src/HOL/Number_Theory/Binomial.thy
author haftmann
Sun, 09 Nov 2014 10:03:18 +0100
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(*  Title:      HOL/Number_Theory/Binomial.thy
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    Authors:    Lawrence C. Paulson, Jeremy Avigad, Tobias Nipkow
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Defines the "choose" function, and establishes basic properties.
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*)
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section {* Binomial *}
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theory Binomial
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imports Cong Fact Complex_Main
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begin
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text {* This development is based on the work of Andy Gordon and
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  Florian Kammueller. *}
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subsection {* Basic definitions and lemmas *}
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primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
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where
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  "0 choose k = (if k = 0 then 1 else 0)"
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| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
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lemma binomial_n_0 [simp]: "(n choose 0) = 1"
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  by (cases n) simp_all
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lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
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  by simp
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lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
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  by simp
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lemma choose_reduce_nat: 
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  "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
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    (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
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  by (metis Suc_diff_1 binomial.simps(2) neq0_conv)
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lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
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  by (induct n arbitrary: k) auto
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declare binomial.simps [simp del]
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lemma binomial_n_n [simp]: "n choose n = 1"
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  by (induct n) (simp_all add: binomial_eq_0)
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lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
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  by (induct n) simp_all
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lemma binomial_1 [simp]: "n choose Suc 0 = n"
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  by (induct n) simp_all
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lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
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  by (induct n k rule: diff_induct) simp_all
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lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
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  by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
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lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
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  by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
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lemma Suc_times_binomial_eq:
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  "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
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  apply (induct n arbitrary: k, simp add: binomial.simps)
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  apply (case_tac k)
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   apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
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  done
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text{*The absorption property*}
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lemma Suc_times_binomial:
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  "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
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  using Suc_times_binomial_eq by auto
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text{*This is the well-known version of absorption, but it's harder to use because of the
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  need to reason about division.*}
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lemma binomial_Suc_Suc_eq_times:
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    "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
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  by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
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text{*Another version of absorption, with -1 instead of Suc.*}
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lemma times_binomial_minus1_eq:
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  "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
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  using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
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  by (auto split add: nat_diff_split)
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subsection {* Combinatorial theorems involving @{text "choose"} *}
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text {*By Florian Kamm\"uller, tidied by LCP.*}
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lemma card_s_0_eq_empty: "finite A \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 1"
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  by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
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lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
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    {s. s \<subseteq> insert x M \<and> card s = Suc k} =
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    {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}"
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  apply safe
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     apply (auto intro: finite_subset [THEN card_insert_disjoint])
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  by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if 
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     card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff)
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lemma finite_bex_subset [simp]:
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  assumes "finite B"
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    and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
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  shows "finite {x. \<exists>A \<subseteq> B. P x A}"
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  by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets)
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text{*There are as many subsets of @{term A} having cardinality @{term k}
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 as there are sets obtained from the former by inserting a fixed element
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 @{term x} into each.*}
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lemma constr_bij:
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   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow>
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    card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
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    card {B. B \<subseteq> A & card(B) = k}"
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  apply (rule card_bij_eq [where f = "\<lambda>s. s - {x}" and g = "insert x"])
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  apply (auto elim!: equalityE simp add: inj_on_def)
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  apply (metis card_Diff_singleton_if finite_subset in_mono)
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  done
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   118
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   119
text {*
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   120
  Main theorem: combinatorial statement about number of subsets of a set.
