1(*  Title:      HOL/Factorial.thy
2    Author:     Jacques D. Fleuriot
3    Author:     Lawrence C Paulson
4    Author:     Jeremy Avigad
5    Author:     Chaitanya Mangla
6    Author:     Manuel Eberl
7*)
8
9section \<open>Factorial Function, Rising Factorials\<close>
10
11theory Factorial
12  imports Groups_List
13begin
14
15subsection \<open>Factorial Function\<close>
16
17context semiring_char_0
18begin
19
20definition fact :: "nat \<Rightarrow> 'a"
21  where fact_prod: "fact n = of_nat (\<Prod>{1..n})"
22
23lemma fact_prod_Suc: "fact n = of_nat (prod Suc {0..<n})"
24  by (cases n)
25    (simp_all add: fact_prod prod.atLeast_Suc_atMost_Suc_shift
26      atLeastLessThanSuc_atLeastAtMost)
27
28lemma fact_prod_rev: "fact n = of_nat (\<Prod>i = 0..<n. n - i)"
29  using prod.atLeastAtMost_rev [of "\<lambda>i. i" 1 n]
30  by (cases n)
31    (simp_all add: fact_prod_Suc prod.atLeast_Suc_atMost_Suc_shift
32      atLeastLessThanSuc_atLeastAtMost)
33
34lemma fact_0 [simp]: "fact 0 = 1"
35  by (simp add: fact_prod)
36
37lemma fact_1 [simp]: "fact 1 = 1"
38  by (simp add: fact_prod)
39
40lemma fact_Suc_0 [simp]: "fact (Suc 0) = 1"
41  by (simp add: fact_prod)
42
43lemma fact_Suc [simp]: "fact (Suc n) = of_nat (Suc n) * fact n"
44  by (simp add: fact_prod atLeastAtMostSuc_conv algebra_simps)
45
46lemma fact_2 [simp]: "fact 2 = 2"
47  by (simp add: numeral_2_eq_2)
48
49lemma fact_split: "k \<le> n \<Longrightarrow> fact n = of_nat (prod Suc {n - k..<n}) * fact (n - k)"
50  by (simp add: fact_prod_Suc prod.union_disjoint [symmetric]
51    ivl_disj_un ac_simps of_nat_mult [symmetric])
52
53end
54
55lemma of_nat_fact [simp]: "of_nat (fact n) = fact n"
56  by (simp add: fact_prod)
57
58lemma of_int_fact [simp]: "of_int (fact n) = fact n"
59  by (simp only: fact_prod of_int_of_nat_eq)
60
61lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
62  by (cases n) auto
63
64lemma fact_nonzero [simp]: "fact n \<noteq> (0::'a::{semiring_char_0,semiring_no_zero_divisors})"
65  apply (induct n)
66  apply auto
67  using of_nat_eq_0_iff
68  apply fastforce
69  done
70
71lemma fact_mono_nat: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: nat)"
72  by (induct n) (auto simp: le_Suc_eq)
73
74lemma fact_in_Nats: "fact n \<in> \<nat>"
75  by (induct n) auto
76
77lemma fact_in_Ints: "fact n \<in> \<int>"
78  by (induct n) auto
79
80context
81  assumes "SORT_CONSTRAINT('a::linordered_semidom)"
82begin
83
84lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
85  by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
86
87lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
88  by (metis le0 fact_0 fact_mono)
89
90lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
91  using fact_ge_1 less_le_trans zero_less_one by blast
92
93lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
94  by (simp add: less_imp_le)
95
96lemma fact_not_neg [simp]: "\<not> fact n < (0 :: 'a)"
97  by (simp add: not_less_iff_gr_or_eq)
98
99lemma fact_le_power: "fact n \<le> (of_nat (n^n) :: 'a)"
100proof (induct n)
101  case 0
102  then show ?case by simp
103next
104  case (Suc n)
105  then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
106    by (rule order_trans) (simp add: power_mono del: of_nat_power)
107  have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
108    by (simp add: algebra_simps)
109  also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n ^ n)"
110    by (simp add: * ordered_comm_semiring_class.comm_mult_left_mono del: of_nat_power)
111  also have "\<dots> \<le> of_nat (Suc n ^ Suc n)"
112    by (metis of_nat_mult order_refl power_Suc)
113  finally show ?case .
