamma(z) * cgamma(Cx(&1) - z) * csin(Cx pi * z)` THEN
1157 SUBGOAL_THEN `!z. g(z + Cx(&1)):complex = g(z)` ASSUME_TAC THENL
1158 [GEN_TAC THEN EXPAND_TAC "g" THEN
1159 MATCH_MP_TAC(MESON[] `(p <=> p') /\ a = a' /\ b = b'
1160 ==> (if p then a else b) = (if p' then a' else b')`) THEN
1161 REWRITE_TAC[COMPLEX_RING `z + Cx(&1) = w <=> z = w - Cx(&1)`] THEN
1163 [REWRITE_TAC[GSYM CX_SUB] THEN
1164 MESON_TAC[INTEGER_CLOSED; REAL_ARITH `(n + &1) - &1 = n`];
1165 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [CGAMMA_RECURRENCE_ALT] THEN
1166 REWRITE_TAC[COMPLEX_RING `a - (z + a):complex = --z`] THEN
1167 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV)
1168 [CGAMMA_RECURRENCE_ALT] THEN
1169 REWRITE_TAC[COMPLEX_ADD_LDISTRIB; COMPLEX_MUL_RID; CSIN_ADD] THEN
1170 REWRITE_TAC[GSYM CX_SIN; GSYM CX_COS; SIN_PI; COS_PI] THEN
1171 REWRITE_TAC[COMPLEX_RING `--z + Cx(&1) = Cx(&1) - z`] THEN
1172 REWRITE_TAC[complex_div; COMPLEX_INV_NEG; COMPLEX_MUL_AC;
1173 CX_NEG; COMPLEX_MUL_LID; COMPLEX_MUL_LZERO] THEN
1174 REWRITE_TAC[COMPLEX_ADD_RID; COMPLEX_MUL_LNEG; COMPLEX_MUL_RNEG] THEN
1175 REWRITE_TAC[COMPLEX_MUL_LID; COMPLEX_NEG_NEG] THEN
1176 REWRITE_TAC[COMPLEX_MUL_AC]];
1178 SUBGOAL_THEN `!n z. integer n ==> g(z + Cx n):complex = g z` ASSUME_TAC THENL
1179 [SUBGOAL_THEN `!n z. g(z + Cx(&n)):complex = g z` ASSUME_TAC THENL
1181 ASM_REWRITE_TAC[GSYM REAL_OF_NUM_SUC; COMPLEX_ADD_RID] THEN
1182 ASM_REWRITE_TAC[CX_ADD; COMPLEX_ADD_ASSOC];
1183 REWRITE_TAC[integer] THEN REPEAT STRIP_TAC THEN
1184 FIRST_X_ASSUM(DISJ_CASES_THEN SUBST1_TAC o MATCH_MP
1185 (REAL_ARITH `abs x = a ==> x = a \/ x = --a`)) THEN
1186 ASM_REWRITE_TAC[CX_NEG; GSYM complex_sub] THEN
1187 ASM_MESON_TAC[COMPLEX_RING `(z - w) + w:complex = z`]];
1189 SUBGOAL_THEN `!z. ~(?n. integer n /\ z = Cx n)
1190 ==> g complex_differentiable (at z)`
1192 [REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
1193 EXISTS_TAC `\z. cgamma z * cgamma(Cx(&1) - z) * csin(Cx pi * z)` THEN
1194 SUBGOAL_THEN `closed {z | ?n. integer n /\ z = Cx n}` MP_TAC THENL
1195 [MATCH_MP_TAC DISCRETE_IMP_CLOSED THEN EXISTS_TAC `&1` THEN
1196 REWRITE_TAC[REAL_LT_01; IN_ELIM_THM] THEN MESON_TAC[lemma; dist];
1197 REWRITE_TAC[closed; OPEN_CONTAINS_BALL; IN_UNIV; IN_DIFF]] THEN
1198 DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN
1199 ASM_REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN
1200 REWRITE_TAC[SUBSET; IN_BALL; IN_DIFF; IN_ELIM_THM; IN_UNIV] THEN
1201 X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1202 ONCE_REWRITE_TAC[DIST_SYM] THEN EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
1204 REPEAT(MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_MUL_AT THEN CONJ_TAC) THENL
1206 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
1207 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_COMPOSE_AT THEN
1210 (MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_AT_CGAMMA ORELSE
1211 COMPLEX_DIFFERENTIABLE_TAC) THEN
1212 REWRITE_TAC[COMPLEX_RING `z + a:complex = b <=> z = b - a`;
1213 COMPLEX_RING `Cx(&1) - z = a - b <=> z = b - a + Cx(&1)`] THEN
1214 REWRITE_TAC[GSYM CX_ADD; GSYM CX_SUB] THEN ASM_MESON_TAC[INTEGER_CLOSED];
1216 SUBGOAL_THEN `g complex_differentiable at (Cx(&0))` ASSUME_TAC THENL
1217 [MATCH_MP_TAC HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT THEN
1218 EXISTS_TAC `ball(Cx(&0),&1)` THEN
1219 REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN
1220 MATCH_MP_TAC NO_ISOLATED_SINGULARITY THEN EXISTS_TAC `{Cx(&0)}` THEN
1221 REWRITE_TAC[OPEN_BALL; FINITE_SING] THEN CONJ_TAC THENL
1223 REWRITE_TAC[HOLOMORPHIC_ON_DIFFERENTIABLE; IN_DIFF; IN_SING] THEN
1224 X_GEN_TAC `z:complex` THEN REWRITE_TAC[COMPLEX_IN_BALL_0] THEN
1225 STRIP_TAC THEN MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_AT_WITHIN THEN
1226 FIRST_X_ASSUM MATCH_MP_TAC THEN UNDISCH_TAC `~(z = Cx(&0))` THEN
1227 REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_TAC THEN MATCH_MP_TAC lemma THEN
1228 ASM_REWRITE_TAC[dist; COMPLEX_SUB_RZERO] THEN
1229 MESON_TAC[INTEGER_CLOSED]] THEN
1230 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
1231 X_GEN_TAC `z:complex` THEN REWRITE_TAC[COMPLEX_IN_BALL_0] THEN
1232 DISCH_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
1233 ASM_CASES_TAC `z = Cx(&0)` THENL
1235 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT THEN
1236 FIRST_X_ASSUM MATCH_MP_TAC THEN UNDISCH_TAC `~(z = Cx(&0))` THEN
1237 REWRITE_TAC[CONTRAPOS_THM] THEN DISCH_TAC THEN MATCH_MP_TAC lemma THEN
1238 ASM_REWRITE_TAC[dist; COMPLEX_SUB_RZERO] THEN
1239 MESON_TAC[INTEGER_CLOSED]] THEN
1240 FIRST_X_ASSUM SUBST_ALL_TAC THEN REWRITE_TAC[CONTINUOUS_AT] THEN
1242 REWRITE_TAC[MESON[INTEGER_CLOSED] `?n. integer n /\ Cx(&0) = Cx(n)`] THEN
1243 MATCH_MP_TAC LIM_TRANSFORM_WITHIN_OPEN THEN
1244 EXISTS_TAC `\z. Cx pi * (z * cgamma(z)) * cgamma(Cx(&1) - z) *
1245 csin(Cx pi * z) / (Cx pi * z)` THEN
1246 EXISTS_TAC `ball(Cx(&0),&1)` THEN
1247 REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN CONJ_TAC THENL
1248 [X_GEN_TAC `w:complex` THEN REWRITE_TAC[COMPLEX_IN_BALL_0] THEN
1249 STRIP_TAC THEN COND_CASES_TAC THENL
1250 [UNDISCH_TAC `~(w = Cx(&0))` THEN
1251 MATCH_MP_TAC(TAUT `p ==> ~p ==> q`) THEN MATCH_MP_TAC lemma THEN
1252 ASM_REWRITE_TAC[dist; COMPLEX_SUB_RZERO] THEN
1253 MESON_TAC[INTEGER_CLOSED];
1254 UNDISCH_TAC `~(w = Cx(&0))` THEN MP_TAC PI_NZ THEN
1255 REWRITE_TAC[GSYM CX_INJ] THEN CONV_TAC COMPLEX_FIELD];
1256 GEN_REWRITE_TAC LAND_CONV [COMPLEX_RING
1257 `p = p * Cx(&1) * Cx(&1) * Cx(&1)`] THEN
1258 MATCH_MP_TAC LIM_COMPLEX_LMUL THEN MATCH_MP_TAC LIM_COMPLEX_MUL THEN
1260 [MP_TAC(SPEC `0` CGAMMA_SIMPLE_POLES) THEN
1261 REWRITE_TAC[FACT; COMPLEX_DIV_1; complex_pow; COMPLEX_ADD_RID] THEN
1262 REWRITE_TAC[COMPLEX_NEG_0];
1264 MATCH_MP_TAC LIM_COMPLEX_MUL THEN CONJ_TAC THENL
1265 [SUBGOAL_THEN `(cgamma o (\z. Cx(&1) - z)) continuous (at (Cx(&0)))`
1267 [MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN
1268 CONJ_TAC THENL [CONTINUOUS_TAC; ALL_TAC] THEN
1269 MATCH_MP_TAC CONTINUOUS_AT_CGAMMA THEN
1270 REWRITE_TAC[GSYM CX_ADD; GSYM CX_SUB; CX_INJ] THEN
1272 REWRITE_TAC[CONTINUOUS_AT; o_DEF; COMPLEX_SUB_RZERO; CGAMMA_1]];
1274 `(\z. csin(Cx pi * z) / (Cx pi * z)) =
1275 (\z. csin z / z) o (\w. Cx pi * w)`
1276 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
1277 MATCH_MP_TAC LIM_COMPOSE_AT THEN
1278 EXISTS_TAC `Cx(&0)` THEN REWRITE_TAC[LIM_CSIN_OVER_X] THEN
1279 SIMP_TAC[EVENTUALLY_AT; COMPLEX_ENTIRE; CX_INJ; PI_NZ;
1281 CONJ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_01]] THEN
1282 LIM_TAC THEN CONV_TAC COMPLEX_RING]];
1284 SUBGOAL_THEN `g holomorphic_on (:complex)` ASSUME_TAC THENL
1285 [REWRITE_TAC[HOLOMORPHIC_ON_DIFFERENTIABLE; WITHIN_UNIV; IN_UNIV] THEN
1286 X_GEN_TAC `z:complex` THEN ASM_CASES_TAC `?n. integer n /\ z = Cx n` THEN
1288 FIRST_X_ASSUM(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN
1289 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
1290 EXISTS_TAC `(g:complex->complex) o (\z. z - Cx n)` THEN
1291 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN CONJ_TAC THENL
1292 [REWRITE_TAC[o_THM] THEN
1293 ASM_MESON_TAC[COMPLEX_RING `(z - w) + w:complex = z`];
1294 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_COMPOSE_AT THEN
1295 ASM_REWRITE_TAC[COMPLEX_SUB_REFL] THEN
1296 COMPLEX_DIFFERENTIABLE_TAC];
1299 `!z. g(z / Cx(&2)) * g((z + Cx(&1)) / Cx(&2)) =
1300 cgamma(Cx(&1 / &2)) pow 2 * g(z)`
1302 [MATCH_MP_TAC(SET_RULE
1303 `!s. s = UNIV /\ (!x. x IN s ==> P x) ==> !x. P x`) THEN
1304 EXISTS_TAC `closure {z | !n. ~(integer n /\ z = Cx n)}` THEN
1306 [REWRITE_TAC[CLOSURE_INTERIOR] THEN
1307 MATCH_MP_TAC(SET_RULE `s = {} ==> t DIFF s = t`) THEN
1308 MATCH_MP_TAC COUNTABLE_EMPTY_INTERIOR THEN
1309 MATCH_MP_TAC DISCRETE_IMP_COUNTABLE THEN
1310 REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_THM] THEN
1311 REPEAT STRIP_TAC THEN EXISTS_TAC `&1` THEN
1312 REWRITE_TAC[GSYM REAL_NOT_LT; GSYM dist; REAL_LT_01] THEN
1313 ASM_MESON_TAC[lemma];
1315 ONCE_REWRITE_TAC[GSYM COMPLEX_SUB_0] THEN REWRITE_TAC[GSYM IN_SING] THEN
1316 MATCH_MP_TAC FORALL_IN_CLOSURE THEN REWRITE_TAC[CLOSED_SING] THEN
1318 [MATCH_MP_TAC CONTINUOUS_ON_SUB THEN CONJ_TAC THEN
1319 MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN CONJ_TAC THEN
1320 REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
1321 TRY(GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
1322 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC) THEN
1323 FIRST_ASSUM(MP_TAC o MATCH_MP HOLOMORPHIC_ON_IMP_CONTINUOUS_ON) THEN
1324 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
1325 REWRITE_TAC[IN_UNIV; WITHIN_UNIV] THEN REPEAT STRIP_TAC THEN
1326 MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN ASM_REWRITE_TAC[] THEN
1327 REWRITE_TAC[complex_div] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN
1328 CONJ_TAC THEN CONTINUOUS_TAC;
1330 X_GEN_TAC `z:complex` THEN
1331 REWRITE_TAC[IN_ELIM_THM; IN_SING; COMPLEX_SUB_0] THEN DISCH_TAC THEN
1332 EXPAND_TAC "g" THEN REWRITE_TAC[] THEN
1333 REWRITE_TAC[COMPLEX_RING `z / Cx(&2) = w <=> z = Cx(&2) * w`] THEN
1334 REWRITE_TAC[COMPLEX_RING `z + Cx(&1) = w <=> z = w - Cx(&1)`] THEN
1335 REWRITE_TAC[GSYM CX_MUL; GSYM CX_SUB] THEN
1336 REPEAT(COND_CASES_TAC THENL [ASM_MESON_TAC[INTEGER_CLOSED]; ALL_TAC]) THEN
1337 REWRITE_TAC[COMPLEX_RING
1338 `(a * b * c) * (d * e * f):complex =
1339 (a * d) * (e * b) * (c * f)`] THEN
1340 REWRITE_TAC[COMPLEX_RING
1341 `Cx(&1) - (z + Cx(&1)) / Cx(&2) = (Cx(&1) - z) / Cx(&2)`] THEN
1342 REWRITE_TAC[COMPLEX_RING
1343 `(z + Cx(&1)) / Cx(&2) = z / Cx(&2) + Cx(&1) / Cx(&2) /\
1344 Cx(&1) - z / Cx(&2) = ((Cx(&1) - z) / Cx(&2)) + Cx(&1) / Cx(&2)`] THEN
1345 REWRITE_TAC[CGAMMA_LEGENDRE_ALT] THEN
1346 MP_TAC(ISPEC `Cx(&1 / &2) * z * Cx pi` CSIN_DOUBLE) THEN
1347 MP_TAC(SPECL [`Cx(&2)`; `z:complex`; `--z:complex`] CPOW_ADD) THEN
1348 REWRITE_TAC[COMPLEX_ADD_RINV] THEN
1349 CONV_TAC(DEPTH_BINOP_CONV `==>`
1350 (BINOP_CONV(DEPTH_BINOP_CONV `complex_mul`
1351 (RAND_CONV COMPLEX_POLY_CONV)))) THEN
1352 REWRITE_TAC[CPOW_ADD; CX_NEG; COMPLEX_MUL_LNEG; COMPLEX_MUL_LID] THEN
1353 REWRITE_TAC[CPOW_N; CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ; CSIN_ADD] THEN
1354 REWRITE_TAC[GSYM CX_MUL; GSYM CX_SIN; GSYM CX_COS; SIN_PI2; COS_PI2;
1355 REAL_ARITH `&1 / &2 * x = x / &2`] THEN
1356 CONV_TAC COMPLEX_RING;
1359 `?h. h holomorphic_on (:complex) /\
1360 !z. z IN (:complex) ==> g z = cexp(h z)`
1362 [MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]
1363 CONTRACTIBLE_IMP_HOLOMORPHIC_LOG) THEN
1364 ASM_REWRITE_TAC[CONTRACTIBLE_UNIV; IN_UNIV] THEN
1365 X_GEN_TAC `z:complex` THEN EXPAND_TAC "g" THEN
1366 COND_CASES_TAC THEN REWRITE_TAC[CX_INJ; PI_NZ] THEN
1367 REWRITE_TAC[CGAMMA_EQ_0; CSIN_EQ_0; COMPLEX_ENTIRE] THEN
1368 REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM] CX_MUL] THEN
1369 REWRITE_TAC[COMPLEX_RING `a - z + b = Cx(&0) <=> z = a + b`] THEN
1370 REWRITE_TAC[COMPLEX_EQ_MUL_LCANCEL; CX_INJ; PI_NZ] THEN
1371 REWRITE_TAC[COMPLEX_RING `z + a = Cx(&0) <=> z = --a`] THEN
1372 REWRITE_TAC[GSYM CX_ADD; GSYM CX_NEG] THEN ASM_MESON_TAC[INTEGER_CLOSED];
1373 REWRITE_TAC[IN_UNIV] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM)] THEN
1374 MP_TAC(ISPECL [`h:complex->complex`; `(:complex)`]
1375 HOLOMORPHIC_ON_OPEN) THEN
1376 ASM_REWRITE_TAC[OPEN_UNIV; IN_UNIV; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
1377 X_GEN_TAC `h':complex->complex` THEN DISCH_TAC THEN
1379 `!z. (h'(z / Cx(&2)) + h'((z + Cx(&1)) / Cx(&2))) / Cx(&2) = h'(z)`
1381 [X_GEN_TAC `z:complex` THEN MATCH_MP_TAC
1382 (COMPLEX_RING `!a. ~(a = Cx(&0)) /\ a * x = a * y ==> x = y`) THEN
1383 EXISTS_TAC `g(z / Cx(&2)) * g((z + Cx(&1)) / Cx(&2))` THEN
1384 REWRITE_TAC[COMPLEX_ENTIRE] THEN
1385 CONJ_TAC THENL [ASM_MESON_TAC[CEXP_NZ]; ALL_TAC] THEN
1386 MATCH_MP_TAC COMPLEX_DERIVATIVE_UNIQUE_AT THEN
1387 EXISTS_TAC `\z. g (z / Cx(&2)) * g ((z + Cx(&1)) / Cx(&2)):complex` THEN
1388 EXISTS_TAC `z:complex` THEN CONJ_TAC THENL
1389 [REWRITE_TAC[]; ASM_REWRITE_TAC[]] THEN
1390 FIRST_ASSUM(SUBST1_TAC o SYM o GEN_REWRITE_RULE I [GSYM FUN_EQ_THM]) THEN
1392 ASM GEN_COMPLEX_DIFF_TAC [] THEN
1393 (CONJ_TAC THENL [ASM_REWRITE_TAC[]; CONV_TAC COMPLEX_RING]);
1395 MP_TAC(ISPECL [`h:complex->complex`; `h':complex->complex`; `(:complex)`]
1396 HOLOMORPHIC_DERIVATIVE) THEN
1397 ASM_REWRITE_TAC[OPEN_UNIV] THEN DISCH_TAC THEN
1398 MP_TAC(ISPECL [`h':complex->complex`; `(:complex)`]
1399 HOLOMORPHIC_ON_OPEN) THEN
1400 ASM_REWRITE_TAC[OPEN_UNIV; IN_UNIV; SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
1401 X_GEN_TAC `h'':complex->complex` THEN DISCH_TAC THEN
1403 `!z. (h''(z / Cx(&2)) + h''((z + Cx(&1)) / Cx(&2))) / Cx(&4) = h''(z)`
1405 [X_GEN_TAC `z:complex` THEN
1406 MATCH_MP_TAC COMPLEX_DERIVATIVE_UNIQUE_AT THEN
1407 EXISTS_TAC `\z. (h'(z / Cx(&2)) + h'((z + Cx(&1)) / Cx(&2))) / Cx(&2)` THEN
1408 EXISTS_TAC `z:complex` THEN CONJ_TAC THENL
1409 [REWRITE_TAC[]; ASM_REWRITE_TAC[]] THEN
1410 ASM GEN_COMPLEX_DIFF_TAC [] THEN
1411 (CONJ_TAC THENL [ASM_REWRITE_TAC[ETA_AX]; CONV_TAC COMPLEX_RING]);
1413 MP_TAC(ISPECL [`h':complex->complex`; `h'':complex->complex`; `(:complex)`]
1414 HOLOMORPHIC_DERIVATIVE) THEN
1415 ASM_REWRITE_TAC[OPEN_UNIV] THEN DISCH_TAC THEN
1416 SUBGOAL_THEN `!z. z IN (:complex) ==> h''(z) = Cx(&0)` MP_TAC THENL
1417 [MATCH_MP_TAC ANALYTIC_CONTINUATION THEN
1418 EXISTS_TAC `cball(Cx(&0),&1)` THEN EXISTS_TAC `Cx(&0)` THEN
1419 ASM_REWRITE_TAC[CONNECTED_UNIV; SUBSET_UNIV; IN_UNIV] THEN
1420 SIMP_TAC[INTERIOR_LIMIT_POINT; INTERIOR_CBALL; CENTRE_IN_BALL;
1421 OPEN_UNIV; REAL_LT_01] THEN
1422 MP_TAC(ISPECL [`\z. norm((h'':complex->complex) z)`; `cball(Cx(&0),&1)`]
1423 CONTINUOUS_ATTAINS_SUP) THEN
1424 ASM_REWRITE_TAC[COMPLEX_IN_CBALL_0; COMPACT_CBALL; CBALL_EQ_EMPTY] THEN
1425 REWRITE_TAC[o_DEF] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ANTS_TAC THENL
1426 [MATCH_MP_TAC CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN
1427 ASM_MESON_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON; CONTINUOUS_ON_SUBSET;
1429 DISCH_THEN(X_CHOOSE_THEN `w:complex` STRIP_ASSUME_TAC) THEN
1430 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN
1431 MATCH_MP_TAC(NORM_ARITH
1432 `!w:complex. norm(w) <= norm(w) / &2 /\ norm(z) <= norm(w)
1433 ==> z = vec 0`) THEN
1434 EXISTS_TAC `(h'':complex->complex) w` THEN ASM_SIMP_TAC[] THEN
1435 FIRST_X_ASSUM(fun th ->
1436 GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM th]) THEN
1437 REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
1438 MATCH_MP_TAC(NORM_ARITH
1439 `norm(a) <= e /\ norm(b) <= e ==> norm(a + b) / &4 <= e / &2`) THEN
1440 CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
1441 REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
1442 UNDISCH_TAC `norm(w:complex) <= &1` THEN
1443 MP_TAC(SPEC `&1` COMPLEX_NORM_CX) THEN CONV_TAC NORM_ARITH];
1444 REWRITE_TAC[IN_UNIV] THEN DISCH_TAC] THEN
1445 MP_TAC(ISPECL [`h':complex->complex`; `(:complex)`]
1446 HAS_COMPLEX_DERIVATIVE_ZERO_CONSTANT) THEN
1447 REWRITE_TAC[CONVEX_UNIV; IN_UNIV] THEN ANTS_TAC THENL
1448 [ASM_MESON_TAC[HAS_COMPLEX_DERIVATIVE_AT_WITHIN]; ALL_TAC] THEN
1449 DISCH_THEN(X_CHOOSE_TAC `a:complex`) THEN
1450 MP_TAC(ISPECL [`\z. (h:complex->complex) z - a * z`; `(:complex)`]
1451 HAS_COMPLEX_DERIVATIVE_ZERO_CONSTANT) THEN
1452 REWRITE_TAC[CONVEX_UNIV; IN_UNIV] THEN ANTS_TAC THENL
1453 [GEN_TAC THEN ASM GEN_COMPLEX_DIFF_TAC[] THEN
1454 ASM_REWRITE_TAC[] THEN CONV_TAC COMPLEX_RING;
1455 REWRITE_TAC[COMPLEX_RING `a - b:complex = c <=> a = b + c`] THEN
1456 DISCH_THEN(X_CHOOSE_TAC `b:complex`)] THEN
1457 FIRST_X_ASSUM(MP_TAC o SPECL [`&1`; `Cx(&0)`]) THEN
1458 MP_TAC(ASSUME `!z:complex. cexp (h z) = g z`) THEN
1459 DISCH_THEN(fun th -> REWRITE_TAC[GSYM th]) THEN
1460 REWRITE_TAC[ ASSUME`!x. (h:complex->complex) x = a * x + b`] THEN
1461 REWRITE_TAC[INTEGER_CLOSED; COMPLEX_ADD_LID; COMPLEX_MUL_RZERO] THEN
1462 REWRITE_TAC[CEXP_ADD; COMPLEX_MUL_RID] THEN
1463 REWRITE_TAC[COMPLEX_RING `a * b = b <=> a = Cx(&1) \/ b = Cx(&0)`] THEN
1464 REWRITE_TAC[CEXP_NZ; CEXP_EQ_1] THEN
1465 DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
1466 (X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC)) THEN
1467 UNDISCH_TAC `!z:complex. cexp(h z) = g z` THEN
1468 ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `n = &0` THENL
1469 [SUBST1_TAC(SYM(SPEC `a:complex` COMPLEX)) THEN