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   121
*}
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   122
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   123
theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   124
proof (induct k arbitrary: A)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   125
  case 0 then show ?case by (simp add: card_s_0_eq_empty)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   126
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   127
  case (Suc k)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   128
  show ?case using `finite A`
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   129
  proof (induct A)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   130
    case empty show ?case by (simp add: card_s_0_eq_empty)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   131
  next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   132
    case (insert x A)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   133
    then show ?case using Suc.hyps
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   134
      apply (simp add: card_s_0_eq_empty choose_deconstruct)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   135
      apply (subst card_Un_disjoint)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   136
         prefer 4 apply (force simp add: constr_bij)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   137
        prefer 3 apply force
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   138
       prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
55143
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 55130
diff changeset
   139
         finite_subset [of _ "Pow (insert x F)" for F])
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   140
      apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   141
      done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   142
  qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   143
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   144
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   145
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   146
subsection {* The binomial theorem (courtesy of Tobias Nipkow): *}
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   147
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   148
text{* Avigad's version, generalized to any commutative ring *}
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   149
theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n = 
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   150
  (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   151
proof (induct n)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   152
  case 0 then show "?P 0" by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   153
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   154
  case (Suc n)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   155
  have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   156
    by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   157
  have decomp2: "{0..n} = {0} Un {1..n}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   158
    by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   159
  have "(a+b)^(n+1) = 
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   160
      (a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   161
    using Suc.hyps by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   162
  also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   163
                   b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   164
    by (rule distrib)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   165
  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   166
                  (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   167
    by (auto simp add: setsum_right_distrib ac_simps)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   168
  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   169
                  (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   170
    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps  
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   171
        del:setsum_cl_ivl_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   172
  also have "\<dots> = a^(n+1) + b^(n+1) +
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   173
                  (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   174
                  (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   175
    by (simp add: decomp2)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   176
  also have
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   177
      "\<dots> = a^(n+1) + b^(n+1) + 
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   178
            (\<Sum>k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   179
    by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   180
  also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   181
    using decomp by (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   182
  finally show "?P (Suc n)" by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   183
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   184
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   185
text{* Original version for the naturals *}
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   186
corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   187
    using binomial_ring [of "int a" "int b" n]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   188
  by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   189
           of_nat_setsum [symmetric]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   190
           of_nat_eq_iff of_nat_id)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   191
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   192
lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   193
  using binomial [of 1 "1" n]
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   194
  by (simp add: numeral_2_eq_2)
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   195
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   196
lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   197
  by (induct n) auto
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   198
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   199
lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   200
  by (induct n) auto
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   201
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   202
lemma natsum_reverse_index:
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   203
  fixes m::nat
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   204
  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)"
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   205
  by (rule setsum.reindex_bij_witness[where i="\<lambda>k. m+n-k" and j="\<lambda>k. m+n-k"]) auto
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   206
58841
e16712bb1d41 Some comments and a new version of a result
paulson <lp15@cam.ac.uk>
parents: 58833
diff changeset
   207
text{*NW diagonal sum property*}
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   208
lemma sum_choose_diagonal:
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   209
  assumes "m\<le>n" shows "(\<Sum>k=0..m. (n-k) choose (m-k)) = Suc n choose m"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   210
proof -
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   211
  have "(\<Sum>k=0..m. (n-k) choose (m-k)) = (\<Sum>k=0..m. (n-m+k) choose k)"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   212
    by (rule natsum_reverse_index) (simp add: assms)
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   213
  also have "... = Suc (n-m+m) choose m"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   214
    by (rule sum_choose_lower)
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   215
  also have "... = Suc n choose m" using assms
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   216
    by simp
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   217
  finally show ?thesis .
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   218
qed
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   219
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   220
subsection{* Pochhammer's symbol : generalized rising factorial *}
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   221
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   222
text {* See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"} *}
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   223
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   224
definition "pochhammer (a::'a::comm_semiring_1) n =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   225
  (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   226
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   227
lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   228
  by (simp add: pochhammer_def)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   229
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   230
lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   231
  by (simp add: pochhammer_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   232
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   233
lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   234
  by (simp add: pochhammer_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   235
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   236
lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   237
  by (simp add: pochhammer_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   238
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   239
lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   240
proof -
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   241
  have "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   242
  then show ?thesis by (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   243
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   244
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   245
lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   246
proof -
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   247
  have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   248
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   249
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   250
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   251
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   252
lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   253
proof (cases n)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   254
  case 0
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   255
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   256
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   257
  case (Suc n)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   258
  show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   259
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   260
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   261
lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   262
proof (cases "n = 0")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   263
