src/HOL/Binomial.thy
 author wenzelm Tue Sep 01 22:32:58 2015 +0200 (2015-09-01) changeset 61076 bdc1e2f0a86a parent 60758 d8d85a8172b5 child 61531 ab2e862263e7 permissions -rw-r--r--
eliminated \<Colon>;
 lp15@59669 ` 1` ```(* Title : Binomial.thy ``` paulson@12196 ` 2` ``` Author : Jacques D. Fleuriot ``` paulson@12196 ` 3` ``` Copyright : 1998 University of Cambridge ``` paulson@15094 ` 4` ``` Conversion to Isar and new proofs by Lawrence C Paulson, 2004 ``` avigad@32036 ` 5` ``` The integer version of factorial and other additions by Jeremy Avigad. ``` paulson@12196 ` 6` ```*) ``` paulson@12196 ` 7` wenzelm@60758 ` 8` ```section\Factorial Function, Binomial Coefficients and Binomial Theorem\ ``` paulson@15094 ` 9` lp15@59669 ` 10` ```theory Binomial ``` haftmann@33319 ` 11` ```imports Main ``` nipkow@15131 ` 12` ```begin ``` paulson@15094 ` 13` wenzelm@60758 ` 14` ```subsection \Factorial\ ``` lp15@59730 ` 15` lp15@59733 ` 16` ```fun fact :: "nat \ ('a::semiring_char_0)" ``` lp15@59730 ` 17` ``` where "fact 0 = 1" | "fact (Suc n) = of_nat (Suc n) * fact n" ``` avigad@32036 ` 18` lp15@59730 ` 19` ```lemmas fact_Suc = fact.simps(2) ``` lp15@59730 ` 20` lp15@59730 ` 21` ```lemma fact_1 [simp]: "fact 1 = 1" ``` lp15@59730 ` 22` ``` by simp ``` lp15@59730 ` 23` lp15@59730 ` 24` ```lemma fact_Suc_0 [simp]: "fact (Suc 0) = Suc 0" ``` lp15@59730 ` 25` ``` by simp ``` avigad@32036 ` 26` lp15@59730 ` 27` ```lemma of_nat_fact [simp]: "of_nat (fact n) = fact n" ``` lp15@59730 ` 28` ``` by (induct n) (auto simp add: algebra_simps of_nat_mult) ``` lp15@59730 ` 29` ``` ``` lp15@59730 ` 30` ```lemma fact_reduce: "n > 0 \ fact n = of_nat n * fact (n - 1)" ``` lp15@59730 ` 31` ``` by (cases n) auto ``` avigad@32036 ` 32` lp15@59733 ` 33` ```lemma fact_nonzero [simp]: "fact n \ (0::'a::{semiring_char_0,semiring_no_zero_divisors})" ``` lp15@59730 ` 34` ``` apply (induct n) ``` lp15@59730 ` 35` ``` apply auto ``` lp15@59730 ` 36` ``` using of_nat_eq_0_iff by fastforce ``` lp15@59730 ` 37` lp15@59730 ` 38` ```lemma fact_mono_nat: "m \ n \ fact m \ (fact n :: nat)" ``` lp15@59730 ` 39` ``` by (induct n) (auto simp: le_Suc_eq) ``` avigad@32036 ` 40` lp15@59730 ` 41` ```context ``` wenzelm@60241 ` 42` ``` assumes "SORT_CONSTRAINT('a::linordered_semidom)" ``` lp15@59667 ` 43` ```begin ``` lp15@59730 ` 44` ``` ``` lp15@59730 ` 45` ``` lemma fact_mono: "m \ n \ fact m \ (fact n :: 'a)" ``` lp15@59730 ` 46` ``` by (metis of_nat_fact of_nat_le_iff fact_mono_nat) ``` lp15@59730 ` 47` ``` ``` lp15@59730 ` 48` ``` lemma fact_ge_1 [simp]: "fact n \ (1 :: 'a)" ``` lp15@59730 ` 49` ``` by (metis le0 fact.simps(1) fact_mono) ``` lp15@59730 ` 50` ``` ``` lp15@59730 ` 51` ``` lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)" ``` lp15@59730 ` 52` ``` using fact_ge_1 less_le_trans zero_less_one by blast ``` lp15@59730 ` 53` ``` ``` lp15@59730 ` 54` ``` lemma fact_ge_zero [simp]: "fact n \ (0 :: 'a)" ``` lp15@59730 ` 55` ``` by (simp add: less_imp_le) ``` avigad@32036 ` 56` lp15@59730 ` 57` ``` lemma fact_not_neg [simp]: "~ (fact n < (0 :: 'a))" ``` lp15@59730 ` 58` ``` by (simp add: not_less_iff_gr_or_eq) ``` lp15@59730 ` 59` ``` ``` lp15@59730 ` 60` ``` lemma fact_le_power: ``` lp15@59730 ` 61` ``` "fact n \ (of_nat (n^n) ::'a)" ``` lp15@59730 ` 62` ``` proof (induct n) ``` lp15@59730 ` 63` ``` case (Suc n) ``` lp15@59730 ` 64` ``` then have *: "fact n \ (of_nat (Suc n ^ n) ::'a)" ``` lp15@59730 ` 65` ``` by (rule order_trans) (simp add: power_mono) ``` lp15@59730 ` 66` ``` have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)" ``` lp15@59730 ` 67` ``` by (simp add: algebra_simps) ``` lp15@59730 ` 68` ``` also have "... \ (of_nat (Suc n) * of_nat (Suc n ^ n) ::'a)" ``` lp15@59730 ` 69` ``` by (simp add: "*" ordered_comm_semiring_class.comm_mult_left_mono) ``` lp15@59730 ` 70` ``` also have "... \ (of_nat (Suc n ^ Suc n) ::'a)" ``` lp15@59730 ` 71` ``` by (metis of_nat_mult order_refl power_Suc) ``` lp15@59730 ` 72` ``` finally show ?case . ``` lp15@59730 ` 73` ``` qed simp ``` avigad@32036 ` 74` avigad@32036 ` 75` ```end ``` avigad@32036 ` 76` wenzelm@60758 ` 77` ```text\Note that @{term "fact 0 = fact 1"}\ ``` lp15@59730 ` 78` ```lemma fact_less_mono_nat: "\0 < m; m < n\ \ fact m < (fact n :: nat)" ``` lp15@59730 ` 79` ``` by (induct n) (auto simp: less_Suc_eq) ``` avigad@32036 ` 80` lp15@59730 ` 81` ```lemma fact_less_mono: ``` wenzelm@60241 ` 82` ``` "\0 < m; m < n\ \ fact m < (fact n :: 'a::linordered_semidom)" ``` lp15@59730 ` 83` ``` by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat) ``` avigad@32036 ` 84` lp15@59730 ` 85` ```lemma fact_ge_Suc_0_nat [simp]: "fact n \ Suc 0" ``` lp15@59730 ` 86` ``` by (metis One_nat_def fact_ge_1) ``` avigad@32036 ` 87` lp15@59730 ` 88` ```lemma dvd_fact: ``` lp15@59730 ` 89` ``` shows "1 \ m \ m \ n \ m dvd fact n" ``` lp15@59730 ` 90` ``` by (induct n) (auto simp: dvdI le_Suc_eq) ``` avigad@32036 ` 91` lp15@59730 ` 92` ```lemma fact_altdef_nat: "fact n = \{1..n}" ``` lp15@59730 ` 93` ``` by (induct n) (auto simp: atLeastAtMostSuc_conv) ``` avigad@32036 ` 94` lp15@59730 ` 95` ```lemma fact_altdef: "fact n = setprod of_nat {1..n}" ``` lp15@59730 ` 96` ``` by (induct n) (auto simp: atLeastAtMostSuc_conv) ``` paulson@15094 ` 97` lp15@59730 ` 98` ```lemma fact_dvd: "n \ m \ fact n dvd (fact m :: 'a :: {semiring_div,linordered_semidom})" ``` lp15@59730 ` 99` ``` by (induct m) (auto simp: le_Suc_eq) ``` avigad@32036 ` 100` lp15@59730 ` 101` ```lemma fact_mod: "m \ n \ fact n mod (fact m :: 'a :: {semiring_div,linordered_semidom}) = 0" ``` lp15@59730 ` 102` ``` by (auto simp add: fact_dvd) ``` bulwahn@40033 ` 103` bulwahn@40033 ` 104` ```lemma fact_div_fact: ``` lp15@59730 ` 105` ``` assumes "m \ n" ``` bulwahn@40033 ` 106` ``` shows "(fact m) div (fact n) = \{n + 1..m}" ``` bulwahn@40033 ` 107` ```proof - ``` bulwahn@40033 ` 108` ``` obtain d where "d = m - n" by auto ``` bulwahn@40033 ` 109` ``` from assms this have "m = n + d" by auto ``` bulwahn@40033 ` 110` ``` have "fact (n + d) div (fact n) = \{n + 1..n + d}" ``` bulwahn@40033 ` 111` ``` proof (induct d) ``` bulwahn@40033 ` 112` ``` case 0 ``` bulwahn@40033 ` 113` ``` show ?case by simp ``` bulwahn@40033 ` 114` ``` next ``` bulwahn@40033 ` 115` ``` case (Suc d') ``` bulwahn@40033 ` 116` ``` have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n" ``` bulwahn@40033 ` 117` ``` by simp ``` lp15@59667 ` 118` ``` also from Suc.hyps have "... = Suc (n + d') * \{n + 1..n + d'}" ``` bulwahn@40033 ` 119` ``` unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod) ``` bulwahn@40033 ` 120` ``` also have "... = \{n + 1..n + Suc d'}" ``` lp15@59730 ` 121` ``` by (simp add: atLeastAtMostSuc_conv) ``` bulwahn@40033 ` 122` ``` finally show ?case . ``` bulwahn@40033 ` 123` ``` qed ``` wenzelm@60758 ` 124` ``` from this \m = n + d\ show ?thesis by simp ``` bulwahn@40033 ` 125` ```qed ``` bulwahn@40033 ` 126` lp15@59730 ` 127` ```lemma fact_num_eq_if: ``` lp15@59730 ` 128` ``` "fact m = (if m=0 then 1 else of_nat m * fact (m - 1))" ``` avigad@32036 ` 129` ```by (cases m) auto ``` avigad@32036 ` 130` lp15@59730 ` 131` ```lemma fact_eq_rev_setprod_nat: "fact k = (\i n" shows "fact n div fact (n - r) \ n ^ r" ``` hoelzl@50240 ` 137` ```proof - ``` hoelzl@50240 ` 138` ``` have "\r. r \ n \ \{n - r..n} = (n - r) * \{Suc (n - r)..n}" ``` haftmann@57418 ` 139` ``` by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL) ``` hoelzl@50240 ` 140` ``` with assms show ?thesis ``` hoelzl@50240 ` 141` ``` by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono) ``` hoelzl@50240 ` 142` ```qed ``` hoelzl@50240 ` 143` wenzelm@60758 ` 144` ```lemma fact_numeral: --\Evaluation for specific numerals\ ``` lp15@57113 ` 145` ``` "fact (numeral k) = (numeral k) * (fact (pred_numeral k))" ``` lp15@59730 ` 146` ``` by (metis fact.simps(2) numeral_eq_Suc of_nat_numeral) ``` lp15@57113 ` 147` lp15@59658 ` 148` wenzelm@60758 ` 149` ```text \This development is based on the work of Andy Gordon and ``` wenzelm@60758 ` 150` ``` Florian Kammueller.