(* Title : Binomial.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
The integer version of factorial and other additions by Jeremy Avigad.
*)
section\<open>Factorial Function, Binomial Coefficients and Binomial Theorem\<close>
theory Binomial
imports Main
begin
subsection \<open>Factorial\<close>
fun fact :: "nat \<Rightarrow> ('a::semiring_char_0)"
where "fact 0 = 1" | "fact (Suc n) = of_nat (Suc n) * fact n"
lemmas fact_Suc = fact.simps(2)
lemma fact_1 [simp]: "fact 1 = 1"
by simp
lemma fact_Suc_0 [simp]: "fact (Suc 0) = Suc 0"
by simp
lemma of_nat_fact [simp]: "of_nat (fact n) = fact n"
by (induct n) (auto simp add: algebra_simps of_nat_mult)
lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
by (cases n) auto
lemma fact_nonzero [simp]: "fact n \<noteq> (0::'a::{semiring_char_0,semiring_no_zero_divisors})"
apply (induct n)
apply auto
using of_nat_eq_0_iff by fastforce
lemma fact_mono_nat: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: nat)"
by (induct n) (auto simp: le_Suc_eq)
context
assumes "SORT_CONSTRAINT('a::linordered_semidom)"
begin
lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
by (metis le0 fact.simps(1) fact_mono)
lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
using fact_ge_1 less_le_trans zero_less_one by blast
lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
by (simp add: less_imp_le)
lemma fact_not_neg [simp]: "~ (fact n < (0 :: 'a))"
by (simp add: not_less_iff_gr_or_eq)
lemma fact_le_power:
"fact n \<le> (of_nat (n^n) ::'a)"
proof (induct n)
case (Suc n)
then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
by (rule order_trans) (simp add: power_mono)
have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
by (simp add: algebra_simps)
also have "... \<le> (of_nat (Suc n) * of_nat (Suc n ^ n) ::'a)"
by (simp add: "*" ordered_comm_semiring_class.comm_mult_left_mono)
also have "... \<le> (of_nat (Suc n ^ Suc n) ::'a)"
by (metis of_nat_mult order_refl power_Suc)
finally show ?case .
qed simp
end
text\<open>Note that @{term "fact 0 = fact 1"}\<close>
lemma fact_less_mono_nat: "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: nat)"
by (induct n) (auto simp: less_Suc_eq)
lemma fact_less_mono:
"\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: 'a::linordered_semidom)"
by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat)
lemma fact_ge_Suc_0_nat [simp]: "fact n \<ge> Suc 0"
by (metis One_nat_def fact_ge_1)
lemma dvd_fact:
shows "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
by (induct n) (auto simp: dvdI le_Suc_eq)
lemma fact_altdef_nat: "fact n = \<Prod>{1..n}"
by (induct n) (auto simp: atLeastAtMostSuc_conv)
lemma fact_altdef: "fact n = setprod of_nat {1..n}"
by (induct n) (auto simp: atLeastAtMostSuc_conv)
lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a :: {semiring_div,linordered_semidom})"
by (induct m) (auto simp: le_Suc_eq)
lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a :: {semiring_div,linordered_semidom}) = 0"
by (auto simp add: fact_dvd)
lemma fact_div_fact:
assumes "m \<ge> n"
shows "(fact m) div (fact n) = \<Prod>{n + 1..m}"
proof -
obtain d where "d = m - n" by auto
from assms this have "m = n + d" by auto
have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
proof (induct d)
case 0
show ?case by simp
next
case (Suc d')
have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
by simp
also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}"
unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
also have "... = \<Prod>{n + 1..n + Suc d'}"
by (simp add: atLeastAtMostSuc_conv)
finally show ?case .