  case True
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   264
  then show ?thesis by (simp add: pochhammer_Suc_setprod)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   265
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   266
  case False
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   267
  have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   268
  have eq: "insert 0 {1 .. n} = {0..n}" by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   269
  have **: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) = (\<Prod>n\<in>{0\<Colon>nat..n - 1}. a + 1 + of_nat n)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   270
    apply (rule setprod.reindex_cong [where l = Suc])
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   271
    using False
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   272
    apply (auto simp add: fun_eq_iff field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   273
    done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   274
  show ?thesis
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   275
    apply (simp add: pochhammer_def)
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   276
    unfolding setprod.insert [OF *, unfolded eq]
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   277
    using ** apply (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   278
    done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   279
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   280
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   281
lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   282
  unfolding fact_altdef_nat
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   283
  apply (cases n)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   284
   apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   285
  apply (rule setprod.reindex_cong [where l = Suc])
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   286
    apply (auto simp add: fun_eq_iff)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   287
  done
44872
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   288
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   289
lemma pochhammer_of_nat_eq_0_lemma:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   290
  assumes "k > n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   291
  shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   292
proof (cases "n = 0")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   293
  case True
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   294
  then show ?thesis
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   295
    using assms by (cases k) (simp_all add: pochhammer_rec)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   296
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   297
  case False
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   298
  from assms obtain h where "k = Suc h" by (cases k) auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   299
  then show ?thesis
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   300
    by (simp add: pochhammer_Suc_setprod)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   301
       (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1))
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   302
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   303
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   304
lemma pochhammer_of_nat_eq_0_lemma':
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   305
  assumes kn: "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   306
  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 0"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   307
proof (cases k)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   308
  case 0
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   309
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   310
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   311
  case (Suc h)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   312
  then show ?thesis
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   313
    apply (simp add: pochhammer_Suc_setprod)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   314
    using Suc kn apply (auto simp add: algebra_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   315
    done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   316
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   317
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   318
lemma pochhammer_of_nat_eq_0_iff:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   319
  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   320
  (is "?l = ?r")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   321
  using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   322
    pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   323
  by (auto simp add: not_le[symmetric])
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   324
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   325
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   326
lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   327
  apply (auto simp add: pochhammer_of_nat_eq_0_iff)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   328
  apply (cases n)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   329
   apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   330
  apply (metis leD not_less_eq)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   331
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   332
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   333
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   334
lemma pochhammer_eq_0_mono:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   335
  "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   336
  unfolding pochhammer_eq_0_iff by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   337
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   338
lemma pochhammer_neq_0_mono:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   339
  "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   340
  unfolding pochhammer_eq_0_iff by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   341
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   342
lemma pochhammer_minus:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   343
  assumes kn: "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   344
  shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   345
proof (cases k)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   346
  case 0
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   347
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   348
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   349
  case (Suc h)
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   350
  have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i=0..h. - 1)"
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   351
    using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   352
    by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   353
  show ?thesis
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   354
    unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric]
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   355
    by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"])
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   356
       (auto simp: of_nat_diff)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   357
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   358
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   359
lemma pochhammer_minus':
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   360
  assumes kn: "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   361
  shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   362
  unfolding pochhammer_minus[OF kn, where b=b]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   363
  unfolding mult.assoc[symmetric]
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   364
  unfolding power_add[symmetric]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   365
  by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   366
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   367
lemma pochhammer_same: "pochhammer (- of_nat n) n =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   368
    ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   369
  unfolding pochhammer_minus[OF le_refl[of n]]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   370
  by (simp add: of_nat_diff pochhammer_fact)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   371
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   372
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   373
subsection{* Generalized binomial coefficients *}
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   374
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   375
definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   376
  where "a gchoose n =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   377
    (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   378
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   379
lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   380
  apply (simp_all add: gbinomial_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   381
  apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   382
   apply (simp del:setprod_zero_iff)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   383
  apply simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   384
  done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   385
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   386
lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   387
proof (cases "n = 0")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   388
  case True
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   389
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   390
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   391
  case False
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   392
  from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   393
  have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   394
    by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   395
  from False show ?thesis
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   396
    by (simp add: pochhammer_def gbinomial_def field_simps
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   397
      eq setprod.distrib[symmetric])
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   398
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   399
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   400
lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   401
proof (induct n arbitrary: k rule: nat_less_induct)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   402
  fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   403
                      fact m" and kn: "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   404
  let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   405
  { assume "n=0" then have ?ths using kn by simp }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   406