1470 ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO; GSYM CX_DEF] THEN
1471 REWRITE_TAC[COMPLEX_MUL_LZERO; COMPLEX_ADD_LID] THEN
1472 DISCH_THEN(ASSUME_TAC o GSYM) THEN X_GEN_TAC `z:complex` THEN
1473 ASM_CASES_TAC `?n. integer n /\ z = Cx n` THENL
1474 [REWRITE_TAC[complex_div] THEN
1475 MATCH_MP_TAC(COMPLEX_RING
1476 `z = Cx(&0) /\ ((w = Cx(&0)) \/ (y = Cx(&0)))
1477 ==> w * y = p * z`) THEN
1478 REWRITE_TAC[CGAMMA_EQ_0; CSIN_EQ_0; COMPLEX_INV_EQ_0] THEN
1479 CONJ_TAC THENL [ASM_MESON_TAC[REAL_MUL_SYM; CX_MUL]; ALL_TAC] THEN
1480 REWRITE_TAC[OR_EXISTS_THM; GSYM COMPLEX_ADD_ASSOC] THEN
1481 REWRITE_TAC[COMPLEX_RING `c - z + n = Cx(&0) <=> z = n + c`] THEN
1482 FIRST_X_ASSUM(X_CHOOSE_THEN `m:real`
1483 (CONJUNCTS_THEN2 MP_TAC SUBST1_TAC)) THEN
1484 REWRITE_TAC[integer] THEN DISCH_THEN(X_CHOOSE_THEN `p:num`
1485 (MP_TAC o MATCH_MP (REAL_ARITH `abs x = n ==> x = n \/ x = --n`))) THEN
1486 DISCH_THEN(DISJ_CASES_THEN SUBST_ALL_TAC) THEN
1487 REWRITE_TAC[GSYM CX_ADD; CX_INJ; REAL_OF_NUM_ADD; REAL_OF_NUM_EQ;
1488 REAL_ARITH `--p + n = &0 <=> p = n`] THEN
1489 REWRITE_TAC[EXISTS_OR_THM; GSYM EXISTS_REFL] THEN
1490 REWRITE_TAC[ADD_EQ_0; RIGHT_EXISTS_AND_THM; EXISTS_REFL] THEN
1491 MESON_TAC[num_CASES; ADD1];
1492 SUBGOAL_THEN `(g:complex->complex) z = g(Cx(&0))` MP_TAC THENL
1493 [ASM_REWRITE_TAC[]; EXPAND_TAC "g"] THEN
1494 ASM_REWRITE_TAC[] THEN
1495 COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[INTEGER_CLOSED]] THEN
1496 MP_TAC PI_NZ THEN REWRITE_TAC[GSYM CX_INJ] THEN CONV_TAC COMPLEX_FIELD];
1497 DISCH_THEN(fun th ->
1498 MP_TAC(SPEC `Cx(inv(&4 * n))` th) THEN MP_TAC(SPEC `Cx(&0)` th)) THEN
1499 REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_ADD_LID] THEN
1500 SUBGOAL_THEN `g(Cx(&0)) = Cx pi` SUBST1_TAC THENL
1501 [ASM_MESON_TAC[INTEGER_CLOSED]; ALL_TAC] THEN
1502 REWRITE_TAC[CEXP_ADD] THEN DISCH_THEN SUBST1_TAC THEN
1503 SUBST1_TAC(SYM(SPEC `a:complex` COMPLEX)) THEN
1504 ASM_REWRITE_TAC[] THEN
1505 GEN_REWRITE_TAC (funpow 3 LAND_CONV o RAND_CONV) [complex_mul] THEN
1506 REWRITE_TAC[RE; IM; REAL_MUL_LZERO; IM_CX; RE_CX] THEN
1507 REWRITE_TAC[CEXP_COMPLEX; REAL_ADD_LID] THEN ASM_SIMP_TAC[REAL_FIELD
1508 `~(n = &0) ==> (&2 * n * pi) * inv(&4 * n) = pi / &2`] THEN
1509 REWRITE_TAC[SIN_PI2; COS_PI2; GSYM ii] THEN
1510 REWRITE_TAC[REAL_MUL_RZERO; REAL_SUB_REFL; REAL_EXP_0] THEN
1511 REWRITE_TAC[COMPLEX_MUL_LID] THEN
1512 MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN EXPAND_TAC "g" THEN
1513 ASM_SIMP_TAC[CX_INJ; REAL_FIELD
1514 `~(n = &0) ==> (inv(&4 * n) = m <=> &4 * m * n = &1)`] THEN
1515 COND_CASES_TAC THENL
1516 [FIRST_X_ASSUM(X_CHOOSE_THEN `m:real` STRIP_ASSUME_TAC) THEN
1517 FIRST_X_ASSUM(MP_TAC o AP_TERM `abs`) THEN
1518 REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NUM] THEN
1519 REPEAT(FIRST_X_ASSUM(CHOOSE_TAC o GEN_REWRITE_RULE I [integer])) THEN
1520 ASM_REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_EQ; MULT_EQ_1] THEN
1521 CONV_TAC NUM_REDUCE_CONV;
1522 MATCH_MP_TAC(MESON[] `real y /\ ~real x ==> ~(x = y)`) THEN
1523 SIMP_TAC[GSYM CX_SUB; GSYM CX_MUL; REAL_CX; REAL_SIN; REAL_GAMMA;
1524 REAL_MUL; ONCE_REWRITE_RULE[COMPLEX_MUL_SYM] REAL_MUL_CX] THEN
1525 REWRITE_TAC[PI_NZ; real; IM_II] THEN ARITH_TAC]]);;
1527 let CGAMMA_HALF = prove
1528 (`cgamma(Cx(&1) / Cx(&2)) = Cx(sqrt pi)`,
1529 MP_TAC(SPEC `Cx(&1) / Cx(&2)` CGAMMA_REFLECTION) THEN
1530 REWRITE_TAC[COMPLEX_RING `Cx(&1) - Cx(&1) / Cx(&2) = Cx(&1) / Cx(&2)`] THEN
1531 REWRITE_TAC[GSYM CX_DIV; GSYM CX_MUL; GSYM CX_SIN] THEN
1532 REWRITE_TAC[REAL_ARITH `x * &1 / &2 = x / &2`; SIN_PI2; REAL_DIV_1] THEN
1533 SUBGOAL_THEN `Cx pi = Cx(sqrt pi) pow 2` SUBST1_TAC THENL
1534 [REWRITE_TAC[GSYM CX_POW] THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
1535 REWRITE_TAC[SQRT_POW2; PI_POS_LE];
1536 REWRITE_TAC[COMPLEX_RING
1537 `a * a:complex = b pow 2 <=> a = b \/ a = --b`] THEN
1538 STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
1539 MP_TAC(SPEC `Cx(&1 / &2)` RE_POS_CGAMMA_REAL) THEN
1540 ASM_REWRITE_TAC[REAL_CX; RE_CX; RE_NEG] THEN
1541 CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
1542 REWRITE_TAC[REAL_ARITH `~(&0 <= --x) <=> &0 < x`] THEN
1543 MATCH_MP_TAC SQRT_POS_LT THEN REWRITE_TAC[PI_POS]]);;
1545 let CGAMMA_LEGENDRE = prove
1546 (`!z. cgamma(z) * cgamma(z + Cx(&1) / Cx(&2)) =
1547 Cx(&2) cpow (Cx(&1) - Cx(&2) * z) * Cx(sqrt pi) * cgamma(Cx(&2) * z)`,
1548 REWRITE_TAC[CGAMMA_LEGENDRE_ALT; CGAMMA_HALF]);;
1550 (* ------------------------------------------------------------------------- *)
1551 (* Thw Weierstrass product definition. *)
1552 (* ------------------------------------------------------------------------- *)
1554 let CGAMMA_WEIERSTRASS = prove
1555 (`!z. ((\n. cexp(--(Cx euler_mascheroni) * z) / z *
1556 cproduct(1..n) (\k. cexp(z / Cx(&k)) / (Cx(&1) + z / Cx(&k))))
1557 --> cgamma z) sequentially`,
1558 GEN_TAC THEN SIMP_TAC[complex_div; CPRODUCT_MUL; FINITE_NUMSEG] THEN
1559 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
1560 EXISTS_TAC `\n. (cexp(--Cx euler_mascheroni * z) *
1561 cproduct(1..n) (\k. cexp(z * inv(Cx(&k)))) *
1562 cexp(--Cx(log(&n)) * z)) *
1563 (Cx(&n) cpow z / z *
1564 cproduct (1..n) (\k. inv(Cx(&1) + z * inv(Cx(&k)))))` THEN
1565 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN CONJ_TAC THENL
1566 [EXISTS_TAC `1` THEN REPEAT STRIP_TAC THEN REWRITE_TAC[complex_div] THEN
1567 MATCH_MP_TAC(COMPLEX_RING
1568 `c * c' = Cx(&1) ==> (a * b * c) * (c' * d) * e = ((a * d) * b * e)`) THEN
1569 ASM_SIMP_TAC[cpow; CX_INJ; REAL_OF_NUM_EQ; LE_1; GSYM CEXP_ADD] THEN
1570 REWRITE_TAC[GSYM CEXP_0] THEN AP_TERM_TAC THEN
1571 ASM_SIMP_TAC[CX_LOG; REAL_OF_NUM_LT; LE_1] THEN CONV_TAC COMPLEX_RING;
1573 GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_MUL_LID] THEN
1574 MATCH_MP_TAC LIM_COMPLEX_MUL THEN CONJ_TAC THENL
1575 [SIMP_TAC[GSYM CEXP_VSUM; FINITE_NUMSEG; GSYM CEXP_ADD] THEN
1576 REWRITE_TAC[GSYM CEXP_0] THEN
1577 MATCH_MP_TAC(ISPEC `cexp` LIM_CONTINUOUS_FUNCTION) THEN
1578 SIMP_TAC[CONTINUOUS_AT_CEXP; VSUM_COMPLEX_LMUL; FINITE_NUMSEG] THEN
1579 REWRITE_TAC[COMPLEX_RING
1580 `--w * z + z * x + --y * z:complex = z * ((x - y) - w)`] THEN
1581 MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
1582 REWRITE_TAC[GSYM LIM_NULL_COMPLEX] THEN
1583 REWRITE_TAC[GSYM CX_INV; VSUM_CX] THEN
1584 REWRITE_TAC[GSYM CX_SUB; REWRITE_RULE[o_DEF] (GSYM REALLIM_COMPLEX)] THEN
1585 REWRITE_TAC[EULER_MASCHERONI];
1586 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC
1587 `\n. (Cx(&n) cpow z * Cx(&(FACT n))) /
1588 cproduct(0..n) (\m. z + Cx(&m))` THEN
1589 REWRITE_TAC[CGAMMA] THEN
1590 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
1591 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
1592 SIMP_TAC[CPRODUCT_CLAUSES_LEFT; LE_0] THEN
1593 REWRITE_TAC[COMPLEX_ADD_RID; ADD_CLAUSES] THEN
1594 REWRITE_TAC[complex_div; COMPLEX_INV_MUL; GSYM COMPLEX_MUL_ASSOC] THEN
1595 AP_TERM_TAC THEN GEN_REWRITE_TAC LAND_CONV
1596 [COMPLEX_RING `a * b * c:complex = b * a * c`] THEN
1597 AP_TERM_TAC THEN SIMP_TAC[CPRODUCT_INV; FINITE_NUMSEG] THEN
1598 GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM COMPLEX_INV_INV] THEN
1599 REWRITE_TAC[GSYM COMPLEX_INV_MUL] THEN AP_TERM_TAC THEN
1600 MATCH_MP_TAC(COMPLEX_FIELD
1601 `~(z = Cx(&0)) /\ z * x = y ==> inv z * y = x`) THEN
1602 REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; FACT_NZ] THEN
1603 CONV_TAC SYM_CONV THEN SPEC_TAC(`n:num`,`p:num`) THEN
1604 INDUCT_TAC THEN REWRITE_TAC[FACT; CPRODUCT_CLAUSES_NUMSEG; ARITH] THEN
1605 REWRITE_TAC[ARITH_RULE `1 <= SUC n`; COMPLEX_MUL_LID] THEN
1606 ASM_REWRITE_TAC[CX_MUL; GSYM REAL_OF_NUM_MUL] THEN
1607 MATCH_MP_TAC(COMPLEX_RING
1608 `d * e:complex = c ==> (a * b) * c = (d * a) * b * e`) THEN
1609 MATCH_MP_TAC(COMPLEX_FIELD
1610 `~(p = Cx(&0)) ==> p * (Cx(&1) + z * inv p) = z + p`) THEN
1611 REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; NOT_SUC]]);;
1613 (* ------------------------------------------------------------------------- *)
1614 (* Sometimes the reciprocal function is convenient, since it's entire. *)
1615 (* ------------------------------------------------------------------------- *)
1617 let COMPLEX_DIFFERENTIABLE_AT_RECIP_CGAMMA = prove
1618 (`!z. (inv o cgamma) complex_differentiable (at z)`,
1619 GEN_TAC THEN ASM_CASES_TAC `!n. ~(z + Cx(&n) = Cx(&0))` THENL
1620 [MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_COMPOSE_AT THEN
1621 ASM_SIMP_TAC[COMPLEX_DIFFERENTIABLE_AT_CGAMMA] THEN
1622 GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
1623 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_INV_AT THEN
1624 REWRITE_TAC[COMPLEX_DIFFERENTIABLE_ID; CGAMMA_EQ_0] THEN
1626 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_FORALL_THM]) THEN
1627 REWRITE_TAC[COMPLEX_RING `z + w = Cx(&0) <=> z = --w`] THEN
1628 DISCH_THEN(X_CHOOSE_THEN `n:num` SUBST1_TAC) THEN
1629 REWRITE_TAC[complex_differentiable; HAS_COMPLEX_DERIVATIVE_AT] THEN
1630 REWRITE_TAC[o_DEF; CGAMMA_POLES; COMPLEX_INV_0; complex_div;
1631 COMPLEX_MUL_LZERO; COMPLEX_SUB_RZERO] THEN
1632 SIMP_TAC[GSYM COMPLEX_INV_MUL; COMPLEX_RING `x - --z:complex = x + z`] THEN
1633 EXISTS_TAC `inv(--Cx(&1) pow n / Cx(&(FACT n)))` THEN
1634 MATCH_MP_TAC LIM_COMPLEX_INV THEN
1635 REWRITE_TAC[COMPLEX_DIV_EQ_0; COMPLEX_POW_EQ_0; CX_INJ] THEN
1636 REWRITE_TAC[REAL_OF_NUM_EQ; FACT_NZ] THEN
1637 ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN
1638 REWRITE_TAC[CGAMMA_SIMPLE_POLES] THEN CONV_TAC COMPLEX_RING]);;
1640 let COMPLEX_DIFFERENTIABLE_WITHIN_RECIP_CGAMMA = prove
1641 (`!z s. (inv o cgamma) complex_differentiable at z within s`,
1642 SIMP_TAC[COMPLEX_DIFFERENTIABLE_AT_WITHIN;
1643 COMPLEX_DIFFERENTIABLE_AT_RECIP_CGAMMA]);;
1645 let HOLOMORPHIC_ON_RECIP_CGAMMA = prove
1646 (`!s. (inv o cgamma) holomorphic_on s`,
1647 REWRITE_TAC[HOLOMORPHIC_ON_DIFFERENTIABLE] THEN
1648 REWRITE_TAC[COMPLEX_DIFFERENTIABLE_WITHIN_RECIP_CGAMMA]);;
1650 let CONTINUOUS_AT_RECIP_CGAMMA = prove
1651 (`!z. (inv o cgamma) continuous at z`,
1652 SIMP_TAC[COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT;
1653 COMPLEX_DIFFERENTIABLE_AT_RECIP_CGAMMA]);;
1655 let CONTINUOUS_WITHIN_RECIP_CGAMMA = prove
1656 (`!z s. (inv o cgamma) continuous at z within s`,
1657 SIMP_TAC[COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN;
1658 COMPLEX_DIFFERENTIABLE_WITHIN_RECIP_CGAMMA]);;
1660 let CONTINUOUS_ON_RECIP_CGAMMA = prove
1661 (`!s. (inv o cgamma) continuous_on s`,
1662 SIMP_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON; HOLOMORPHIC_ON_RECIP_CGAMMA]);;
1664 let RECIP_CGAMMA = prove
1665 (`!z. ((\n. cproduct(0..n) (\m. z + Cx(&m)) /
1666 (Cx(&n) cpow z * Cx(&(FACT n)))) --> inv(cgamma z))
1668 GEN_TAC THEN ASM_CASES_TAC `!n. ~(z + Cx(&n) = Cx(&0))` THENL
1669 [ONCE_REWRITE_TAC[GSYM COMPLEX_INV_INV] THEN
1670 MATCH_MP_TAC LIM_COMPLEX_INV THEN
1671 REWRITE_TAC[COMPLEX_INV_INV; CGAMMA; COMPLEX_INV_DIV; CGAMMA_EQ_0] THEN
1673 SUBGOAL_THEN `cgamma z = Cx(&0)` SUBST1_TAC THENL
1674 [ASM_MESON_TAC[CGAMMA_EQ_0]; REWRITE_TAC[COMPLEX_INV_0]] THEN
1675 MATCH_MP_TAC LIM_EVENTUALLY THEN
1676 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; COMPLEX_DIV_EQ_0; COMPLEX_ENTIRE] THEN
1677 SIMP_TAC[CPRODUCT_EQ_0; IN_NUMSEG; FINITE_NUMSEG; LE_0] THEN
1680 let RECIP_CGAMMA_WEIERSTRASS = prove
1681 (`!n. ((\n. (z * cexp(Cx euler_mascheroni * z) *
1682 cproduct(1..n) (\k. (Cx(&1) + z / Cx(&k)) / cexp(z / Cx(&k)))))
1683 --> inv(cgamma z)) sequentially`,
1684 GEN_TAC THEN ASM_CASES_TAC `?n. z + Cx(&n) = Cx(&0)` THENL
1685 [FIRST_X_ASSUM(X_CHOOSE_TAC `N:num`) THEN
1686 MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
1687 EXISTS_TAC `N:num` THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[GE] THEN
1688 DISCH_TAC THEN SUBGOAL_THEN `cgamma(z) = Cx(&0)` SUBST1_TAC THENL
1689 [ASM_MESON_TAC[CGAMMA_EQ_0]; REWRITE_TAC[COMPLEX_INV_0]] THEN
1690 REWRITE_TAC[COMPLEX_ENTIRE; COMPLEX_DIV_EQ_0] THEN
1691 ASM_CASES_TAC `N = 0` THENL [ASM_MESON_TAC[COMPLEX_ADD_RID]; ALL_TAC] THEN
1692 REPEAT DISJ2_TAC THEN
1693 SIMP_TAC[CPRODUCT_EQ_0; FINITE_NUMSEG; IN_NUMSEG] THEN
1694 EXISTS_TAC `N:num` THEN ASM_SIMP_TAC[LE_1; COMPLEX_DIV_EQ_0] THEN
1695 DISJ1_TAC THEN MATCH_MP_TAC(COMPLEX_FIELD
1696 `x + n = Cx(&0) /\ ~(n = Cx(&0)) ==> Cx(&1) + x / n = Cx(&0)`) THEN
1697 ASM_REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ];
1698 GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV o ABS_CONV)
1699 [GSYM COMPLEX_INV_INV] THEN
1700 MATCH_MP_TAC LIM_COMPLEX_INV THEN
1701 ASM_REWRITE_TAC[CGAMMA_EQ_0] THEN
1702 SIMP_TAC[COMPLEX_INV_MUL; GSYM CEXP_NEG; GSYM CPRODUCT_INV;
1703 FINITE_NUMSEG; GSYM COMPLEX_MUL_LNEG; COMPLEX_INV_DIV] THEN
1704 REWRITE_TAC[SIMPLE_COMPLEX_ARITH `inv z * a * b:complex = a / z * b`] THEN
1705 REWRITE_TAC[CGAMMA_WEIERSTRASS]]);;
1707 (* ------------------------------------------------------------------------- *)
1708 (* The real gamma function. *)
1709 (* ------------------------------------------------------------------------- *)
1711 let gamma = new_definition
1712 `gamma(x) = Re(cgamma(Cx x))`;;
1714 let CX_GAMMA = prove
1715 (`!x. Cx(gamma x) = cgamma(Cx x)`,
1716 REWRITE_TAC[gamma] THEN MESON_TAC[REAL; REAL_CX; REAL_GAMMA]);;
1719 (`!x. ((\n. (&n rpow x * &(FACT n)) / product(0..n) (\m. x + &m))
1720 ---> gamma(x)) sequentially`,
1721 REWRITE_TAC[REALLIM_COMPLEX; o_DEF; CX_GAMMA; CX_DIV] THEN
1722 X_GEN_TAC `x:real` THEN MP_TAC(SPEC `Cx x` CGAMMA) THEN
1723 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
1724 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
1725 SIMP_TAC[CX_MUL; CX_PRODUCT; FINITE_NUMSEG; CX_ADD; cpow; rpow;
1726 CX_INJ; REAL_OF_NUM_EQ; REAL_OF_NUM_LT; LE_1; CX_LOG; CX_EXP]);;
1728 let GAMMA_EQ_0 = prove
1729 (`!x. gamma(x) = &0 <=> ?n. x + &n = &0`,
1730 REWRITE_TAC[GSYM CX_INJ; CX_ADD; CX_GAMMA; CGAMMA_EQ_0]);;
1732 let REAL_DIFFERENTIABLE_AT_GAMMA = prove
1733 (`!x. (!n. ~(x + &n = &0)) ==> gamma real_differentiable atreal x`,
1734 REPEAT STRIP_TAC THEN
1735 MP_TAC(SPEC `Cx x` COMPLEX_DIFFERENTIABLE_AT_CGAMMA) THEN
1736 ASM_REWRITE_TAC[GSYM CX_ADD; CX_INJ; complex_differentiable] THEN
1737 REWRITE_TAC[HAS_COMPLEX_DERIVATIVE_AT; real_differentiable] THEN
1738 REWRITE_TAC[GSYM CX_GAMMA; HAS_REAL_DERIVATIVE_ATREAL] THEN
1739 REWRITE_TAC[REALLIM_COMPLEX; o_DEF] THEN
1740 DISCH_THEN(X_CHOOSE_TAC `f':complex`) THEN
1741 SUBGOAL_THEN `real f'` ASSUME_TAC THENL
1742 [MATCH_MP_TAC(ISPEC `at (Cx x) within real` REAL_LIM) THEN
1743 EXISTS_TAC `\z. (cgamma z - Cx(gamma x)) / (z - Cx x)` THEN
1744 ASM_SIMP_TAC[LIM_AT_WITHIN] THEN REWRITE_TAC[WITHIN; AT] THEN
1745 SIMP_TAC[TRIVIAL_LIMIT_WITHIN_CONVEX; CONVEX_REAL] THEN
1746 REWRITE_TAC[TRIVIAL_LIMIT_WITHIN_REAL; REAL_CX] THEN
1747 REWRITE_TAC[SET_RULE `p /\ x IN real <=> real x /\ p`] THEN
1748 SIMP_TAC[REAL_EXISTS; IMP_CONJ; LEFT_IMP_EXISTS_THM] THEN
1749 REWRITE_TAC[GSYM REAL_EXISTS; GSYM CX_SUB; GSYM CX_DIV;
1750 GSYM CX_GAMMA; REAL_CX] THEN
1751 REPEAT(EXISTS_TAC `Cx(x + &1)`) THEN
1752 REWRITE_TAC[REAL_LE_REFL; REAL_CX; DIST_CX] THEN REAL_ARITH_TAC;
1753 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_EXISTS]) THEN
1754 DISCH_THEN(X_CHOOSE_THEN `f':real` SUBST_ALL_TAC) THEN
1755 EXISTS_TAC `f':real` THEN REWRITE_TAC[LIM_ATREAL_ATCOMPLEX] THEN
1756 FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_AT]) THEN
1757 REWRITE_TAC[LIM_WITHIN; o_DEF; CX_SUB; CX_DIV; CX_GAMMA] THEN
1758 SIMP_TAC[IMP_CONJ; IN; REAL_EXISTS; LEFT_IMP_EXISTS_THM; RE_CX] THEN
1761 let REAL_DIFFERENTIABLE_WITHIN_GAMMA = prove
1762 (`!x s. (!n. ~(x + &n = &0)) ==> gamma real_differentiable atreal x within s`,
1763 SIMP_TAC[REAL_DIFFERENTIABLE_AT_GAMMA;
1764 REAL_DIFFERENTIABLE_ATREAL_WITHIN]);;
1766 let REAL_DIFFERENTIABLE_ON_GAMMA = prove
1767 (`!s. s SUBSET {x | !n. ~(x + &n = &0)} ==> gamma real_differentiable_on s`,
1768 SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_DIFFERENTIABLE_ON_DIFFERENTIABLE;
1769 REAL_DIFFERENTIABLE_WITHIN_GAMMA]);;
1771 let REAL_CONTINUOUS_ATREAL_GAMMA = prove
1772 (`!x. (!n. ~(x + &n = &0)) ==> gamma real_continuous atreal x`,
1773 REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_COMPLEX_CONTINUOUS_ATREAL] THEN
1774 REWRITE_TAC[o_DEF; CX_GAMMA] THEN
1775 GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
1776 MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN
1777 SIMP_TAC[LINEAR_CONTINUOUS_WITHIN; REWRITE_RULE[o_DEF] LINEAR_CX_RE] THEN
1778 MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