qed
from this \<open>m = n + d\<close> show ?thesis by simp
qed
lemma fact_num_eq_if:
"fact m = (if m=0 then 1 else of_nat m * fact (m - 1))"
by (cases m) auto
lemma fact_eq_rev_setprod_nat: "fact k = (\<Prod>i<k. k - i)"
unfolding fact_altdef_nat
by (rule setprod.reindex_bij_witness[where i="\<lambda>i. k - i" and j="\<lambda>i. k - i"]) auto
lemma fact_div_fact_le_pow:
assumes "r \<le> n" shows "fact n div fact (n - r) \<le> n ^ r"
proof -
have "\<And>r. r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}"
by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
with assms show ?thesis
by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
qed
lemma fact_numeral: --\<open>Evaluation for specific numerals\<close>
"fact (numeral k) = (numeral k) * (fact (pred_numeral k))"
by (metis fact.simps(2) numeral_eq_Suc of_nat_numeral)
text \<open>This development is based on the work of Andy Gordon and
Florian Kammueller.\<close>
subsection \<open>Basic definitions and lemmas\<close>
primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
where
"0 choose k = (if k = 0 then 1 else 0)"
| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
lemma binomial_n_0 [simp]: "(n choose 0) = 1"
by (cases n) simp_all
lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
by simp
lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
by simp
lemma choose_reduce_nat:
"0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
(n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
by (metis Suc_diff_1 binomial.simps(2) neq0_conv)
lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
by (induct n arbitrary: k) auto
declare binomial.simps [simp del]
lemma binomial_n_n [simp]: "n choose n = 1"
by (induct n) (simp_all add: binomial_eq_0)
lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
by (induct n) simp_all
lemma binomial_1 [simp]: "n choose Suc 0 = n"
by (induct n) simp_all
lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
by (induct n k rule: diff_induct) simp_all
lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
lemma Suc_times_binomial_eq:
"Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
apply (induct n arbitrary: k, simp add: binomial.simps)
apply (case_tac k)
apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
done
lemma binomial_le_pow2: "n choose k \<le> 2^n"
apply (induction n arbitrary: k)
apply (simp add: binomial.simps)
apply (case_tac k)
apply (auto simp: power_Suc)
by (simp add: add_le_mono mult_2)
text\<open>The absorption property\<close>
lemma Suc_times_binomial:
"Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
using Suc_times_binomial_eq by auto
text\<open>This is the well-known version of absorption, but it's harder to use because of the
need to reason about division.\<close>
lemma binomial_Suc_Suc_eq_times:
"(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
text\<open>Another version of absorption, with -1 instead of Suc.\<close>
lemma times_binomial_minus1_eq:
"0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
by (auto split add: nat_diff_split)
subsection \<open>Combinatorial theorems involving @{text "choose"}\<close>
text \<open>By Florian Kamm\"uller, tidied by LCP.\<close>
lemma card_s_0_eq_empty: "finite A \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 1"
by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
{s. s \<subseteq> insert x M \<and> card s = Suc k} =
{s. s \<subseteq> M \<and> card s = Suc k} \<union> {s. \<exists>t. t \<subseteq> M \<and> card t = k \<and> s = insert x t}"
apply safe
apply (auto intro: finite_subset [THEN card_insert_disjoint])
by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if
card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff)
lemma finite_bex_subset [simp]:
assumes "finite B"
and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
shows "finite {x. \<exists>A \<subseteq> B. P x A}"
by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets)
text\<open>There are as many subsets of @{term A} having cardinality @{term k}
as there are sets obtained from the former by inserting a fixed element
@{term x} into each.\<close>
lemma constr_bij:
"finite A \<Longrightarrow> x \<notin> A \<Longrightarrow>
card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
card {B. B \<subseteq> A & card(B) = k}"
apply (rule card_bij_eq [where f = "\<lambda>s. s - {x}" and g = "insert x"])
apply (auto elim!: equalityE simp add: inj_on_def)
apply (metis card_Diff_singleton_if finite_subset in_mono)
done
text \<open>
Main theorem: combinatorial statement about number of subsets of a set.