\ ``` lp15@59658 ` 151` wenzelm@60758 ` 152` ```subsection \Basic definitions and lemmas\ ``` lp15@59658 ` 153` lp15@59658 ` 154` ```primrec binomial :: "nat \ nat \ nat" (infixl "choose" 65) ``` lp15@59658 ` 155` ```where ``` lp15@59658 ` 156` ``` "0 choose k = (if k = 0 then 1 else 0)" ``` lp15@59658 ` 157` ```| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" ``` lp15@59658 ` 158` lp15@59658 ` 159` ```lemma binomial_n_0 [simp]: "(n choose 0) = 1" ``` lp15@59658 ` 160` ``` by (cases n) simp_all ``` lp15@59658 ` 161` lp15@59658 ` 162` ```lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" ``` lp15@59658 ` 163` ``` by simp ``` lp15@59658 ` 164` lp15@59658 ` 165` ```lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" ``` lp15@59658 ` 166` ``` by simp ``` lp15@59658 ` 167` lp15@59667 ` 168` ```lemma choose_reduce_nat: ``` lp15@59658 ` 169` ``` "0 < (n::nat) \ 0 < k \ ``` lp15@59658 ` 170` ``` (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)" ``` lp15@59658 ` 171` ``` by (metis Suc_diff_1 binomial.simps(2) neq0_conv) ``` lp15@59658 ` 172` lp15@59658 ` 173` ```lemma binomial_eq_0: "n < k \ n choose k = 0" ``` lp15@59658 ` 174` ``` by (induct n arbitrary: k) auto ``` lp15@59658 ` 175` lp15@59658 ` 176` ```declare binomial.simps [simp del] ``` lp15@59658 ` 177` lp15@59658 ` 178` ```lemma binomial_n_n [simp]: "n choose n = 1" ``` lp15@59658 ` 179` ``` by (induct n) (simp_all add: binomial_eq_0) ``` lp15@59658 ` 180` lp15@59658 ` 181` ```lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n" ``` lp15@59658 ` 182` ``` by (induct n) simp_all ``` lp15@59658 ` 183` lp15@59658 ` 184` ```lemma binomial_1 [simp]: "n choose Suc 0 = n" ``` lp15@59658 ` 185` ``` by (induct n) simp_all ``` lp15@59658 ` 186` lp15@59658 ` 187` ```lemma zero_less_binomial: "k \ n \ n choose k > 0" ``` lp15@59658 ` 188` ``` by (induct n k rule: diff_induct) simp_all ``` lp15@59658 ` 189` lp15@59658 ` 190` ```lemma binomial_eq_0_iff [simp]: "n choose k = 0 \ n < k" ``` lp15@59658 ` 191` ``` by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial) ``` lp15@59658 ` 192` lp15@59658 ` 193` ```lemma zero_less_binomial_iff [simp]: "n choose k > 0 \ k \ n" ``` lp15@59658 ` 194` ``` by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial) ``` lp15@59658 ` 195` lp15@59658 ` 196` ```lemma Suc_times_binomial_eq: ``` lp15@59658 ` 197` ``` "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" ``` lp15@59658 ` 198` ``` apply (induct n arbitrary: k, simp add: binomial.simps) ``` lp15@59658 ` 199` ``` apply (case_tac k) ``` lp15@59658 ` 200` ``` apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) ``` lp15@59658 ` 201` ``` done ``` lp15@59658 ` 202` lp15@60141 ` 203` ```lemma binomial_le_pow2: "n choose k \ 2^n" ``` lp15@60141 ` 204` ``` apply (induction n arbitrary: k) ``` lp15@60141 ` 205` ``` apply (simp add: binomial.simps) ``` lp15@60141 ` 206` ``` apply (case_tac k) ``` lp15@60141 ` 207` ``` apply (auto simp: power_Suc) ``` lp15@60141 ` 208` ``` by (simp add: add_le_mono mult_2) ``` lp15@60141 ` 209` wenzelm@60758 ` 210` ```text\The absorption property\ ``` lp15@59658 ` 211` ```lemma Suc_times_binomial: ``` lp15@59658 ` 212` ``` "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)" ``` lp15@59658 ` 213` ``` using Suc_times_binomial_eq by auto ``` lp15@59658 ` 214` wenzelm@60758 ` 215` ```text\This is the well-known version of absorption, but it's harder to use because of the ``` wenzelm@60758 ` 216` ``` need to reason about division.\ ``` lp15@59658 ` 217` ```lemma binomial_Suc_Suc_eq_times: ``` lp15@59658 ` 218` ``` "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" ``` lp15@59658 ` 219` ``` by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right) ``` lp15@59658 ` 220` wenzelm@60758 ` 221` ```text\Another version of absorption, with -1 instead of Suc.\ ``` lp15@59658 ` 222` ```lemma times_binomial_minus1_eq: ``` lp15@59658 ` 223` ``` "0 < k \ k * (n choose k) = n * ((n - 1) choose (k - 1))" ``` lp15@59658 ` 224` ``` using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"] ``` lp15@59658 ` 225` ``` by (auto split add: nat_diff_split) ``` lp15@59658 ` 226` lp15@59658 ` 227` wenzelm@60758 ` 228` ```subsection \Combinatorial theorems involving @{text "choose"}\ ``` lp15@59658 ` 229` wenzelm@60758 ` 230` ```text \By Florian Kamm\"uller, tidied by LCP.\ ``` lp15@59658 ` 231` lp15@59658 ` 232` ```lemma card_s_0_eq_empty: "finite A \ card {B. B \ A & card B = 0} = 1" ``` lp15@59658 ` 233` ``` by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) ``` lp15@59658 ` 234` lp15@59658 ` 235` ```lemma choose_deconstruct: "finite M \ x \ M \ ``` lp15@59658 ` 236` ``` {s. s \ insert x M \ card s = Suc k} = ``` lp15@59658 ` 237` ``` {s. s \ M \ card s = Suc k} \ {s. \t. t \ M \ card t = k \ s = insert x t}" ``` lp15@59658 ` 238` ``` apply safe ``` lp15@59658 ` 239` ``` apply (auto intro: finite_subset [THEN card_insert_disjoint]) ``` lp15@59667 ` 240` ``` by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if ``` lp15@59658 ` 241` ``` card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff) ``` lp15@59658 ` 242` lp15@59658 ` 243` ```lemma finite_bex_subset [simp]: ``` lp15@59658 ` 244` ``` assumes "finite B" ``` lp15@59658 ` 245` ``` and "\A. A \ B \ finite {x. P x A}" ``` lp15@59658 ` 246` ``` shows "finite {x. \A \ B. P x A}" ``` lp15@59658 ` 247` ``` by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets) ``` lp15@59658 ` 248` wenzelm@60758 ` 249` ```text\There are as many subsets of @{term A} having cardinality @{term k} ``` lp15@59658 ` 250` ``` as there are sets obtained from the former by inserting a fixed element ``` wenzelm@60758 ` 251` ``` @{term x} into each.\ ``` lp15@59658 ` 252` ```lemma constr_bij: ``` lp15@59658 ` 253` ``` "finite A \ x \ A \ ``` lp15@59658 ` 254` ``` card {B. \C. C \ A \ card C = k \ B = insert x C} = ``` lp15@59658 ` 255` ``` card {B. B \ A & card(B) = k}" ``` lp15@59658 ` 256` ``` apply (rule card_bij_eq [where f = "\s. s - {x}" and g = "insert x"]) ``` lp15@59658 ` 257` ``` apply (auto elim!: equalityE simp add: inj_on_def) ``` lp15@59658 ` 258` ``` apply (metis card_Diff_singleton_if finite_subset in_mono) ``` lp15@59658 ` 259` ``` done ``` lp15@59658 ` 260` wenzelm@60758 ` 261` ```text \ ``` lp15@59658 ` 262` ``` Main theorem: combinatorial statement about number of subsets of a set. ``` wenzelm@60758 ` 263` ```\ ``` lp15@59658 ` 264` lp15@59658 ` 265` ```theorem n_subsets: "finite A \ card {B. B \ A \ card B = k} = (card A choose k)" ``` lp15@59658 ` 266` ```proof (induct k arbitrary: A) ``` lp15@59658 ` 267` ``` case 0 then show ?case by (simp add: card_s_0_eq_empty) ``` lp15@59658 ` 268` ```next ``` lp15@59658 ` 269` ``` case (Suc k) ``` wenzelm@60758 ` 270` ``` show ?case using \finite A\ ``` lp15@59658 ` 271` ``` proof (induct A) ``` lp15@59658 ` 272` ``` case empty show ?case by (simp add: card_s_0_eq_empty) ``` lp15@59658 ` 273` ``` next ``` lp15@59658 ` 274` ``` case (insert