1779 MATCH_MP_TAC CONTINUOUS_AT_CGAMMA THEN
1780 ASM_REWRITE_TAC[GSYM CX_ADD; RE_CX; CX_INJ]);;
1782 let REAL_CONTINUOUS_WITHIN_GAMMA = prove
1783 (`!x s. (!n. ~(x + &n = &0)) ==> gamma real_continuous atreal x within s`,
1784 SIMP_TAC[REAL_CONTINUOUS_ATREAL_WITHINREAL; REAL_CONTINUOUS_ATREAL_GAMMA]);;
1786 let REAL_CONTINUOUS_ON_GAMMA = prove
1787 (`!s. s SUBSET {x | !n. ~(x + &n = &0)} ==> gamma real_continuous_on s`,
1788 SIMP_TAC[SUBSET; IN_ELIM_THM; REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN;
1789 REAL_CONTINUOUS_WITHIN_GAMMA]);;
1791 let GAMMA_RECURRENCE_ALT = prove
1792 (`!x. gamma(x) = gamma(x + &1) / x`,
1793 REWRITE_TAC[GSYM CX_INJ; CX_DIV; CX_GAMMA; CX_ADD] THEN
1794 REWRITE_TAC[GSYM CGAMMA_RECURRENCE_ALT]);;
1798 REWRITE_TAC[GSYM CX_INJ; CX_GAMMA; CGAMMA_1]);;
1800 let GAMMA_RECURRENCE = prove
1801 (`!x. gamma(x + &1) = if x = &0 then &1 else x * gamma(x)`,
1802 REWRITE_TAC[GSYM CX_INJ; CX_GAMMA] THEN
1803 GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [COND_RAND] THEN
1804 REWRITE_TAC[CX_MUL; CX_GAMMA; CX_ADD; GSYM CGAMMA_RECURRENCE]);;
1806 let GAMMA_FACT = prove
1807 (`!n. gamma(&(n + 1)) = &(FACT n)`,
1808 REWRITE_TAC[GSYM CX_INJ; CX_GAMMA; CGAMMA_FACT]);;
1810 let GAMMA_POLES = prove
1811 (`!n. gamma(--(&n)) = &0`,
1812 REWRITE_TAC[GSYM CX_INJ; CX_GAMMA; CGAMMA_POLES; CX_NEG]);;
1814 let GAMMA_SIMPLE_POLES = prove
1815 (`!n. ((\x. (x + &n) * gamma x) ---> -- &1 pow n / &(FACT n))
1817 REWRITE_TAC[REALLIM_COMPLEX; o_DEF; LIM_ATREAL_ATCOMPLEX] THEN
1819 REWRITE_TAC[CX_MUL; CX_ADD; CX_GAMMA; CX_DIV; CX_POW; CX_NEG] THEN
1821 `(\z. (Cx(Re z) + Cx(&n)) * cgamma(Cx(Re z))) =
1822 (\z. (z + Cx(&n)) * cgamma z) o (Cx o Re)`
1823 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
1824 MATCH_MP_TAC LIM_COMPOSE_WITHIN THEN
1825 MAP_EVERY EXISTS_TAC [`(:complex)`; `--Cx(&n)`] THEN
1826 REWRITE_TAC[CGAMMA_SIMPLE_POLES; WITHIN_UNIV] THEN
1827 REWRITE_TAC[EVENTUALLY_WITHIN; o_DEF; IN_UNIV; GSYM DIST_NZ] THEN
1828 REWRITE_TAC[real; IN; GSYM CX_NEG; CX_INJ] THEN
1829 REWRITE_TAC[COMPLEX_EQ; RE_CX; IM_CX] THEN
1830 CONJ_TAC THENL [ALL_TAC; MESON_TAC[REAL_LT_01]] THEN
1831 MATCH_MP_TAC LIM_AT_WITHIN THEN LIM_TAC THEN REWRITE_TAC[RE_CX] THEN
1832 MATCH_MP_TAC LINEAR_CONTINUOUS_AT THEN
1833 REWRITE_TAC[REWRITE_RULE[o_DEF] LINEAR_CX_RE]);;
1835 let GAMMA_POS_LE = prove
1836 (`!x. &0 <= x ==> &0 <= gamma x`,
1837 SIMP_TAC[gamma; RE_POS_CGAMMA_REAL; RE_CX; REAL_CX]);;
1839 let GAMMA_POS_LT = prove
1840 (`!x. &0 < x ==> &0 < gamma x`,
1841 SIMP_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`; GAMMA_POS_LE] THEN
1842 REWRITE_TAC[GAMMA_EQ_0] THEN REAL_ARITH_TAC);;
1844 let GAMMA_LEGENDRE_ALT = prove
1845 (`!x. gamma(x) * gamma(x + &1 / &2) =
1846 &2 rpow (&1 - &2 * x) * gamma(&1 / &2) * gamma(&2 * x)`,
1847 REWRITE_TAC[GSYM CX_INJ; CX_GAMMA; CX_ADD;
1848 CX_DIV; CGAMMA_LEGENDRE_ALT; CX_MUL] THEN
1849 REWRITE_TAC[cpow; rpow; CX_INJ; REAL_OF_NUM_LT; REAL_OF_NUM_EQ; ARITH] THEN
1850 SIMP_TAC[CX_MUL; CX_EXP; CX_SUB; CX_LOG; REAL_OF_NUM_LT; ARITH]);;
1852 let GAMMA_REFLECTION = prove
1853 (`!x. gamma(x) * gamma(&1 - x) = pi / sin(pi * x)`,
1854 SIMP_TAC[GSYM CX_INJ; CX_MUL; CX_DIV; CX_GAMMA; CX_SIN; CX_SUB] THEN
1855 REWRITE_TAC[CGAMMA_REFLECTION]);;
1857 let GAMMA_HALF = prove
1858 (`gamma(&1 / &2) = sqrt pi`,
1859 REWRITE_TAC[GSYM CX_INJ; CX_DIV; CX_GAMMA; CGAMMA_HALF]);;
1861 let GAMMA_LEGENDRE = prove
1862 (`!x. gamma(x) * gamma(x + &1 / &2) =
1863 &2 rpow (&1 - &2 * x) * sqrt pi * gamma(&2 * x)`,
1864 REWRITE_TAC[GAMMA_LEGENDRE_ALT; GAMMA_HALF]);;
1866 let GAMMA_WEIERSTRASS = prove
1867 (`!x. ((\n. exp(--(euler_mascheroni) * x) / x *
1868 product(1..n) (\k. exp(x / &k) / (&1 + x / &k)))
1869 ---> gamma x) sequentially`,
1870 REWRITE_TAC[REALLIM_COMPLEX; o_DEF; CX_GAMMA; CX_DIV] THEN
1871 X_GEN_TAC `x:real` THEN MP_TAC(SPEC `Cx x` CGAMMA_WEIERSTRASS) THEN
1872 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
1873 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
1874 SIMP_TAC[CX_MUL; CX_DIV; CX_EXP; CX_NEG; CX_PRODUCT; FINITE_NUMSEG;
1877 (* ------------------------------------------------------------------------- *)
1878 (* Characterization of the real gamma function using log-convexity. *)
1879 (* ------------------------------------------------------------------------- *)
1881 let REAL_LOG_CONVEX_GAMMA = prove
1882 (`!s. (!x. x IN s ==> &0 < x) ==> gamma real_log_convex_on s`,
1883 REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LOG_CONVEX_ON_SUBSET THEN
1884 EXISTS_TAC `{x | &0 < x}` THEN
1885 ASM_REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN POP_ASSUM(K ALL_TAC) THEN
1886 MATCH_MP_TAC(ISPEC `sequentially` REAL_LOG_CONVEX_LIM) THEN
1887 EXISTS_TAC `\n x. (&n rpow x * &(FACT n)) / product(0..n) (\m. x + &m)` THEN
1888 REWRITE_TAC[GAMMA; TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN
1889 EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
1890 REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC] THEN
1891 REPEAT(MATCH_MP_TAC REAL_LOG_CONVEX_MUL THEN CONJ_TAC) THEN
1892 SIMP_TAC[REAL_LOG_CONVEX_CONST; REAL_POS] THEN
1893 ASM_SIMP_TAC[REAL_LOG_CONVEX_RPOW_RIGHT; REAL_OF_NUM_LT; LE_1] THEN
1894 SIMP_TAC[GSYM PRODUCT_INV; FINITE_NUMSEG] THEN
1895 MATCH_MP_TAC REAL_LOG_CONVEX_PRODUCT THEN
1896 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN X_GEN_TAC `k:num` THEN
1898 MATCH_MP_TAC MIDPOINT_REAL_LOG_CONVEX THEN
1899 ASM_SIMP_TAC[IN_ELIM_THM; REAL_LT_INV_EQ; REAL_LTE_ADD; REAL_POS] THEN
1900 REPEAT CONJ_TAC THENL
1901 [MATCH_MP_TAC REAL_DIFFERENTIABLE_ON_IMP_REAL_CONTINUOUS_ON THEN
1902 REWRITE_TAC[REAL_DIFFERENTIABLE_ON_DIFFERENTIABLE; IN_ELIM_THM] THEN
1903 REPEAT STRIP_TAC THEN REAL_DIFFERENTIABLE_TAC THEN ASM_REAL_ARITH_TAC;
1904 REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC;
1905 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_INV_MUL; REAL_POW_INV] THEN
1906 MATCH_MP_TAC REAL_LE_INV2 THEN
1907 ASM_SIMP_TAC[REAL_LT_MUL; REAL_LTE_ADD; REAL_POS] THEN
1908 REWRITE_TAC[REAL_LE_POW_2; REAL_ARITH
1909 `(x + &k) * (y + &k) <= ((x + y) / &2 + &k) pow 2 <=>
1910 &0 <= (x - y) pow 2`]]);;
1912 let GAMMA_REAL_LOG_CONVEX_UNIQUE = prove
1914 f(&1) = &1 /\ (!x. &0 < x ==> f(x + &1) = x * f(x)) /\
1915 (!x. &0 < x ==> &0 < f x) /\ f real_log_convex_on {x | &0 < x}
1916 ==> !x. &0 < x ==> f x = gamma x`,
1917 GEN_TAC THEN STRIP_TAC THEN
1918 SUBGOAL_THEN `!x. &0 < x /\ x <= &1 ==> f x = gamma x` ASSUME_TAC THENL
1921 `!y. &0 < y /\ y <= &1 ==> !n. f(&n + y) = gamma(&n + y)`
1923 [GEN_TAC THEN STRIP_TAC THEN
1924 INDUCT_TAC THEN ASM_SIMP_TAC[REAL_ADD_LID] THEN
1925 REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN
1926 REWRITE_TAC[REAL_ARITH `(n + &1) + x = (n + x) + &1`] THEN
1927 ASM_SIMP_TAC[GAMMA_RECURRENCE; REAL_LET_ADD; REAL_POS] THEN
1929 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
1930 MP_TAC(SPEC `x:real` FLOOR_POS) THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
1931 DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN
1932 MP_TAC(SPEC `x:real` FLOOR_FRAC) THEN ASM_REWRITE_TAC[] THEN
1933 ASM_CASES_TAC `frac x = &0` THEN ASM_REWRITE_TAC[REAL_LT_LE] THENL
1934 [ASM_CASES_TAC `n = 0` THEN
1935 ASM_SIMP_TAC[REAL_ADD_RID; REAL_LT_IMP_NZ] THEN
1936 SUBGOAL_THEN `&(n - 1) + &1 = &n`
1937 (fun th -> ASM_MESON_TAC[th; REAL_LE_REFL; REAL_LT_01]) THEN
1938 ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; LE_1] THEN REAL_ARITH_TAC;
1939 STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `frac x`) THEN
1940 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ASM_MESON_TAC[]]]]] THEN
1941 X_GEN_TAC `x:real` THEN STRIP_TAC THEN
1942 ASM_CASES_TAC `x = &1` THEN ASM_REWRITE_TAC[GAMMA_1] THEN
1943 SUBGOAL_THEN `&0 < x` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
1944 MATCH_MP_TAC(REAL_FIELD `&0 < g /\ f / g = &1 ==> f = g`) THEN
1945 ASM_SIMP_TAC[GAMMA_POS_LT] THEN
1946 MATCH_MP_TAC(ISPEC `sequentially` REALLIM_UNIQUE) THEN
1948 `\n. f(x) / ((&n rpow x * &(FACT n)) / product (0..n) (\m. x + &m))` THEN
1949 REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
1950 ASM_SIMP_TAC[REALLIM_DIV; REALLIM_CONST; GAMMA_POS_LT;
1951 GAMMA; REAL_LT_IMP_NZ] THEN
1952 ONCE_REWRITE_TAC[REALLIM_NULL] THEN
1953 MATCH_MP_TAC REALLIM_NULL_COMPARISON THEN
1954 EXISTS_TAC `\n. x * inv(&n)` THEN
1955 SIMP_TAC[REALLIM_NULL_LMUL; REALLIM_1_OVER_N] THEN
1956 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `2` THEN
1959 ==> log(f(&n)) - log(f(&n - &1))
1960 <= (log(f(&n + x)) - log(f(&n))) / x /\
1961 (log(f(&n + x)) - log(f(&n))) / x
1962 <= log(f(&n + &1)) - log(f(&n))`
1964 [MP_TAC(SPECL [`f:real->real`; `{x | &0 < x}`] REAL_LOG_CONVEX_ON) THEN
1965 ASM_REWRITE_TAC[IN_ELIM_THM] THEN ANTS_TAC THENL
1966 [REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC;
1968 MAP_EVERY (MP_TAC o SPECL [`log o f:real->real`; `{x | &0 < x}`])
1969 [REAL_CONVEX_ON_LEFT_SECANT; REAL_CONVEX_ON_RIGHT_SECANT] THEN
1970 ASM_REWRITE_TAC[] THEN
1971 DISCH_THEN(LABEL_TAC "L") THEN DISCH_THEN(LABEL_TAC "R") THEN
1972 REPEAT STRIP_TAC THENL
1973 [USE_THEN "L" (MP_TAC o SPECL [`&n - &1`; `&n + x`; `&n`]) THEN
1974 USE_THEN "R" (MP_TAC o SPECL [`&n - &1`; `&n + x`; `&n`]) THEN
1975 REWRITE_TAC[IN_ELIM_THM; IN_REAL_SEGMENT; o_THM] THEN
1976 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
1977 REWRITE_TAC[REAL_ARITH `abs(x - (x - &1)) = &1`; REAL_DIV_1] THEN
1978 DISCH_TAC THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
1979 ASM_SIMP_TAC[REAL_ARITH `&0 < x ==> abs((n + x) - n) = x`] THEN
1981 ASM_CASES_TAC `x = &1` THEN
1982 ASM_REWRITE_TAC[REAL_LE_REFL; REAL_DIV_1] THEN
1983 USE_THEN "R" (MP_TAC o SPECL [`&n`; `&n + &1`; `&n + x`]) THEN
1984 REWRITE_TAC[IN_ELIM_THM; IN_REAL_SEGMENT; o_THM] THEN
1985 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
1986 REWRITE_TAC[REAL_ARITH `abs((n + &1) - n) = &1`; REAL_DIV_1] THEN
1987 ASM_SIMP_TAC[REAL_ARITH `&0 < x ==> abs((n + x) - n) = x`] THEN
1988 ASM_REAL_ARITH_TAC];
1990 ASM_SIMP_TAC[GSYM LOG_DIV; REAL_LT_MUL;
1991 REAL_ARITH `&2 <= x ==> &0 < x /\ &0 < x + &1 /\ &0 < x - &1`;
1992 REAL_FIELD `&0 < x ==> (n * x) / x = n`] THEN
1994 `!n. &2 <= n ==> f(n - &1) = f(n) / (n - &1)` (fun th -> SIMP_TAC[th])
1996 [REPEAT STRIP_TAC THEN
1997 SUBGOAL_THEN `(f:real->real) n = f((n - &1) + &1)` SUBST1_TAC THENL
1998 [AP_TERM_TAC THEN REAL_ARITH_TAC;
1999 ASM_SIMP_TAC[REAL_ARITH `&2 <= n ==> &0 < n - &1`] THEN
2000 UNDISCH_TAC `&2 <= n` THEN CONV_TAC REAL_FIELD];
2001 ASM_SIMP_TAC[REAL_ARITH `&2 <= x ==> &0 < x`; REAL_FIELD
2002 `&0 < x /\ &2 <= y ==> x / (x / (y - &1)) = y - &1`] THEN
2003 ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ]] THEN
2004 SIMP_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_SUB;
2005 ARITH_RULE `2 <= n ==> 1 <= n`] THEN
2006 REWRITE_TAC[REAL_LE_SUB_LADD; REAL_LE_SUB_RADD] THEN
2008 `!n. 0 < n ==> f(&n) = &(FACT(n - 1))`
2009 (fun th -> SIMP_TAC[ARITH_RULE `2 <= n ==> 0 < n /\ 0 < n - 1`; th])
2011 [INDUCT_TAC THEN REWRITE_TAC[ARITH; LT_0] THEN
2012 ASM_CASES_TAC `n = 0` THEN
2013 ASM_REWRITE_TAC[REAL_ADD_LID; FACT; ARITH] THEN
2014 ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> SUC n - 1 = SUC(n - 1)`] THEN
2015 ASM_SIMP_TAC[REAL_OF_NUM_LT; GSYM REAL_OF_NUM_SUC; LE_1; FACT] THEN
2016 ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> SUC(n - 1) = n`] THEN
2017 REWRITE_TAC[REAL_OF_NUM_MUL; MULT_SYM];
2019 GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV o RAND_CONV o BINOP_CONV)
2020 [GSYM REAL_EXP_MONO_LE] THEN
2021 REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN
2022 ASM_SIMP_TAC[REAL_EXP_ADD; EXP_LOG; REAL_OF_NUM_LT; LE_1; FACT_NZ;
2023 ARITH_RULE `&2 <= &n /\ &0 < x ==> &0 < &n + x`] THEN
2024 REWRITE_TAC[REAL_OF_NUM_LE] THEN
2026 `!n. 0 < n ==> f(&n + x) = product(0..n-1) (\k. x + &k) * f(x)`
2027 (fun th -> SIMP_TAC[ARITH_RULE `2 <= n ==> 0 < n`; th])
2029 [INDUCT_TAC THEN REWRITE_TAC[ARITH] THEN
2030 DISCH_TAC THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN
2031 ASM_CASES_TAC `n = 0` THENL
2032 [ASM_SIMP_TAC[REAL_ARITH `(&0 + &1) + x = x + &1`; ARITH] THEN
2033 REWRITE_TAC[PRODUCT_SING_NUMSEG; REAL_ADD_RID];
2034 ASM_SIMP_TAC[REAL_ARITH `(n + &1) + x = (n + x) + &1`;
2035 REAL_LT_ADD; REAL_OF_NUM_LT; LE_1] THEN
2036 ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> SUC n - 1 = SUC(n - 1)`] THEN
2037 REWRITE_TAC[PRODUCT_CLAUSES_NUMSEG; LE_0] THEN
2038 ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> SUC(n - 1) = n`] THEN
2039 REWRITE_TAC[REAL_MUL_AC; REAL_ADD_AC]];
2041 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
2042 MATCH_MP_TAC(REAL_ARITH
2043 `&1 <= x /\ x <= &1 + e ==> abs(x - &1) <= e`) THEN
2044 ASM_SIMP_TAC[REAL_OF_NUM_LE; REAL_FIELD
2045 `&2 <= n ==> &1 + x * inv n = (x + n) / n`] THEN
2046 REWRITE_TAC[real_div; REAL_INV_MUL; REAL_INV_INV] THEN
2047 ONCE_REWRITE_TAC[REAL_ARITH `f * (r * n) * p:real = (p * f) * r * n`] THEN
2048 REWRITE_TAC[GSYM REAL_INV_MUL] THEN REWRITE_TAC[GSYM real_div] THEN
2049 SUBGOAL_THEN `&0 < &n rpow x * &(FACT n)` ASSUME_TAC THENL
2050 [ASM_SIMP_TAC[REAL_LT_MUL; REAL_OF_NUM_LT; LE_1; FACT_NZ;
2051 RPOW_POS_LT; ARITH_RULE `2 <= x ==> 0 < x`];
2054 [ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_MUL_LID] THEN
2055 FIRST_X_ASSUM(MP_TAC o SPEC `n + 1`) THEN
2056 ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_THEN(MP_TAC o CONJUNCT1)] THEN
2057 REWRITE_TAC[ADD_SUB] THEN
2058 ASM_SIMP_TAC[rpow; REAL_OF_NUM_LE; REAL_ARITH `&2 <= x ==> &0 < x`] THEN
2059 REWRITE_TAC[REAL_MUL_AC];
2060 ASM_SIMP_TAC[PRODUCT_CLAUSES_RIGHT; LE_0;
2061 ARITH_RULE `2 <= n ==> 0 < n`] THEN
2062 REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC] THEN
2063 GEN_REWRITE_TAC LAND_CONV
2064 [REAL_ARITH `a * b * c * d:real = b * (a * c) * d`] THEN
2065 MATCH_MP_TAC REAL_LE_LMUL THEN
2066 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2067 ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ] THEN
2068 FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[] THEN
2069 DISCH_THEN(MP_TAC o CONJUNCT2) THEN
2070 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
2071 SUBGOAL_THEN `FACT n = FACT(SUC(n - 1))` SUBST1_TAC THENL
2072 [AP_TERM_TAC THEN ASM_ARITH_TAC; REWRITE_TAC[FACT]] THEN
2073 ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> SUC(n - 1) = n`] THEN
2074 MATCH_MP_TAC REAL_EQ_IMP_LE THEN
2075 SUBGOAL_THEN `&0 < &n` MP_TAC THENL
2076 [REWRITE_TAC[REAL_OF_NUM_LT] THEN ASM_ARITH_TAC;
2077 SIMP_TAC[rpow; GSYM REAL_OF_NUM_MUL; REAL_MUL_AC] THEN
2078 CONV_TAC REAL_FIELD]]);;
2080 (* ------------------------------------------------------------------------- *)
2081 (* The integral definition, the current usual one and Euler's original one. *)
2082 (* ------------------------------------------------------------------------- *)
2084 let EULER_HAS_INTEGRAL_CGAMMA = prove
2086 ==> ((\t. Cx(drop t) cpow (z - Cx(&1)) / cexp(Cx(drop t)))
2087 has_integral cgamma(z))
2088 {t | &0 <= drop t}`,
2090 (`!z a b. &0 < Re z /\ &0 <= drop a
2091 ==> (\t. Cx(drop t) cpow z) continuous_on interval [a,b]`,
2092 REPEAT STRIP_TAC THEN
2093 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
2094 X_GEN_TAC `t:real^1` THEN REWRITE_TAC[IN_INTERVAL_1] THEN STRIP_TAC THEN
2095 ASM_CASES_TAC `t:real^1 = vec 0` THENL
2096 [ASM_SIMP_TAC[CONTINUOUS_WITHIN; cpow; CX_INJ; GSYM LIFT_EQ; LIFT_NUM;
2098 ONCE_REWRITE_TAC[LIM_NULL_COMPLEX_NORM] THEN
2099 REWRITE_TAC[NORM_CEXP] THEN
2100 MATCH_MP_TAC LIM_TRANSFORM_WITHIN THEN
2101 EXISTS_TAC `\t. Cx(exp(Re z * log(drop t)))` THEN
2102 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN CONJ_TAC THENL
2103 [X_GEN_TAC `u:real^1` THEN
2104 REWRITE_TAC[IN_INTERVAL_1; DIST_0; NORM_POS_LT; GSYM DROP_EQ;
2105 LIFT_DROP; DROP_VEC] THEN
2106 STRIP_TAC THEN SUBGOAL_THEN `&0 < drop u`
2107 (fun th -> SIMP_TAC[GSYM CX_LOG; RE_MUL_CX; th]) THEN
2109 REWRITE_TAC[LIM_WITHIN; IN_INTERVAL_1; DIST_CX; REAL_SUB_RZERO] THEN
2110 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2111 EXISTS_TAC `e rpow inv(Re z)` THEN
2112 ASM_SIMP_TAC[RPOW_POS_LT; DIST_0; REAL_ABS_EXP] THEN
2113 X_GEN_TAC `u:real^1` THEN STRIP_TAC THEN
2114 SUBGOAL_THEN `e = exp(log e)` SUBST1_TAC THENL
2115 [ASM_MESON_TAC[EXP_LOG]; REWRITE_TAC[REAL_EXP_MONO_LT]] THEN
2116 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
2117 ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
2118 REWRITE_TAC[REAL_ARITH `x / y:real = inv y * x`] THEN