114qed
115
116end
117
118lemma fact_less_mono_nat: "0 < m \<Longrightarrow> m < n \<Longrightarrow> fact m < (fact n :: nat)"
119  by (induct n) (auto simp: less_Suc_eq)
120
121lemma fact_less_mono: "0 < m \<Longrightarrow> m < n \<Longrightarrow> fact m < (fact n :: 'a::linordered_semidom)"
122  by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat)
123
124lemma fact_ge_Suc_0_nat [simp]: "fact n \<ge> Suc 0"
125  by (metis One_nat_def fact_ge_1)
126
127lemma dvd_fact: "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
128  by (induct n) (auto simp: dvdI le_Suc_eq)
129
130lemma fact_ge_self: "fact n \<ge> n"
131  by (cases "n = 0") (simp_all add: dvd_imp_le dvd_fact)
132
133lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a::linordered_semidom)"
134  by (induct m) (auto simp: le_Suc_eq)
135
136lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a::{semidom_modulo, linordered_semidom}) = 0"
137  by (simp add: mod_eq_0_iff_dvd fact_dvd)
138
139lemma fact_div_fact:
140  assumes "m \<ge> n"
141  shows "fact m div fact n = \<Prod>{n + 1..m}"
142proof -
143  obtain d where "d = m - n"
144    by auto
145  with assms have "m = n + d"
146    by auto
147  have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
148  proof (induct d)
149    case 0
150    show ?case by simp
151  next
152    case (Suc d')
153    have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
154      by simp
155    also from Suc.hyps have "\<dots> = Suc (n + d') * \<Prod>{n + 1..n + d'}"
156      unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
157    also have "\<dots> = \<Prod>{n + 1..n + Suc d'}"
158      by (simp add: atLeastAtMostSuc_conv)
159    finally show ?case .
160  qed
161  with \<open>m = n + d\<close> show ?thesis by simp
162qed
163
164lemma fact_num_eq_if: "fact m = (if m = 0 then 1 else of_nat m * fact (m - 1))"
165  by (cases m) auto
166
167lemma fact_div_fact_le_pow:
168  assumes "r \<le> n"
169  shows "fact n div fact (n - r) \<le> n ^ r"
170proof -
171  have "r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}" for r
172    by (subst prod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
173  with assms show ?thesis
174    by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
175qed
176
177lemma fact_numeral: "fact (numeral k) = numeral k * fact (pred_numeral k)"
178  \<comment> \<open>Evaluation for specific numerals\<close>
179  by (metis fact_Suc numeral_eq_Suc of_nat_numeral)
180
181
182
183subsection \<open>Pochhammer's symbol: generalized rising factorial\<close>
184
185text \<open>See \<^url>\<open>https://en.wikipedia.org/wiki/Pochhammer_symbol\<close>.\<close>
186
187context comm_semiring_1
188begin
189
190definition pochhammer :: "'a \<Rightarrow> nat \<Rightarrow> 'a"
191  where pochhammer_prod: "pochhammer a n = prod (\<lambda>i. a + of_nat i) {0..<n}"
192
193lemma pochhammer_prod_rev: "pochhammer a n = prod (\<lambda>i. a + of_nat (n - i)) {1..n}"
194  using prod.atLeastLessThan_rev_at_least_Suc_atMost [of "\<lambda>i. a + of_nat i" 0 n]
195  by (simp add: pochhammer_prod)
196
197lemma pochhammer_Suc_prod: "pochhammer a (Suc n) = prod (\<lambda>i. a + of_nat i) {0..n}"
198  by (simp add: pochhammer_prod atLeastLessThanSuc_atLeastAtMost)
199
200lemma pochhammer_Suc_prod_rev: "pochhammer a (Suc n) = prod (\<lambda>i. a + of_nat (n - i)) {0..n}"
201  by (simp add: pochhammer_prod_rev prod.atLeast_Suc_atMost_Suc_shift)
202
203lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
204  by (simp add: pochhammer_prod)
205
206lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
207  by (simp add: pochhammer_prod lessThan_Suc)
208
209lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
210  by (simp add: pochhammer_prod lessThan_Suc)