\<close>
theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
proof (induct k arbitrary: A)
case 0 then show ?case by (simp add: card_s_0_eq_empty)
next
case (Suc k)
show ?case using \<open>finite A\<close>
proof (induct A)
case empty show ?case by (simp add: card_s_0_eq_empty)
next
case (insert x A)
then show ?case using Suc.hyps
apply (simp add: card_s_0_eq_empty choose_deconstruct)
apply (subst card_Un_disjoint)
prefer 4 apply (force simp add: constr_bij)
prefer 3 apply force
prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
finite_subset [of _ "Pow (insert x F)" for F])
apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
done
qed
qed
subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close>
text\<open>Avigad's version, generalized to any commutative ring\<close>
theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n =
(\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
proof (induct n)
case 0 then show "?P 0" by simp
next
case (Suc n)
have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
by auto
have decomp2: "{0..n} = {0} Un {1..n}"
by auto
have "(a+b)^(n+1) =
(a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
using Suc.hyps by simp
also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
by (rule distrib_right)
also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
by (auto simp add: setsum_right_distrib ac_simps)
also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
(\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps
del:setsum_cl_ivl_Suc)
also have "\<dots> = a^(n+1) + b^(n+1) +
(\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
(\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
by (simp add: decomp2)
also have
"\<dots> = a^(n+1) + b^(n+1) +
(\<Sum>k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat)
also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
using decomp by (simp add: field_simps)
finally show "?P (Suc n)" by simp
qed
text\<open>Original version for the naturals\<close>
corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
using binomial_ring [of "int a" "int b" n]
by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
of_nat_setsum [symmetric]
of_nat_eq_iff of_nat_id)
lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
proof (induct n arbitrary: k rule: nat_less_induct)
fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
fact m" and kn: "k \<le> n"
let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
{ assume "n=0" then have ?ths using kn by simp }
moreover
{ assume "k=0" then have ?ths using kn by simp }
moreover
{ assume nk: "n=k" then have ?ths by simp }
moreover
{ fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
from n have mn: "m < n" by arith
from hm have hm': "h \<le> m" by arith
from hm h n kn have km: "k \<le> m" by arith
have "m - h = Suc (m - Suc h)" using h km hm by arith
with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
by simp
from n h th0
have "fact k * fact (n - k) * (n choose k) =
k * (fact h * fact (m - h) * (m choose h)) +
(m - h) * (fact k * fact (m - k) * (m choose k))"
by (simp add: field_simps)
also have "\<dots> = (k + (m - h)) * fact m"
using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
by (simp add: field_simps)
finally have ?ths using h n km by simp }
moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
using kn by presburger
ultimately show ?ths by blast
qed
lemma binomial_fact:
assumes kn: "k \<le> n"
shows "(of_nat (n choose k) :: 'a::field_char_0) =
(fact n) / (fact k * fact(n - k))"
using binomial_fact_lemma[OF kn]
apply (simp add: field_simps)
by (metis mult.commute of_nat_fact of_nat_mult)
lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
using binomial [of 1 "1" n]
by (simp add: numeral_2_eq_2)
lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
by (induct n) auto
lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
by (induct n) auto
lemma natsum_reverse_index:
fixes m::nat
shows "(\<And>k. m \<le> k \<Longrightarrow> k \<le> n \<Longrightarrow> g k = f (m + n - k)) \<Longrightarrow> (\<Sum>k=m..n. f k) = (\<Sum>k=m..n. g k)"
by (rule setsum.reindex_bij_witness[where i="\<lambda>k. m+n-k" and j="\<lambda>k. m+n-k"]) auto
text\<open>NW diagonal sum property\<close>
lemma sum_choose_diagonal:
assumes "m\<le>n" shows "(\<Sum>k=0..m. (n-k) choose (m-k)) = Suc n choose m"
proof -
have "(\<Sum>k=0..m. (n-k) choose (m-k)) = (\<Sum>k=0..m. (n-m+k) choose k)"
by (rule natsum_reverse_index) (simp add: assms)
also have "... = Suc (n-m+m) choose m"
by (rule sum_choose_lower)
also have "... = Suc n choose m" using assms
by simp
finally show ?thesis .