2119 ASM_SIMP_TAC[GSYM LOG_RPOW] THEN MATCH_MP_TAC LOG_MONO_LT_IMP THEN
2120 SUBGOAL_THEN `norm u = drop u` (fun th -> ASM_MESON_TAC[th]) THEN
2121 REWRITE_TAC[NORM_REAL; GSYM drop] THEN ASM_REAL_ARITH_TAC];
2122 MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN THEN
2124 `(\a. Cx(drop a) cpow z) = (\w. w cpow z) o Cx o drop`
2125 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2126 MATCH_MP_TAC DIFFERENTIABLE_CHAIN_WITHIN THEN CONJ_TAC THENL
2127 [MATCH_MP_TAC DIFFERENTIABLE_LINEAR THEN
2128 REWRITE_TAC[linear; o_DEF; DROP_ADD; DROP_CMUL; CX_ADD;
2129 COMPLEX_CMUL; GSYM CX_MUL];
2130 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_IMP_DIFFERENTIABLE THEN
2131 COMPLEX_DIFFERENTIABLE_TAC THEN DISCH_THEN(K ALL_TAC) THEN
2132 REWRITE_TAC[o_DEF; RE_CX] THEN
2133 ASM_REWRITE_TAC[REAL_LT_LE; GSYM LIFT_EQ; LIFT_NUM; LIFT_DROP] THEN
2134 ASM_REAL_ARITH_TAC]]) in
2137 ==> ((\t. Cx(drop t) cpow (z - Cx(&1)) * Cx(&1 - drop t) pow n)
2138 has_integral Cx(&(FACT n)) / cproduct (0..n) (\m. z + Cx(&m)))
2139 (interval[vec 0,vec 1])`,
2140 INDUCT_TAC THEN X_GEN_TAC `z:complex` THEN
2141 ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[RE_CX; REAL_LT_REFL] THEN
2143 [REWRITE_TAC[complex_pow; COMPLEX_MUL_RID; CPRODUCT_CLAUSES_NUMSEG] THEN
2145 [`\t. Cx(drop t) cpow z / z`;
2146 `\t. Cx(drop t) cpow (z - Cx(&1))`;
2147 `vec 0:real^1`; `vec 1:real^1`]
2148 FUNDAMENTAL_THEOREM_OF_CALCULUS_INTERIOR) THEN
2149 REWRITE_TAC[DROP_VEC; CPOW_1; CPOW_0; FACT; complex_div] THEN
2150 REWRITE_TAC[COMPLEX_MUL_LZERO; COMPLEX_ADD_RID; COMPLEX_SUB_RZERO] THEN
2151 DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[REAL_POS] THEN CONJ_TAC THENL
2152 [MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_RMUL THEN
2153 MATCH_MP_TAC lemma0 THEN ASM_REWRITE_TAC[DROP_VEC; REAL_POS];
2154 REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN REPEAT STRIP_TAC THEN
2155 REWRITE_TAC[CX_SUB] THEN
2156 MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_REAL_COMPLEX THEN
2157 COMPLEX_DIFF_TAC THEN ASM_REWRITE_TAC[RE_CX] THEN
2158 UNDISCH_TAC `~(z = Cx(&0))` THEN CONV_TAC COMPLEX_FIELD];
2159 FIRST_X_ASSUM(MP_TAC o SPEC `z + Cx(&1)`) THEN
2160 REWRITE_TAC[RE_ADD; RE_CX; COMPLEX_RING `(z + Cx(&1)) - Cx(&1) = z`] THEN
2161 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_TAC] THEN
2164 `\t. Cx(drop t) cpow z`;
2165 `\t. Cx(&1) / Cx(&n + &1) * Cx(&1 - drop t) pow (n + 1)`;
2166 `\t. z * Cx(drop t) cpow (z - Cx(&1))`;
2167 `\t. --(Cx(&1 - drop t) pow n)`;
2168 `vec 0:real^1`; `vec 1:real^1`;
2170 `Cx(&(FACT n)) / cproduct (0..n) (\m. (z + Cx(&1)) + Cx(&m))`]
2171 INTEGRATION_BY_PARTS) THEN
2172 REWRITE_TAC[BILINEAR_COMPLEX_MUL; DROP_VEC;
2173 REAL_POS; COUNTABLE_SING] THEN
2174 REWRITE_TAC[CPOW_0; REAL_SUB_REFL; COMPLEX_POW_ZERO;
2175 ADD_EQ_0; ARITH] THEN
2176 REWRITE_TAC[COMPLEX_MUL_LZERO; COMPLEX_MUL_RZERO; COMPLEX_SUB_LZERO] THEN
2177 REWRITE_TAC[COMPLEX_MUL_RNEG; COMPLEX_RING `--Cx(&0) - y = --y`] THEN
2179 [REPEAT CONJ_TAC THENL
2180 [MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN
2181 ASM_SIMP_TAC[lemma0; DROP_VEC; REAL_POS] THEN
2182 MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_LMUL THEN
2183 MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_POW THEN
2184 REWRITE_TAC[CONTINUOUS_ON_CX_LIFT; LIFT_SUB; LIFT_DROP] THEN
2185 SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_ID; CONTINUOUS_ON_CONST];
2186 SIMP_TAC[IN_INTERVAL_1; DROP_VEC; IN_DIFF;
2187 IN_SING; GSYM DROP_EQ] THEN
2188 REPEAT STRIP_TAC THEN REWRITE_TAC[CX_SUB] THEN
2189 MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_REAL_COMPLEX THEN
2190 COMPLEX_DIFF_TAC THEN
2191 ASM_REWRITE_TAC[RE_CX; COMPLEX_MUL_LID] THEN
2192 REWRITE_TAC[REAL_OF_NUM_ADD; ADD_SUB; COMPLEX_MUL_ASSOC] THEN
2193 MP_TAC(ARITH_RULE `~(n + 1 = 0)`) THEN
2194 REWRITE_TAC[GSYM REAL_OF_NUM_EQ; GSYM CX_INJ] THEN
2195 CONV_TAC COMPLEX_FIELD;
2196 MATCH_MP_TAC HAS_INTEGRAL_NEG THEN ASM_REWRITE_TAC[]];
2197 DISCH_THEN(MP_TAC o SPEC `Cx(&n + &1) / z` o
2198 MATCH_MP HAS_INTEGRAL_COMPLEX_LMUL) THEN
2199 ASM_SIMP_TAC[REAL_ARITH `~(&n + &1 = &0)`; CX_INJ; COMPLEX_FIELD
2200 `~(n = Cx(&0)) /\ ~(z = Cx(&0))
2201 ==> n / z * (z * p) * Cx(&1) / n * q = p * q`] THEN
2202 REWRITE_TAC[GSYM ADD1] THEN MATCH_MP_TAC EQ_IMP THEN
2203 AP_THM_TAC THEN AP_TERM_TAC THEN
2204 SIMP_TAC[FACT; GSYM REAL_OF_NUM_MUL; CX_MUL; GSYM REAL_OF_NUM_SUC] THEN
2205 REWRITE_TAC[complex_div; GSYM COMPLEX_MUL_ASSOC] THEN AP_TERM_TAC THEN
2206 SIMP_TAC[CPRODUCT_CLAUSES_LEFT; ARITH_RULE `0 <= SUC n`] THEN
2207 REWRITE_TAC[ADD1; COMPLEX_INV_MUL; COMPLEX_ADD_RID] THEN
2208 REWRITE_TAC[ISPECL [`f:num->complex`; `m:num`; `1`]
2209 CPRODUCT_OFFSET] THEN
2210 REWRITE_TAC[GSYM COMPLEX_ADD_ASSOC; GSYM CX_ADD; REAL_OF_NUM_ADD] THEN
2211 REWRITE_TAC[ARITH_RULE `n + 1 = 1 + n`; COMPLEX_MUL_AC]]]) in
2214 ==> ((\t. if drop t <= &n
2215 then Cx(drop t) cpow (z - Cx(&1)) *
2216 Cx(&1 - drop t / &n) pow n
2218 has_integral (Cx(&n) cpow z * Cx(&(FACT n))) /
2219 cproduct (0..n) (\m. z + Cx(&m)))
2220 {t | &0 <= drop t}`,
2221 REPEAT STRIP_TAC THEN ASM_CASES_TAC `n = 0` THENL
2222 [ASM_REWRITE_TAC[CPOW_0; COMPLEX_MUL_LZERO; complex_div] THEN
2223 MATCH_MP_TAC HAS_INTEGRAL_SPIKE THEN
2224 MAP_EVERY EXISTS_TAC [`\t:real^1. Cx(&0)`; `{vec 0:real^1}`] THEN
2225 REWRITE_TAC[GSYM COMPLEX_VEC_0; HAS_INTEGRAL_0; NEGLIGIBLE_SING] THEN
2226 REWRITE_TAC[IN_DIFF; IN_SING; GSYM DROP_EQ; DROP_VEC; IN_ELIM_THM] THEN
2228 REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN
2229 ONCE_REWRITE_TAC[SET_RULE `drop x <= b <=> x IN {x | drop x <= b}`] THEN
2230 REWRITE_TAC[HAS_INTEGRAL_RESTRICT_INTER] THEN
2231 FIRST_ASSUM(MP_TAC o SPEC `n:num` o MATCH_MP lemma1) THEN
2232 DISCH_THEN(MP_TAC o SPECL [`inv(&n)`; `vec 0:real^1`] o
2233 MATCH_MP (REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_AFFINITY)) THEN
2234 REWRITE_TAC[DIMINDEX_1; REAL_POW_1] THEN
2235 ASM_SIMP_TAC[REAL_INV_EQ_0; REAL_OF_NUM_EQ; LE_1; REAL_ABS_INV] THEN
2236 REWRITE_TAC[IMAGE_AFFINITY_INTERVAL; REAL_INV_INV; REAL_POS] THEN
2237 REWRITE_TAC[UNIT_INTERVAL_NONEMPTY; VECTOR_ADD_RID; REAL_ABS_NUM] THEN
2238 REWRITE_TAC[VECTOR_MUL_RZERO; VECTOR_NEG_0; VECTOR_ADD_RID] THEN
2239 REWRITE_TAC[DROP_CMUL; COMPLEX_CMUL] THEN
2240 DISCH_THEN(MP_TAC o SPEC `Cx(&n) cpow (z - Cx(&1))` o
2241 MATCH_MP HAS_INTEGRAL_COMPLEX_LMUL) THEN
2242 REWRITE_TAC[COMPLEX_MUL_ASSOC;
2243 GSYM(ONCE_REWRITE_RULE[COMPLEX_MUL_SYM] CPOW_SUC)] THEN
2244 REWRITE_TAC[complex_div; GSYM COMPLEX_MUL_ASSOC; GSYM LIFT_EQ_CMUL] THEN
2245 REWRITE_TAC[ONCE_REWRITE_RULE[INTER_COMM] INTER; IN_ELIM_THM] THEN
2246 REWRITE_TAC[GSYM DROP_VEC; LIFT_DROP; GSYM IN_INTERVAL_1] THEN
2247 REWRITE_TAC[DROP_VEC; SET_RULE `{x | x IN s} = s`] THEN
2248 REWRITE_TAC[COMPLEX_RING `(z - Cx(&1)) + Cx(&1) = z`] THEN
2249 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_EQ) THEN
2250 REWRITE_TAC[FORALL_LIFT; IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN
2251 X_GEN_TAC `t:real` THEN STRIP_TAC THEN
2252 REWRITE_TAC[COMPLEX_MUL_ASSOC; REAL_ARITH `inv x * y:real = y / x`] THEN
2253 AP_THM_TAC THEN AP_TERM_TAC THEN IMP_REWRITE_TAC[GSYM CPOW_MUL_REAL] THEN
2254 ASM_SIMP_TAC[REAL_CX; RE_CX; REAL_LE_DIV; REAL_POS; GSYM CX_MUL] THEN
2255 ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_OF_NUM_EQ; LE_1]]) in
2258 ==> (\x:real^N. lift(Re(f x))) integrable_on s /\
2259 integral s (\x. lift(Re(f x))) = lift(Re(integral s f))`,
2260 SUBGOAL_THEN `!z. lift(Re z) = (lift o Re) z`
2261 (fun th -> REWRITE_TAC[th]) THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2262 ONCE_REWRITE_TAC[GSYM o_DEF] THEN REPEAT STRIP_TAC THENL
2263 [MATCH_MP_TAC INTEGRABLE_LINEAR; MATCH_MP_TAC INTEGRAL_LINEAR] THEN
2264 ASM_REWRITE_TAC[linear; COMPLEX_CMUL; o_THM; RE_ADD; RE_MUL_CX] THEN
2265 REWRITE_TAC[LIFT_CMUL; LIFT_ADD]) in
2266 REPEAT STRIP_TAC THEN
2268 [`\n t. if drop t <= &n
2269 then Cx(drop t) cpow (z - Cx(&1)) * Cx(&1 - drop t / &n) pow n
2271 `\t. Cx(drop t) cpow (z - Cx(&1)) / cexp(Cx(drop t))`;
2272 `\t. lift(Re(Cx(drop t) cpow (Cx(Re z) - Cx(&1)) / cexp(Cx(drop t))))`;
2273 `{t | &0 <= drop t}`] DOMINATED_CONVERGENCE) THEN
2274 ASM_SIMP_TAC[REWRITE_RULE[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] lemma2] THEN
2277 SIMP_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN
2278 MESON_TAC[CGAMMA; LIM_UNIQUE; TRIVIAL_LIMIT_SEQUENTIALLY]] THEN
2279 REWRITE_TAC[IN_ELIM_THM; LIFT_DROP] THEN REPEAT CONJ_TAC THENL
2281 REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN
2282 MAP_EVERY X_GEN_TAC [`n:num`; `t:real`] THEN DISCH_TAC THEN
2283 REWRITE_TAC[cpow; CX_INJ] THEN ASM_CASES_TAC `t = &0` THEN
2284 ASM_REWRITE_TAC[COMPLEX_MUL_LZERO; COND_ID; RE_CX; CEXP_0; COMPLEX_DIV_1;
2285 COMPLEX_NORM_0; REAL_LE_REFL] THEN
2286 ASM_SIMP_TAC[GSYM CX_LOG; REAL_LT_LE; GSYM CX_EXP; GSYM CX_SUB;
2287 GSYM CX_MUL; RE_CX; GSYM CX_DIV; GSYM REAL_EXP_SUB] THEN
2288 COND_CASES_TAC THEN REWRITE_TAC[COMPLEX_NORM_0; REAL_EXP_POS_LE] THEN
2289 REWRITE_TAC[COMPLEX_NORM_MUL; REAL_EXP_SUB; NORM_CEXP] THEN
2290 REWRITE_TAC[RE_MUL_CX; RE_SUB; RE_CX; COMPLEX_NORM_POW] THEN
2291 REWRITE_TAC[real_div] THEN SIMP_TAC[REAL_LE_LMUL_EQ; REAL_EXP_POS_LT] THEN
2292 REWRITE_TAC[COMPLEX_NORM_CX] THEN
2293 SUBGOAL_THEN `~(&n = &0)` MP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2294 REWRITE_TAC[GSYM real_div; REAL_OF_NUM_EQ] THEN DISCH_TAC THEN
2295 ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LE_LDIV_EQ; REAL_MUL_LID;
2296 REAL_OF_NUM_LT; LE_1] THEN
2297 TRANS_TAC REAL_LE_TRANS `exp(--t / &n) pow n` THEN CONJ_TAC THENL
2298 [MATCH_MP_TAC REAL_POW_LE2 THEN
2299 ASM_SIMP_TAC[REAL_SUB_LE; REAL_LE_LDIV_EQ; REAL_MUL_LID;
2300 REAL_OF_NUM_LT; LE_1] THEN
2301 REWRITE_TAC[REAL_ARITH `&1 - t / n = &1 + --t / n`; REAL_EXP_LE_X];
2302 REWRITE_TAC[GSYM REAL_EXP_N; GSYM REAL_EXP_NEG] THEN
2303 ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_OF_NUM_EQ; REAL_LE_REFL]];
2304 REWRITE_TAC[FORALL_LIFT; LIFT_DROP] THEN X_GEN_TAC `t:real` THEN
2306 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
2307 EXISTS_TAC `\k. Cx t cpow (z - Cx(&1)) * Cx(&1 - t / &k) pow k` THEN
2309 [MP_TAC(SPEC `t + &1` REAL_ARCH_SIMPLE) THEN
2310 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; GSYM REAL_OF_NUM_LE] THEN
2311 MATCH_MP_TAC MONO_EXISTS THEN REAL_ARITH_TAC;
2312 REWRITE_TAC[complex_div] THEN MATCH_MP_TAC LIM_COMPLEX_LMUL THEN
2313 REWRITE_TAC[CX_SUB; CX_DIV; GSYM CEXP_NEG] THEN
2314 REWRITE_TAC[SIMPLE_COMPLEX_ARITH `a - x / y:complex = a + --x / y`] THEN
2315 REWRITE_TAC[CEXP_LIMIT]]] THEN
2317 [`\n t. lift(Re(if drop t <= &n
2318 then Cx(drop t) cpow (Cx(Re z) - Cx(&1)) *
2319 Cx(&1 - drop t / &n) pow n
2321 `\t. lift(Re(Cx(drop t) cpow (Cx(Re z) - Cx(&1)) / cexp(Cx(drop t))))`;
2322 `{t | &0 <= drop t}`]
2323 MONOTONE_CONVERGENCE_INCREASING) THEN
2324 REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
2325 ASM_SIMP_TAC[lemma3; REWRITE_RULE[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] lemma2;
2326 RE_CX; bounded; FORALL_IN_GSPEC] THEN
2327 REWRITE_TAC[IN_UNIV; FORALL_LIFT; LIFT_DROP; NORM_LIFT] THEN
2328 REPEAT CONJ_TAC THENL
2329 [MAP_EVERY X_GEN_TAC [`n:num`; `t:real`] THEN DISCH_TAC THEN
2330 ASM_CASES_TAC `t = &0` THEN
2331 ASM_SIMP_TAC[CPOW_0; COMPLEX_MUL_LZERO; COND_ID; REAL_LE_REFL] THEN
2332 ASM_REWRITE_TAC[cpow; CX_INJ] THEN
2333 ASM_SIMP_TAC[cpow; CX_INJ; GSYM CX_SUB; GSYM CX_EXP; GSYM CX_LOG;
2334 REAL_LT_LE; GSYM CX_MUL; GSYM CX_DIV; RE_CX; GSYM CX_POW] THEN
2335 REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN ASM_CASES_TAC `t <= &n` THEN
2336 ASM_SIMP_TAC[REAL_ARITH `x <= n ==> x <= n + &1`] THENL
2337 [SIMP_TAC[RE_CX; REAL_LE_LMUL_EQ; REAL_EXP_POS_LT] THEN
2338 REWRITE_TAC[GSYM RPOW_POW; GSYM REAL_OF_NUM_SUC] THEN
2339 MATCH_MP_TAC REAL_EXP_LIMIT_RPOW_LE THEN ASM_REAL_ARITH_TAC;
2340 COND_CASES_TAC THEN REWRITE_TAC[REAL_LE_REFL; RE_CX] THEN
2341 MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_EXP_POS_LE] THEN
2342 MATCH_MP_TAC REAL_POW_LE THEN REWRITE_TAC[REAL_SUB_LE] THEN
2343 SIMP_TAC[REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
2344 ASM_REAL_ARITH_TAC];
2345 X_GEN_TAC `t:real` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
2346 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC
2347 `\k. lift(Re(Cx t cpow (Cx(Re z) - Cx(&1)) *
2348 Cx(&1 - t / &k) pow k))` THEN
2350 [MP_TAC(SPEC `t + &1` REAL_ARCH_SIMPLE) THEN
2351 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; GSYM REAL_OF_NUM_LE] THEN
2352 MATCH_MP_TAC MONO_EXISTS THEN REAL_ARITH_TAC;
2353 REWRITE_TAC[cpow; CX_INJ] THEN
2354 ASM_CASES_TAC `t = &0` THEN
2355 ASM_REWRITE_TAC[COMPLEX_MUL_LZERO; complex_div; LIM_CONST] THEN
2356 ASM_SIMP_TAC[cpow; CX_INJ; GSYM CX_SUB; GSYM CX_EXP; GSYM CX_LOG;
2357 REAL_LT_LE; GSYM CX_MUL; GSYM CX_INV; RE_CX; GSYM CX_POW] THEN
2358 REWRITE_TAC[real_div; LIFT_CMUL; RE_CX] THEN MATCH_MP_TAC LIM_CMUL THEN
2359 REWRITE_TAC[REAL_ARITH `&1 - t * inv x = &1 + --t / x`] THEN
2360 REWRITE_TAC[GSYM REAL_EXP_NEG] THEN
2361 REWRITE_TAC[GSYM(REWRITE_RULE[o_DEF] TENDSTO_REAL)] THEN
2362 REWRITE_TAC[EXP_LIMIT]];
2363 MP_TAC(MATCH_MP CONVERGENT_IMP_BOUNDED (SPEC `Cx(Re z)` CGAMMA)) THEN
2364 REWRITE_TAC[bounded; FORALL_IN_IMAGE; IN_UNIV] THEN
2365 MESON_TAC[COMPLEX_NORM_GE_RE_IM; REAL_LE_TRANS]]);;
2367 let EULER_INTEGRAL = prove
2369 ==> integral {t | &0 <= drop t}
2370 (\t. Cx(drop t) cpow (z - Cx(&1)) / cexp(Cx(drop t))) =
2372 MESON_TAC[INTEGRAL_UNIQUE; EULER_HAS_INTEGRAL_CGAMMA]);;
2374 let EULER_INTEGRABLE = prove
2376 ==> (\t. Cx(drop t) cpow (z - Cx(&1)) / cexp(Cx(drop t))) integrable_on
2377 {t | &0 <= drop t}`,
2378 MESON_TAC[EULER_HAS_INTEGRAL_CGAMMA; integrable_on]);;
2380 let EULER_HAS_REAL_INTEGRAL_GAMMA = prove
2382 ==> ((\t. t rpow (x - &1) / exp t) has_real_integral gamma(x))
2384 REPEAT STRIP_TAC THEN REWRITE_TAC[has_real_integral] THEN
2385 MP_TAC(SPEC `Cx x` EULER_HAS_INTEGRAL_CGAMMA) THEN
2386 ASM_REWRITE_TAC[gamma; o_DEF; RE_CX] THEN
2387 DISCH_THEN(MP_TAC o ISPEC `lift o Re` o MATCH_MP
2388 (REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_LINEAR)) THEN
2389 REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL
2390 [REWRITE_TAC[linear; RE_ADD; LIFT_ADD; COMPLEX_CMUL; RE_MUL_CX] THEN
2391 REWRITE_TAC[LIFT_CMUL];
2393 SUBGOAL_THEN `IMAGE lift {t | &0 <= t} = {t | &0 <= drop t}`
2395 [MATCH_MP_TAC SURJECTIVE_IMAGE_EQ THEN
2396 REWRITE_TAC[IN_ELIM_THM; LIFT_DROP] THEN MESON_TAC[LIFT_DROP];
2398 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC]
2399 HAS_INTEGRAL_SPIKE)) THEN
2400 EXISTS_TAC `{vec 0:real^1}` THEN
2401 REWRITE_TAC[NEGLIGIBLE_SING; FORALL_LIFT; IN_DIFF; IN_ELIM_THM;
2402 LIFT_DROP; IN_SING; GSYM DROP_EQ; DROP_VEC] THEN
2403 SIMP_TAC[cpow; CX_INJ; rpow; REAL_LT_LE; GSYM CX_LOG; GSYM CX_SUB;
2404 GSYM CX_EXP; GSYM CX_MUL; RE_CX; GSYM CX_DIV]);;
2406 let EULER_REAL_INTEGRAL = prove
2408 ==> real_integral {t | &0 <= t} (\t. t rpow (x - &1) / exp t) =
2410 MESON_TAC[REAL_INTEGRAL_UNIQUE; EULER_HAS_REAL_INTEGRAL_GAMMA]);;
2412 let EULER_REAL_INTEGRABLE = prove
2414 ==> (\t. t rpow (x - &1) / exp t) real_integrable_on {t | &0 <= t}`,
2415 MESON_TAC[EULER_HAS_REAL_INTEGRAL_GAMMA; real_integrable_on]);;
2417 let EULER_ORIGINAL_REAL_INTEGRABLE = prove
2419 ==> (\t. (--log t) rpow (x - &1))
2420 real_integrable_on real_interval[&0,&1]`,
2423 ==> ((\t. (--log t) rpow (x - &1))
2424 real_integrable_on real_interval[&0,&1] <=>
2425 (\t. (--log t) rpow x)
2426 real_integrable_on real_interval[&0,&1])`
2428 [GEN_TAC THEN ASM_CASES_TAC `x = &0` THEN
2429 ASM_REWRITE_TAC[REAL_LT_REFL] THEN DISCH_TAC THEN MP_TAC(ISPECL
2430 [`\t. inv(x) * (--log t) rpow x`;
2432 `\t. --(&1) / t * (--log t) rpow (x - &1)`;
2433 `\t:real. &1`; `&0`; `&1`; `{&0,&1}`]
2434 REAL_INTEGRABLE_BY_PARTS_EQ) THEN
2435 REWRITE_TAC[COUNTABLE_INSERT; COUNTABLE_EMPTY; REAL_MUL_RID] THEN
2438 [REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
2439 MATCH_MP_TAC REAL_CONTINUOUS_ON_LMUL THEN
2440 REWRITE_TAC[REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
2441 X_GEN_TAC `t:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
2443 FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP (REAL_ARITH
2444 `&0 <= x ==> &0 < x \/ x = &0`))
2446 [MATCH_MP_TAC REAL_CONTINUOUS_MUL THEN
2447 REWRITE_TAC[REAL_CONTINUOUS_WITHIN_ID] THEN
2448 MATCH_MP_TAC REAL_CONTINUOUS_ATREAL_WITHINREAL THEN
2450 `(\y. --log(y) rpow x) = (\y. y rpow x) o (\x. --x) o log`