  moreover
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   407
  { assume "k=0" then have ?ths using kn by simp }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   408
  moreover
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   409
  { assume nk: "n=k" then have ?ths by simp }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   410
  moreover
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   411
  { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   412
    from n have mn: "m < n" by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   413
    from hm have hm': "h \<le> m" by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   414
    from hm h n kn have km: "k \<le> m" by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   415
    have "m - h = Suc (m - Suc h)" using  h km hm by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   416
    with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   417
      by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   418
    from n h th0
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   419
    have "fact k * fact (n - k) * (n choose k) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   420
        k * (fact h * fact (m - h) * (m choose h)) + 
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   421
        (m - h) * (fact k * fact (m - k) * (m choose k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   422
      by (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   423
    also have "\<dots> = (k + (m - h)) * fact m"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   424
      using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   425
      by (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   426
    finally have ?ths using h n km by simp }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   427
  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   428
    using kn by presburger
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   429
  ultimately show ?ths by blast
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   430
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   431
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   432
lemma binomial_fact:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   433
  assumes kn: "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   434
  shows "(of_nat (n choose k) :: 'a::field_char_0) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   435
    of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   436
  using binomial_fact_lemma[OF kn]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   437
  by (simp add: field_simps of_nat_mult [symmetric])
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   438
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   439
lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   440
proof -
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   441
  { assume kn: "k > n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   442
    then have ?thesis
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   443
      by (subst binomial_eq_0[OF kn]) 
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   444
         (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   445
  moreover
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   446
  { assume "k=0" then have ?thesis by simp }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   447
  moreover
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   448
  { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   449
    from k0 obtain h where h: "k = Suc h" by (cases k) auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   450
    from h
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   451
    have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   452
      by (subst setprod_constant) auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   453
    have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   454
        using h kn
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   455
      by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   456
         (auto simp: of_nat_diff)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   457
    have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   458
        "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   459
        eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   460
      using h kn by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   461
    from eq[symmetric]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   462
    have ?thesis using kn
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   463
      apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   464
        gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   465
      apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   466
        of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc)
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   467
      unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   468
      unfolding mult.assoc[symmetric]
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   469
      unfolding setprod.distrib[symmetric]
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   470
      apply simp
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   471
      apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 56178
diff changeset
   472
      apply (auto simp: of_nat_diff)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   473
      done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   474
  }
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   475
  moreover
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   476
  have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   477
  ultimately show ?thesis by blast
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   478
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   479
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   480
lemma gbinomial_1[simp]: "a gchoose 1 = a"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   481
  by (simp add: gbinomial_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   482
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   483
lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   484
  by (simp add: gbinomial_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   485
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   486
lemma gbinomial_mult_1:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   487
  "a * (a gchoose n) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   488
    of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?r")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   489
proof -
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   490
  have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   491
    unfolding gbinomial_pochhammer
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   492
      pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   493
    by (simp add:  field_simps del: of_nat_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   494
  also have "\<dots> = ?l" unfolding gbinomial_pochhammer
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   495
    by (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   496
  finally show ?thesis ..
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   497
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   498
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   499
lemma gbinomial_mult_1':
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   500
    "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   501
  by (simp add: mult.commute gbinomial_mult_1)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   502
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   503
lemma gbinomial_Suc:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   504
    "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   505
  by (simp add: gbinomial_def)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   506
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   507
lemma gbinomial_mult_fact:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   508
  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   509
    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   510
  by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   511
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   512
lemma gbinomial_mult_fact':
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   513
  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   514
    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   515
  using gbinomial_mult_fact[of k a]
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   516
  by (subst mult.commute)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   517
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   518
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   519
lemma gbinomial_Suc_Suc:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   520
  "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   521
proof (cases k)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   522
  case 0
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   523
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   524
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   525
  case (Suc h)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   526
  have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   527
    apply (rule setprod.reindex_cong [where l = Suc])
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   528
      using Suc
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   529
      apply auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   530
    done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   531
  have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   532
    ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   533
    apply (simp add: Suc field_simps del: fact_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   534
    unfolding gbinomial_mult_fact'
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   535
    apply (subst fact_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   536
    unfolding of_nat_mult
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   537
    apply (subst mult.commute)
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   538
    unfolding mult.assoc
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   539
    unfolding gbinomial_mult_fact
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   540
    apply (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   541
    done
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   542
  also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   543
    unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   544
    by (simp add: field_simps Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   545
  also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   546
    using eq0
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   547
    by (simp add: Suc setprod_nat_ivl_1_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   548
  also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   549
    unfolding gbinomial_mult_fact ..