x A) ``` lp15@59658 ` 275` ``` then show ?case using Suc.hyps ``` lp15@59658 ` 276` ``` apply (simp add: card_s_0_eq_empty choose_deconstruct) ``` lp15@59658 ` 277` ``` apply (subst card_Un_disjoint) ``` lp15@59658 ` 278` ``` prefer 4 apply (force simp add: constr_bij) ``` lp15@59658 ` 279` ``` prefer 3 apply force ``` lp15@59658 ` 280` ``` prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] ``` lp15@59658 ` 281` ``` finite_subset [of _ "Pow (insert x F)" for F]) ``` lp15@59658 ` 282` ``` apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) ``` lp15@59658 ` 283` ``` done ``` lp15@59658 ` 284` ``` qed ``` lp15@59658 ` 285` ```qed ``` lp15@59658 ` 286` lp15@59658 ` 287` wenzelm@60758 ` 288` ```subsection \The binomial theorem (courtesy of Tobias Nipkow):\ ``` lp15@59658 ` 289` wenzelm@60758 ` 290` ```text\Avigad's version, generalized to any commutative ring\ ``` lp15@59667 ` 291` ```theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n = ``` lp15@59658 ` 292` ``` (\k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n") ``` lp15@59658 ` 293` ```proof (induct n) ``` lp15@59658 ` 294` ``` case 0 then show "?P 0" by simp ``` lp15@59658 ` 295` ```next ``` lp15@59658 ` 296` ``` case (Suc n) ``` lp15@59658 ` 297` ``` have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}" ``` lp15@59658 ` 298` ``` by auto ``` lp15@59658 ` 299` ``` have decomp2: "{0..n} = {0} Un {1..n}" ``` lp15@59658 ` 300` ``` by auto ``` lp15@59667 ` 301` ``` have "(a+b)^(n+1) = ``` lp15@59658 ` 302` ``` (a+b) * (\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" ``` lp15@59658 ` 303` ``` using Suc.hyps by simp ``` lp15@59658 ` 304` ``` also have "\ = a*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k)) + ``` lp15@59658 ` 305` ``` b*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" ``` lp15@59658 ` 306` ``` by (rule distrib_right) ``` lp15@59658 ` 307` ``` also have "\ = (\k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + ``` lp15@59658 ` 308` ``` (\k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))" ``` lp15@59658 ` 309` ``` by (auto simp add: setsum_right_distrib ac_simps) ``` lp15@59658 ` 310` ``` also have "\ = (\k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) + ``` lp15@59658 ` 311` ``` (\k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))" ``` lp15@59667 ` 312` ``` by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps ``` lp15@59658 ` 313` ``` del:setsum_cl_ivl_Suc) ``` lp15@59658 ` 314` ``` also have "\ = a^(n+1) + b^(n+1) + ``` lp15@59658 ` 315` ``` (\k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) + ``` lp15@59658 ` 316` ``` (\k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))" ``` lp15@59658 ` 317` ``` by (simp add: decomp2) ``` lp15@59658 ` 318` ``` also have ``` lp15@59667 ` 319` ``` "\ = a^(n+1) + b^(n+1) + ``` lp15@59658 ` 320` ``` (\k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))" ``` lp15@59658 ` 321` ``` by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat) ``` lp15@59658 ` 322` ``` also have "\ = (\k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))" ``` lp15@59658 ` 323` ``` using decomp by (simp add: field_simps) ``` lp15@59658 ` 324` ``` finally show "?P (Suc n)" by simp ``` lp15@59658 ` 325` ```qed ``` lp15@59658 ` 326` wenzelm@60758 ` 327` ```text\Original version for the naturals\ ``` lp15@59658 ` 328` ```corollary binomial: "(a+b::nat)^n = (\k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" ``` lp15@59658 ` 329` ``` using binomial_ring [of "int a" "int b" n] ``` lp15@59658 ` 330` ``` by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric] ``` lp15@59658 ` 331` ``` of_nat_setsum [symmetric] ``` lp15@59658 ` 332` ``` of_nat_eq_iff of_nat_id) ``` lp15@59658 ` 333` lp15@59658 ` 334` ```lemma binomial_fact_lemma: "k \ n \ fact k * fact (n - k) * (n choose k) = fact n" ``` lp15@59658 ` 335` ```proof (induct n arbitrary: k rule: nat_less_induct) ``` lp15@59658 ` 336` ``` fix n k assume H: "\mx\m. fact x * fact (m - x) * (m choose x) = ``` lp15@59658 ` 337` ``` fact m" and kn: "k \ n" ``` lp15@59658 ` 338` ``` let ?ths = "fact k * fact (n - k) * (n choose k) = fact n" ``` lp15@59658 ` 339` ``` { assume "n=0" then have ?ths using kn by simp } ``` lp15@59658 ` 340` ``` moreover ``` lp15@59658 ` 341` ``` { assume "k=0" then have ?ths using kn by simp } ``` lp15@59658 ` 342` ``` moreover ``` lp15@59658 ` 343` ``` { assume nk: "n=k" then have ?ths by simp } ``` lp15@59658 ` 344` ``` moreover ``` lp15@59658 ` 345` ``` { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m" ``` lp15@59658 ` 346` ``` from n have mn: "m < n" by arith ``` lp15@59658 ` 347` ``` from hm have hm': "h \ m" by arith ``` lp15@59658 ` 348` ``` from hm h n kn have km: "k \ m" by arith ``` lp15@59658 ` 349` ``` have "m - h = Suc (m - Suc h)" using h km hm by arith ``` lp15@59658 ` 350` ``` with km h have th0: "fact (m - h) = (m - h) * fact (m - k)" ``` lp15@59658 ` 351` ``` by simp ``` lp15@59658 ` 352` ``` from n h th0 ``` lp15@59658 ` 353` ``` have "fact k * fact (n - k) * (n choose k) = ``` lp15@59667 ` 354` ``` k * (fact h * fact (m - h) * (m choose h)) + ``` lp15@59658 ` 355` ``` (m - h) * (fact k * fact (m - k) * (m choose k))" ``` lp15@59658 ` 356` ``` by (simp add: field_simps) ``` lp15@59658 ` 357` ``` also have "\ = (k + (m - h)) * fact m" ``` lp15@59658 ` 358` ``` using H[rule_format, OF mn hm'] H[rule_format, OF mn km] ``` lp15@59658 ` 359` ``` by (simp add: field_simps) ``` lp15@59658 ` 360` ``` finally have ?ths using h n km by simp } ``` lp15@59658 ` 361` ``` moreover have "n=0 \ k = 0 \ k = n \ (\m h. n = Suc m \ k = Suc h \ h < m)" ``` lp15@59658 ` 362` ``` using kn by presburger ``` lp15@59658 ` 363` ``` ultimately show ?ths by blast ``` lp15@59658 ` 364` ```qed ``` lp15@59658 ` 365` lp15@59658 ` 366` ```lemma binomial_fact: ``` lp15@59658 ` 367` ``` assumes kn: "k \ n" ``` lp15@59730 ` 368` ``` shows "(of_nat (n choose k) :: 'a::field_char_0) = ``` lp15@59730 ` 369` ``` (fact n) / (fact k * fact(n - k))" ``` lp15@59658 ` 370` ``` using binomial_fact_lemma[OF kn] ``` lp15@59730 ` 371` ``` apply (simp add: field_simps) ``` lp15@59730 ` 372` ``` by (metis mult.commute of_nat_fact of_nat_mult) ``` lp15@59658 ` 373` lp15@59667 ` 374` ```lemma choose_row_sum: "(\k=0..n. n choose k) = 2^n" ``` lp15@59667 ` 375` ``` using binomial [of 1 "1" n] ``` lp15@59667 ` 376` ``` by (simp add: numeral_2_eq_2) ``` lp15@59667 ` 377` lp15@59667 ` 378` ```lemma sum_choose_lower: "(\k=0..n. (r+k) choose k) = Suc (r+n) choose n" ``` lp15@59667 ` 379` ``` by (induct n) auto ``` lp15@59667 ` 380` lp15@59667 ` 381` ```lemma sum_choose_upper: "(\k=0..n. k choose m) = Suc n choose Suc m" ``` lp15@59667 ` 382` ``` by (induct n) auto ``` lp15@59667 ` 383` lp15@59667 ` 384` ```lemma natsum_reverse_index: ``` lp15@59667 ` 385` ``` fixes m::nat ``` lp15@59667 ` 386` ``` shows "(\k. m \ k \ k \ n \ g k = f (m + n - k)) \ (\k=m..n. f k) = (\k=m..n. g k)" ``` lp15@59667 ` 387` ``` by (rule setsum.reindex_bij_witness[where i="\k. m+n-k" and j="\k. m+n-k"]) auto ``` lp15@59667 ` 388` wenzelm@60758 ` 389` ```text\NW diagonal sum property\ ``` lp15@59667 ` 390` ```lemma sum_choose_diagonal: ``` lp15@59667 ` 391` ``` assumes "m\n" shows "(\k=0..m. (n-k) choose (m-k)) = Suc n choose m" ``` lp15@59667 ` 392` ```proof - ``` lp15@59667 ` 393` ``` have "(\k=0..m. (n-k) choose (m-k)) = (\k=0..m. (n-m+k) choose k)" ``` lp15@59667 ` 394` ``` by (rule natsum_reverse_index) (simp add: assms) ``` lp15@59667 ` 395` ``` also have "... = Suc (n-m+m) choose m" ``` lp15@59667 ` 396` ``` by (rule sum_choose_lower) ``` lp15@59667 ` 397` ``` also have "... = Suc n choose m" using assms ``` lp15@59667 ` 398` ``` by simp ``` lp15@59667 ` 399` ``` finally show ?thesis . ``` lp15@59667 ` 400` ```qed ``` lp15@59667 ` 401` wenzelm@60758 ` 402` ```subsection\Pochhammer's symbol : generalized rising factorial\ ``` lp15@59667 ` 403` wenzelm@60758 ` 404` ```text \See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"}\ ``` lp15@59667 ` 405` lp15@59667 ` 406` ```definition "pochhammer (a::'a::comm_semiring_1) n = ``` lp15@59667 ` 407` ``` (if n = 0 then 1 else setprod (\n. a + of_nat n) {0 .. n - 1})" ``` lp15@59667 ` 408` lp15@59667 ` 409` ```lemma pochhammer_0 [simp]: "pochhammer a 0 = 1" ``` lp15@59667 ` 410` ``` by (simp add: pochhammer_def) ``` lp15@59667 ` 411` lp15@59667 ` 412` ```lemma pochhammer_1 [simp]: "pochhammer a 1 = a" ``` lp15@59667 ` 413` ``` by (simp add: pochhammer_def) ``` lp15@59667 ` 414` lp15@59667 ` 415` ```lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a" ``` lp15@59667 ` 416` ``` by (simp add: pochhammer_def) ``` lp15@59667 ` 417` lp15@59667 ` 418` ```lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\n. a + of_nat n) {0 .. n}" ``` lp15@59667 ` 419` ``` by (simp add: pochhammer_def) ``` lp15@59667 ` 420` lp15@59667 ` 421` ```lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)" ``` lp15@59667 ` 422` ```proof - ``` lp15@59667 ` 423` ``` have "{0..Suc n} = {0..n} \ {Suc n}" by auto ``` lp15@59667 ` 424` ``` then show ?thesis by (simp add: field_simps) ``` lp15@59667 ` 425` ```qed ``` lp15@59667 ` 426` lp15@59667 ` 427` ```lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}" ``` lp15@59667 ` 428` ```proof - ``` lp15@59667 ` 429` ``` have "{0..Suc n} = {0} \ {1 .. Suc n}" by auto ``` lp15@59667 ` 430` ``` then show ?thesis by simp ``` lp15@59667 ` 431` ```qed ``` lp15@59667 ` 432` lp15@59667 ` 433` lp15@59667 ` 434` ```lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)" ``` lp15@59667 ` 435` ```proof (cases n) ``` lp15@59667 ` 436` ``` case 0 ``` lp15@59667 ` 437` ``` then show ?thesis by simp ``` lp15@59667 ` 438` ```next ``` lp15@59667 ` 439` ``` case (Suc n) ``` lp15@59667 ` 440` ``` show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc .. ``` lp15@59667 ` 441` ```qed ``` lp15@59667 ` 442` lp15@59667 ` 443` ```lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n" ``` lp15@59667 ` 444` ```proof (cases "n = 0") ``` lp15@59667 ` 445` ``` case True ``` lp15@59667 ` 446` ``` then show ?thesis by (simp add: pochhammer_Suc_setprod) ``` lp15@59667 ` 447` ```next ``` lp15@59667 ` 448` ``` case False ``` lp15@59667 ` 449` ``` have *: "finite {1 .. n}" "0 \ {1 .. n}" by auto ``` lp15@59667 ` 450` ``` have eq: "insert 0 {1 .. n} = {0..n}" by auto ``` wenzelm@61076 ` 451` ``` have **: "(\n\{1::nat..n}. a + of_nat n) = (\n\{0::nat..n - 1}. a + 1 + of_nat n)" ``` lp15@59667 ` 452` ``` apply (rule setprod.reindex_cong [where l = Suc]) ``` lp15@59667 ` 453` ``` using False ``` lp15@59667 ` 454` ``` apply (auto simp add: fun_eq_iff field_simps) ``` lp15@59667 ` 455` ``` done ``` lp15@59667 ` 456` ``` show ?thesis ``` lp15@59667 ` 457` ``` apply (simp add: pochhammer_def) ``` lp15@59667 ` 458` ``` unfolding setprod.insert [OF *, unfolded eq] ``` lp15@59667 ` 459` ``` using ** apply (simp add: field_simps) ``` lp15@59667 ` 460` ``` done ``` lp15@59667 ` 461` ```qed ``` lp15@59667 ` 462` lp15@59730 ` 463` ```lemma pochhammer_fact: "fact n = pochhammer 1 n" ``` lp15@59730 ` 464` ``` unfolding fact_altdef ``` lp15@59667 ` 465` ``` apply (cases n) ``` lp15@59667 ` 466` ``` apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod) ``` lp15@59667 ` 467` ``` apply (rule setprod.reindex_cong [where l = Suc]) ``` lp15@59667 ` 468` ``` apply (auto simp add: fun_eq_iff) ``` lp15@59667 ` 469` ``` done ``` lp15@59667 ` 470` lp15@59667 ` 471` ```lemma pochhammer_of_nat_eq_0_lemma: ``` lp15@59667 ` 472` ``` assumes "k > n" ``` lp15@59667 ` 473` ``` shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0" ``` lp15@59667 ` 474` ```proof (cases "n = 0") ``` lp15@59667 ` 475` ``` case True ``` lp15@59667 ` 476` ``` then show ?thesis ``` lp15@59667 ` 477` ``` using assms by (cases k) (simp_all add: pochhammer_rec) ``` lp15@59667 ` 478` ```next ``` lp15@59667 ` 479` ``` case False ``` lp15@59667 ` 480` ``` from assms obtain h where "k = Suc h" by (cases k) auto ``` lp15@59667 ` 481` ``` then show ?thesis ``` lp15@59667 ` 482` ``` by (simp add: pochhammer_Suc_setprod) ``` lp15@59667 ` 483` ``` (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1)) ``` lp15@59667 ` 484` ```qed ``` lp15@59667 ` 485` lp15@59667 ` 486` ```lemma pochhammer_of_nat_eq_0_lemma': ``` lp15@59667 ` 487` ``` assumes kn: "k \ n" ``` lp15@59667 ` 488` ``` shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \ 0" ``` lp15@59667 ` 489` ```proof (cases k) ``` lp15@59667 ` 490` ``` case 0 ``` lp15@59667 ` 491` ``` then show ?thesis by simp ``` lp15@59667 ` 492` ```next ``` lp15@59667 ` 493` ``` case (Suc h) ``` lp15@59667 ` 494` ``` then show ?thesis ``` lp15@59667 ` 495` ``` apply (simp add: pochhammer_Suc_setprod) ``` lp15@59667 ` 496` ``` using Suc kn apply (auto simp add: algebra_simps) ``` lp15@59667 ` 497` ``` done ``` lp15@59667 ` 498` ```qed ``` lp15@59667 ` 499` lp15@59667 ` 500` ```lemma pochhammer_of_nat_eq_0_iff: ``` lp15@59667 ` 501` ``` shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \ k > n" ``` lp15@59667 ` 502` ``` (is "?l = ?r") ``` lp15@59667 ` 503` ``` using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a] ``` lp15@59667 ` 504` ``` pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a] ``` lp15@59667 ` 505` ``` by (auto simp add: not_le[symmetric]) ``` lp15@59667 ` 506` lp15@59667 ` 507` ```lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \ (\k < n. a = - of_nat k)" ``` lp15@59667 ` 508` ``` apply (auto simp add: pochhammer_of_nat_eq_0_iff) ``` lp15@59667 ` 509` ``` apply (cases n) ``` lp15@59667 ` 510` ``` apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0) ``` lp15@59667 ` 511` ``` apply (metis leD not_less_eq) ``` lp15@59667 ` 512` ``` done ``` lp15@59667 ` 513` lp15@59667 ` 514` ```lemma pochhammer_eq_0_mono: ``` lp15@59667 ` 515` ``` "pochhammer a n = (0::'a::field_char_0) \ m \ n \ pochhammer a m = 0" ``` lp15@59667 ` 516` ``` unfolding pochhammer_eq_0_iff by auto ``` lp15@59667 ` 517` lp15@59667 ` 518` ```lemma pochhammer_neq_0_mono: ``` lp15@59667 ` 519` ``` "pochhammer a m \ (0::'a::field_char_0) \ m \ n \ pochhammer a n \ 0" ``` lp15@59667 ` 520` ``` unfolding pochhammer_eq_0_iff by auto ``` lp15@59667 ` 521` lp15@59667 ` 522` ```lemma pochhammer_minus: ``` lp15@59862 ` 523` ``` "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k" ``` lp15@59667 ` 524` ```proof (cases k) ``` lp15@59667 ` 525` ``` case 0 ``` lp15@59667 ` 526` ``` then show ?thesis by simp ``` lp15@59667 ` 527` ```next ``` lp15@59667 ` 528` ``` case (Suc h) ``` lp15@59667 ` 529` ``` have eq: "((- 1) ^ Suc h :: 'a) = (\i=0..h. - 1)" ``` lp15@59667 ` 530` ``` using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"] ``` lp15@59667 ` 531` ``` by auto ``` lp15@59667 ` 532` ``` show ?thesis ``` lp15@59667 ` 533` ``` unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric] ``` lp15@59667 ` 534` ``` by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"]) ``` lp15@59667 ` 535` ``` (auto simp: of_nat_diff) ``` lp15@59667 ` 536` ```qed ``` lp15@59667 ` 537` lp15@59667 ` 538` ```lemma