211
212lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
213  by (simp add: pochhammer_prod atLeast0_lessThan_Suc ac_simps)
214
215end
216
217lemma pochhammer_nonneg:
218  fixes x :: "'a :: linordered_semidom"
219  shows "x > 0 \<Longrightarrow> pochhammer x n \<ge> 0"
220  by (induction n) (auto simp: pochhammer_Suc intro!: mult_nonneg_nonneg add_nonneg_nonneg)
221
222lemma pochhammer_pos:
223  fixes x :: "'a :: linordered_semidom"
224  shows "x > 0 \<Longrightarrow> pochhammer x n > 0"
225  by (induction n) (auto simp: pochhammer_Suc intro!: mult_pos_pos add_pos_nonneg)
226
227lemma pochhammer_of_nat: "pochhammer (of_nat x) n = of_nat (pochhammer x n)"
228  by (simp add: pochhammer_prod)
229
230lemma pochhammer_of_int: "pochhammer (of_int x) n = of_int (pochhammer x n)"
231  by (simp add: pochhammer_prod)
232
233lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
234  by (simp add: pochhammer_prod prod.atLeast0_lessThan_Suc_shift ac_simps)
235
236lemma pochhammer_rec': "pochhammer z (Suc n) = (z + of_nat n) * pochhammer z n"
237  by (simp add: pochhammer_prod prod.atLeast0_lessThan_Suc ac_simps)
238
239lemma pochhammer_fact: "fact n = pochhammer 1 n"
240  by (simp add: pochhammer_prod fact_prod_Suc)
241
242lemma pochhammer_of_nat_eq_0_lemma: "k > n \<Longrightarrow> pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
243  by (auto simp add: pochhammer_prod)
244
245lemma pochhammer_of_nat_eq_0_lemma':
246  assumes kn: "k \<le> n"
247  shows "pochhammer (- (of_nat n :: 'a::{idom,ring_char_0})) k \<noteq> 0"
248proof (cases k)
249  case 0
250  then show ?thesis by simp
251next
252  case (Suc h)
253  then show ?thesis
254    apply (simp add: pochhammer_Suc_prod)
255    using Suc kn
256    apply (auto simp add: algebra_simps)
257    done
258qed
259
260lemma pochhammer_of_nat_eq_0_iff:
261  "pochhammer (- (of_nat n :: 'a::{idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
262  (is "?l = ?r")
263  using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
264    pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
265  by (auto simp add: not_le[symmetric])
266
267lemma pochhammer_0_left:
268  "pochhammer 0 n = (if n = 0 then 1 else 0)"
269  by (induction n) (simp_all add: pochhammer_rec)
270
271lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
272  by (auto simp add: pochhammer_prod eq_neg_iff_add_eq_0)
273
274lemma pochhammer_eq_0_mono:
275  "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
276  unfolding pochhammer_eq_0_iff by auto
277
278lemma pochhammer_neq_0_mono:
279  "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
280  unfolding pochhammer_eq_0_iff by auto
281
282lemma pochhammer_minus:
283  "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
284proof (cases k)
285  case 0
286  then show ?thesis by simp
287next
288  case (Suc h)
289  have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i = 0..h. - 1)"
290    using prod_constant [where A="{0.. h}" and y="- 1 :: 'a"]
291    by auto
292  with Suc show ?thesis
293    using pochhammer_Suc_prod_rev [of "b - of_nat k + 1"] 
294    by (auto simp add: pochhammer_Suc_prod prod.distrib [symmetric] eq of_nat_diff simp del: prod_constant)
295qed
296
297lemma pochhammer_minus':
298  "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
299  by (simp add: pochhammer_minus)
300
301lemma pochhammer_same: "pochhammer (- of_nat n) n =
302    ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * fact n"
303  unfolding pochhammer_minus
304  by (simp add: of_nat_diff pochhammer_fact)
305
306lemma pochhammer_product': "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
307proof (induct n arbitrary: z)
308  case 0
309  then show ?case by simp
310next
311  case (Suc n z)