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   550
  finally show ?thesis by (simp del: fact_Suc)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   551
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   552
58833
09974789e483 choose_reduce_nat: re-ordered operands
paulson <lp15@cam.ac.uk>
parents: 58713
diff changeset
   553
lemma gbinomial_reduce_nat:
09974789e483 choose_reduce_nat: re-ordered operands
paulson <lp15@cam.ac.uk>
parents: 58713
diff changeset
   554
  "0 < k \<Longrightarrow> (a::'a::field_char_0) gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
09974789e483 choose_reduce_nat: re-ordered operands
paulson <lp15@cam.ac.uk>
parents: 58713
diff changeset
   555
by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
09974789e483 choose_reduce_nat: re-ordered operands
paulson <lp15@cam.ac.uk>
parents: 58713
diff changeset
   556
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   557
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   558
lemma binomial_symmetric:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   559
  assumes kn: "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   560
  shows "n choose k = n choose (n - k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   561
proof-
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   562
  from kn have kn': "n - k \<le> n" by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   563
  from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   564
  have "fact k * fact (n - k) * (n choose k) =
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   565
    fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   566
  then show ?thesis using kn by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   567
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   568
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   569
(* Contributed by Manuel Eberl *)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   570
(* Alternative definition of the binomial coefficient as \<Prod>i<k. (n - i) / (k - i) *)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   571
lemma binomial_altdef_of_nat:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   572
  fixes n k :: nat
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   573
    and x :: "'a :: {field_char_0,field_inverse_zero}"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   574
  assumes "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   575
  shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   576
proof (cases "0 < k")
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   577
  case True
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   578
  then have "(of_nat (n choose k) :: 'a) = (\<Prod>i<k. of_nat n - of_nat i) / of_nat (fact k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   579
    unfolding binomial_gbinomial gbinomial_def
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   580
    by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   581
  also have "\<dots> = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   582
    using `k \<le> n` unfolding fact_eq_rev_setprod_nat of_nat_setprod
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   583
    by (auto simp add: setprod_dividef intro!: setprod.cong of_nat_diff[symmetric])
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   584
  finally show ?thesis .
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   585
next
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   586
  case False
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   587
  then show ?thesis by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   588
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   589
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   590
lemma binomial_ge_n_over_k_pow_k:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   591
  fixes k n :: nat
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   592
    and x :: "'a :: linordered_field_inverse_zero"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   593
  assumes "0 < k"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   594
    and "k \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   595
  shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   596
proof -
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   597
  have "(of_nat n / of_nat k :: 'a) ^ k = (\<Prod>i<k. of_nat n / of_nat k :: 'a)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   598
    by (simp add: setprod_constant)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   599
  also have "\<dots> \<le> of_nat (n choose k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   600
    unfolding binomial_altdef_of_nat[OF `k\<le>n`]
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   601
  proof (safe intro!: setprod_mono)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   602
    fix i :: nat
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   603
    assume  "i < k"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   604
    from assms have "n * i \<ge> i * k" by simp
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   605
    then have "n * k - n * i \<le> n * k - i * k" by arith
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   606
    then have "n * (k - i) \<le> (n - i) * k"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   607
      by (simp add: diff_mult_distrib2 mult.commute)
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   608
    then have "of_nat n * of_nat (k - i) \<le> of_nat (n - i) * (of_nat k :: 'a)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   609
      unfolding of_nat_mult[symmetric] of_nat_le_iff .
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   610
    with assms show "of_nat n / of_nat k \<le> of_nat (n - i) / (of_nat (k - i) :: 'a)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   611
      using `i < k` by (simp add: field_simps)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   612
  qed (simp add: zero_le_divide_iff)
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   613
  finally show ?thesis .