2451 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2452 REPEAT(MATCH_MP_TAC REAL_CONTINUOUS_ATREAL_COMPOSE THEN
2454 ASM_SIMP_TAC[REAL_CONTINUOUS_NEG; REAL_CONTINUOUS_AT_ID;
2455 REAL_CONTINUOUS_AT_LOG] THEN
2456 MATCH_MP_TAC REAL_CONTINUOUS_AT_RPOW THEN
2457 ASM_SIMP_TAC[REAL_LE_LT];
2458 FIRST_X_ASSUM SUBST_ALL_TAC THEN
2459 REWRITE_TAC[REAL_CONTINUOUS_WITHINREAL; REAL_MUL_RZERO] THEN
2460 MATCH_MP_TAC REALLIM_TRANSFORM_EVENTUALLY THEN
2461 EXISTS_TAC `\y. (--x * log(y rpow inv x) * y rpow inv x) rpow x` THEN
2463 [REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN
2464 REWRITE_TAC[REAL_ARITH `&0 < abs(x - a) <=> ~(x = a)`] THEN
2465 EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN
2466 X_GEN_TAC `y:real` THEN REWRITE_TAC[IN_REAL_INTERVAL] THEN
2468 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL
2469 [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2470 ASM_SIMP_TAC[LOG_RPOW; REAL_LT_INV_EQ; REAL_FIELD
2471 `~(x = &0) ==> --x * (inv x * y) * z = --y * z`] THEN
2472 ASM_SIMP_TAC[RPOW_MUL; RPOW_RPOW; REAL_MUL_LINV] THEN
2473 REWRITE_TAC[RPOW_POW; REAL_POW_1];
2474 SUBGOAL_THEN `&0 = &0 rpow x`
2475 (fun th -> GEN_REWRITE_TAC LAND_CONV [th])
2476 THENL [ASM_SIMP_TAC[rpow; REAL_LT_IMP_NZ; REAL_LT_REFL];
2478 MATCH_MP_TAC(REWRITE_RULE[] (ISPEC `\y. y rpow x`
2479 REALLIM_REAL_CONTINUOUS_FUNCTION)) THEN
2480 ASM_SIMP_TAC[REAL_CONTINUOUS_AT_RPOW; REAL_LT_IMP_LE] THEN
2481 MATCH_MP_TAC REALLIM_NULL_LMUL THEN
2483 `(\y. log (y rpow inv x) * y rpow inv x) =
2484 (\y. log y * y) o (\y. y rpow inv x)`
2485 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2486 MATCH_MP_TAC REALLIM_COMPOSE_WITHIN THEN
2487 EXISTS_TAC `{x | &0 <= x}` THEN EXISTS_TAC `&0 rpow inv x` THEN
2488 ASM_REWRITE_TAC[REAL_ENTIRE; RPOW_EQ_0; REAL_INV_EQ_0] THEN
2489 REPEAT CONJ_TAC THENL
2490 [MATCH_MP_TAC(REWRITE_RULE[] (ISPEC `\y. y rpow x`
2491 REALLIM_REAL_CONTINUOUS_FUNCTION)) THEN
2492 ASM_SIMP_TAC[REAL_CONTINUOUS_AT_RPOW; REAL_LT_IMP_LE;
2493 REAL_LE_INV_EQ] THEN
2494 MP_TAC(ISPEC `&0` REAL_CONTINUOUS_AT_ID) THEN
2495 SIMP_TAC[REAL_CONTINUOUS_ATREAL; REALLIM_ATREAL_WITHINREAL];
2496 REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN
2497 SIMP_TAC[RPOW_POS_LE; IN_REAL_INTERVAL; IN_ELIM_THM] THEN
2498 MESON_TAC[REAL_LT_01];
2499 ASM_REWRITE_TAC[RPOW_ZERO; REAL_INV_EQ_0] THEN
2500 ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
2501 REWRITE_TAC[REALLIM_X_TIMES_LOG]]]];
2502 REWRITE_TAC[IN_REAL_INTERVAL; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN
2503 X_GEN_TAC `t:real` THEN REWRITE_TAC[DE_MORGAN_THM] THEN STRIP_TAC THEN
2504 CONJ_TAC THEN REAL_DIFF_TAC THEN ASM_REWRITE_TAC[] THEN
2505 ASM_SIMP_TAC[GSYM LOG_INV; REAL_INV_1_LT; LOG_POS_LT] THEN
2506 REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD];
2507 ASM_REWRITE_TAC[REAL_INTEGRABLE_LMUL_EQ; REAL_INV_EQ_0] THEN
2508 DISCH_THEN SUBST1_TAC THEN
2509 REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC] THEN
2510 REWRITE_TAC[REAL_INTEGRABLE_LMUL_EQ] THEN
2511 CONV_TAC REAL_RAT_REDUCE_CONV THEN
2512 MATCH_MP_TAC REAL_INTEGRABLE_SPIKE_EQ THEN EXISTS_TAC `{&0}` THEN
2513 REWRITE_TAC[REAL_NEGLIGIBLE_SING; IN_DIFF; IN_SING] THEN
2514 CONV_TAC REAL_FIELD];
2517 `!n. (\t. --log t pow n) real_integrable_on real_interval[&0,&1]`
2519 [MATCH_MP_TAC num_INDUCTION THEN
2520 REWRITE_TAC[CONJUNCT1 real_pow; REAL_INTEGRABLE_CONST] THEN
2521 REWRITE_TAC[GSYM RPOW_POW] THEN X_GEN_TAC `n:num` THEN
2522 SUBGOAL_THEN `&n = &(SUC n) - &1` SUBST1_TAC THENL
2523 [REWRITE_TAC[GSYM REAL_OF_NUM_SUC] THEN REAL_ARITH_TAC;
2524 MATCH_MP_TAC EQ_IMP THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
2525 REWRITE_TAC[REAL_OF_NUM_LT; LT_0]];
2529 ==> (\t. --log t rpow x) real_integrable_on real_interval[&0,&1]`
2530 (fun th -> ASM_MESON_TAC[th]) THEN
2531 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
2532 MP_TAC(SPEC `x:real` REAL_ARCH_SIMPLE) THEN
2533 DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN
2534 MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_AE_BY_INTEGRABLE_IMP_INTEGRABLE THEN
2535 EXISTS_TAC `\t. max (&1) (--log t pow n)` THEN EXISTS_TAC `{&0}` THEN
2536 REWRITE_TAC[REAL_NEGLIGIBLE_SING] THEN REPEAT CONJ_TAC THENL
2537 [MATCH_MP_TAC REAL_MEASURABLE_ON_RPOW THEN
2538 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC INTEGRABLE_IMP_REAL_MEASURABLE THEN
2539 FIRST_X_ASSUM(MP_TAC o SPEC `1`) THEN REWRITE_TAC[REAL_POW_1];
2540 MATCH_MP_TAC ABSOLUTELY_REAL_INTEGRABLE_IMP_INTEGRABLE THEN
2541 MATCH_MP_TAC ABSOLUTELY_REAL_INTEGRABLE_MAX THEN
2542 REWRITE_TAC[ABSOLUTELY_REAL_INTEGRABLE_CONST] THEN MATCH_MP_TAC
2543 ABSOLUTELY_REAL_INTEGRABLE_ABSOLUTELY_REAL_INTEGRABLE_LBOUND THEN
2544 EXISTS_TAC `\x:real. min (--log(&0) pow n) (&0)` THEN
2545 ASM_REWRITE_TAC[ABSOLUTELY_REAL_INTEGRABLE_CONST] THEN
2546 REWRITE_TAC[IN_REAL_INTERVAL] THEN X_GEN_TAC `t:real` THEN
2547 STRIP_TAC THEN ASM_CASES_TAC `t = &0` THEN ASM_REWRITE_TAC[] THENL
2548 [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2549 SUBGOAL_THEN `&0 < t` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2550 MATCH_MP_TAC(REAL_ARITH `b <= c ==> min a b <= c`) THEN
2551 MATCH_MP_TAC REAL_POW_LE THEN ASM_SIMP_TAC[GSYM LOG_INV] THEN
2552 MATCH_MP_TAC LOG_POS THEN MATCH_MP_TAC REAL_INV_1_LE THEN
2554 REWRITE_TAC[IN_REAL_INTERVAL; IN_DIFF; IN_SING] THEN
2555 X_GEN_TAC `t:real` THEN STRIP_TAC THEN
2556 SUBGOAL_THEN `&0 < t` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2557 SUBGOAL_THEN `&0 <= --log t` ASSUME_TAC THENL
2558 [ASM_SIMP_TAC[GSYM LOG_INV] THEN MATCH_MP_TAC LOG_POS THEN
2559 MATCH_MP_TAC REAL_INV_1_LE THEN ASM_REAL_ARITH_TAC;
2560 ASM_SIMP_TAC[real_abs; RPOW_POS_LE]] THEN
2561 ASM_CASES_TAC `--(log t) <= &1` THENL
2562 [MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= max y z`) THEN
2563 MATCH_MP_TAC RPOW_1_LE THEN ASM_REAL_ARITH_TAC;
2564 MATCH_MP_TAC(REAL_ARITH `x <= z ==> x <= max y z`) THEN
2565 REWRITE_TAC[GSYM RPOW_POW] THEN MATCH_MP_TAC RPOW_MONO THEN
2566 ASM_REAL_ARITH_TAC]]);;
2568 let EULER_ORIGINAL_INTEGRABLE = prove
2570 ==> (\t. (--clog(Cx(drop t))) cpow (z - Cx(&1)))
2571 integrable_on interval[vec 0,vec 1]`,
2572 REPEAT STRIP_TAC THEN
2573 MATCH_MP_TAC MEASURABLE_BOUNDED_AE_BY_INTEGRABLE_IMP_INTEGRABLE THEN
2574 EXISTS_TAC `\t. lift((--log(drop t)) rpow (Re z - &1))` THEN
2575 EXISTS_TAC `{vec 0:real^1,vec 1}` THEN
2576 REWRITE_TAC[NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY] THEN REPEAT CONJ_TAC THENL
2578 CONTINUOUS_AE_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN
2579 EXISTS_TAC `{vec 0:real^1,vec 1}` THEN
2580 REWRITE_TAC[NEGLIGIBLE_INSERT; NEGLIGIBLE_EMPTY;
2581 LEBESGUE_MEASURABLE_INTERVAL] THEN
2583 `(\t. --clog (Cx(drop t)) cpow (z - Cx(&1))) =
2584 ((\w. w cpow (z - Cx(&1))) o (--) o clog) o (\x. Cx(drop x))`
2585 SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
2586 MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
2587 SIMP_TAC[CONTINUOUS_ON_CX_DROP; CONTINUOUS_ON_ID] THEN
2588 MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN
2589 REWRITE_TAC[IN_DIFF; IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM;
2590 IMP_CONJ; FORALL_IN_IMAGE] THEN
2591 REWRITE_TAC[FORALL_LIFT; IN_INTERVAL_1; LIFT_DROP] THEN
2592 REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP; DROP_VEC; o_DEF] THEN
2593 REPEAT STRIP_TAC THEN
2594 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT THEN
2595 COMPLEX_DIFFERENTIABLE_TAC THEN
2596 ASM_SIMP_TAC[GSYM CX_LOG; RE_CX; REAL_LT_LE; RE_NEG; GSYM LOG_INV] THEN
2597 DISCH_TAC THEN REWRITE_TAC[GSYM REAL_LT_LE] THEN
2598 MATCH_MP_TAC LOG_POS_LT THEN MATCH_MP_TAC REAL_INV_1_LT THEN
2600 REWRITE_TAC[INTERVAL_REAL_INTERVAL; DROP_VEC;
2601 REWRITE_RULE[o_DEF] (GSYM REAL_INTEGRABLE_ON)] THEN
2602 MATCH_MP_TAC EULER_ORIGINAL_REAL_INTEGRABLE THEN ASM_REWRITE_TAC[];
2604 REWRITE_TAC[FORALL_LIFT; IN_DIFF; IN_INSERT; GSYM DROP_EQ; LIFT_DROP;
2605 IN_INTERVAL_1; DROP_VEC; NOT_IN_EMPTY] THEN
2606 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
2607 SUBGOAL_THEN `&0 < x /\ x < &1` STRIP_ASSUME_TAC THENL
2608 [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2609 ASM_SIMP_TAC[GSYM CX_LOG; cpow; rpow; GSYM CX_NEG; REAL_LT_IMP_NZ; RE_CX;
2610 REAL_LT_IMP_LE; COMPLEX_NORM_MUL; NORM_CEXP; RE_SUB; CX_INJ;
2611 REAL_LE_REFL; RE_MUL_CX; GSYM LOG_INV; LOG_POS_LT; REAL_INV_1_LT]]);;
2613 let EULER_ORIGINAL_HAS_INTEGRAL_CGAMMA = prove
2615 ==> ((\t. (--clog(Cx(drop t))) cpow (z - Cx(&1)))
2616 has_integral cgamma(z)) (interval[vec 0,vec 1])`,
2617 REPEAT STRIP_TAC THEN
2618 FIRST_ASSUM(ASSUME_TAC o MATCH_MP EULER_ORIGINAL_INTEGRABLE) THEN
2619 FIRST_ASSUM(MP_TAC o MATCH_MP INDEFINITE_INTEGRAL_CONTINUOUS_LEFT) THEN
2620 REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
2621 DISCH_THEN(MP_TAC o SPEC `vec 0:real^1`) THEN
2622 REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; CONTINUOUS_WITHIN] THEN DISCH_TAC THEN
2623 ASM_REWRITE_TAC[HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN
2624 FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (ONCE_REWRITE_RULE
2625 [TAUT `a /\ b /\ c ==> d <=> b ==> a /\ c ==> d`] LIM_UNIQUE)) THEN
2626 REWRITE_TAC[TRIVIAL_LIMIT_WITHIN] THEN CONJ_TAC THENL
2627 [MATCH_MP_TAC CONNECTED_IMP_PERFECT THEN
2628 REWRITE_TAC[ENDS_IN_UNIT_INTERVAL; CONNECTED_INTERVAL] THEN
2629 MATCH_MP_TAC(SET_RULE
2630 `{a,b} SUBSET interval[a,b] /\ ~(a = b)
2631 ==> ~(?x. interval[a,b] = {x})`) THEN
2632 REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; ENDS_IN_UNIT_INTERVAL] THEN
2633 REWRITE_TAC[VEC_EQ; ARITH_EQ];
2635 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
2636 EXISTS_TAC `\x. integral(interval[vec 0,lift(--log(drop x))])
2637 (\t. Cx(drop t) cpow (z - Cx(&1)) / cexp(Cx(drop t)))` THEN
2638 REWRITE_TAC[EVENTUALLY_WITHIN] THEN CONJ_TAC THENL
2639 [EXISTS_TAC `&1` THEN
2640 REWRITE_TAC[REAL_LT_01; DIST_0; NORM_POS_LT; IN_INTERVAL_1; DROP_VEC;
2641 GSYM DROP_EQ; DROP_VEC; FORALL_LIFT; LIFT_DROP] THEN
2642 X_GEN_TAC `x:real` THEN STRIP_TAC THEN
2643 SUBGOAL_THEN `&0 < x` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2645 [`\t. (--clog(Cx(drop(--t)))) cpow (z - Cx(&1))`;
2646 `\t. --lift(exp(--drop t))`;
2647 `\t. lift(exp(--drop t))`;
2648 `vec 0:real^1`; `lift(--log x):real^1`; `--vec 1:real^1`; `vec 0:real^1`;
2649 `{vec 0:real^1,vec 1}`]
2650 HAS_INTEGRAL_SUBSTITUTION_STRONG) THEN
2651 REWRITE_TAC[LIFT_DROP; DROP_VEC; GSYM DROP_NEG] THEN
2652 REWRITE_TAC[GSYM REFLECT_INTERVAL; GSYM INTEGRAL_REFLECT_GEN] THEN
2653 REWRITE_TAC[DROP_NEG; LIFT_DROP; REAL_LE_NEG2; REAL_EXP_MONO_LE] THEN
2654 REWRITE_TAC[REAL_NEG_NEG; DROP_VEC; REAL_EXP_0; REAL_NEG_0] THEN
2655 SIMP_TAC[GSYM CX_LOG; REAL_EXP_POS_LT; LOG_EXP] THEN ANTS_TAC THENL
2656 [REWRITE_TAC[REAL_POS; COUNTABLE_INSERT; COUNTABLE_EMPTY] THEN
2657 CONV_TAC REAL_RAT_REDUCE_CONV THEN
2658 ASM_SIMP_TAC[GSYM LOG_INV; LOG_POS; REAL_INV_1_LE] THEN
2659 REWRITE_TAC[GSYM DROP_NEG; INTEGRABLE_REFLECT_GEN] THEN
2660 REWRITE_TAC[REFLECT_INTERVAL; VECTOR_NEG_NEG; VECTOR_NEG_0] THEN
2661 ASM_SIMP_TAC[EULER_ORIGINAL_INTEGRABLE] THEN REPEAT CONJ_TAC THENL
2662 [REWRITE_TAC[GSYM LIFT_NEG; DROP_NEG] THEN
2663 REWRITE_TAC[GSYM LIFT_NUM; GSYM IMAGE_LIFT_REAL_INTERVAL] THEN
2664 REWRITE_TAC[REWRITE_RULE[o_DEF] (GSYM REAL_CONTINUOUS_ON)] THEN
2665 MATCH_MP_TAC REAL_DIFFERENTIABLE_ON_IMP_REAL_CONTINUOUS_ON THEN
2666 REWRITE_TAC[REAL_DIFFERENTIABLE_ON_DIFFERENTIABLE] THEN
2667 REPEAT STRIP_TAC THEN REAL_DIFFERENTIABLE_TAC;
2668 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_VEC;
2669 FORALL_LIFT; LIFT_DROP; DROP_NEG] THEN
2670 X_GEN_TAC `y:real` THEN STRIP_TAC THEN
2671 REWRITE_TAC[REAL_ARITH `--x <= &0 <=> &0 <= x`; REAL_EXP_POS_LE] THEN
2672 REWRITE_TAC[REAL_EXP_NEG; REAL_LE_NEG2] THEN
2673 MATCH_MP_TAC REAL_INV_LE_1 THEN
2674 GEN_REWRITE_TAC LAND_CONV [GSYM REAL_EXP_0] THEN
2675 ASM_REWRITE_TAC[REAL_EXP_MONO_LE];
2676 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_VEC;
2677 FORALL_LIFT; LIFT_DROP; DROP_NEG; IN_DIFF; IN_INSERT;
2678 NOT_IN_EMPTY; GSYM DROP_EQ; DE_MORGAN_THM] THEN
2679 X_GEN_TAC `y:real` THEN REPEAT STRIP_TAC THENL
2680 [MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_AT_WITHIN THEN
2681 REWRITE_TAC[GSYM LIFT_NEG; REWRITE_RULE[o_DEF]
2682 (GSYM HAS_REAL_VECTOR_DERIVATIVE_AT)] THEN
2683 REAL_DIFF_TAC THEN REAL_ARITH_TAC;
2684 MATCH_MP_TAC CONTINUOUS_AT_WITHIN THEN
2686 `(\t. --clog(Cx(--drop t)) cpow (z - Cx(&1))) =
2687 (\w. --clog w cpow (z - Cx(&1))) o (\t. Cx(drop(--t)))`
2688 SUBST1_TAC THENL [REWRITE_TAC[o_DEF; DROP_NEG]; ALL_TAC] THEN
2689 MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN CONJ_TAC THENL
2690 [MATCH_MP_TAC CONTINUOUS_CX_DROP THEN
2691 GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN
2692 SIMP_TAC[CONTINUOUS_NEG; CONTINUOUS_AT_ID];
2693 MATCH_MP_TAC COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT THEN
2694 COMPLEX_DIFFERENTIABLE_TAC THEN
2695 REWRITE_TAC[RE_CX; VECTOR_NEG_NEG; LIFT_DROP] THEN
2696 SIMP_TAC[REAL_EXP_POS_LT; GSYM CX_LOG; LOG_EXP] THEN
2697 REWRITE_TAC[RE_NEG; RE_CX; REAL_NEG_NEG] THEN
2698 ASM_REAL_ARITH_TAC]]];
2699 DISCH_THEN(MP_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN
2700 ASM_SIMP_TAC[EXP_LOG; LIFT_NUM; COMPLEX_CMUL; CX_EXP; CEXP_NEG; CX_NEG;
2701 COMPLEX_NEG_NEG] THEN
2702 DISCH_THEN(SUBST1_TAC o SYM) THEN
2703 ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN REWRITE_TAC[complex_div]];
2704 FIRST_ASSUM(MP_TAC o MATCH_MP EULER_HAS_INTEGRAL_CGAMMA) THEN
2705 GEN_REWRITE_TAC LAND_CONV [HAS_INTEGRAL_ALT] THEN
2706 REWRITE_TAC[INTEGRABLE_RESTRICT_INTER; INTEGRAL_RESTRICT_INTER] THEN
2707 DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN REWRITE_TAC[LIM_WITHIN] THEN
2708 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
2709 FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
2710 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
2711 EXISTS_TAC `exp(--B)` THEN ASM_REWRITE_TAC[REAL_EXP_POS_LT] THEN
2712 REWRITE_TAC[REAL_LT_01; DIST_0; NORM_POS_LT; IN_INTERVAL_1; DROP_VEC;
2713 GSYM DROP_EQ; DROP_VEC; FORALL_LIFT; LIFT_DROP; NORM_LIFT] THEN
2714 X_GEN_TAC `x:real` THEN STRIP_TAC THEN
2715 SUBGOAL_THEN `&0 < x` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2716 FIRST_X_ASSUM(MP_TAC o SPECL [`--lift B`; `lift(--log x)`]) THEN
2718 [REWRITE_TAC[BALL_1; SUBSET_INTERVAL_1; DROP_ADD; DROP_SUB; DROP_VEC] THEN
2719 REWRITE_TAC[LIFT_DROP; DROP_NEG] THEN DISJ2_TAC THEN
2720 REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
2721 REWRITE_TAC[REAL_ADD_LID] THEN
2722 ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LE] THEN
2723 ASM_SIMP_TAC[REAL_EXP_NEG; EXP_LOG] THEN
2724 GEN_REWRITE_TAC LAND_CONV [GSYM REAL_INV_INV] THEN
2725 ONCE_REWRITE_TAC[GSYM REAL_EXP_NEG] THEN
2726 MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REAL_ARITH_TAC;
2727 MATCH_MP_TAC(NORM_ARITH
2728 `i = j ==> norm(i - g) < e ==> dist(j,g) < e`) THEN
2729 AP_THM_TAC THEN AP_TERM_TAC THEN
2730 REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; IN_INTERVAL_1] THEN
2731 REWRITE_TAC[DROP_NEG; LIFT_DROP; DROP_VEC] THEN ASM_REAL_ARITH_TAC]]);;
2733 let EULER_ORIGINAL_INTEGRAL = prove
2735 ==> integral (interval[vec 0,vec 1])
2736 (\t. (--clog(Cx(drop t))) cpow (z - Cx(&1))) =
2738 REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN
2739 MATCH_MP_TAC EULER_ORIGINAL_HAS_INTEGRAL_CGAMMA THEN
2740 ASM_REWRITE_TAC[]);;
2742 let EULER_ORIGINAL_HAS_REAL_INTEGRAL_GAMMA = prove
2744 ==> ((\t. (--log t) rpow (x - &1)) has_real_integral gamma(x))
2745 (real_interval[&0,&1])`,
2746 REPEAT STRIP_TAC THEN REWRITE_TAC[has_real_integral] THEN
2747 MP_TAC(SPEC `Cx x` EULER_ORIGINAL_HAS_INTEGRAL_CGAMMA) THEN
2748 ASM_REWRITE_TAC[gamma; o_DEF; RE_CX] THEN
2749 DISCH_THEN(MP_TAC o ISPEC `lift o Re` o MATCH_MP
2750 (REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_LINEAR)) THEN
2751 REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL
2752 [REWRITE_TAC[linear; RE_ADD; LIFT_ADD; COMPLEX_CMUL; RE_MUL_CX] THEN
2753 REWRITE_TAC[LIFT_CMUL];
2754 REWRITE_TAC[IMAGE_LIFT_REAL_INTERVAL; LIFT_NUM]] THEN
2755 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ] (REWRITE_RULE[CONJ_ASSOC]
2756 HAS_INTEGRAL_SPIKE)) THEN
2757 EXISTS_TAC `{vec 0:real^1,vec 1}` THEN
2758 REWRITE_TAC[NEGLIGIBLE_INSERT; FORALL_LIFT; IN_INTERVAL_1; DE_MORGAN_THM;
2759 LIFT_DROP; IN_INSERT; NOT_IN_EMPTY; GSYM DROP_EQ; DROP_VEC; IN_DIFF] THEN