pochhammer_minus': ``` lp15@59862 ` 539` ``` "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k" ``` lp15@59862 ` 540` ``` unfolding pochhammer_minus[where b=b] ``` lp15@59667 ` 541` ``` unfolding mult.assoc[symmetric] ``` lp15@59667 ` 542` ``` unfolding power_add[symmetric] ``` lp15@59667 ` 543` ``` by simp ``` lp15@59667 ` 544` lp15@59667 ` 545` ```lemma pochhammer_same: "pochhammer (- of_nat n) n = ``` lp15@59730 ` 546` ``` ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * (fact n)" ``` lp15@59862 ` 547` ``` unfolding pochhammer_minus ``` lp15@59667 ` 548` ``` by (simp add: of_nat_diff pochhammer_fact) ``` lp15@59667 ` 549` lp15@59667 ` 550` wenzelm@60758 ` 551` ```subsection\Generalized binomial coefficients\ ``` lp15@59667 ` 552` lp15@59667 ` 553` ```definition gbinomial :: "'a::field_char_0 \ nat \ 'a" (infixl "gchoose" 65) ``` lp15@59667 ` 554` ``` where "a gchoose n = ``` lp15@59730 ` 555` ``` (if n = 0 then 1 else (setprod (\i. a - of_nat i) {0 .. n - 1}) / (fact n))" ``` lp15@59667 ` 556` lp15@59667 ` 557` ```lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0" ``` haftmann@59867 ` 558` ``` by (simp_all add: gbinomial_def) ``` lp15@59667 ` 559` lp15@59730 ` 560` ```lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / (fact n)" ``` lp15@59667 ` 561` ```proof (cases "n = 0") ``` lp15@59667 ` 562` ``` case True ``` lp15@59667 ` 563` ``` then show ?thesis by simp ``` lp15@59667 ` 564` ```next ``` lp15@59667 ` 565` ``` case False ``` lp15@59667 ` 566` ``` from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"] ``` wenzelm@61076 ` 567` ``` have eq: "(- (1::'a)) ^ n = setprod (\i. - 1) {0 .. n - 1}" ``` lp15@59667 ` 568` ``` by auto ``` lp15@59667 ` 569` ``` from False show ?thesis ``` lp15@59667 ` 570` ``` by (simp add: pochhammer_def gbinomial_def field_simps ``` lp15@59667 ` 571` ``` eq setprod.distrib[symmetric]) ``` lp15@59667 ` 572` ```qed ``` lp15@59667 ` 573` lp15@59730 ` 574` ```lemma binomial_gbinomial: ``` lp15@59730 ` 575` ``` "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)" ``` lp15@59667 ` 576` ```proof - ``` lp15@59667 ` 577` ``` { assume kn: "k > n" ``` lp15@59667 ` 578` ``` then have ?thesis ``` lp15@59667 ` 579` ``` by (subst binomial_eq_0[OF kn]) ``` lp15@59667 ` 580` ``` (simp add: gbinomial_pochhammer field_simps pochhammer_of_nat_eq_0_iff) } ``` lp15@59667 ` 581` ``` moreover ``` lp15@59667 ` 582` ``` { assume "k=0" then have ?thesis by simp } ``` lp15@59667 ` 583` ``` moreover ``` lp15@59667 ` 584` ``` { assume kn: "k \ n" and k0: "k\ 0" ``` lp15@59667 ` 585` ``` from k0 obtain h where h: "k = Suc h" by (cases k) auto ``` lp15@59667 ` 586` ``` from h ``` lp15@59667 ` 587` ``` have eq:"(- 1 :: 'a) ^ k = setprod (\i. - 1) {0..h}" ``` lp15@59667 ` 588` ``` by (subst setprod_constant) auto ``` lp15@59667 ` 589` ``` have eq': "(\i\{0..h}. of_nat n + - (of_nat i :: 'a)) = (\i\{n - h..n}. of_nat i)" ``` lp15@59667 ` 590` ``` using h kn ``` lp15@59667 ` 591` ``` by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"]) ``` lp15@59667 ` 592` ``` (auto simp: of_nat_diff) ``` lp15@59667 ` 593` ``` have th0: "finite {1..n - Suc h}" "finite {n - h .. n}" ``` lp15@59667 ` 594` ``` "{1..n - Suc h} \ {n - h .. n} = {}" and ``` lp15@59667 ` 595` ``` eq3: "{1..n - Suc h} \ {n - h .. n} = {1..n}" ``` lp15@59667 ` 596` ``` using h kn by auto ``` lp15@59667 ` 597` ``` from eq[symmetric] ``` lp15@59667 ` 598` ``` have ?thesis using kn ``` lp15@59667 ` 599` ``` apply (simp add: binomial_fact[OF kn, where ?'a = 'a] ``` lp15@59667 ` 600` ``` gbinomial_pochhammer field_simps pochhammer_Suc_setprod) ``` lp15@59730 ` 601` ``` apply (simp add: pochhammer_Suc_setprod fact_altdef h ``` lp15@59667 ` 602` ``` of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc) ``` lp15@59667 ` 603` ``` unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \ 'a"] eq[unfolded h] ``` lp15@59730 ` 604` ``` unfolding mult.assoc ``` lp15@59667 ` 605` ``` unfolding setprod.distrib[symmetric] ``` lp15@59667 ` 606` ``` apply simp ``` lp15@59667 ` 607` ``` apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"]) ``` lp15@59667 ` 608` ``` apply (auto simp: of_nat_diff) ``` lp15@59667 ` 609` ``` done ``` lp15@59667 ` 610` ``` } ``` lp15@59667 ` 611` ``` moreover ``` lp15@59667 ` 612` ``` have "k > n \ k = 0 \ (k \ n \ k \ 0)" by arith ``` lp15@59667 ` 613` ``` ultimately show ?thesis by blast ``` lp15@59667 ` 614` ```qed ``` lp15@59667 ` 615` lp15@59667 ` 616` ```lemma gbinomial_1[simp]: "a gchoose 1 = a" ``` lp15@59667 ` 617` ``` by (simp add: gbinomial_def) ``` lp15@59667 ` 618` lp15@59667 ` 619` ```lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a" ``` lp15@59667 ` 620` ``` by (simp add: gbinomial_def) ``` lp15@59667 ` 621` lp15@59667 ` 622` ```lemma gbinomial_mult_1: ``` lp15@59730 ` 623` ``` fixes a :: "'a :: field_char_0" ``` lp15@59730 ` 624` ``` shows "a * (a gchoose n) = ``` lp15@59667 ` 625` ``` of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r") ``` lp15@59667 ` 626` ```proof - ``` lp15@59730 ` 627` ``` have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))" ``` lp15@59667 ` 628` ``` unfolding gbinomial_pochhammer ``` lp15@59730 ` 629` ``` pochhammer_Suc of_nat_mult right_diff_distrib power_Suc ``` lp15@59730 ` 630` ``` apply (simp del: of_nat_Suc fact.simps) ``` lp15@59730 ` 631` ``` apply (auto simp add: field_simps simp del: of_nat_Suc) ``` lp15@59730 ` 632` ``` done ``` lp15@59667 ` 633` ``` also have "\ = ?l" unfolding gbinomial_pochhammer ``` lp15@59667 ` 634` ``` by (simp add: field_simps) ``` lp15@59667 ` 635` ``` finally show ?thesis .. ``` lp15@59667 ` 636` ```qed ``` lp15@59667 ` 637` lp15@59667 ` 638` ```lemma gbinomial_mult_1': ``` lp15@59730 ` 639` ``` fixes a :: "'a :: field_char_0" ``` lp15@59730 ` 640` ``` shows "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" ``` lp15@59667 ` 641` ``` by (simp add: mult.commute gbinomial_mult_1) ``` lp15@59667 ` 642` lp15@59667 ` 643` ```lemma gbinomial_Suc: ``` lp15@59730 ` 644` ``` "a gchoose (Suc k) = (setprod (\i. a - of_nat i) {0 .. k}) / (fact (Suc k))" ``` lp15@59667 ` 645` ``` by (simp add: gbinomial_def) ``` lp15@59667 ` 646` lp15@59667 ` 647` ```lemma gbinomial_mult_fact: ``` lp15@59730 ` 648` ``` fixes a :: "'a::field_char_0" ``` lp15@59730 ` 649` ``` shows ``` lp15@59730 ` 650` ``` "fact (Suc k) * (a gchoose (Suc k)) = ``` lp15@59667 ` 651` ``` (setprod (\i. a - of_nat i) {0 .. k})" ``` lp15@59730 ` 652` ``` by (simp_all add: gbinomial_Suc field_simps del: fact.simps) ``` lp15@59667 ` 653` lp15@59667 ` 654` ```lemma gbinomial_mult_fact': ``` lp15@59730 ` 655` ``` fixes a :: "'a::field_char_0" ``` lp15@59730 ` 656` ``` shows "(a gchoose (Suc k)) * fact (Suc k) = (setprod (\i. a - of_nat i) {0 .. k})" ``` lp15@59667 ` 657` ``` using gbinomial_mult_fact[of k a] ``` lp15@59667 ` 658` ``` by (subst mult.commute) ``` lp15@59667 ` 659` lp15@59667 ` 660` ```lemma gbinomial_Suc_Suc: ``` lp15@59730 ` 661` ``` fixes a :: "'a :: field_char_0" ``` lp15@59730 ` 662` ``` shows "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))" ``` lp15@59667 ` 663` ```proof (cases k) ``` lp15@59667 ` 664` ``` case 0 ``` lp15@59667 ` 665` ``` then show ?thesis by simp ``` lp15@59667 ` 666` ```next ``` lp15@59667 ` 667` ``` case (Suc h) ``` lp15@59667 ` 668` ``` have eq0: "(\i\{1..k}. (a + 1) - of_nat i) = (\i\{0..h}. a - of_nat i)" ``` lp15@59667 ` 669` ``` apply (rule setprod.reindex_cong [where l = Suc]) ``` lp15@59667 ` 670` ``` using Suc ``` lp15@59667 ` 671` ``` apply