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   614
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   615
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   616
lemma binomial_le_pow:
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   617
  assumes "r \<le> n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   618
  shows "n choose r \<le> n ^ r"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   619
proof -
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   620
  have "n choose r \<le> fact n div fact (n - r)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   621
    using `r \<le> n` by (subst binomial_fact_lemma[symmetric]) auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   622
  with fact_div_fact_le_pow [OF assms] show ?thesis by auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   623
qed
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   624
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   625
lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   626
    n choose k = fact n div (fact k * fact (n - k))"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   627
 by (subst binomial_fact_lemma [symmetric]) auto
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   628
58713
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   629
lemma choose_dvd_nat: "(k::nat) \<le> n \<Longrightarrow> fact k * fact (n - k) dvd fact n"
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   630
by (metis binomial_fact_lemma dvd_def)
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   631
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   632
lemma choose_dvd_int: 
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   633
  assumes "(0::int) <= k" and "k <= n"
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   634
  shows "fact k * fact (n - k) dvd fact n"
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   635
  apply (subst tsub_eq [symmetric], rule assms)
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   636
  apply (rule choose_dvd_nat [transferred])
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   637
  using assms apply auto
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   638
  done
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   639
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   640
lemma fact_fact_dvd_fact: fixes k::nat shows "fact k * fact n dvd fact (n + k)"
58713
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   641
by (metis add.commute add_diff_cancel_left' choose_dvd_nat le_add2)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   642
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   643
lemma choose_mult_lemma:
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   644
     "((m+r+k) choose (m+k)) * ((m+k) choose k) = ((m+r+k) choose k) * ((m+r) choose m)"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   645
proof -
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   646
  have "((m+r+k) choose (m+k)) * ((m+k) choose k) =
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   647
        fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   648
    by (simp add: assms binomial_altdef_nat)
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   649
  also have "... = fact (m+r+k) div (fact r * (fact k * fact m))"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   650
    apply (subst div_mult_div_if_dvd)
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   651
    apply (auto simp: fact_fact_dvd_fact)
58713
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   652
    apply (metis add.assoc add.commute fact_fact_dvd_fact)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   653
    done
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   654
  also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   655
    apply (subst div_mult_div_if_dvd [symmetric])
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   656
    apply (auto simp: fact_fact_dvd_fact)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   657
    apply (metis dvd_trans dvd.dual_order.refl fact_fact_dvd_fact mult_dvd_mono mult.left_commute)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   658
    done
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   659
  also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   660
    apply (subst div_mult_div_if_dvd)
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   661
    apply (auto simp: fact_fact_dvd_fact)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   662
    apply(metis mult.left_commute)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   663
    done
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   664
  finally show ?thesis
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   665
    by (simp add: binomial_altdef_nat mult.commute)
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   666
qed
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   667
58917
a3be9a47e2d7 Tidying up. Removing unnecessary conditions from some theorems.
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   668
text{*The "Subset of a Subset" identity*}
56178
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   669
lemma choose_mult:
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   670
  assumes "k\<le>m" "m\<le>n"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   671
    shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))"
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   672
using assms choose_mult_lemma [of "m-k" "n-m" k]
2a6f58938573 a few new theorems
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
   673
by simp
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   674
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   675
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   676
subsection {* Binomial coefficients *}
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   677
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   678
lemma choose_one: "(n::nat) choose 1 = n"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   679
  by simp
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   680
58713
572a5a870c84 tweaked
paulson <lp15@cam.ac.uk>
parents: 58410
diff changeset
   681
(*FIXME: messy and apparently unused*)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   682
lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow> 
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   683
    (ALL n. P (Suc n) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (Suc k) \<longrightarrow>
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   684
    P (Suc n) (Suc k))) \<longrightarrow> (ALL k <= n. P n k)"
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   685
  apply (induct n)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   686
  apply auto
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   687
  apply (case_tac "k = 0")
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   688
  apply auto
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   689
  apply (case_tac "k = Suc n")
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   690
  apply auto
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   691
  apply (metis Suc_le_eq fact_nat.cases le_Suc_eq le_eq_less_or_eq)
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 41340
diff changeset
   692
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   693
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   694
lemma card_UNION:
51292
8a635bf2c86c use lemma from Big_Operators
Andreas Lochbihler
parents: 51291
diff changeset
   695
  assumes "finite A" and "\<forall>k \<in> A. finite k"
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   696
  shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   697
  (is "?lhs = ?rhs")
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   698
proof -
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   699
  have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))" by simp
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   700
  also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs")
51292
8a635bf2c86c use lemma from Big_Operators
Andreas Lochbihler
parents: 51291
diff changeset
   701
    by(subst setsum_right_distrib) simp
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   702
  also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   703
    using assms by(subst setsum.Sigma)(auto)
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   704