2760 REWRITE_TAC[NEGLIGIBLE_EMPTY] THEN X_GEN_TAC `y:real` THEN STRIP_TAC THEN
2761 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2762 SUBGOAL_THEN `&0 < --log y` ASSUME_TAC THENL
2763 [ASM_SIMP_TAC[GSYM LOG_INV] THEN MATCH_MP_TAC LOG_POS_LT THEN
2764 MATCH_MP_TAC REAL_INV_1_LT THEN ASM_REAL_ARITH_TAC;
2765 ASM_SIMP_TAC[GSYM CX_LOG; cpow; rpow; COMPLEX_NEG_EQ_0; CX_INJ;
2766 REAL_LT_IMP_NZ] THEN
2767 ASM_SIMP_TAC[GSYM CX_NEG; GSYM CX_LOG; GSYM CX_SUB; GSYM CX_MUL] THEN
2768 COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2769 REWRITE_TAC[GSYM CX_EXP; RE_CX]]);;
2771 let EULER_ORIGINAL_REAL_INTEGRAL = prove
2773 ==> real_integral (real_interval[&0,&1])
2774 (\t. (--log t) rpow (x - &1)) =
2776 MESON_TAC[REAL_INTEGRAL_UNIQUE; EULER_ORIGINAL_HAS_REAL_INTEGRAL_GAMMA]);;
2778 (* ------------------------------------------------------------------------- *)
2779 (* Stirling's approximation. *)
2780 (* ------------------------------------------------------------------------- *)
2782 let LGAMMA_STIRLING_INTEGRALS_EXIST,LGAMMA_STIRLING = (CONJ_PAIR o prove)
2783 (`(!z n. 1 <= n /\ (Im z = &0 ==> &0 < Re z)
2784 ==> (\t. Cx(bernoulli n (frac (drop t))) / (z + Cx(drop t)) pow n)
2785 integrable_on {t | &0 <= drop t}) /\
2787 (Im z = &0 ==> &0 < Re z)
2789 ((z - Cx(&1) / Cx(&2)) * clog(z) - z + Cx(log(&2 * pi) / &2)) +
2790 vsum(1..p) (\k. Cx(bernoulli (2 * k) (&0) /
2791 (&4 * &k pow 2 - &2 * &k)) / z pow (2 * k - 1)) -
2792 integral {t | &0 <= drop t}
2793 (\t. Cx(bernoulli (2 * p + 1) (frac(drop t))) /
2794 (z + Cx(drop t)) pow (2 * p + 1)) /
2797 (`!p n z. (Im z = &0 ==> &0 < Re z)
2798 ==> (\x. Cx(bernoulli (2 * p + 1) (frac(drop x))) *
2799 Cx(&(FACT(2 * p))) / (z + Cx(drop x)) pow (2 * p + 1))
2800 integrable_on interval [lift(&0),lift(&n)] /\
2801 vsum(0..n) (\m. clog(Cx(&m) + z)) =
2802 (z + Cx(&n) + Cx(&1) / Cx(&2)) * clog(z + Cx(&n)) -
2803 (z - Cx(&1) / Cx(&2)) * clog(z) - Cx(&n) +
2805 (\k. Cx(bernoulli (2 * k) (&0) / &(FACT(2 * k))) *
2806 (Cx(&(FACT(2 * k - 2))) / (z + Cx(&n)) pow (2 * k - 1) -
2807 Cx(&(FACT(2 * k - 2))) / z pow (2 * k - 1))) +
2808 integral (interval[lift(&0),lift(&n)])
2809 (\x. Cx(bernoulli (2 * p + 1) (frac(drop x))) *
2810 Cx(&(FACT(2 * p))) / (z + Cx(drop x)) pow (2 * p + 1)) /
2811 Cx(&(FACT(2 * p + 1)))`,
2812 REPEAT GEN_TAC THEN STRIP_TAC THEN
2813 REWRITE_TAC[COMPLEX_RING `Cx(&n) + z = z + Cx(&n)`] THEN MP_TAC(ISPECL
2814 [`\n w. if n = 0 then (z + w) * clog(z + w) - (z + w)
2815 else if n = 1 then clog(z + w)
2816 else Cx(--(&1) pow n * &(FACT(n - 2))) /
2817 (z + w) pow (n - 1)`;
2818 `0`; `n:num`; `p:num`] COMPLEX_EULER_MACLAURIN_ANTIDERIVATIVE) THEN
2819 ASM_REWRITE_TAC[ADD_EQ_0; MULT_EQ_0; MULT_EQ_1] THEN
2820 CONV_TAC NUM_REDUCE_CONV THEN ANTS_TAC THENL
2821 [REWRITE_TAC[IN_REAL_INTERVAL; LE_0] THEN
2822 MAP_EVERY X_GEN_TAC [`k:num`; `x:real`] THEN REPEAT STRIP_TAC THEN
2823 SUBGOAL_THEN `~(z + Cx x = Cx(&0))` ASSUME_TAC THENL
2824 [REWRITE_TAC[COMPLEX_EQ; IM_ADD; RE_ADD; IM_CX; RE_CX] THEN
2827 REWRITE_TAC[o_THM; ARITH_RULE `k + 1 = 1 <=> k = 0`] THEN
2828 ASM_CASES_TAC `k = 0` THEN ASM_REWRITE_TAC[] THENL
2829 [COMPLEX_DIFF_TAC THEN CONJ_TAC THENL
2830 [REWRITE_TAC[RE_ADD; IM_ADD; RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC;
2831 UNDISCH_TAC `~(z + Cx x = Cx(&0))` THEN CONV_TAC COMPLEX_FIELD];
2833 ASM_CASES_TAC `k = 1` THEN ASM_REWRITE_TAC[] THEN COMPLEX_DIFF_TAC THEN
2834 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
2835 REWRITE_TAC[complex_div; COMPLEX_ADD_LID; COMPLEX_MUL_LID;
2836 COMPLEX_POW_1; IM_ADD; RE_ADD; IM_CX; RE_CX] THEN
2837 (CONJ_TAC ORELSE ASM_REAL_ARITH_TAC) THEN
2838 ASM_REWRITE_TAC[COMPLEX_POW_EQ_0] THEN
2839 REWRITE_TAC[REAL_POW_ADD; REAL_POW_1; COMPLEX_MUL_LZERO] THEN
2840 REWRITE_TAC[REAL_MUL_RNEG; REAL_MUL_RID; REAL_NEG_NEG; REAL_MUL_LNEG] THEN
2841 REWRITE_TAC[COMPLEX_MUL_RID; CX_NEG; COMPLEX_POW_POW] THEN
2842 REWRITE_TAC[GSYM COMPLEX_MUL_ASSOC; COMPLEX_SUB_LZERO; COMPLEX_NEG_NEG;
2843 COMPLEX_MUL_LNEG] THEN
2844 REWRITE_TAC[GSYM complex_div] THEN AP_TERM_TAC THEN
2845 FIRST_X_ASSUM(K ALL_TAC o check (is_imp o concl)) THEN
2846 ASM_SIMP_TAC[GSYM complex_div; COMPLEX_DIV_POW2] THEN COND_CASES_TAC THENL
2847 [MATCH_MP_TAC(TAUT `F ==> p`) THEN ASM_ARITH_TAC; ALL_TAC] THEN
2848 ASM_SIMP_TAC[ADD_SUB; ARITH_RULE
2849 `~(k = 0) /\ ~(k = 1) ==> (k - 1) * 2 - (k - 1 - 1) = k`] THEN
2850 REWRITE_TAC[COMPLEX_MUL_ASSOC; complex_div] THEN
2851 AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM CX_MUL] THEN
2852 AP_TERM_TAC THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN AP_TERM_TAC THEN
2853 ASM_SIMP_TAC[ARITH_RULE `~(k = 0) ==> (k + 1) - 2 = k - 1`] THEN
2854 ASM_SIMP_TAC[ARITH_RULE
2855 `~(k = 0) /\ ~(k = 1) ==> k - 1 = SUC(k - 2)`] THEN
2856 REWRITE_TAC[FACT; REAL_OF_NUM_MUL] THEN REWRITE_TAC[MULT_SYM];
2857 REWRITE_TAC[ARITH_RULE `~(2 * p + 2 = 1)`] THEN
2858 DISCH_THEN(CONJUNCTS_THEN2 MP_TAC SUBST1_TAC)] THEN
2860 REWRITE_TAC[ARITH_RULE `(2 * p + 2) - 1 = 2 * p + 1`; ADD_SUB] THEN
2861 REWRITE_TAC[REAL_POW_NEG; EVEN_ADD; EVEN_MULT; ARITH_EVEN] THEN
2862 REWRITE_TAC[REAL_POW_ONE; REAL_MUL_LID] THEN
2863 DISCH_TAC THEN ASM_REWRITE_TAC[COMPLEX_ADD_RID] THEN
2864 CONV_TAC COMPLEX_RING) in
2866 (`!z n. 2 <= n /\ (Im z = &0 ==> &0 < Re z)
2867 ==> (\t. inv(z + Cx(drop t)) pow n)
2868 absolutely_integrable_on {t | &0 <= drop t}`,
2869 REPEAT STRIP_TAC THEN REWRITE_TAC[COMPLEX_POW_INV] THEN
2871 MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE THEN
2872 EXISTS_TAC `\t. lift(inv(max (abs(Im z)) (Re z + drop t)) pow n)` THEN
2873 REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
2874 [MATCH_MP_TAC CONTINUOUS_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET THEN
2876 [MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_INV THEN
2877 SIMP_TAC[CONTINUOUS_ON_COMPLEX_POW; CONTINUOUS_ON_ADD;
2878 CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID; CONTINUOUS_ON_CX_DROP] THEN
2879 REWRITE_TAC[IN_ELIM_THM; GSYM FORALL_DROP] THEN
2880 REWRITE_TAC[COMPLEX_POW_EQ_0] THEN
2881 REWRITE_TAC[COMPLEX_EQ; RE_ADD; IM_ADD; RE_CX; IM_CX] THEN
2883 MATCH_MP_TAC LEBESGUE_MEASURABLE_CONVEX THEN
2884 MATCH_MP_TAC IS_INTERVAL_CONVEX THEN
2885 REWRITE_TAC[IS_INTERVAL_1; IN_ELIM_THM] THEN REAL_ARITH_TAC];
2887 REWRITE_TAC[FORALL_IN_GSPEC; GSYM FORALL_DROP; LIFT_DROP] THEN
2888 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
2889 REWRITE_TAC[GSYM COMPLEX_POW_INV; COMPLEX_NORM_POW] THEN
2890 MATCH_MP_TAC REAL_POW_LE2 THEN
2891 REWRITE_TAC[NORM_POS_LE; COMPLEX_NORM_INV] THEN
2892 MATCH_MP_TAC REAL_LE_INV2 THEN
2893 MP_TAC(SPEC `z + Cx x` COMPLEX_NORM_GE_RE_IM) THEN
2894 REWRITE_TAC[RE_ADD; IM_ADD; RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC] THEN
2895 REWRITE_TAC[REAL_ARITH `max m n = if n <= m then m else n`] THEN
2896 REWRITE_TAC[COND_RAND; COND_RATOR] THEN
2897 MATCH_MP_TAC INTEGRABLE_CASES THEN REWRITE_TAC[IN_ELIM_THM] THEN
2900 `{t | &0 <= drop t /\ Re z + drop t <= abs (Im z)} =
2901 interval[vec 0,lift(abs(Im z) - Re z)]`
2902 (fun th -> REWRITE_TAC[th; INTEGRABLE_CONST]) THEN
2903 SIMP_TAC[EXTENSION; IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC; LIFT_DROP] THEN
2906 MATCH_MP_TAC(REWRITE_RULE[IMP_IMP] INTEGRABLE_SPIKE_SET) THEN
2908 `{t | abs(Im z) - Re z <= drop t} INTER {t | &0 <= drop t}` THEN
2910 [MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN
2911 EXISTS_TAC `{lift(abs(Im z) - Re z)}` THEN
2912 REWRITE_TAC[NEGLIGIBLE_SING] THEN
2913 REWRITE_TAC[SUBSET; IN_DIFF; IN_UNION;
2914 IN_ELIM_THM; IN_SING; IN_INTER] THEN
2915 REWRITE_TAC[GSYM DROP_EQ; LIFT_DROP] THEN ASM_REAL_ARITH_TAC;
2917 REWRITE_TAC[GSYM INTEGRABLE_RESTRICT_INTER] THEN
2918 REWRITE_TAC[integrable_on] THEN
2919 ONCE_REWRITE_TAC[HAS_INTEGRAL_LIM_AT_POSINFINITY] THEN
2920 REWRITE_TAC[INTEGRABLE_RESTRICT_INTER; INTEGRAL_RESTRICT_INTER] THEN
2922 `!a. {t | abs(Im z) - Re z <= drop t} INTER interval[vec 0,a] =
2923 interval[lift(max (&0) (abs (Im z) - Re z)),a]`
2924 (fun th -> REWRITE_TAC[th])
2926 [REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM; IN_INTERVAL_1] THEN
2927 REWRITE_TAC[LIFT_DROP; DROP_VEC] THEN REAL_ARITH_TAC;
2930 `!b. &0 <= drop b /\ abs (Im z) - Re z <= drop b
2931 ==> ((\x. lift(inv (Re z + drop x) pow n)) has_integral
2933 (inv((&1 - &n) * (Re z + drop b) pow (n - 1)) -
2935 (Re z + max(&0) (abs(Im z) - Re z)) pow (n - 1))))
2936 (interval[lift(max (&0) (abs (Im z) - Re z)),b])`
2938 [REPEAT STRIP_TAC THEN
2940 [`\x. lift(inv((&1 - &n) * (Re z + drop x) pow (n - 1)))`;
2941 `\x. lift(inv(Re z + drop x) pow n)`;
2942 `lift(max (&0) (abs (Im z) - Re z))`;
2943 `b:real^1`] FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
2944 ASM_REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; REAL_MAX_LE] THEN
2946 [REWRITE_TAC[FORALL_LIFT; LIFT_DROP; INTERVAL_REAL_INTERVAL] THEN
2947 REWRITE_TAC[REWRITE_RULE[o_DEF]
2948 (GSYM HAS_REAL_VECTOR_DERIVATIVE_WITHIN)] THEN
2949 REWRITE_TAC[REAL_INV_MUL; REAL_INV_POW] THEN REPEAT STRIP_TAC THEN
2951 ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; ARITH_RULE `2 <= n ==> 1 <= n`] THEN
2952 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
2953 ASM_SIMP_TAC[real_div; REAL_OF_NUM_LE; REAL_FIELD
2954 `&2 <= x ==> inv(&1 - x) * (x - &1) * y * --(&0 + &1) * p =
2956 REWRITE_TAC[GSYM REAL_INV_MUL; GSYM REAL_INV_POW] THEN
2957 AP_TERM_TAC THEN REWRITE_TAC[GSYM REAL_POW_ADD] THEN
2958 AP_TERM_TAC THEN ASM_ARITH_TAC;
2959 REWRITE_TAC[LIFT_DROP; GSYM LIFT_SUB]];
2961 EXISTS_TAC `lift(inv
2963 (Re z + max (&0) (abs(Im z) - Re z)) pow (n - 1)))` THEN
2965 [X_GEN_TAC `b:real^1` THEN
2966 ASM_CASES_TAC `&0 <= drop b /\ abs (Im z) - Re z <= drop b` THENL
2967 [ASM_MESON_TAC[integrable_on]; MATCH_MP_TAC INTEGRABLE_ON_NULL] THEN
2968 REWRITE_TAC[CONTENT_EQ_0_1; LIFT_DROP] THEN ASM_REAL_ARITH_TAC;
2970 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
2972 `\b. lift(inv((&1 - &n) * (Re z + b) pow (n - 1)) -
2974 (Re z + max (&0) (abs(Im z) - Re z)) pow (n - 1)))` THEN
2976 [REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN
2977 EXISTS_TAC `max(&0) (abs(Im z) - Re z)` THEN
2978 REWRITE_TAC[FORALL_DROP; LIFT_DROP] THEN
2979 REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
2980 MATCH_MP_TAC INTEGRAL_UNIQUE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
2983 REWRITE_TAC[LIFT_SUB; VECTOR_SUB] THEN
2984 GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_LID] THEN
2985 MATCH_MP_TAC LIM_ADD THEN
2986 REWRITE_TAC[GSYM LIFT_NEG; GSYM REAL_INV_NEG; GSYM REAL_MUL_LNEG] THEN
2987 REWRITE_TAC[REAL_NEG_SUB; LIM_CONST] THEN
2988 REWRITE_TAC[REAL_INV_MUL; LIFT_CMUL] THEN
2989 MATCH_MP_TAC LIM_NULL_CMUL THEN
2990 REWRITE_TAC[GSYM LIFT_NUM; GSYM LIM_CX_LIFT] THEN
2991 REWRITE_TAC[CX_INV; CX_POW; GSYM COMPLEX_POW_INV; CX_ADD] THEN
2992 ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN
2993 MATCH_MP_TAC LIM_INV_X_POW_OFFSET THEN ASM_ARITH_TAC) in
2995 (`!z n. 2 <= n /\ (Im z = &0 ==> &0 < Re z)
2996 ==> (\t. Cx(bernoulli n (frac (drop t))) / (z + Cx(drop t)) pow n)
2997 integrable_on {t | &0 <= drop t}`,
2998 REPEAT STRIP_TAC THEN
2999 REWRITE_TAC[complex_div; GSYM COMPLEX_CMUL] THEN
3000 MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN
3001 MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_ON_MUL_BERNOULLI_FRAC THEN
3002 ASM_SIMP_TAC[GSYM COMPLEX_POW_INV; lemma2]) in
3004 (`!z p. (Im z = &0 ==> &0 < Re z) /\ 1 <= p
3005 ==> ((\t. Cx(bernoulli 1 (frac(drop t))) / (z + Cx(drop t)))
3007 (integral {t | &0 <= drop t}
3008 (\t. Cx(bernoulli (2 * p + 1) (frac(drop t))) /
3009 (z + Cx(drop t)) pow (2 * p + 1)) /
3011 vsum(1..p) (\k. Cx(bernoulli (2 * k) (&0) /
3012 (&4 * &k pow 2 - &2 * &k)) /
3013 z pow (2 * k - 1))))
3014 {t | &0 <= drop t}`,
3015 REPEAT STRIP_TAC THEN
3016 MATCH_MP_TAC HAS_INTEGRAL_LIM_SEQUENTIALLY THEN CONJ_TAC THENL
3017 [REWRITE_TAC[o_DEF; LIFT_DROP; complex_div; COMPLEX_VEC_0] THEN
3018 MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL_BOUNDED THEN
3019 EXISTS_TAC `&1 / &2` THEN CONJ_TAC THENL
3020 [MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `x:real` THEN
3021 REWRITE_TAC[BERNOULLI_CONV `bernoulli 1 x`] THEN DISJ1_TAC THEN
3022 REWRITE_TAC[COMPLEX_NORM_CX] THEN
3023 MP_TAC(SPEC `x:real` FLOOR_FRAC) THEN REAL_ARITH_TAC;
3024 MATCH_MP_TAC(REWRITE_RULE[o_DEF] LIM_INFINITY_POSINFINITY_CX) THEN
3025 ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN REWRITE_TAC[LIM_INV_Z_OFFSET]];
3027 MP_TAC(GEN `n:num` (CONJ
3028 (SPECL [`0`; `n:num`; `z:complex`] lemma1)
3029 (SPECL [`p:num`; `n:num`; `z:complex`] lemma1))) THEN
3030 ASM_REWRITE_TAC[FORALL_AND_THM] THEN ONCE_REWRITE_TAC[IMP_CONJ] THEN
3031 REWRITE_TAC[VSUM_CLAUSES_NUMSEG] THEN CONV_TAC NUM_REDUCE_CONV THEN
3032 REWRITE_TAC[SIMPLE_COMPLEX_ARITH `a * Cx(&1) / b = a / b`] THEN
3033 REWRITE_TAC[LIFT_NUM; COMPLEX_POW_1; COMPLEX_DIV_1] THEN
3034 DISCH_THEN(fun th -> ONCE_REWRITE_TAC[th]) THEN
3035 REWRITE_TAC[VECTOR_ARITH `a + vec 0 + x:real^N = a + y <=> x = y`] THEN
3036 DISCH_THEN(fun th -> ONCE_REWRITE_TAC[th]) THEN
3037 GEN_REWRITE_TAC LAND_CONV [COMPLEX_RING `a - b:complex = --b + a`] THEN
3038 MATCH_MP_TAC LIM_ADD THEN CONJ_TAC THENL
3039 [REWRITE_TAC[GSYM VSUM_NEG] THEN MATCH_MP_TAC LIM_VSUM THEN
3040 REWRITE_TAC[FINITE_NUMSEG; IN_NUMSEG] THEN
3041 X_GEN_TAC `k:num` THEN STRIP_TAC THEN
3042 REWRITE_TAC[CX_DIV; complex_div; GSYM COMPLEX_MUL_RNEG] THEN
3043 REWRITE_TAC[GSYM COMPLEX_MUL_ASSOC] THEN
3044 MATCH_MP_TAC LIM_COMPLEX_LMUL THEN
3045 REWRITE_TAC[SIMPLE_COMPLEX_ARITH
3046 `inv x * (y * w - y * z):complex = y / x * (w - z)`] THEN
3048 `Cx(&(FACT(2 * k - 2))) / Cx(&(FACT(2 * k))) =
3049 inv(Cx(&4 * &k pow 2 - &2 * &k))`
3050 (fun th -> ONCE_REWRITE_TAC[th])
3052 [MATCH_MP_TAC(COMPLEX_FIELD
3053 `~(y = Cx(&0)) /\ ~(z = Cx(&0)) /\ x * z = y
3054 ==> x / y = inv(z)`) THEN
3055 REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; FACT_NZ] THEN
3056 REWRITE_TAC[REAL_ENTIRE; REAL_ARITH
3057 `&4 * x pow 2 - &2 * x = (&2 * x) * (&2 * x - &1)`] THEN
3058 ASM_SIMP_TAC[ARITH_RULE `1 <= k ==> 1 <= 2 * k`; REAL_OF_NUM_SUB;
3059 REAL_OF_NUM_MUL; GSYM CX_MUL; CX_INJ; REAL_OF_NUM_EQ] THEN
3060 CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
3061 UNDISCH_TAC `1 <= k` THEN SPEC_TAC(`k:num`,`k:num`) THEN
3062 INDUCT_TAC THEN REWRITE_TAC[ADD_CLAUSES; MULT_CLAUSES] THEN
3063 CONV_TAC NUM_REDUCE_CONV THEN DISCH_THEN (K ALL_TAC) THEN
3064 REWRITE_TAC[ARITH_RULE `(2 + k) - 2 = k /\ (2 + k) - 1 = k + 1`] THEN
3065 REWRITE_TAC[ARITH_RULE `2 + k = SUC(SUC k)`; FACT] THEN ARITH_TAC;
3066 MATCH_MP_TAC LIM_COMPLEX_LMUL THEN
3067 ONCE_REWRITE_TAC[COMPLEX_RING `--z = Cx(&0) - z`] THEN
3068 MATCH_MP_TAC LIM_SUB THEN REWRITE_TAC[LIM_CONST] THEN
3069 REWRITE_TAC[GSYM COMPLEX_POW_INV] THEN
3070 ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN
3071 MATCH_MP_TAC LIM_INV_N_POW_OFFSET THEN ASM_ARITH_TAC];
3072 MP_TAC(SPECL [`z:complex`; `2 * p + 1`] lemma3) THEN
3073 ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC] THEN
3074 FIRST_ASSUM(MP_TAC o SPEC `Cx(&(FACT(2 * p)))` o
3075 MATCH_MP INTEGRABLE_COMPLEX_LMUL) THEN
3076 DISCH_THEN(MP_TAC o MATCH_MP INTEGRABLE_INTEGRAL) THEN
3077 ASM_SIMP_TAC[INTEGRAL_COMPLEX_LMUL] THEN
3078 REWRITE_TAC[HAS_INTEGRAL_LIM_AT_POSINFINITY] THEN
3080 MATCH_MP LIM_POSINFINITY_SEQUENTIALLY o CONJUNCT2) THEN
3081 DISCH_THEN(MP_TAC o SPEC `inv(Cx(&(FACT(2 * p + 1))))` o
3082 MATCH_MP LIM_COMPLEX_RMUL) THEN
3083 MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THENL
3084 [REWRITE_TAC[LIFT_NUM; FUN_EQ_THM] THEN
3085 X_GEN_TAC `n:num` THEN REWRITE_TAC[complex_div] THEN
3086 AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC INTEGRAL_EQ THEN
3087 REWRITE_TAC[complex_div; COMPLEX_MUL_AC];
3089 REWRITE_TAC[complex_div] THEN MATCH_MP_TAC(COMPLEX_RING
3090 `x * y:complex = z ==> (x * i) * y = i * z`) THEN
3091 REWRITE_TAC[GSYM ADD1; FACT; GSYM REAL_OF_NUM_MUL; CX_MUL] THEN
3092 MATCH_MP_TAC(COMPLEX_FIELD
3093 `~(a = Cx(&0)) /\ ~(b = Cx(&0)) ==> a * inv(b * a) = inv b`) THEN
3094 REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; FACT_NZ; NOT_SUC]]]) in
3096 (`!z n. 1 <= n /\ (Im z = &0 ==> &0 < Re z)
3097 ==> (\t. Cx(bernoulli n (frac (drop t))) / (z + Cx(drop t)) pow n)
3098 integrable_on {t | &0 <= drop t}`,
3099 REPEAT STRIP_TAC THEN FIRST_X_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE
3100 `1 <= n ==> n = 1 \/ 2 <= n`)) THEN
3101 ASM_SIMP_TAC[lemma3; COMPLEX_POW_1] THEN