qed
subsection\<open>Pochhammer's symbol : generalized rising factorial\<close>
text \<open>See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"}\<close>
definition "pochhammer (a::'a::comm_semiring_1) n =
(if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
by (simp add: pochhammer_def)
lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
by (simp add: pochhammer_def)
lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
by (simp add: pochhammer_def)
lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
by (simp add: pochhammer_def)
lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
proof -
have "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
then show ?thesis by (simp add: field_simps)
qed
lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
proof -
have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
then show ?thesis by simp
qed
lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
proof (cases n)
case 0
then show ?thesis by simp
next
case (Suc n)
show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
qed
lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
proof (cases "n = 0")
case True
then show ?thesis by (simp add: pochhammer_Suc_setprod)
next
case False
have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
have eq: "insert 0 {1 .. n} = {0..n}" by auto
have **: "(\<Prod>n\<in>{1::nat..n}. a + of_nat n) = (\<Prod>n\<in>{0::nat..n - 1}. a + 1 + of_nat n)"
apply (rule setprod.reindex_cong [where l = Suc])
using False
apply (auto simp add: fun_eq_iff field_simps)
done
show ?thesis
apply (simp add: pochhammer_def)
unfolding setprod.insert [OF *, unfolded eq]
using ** apply (simp add: field_simps)
done
qed
lemma pochhammer_fact: "fact n = pochhammer 1 n"
unfolding fact_altdef
apply (cases n)
apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
apply (rule setprod.reindex_cong [where l = Suc])
apply (auto simp add: fun_eq_iff)
done
lemma pochhammer_of_nat_eq_0_lemma:
assumes "k > n"
shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
proof (cases "n = 0")
case True
then show ?thesis
using assms by (cases k) (simp_all add: pochhammer_rec)
next
case False
from assms obtain h where "k = Suc h" by (cases k) auto
then show ?thesis
by (simp add: pochhammer_Suc_setprod)
(metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1))
qed
lemma pochhammer_of_nat_eq_0_lemma':
assumes kn: "k \<le> n"
shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 0"
proof (cases k)
case 0
then show ?thesis by simp
next
case (Suc h)
then show ?thesis
apply (simp add: pochhammer_Suc_setprod)
using Suc kn apply (auto simp add: algebra_simps)
done
qed
lemma pochhammer_of_nat_eq_0_iff:
shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
(is "?l = ?r")
using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
by (auto simp add: not_le[symmetric])
lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
apply (auto simp add: pochhammer_of_nat_eq_0_iff)
apply (cases n)
apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)
apply (metis leD not_less_eq)
done
lemma pochhammer_eq_0_mono:
"pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
unfolding pochhammer_eq_0_iff by auto
lemma pochhammer_neq_0_mono:
"pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
unfolding pochhammer_eq_0_iff by auto
lemma pochhammer_minus:
"pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
proof (cases k)
case 0
then show ?thesis by simp
next
case (Suc h)
have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i=0..h. - 1)"
using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
by auto
show ?thesis
unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric]
by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"])
(auto simp: of_nat_diff)
qed
lemma pochhammer_minus':
"pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
unfolding pochhammer_minus[where b=b]
unfolding mult.assoc[symmetric]
unfolding power_add[symmetric]
by simp
lemma pochhammer_same: "pochhammer (- of_nat n) n =
((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * (fact n)"
unfolding pochhammer_minus
by (simp add: of_nat_diff pochhammer_fact)
subsection\<open>Generalized binomial coefficients\<close>
definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
where "a gchoose n =
(if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / (fact n))"
lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
by (simp_all add: gbinomial_def)
lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / (fact n)"
proof (cases "n = 0")
case True
then show ?thesis by simp
next
case False
from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
have eq: "(- (1::'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
by auto
from False show ?thesis
by (simp add: pochhammer_def gbinomial_def field_simps
eq setprod.distrib[symmetric])
qed
lemma binomial_gbinomial:
"of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)"
proof -
{ assume kn: "k > n"
then have ?thesis
by (subst binomial_eq_0[OF kn])
(simp add: gbinomial_pochhammer field_simps pochhammer_of_nat_eq_0_iff) }
moreover
{ assume "k=0" then have ?thesis by simp }
moreover
{ assume kn: "k \<le> n" and k0: "k\<noteq> 0"
from k0 obtain h where h: "k = Suc h" by (cases k) auto
from h
have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
by (subst setprod_constant) auto
have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
using h kn
by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
(auto simp: of_nat_diff)
have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
"{1..n - Suc h} \<inter> {n - h .. n} = {}" and
eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
using h kn by auto
from eq[symmetric]
have ?thesis using kn
apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
apply (simp add: pochhammer_Suc_setprod fact_altdef h
of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc)
unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
unfolding mult.assoc
unfolding setprod.distrib[symmetric]
apply simp
apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
apply (auto simp: of_nat_diff)
done
}
moreover
have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
ultimately show ?thesis by blast
qed
lemma gbinomial_1[simp]: "a gchoose 1 = a"
by (simp add: gbinomial_def)
lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
by (simp add: gbinomial_def)
lemma gbinomial_mult_1:
fixes a :: "'a :: field_char_0"
shows "a * (a gchoose n) =
of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r")
proof -
have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat n))"
unfolding gbinomial_pochhammer
pochhammer_Suc of_nat_mult right_diff_distrib power_Suc
apply (simp del: of_nat_Suc fact.simps)
apply (auto simp add: field_simps simp del: of_nat_Suc)
done
also have "\<dots> = ?l" unfolding gbinomial_pochhammer
by (simp add: field_simps)
finally show ?thesis ..