312  have "pochhammer z (Suc n) * pochhammer (z + of_nat (Suc n)) m =
313      z * (pochhammer (z + 1) n * pochhammer (z + 1 + of_nat n) m)"
314    by (simp add: pochhammer_rec ac_simps)
315  also note Suc[symmetric]
316  also have "z * pochhammer (z + 1) (n + m) = pochhammer z (Suc (n + m))"
317    by (subst pochhammer_rec) simp
318  finally show ?case
319    by simp
320qed
321
322lemma pochhammer_product:
323  "m \<le> n \<Longrightarrow> pochhammer z n = pochhammer z m * pochhammer (z + of_nat m) (n - m)"
324  using pochhammer_product'[of z m "n - m"] by simp
325
326lemma pochhammer_times_pochhammer_half:
327  fixes z :: "'a::field_char_0"
328  shows "pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\<Prod>k=0..2*n+1. z + of_nat k / 2)"
329proof (induct n)
330  case 0
331  then show ?case
332    by (simp add: atLeast0_atMost_Suc)
333next
334  case (Suc n)
335  define n' where "n' = Suc n"
336  have "pochhammer z (Suc n') * pochhammer (z + 1 / 2) (Suc n') =
337      (pochhammer z n' * pochhammer (z + 1 / 2) n') * ((z + of_nat n') * (z + 1/2 + of_nat n'))"
338    (is "_ = _ * ?A")
339    by (simp_all add: pochhammer_rec' mult_ac)
340  also have "?A = (z + of_nat (Suc (2 * n + 1)) / 2) * (z + of_nat (Suc (Suc (2 * n + 1))) / 2)"
341    (is "_ = ?B")
342    by (simp add: field_simps n'_def)
343  also note Suc[folded n'_def]
344  also have "(\<Prod>k=0..2 * n + 1. z + of_nat k / 2) * ?B = (\<Prod>k=0..2 * Suc n + 1. z + of_nat k / 2)"
345    by (simp add: atLeast0_atMost_Suc)
346  finally show ?case
347    by (simp add: n'_def)
348qed
349
350lemma pochhammer_double:
351  fixes z :: "'a::field_char_0"
352  shows "pochhammer (2 * z) (2 * n) = of_nat (2^(2*n)) * pochhammer z n * pochhammer (z+1/2) n"
353proof (induct n)
354  case 0
355  then show ?case by simp
356next
357  case (Suc n)
358  have "pochhammer (2 * z) (2 * (Suc n)) = pochhammer (2 * z) (2 * n) *
359      (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1)"
360    by (simp add: pochhammer_rec' ac_simps)
361  also note Suc
362  also have "of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + 1/2) n *
363        (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1) =
364      of_nat (2 ^ (2 * (Suc n))) * pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n)"
365    by (simp add: field_simps pochhammer_rec')
366  finally show ?case .
367qed
368
369lemma fact_double:
370  "fact (2 * n) = (2 ^ (2 * n) * pochhammer (1 / 2) n * fact n :: 'a::field_char_0)"
371  using pochhammer_double[of "1/2::'a" n] by (simp add: pochhammer_fact)
372
373lemma pochhammer_absorb_comp: "(r - of_nat k) * pochhammer (- r) k = r * pochhammer (-r + 1) k"
374  (is "?lhs = ?rhs")
375  for r :: "'a::comm_ring_1"
376proof -
377  have "?lhs = - pochhammer (- r) (Suc k)"
378    by (subst pochhammer_rec') (simp add: algebra_simps)
379  also have "\<dots> = ?rhs"
380    by (subst pochhammer_rec) simp
381  finally show ?thesis .
382qed
383
384
385subsection \<open>Misc\<close>
386
387lemma fact_code [code]:
388  "fact n = (of_nat (fold_atLeastAtMost_nat (( * )) 2 n 1) :: 'a::semiring_char_0)"
389proof -
390  have "fact n = (of_nat (\<Prod>{1..n}) :: 'a)"
391    by (simp add: fact_prod)
392  also have "\<Prod>{1..n} = \<Prod>{2..n}"
393    by (intro prod.mono_neutral_right) auto
394  also have "\<dots> = fold_atLeastAtMost_nat (( * )) 2 n 1"
395    by (simp add: prod_atLeastAtMost_code)
396  finally show ?thesis .
397qed
398
399lemma pochhammer_code [code]:
400  "pochhammer a n =
401    (if n = 0 then 1
402     else fold_atLeastAtMost_nat (\<lambda>n acc. (a + of_nat n) * acc) 0 (n - 1) 1)"
403  by (cases n)
404    (simp_all add: pochhammer_prod prod_atLeastAtMost_code [symmetric]
405      atLeastLessThanSuc_atLeastAtMost)
406
407end
408