3102 REWRITE_TAC[integrable_on] THEN
3103 ASM_MESON_TAC[LE_REFL; lemma4]) in
3105 (`!z. (Im z = &0 ==> &0 < Re z)
3107 ((z - Cx(&1) / Cx(&2)) * clog(z) - z + Cx(&1)) +
3108 (integral {t | &0 <= drop t}
3109 (\t. Cx(bernoulli 1 (frac(drop t))) /
3110 (Cx(&1) + Cx(drop t))) -
3111 integral {t | &0 <= drop t}
3112 (\t. Cx(bernoulli 1 (frac(drop t))) /
3113 (z + Cx(drop t))))`,
3114 REPEAT STRIP_TAC THEN
3115 SUBGOAL_THEN `!n. ~(z + Cx(&n) = Cx(&0))` ASSUME_TAC THENL
3116 [REWRITE_TAC[COMPLEX_EQ; IM_ADD; RE_ADD; IM_CX; RE_CX] THEN
3119 SUBGOAL_THEN `~(z = Cx(&0))` ASSUME_TAC THENL
3120 [ASM_MESON_TAC[COMPLEX_ADD_RID]; ALL_TAC] THEN
3121 FIRST_ASSUM(MP_TAC o MATCH_MP LGAMMA_ALT) THEN
3122 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT]
3123 (REWRITE_RULE[TRIVIAL_LIMIT_SEQUENTIALLY]
3124 (ISPEC `sequentially` LIM_UNIQUE))) THEN
3125 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC
3126 `\n. z * clog (Cx(&n)) - clog (Cx(&n + &1)) +
3127 ((Cx(&1) + Cx(&n) + Cx(&1) / Cx(&2)) * clog (Cx(&1) + Cx(&n)) -
3128 (Cx(&1) - Cx(&1) / Cx(&2)) * clog (Cx(&1)) -
3130 integral (interval [lift (&0),lift (&n)])
3131 (\x. Cx(bernoulli 1 (frac (drop x))) *
3132 Cx(&1) / (Cx(&1) + Cx(drop x)) pow 1) /
3134 ((z + Cx(&n) + Cx(&1) / Cx(&2)) * clog (z + Cx(&n)) -
3135 (z - Cx(&1) / Cx(&2)) * clog z -
3137 integral (interval [lift (&0),lift (&n)])
3138 (\x. Cx(bernoulli 1 (frac (drop x))) *
3139 Cx(&1) / (z + Cx(drop x)) pow 1) /
3141 REWRITE_TAC[] THEN CONJ_TAC THENL
3142 [REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
3143 EXISTS_TAC `1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
3145 `clog(Cx(&(FACT n))) =
3146 vsum(0..n) (\m. clog(Cx(&m) + Cx(&1))) - clog(Cx(&n + &1))`
3148 [REWRITE_TAC[REAL_OF_NUM_ADD; GSYM CX_ADD] THEN
3149 REWRITE_TAC[GSYM(SPEC `1` VSUM_OFFSET); ADD_CLAUSES] THEN
3150 SIMP_TAC[GSYM NPRODUCT_FACT; REAL_OF_NUM_NPRODUCT; FINITE_NUMSEG;
3151 GSYM CX_LOG; LOG_PRODUCT; PRODUCT_POS_LT; IN_NUMSEG;
3152 REAL_OF_NUM_LT; LE_1; GSYM VSUM_CX] THEN
3153 REWRITE_TAC[GSYM ADD1; VSUM_CLAUSES_NUMSEG] THEN
3154 REWRITE_TAC[ARITH_RULE `1 <= SUC n`; REAL_OF_NUM_SUC] THEN
3155 SIMP_TAC[CX_LOG; REAL_OF_NUM_LT; LT_0] THEN
3156 REWRITE_TAC[COMPLEX_RING `(a + b) - b:complex = a`];
3157 ONCE_REWRITE_TAC[COMPLEX_RING
3158 `(a + b - c) - d:complex = (a - c) + (b - d)`]] THEN
3159 MP_TAC(SPECL [`0`; `n:num`] lemma1) THEN
3160 CONV_TAC NUM_REDUCE_CONV THEN
3161 ASM_REWRITE_TAC[VSUM_CLAUSES_NUMSEG; ARITH; VECTOR_ADD_LID] THEN
3162 ASM_SIMP_TAC[RE_CX; REAL_LT_01];
3164 REWRITE_TAC[COMPLEX_POW_1; COMPLEX_DIV_1] THEN
3165 ONCE_REWRITE_TAC[COMPLEX_RING
3166 `a + (b - c - n + x) - (d - e - n + y):complex =
3167 (a + (b - c) - (d - e)) + (x - y)`] THEN
3168 MATCH_MP_TAC LIM_ADD THEN CONJ_TAC THENL
3169 [REWRITE_TAC[CLOG_1; COMPLEX_MUL_RZERO; COMPLEX_SUB_RZERO] THEN
3170 ONCE_REWRITE_TAC[LIM_NULL_COMPLEX] THEN REWRITE_TAC[] THEN
3171 ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN REWRITE_TAC[CX_ADD] THEN
3172 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC
3173 `\n. z * (clog(Cx(&n)) - clog(z + Cx(&n))) +
3174 (Cx(&n) + Cx(&1) / Cx(&2)) *
3175 (clog(Cx(&1) + Cx(&n)) - clog(z + Cx(&n))) -
3178 [MATCH_MP_TAC ALWAYS_EVENTUALLY THEN CONV_TAC COMPLEX_RING;
3180 MATCH_MP_TAC LIM_NULL_COMPLEX_ADD THEN CONJ_TAC THENL
3181 [MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
3182 GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV o BINDER_CONV o
3183 LAND_CONV o RAND_CONV) [GSYM COMPLEX_ADD_LID] THEN
3184 ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN REWRITE_TAC[LIM_SUB_CLOG];
3185 ONCE_REWRITE_TAC[COMPLEX_RING
3186 `(a + h) * x - y:complex = h * x + a * x - y`] THEN
3187 MATCH_MP_TAC LIM_NULL_COMPLEX_ADD THEN REWRITE_TAC[] THEN
3188 ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN
3189 SIMP_TAC[LIM_NULL_COMPLEX_LMUL; LIM_SUB_CLOG] THEN
3190 REWRITE_TAC[GSYM LIM_NULL_COMPLEX; LIM_N_MUL_SUB_CLOG]];
3191 REWRITE_TAC[complex_div; COMPLEX_MUL_LID] THEN
3192 REWRITE_TAC[LIFT_NUM] THEN MATCH_MP_TAC LIM_SUB THEN
3193 CONJ_TAC THEN REWRITE_TAC[GSYM LIFT_NUM] THEN
3194 MATCH_MP_TAC LIM_POSINFINITY_SEQUENTIALLY THEN
3195 REWRITE_TAC[LIFT_NUM] THEN
3196 MATCH_MP_TAC(MATCH_MP (TAUT `(p <=> q /\ r) ==> (p ==> r)`)
3197 (SPEC_ALL HAS_INTEGRAL_LIM_AT_POSINFINITY)) THEN
3198 REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL; GSYM complex_div] THEN
3199 ONCE_REWRITE_TAC[COMPLEX_RING `z + x:complex = (z + x) pow 1`] THEN
3200 MATCH_MP_TAC lemma5 THEN
3201 ASM_REWRITE_TAC[LE_REFL; RE_CX; REAL_LT_01]]) in
3203 (`((\y. integral {t | &0 <= drop t}
3204 (\t. Cx (bernoulli 1 (frac (drop t))) / (ii * Cx y + Cx(drop t))))
3205 --> Cx(&0)) at_posinfinity`,
3206 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
3208 `\y. integral {t | &0 <= drop t}
3209 (\t. Cx(bernoulli 3 (frac (drop t))) /
3210 (ii * Cx y + Cx(drop t)) pow 3) / Cx(&3) -
3211 Cx(bernoulli 2 (&0) / &2) / (ii * Cx y)` THEN
3213 [REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN
3214 EXISTS_TAC `&1` THEN X_GEN_TAC `y:real` THEN DISCH_TAC THEN
3215 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
3216 MP_TAC(ISPECL [`ii * Cx y`; `1`] lemma4) THEN
3217 ASM_SIMP_TAC[IM_MUL_II; RE_CX; REAL_LT_IMP_NZ; LE_REFL] THEN
3218 REWRITE_TAC[VSUM_SING_NUMSEG] THEN
3219 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
3220 REWRITE_TAC[COMPLEX_POW_1] THEN
3221 DISCH_THEN(SUBST1_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN REFL_TAC;
3223 MATCH_MP_TAC LIM_NULL_COMPLEX_SUB THEN CONJ_TAC THENL
3225 REWRITE_TAC[complex_div] THEN MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
3226 REWRITE_TAC[COMPLEX_INV_MUL] THEN MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
3227 REWRITE_TAC[LIM_INV_X]] THEN
3228 ONCE_REWRITE_TAC[complex_div] THEN MATCH_MP_TAC LIM_NULL_COMPLEX_RMUL THEN
3229 MATCH_MP_TAC LIM_NULL_COMPARISON_COMPLEX THEN
3231 `\y. Cx(&1 / &2 / y) *
3232 integral {t | &0 <= drop t}
3233 (\t. inv(Cx(&1) + Cx(drop t)) pow 2)` THEN
3236 MATCH_MP_TAC LIM_NULL_COMPLEX_RMUL THEN
3237 REWRITE_TAC[real_div; CX_MUL] THEN
3238 MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL THEN
3239 REWRITE_TAC[LIM_INV_X; CX_INV]] THEN
3240 REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN
3241 EXISTS_TAC `&1` THEN X_GEN_TAC `y:real` THEN DISCH_TAC THEN
3242 SIMP_TAC[GSYM INTEGRAL_COMPLEX_LMUL; lemma2;
3243 ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE; ARITH;
3244 RE_CX; REAL_OF_NUM_LT] THEN
3245 MATCH_MP_TAC(REAL_ARITH
3246 `abs(Re z) <= norm z /\ x <= Re z ==> x <= norm z`) THEN
3247 REWRITE_TAC[COMPLEX_NORM_GE_RE_IM] THEN REWRITE_TAC[RE_DEF] THEN
3248 MATCH_MP_TAC INTEGRAL_NORM_BOUND_INTEGRAL_COMPONENT THEN
3249 REWRITE_TAC[DIMINDEX_2; ARITH] THEN REPEAT CONJ_TAC THENL
3250 [MATCH_MP_TAC lemma5 THEN
3251 REWRITE_TAC[ARITH; IM_MUL_II; RE_CX] THEN ASM_REAL_ARITH_TAC;
3252 MATCH_MP_TAC INTEGRABLE_COMPLEX_LMUL THEN
3253 MATCH_MP_TAC ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE THEN
3254 MATCH_MP_TAC lemma2 THEN
3255 REWRITE_TAC[RE_CX; REAL_LT_01] THEN ARITH_TAC;
3257 REWRITE_TAC[FORALL_LIFT; IN_ELIM_THM; LIFT_DROP; GSYM RE_DEF] THEN
3258 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
3259 REWRITE_TAC[GSYM CX_ADD; GSYM CX_INV; GSYM CX_MUL; GSYM CX_POW] THEN
3260 REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; RE_CX] THEN
3261 REWRITE_TAC[REAL_ARITH `&1 / &2 / y * x = (&1 / &4) * (&2 / y * x)`] THEN
3262 REWRITE_TAC[real_div] THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
3263 REWRITE_TAC[REAL_ABS_POS; REAL_LE_INV_EQ; NORM_POS_LE] THEN CONJ_TAC THENL
3264 [MP_TAC(SPECL [`3`; `frac x`] BERNOULLI_BOUND) THEN
3265 CONV_TAC NUM_REDUCE_CONV THEN
3266 REWRITE_TAC[BERNOULLI_CONV `bernoulli 2 (&0)`] THEN
3267 ANTS_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN
3268 SIMP_TAC[IN_REAL_INTERVAL; REAL_LT_IMP_LE; FLOOR_FRAC];
3270 REWRITE_TAC[REAL_POW_INV; GSYM REAL_INV_MUL; GSYM REAL_MUL_ASSOC] THEN
3271 GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [GSYM REAL_INV_INV] THEN
3272 REWRITE_TAC[GSYM REAL_INV_MUL] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
3274 [MATCH_MP_TAC REAL_LT_MUL THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
3275 MATCH_MP_TAC REAL_LT_MUL THEN
3276 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC REAL_POW_LT] THEN
3279 REWRITE_TAC[COMPLEX_NORM_POW; REAL_ARITH
3280 `x pow 3 = (x:real) * x pow 2`] THEN
3281 ONCE_REWRITE_TAC[REAL_ARITH `inv(&2) * y * x = y * x / &2`] THEN
3282 MATCH_MP_TAC REAL_LE_MUL2 THEN REPEAT CONJ_TAC THENL
3283 [ASM_REAL_ARITH_TAC;
3284 W(MP_TAC o PART_MATCH (rand o rand) COMPLEX_NORM_GE_RE_IM o
3286 REWRITE_TAC[IM_ADD; IM_MUL_II; RE_CX; IM_CX] THEN REAL_ARITH_TAC;
3287 REWRITE_TAC[REAL_ARITH `&0 <= x / &2 <=> &0 <= x`] THEN
3288 MATCH_MP_TAC REAL_POW_LE THEN ASM_REAL_ARITH_TAC;
3290 REWRITE_TAC[COMPLEX_SQNORM] THEN
3291 REWRITE_TAC[RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; IM_CX; RE_CX] THEN
3292 REWRITE_TAC[REAL_ADD_LID; REAL_ADD_RID; REAL_NEG_0] THEN
3293 MATCH_MP_TAC(REAL_ARITH
3294 `&1 pow 2 <= y pow 2 /\ &0 <= (x - &1) pow 2
3295 ==> (&1 + x) pow 2 / &2 <= x pow 2 + y pow 2`) THEN
3296 REWRITE_TAC[REAL_LE_POW_2] THEN MATCH_MP_TAC REAL_POW_LE2 THEN
3297 ASM_REAL_ARITH_TAC]) in
3299 (`integral {t | &0 <= drop t}
3300 (\t. Cx (bernoulli 1 (frac (drop t))) / (Cx(&1) + Cx(drop t))) =
3301 Cx(log(&2 * pi) / &2 - &1)`,
3302 MATCH_MP_TAC(MESON[REAL]
3303 `real z /\ Re z = y ==> z = Cx y`) THEN
3305 [MATCH_MP_TAC REAL_COMPLEX_INTEGRAL THEN CONJ_TAC THENL
3306 [ONCE_REWRITE_TAC[COMPLEX_RING `z + x:complex = (z + x) pow 1`] THEN
3307 MATCH_MP_TAC lemma5 THEN
3308 ASM_REWRITE_TAC[LE_REFL; RE_CX; REAL_LT_01];
3309 REWRITE_TAC[GSYM CX_ADD; GSYM CX_DIV; REAL_CX]];
3310 GEN_REWRITE_TAC I [GSYM CX_INJ]] THEN
3311 MATCH_MP_TAC(ISPEC `at_posinfinity` LIM_UNIQUE) THEN
3313 `\y:real. Cx(Re(integral {t | &0 <= drop t}
3314 (\t. Cx(bernoulli 1 (frac (drop t))) / (Cx(&1) + Cx(drop t)))))` THEN
3315 REWRITE_TAC[TRIVIAL_LIMIT_AT_POSINFINITY; LIM_CONST] THEN
3316 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
3318 `\y. Cx(Re(lgamma(ii * Cx y) -
3319 ((ii * Cx y - Cx(&1) / Cx(&2)) * clog(ii * Cx y) -
3320 ii * Cx y + Cx(&1)) +
3321 integral {t | &0 <= drop t}
3322 (\t. Cx(bernoulli 1 (frac(drop t))) /
3323 (ii * Cx y + Cx(drop t)))))` THEN
3324 REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN CONJ_TAC THENL
3325 [EXISTS_TAC `&1` THEN X_GEN_TAC `y:real` THEN DISCH_TAC THEN
3326 AP_TERM_TAC THEN AP_TERM_TAC THEN
3327 MP_TAC(SPEC `ii * Cx y` lemma6) THEN
3328 ASM_SIMP_TAC[IM_MUL_II; RE_CX; REAL_LT_IMP_NZ] THEN
3329 ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN
3330 REWRITE_TAC[COMPLEX_RING `(s + i - j) - s + j:complex = i`];
3332 REWRITE_TAC[RE_ADD; RE_SUB; CX_ADD; CX_SUB] THEN
3333 GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_ADD_RID] THEN
3334 MATCH_MP_TAC LIM_ADD THEN CONJ_TAC THENL
3336 MATCH_MP_TAC LIM_NULL_COMPARISON_COMPLEX THEN
3337 EXISTS_TAC `\y. integral {t | &0 <= drop t}
3338 (\t. Cx(bernoulli 1 (frac(drop t))) / (ii * Cx y + Cx(drop t)))` THEN
3339 REWRITE_TAC[lemma7] THEN MATCH_MP_TAC ALWAYS_EVENTUALLY THEN
3340 REWRITE_TAC[COMPLEX_NORM_CX; COMPLEX_NORM_GE_RE_IM]] THEN
3341 REWRITE_TAC[RE_MUL_II; IM_CX; REAL_NEG_0; COMPLEX_SUB_RZERO; RE_CX] THEN
3342 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
3344 `\y. Cx(log(norm(cgamma(ii * Cx y)))) -
3345 (Cx(--(pi * y + log y) / &2) + Cx(&1))` THEN
3346 REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN CONJ_TAC THENL
3347 [EXISTS_TAC `&1` THEN X_GEN_TAC `y:real` THEN DISCH_TAC THEN
3348 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
3350 [MP_TAC(SPEC `ii * Cx y` CGAMMA_LGAMMA) THEN COND_CASES_TAC THENL
3351 [FIRST_X_ASSUM(CHOOSE_THEN (MP_TAC o AP_TERM `Im`)) THEN
3352 REWRITE_TAC[IM_ADD; IM_MUL_II; IM_CX; RE_CX] THEN ASM_REAL_ARITH_TAC;
3353 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[NORM_CEXP; LOG_EXP]];
3354 AP_THM_TAC THEN AP_TERM_TAC THEN
3355 AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
3356 REWRITE_TAC[COMPLEX_SUB_RDISTRIB; RE_SUB; GSYM CX_DIV] THEN
3357 REWRITE_TAC[GSYM COMPLEX_MUL_ASSOC; RE_MUL_II; IM_MUL_II; RE_MUL_CX;
3359 ASM_SIMP_TAC[RE_CX; IM_CX; REAL_LT_IMP_LE; CX_INJ; REAL_LT_IMP_NZ;
3360 GSYM CX_LOG; CLOG_MUL_II] THEN
3361 REWRITE_TAC[RE_ADD; IM_ADD; RE_CX; IM_CX; RE_MUL_II; IM_MUL_II] THEN
3364 MATCH_MP_TAC LIM_TRANSFORM_EVENTUALLY THEN
3366 `\y. Cx((log(&2 * pi) - log(&1 - inv(exp(pi * y) pow 2))) / &2 - &1)` THEN
3367 REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN CONJ_TAC THENL
3368 [EXISTS_TAC `&1` THEN X_GEN_TAC `y:real` THEN DISCH_TAC THEN
3369 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
3370 MP_TAC(SPEC `ii * Cx y + Cx(&1)` CGAMMA_REFLECTION) THEN
3371 REWRITE_TAC[CGAMMA_RECURRENCE; COMPLEX_ENTIRE; II_NZ; CX_INJ] THEN
3372 ASM_SIMP_TAC[REAL_LT_IMP_NZ] THEN
3373 REWRITE_TAC[COMPLEX_RING `Cx(&1) - (z + Cx(&1)) = --z`] THEN
3374 MP_TAC(SPEC `cgamma(ii * Cx y)` COMPLEX_NORM_POW_2) THEN
3375 REWRITE_TAC[CNJ_MUL; CNJ_II; CNJ_CX; CNJ_CGAMMA] THEN
3376 REWRITE_TAC[COMPLEX_MUL_LNEG; GSYM COMPLEX_MUL_ASSOC] THEN
3377 DISCH_THEN(SUBST1_TAC o SYM) THEN
3378 REWRITE_TAC[COMPLEX_RING `w * (z + Cx(&1)) = w * z + w`] THEN
3379 REWRITE_TAC[CSIN_ADD; GSYM CX_SIN; GSYM CX_COS; SIN_PI; COS_PI] THEN
3380 REWRITE_TAC[COMPLEX_MUL_RZERO; CX_NEG; COMPLEX_MUL_RNEG] THEN
3381 REWRITE_TAC[COMPLEX_ADD_RID; COMPLEX_MUL_RID] THEN
3382 REWRITE_TAC[csin; COMPLEX_RING `--ii * x * ii * y = x * y`] THEN
3383 REWRITE_TAC[COMPLEX_RING `ii * x * ii * y = --(x * y)`] THEN
3384 REWRITE_TAC[complex_div; COMPLEX_INV_INV; COMPLEX_INV_MUL;
3385 COMPLEX_INV_NEG; COMPLEX_MUL_RNEG] THEN
3386 ASM_SIMP_TAC[CX_INJ; REAL_LT_IMP_NZ; COMPLEX_FIELD
3388 ==> (ii * y * z = --(p * i * Cx(&2) * ii) <=>
3389 z = --(Cx(&2) * p) * inv y * i)`] THEN
3390 REWRITE_TAC[GSYM COMPLEX_INV_MUL; GSYM CX_MUL;
3391 GSYM CX_EXP; CEXP_NEG] THEN
3392 REWRITE_TAC[COMPLEX_MUL_LNEG; GSYM COMPLEX_MUL_RNEG;
3393 COMPLEX_NEG_INV] THEN
3394 REWRITE_TAC[COMPLEX_NEG_SUB] THEN
3395 REWRITE_TAC[GSYM CX_INV; GSYM CX_SUB; GSYM CX_MUL; GSYM CX_INV] THEN
3396 REWRITE_TAC[CX_INJ; GSYM CX_POW] THEN
3397 DISCH_THEN(MP_TAC o AP_TERM `sqrt`) THEN
3398 REWRITE_TAC[POW_2_SQRT_ABS; REAL_ABS_NORM] THEN
3399 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[REAL_INV_MUL] THEN
3400 SUBGOAL_THEN `&0 < exp(pi * y) - inv(exp(pi * y))` ASSUME_TAC THENL
3401 [MATCH_MP_TAC(REAL_ARITH `x < &1 /\ &1 < y ==> &0 < y - x`) THEN
3402 CONJ_TAC THENL [MATCH_MP_TAC REAL_INV_LT_1; ALL_TAC] THEN
3403 MATCH_MP_TAC REAL_EXP_LT_1 THEN MATCH_MP_TAC REAL_LT_MUL THEN
3404 ASM_REWRITE_TAC[PI_POS];
3405 ASM_SIMP_TAC[LOG_MUL; REAL_LT_INV_EQ; PI_POS; REAL_LT_MUL;
3406 REAL_ARITH `&0 < &2 * x <=> &0 < x`; LOG_SQRT; LOG_INV]] THEN
3407 REWRITE_TAC[GSYM CX_ADD; GSYM CX_SUB] THEN
3408 REWRITE_TAC[REAL_ARITH
3409 `(l2 + --l + x) / &2 - (--(py + l) / &2 + &1) =
3410 (l2 + py + x) / &2 - &1`] THEN
3411 SIMP_TAC[REAL_EXP_NZ; REAL_FIELD
3412 `~(e = &0) ==> e - inv e = e * (&1 - inv(e pow 2))`] THEN
3413 SUBGOAL_THEN `inv (exp (pi * y) pow 2) < &1` ASSUME_TAC THENL
3414 [MATCH_MP_TAC REAL_INV_LT_1 THEN
3415 MATCH_MP_TAC REAL_POW_LT_1 THEN
3416 REWRITE_TAC[ARITH] THEN
3417 MATCH_MP_TAC REAL_EXP_LT_1 THEN
3418 MATCH_MP_TAC REAL_LT_MUL THEN
3419 ASM_SIMP_TAC[REAL_LT_MUL; PI_POS];
3420 ASM_SIMP_TAC[LOG_MUL; REAL_EXP_POS_LT; REAL_SUB_LT; LOG_EXP]] THEN
3421 REWRITE_TAC[CX_INJ] THEN REAL_ARITH_TAC;
3423 REWRITE_TAC[CX_SUB; CX_DIV] THEN MATCH_MP_TAC LIM_SUB THEN
3424 REWRITE_TAC[LIM_CONST] THEN
3425 MATCH_MP_TAC LIM_COMPLEX_DIV THEN REWRITE_TAC[LIM_CONST] THEN
3426 CONJ_TAC THENL [ALL_TAC; CONV_TAC COMPLEX_RING] THEN
3427 GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_SUB_RZERO] THEN
3428 MATCH_MP_TAC LIM_SUB THEN REWRITE_TAC[LIM_CONST] THEN
3430 [`clog`; `at_posinfinity`;
3431 `\y. Cx(&1 - inv(exp(pi * y) pow 2))`;
3432 `Cx(&1)`] LIM_CONTINUOUS_FUNCTION) THEN
3433 REWRITE_TAC[CLOG_1] THEN ANTS_TAC THENL
3434 [SIMP_TAC[CONTINUOUS_AT_CLOG; RE_CX; REAL_LT_01; CX_SUB] THEN
3435 GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_SUB_RZERO] THEN
3436 MATCH_MP_TAC LIM_SUB THEN REWRITE_TAC[LIM_CONST] THEN
3437 REWRITE_TAC[CX_INV; CX_EXP; CX_POW; GSYM COMPLEX_POW_INV] THEN
3438 MATCH_MP_TAC LIM_NULL_COMPLEX_POW THEN CONV_TAC NUM_REDUCE_CONV THEN
3439 REWRITE_TAC[LIM_AT_POSINFINITY] THEN
3440 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