qed
lemma gbinomial_mult_1':
fixes a :: "'a :: field_char_0"
shows "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
by (simp add: mult.commute gbinomial_mult_1)
lemma gbinomial_Suc:
"a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / (fact (Suc k))"
by (simp add: gbinomial_def)
lemma gbinomial_mult_fact:
fixes a :: "'a::field_char_0"
shows
"fact (Suc k) * (a gchoose (Suc k)) =
(setprod (\<lambda>i. a - of_nat i) {0 .. k})"
by (simp_all add: gbinomial_Suc field_simps del: fact.simps)
lemma gbinomial_mult_fact':
fixes a :: "'a::field_char_0"
shows "(a gchoose (Suc k)) * fact (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
using gbinomial_mult_fact[of k a]
by (subst mult.commute)
lemma gbinomial_Suc_Suc:
fixes a :: "'a :: field_char_0"
shows "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
proof (cases k)
case 0
then show ?thesis by simp
next
case (Suc h)
have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
apply (rule setprod.reindex_cong [where l = Suc])
using Suc
apply auto
done
have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
(a gchoose Suc h) * (fact (Suc (Suc h))) +
(a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
by (simp add: Suc field_simps del: fact.simps)
also have "... = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) +
(\<Prod>i = 0..Suc h. a - of_nat i)"
by (metis fact.simps(2) gbinomial_mult_fact' of_nat_fact of_nat_id)
also have "... = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) +
(\<Prod>i = 0..Suc h. a - of_nat i)"
by (simp only: fact.simps(2) mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
also have "... = of_nat (Suc (Suc h)) * (\<Prod>i = 0..h. a - of_nat i) +
(\<Prod>i = 0..Suc h. a - of_nat i)"
by (metis gbinomial_mult_fact mult.commute)
also have "... = (\<Prod>i = 0..Suc h. a - of_nat i) +
(of_nat h * (\<Prod>i = 0..h. a - of_nat i) + 2 * (\<Prod>i = 0..h. a - of_nat i))"
by (simp add: field_simps)
also have "... =
((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0::nat..Suc h}. a - of_nat i)"
unfolding gbinomial_mult_fact'
by (simp add: comm_semiring_class.distrib field_simps Suc)
also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
by (simp add: field_simps Suc)
also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
using eq0
by (simp add: Suc setprod_nat_ivl_1_Suc)
also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
unfolding gbinomial_mult_fact ..
finally show ?thesis
by (metis fact_nonzero mult_cancel_left)
qed
lemma gbinomial_reduce_nat:
fixes a :: "'a :: field_char_0"
shows "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
lemma gchoose_row_sum_weighted:
fixes r :: "'a::field_char_0"
shows "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
proof (induct m)
case 0 show ?case by simp
next
case (Suc m)
from Suc show ?case
by (simp add: algebra_simps distrib gbinomial_mult_1)
qed
lemma binomial_symmetric:
assumes kn: "k \<le> n"
shows "n choose k = n choose (n - k)"
proof-
from kn have kn': "n - k \<le> n" by arith
from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
have "fact k * fact (n - k) * (n choose k) =
fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
then show ?thesis using kn by simp
qed
text\<open>Contributed by Manuel Eberl, generalised by LCP.
Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}\<close>
lemma gbinomial_altdef_of_nat:
fixes k :: nat
and x :: "'a :: {field_char_0,field}"
shows "x gchoose k = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
proof -
have "(x gchoose k) = (\<Prod>i<k. x - of_nat i) / of_nat (fact k)"
unfolding gbinomial_def
by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
also have "\<dots> = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
unfolding fact_eq_rev_setprod_nat of_nat_setprod
by (auto simp add: setprod_dividef intro!: setprod.cong of_nat_diff[symmetric])
finally show ?thesis .
qed
lemma gbinomial_ge_n_over_k_pow_k:
fixes k :: nat
and x :: "'a :: linordered_field"
assumes "of_nat k \<le> x"
shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose k"
proof -
have x: "0 \<le> x"
using assms of_nat_0_le_iff order_trans by blast
have "(x / of_nat k :: 'a) ^ k = (\<Prod>i<k. x / of_nat k :: 'a)"
by (simp add: setprod_constant)
also have "\<dots> \<le> x gchoose k"
unfolding gbinomial_altdef_of_nat
proof (safe intro!: setprod_mono)
fix i :: nat
assume ik: "i < k"
from assms have "x * of_nat i \<ge> of_nat (i * k)"
by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)" by arith
then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
using ik
by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult)
then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
unfolding of_nat_mult[symmetric] of_nat_le_iff .
with assms show "x / of_nat k \<le> (x - of_nat i) / (of_nat (k - i) :: 'a)"
using \<open>i < k\<close> by (simp add: field_simps)
qed (simp add: x zero_le_divide_iff)
finally show ?thesis .
qed
text\<open>Versions of the theorems above for the natural-number version of "choose"\<close>
lemma binomial_altdef_of_nat:
fixes n k :: nat
and x :: "'a :: {field_char_0,field}" --\<open>the point is to constrain @{typ 'a}\<close>
assumes "k \<le> n"
shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
using assms
by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
lemma binomial_ge_n_over_k_pow_k:
fixes k n :: nat
and x :: "'a :: linordered_field"
assumes "k \<le> n"
shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
by (simp add: assms gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
lemma binomial_le_pow:
assumes "r \<le> n"
shows "n choose r \<le> n ^ r"
proof -
have "n choose r \<le> fact n div fact (n - r)"
using \<open>r \<le> n\<close> by (subst binomial_fact_lemma[symmetric]) auto
with fact_div_fact_le_pow [OF assms] show ?thesis by auto
qed
lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
n choose k = fact n div (fact k * fact (n - k))"
by (subst binomial_fact_lemma [symmetric]) auto
lemma choose_dvd: "k \<le> n \<Longrightarrow> fact k * fact (n - k) dvd (fact n :: 'a :: {semiring_div,linordered_semidom})"
unfolding dvd_def
apply (rule exI [where x="of_nat (n choose k)"])
using binomial_fact_lemma [of k n, THEN arg_cong [where f = of_nat and 'b='a]]
apply (auto simp: of_nat_mult)
done
lemma fact_fact_dvd_fact:
"fact k * fact n dvd (fact (k+n) :: 'a :: {semiring_div,linordered_semidom})"
by (metis add.commute add_diff_cancel_left' choose_dvd le_add2)
lemma choose_mult_lemma:
"((m+r+k) choose (m+k)) * ((m+k) choose k) = ((m+r+k) choose k) * ((m+r) choose m)"
proof -
have "((m+r+k) choose (m+k)) * ((m+k) choose k) =
fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))"
by (simp add: assms binomial_altdef_nat)
also have "... = fact (m+r+k) div (fact r * (fact k * fact m))"
apply (subst div_mult_div_if_dvd)
apply (auto simp: algebra_simps fact_fact_dvd_fact)
apply (metis add.assoc add.commute fact_fact_dvd_fact)
done
also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))"
apply (subst div_mult_div_if_dvd [symmetric])
apply (auto simp add: algebra_simps)
apply (metis fact_fact_dvd_fact dvd.order.trans nat_mult_dvd_cancel_disj)
done
also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))"
apply (subst div_mult_div_if_dvd)
apply (auto simp: fact_fact_dvd_fact algebra_simps)
done
finally show ?thesis
by (simp add: binomial_altdef_nat mult.commute)
qed
text\<open>The "Subset of a Subset" identity\<close>
