src/HOL/Library/Binomial.thy
 author chaieb Mon Jun 11 11:06:04 2007 +0200 (2007-06-11) changeset 23315 df3a7e9ebadb parent 21263 de65ce2bfb32 child 25112 98824cc791c0 permissions -rw-r--r--
tuned Proof
```     1 (*  Title:      HOL/Binomial.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson
```
```     4     Copyright   1997  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Binomial Coefficients *}
```
```     8
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```     9 theory Binomial
```
```    10 imports Main
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```    11 begin
```
```    12
```
```    13 text {* This development is based on the work of Andy Gordon and
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```    14   Florian Kammueller. *}
```
```    15
```
```    16 consts
```
```    17   binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat"      (infixl "choose" 65)
```
```    18 primrec
```
```    19   binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
```
```    20   binomial_Suc: "(Suc n choose k) =
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```    21                  (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
```
```    22
```
```    23 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
```
```    24   by (cases n) simp_all
```
```    25
```
```    26 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
```
```    27   by simp
```
```    28
```
```    29 lemma binomial_Suc_Suc [simp]:
```
```    30     "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
```
```    31   by simp
```
```    32
```
```    33 lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
```
```    34   by (induct n) auto
```
```    35
```
```    36 declare binomial_0 [simp del] binomial_Suc [simp del]
```
```    37
```
```    38 lemma binomial_n_n [simp]: "(n choose n) = 1"
```
```    39   by (induct n) (simp_all add: binomial_eq_0)
```
```    40
```
```    41 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
```
```    42   by (induct n) simp_all
```
```    43
```
```    44 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
```
```    45   by (induct n) simp_all
```
```    46
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```    47 lemma zero_less_binomial: "k \<le> n ==> 0 < (n choose k)"
```
```    48   by (induct n k rule: diff_induct) simp_all
```
```    49
```
```    50 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
```
```    51   apply (safe intro!: binomial_eq_0)
```
```    52   apply (erule contrapos_pp)
```
```    53   apply (simp add: zero_less_binomial)
```
```    54   done
```
```    55
```
```    56 lemma zero_less_binomial_iff: "(0 < n choose k) = (k\<le>n)"
```
```    57   by (simp add: linorder_not_less [symmetric] binomial_eq_0_iff [symmetric])
```
```    58
```
```    59 (*Might be more useful if re-oriented*)
```
```    60 lemma Suc_times_binomial_eq:
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```    61     "!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
```
```    62   apply (induct n)
```
```    63   apply (simp add: binomial_0)
```
```    64   apply (case_tac k)
```
```    65   apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
```
```    66     binomial_eq_0)
```
```    67   done
```
```    68
```
```    69 text{*This is the well-known version, but it's harder to use because of the
```
```    70   need to reason about division.*}
```
```    71 lemma binomial_Suc_Suc_eq_times:
```
```    72     "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
```
```    73   by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
```
```    74     del: mult_Suc mult_Suc_right)
```
```    75
```
```    76 text{*Another version, with -1 instead of Suc.*}
```
```    77 lemma times_binomial_minus1_eq:
```
```    78     "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
```
```    79   apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
```
```    80   apply (simp split add: nat_diff_split, auto)
```
```    81   done
```
```    82
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```    83
```
```    84 subsubsection {* Theorems about @{text "choose"} *}
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```    85
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```    86 text {*
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```    87   \medskip Basic theorem about @{text "choose"}.  By Florian
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```    88   Kamm\"uller, tidied by LCP.
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```    89 *}
```
```    90
```
```    91 lemma card_s_0_eq_empty:
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```    92     "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
```
```    93   apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
```
```    94   apply (simp cong add: rev_conj_cong)
```
```    95   done
```
```    96
```
```    97 lemma choose_deconstruct: "finite M ==> x \<notin> M
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```    98   ==> {s. s <= insert x M & card(s) = Suc k}
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```    99        = {s. s <= M & card(s) = Suc k} Un
```
```   100          {s. EX t. t <= M & card(t) = k & s = insert x t}"
```
```   101   apply safe
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```   102    apply (auto intro: finite_subset [THEN card_insert_disjoint])
```
```   103   apply (drule_tac x = "xa - {x}" in spec)
```
```   104   apply (subgoal_tac "x \<notin> xa", auto)
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```   105   apply (erule rev_mp, subst card_Diff_singleton)
```
```   106   apply (auto intro: finite_subset)
```
```   107   done
```
```   108
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```   109 text{*There are as many subsets of @{term A} having cardinality @{term k}
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```   110  as there are sets obtained from the former by inserting a fixed element
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```   111  @{term x} into each.*}
```
```   112 lemma constr_bij:
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```   113    "[|finite A; x \<notin> A|] ==>
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```   114     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
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```   115     card {B. B <= A & card(B) = k}"
```
```   116   apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
```
```   117        apply (auto elim!: equalityE simp add: inj_on_def)
```
```   118     apply (subst Diff_insert0, auto)
```
```   119    txt {* finiteness of the two sets *}
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```   120    apply (rule_tac [2] B = "Pow (A)" in finite_subset)
```
```   121    apply (rule_tac B = "Pow (insert x A)" in finite_subset)
```
```   122    apply fast+
```
```   123   done
```
```   124
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```   125 text {*
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```   126   Main theorem: combinatorial statement about number of subsets of a set.
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```   127 *}
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```   128
```
```   129 lemma n_sub_lemma:
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```   130     "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```   131   apply (induct k)
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```   132    apply (simp add: card_s_0_eq_empty, atomize)
```
```   133   apply (rotate_tac -1, erule finite_induct)
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```   134    apply (simp_all (no_asm_simp) cong add: conj_cong
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```   135      add: card_s_0_eq_empty choose_deconstruct)
```
```   136   apply (subst card_Un_disjoint)
```
```   137      prefer 4 apply (force simp add: constr_bij)
```
```   138     prefer 3 apply force
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```   139    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
```
```   140      finite_subset [of _ "Pow (insert x F)", standard])
```
```   141   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
```
```   142   done
```
```   143
```
```   144 theorem n_subsets:
```
```   145     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```   146   by (simp add: n_sub_lemma)
```
```   147
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```   148
```
```   149 text{* The binomial theorem (courtesy of Tobias Nipkow): *}
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```   150
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```   151 theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```   152 proof (induct n)
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```   153   case 0 thus ?case by simp
```
```   154 next
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```   155   case (Suc n)
```
```   156   have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"
```
```   157     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   158   have decomp2: "{0..n} = {0} \<union> {1..n}"
```
```   159     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   160   have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```   161     using Suc by simp
```
```   162   also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
```
```   163                    b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```   164     by (rule nat_distrib)
```
```   165   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
```
```   166                   (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
```
```   167     by (simp add: setsum_right_distrib mult_ac)
```
```   168   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
```
```   169                   (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
```
```   170     by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
```
```   171              del:setsum_cl_ivl_Suc)
```
```   172   also have "\<dots> = a^(n+1) + b^(n+1) +
```
```   173                   (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
```
```   174                   (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"
```
```   175     by (simp add: decomp2)
```
```   176   also have
```
```   177       "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
```
```   178     by (simp add: nat_distrib setsum_addf binomial.simps)
```
```   179   also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
```
```   180     using decomp by simp
```
```   181   finally show ?case by simp
```
```   182 qed
```
```   183
```
```   184 end
```