--- a/src/HOL/Binomial.thy Wed Nov 08 22:24:54 2006 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,189 +0,0 @@
-(* Title: HOL/Binomial.thy
- ID: $Id$
- Author: Lawrence C Paulson
- Copyright 1997 University of Cambridge
-*)
-
-header{*Binomial Coefficients*}
-
-theory Binomial
-imports GCD
-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 [rule_format]: "\<forall>k. n < k --> (n choose k) = 0"
-apply (induct "n")
-apply auto
-done
-
-declare binomial_0 [simp del] binomial_Suc [simp del]
-
-lemma binomial_n_n [simp]: "(n choose n) = 1"
-apply (induct "n")
-apply (simp_all add: binomial_eq_0)
-done
-
-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 [rule_format]: "k \<le> n --> 0 < (n choose k)"
-by (rule_tac m = n and n = k in 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: "(0 < n choose k) = (k\<le>n)"
-by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
-
-(*Might be more useful if re-oriented*)
-lemma Suc_times_binomial_eq [rule_format]:
- "\<forall>k. k \<le> n --> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
-apply (induct "n")
-apply (simp add: binomial_0, clarify)
-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