lemma choose_mult:
assumes "k\<le>m" "m\<le>n"
shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))"
using assms choose_mult_lemma [of "m-k" "n-m" k]
by simp
subsection \<open>Binomial coefficients\<close>
lemma choose_one: "(n::nat) choose 1 = n"
by simp
(*FIXME: messy and apparently unused*)
lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow>
(ALL n. P (Suc n) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (Suc k) \<longrightarrow>
P (Suc n) (Suc k))) \<longrightarrow> (ALL k <= n. P n k)"
apply (induct n)
apply auto
apply (case_tac "k = 0")
apply auto
apply (case_tac "k = Suc n")
apply auto
apply (metis Suc_le_eq fact.cases le_Suc_eq le_eq_less_or_eq)
done
lemma card_UNION:
assumes "finite A" and "\<forall>k \<in> A. finite k"
shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
(is "?lhs = ?rhs")
proof -
have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))" by simp
also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs")
by(subst setsum_right_distrib) simp
also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
using assms by(subst setsum.Sigma)(auto)
also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:UNIV. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
by (rule setsum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:\<Union>A. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
using assms by(auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
also have "\<dots> = (\<Sum>x\<in>\<Union>A. (\<Sum>I|I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I. (- 1) ^ (card I + 1)))"
using assms by(subst setsum.Sigma) auto
also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "setsum ?lhs _ = _")
proof(rule setsum.cong[OF refl])
fix x
assume x: "x \<in> \<Union>A"
def K \<equiv> "{X \<in> A. x \<in> X}"
with \<open>finite A\<close> have K: "finite K" by auto
let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
using assms by(auto intro!: inj_onI)
moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
using assms by(auto intro!: rev_image_eqI[where x="(card a, a)" for a]
simp add: card_gt_0_iff[folded Suc_le_eq]
dest: finite_subset intro: card_mono)
ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
by (rule setsum.reindex_cong [where l = snd]) fastforce
also have "\<dots> = (\<Sum>i=1..card A. (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. (- 1) ^ (i + 1)))"
using assms by(subst setsum.Sigma) auto
also have "\<dots> = (\<Sum>i=1..card A. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1))"
by(subst setsum_right_distrib) simp
also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))" (is "_ = ?rhs")
proof(rule setsum.mono_neutral_cong_right[rule_format])
show "{1..card K} \<subseteq> {1..card A}" using \<open>finite A\<close>
by(auto simp add: K_def intro: card_mono)
next
fix i
assume "i \<in> {1..card A} - {1..card K}"
hence i: "i \<le> card A" "card K < i" by auto
have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}"
by(auto simp add: K_def)
also have "\<dots> = {}" using \<open>finite A\<close> i
by(auto simp add: K_def dest: card_mono[rotated 1])
finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
by(simp only:) simp
next
fix i
have "(\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1) = (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)"
(is "?lhs = ?rhs")
by(rule setsum.cong)(auto simp add: K_def)
thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp
qed simp
also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}" using assms
by(auto simp add: card_eq_0_iff K_def dest: finite_subset)
hence "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
by(subst (2) setsum_head_Suc)(simp_all )
also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
using K by(subst n_subsets[symmetric]) simp_all
also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
by(subst setsum_right_distrib[symmetric]) simp
also have "\<dots> = - ((-1 + 1) ^ card K) + 1"
by(subst binomial_ring)(simp add: ac_simps)
also have "\<dots> = 1" using x K by(auto simp add: K_def card_gt_0_iff)
finally show "?lhs x = 1" .
qed
also have "nat \<dots> = card (\<Union>A)" by simp
finally show ?thesis ..