3441 EXISTS_TAC `&1 + inv(e)` THEN REWRITE_TAC[dist; real_ge] THEN
3442 X_GEN_TAC `x:real` THEN DISCH_TAC THEN
3443 REWRITE_TAC[COMPLEX_SUB_RZERO; COMPLEX_NORM_INV] THEN
3444 MATCH_MP_TAC REAL_LT_LINV THEN ASM_REWRITE_TAC[NORM_CEXP; RE_CX] THEN
3445 FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
3447 ==> &1 + pi * x <= z /\ &0 <= x * (pi - &1) ==> e < z`)) THEN
3448 REWRITE_TAC[REAL_EXP_LE_X] THEN MATCH_MP_TAC REAL_LE_MUL THEN
3449 MP_TAC(SPEC `e:real` REAL_LT_INV_EQ) THEN
3450 MP_TAC PI_APPROX_32 THEN ASM_REAL_ARITH_TAC;
3451 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
3452 REWRITE_TAC[EVENTUALLY_AT_POSINFINITY; real_ge] THEN
3453 EXISTS_TAC `&1` THEN X_GEN_TAC `y:real` THEN DISCH_TAC THEN
3454 SUBGOAL_THEN `&0 < y` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
3455 MATCH_MP_TAC(GSYM CX_LOG) THEN REWRITE_TAC[REAL_SUB_LT] THEN
3456 MATCH_MP_TAC REAL_INV_LT_1 THEN MATCH_MP_TAC REAL_POW_LT_1 THEN
3457 REWRITE_TAC[ARITH] THEN MATCH_MP_TAC REAL_EXP_LT_1 THEN
3458 MATCH_MP_TAC REAL_LT_MUL THEN ASM_SIMP_TAC[REAL_LT_MUL; PI_POS]]) in
3459 CONJ_TAC THENL [MATCH_ACCEPT_TAC lemma5; ALL_TAC] THEN
3460 REPEAT STRIP_TAC THEN ASM_SIMP_TAC[lemma6] THEN
3461 REWRITE_TAC[lemma8; CX_SUB] THEN
3462 REWRITE_TAC[COMPLEX_RING
3463 `(x + Cx(&1)) + (a - Cx(&1)) - b = (x + a) - b`] THEN
3464 REWRITE_TAC[complex_sub; GSYM COMPLEX_ADD_ASSOC] THEN AP_TERM_TAC THEN
3465 AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_CASES_TAC `p = 0` THENL
3466 [ASM_REWRITE_TAC[VSUM_CLAUSES_NUMSEG] THEN
3467 CONV_TAC NUM_REDUCE_CONV THEN
3468 REWRITE_TAC[COMPLEX_POW_1; COMPLEX_DIV_1; VECTOR_ADD_LID];
3469 MP_TAC(SPECL [`z:complex`; `p:num`] lemma4) THEN
3470 ASM_SIMP_TAC[LE_1] THEN
3471 DISCH_THEN(SUBST1_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN
3472 REWRITE_TAC[complex_sub; COMPLEX_NEG_ADD; VECTOR_NEG_NEG] THEN
3473 REWRITE_TAC[COMPLEX_ADD_SYM]]);;
3475 let LOG_GAMMA_STIRLING = prove
3478 ((x - &1 / &2) * log(x) - x + log(&2 * pi) / &2) +
3479 sum(1..p) (\k. bernoulli (2 * k) (&0) /
3480 (&4 * &k pow 2 - &2 * &k) / x pow (2 * k - 1)) -
3481 real_integral {t | &0 <= t}
3482 (\t. bernoulli (2 * p + 1) (frac t) /
3483 (x + t) pow (2 * p + 1)) /
3485 REPEAT STRIP_TAC THEN
3486 GEN_REWRITE_TAC I [GSYM REAL_EXP_INJ] THEN
3487 ASM_SIMP_TAC[EXP_LOG; GAMMA_POS_LT] THEN
3488 GEN_REWRITE_TAC I [GSYM CX_INJ] THEN REWRITE_TAC[CX_GAMMA] THEN
3489 MP_TAC(ISPEC `Cx x` CGAMMA_LGAMMA) THEN
3490 COND_CASES_TAC THENL
3491 [FIRST_ASSUM(CHOOSE_THEN (MP_TAC o AP_TERM `Re`)) THEN
3492 REWRITE_TAC[RE_ADD; RE_CX] THEN ASM_REAL_ARITH_TAC;
3494 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[CX_EXP] THEN AP_TERM_TAC THEN
3495 MP_TAC(ISPECL [`Cx x`; `p:num`] LGAMMA_STIRLING) THEN
3496 ASM_REWRITE_TAC[RE_CX; CX_GAMMA] THEN DISCH_THEN SUBST1_TAC THEN
3497 REWRITE_TAC[CX_ADD; CX_SUB; CX_DIV; CX_MUL; GSYM VSUM_CX] THEN
3498 ASM_SIMP_TAC[GSYM CX_LOG; CX_POW] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
3499 AP_THM_TAC THEN AP_TERM_TAC THEN
3500 REWRITE_TAC[COMPLEX_EQ; RE_CX; IM_CX] THEN
3501 REWRITE_TAC[RE_DEF; IM_DEF] THEN CONJ_TAC THEN
3502 W(MP_TAC o PART_MATCH (lhand o rand) INTEGRAL_COMPONENT o lhand o snd) THEN
3503 ASM_SIMP_TAC[LGAMMA_STIRLING_INTEGRALS_EXIST;
3504 ARITH_RULE `1 <= 2 * p + 1`; RE_CX] THEN
3505 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM RE_DEF; GSYM IM_DEF] THEN
3506 REWRITE_TAC[GSYM CX_ADD; GSYM CX_POW; GSYM CX_DIV; RE_CX; IM_CX] THEN
3507 REWRITE_TAC[LIFT_NUM; INTEGRAL_0; DROP_VEC] THEN
3509 `{t | &0 <= drop t} = IMAGE lift {t | &0 <= t}`
3511 [REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM] THEN MESON_TAC[LIFT_DROP];
3512 MATCH_MP_TAC(GSYM(REWRITE_RULE[o_DEF] REAL_INTEGRAL))] THEN
3513 REWRITE_TAC[REAL_INTEGRABLE_ON; o_DEF] THEN
3514 MP_TAC(SPECL [`Cx x`; `2 * p + 1`] LGAMMA_STIRLING_INTEGRALS_EXIST) THEN
3515 ASM_REWRITE_TAC[RE_CX; ARITH_RULE `1 <= 2 * p + 1`] THEN
3516 GEN_REWRITE_TAC LAND_CONV [INTEGRABLE_COMPONENTWISE] THEN
3517 DISCH_THEN(MP_TAC o SPEC `1`) THEN REWRITE_TAC[DIMINDEX_2; ARITH] THEN
3518 REWRITE_TAC[GSYM CX_ADD; GSYM CX_POW; GSYM CX_DIV; RE_CX; GSYM RE_DEF] THEN
3519 MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
3520 REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM] THEN MESON_TAC[LIFT_DROP]);;
3522 (* ------------------------------------------------------------------------- *)
3523 (* Some other mathematical facts that don't directly involve the gamma *)
3524 (* function can now be proved relatively easily: Euler's product for sin, *)
3525 (* Wallis's product for pi, and the Gaussian integral. *)
3526 (* ------------------------------------------------------------------------- *)
3528 let CSIN_PRODUCT = prove
3529 (`!z. ((\n. z * cproduct(1..n) (\m. Cx(&1) - (z / Cx(pi * &m)) pow 2))
3530 --> csin(z)) sequentially`,
3531 GEN_TAC THEN REWRITE_TAC[CX_MUL; complex_div; COMPLEX_INV_MUL] THEN
3532 REWRITE_TAC[COMPLEX_MUL_ASSOC] THEN REWRITE_TAC[GSYM complex_div] THEN
3533 ABBREV_TAC `w = z / Cx pi` THEN
3534 SUBGOAL_THEN `Cx pi * w = z` ASSUME_TAC THENL
3535 [EXPAND_TAC "w" THEN MP_TAC PI_NZ THEN REWRITE_TAC[GSYM CX_INJ] THEN
3536 CONV_TAC COMPLEX_FIELD;
3538 SUBGOAL_THEN `csin (Cx pi * w) = Cx pi * csin(Cx pi * w) / Cx pi`
3540 [MP_TAC PI_NZ THEN REWRITE_TAC[GSYM CX_INJ] THEN CONV_TAC COMPLEX_FIELD;
3541 REWRITE_TAC[GSYM COMPLEX_MUL_ASSOC] THEN
3542 MATCH_MP_TAC LIM_COMPLEX_LMUL THEN
3543 ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
3544 REWRITE_TAC[GSYM CGAMMA_REFLECTION] THEN
3545 REWRITE_TAC[COMPLEX_INV_DIV; COMPLEX_INV_MUL] THEN
3546 POP_ASSUM_LIST(K ALL_TAC) THEN SPEC_TAC(`w:complex`,`z:complex`)]] THEN
3547 X_GEN_TAC `z:complex` THEN
3548 MP_TAC(SPEC `z:complex` RECIP_CGAMMA) THEN
3549 MP_TAC(SPEC `Cx(&1) - z` RECIP_CGAMMA) THEN
3550 REWRITE_TAC[COMPLEX_RING `Cx(&1) - z + m = (m + Cx(&1)) - z`] THEN
3551 REWRITE_TAC[GSYM CX_ADD; REAL_OF_NUM_ADD] THEN
3553 (ISPECL [`f:num->complex`; `m:num`; `1`] CPRODUCT_OFFSET)] THEN
3554 SIMP_TAC[CPRODUCT_CLAUSES_LEFT; LE_0] THEN
3555 REWRITE_TAC[ADD_CLAUSES; COMPLEX_ADD_RID; GSYM IMP_CONJ_ALT] THEN
3556 SIMP_TAC[CPRODUCT_CLAUSES_RIGHT; ARITH_RULE `0 < n + 1 /\ 1 <= n + 1`] THEN
3557 DISCH_THEN(MP_TAC o MATCH_MP LIM_COMPLEX_MUL) THEN
3558 REWRITE_TAC[GSYM REAL_OF_NUM_ADD; CX_ADD; ADD_SUB] THEN
3559 SUBGOAL_THEN `((\n. Cx(&n) / (Cx(&n) + Cx(&1) - z)) --> Cx(&1)) sequentially`
3561 [GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_INV_1] THEN
3562 ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
3563 MATCH_MP_TAC LIM_COMPLEX_INV THEN
3564 CONJ_TAC THENL [ALL_TAC; CONV_TAC COMPLEX_RING] THEN
3565 REWRITE_TAC[complex_div; COMPLEX_ADD_RDISTRIB] THEN
3566 GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_ADD_RID] THEN
3567 MATCH_MP_TAC LIM_ADD THEN
3568 SIMP_TAC[LIM_INV_N; LIM_NULL_COMPLEX_LMUL] THEN
3569 MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
3571 SIMP_TAC[COMPLEX_MUL_RINV; CX_INJ; REAL_OF_NUM_EQ; LE_1];
3572 REWRITE_TAC[IMP_IMP] THEN
3573 DISCH_THEN(MP_TAC o MATCH_MP LIM_COMPLEX_MUL)] THEN
3574 REWRITE_TAC[COMPLEX_MUL_LID] THEN
3575 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN
3576 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN
3577 MP_TAC(ISPEC `norm(z:complex)` REAL_ARCH_SIMPLE) THEN
3578 DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
3579 EXISTS_TAC `MAX 1 N` THEN X_GEN_TAC `n:num` THEN
3580 REWRITE_TAC[ARITH_RULE `MAX a b <= c <=> a <= c /\ b <= c`] THEN
3581 STRIP_TAC THEN REWRITE_TAC[CPOW_SUB; CPOW_N; COMPLEX_POW_1] THEN
3582 ASM_SIMP_TAC[CX_INJ; REAL_OF_NUM_EQ; LE_1] THEN
3583 REWRITE_TAC[complex_div; COMPLEX_INV_MUL; COMPLEX_INV_INV] THEN
3584 REWRITE_TAC[GSYM COMPLEX_MUL_ASSOC] THEN REWRITE_TAC[COMPLEX_RING
3585 `a * b * c * d * e * f * g * h * i * j * k :complex =
3586 (a * i) * (j * e) * (h * b) * c * (d * g) * (f * k)`] THEN
3587 SUBGOAL_THEN `~((Cx(&n) + Cx(&1)) - z = Cx(&0))` ASSUME_TAC THENL
3588 [REWRITE_TAC[COMPLEX_SUB_0; GSYM CX_ADD] THEN
3589 DISCH_THEN(MP_TAC o AP_TERM `norm:complex->real`) THEN
3590 REWRITE_TAC[COMPLEX_NORM_CX] THEN
3591 RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN
3593 REWRITE_TAC[COMPLEX_RING `n + Cx(&1) - z = (n + Cx(&1)) - z`]] THEN
3594 ASM_SIMP_TAC[COMPLEX_MUL_RINV; CX_INJ; REAL_OF_NUM_EQ; CPOW_EQ_0; LE_1] THEN
3595 REWRITE_TAC[COMPLEX_MUL_LID] THEN AP_TERM_TAC THEN
3596 SIMP_TAC[GSYM NPRODUCT_FACT; REAL_OF_NUM_NPRODUCT; CX_PRODUCT; FINITE_NUMSEG;
3597 GSYM CPRODUCT_INV; GSYM CPRODUCT_MUL] THEN
3598 MATCH_MP_TAC CPRODUCT_EQ THEN X_GEN_TAC `k:num` THEN
3599 REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN
3600 SUBGOAL_THEN `~(Cx(&k) = Cx(&0))` MP_TAC THENL
3601 [ASM_SIMP_TAC[CX_INJ; REAL_OF_NUM_EQ; LE_1];
3602 CONV_TAC COMPLEX_FIELD]);;
3604 let SIN_PRODUCT = prove
3605 (`!x. ((\n. x * product(1..n) (\m. &1 - (x / (pi * &m)) pow 2)) ---> sin(x))
3607 GEN_TAC THEN MP_TAC(SPEC `Cx x` CSIN_PRODUCT) THEN
3608 REWRITE_TAC[REALLIM_COMPLEX; o_DEF] THEN
3609 SIMP_TAC[CX_MUL; CX_PRODUCT; FINITE_NUMSEG; CX_SIN] THEN
3610 REWRITE_TAC[CX_SUB; CX_DIV; CX_POW; CX_MUL]);;
3612 let WALLIS_ALT = prove
3613 (`((\n. product(1..n) (\k. (&2 * &k) / (&2 * &k - &1) *
3614 (&2 * &k) / (&2 * &k + &1))) ---> pi / &2)
3616 ONCE_REWRITE_TAC[GSYM REAL_INV_INV] THEN MATCH_MP_TAC REALLIM_INV THEN
3617 CONJ_TAC THENL [ALL_TAC; MP_TAC PI_POS THEN CONV_TAC REAL_FIELD] THEN
3618 MP_TAC(SPEC `pi / &2` SIN_PRODUCT) THEN REWRITE_TAC[SIN_PI2] THEN
3619 REWRITE_TAC[real_div; REAL_INV_MUL; REAL_MUL_ASSOC; REAL_INV_INV] THEN
3620 DISCH_THEN(MP_TAC o SPEC `inv pi * &2` o MATCH_MP REALLIM_LMUL) THEN
3621 SIMP_TAC[REAL_MUL_RID; PI_NZ; REAL_FIELD
3622 `~(pi = &0) ==> (pi * x) * inv pi = x /\
3623 (inv pi * &2) * (pi * inv(&2)) * y = y`] THEN
3624 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REALLIM_TRANSFORM_EVENTUALLY) THEN
3625 REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
3626 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
3627 SIMP_TAC[GSYM PRODUCT_INV; FINITE_NUMSEG] THEN
3628 MATCH_MP_TAC PRODUCT_EQ_NUMSEG THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN
3629 CONV_TAC REAL_FIELD);;
3632 (`((\n. (&2 pow n * &(FACT n)) pow 4 / (&(FACT(2 * n)) * &(FACT(2 * n + 1))))
3633 ---> pi / &2) sequentially`,
3634 MP_TAC WALLIS_ALT THEN
3635 MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REALLIM_TRANSFORM_EVENTUALLY) THEN
3636 MATCH_MP_TAC ALWAYS_EVENTUALLY THEN REWRITE_TAC[] THEN
3637 MATCH_MP_TAC num_INDUCTION THEN REWRITE_TAC[PRODUCT_CLAUSES_NUMSEG] THEN
3638 CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
3639 X_GEN_TAC `n:num` THEN DISCH_TAC THEN
3640 ASM_REWRITE_TAC[ARITH_RULE `1 <= SUC n`] THEN
3641 REWRITE_TAC[GSYM ADD1; ARITH_RULE `2 * SUC n = SUC(SUC(2 * n))`] THEN
3642 REWRITE_TAC[FACT; real_pow; GSYM REAL_OF_NUM_MUL] THEN
3643 REWRITE_TAC[GSYM REAL_OF_NUM_SUC; GSYM REAL_OF_NUM_MUL] THEN
3644 MAP_EVERY (MP_TAC o C SPEC FACT_NZ) [`n:num`; `2 * n`] THEN
3645 REWRITE_TAC[GSYM REAL_OF_NUM_EQ] THEN CONV_TAC REAL_FIELD);;
3647 let GAUSSIAN_INTEGRAL = prove
3648 (`((\x. exp(--(x pow 2))) has_real_integral sqrt pi) (:real)`,
3650 `((\x. exp(--(x pow 2))) has_real_integral gamma(&1 / &2) / &2)
3655 `(:real) = {x | &0 <= x} UNION IMAGE (--) {x | &0 <= x}`
3657 [REWRITE_TAC[EXTENSION; IN_UNION; IN_IMAGE; IN_UNIV; IN_ELIM_THM] THEN
3658 REWRITE_TAC[REAL_ARITH `x:real = --y <=> --x = y`; UNWIND_THM1] THEN
3660 ONCE_REWRITE_TAC[REAL_ARITH `sqrt x = sqrt x / &2 + sqrt x / &2`]] THEN
3661 MATCH_MP_TAC HAS_REAL_INTEGRAL_UNION THEN
3662 REWRITE_TAC[GSYM HAS_REAL_INTEGRAL_REFLECT_GEN] THEN
3663 ASM_SIMP_TAC[REAL_POW_NEG; ARITH; GSYM GAMMA_HALF] THEN
3664 MATCH_MP_TAC REAL_NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{&0}` THEN
3665 REWRITE_TAC[REAL_NEGLIGIBLE_SING; IN_ELIM_THM; SET_RULE
3666 `s INTER IMAGE f s SUBSET {a} <=> !x. x IN s /\ f x IN s ==> f x = a`] THEN
3667 REAL_ARITH_TAC] THEN
3668 MP_TAC(SPEC `&1 / &2` EULER_HAS_REAL_INTEGRAL_GAMMA) THEN
3669 CONV_TAC REAL_RAT_REDUCE_CONV THEN
3670 DISCH_THEN(MP_TAC o SPEC `inv(&2)` o MATCH_MP HAS_REAL_INTEGRAL_RMUL) THEN
3671 REWRITE_TAC[GSYM real_div] THEN
3672 ONCE_REWRITE_TAC[HAS_REAL_INTEGRAL_ALT] THEN
3673 REWRITE_TAC[HAS_REAL_INTEGRAL_RESTRICT_INTER;
3674 REAL_INTEGRABLE_RESTRICT_INTER; REAL_INTEGRAL_RESTRICT_INTER] THEN
3675 REWRITE_TAC[SUBSET_REAL_INTERVAL] THEN
3676 REWRITE_TAC[TAUT `p /\ q <=> ~(p ==> ~q)`] THEN
3677 SIMP_TAC[REAL_ARITH `&0 < B ==> --B <= B /\ ~(B <= --B)`] THEN
3678 REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; INTER; IN_REAL_INTERVAL] THEN
3679 REWRITE_TAC[IN_ELIM_THM; REAL_ARITH
3680 `&0 <= x /\ (a <= x /\ x <= b) <=> max (&0) a <= x /\ x <= b`] THEN
3681 REWRITE_TAC[GSYM real_interval] THEN
3682 ONCE_REWRITE_TAC[REAL_ARITH
3683 `a <= --B /\ --B < B /\ B <= b <=>
3684 max (&0) a = &0 /\ &0 < B /\ a <= --B /\ B <= b`] THEN
3686 REWRITE_TAC[REAL_ARITH
3687 `max (&0) a = &0 /\ &0 < B /\ a <= --B /\ B <= b <=>
3688 a <= --B /\ &0 < B /\ B <= b`] THEN
3689 GEN_REWRITE_TAC (BINOP_CONV o ONCE_DEPTH_CONV) [IMP_CONJ] THEN
3690 REWRITE_TAC[RIGHT_FORALL_IMP_THM; LEFT_FORALL_IMP_THM] THEN
3691 REWRITE_TAC[MESON[REAL_LE_REFL] `?a. a <= B`] THEN
3693 MESON[REAL_ARITH `a <= max a b /\ (&0 <= a ==> max (&0) a = a)`]
3694 `(!a. P (max (&0) a)) <=> (!a. &0 <= a ==> P a)`] THEN
3695 REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN STRIP_TAC THEN
3697 `!a b. &0 <= a /\ a <= b
3698 ==> ((\x. exp(--(x pow 2))) has_real_integral
3699 real_integral (real_interval[a pow 2,b pow 2])
3700 (\t. t rpow (-- &1 / &2) / exp t / &2))
3701 (real_interval[a,b])`
3703 [REPEAT STRIP_TAC THEN
3705 [`\t. t rpow (&1 / &2 - &1) / exp t / &2`;
3706 `\x:real. x pow 2`; `\x. &2 * x`;
3707 `a:real`; `b:real`; `(a:real) pow 2`; `(b:real) pow 2`; `{&0}`]
3708 HAS_REAL_INTEGRAL_SUBSTITUTION_STRONG) THEN
3709 REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
3710 ASM_SIMP_TAC[REAL_LE_REFL; REAL_LE_POW_2; COUNTABLE_SING;
3713 [REPEAT CONJ_TAC THENL
3714 [MATCH_MP_TAC REAL_DIFFERENTIABLE_ON_IMP_REAL_CONTINUOUS_ON THEN
3715 REWRITE_TAC[REAL_DIFFERENTIABLE_ON_DIFFERENTIABLE] THEN
3716 REPEAT STRIP_TAC THEN REAL_DIFFERENTIABLE_TAC;
3717 REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_REAL_INTERVAL] THEN
3718 SIMP_TAC[REAL_LE_POW_2; GSYM REAL_LE_SQUARE_ABS] THEN
3720 REWRITE_TAC[IN_DIFF; IN_SING; IN_REAL_INTERVAL] THEN
3721 REPEAT STRIP_TAC THENL
3722 [REAL_DIFF_TAC THEN CONV_TAC NUM_REDUCE_CONV THEN
3724 MATCH_MP_TAC REAL_DIFFERENTIABLE_IMP_CONTINUOUS_WITHINREAL THEN
3725 REAL_DIFFERENTIABLE_TAC THEN
3726 ASM_REWRITE_TAC[REAL_LT_POW_2; REAL_EXP_NZ]]];
3727 MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ]
3728 (REWRITE_RULE[CONJ_ASSOC] HAS_REAL_INTEGRAL_SPIKE)) THEN
3729 EXISTS_TAC `{&0}` THEN REWRITE_TAC[REAL_NEGLIGIBLE_SING] THEN
3730 X_GEN_TAC `x:real` THEN
3731 REWRITE_TAC[IN_REAL_INTERVAL; IN_DIFF; IN_SING] THEN STRIP_TAC THEN
3732 SUBGOAL_THEN `&0 <= x` ASSUME_TAC THENL
3733 [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
3734 REWRITE_TAC[REAL_EXP_NEG; GSYM RPOW_POW] THEN
3735 ASM_SIMP_TAC[RPOW_RPOW] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
3736 REWRITE_TAC[RPOW_NEG; RPOW_POW; REAL_POW_1] THEN
3737 MP_TAC(SPEC `(x:real) pow 2` REAL_EXP_NZ) THEN
3738 UNDISCH_TAC `~(x = &0)` THEN CONV_TAC REAL_FIELD];
3739 FIRST_X_ASSUM(K ALL_TAC o SPECL [`a:real`; `b:real`]) THEN
3740 DISCH_THEN(LABEL_TAC "*")] THEN
3742 [MAP_EVERY X_GEN_TAC [`a:real`; `b:real`] THEN STRIP_TAC THEN
3743 DISJ_CASES_TAC(REAL_ARITH `b <= a \/ a <= b`) THEN
3744 ASM_SIMP_TAC[REAL_INTEGRABLE_ON_NULL] THEN
3745 REWRITE_TAC[real_integrable_on] THEN ASM_MESON_TAC[];
3747 X_GEN_TAC `e:real` THEN DISCH_TAC THEN
3748 REMOVE_THEN "*" (MP_TAC o SPEC `&0`) THEN
3749 REWRITE_TAC[REAL_LE_REFL; HAS_REAL_INTEGRAL_INTEGRABLE_INTEGRAL] THEN
3750 ONCE_REWRITE_TAC[REAL_ARITH
3751 `&0 < B /\ B <= b <=> &0 < B /\ B <= b /\ &0 <= b`] THEN
3752 SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
3753 FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
3754 DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
3755 EXISTS_TAC `max B (&1)` THEN
3756 ASM_REWRITE_TAC[REAL_LT_MAX; REAL_MAX_LE] THEN
3757 X_GEN_TAC `b:real` THEN STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
3758 ASM_REWRITE_TAC[] THEN TRANS_TAC REAL_LE_TRANS `(b:real) pow 1` THEN
3759 CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
3760 MATCH_MP_TAC REAL_POW_MONO THEN ASM_REWRITE_TAC[ARITH]);;
git://colo12-c703.uibk.ac.at / hl193./.git / blob