auto ``` lp15@59667 ` 672` ``` done ``` lp15@59730 ` 673` ``` have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) = ``` lp15@59730 ` 674` ``` (a gchoose Suc h) * (fact (Suc (Suc h))) + ``` lp15@59730 ` 675` ``` (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))" ``` lp15@59730 ` 676` ``` by (simp add: Suc field_simps del: fact.simps) ``` lp15@59730 ` 677` ``` also have "... = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) + ``` lp15@59730 ` 678` ``` (\i = 0..Suc h. a - of_nat i)" ``` lp15@59730 ` 679` ``` by (metis fact.simps(2) gbinomial_mult_fact' of_nat_fact of_nat_id) ``` lp15@59730 ` 680` ``` also have "... = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) + ``` lp15@59730 ` 681` ``` (\i = 0..Suc h. a - of_nat i)" ``` lp15@59730 ` 682` ``` by (simp only: fact.simps(2) mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult) ``` lp15@59730 ` 683` ``` also have "... = of_nat (Suc (Suc h)) * (\i = 0..h. a - of_nat i) + ``` lp15@59730 ` 684` ``` (\i = 0..Suc h. a - of_nat i)" ``` lp15@59730 ` 685` ``` by (metis gbinomial_mult_fact mult.commute) ``` lp15@59730 ` 686` ``` also have "... = (\i = 0..Suc h. a - of_nat i) + ``` lp15@59730 ` 687` ``` (of_nat h * (\i = 0..h. a - of_nat i) + 2 * (\i = 0..h. a - of_nat i))" ``` lp15@59730 ` 688` ``` by (simp add: field_simps) ``` lp15@59730 ` 689` ``` also have "... = ``` wenzelm@61076 ` 690` ``` ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\i\{0::nat..Suc h}. a - of_nat i)" ``` lp15@59667 ` 691` ``` unfolding gbinomial_mult_fact' ``` lp15@59730 ` 692` ``` by (simp add: comm_semiring_class.distrib field_simps Suc) ``` lp15@59667 ` 693` ``` also have "\ = (\i\{0..h}. a - of_nat i) * (a + 1)" ``` lp15@59667 ` 694` ``` unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc ``` lp15@59667 ` 695` ``` by (simp add: field_simps Suc) ``` lp15@59667 ` 696` ``` also have "\ = (\i\{0..k}. (a + 1) - of_nat i)" ``` lp15@59667 ` 697` ``` using eq0 ``` lp15@59667 ` 698` ``` by (simp add: Suc setprod_nat_ivl_1_Suc) ``` lp15@59730 ` 699` ``` also have "\ = (fact (Suc k)) * ((a + 1) gchoose (Suc k))" ``` lp15@59667 ` 700` ``` unfolding gbinomial_mult_fact .. ``` lp15@59730 ` 701` ``` finally show ?thesis ``` lp15@59730 ` 702` ``` by (metis fact_nonzero mult_cancel_left) ``` lp15@59667 ` 703` ```qed ``` lp15@59667 ` 704` lp15@59667 ` 705` ```lemma gbinomial_reduce_nat: ``` lp15@59730 ` 706` ``` fixes a :: "'a :: field_char_0" ``` lp15@59730 ` 707` ``` shows "0 < k \ a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)" ``` lp15@59730 ` 708` ``` by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc) ``` lp15@59667 ` 709` lp15@60141 ` 710` ```lemma gchoose_row_sum_weighted: ``` lp15@60141 ` 711` ``` fixes r :: "'a::field_char_0" ``` lp15@60141 ` 712` ``` shows "(\k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))" ``` lp15@60141 ` 713` ```proof (induct m) ``` lp15@60141 ` 714` ``` case 0 show ?case by simp ``` lp15@60141 ` 715` ```next ``` lp15@60141 ` 716` ``` case (Suc m) ``` lp15@60141 ` 717` ``` from Suc show ?case ``` lp15@60141 ` 718` ``` by (simp add: algebra_simps distrib gbinomial_mult_1) ``` lp15@60141 ` 719` ```qed ``` lp15@59667 ` 720` lp15@59667 ` 721` ```lemma binomial_symmetric: ``` lp15@59667 ` 722` ``` assumes kn: "k \ n" ``` lp15@59667 ` 723` ``` shows "n choose k = n choose (n - k)" ``` lp15@59667 ` 724` ```proof- ``` lp15@59667 ` 725` ``` from kn have kn': "n - k \ n" by arith ``` lp15@59667 ` 726` ``` from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn'] ``` lp15@59667 ` 727` ``` have "fact k * fact (n - k) * (n choose k) = ``` lp15@59667 ` 728` ``` fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp ``` lp15@59667 ` 729` ``` then show ?thesis using kn by simp ``` lp15@59667 ` 730` ```qed ``` lp15@59667 ` 731` wenzelm@60758 ` 732` ```text\Contributed by Manuel Eberl, generalised by LCP. ``` wenzelm@60758 ` 733` ``` Alternative definition of the binomial coefficient as @{term "\i ``` lp15@59667 ` 734` ```lemma gbinomial_altdef_of_nat: ``` lp15@59667 ` 735` ``` fixes k :: nat ``` haftmann@59867 ` 736` ``` and x :: "'a :: {field_char_0,field}" ``` lp15@59667 ` 737` ``` shows "x gchoose k = (\ii = (\i x" ``` lp15@59667 ` 752` ``` shows "(x / of_nat k :: 'a) ^ k \ x gchoose k" ``` lp15@59667 ` 753` ```proof - ``` lp15@59667 ` 754` ``` have x: "0 \ x" ``` lp15@59667 ` 755` ``` using assms of_nat_0_le_iff order_trans by blast ``` lp15@59667 ` 756` ``` have "(x / of_nat k :: 'a) ^ k = (\i \ x gchoose k" ``` lp15@59667 ` 759` ``` unfolding gbinomial_altdef_of_nat ``` lp15@59667 ` 760` ``` proof (safe intro!: setprod_mono) ``` lp15@59667 ` 761` ``` fix i :: nat ``` lp15@59667 ` 762` ``` assume ik: "i < k" ``` lp15@59667 ` 763` ``` from assms have "x * of_nat i \ of_nat (i * k)" ``` lp15@59667 ` 764` ``` by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult) ``` lp15@59667 ` 765` ``` then have "x * of_nat k - x * of_nat i \ x * of_nat k - of_nat (i * k)" by arith ``` lp15@59667 ` 766` ``` then have "x * of_nat (k - i) \ (x - of_nat i) * of_nat k" ``` lp15@59667 ` 767` ``` using ik ``` lp15@59667 ` 768` ``` by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult) ``` lp15@59667 ` 769` ``` then have "x * of_nat (k - i) \ (x - of_nat i) * (of_nat k :: 'a)" ``` lp15@59667 ` 770` ``` unfolding of_nat_mult[symmetric] of_nat_le_iff . ``` lp15@59667 ` 771` ``` with assms show "x / of_nat k \ (x - of_nat i) / (of_nat (k - i) :: 'a)" ``` wenzelm@60758 ` 772` ``` using \i < k\ by (simp add: field_simps) ``` lp15@59667 ` 773` ``` qed (simp add: x zero_le_divide_iff) ``` lp15@59667 ` 774` ``` finally show ?thesis . ``` lp15@59667 ` 775` ```qed ``` lp15@59667 ` 776` wenzelm@60758 ` 777` ```text\Versions of the theorems above for the natural-number version of "choose"\ ``` lp15@59667 ` 778` ```lemma binomial_altdef_of_nat: ``` lp15@59667 ` 779` ``` fixes n k :: nat ``` wenzelm@60758 ` 780` ``` and x :: "'a :: {field_char_0,field}" --\the point is to constrain @{typ 'a}\ ``` lp15@59667 ` 781` ``` assumes "k \ n" ``` lp15@59667 ` 782` ``` shows "of_nat (n choose k) = (\i n" ``` lp15@59667 ` 790` ``` shows "(of_nat n / of_nat k :: 'a) ^ k \ of_nat (n choose k)" ``` lp15@59667 ` 791` ```by (simp add: assms gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff) ``` lp15@59667 ` 792` lp15@59667 ` 793` ```lemma binomial_le_pow: ``` lp15@59667 ` 794` ``` assumes "r \ n" ``` lp15@59667 ` 795` ``` shows "n choose r \ n ^ r" ``` lp15@59667 ` 796` ```proof - ``` lp15@59667 ` 797` ``` have "n choose r \ fact n div fact (n - r)" ``` wenzelm@60758 ` 798` ``` using \r \ n\ by (subst binomial_fact_lemma[symmetric]) auto ``` lp15@59667 ` 799` ``` with fact_div_fact_le_pow [OF assms] show ?thesis by auto ``` lp15@59667 ` 800` ```qed ``` lp15@59667 ` 801` lp15@59667 ` 802` ```lemma binomial_altdef_nat: "(k::nat) \ n \ ``` lp15@59667 ` 803` ``` n choose k = fact n div (fact k * fact (n - k))" ``` lp15@59667 ` 804` ``` by (subst binomial_fact_lemma [symmetric]) auto ``` lp15@59667 ` 805` lp15@59730 ` 806` ```lemma choose_dvd: "k \ n \ fact k * fact (n - k) dvd (fact n :: 'a :: {semiring_div,linordered_semidom})" ``` lp15@59730 ` 807` ``` unfolding dvd_def ``` lp15@59730 ` 808` ``` apply (rule exI [where x="of_nat (n choose k)"]) ``` lp15@59730 ` 809` ``` using binomial_fact_lemma [of k n, THEN arg_cong [where f = of_nat and 'b='a]] ``` lp15@59730 ` 810` ``` apply (auto simp: of_nat_mult) ``` lp15@59667 ` 811` ``` done ``` lp15@59667 ` 812` lp15@59730 ` 813` ```lemma fact_fact_dvd_fact: ``` lp15@59730 ` 814` ``` "fact k * fact n dvd (fact (k+n) :: 'a :: {semiring_div,linordered_semidom})" ``` lp15@59730 ` 815` ```by (metis add.commute add_diff_cancel_left' choose_dvd le_add2) ``` lp15@59667 ` 816` lp15@59667 ` 817` ```lemma choose_mult_lemma: ``` lp15@59667 ` 818` ``` "((m+r+k) choose (m+k)) * ((m+k) choose k) = ((m+r+k) choose k) * ((m+r) choose m)" ``` lp15@59667 ` 819` ```proof - ``` lp15@59667 ` 820` ``` have "((m+r+k) choose (m+k)) * ((m+k) choose k) = ``` lp15@59667 ` 821` ``` fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))" ``` lp15@59667 ` 822` ``` by (simp add: assms binomial_altdef_nat) ``` lp15@59667 ` 823` ``` also have "... = fact (m+r+k) div (fact r * (fact k * fact m))" ``` lp15@59667 ` 824` ``` apply (subst div_mult_div_if_dvd) ``` lp15@59730 ` 825` ``` apply (auto simp: algebra_simps fact_fact_dvd_fact) ``` lp15@59667 ` 826` ``` apply (metis add.assoc add.commute fact_fact_dvd_fact) ``` lp15@59667 ` 827` ``` done ``` lp15@59667 ` 828` ``` also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))" ``` lp15@59667 ` 829` ``` apply (subst div_mult_div_if_dvd [symmetric]) ``` lp15@59730 ` 830` ``` apply (auto simp add: algebra_simps) ``` lp15@59730 ` 831` ``` apply (metis fact_fact_dvd_fact dvd.order.trans nat_mult_dvd_cancel_disj) ``` lp15@59667 ` 832` ``` done ``` lp15@59667 ` 833` ``` also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))" ``` lp15@59667 ` 834` ``` apply (subst div_mult_div_if_dvd) ``` lp15@59730 ` 835` ``` apply (auto simp: fact_fact_dvd_fact algebra_simps) ``` lp15@59667 ` 836` ``` done ``` lp15@59667 ` 837` ``` finally show ?thesis ``` lp15@59667 ` 838` ``` by (simp add: binomial_altdef_nat mult.commute) ``` lp15@59667 ` 839` ```qed ``` lp15@59667 ` 840` wenzelm@60758 ` 841` ```text\The "Subset of a Subset" identity\ ``` lp15@59667 ` 842` ```lemma choose_mult: ``` lp15@59667 ` 843` ``` assumes "k\m" "m\n" ``` lp15@59667 ` 844` ``` shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))" ``` lp15@59667 ` 845` ```using assms choose_mult_lemma [of "m-k" "n-m" k] ``` lp15@59667 ` 846` ```by simp ``` lp15@59667 ` 847` lp15@59667 ` 848` wenzelm@60758 ` 849` ```subsection \Binomial coefficients\ ``` lp15@59667 ` 850` lp15@59667 ` 851` ```lemma choose_one: "(n::nat) choose 1 = n" ``` lp15@59667 ` 852` ``` by simp ``` lp15@59667 ` 853` lp15@59667 ` 854` ```(*FIXME: messy and apparently unused*) ``` lp15@59667 ` 855` ```lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \ ``` lp15@59667 ` 856` ``` (ALL n. P (Suc n) 0) \ (ALL n. (ALL k < n. P n k \ P n (Suc k) \ ``` lp15@59667 ` 857` ``` P (Suc n) (Suc k))) \ (ALL k <= n. P n k)" ``` lp15@59667 ` 858` ``` apply (induct n) ``` lp15@59667 ` 859` ``` apply auto ``` lp15@59667 ` 860` ``` apply (case_tac "k = 0") ``` lp15@59667 ` 861` ``` apply auto ``` lp15@59667 ` 862` ``` apply (case_tac "k = Suc n") ``` lp15@59667 ` 863` ``` apply auto ``` lp15@59730 ` 864` ``` apply (metis Suc_le_eq fact.cases le_Suc_eq le_eq_less_or_eq) ``` lp15@59667 ` 865` ``` done ``` lp15@59667 ` 866` lp15@59667 ` 867` ```lemma card_UNION: ``` lp15@59667 ` 868` ``` assumes "finite A" and "\k \ A. finite k" ``` lp15@59667 ` 869` ``` shows "card (\A) = nat (\I | I \ A \ I \ {}. (- 1) ^ (card I + 1) * int (card (\I)))" ``` lp15@59667 ` 870` ``` (is "?lhs = ?rhs") ``` lp15@59667 ` 871` ```proof - ``` lp15@59667 ` 872` ``` have "?rhs = nat (\I | I \ A \ I \ {}. (- 1) ^ (card I + 1) * (\_\\I. 1))" by simp ``` lp15@59667 ` 873` ``` also have "\ = nat (\I | I \ A \ I \ {}. (\_\\I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs") ``` lp15@59667 ` 874` ``` by(subst setsum_right_distrib) simp ``` lp15@59667 ` 875` ``` also have "?rhs = (\(I, _)\Sigma {I. I \ A \ I \ {}} Inter. (- 1) ^ (card I + 1))" ``` lp15@59667 ` 876` ``` using assms by(subst setsum.Sigma)(auto) ``` lp15@59667 ` 877` ``` also have "\ = (\(x, I)\(SIGMA x:UNIV. {I. I \ A \ I \ {} \ x \ \I}). (- 1) ^ (card I + 1))" ``` lp15@59667 ` 878` ``` by (rule setsum.reindex_cong [where l = "\(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta) ``` lp15@59667 ` 879` ``` also have "\ = (\(x, I)\(SIGMA x:\A. {I. I \ A \ I \ {} \ x \ \I}). (- 1) ^ (card I + 1))" ``` lp15@59667 ` 880` ``` using assms by(auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\A"]) ``` lp15@59667 ` 881` ``` also have "\ = (\x\\A. (\I|I \ A \ I \ {} \ x \ \I. (- 1) ^ (card I + 1)))" ``` lp15@59667 ` 882` ``` using assms by(subst setsum.Sigma) auto ``` lp15@59667 ` 883` ``` also have "\ = (\_\\A. 1)" (is "setsum ?lhs _ = _") ``` lp15@59667 ` 884` ``` proof(rule setsum.cong[OF refl]) ``` lp15@59667 ` 885` ``` fix x ``` lp15@59667 ` 886` ``` assume x: "x \ \A" ``` lp15@59667 ` 887` ``` def K \ "{X \ A. x \ X}" ``` wenzelm@60758 ` 888` ``` with \finite A\ have K: "finite K" by auto ``` lp15@59667 ` 889` ``` let ?I = "\i. {I. I \ A \ card I = i \ x \ \I}" ``` lp15@59667 ` 890` ``` have "inj_on snd (SIGMA i:{1..card A}. ?I i)" ``` lp15@59667 ` 891` ``` using assms by(auto intro!: inj_onI) ``` lp15@59667 ` 892` ``` moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \ A \ I \ {} \ x \ \I}" ``` lp15@59667 ` 893` ``` using assms by(auto intro!: rev_image_eqI[where x="(card a, a)" for a] ``` lp15@59667 ` 894` ``` simp add: card_gt_0_iff[folded Suc_le_eq] ``` lp15@59667 ` 895` ``` dest: finite_subset intro: card_mono) ``` lp15@59667 ` 896` ``` ultimately have "?lhs x = (\(i, I)\(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))" ``` lp15@59667 ` 897` ``` by (rule setsum.reindex_cong [where l = snd]) fastforce ``` lp15@59667 ` 898` ``` also have "\ = (\i=1..card A. (\I|I \ A \ card I = i \ x \ \I. (- 1) ^ (i + 1)))" ``` lp15@59667 ` 899` ``` using assms by(subst setsum.Sigma) auto ``` lp15@59667 ` 900` ``` also have "\ = (\i=1..card A. (- 1) ^ (i + 1) * (\I|I \ A \ card I = i \ x \ \I. 1))" ``` lp15@59667 ` 901` ``` by(subst setsum_right_distrib) simp ``` lp15@59667 ` 902` ``` also have "\ = (\i=1..card K. (- 1) ^ (i + 1) * (\I|I \ K \ card I = i. 1))" (is "_ = ?rhs") ``` lp15@59667 ` 903` ``` proof(rule setsum.mono_neutral_cong_right[rule_format]) ``` wenzelm@60758 ` 904` ``` show "{1..card K} \ {1..card A}" using \finite A\ ``` lp15@59667 ` 905` ``` by(auto simp add: K_def intro: card_mono) ``` lp15@59667 ` 906` ``` next ``` lp15@59667 ` 907` ``` fix i ``` lp15@59667 ` 908` ``` assume "i \ {1..card A} - {1..card K}" ``` lp15@59667 ` 909` ``` hence i: "i \ card A" "card K < i" by auto ``` lp15@59667 ` 910` ``` have "{I. I \ A \ card I = i \ x \ \I} = {I. I \ K \ card I = i}" ``` lp15@59667 ` 911` ``` by(auto simp add: K_def) ``` wenzelm@60758 ` 912` ``` also have "\ = {}" using \finite A\ i ``` lp15@59667 ` 913` ``` by(auto simp add: K_def dest: card_mono[rotated 1]) ``` lp15@59667 ` 914` ``` finally show "(- 1) ^ (i + 1) * (\I | I \ A \ card I = i \ x \ \I. 1 :: int) = 0" ``` lp15@59667 ` 915` ``` by(simp only:) simp ``` lp15@59667 ` 916` ``` next ``` lp15@59667 ` 917` ``` fix i ``` lp15@59667 ` 918` ``` have "(\I | I \ A \ card I = i \ x \ \I. 1) = (\I | I \ K \ card I = i. 1 :: int)" ``` lp15@59667 ` 919` ``` (is "?lhs = ?rhs") ``` lp15@59667 ` 920` ``` by(rule setsum.cong)(auto simp add: K_def) ``` lp15@59667 ` 921` ``` thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp ``` lp15@59667 ` 922` ``` qed simp ``` lp15@59667 ` 923` ``` also have "{I. I \ K \ card I = 0} = {{}}" using assms ``` lp15@59667 ` 924` ``` by(auto simp add: card_eq_0_iff K_def dest: finite_subset) ``` lp15@59667 ` 925` ``` hence "?rhs = (\i = 