qed
text\<open>The number of nat lists of length @{text m} summing to @{text N} is
@{term "(N + m - 1) choose N"}:\<close>
lemma card_length_listsum_rec:
assumes "m\<ge>1"
shows "card {l::nat list. length l = m \<and> listsum l = N} =
(card {l. length l = (m - 1) \<and> listsum l = N} +
card {l. length l = m \<and> listsum l + 1 = N})"
(is "card ?C = (card ?A + card ?B)")
proof -
let ?A'="{l. length l = m \<and> listsum l = N \<and> hd l = 0}"
let ?B'="{l. length l = m \<and> listsum l = N \<and> hd l \<noteq> 0}"
let ?f ="\<lambda> l. 0#l"
let ?g ="\<lambda> l. (hd l + 1) # tl l"
have 1: "\<And>xs x. xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" by simp
have 2: "\<And>xs. (xs::nat list) \<noteq> [] \<Longrightarrow> listsum(tl xs) = listsum xs - hd xs"
by(auto simp add: neq_Nil_conv)
have f: "bij_betw ?f ?A ?A'"
apply(rule bij_betw_byWitness[where f' = tl])
using assms
by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
have 3: "\<And>xs:: nat list. xs \<noteq> [] \<Longrightarrow> hd xs + (listsum xs - hd xs) = listsum xs"
by (metis 1 listsum_simps(2) 2)
have g: "bij_betw ?g ?B ?B'"
apply(rule bij_betw_byWitness[where f' = "\<lambda> l. (hd l - 1) # tl l"])
using assms
by (auto simp: 2 length_0_conv[symmetric] intro!: 3
simp del: length_greater_0_conv length_0_conv)
{ fix M N :: nat have "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}"
using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto }
note fin = this
have fin_A: "finite ?A" using fin[of _ "N+1"]
by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"],
auto simp: member_le_listsum_nat less_Suc_eq_le)
have fin_B: "finite ?B"
by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"],
auto simp: member_le_listsum_nat less_Suc_eq_le fin)
have uni: "?C = ?A' \<union> ?B'" by auto
have disj: "?A' \<inter> ?B' = {}" by auto
have "card ?C = card(?A' \<union> ?B')" using uni by simp
also have "\<dots> = card ?A + card ?B"
using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
by presburger
finally show ?thesis .
qed
lemma card_length_listsum: --"By Holden Lee, tidied by Tobias Nipkow"
"card {l::nat list. size l = m \<and> listsum l = N} = (N + m - 1) choose N"
proof (cases m)
case 0 then show ?thesis
by (cases N) (auto simp: cong: conj_cong)
next
case (Suc m')
have m: "m\<ge>1" by (simp add: Suc)
then show ?thesis
proof (induct "N + m - 1" arbitrary: N m)
case 0 -- "In the base case, the only solution is [0]."
have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}"
by (auto simp: length_Suc_conv)
have "m=1 \<and> N=0" using 0 by linarith
then show ?case by simp
next
case (Suc k)
have c1: "card {l::nat list. size l = (m - 1) \<and> listsum l = N} =
(N + (m - 1) - 1) choose N"
proof cases
assume "m = 1"
with Suc.hyps have "N\<ge>1" by auto
with \<open>m = 1\<close> show ?thesis by (simp add: binomial_eq_0)
next
assume "m \<noteq> 1" thus ?thesis using Suc by fastforce
qed
from Suc have c2: "card {l::nat list. size l = m \<and> listsum l + 1 = N} =
(if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
proof -
have aux: "\<And>m n. n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" by arith
from Suc have "N>0 \<Longrightarrow>
card {l::nat list. size l = m \<and> listsum l + 1 = N} =
((N - 1) + m - 1) choose (N - 1)" by (simp add: aux)
thus ?thesis by auto
qed
from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> listsum l = N} +
card {l::nat list. size l = m \<and> listsum l + 1 = N}) = (N + m - 1) choose N"
by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
thus ?case using card_length_listsum_rec[OF Suc.prems] by auto
qed
qed
lemma Suc_times_binomial_add: -- \<open>by Lukas Bulwahn\<close>
"Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
proof -
have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b
using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat]
by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc)
have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) =
Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))"
by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd)
also have "\<dots> = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))"
by (simp only: div_mult_mult1)
also have "\<dots> = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))"
using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd)
finally show ?thesis
by (subst (1 2) binomial_altdef_nat)
(simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id)
qed
end