src/HOL/Library/Binomial.thy
 author haftmann Mon Jul 07 08:47:17 2008 +0200 (2008-07-07) changeset 27487 c8a6ce181805 parent 27368 9f90ac19e32b child 29694 2f2558d7bc3e permissions -rw-r--r--
absolute imports of HOL/*.thy theories
```     1 (*  Title:      HOL/Binomial.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson
```
```     4     Copyright   1997  University of Cambridge
```
```     5 *)
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```     6
```
```     7 header {* Binomial Coefficients *}
```
```     8
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```     9 theory Binomial
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```    10 imports Plain "~~/src/HOL/SetInterval"
```
```    11 begin
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```    12
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```    13 text {* This development is based on the work of Andy Gordon and
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```    14   Florian Kammueller. *}
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```    15
```
```    16 consts
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```    17   binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat"      (infixl "choose" 65)
```
```    18 primrec
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```    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
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```    23 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
```
```    24 by (cases n) simp_all
```
```    25
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```    26 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
```
```    27 by simp
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```    28
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```    29 lemma binomial_Suc_Suc [simp]:
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```    30   "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
```
```    31 by simp
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```    32
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```    33 lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
```
```    34 by (induct n) auto
```
```    35
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```    36 declare binomial_0 [simp del] binomial_Suc [simp del]
```
```    37
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```    38 lemma binomial_n_n [simp]: "(n choose n) = 1"
```
```    39 by (induct n) (simp_all add: binomial_eq_0)
```
```    40
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```    41 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
```
```    42 by (induct n) simp_all
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```    43
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```    44 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
```
```    45 by (induct n) simp_all
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```    46
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```    47 lemma zero_less_binomial: "k \<le> n ==> (n choose k) > 0"
```
```    48 by (induct n k rule: diff_induct) simp_all
```
```    49
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```    50 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
```
```    51 apply (safe intro!: binomial_eq_0)
```
```    52 apply (erule contrapos_pp)
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```    53 apply (simp add: zero_less_binomial)
```
```    54 done
```
```    55
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```    56 lemma zero_less_binomial_iff: "(n choose k > 0) = (k\<le>n)"
```
```    57 by(simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric]
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```    58         del:neq0_conv)
```
```    59
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```    60 (*Might be more useful if re-oriented*)
```
```    61 lemma Suc_times_binomial_eq:
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```    62   "!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
```
```    63 apply (induct n)
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```    64 apply (simp add: binomial_0)
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```    65 apply (case_tac k)
```
```    66 apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq
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```    67     binomial_eq_0)
```
```    68 done
```
```    69
```
```    70 text{*This is the well-known version, but it's harder to use because of the
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```    71   need to reason about division.*}
```
```    72 lemma binomial_Suc_Suc_eq_times:
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```    73     "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
```
```    74   by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
```
```    75     del: mult_Suc mult_Suc_right)
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```    76
```
```    77 text{*Another version, with -1 instead of Suc.*}
```
```    78 lemma times_binomial_minus1_eq:
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```    79     "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
```
```    80   apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
```
```    81   apply (simp split add: nat_diff_split, auto)
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```    82   done
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```    83
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```    84
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```    85 subsection {* Theorems about @{text "choose"} *}
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```    86
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```    87 text {*
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```    88   \medskip Basic theorem about @{text "choose"}.  By Florian
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```    89   Kamm\"uller, tidied by LCP.
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```    90 *}
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```    91
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```    92 lemma card_s_0_eq_empty:
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```    93     "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
```
```    94   apply (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
```
```    95   apply (simp cong add: rev_conj_cong)
```
```    96   done
```
```    97
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```    98 lemma choose_deconstruct: "finite M ==> x \<notin> M
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```    99   ==> {s. s <= insert x M & card(s) = Suc k}
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```   100        = {s. s <= M & card(s) = Suc k} Un
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```   101          {s. EX t. t <= M & card(t) = k & s = insert x t}"
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```   102   apply safe
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```   103    apply (auto intro: finite_subset [THEN card_insert_disjoint])
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```   104   apply (drule_tac x = "xa - {x}" in spec)
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```   105   apply (subgoal_tac "x \<notin> xa", auto)
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```   106   apply (erule rev_mp, subst card_Diff_singleton)
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```   107   apply (auto intro: finite_subset)
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```   108   done
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```   109
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```   110 text{*There are as many subsets of @{term A} having cardinality @{term k}
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```   111  as there are sets obtained from the former by inserting a fixed element
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```   112  @{term x} into each.*}
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```   113 lemma constr_bij:
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```   114    "[|finite A; x \<notin> A|] ==>
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```   115     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
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```   116     card {B. B <= A & card(B) = k}"
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```   117   apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
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```   118        apply (auto elim!: equalityE simp add: inj_on_def)
```
```   119     apply (subst Diff_insert0, auto)
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```   120    txt {* finiteness of the two sets *}
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```   121    apply (rule_tac [2] B = "Pow (A)" in finite_subset)
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```   122    apply (rule_tac B = "Pow (insert x A)" in finite_subset)
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```   123    apply fast+
```
```   124   done
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```   125
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```   126 text {*
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```   127   Main theorem: combinatorial statement about number of subsets of a set.
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```   128 *}
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```   129
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```   130 lemma n_sub_lemma:
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```   131     "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
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```   132   apply (induct k)
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```   133    apply (simp add: card_s_0_eq_empty, atomize)
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```   134   apply (rotate_tac -1, erule finite_induct)
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```   135    apply (simp_all (no_asm_simp) cong add: conj_cong
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```   136      add: card_s_0_eq_empty choose_deconstruct)
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```   137   apply (subst card_Un_disjoint)
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```   138      prefer 4 apply (force simp add: constr_bij)
```
```   139     prefer 3 apply force
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```   140    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
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```   141      finite_subset [of _ "Pow (insert x F)", standard])
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```   142   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
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```   143   done
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```   144
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```   145 theorem n_subsets:
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```   146     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
```
```   147   by (simp add: n_sub_lemma)
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```   148
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```   149
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```   150 text{* The binomial theorem (courtesy of Tobias Nipkow): *}
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```   151
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```   152 theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
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```   153 proof (induct n)
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```   154   case 0 thus ?case by simp
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```   155 next
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```   156   case (Suc n)
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```   157   have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"
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```   158     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
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```   159   have decomp2: "{0..n} = {0} \<union> {1..n}"
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```   160     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
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```   161   have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
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```   162     using Suc by simp
```
```   163   also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
```
```   164                    b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
```
```   165     by (rule nat_distrib)
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```   166   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
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```   167                   (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
```
```   168     by (simp add: setsum_right_distrib mult_ac)
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```   169   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
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```   170                   (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
```
```   171     by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
```
```   172              del:setsum_cl_ivl_Suc)
```
```   173   also have "\<dots> = a^(n+1) + b^(n+1) +
```
```   174                   (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
```
```   175                   (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"
```
```   176     by (simp add: decomp2)
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```   177   also have
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```   178       "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
```
```   179     by (simp add: nat_distrib setsum_addf binomial.simps)
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```   180   also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
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```   181     using decomp by simp
```
```   182   finally show ?case by simp
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```   183 qed
```
```   184
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```   185 end
```