0..card K. (- 1) ^ (i + 1) * (\I | I \ K \ card I = i. 1 :: int)) + 1" ``` lp15@59667 ` 926` ``` by(subst (2) setsum_head_Suc)(simp_all ) ``` lp15@59667 ` 927` ``` also have "\ = (\i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1" ``` lp15@59667 ` 928` ``` using K by(subst n_subsets[symmetric]) simp_all ``` lp15@59667 ` 929` ``` also have "\ = - (\i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1" ``` lp15@59667 ` 930` ``` by(subst setsum_right_distrib[symmetric]) simp ``` lp15@59667 ` 931` ``` also have "\ = - ((-1 + 1) ^ card K) + 1" ``` lp15@59667 ` 932` ``` by(subst binomial_ring)(simp add: ac_simps) ``` lp15@59667 ` 933` ``` also have "\ = 1" using x K by(auto simp add: K_def card_gt_0_iff) ``` lp15@59667 ` 934` ``` finally show "?lhs x = 1" . ``` lp15@59667 ` 935` ``` qed ``` lp15@59667 ` 936` ``` also have "nat \ = card (\A)" by simp ``` lp15@59667 ` 937` ``` finally show ?thesis .. ``` lp15@59667 ` 938` ```qed ``` lp15@59667 ` 939` wenzelm@60758 ` 940` ```text\The number of nat lists of length @{text m} summing to @{text N} is ``` wenzelm@60758 ` 941` ```@{term "(N + m - 1) choose N"}:\ ``` lp15@59667 ` 942` lp15@59667 ` 943` ```lemma card_length_listsum_rec: ``` lp15@59667 ` 944` ``` assumes "m\1" ``` lp15@59667 ` 945` ``` shows "card {l::nat list. length l = m \ listsum l = N} = ``` lp15@59667 ` 946` ``` (card {l. length l = (m - 1) \ listsum l = N} + ``` lp15@59667 ` 947` ``` card {l. length l = m \ listsum l + 1 = N})" ``` lp15@59667 ` 948` ``` (is "card ?C = (card ?A + card ?B)") ``` lp15@59667 ` 949` ```proof - ``` lp15@59667 ` 950` ``` let ?A'="{l. length l = m \ listsum l = N \ hd l = 0}" ``` lp15@59667 ` 951` ``` let ?B'="{l. length l = m \ listsum l = N \ hd l \ 0}" ``` lp15@59667 ` 952` ``` let ?f ="\ l. 0#l" ``` lp15@59667 ` 953` ``` let ?g ="\ l. (hd l + 1) # tl l" ``` lp15@59667 ` 954` ``` have 1: "\xs x. xs \ [] \ x = hd xs \ x # tl xs = xs" by simp ``` lp15@59667 ` 955` ``` have 2: "\xs. (xs::nat list) \ [] \ listsum(tl xs) = listsum xs - hd xs" ``` lp15@59667 ` 956` ``` by(auto simp add: neq_Nil_conv) ``` lp15@59667 ` 957` ``` have f: "bij_betw ?f ?A ?A'" ``` lp15@59667 ` 958` ``` apply(rule bij_betw_byWitness[where f' = tl]) ``` lp15@59667 ` 959` ``` using assms ``` lp15@59667 ` 960` ``` by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv) ``` lp15@59667 ` 961` ``` have 3: "\xs:: nat list. xs \ [] \ hd xs + (listsum xs - hd xs) = listsum xs" ``` lp15@59667 ` 962` ``` by (metis 1 listsum_simps(2) 2) ``` lp15@59667 ` 963` ``` have g: "bij_betw ?g ?B ?B'" ``` lp15@59667 ` 964` ``` apply(rule bij_betw_byWitness[where f' = "\ l. (hd l - 1) # tl l"]) ``` lp15@59667 ` 965` ``` using assms ``` lp15@59667 ` 966` ``` by (auto simp: 2 length_0_conv[symmetric] intro!: 3 ``` lp15@59667 ` 967` ``` simp del: length_greater_0_conv length_0_conv) ``` lp15@59667 ` 968` ``` { fix M N :: nat have "finite {xs. size xs = M \ set xs \ {0.. ?B'" by auto ``` lp15@59667 ` 978` ``` have disj: "?A' \ ?B' = {}" by auto ``` lp15@59667 ` 979` ``` have "card ?C = card(?A' \ ?B')" using uni by simp ``` lp15@59667 ` 980` ``` also have "\ = card ?A + card ?B" ``` lp15@59667 ` 981` ``` using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g] ``` lp15@59667 ` 982` ``` bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B ``` lp15@59667 ` 983` ``` by presburger ``` lp15@59667 ` 984` ``` finally show ?thesis . ``` lp15@59667 ` 985` ```qed ``` lp15@59667 ` 986` lp15@59667 ` 987` ```lemma card_length_listsum: --"By Holden Lee, tidied by Tobias Nipkow" ``` lp15@59667 ` 988` ``` "card {l::nat list. size l = m \ listsum l = N} = (N + m - 1) choose N" ``` lp15@59667 ` 989` ```proof (cases m) ``` lp15@59667 ` 990` ``` case 0 then show ?thesis ``` lp15@59667 ` 991` ``` by (cases N) (auto simp: cong: conj_cong) ``` lp15@59667 ` 992` ```next ``` lp15@59667 ` 993` ``` case (Suc m') ``` lp15@59667 ` 994` ``` have m: "m\1" by (simp add: Suc) ``` lp15@59667 ` 995` ``` then show ?thesis ``` lp15@59667 ` 996` ``` proof (induct "N + m - 1" arbitrary: N m) ``` lp15@59667 ` 997` ``` case 0 -- "In the base case, the only solution is [0]." ``` lp15@59667 ` 998` ``` have [simp]: "{l::nat list. length l = Suc 0 \ (\n\set l. n = 0)} = {[0]}" ``` lp15@59667 ` 999` ``` by (auto simp: length_Suc_conv) ``` lp15@59667 ` 1000` ``` have "m=1 \ N=0" using 0 by linarith ``` lp15@59667 ` 1001` ``` then show ?case by simp ``` lp15@59667 ` 1002` ``` next ``` lp15@59667 ` 1003` ``` case (Suc k) ``` lp15@59667 ` 1004` lp15@59667 ` 1005` ``` have c1: "card {l::nat list. size l = (m - 1) \ listsum l = N} = ``` lp15@59667 ` 1006` ``` (N + (m - 1) - 1) choose N" ``` lp15@59667 ` 1007` ``` proof cases ``` lp15@59667 ` 1008` ``` assume "m = 1" ``` lp15@59667 ` 1009` ``` with Suc.hyps have "N\1" by auto ``` wenzelm@60758 ` 1010` ``` with \m = 1\ show ?thesis by (simp add: binomial_eq_0) ``` lp15@59667 ` 1011` ``` next ``` lp15@59667 ` 1012` ``` assume "m \ 1" thus ?thesis using Suc by fastforce ``` lp15@59667 ` 1013` ``` qed ``` lp15@59667 ` 1014` lp15@59667 ` 1015` ``` from Suc have c2: "card {l::nat list. size l = m \ listsum l + 1 = N} = ``` lp15@59667 ` 1016` ``` (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)" ``` lp15@59667 ` 1017` ``` proof - ``` lp15@59667 ` 1018` ``` have aux: "\m n. n > 0 \ Suc m = n \ m = n - 1" by arith ``` lp15@59667 ` 1019` ``` from Suc have "N>0 \ ``` lp15@59667 ` 1020` ``` card {l::nat list. size l = m \ listsum l + 1 = N} = ``` lp15@59667 ` 1021` ``` ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux) ``` lp15@59667 ` 1022` ``` thus ?thesis by auto ``` lp15@59667 ` 1023` ``` qed ``` lp15@59667 ` 1024` lp15@59667 ` 1025` ``` from Suc.prems have "(card {l::nat list. size l = (m - 1) \ listsum l = N} + ``` lp15@59667 ` 1026` ``` card {l::nat list. size l = m \ listsum l + 1 = N}) = (N + m - 1) choose N" ``` lp15@59667 ` 1027` ``` by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def) ``` lp15@59667 ` 1028` ``` thus ?case using card_length_listsum_rec[OF Suc.prems] by auto ``` lp15@59667 ` 1029` ``` qed ``` lp15@59667 ` 1030` ```qed ``` lp15@59667 ` 1031` hoelzl@60604 ` 1032` hoelzl@60604 ` 1033` ```lemma Suc_times_binomial_add: -- \by Lukas Bulwahn\ ``` hoelzl@60604 ` 1034` ``` "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)" ``` hoelzl@60604 ` 1035` ```proof - ``` hoelzl@60604 ` 1036` ``` have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b ``` hoelzl@60604 ` 1037` ``` using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat] ``` hoelzl@60604 ` 1038` ``` by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc) ``` hoelzl@60604 ` 1039` hoelzl@60604 ` 1040` ``` have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) = ``` hoelzl@60604 ` 1041` ``` Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))" ``` hoelzl@60604 ` 1042` ``` by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd) ``` hoelzl@60604 ` 1043` ``` also have "\ = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))" ``` hoelzl@60604 ` 1044` ``` by (simp only: div_mult_mult1) ``` hoelzl@60604 ` 1045` ``` also have "\ = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))" ``` hoelzl@60604 ` 1046` ``` using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd) ``` hoelzl@60604 ` 1047` ``` finally show ?thesis ``` hoelzl@60604 ` 1048` ``` by (subst (1 2) binomial_altdef_nat) ``` hoelzl@60604 ` 1049` ``` (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id) ``` hoelzl@60604 ` 1050` ```qed ``` hoelzl@60604 ` 1051` nipkow@15131 ` 1052` ```end ```