(* Title: HOL/Binomial.thy
ID: $Id$
Author: Lawrence C Paulson
Copyright 1997 University of Cambridge
*)
header {* Binomial Coefficients *}
theory Binomial
imports Main
begin
text {* This development is based on the work of Andy Gordon and
Florian Kammueller. *}
consts
binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
primrec
binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
binomial_Suc: "(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 binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
by (induct n) auto
declare binomial_0 [simp del] binomial_Suc [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 ==> (n choose k) > 0"
by (induct n k rule: diff_induct) simp_all
lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
apply (safe intro!: binomial_eq_0)
apply (erule contrapos_pp)
apply (simp add: zero_less_binomial)
done
lemma zero_less_binomial_iff: "(n choose k > 0) = (k\<le>n)"
by(simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric]
del:neq0_conv)
(*Might be more useful if re-oriented*)
lemma Suc_times_binomial_eq:
"!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
apply (induct n)
apply (simp add: binomial_0)
apply (case_tac k)
apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
binomial_eq_0)
done
text{*This is the well-known version, but it's harder to use because of the
need to reason about division.*}
lemma binomial_Suc_Suc_eq_times:
"k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
del: mult_Suc mult_Suc_right)
text{*Another version, with -1 instead of Suc.*}
lemma times_binomial_minus1_eq:
"[|k \<le> n; 0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
apply (simp split add: nat_diff_split, auto)
done
subsubsection {* Theorems about @{text "choose"} *}
text {*
\medskip Basic theorem about @{text "choose"}. By Florian
Kamm\"uller, tidied by LCP.
*}
lemma card_s_0_eq_empty:
"finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
apply (simp cong add: rev_conj_cong)
done
lemma choose_deconstruct: "finite M ==> x \<notin> M
==> {s. s <= insert x M & card(s) = Suc k}
= {s. s <= M & card(s) = Suc k} Un
{s. EX t. t <= M & card(t) = k & s = insert x t}"
apply safe
apply (auto intro: finite_subset [THEN card_insert_disjoint])
apply (drule_tac x = "xa - {x}" in spec)
apply (subgoal_tac "x \<notin> xa", auto)
apply (erule rev_mp, subst card_Diff_singleton)
apply (auto intro: finite_subset)
done
text{*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.*}
lemma constr_bij:
"[|finite A; x \<notin> A|] ==>
card {B. EX C. C <= A & card(C) = k & B = insert x C} =
card {B. B <= A & card(B) = k}"
apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
apply (auto elim!: equalityE simp add: inj_on_def)
apply (subst Diff_insert0, auto)
txt {* finiteness of the two sets *}
apply (rule_tac [2] B = "Pow (A)" in finite_subset)
apply (rule_tac B = "Pow (insert x A)" in finite_subset)
apply fast+
done
text {*
Main theorem: combinatorial statement about number of subsets of a set.
*}
lemma n_sub_lemma:
"!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
apply (induct k)
apply (simp add: card_s_0_eq_empty, atomize)
apply (rotate_tac -1, erule finite_induct)
apply (simp_all (no_asm_simp) cong add: conj_cong
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)", standard])
apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
done
theorem n_subsets:
"finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
by (simp add: n_sub_lemma)
text{* The binomial theorem (courtesy of Tobias Nipkow): *}
theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
proof (induct n)
case 0 thus ?case by simp
next
case (Suc n)
have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"
by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
have decomp2: "{0..n} = {0} \<union> {1..n}"
by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
using Suc by simp
also have "\<dots> = a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
by (rule nat_distrib)
also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
by (simp add: setsum_right_distrib mult_ac)
also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
(\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
del:setsum_cl_ivl_Suc)
also have "\<dots> = a^(n+1) + b^(n+1) +
(\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
(\<Sum>k=1..n. (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. (n+1 choose k) * a^k * b^(n+1-k))"
by (simp add: nat_distrib setsum_addf binomial.simps)
also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
using decomp by simp
finally show ?case by simp
qed
end