  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))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   705
    by (rule setsum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   706
  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))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   707
    using assms by(auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   708
  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)))" 
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   709
    using assms by(subst setsum.Sigma) auto
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   710
  also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "setsum ?lhs _ = _")
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   711
  proof(rule setsum.cong[OF refl])
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   712
    fix x
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   713
    assume x: "x \<in> \<Union>A"
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   714
    def K \<equiv> "{X \<in> A. x \<in> X}"
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   715
    with `finite A` have K: "finite K" by auto
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   716
    let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   717
    have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   718
      using assms by(auto intro!: inj_onI)
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   719
    moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
55143
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 55130
diff changeset
   720
      using assms by(auto intro!: rev_image_eqI[where x="(card a, a)" for a]
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 55130
diff changeset
   721
        simp add: card_gt_0_iff[folded Suc_le_eq]
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 55130
diff changeset
   722
        dest: finite_subset intro: card_mono)
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   723
    ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   724
      by (rule setsum.reindex_cong [where l = snd]) fastforce
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   725
    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)))"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   726
      using assms by(subst setsum.Sigma) auto
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   727
    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))"
51292
8a635bf2c86c use lemma from Big_Operators
Andreas Lochbihler
parents: 51291
diff changeset
   728
      by(subst setsum_right_distrib) simp
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   729
    also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))" (is "_ = ?rhs")
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   730
    proof(rule setsum.mono_neutral_cong_right[rule_format])
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   731
      show "{1..card K} \<subseteq> {1..card A}" using `finite A`
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   732
        by(auto simp add: K_def intro: card_mono)
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   733
    next
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   734
      fix i
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   735
      assume "i \<in> {1..card A} - {1..card K}"
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   736
      hence i: "i \<le> card A" "card K < i" by auto
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   737
      have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}" 
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   738
        by(auto simp add: K_def)
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   739
      also have "\<dots> = {}" using `finite A` i
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   740
        by(auto simp add: K_def dest: card_mono[rotated 1])
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   741
      finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   742
        by(simp only:) simp
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   743
    next
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   744
      fix i
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   745
      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)"
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   746
        (is "?lhs = ?rhs")
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   747
        by(rule setsum.cong)(auto simp add: K_def)
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   748
      thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   749
    qed simp
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   750
    also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}" using assms
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   751
      by(auto simp add: card_eq_0_iff K_def dest: finite_subset)
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   752
    hence "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   753
      by(subst (2) setsum_head_Suc)(simp_all )
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   754
    also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
55130
70db8d380d62 Restored Suc rather than +1, and using Library/Binimial
paulson <lp15@cam.ac.uk>
parents: 53374
diff changeset
   755
      using K by(subst n_subsets[symmetric]) simp_all
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58194
diff changeset
   756
    also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
51292
8a635bf2c86c use lemma from Big_Operators
Andreas Lochbihler
parents: 51291
diff changeset
   757
      by(subst setsum_right_distrib[symmetric]) simp
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   758
    also have "\<dots> =  - ((-1 + 1) ^ card K) + 1"
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   759
      by(subst binomial_ring)(simp add: ac_simps)
51291
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   760
    also have "\<dots> = 1" using x K by(auto simp add: K_def card_gt_0_iff)
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   761
    finally show "?lhs x = 1" .
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   762
  qed
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   763
  also have "nat \<dots> = card (\<Union>A)" by simp
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   764
  finally show ?thesis ..
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   765
qed
c2b452628afa add inclusion/exclusion lemma
Andreas Lochbihler
parents: 49962
diff changeset
   766
58193
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   767
text{* The number of nat lists of length @{text m} summing to @{text N} is
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   768
@{term "(N + m - 1) choose N"}: *} 
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   769
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   770
lemma card_length_listsum_rec:
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   771
  assumes "m\<ge>1"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   772
  shows "card {l::nat list. length l = m \<and> listsum l = N} =
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   773
    (card {l. length l = (m - 1) \<and> listsum l = N} +
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   774
    card {l. length l = m \<and> listsum l + 1 =  N})"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   775
    (is "card ?C = (card ?A + card ?B)")
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   776
proof - 
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   777
  let ?A'="{l. length l = m \<and> listsum l = N \<and> hd l = 0}"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   778
  let ?B'="{l. length l = m \<and> listsum l = N \<and> hd l \<noteq> 0}"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   779
  let ?f ="\<lambda> l. 0#l"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   780
  let ?g ="\<lambda> l. (hd l + 1) # tl l"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   781
  have 1: "\<And>xs x. xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" by simp
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   782
  have 2: "\<And>xs. (xs::nat list) \<noteq> [] \<Longrightarrow> listsum(tl xs) = listsum xs - hd xs"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   783
    by(auto simp add: neq_Nil_conv)
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   784
  have f: "bij_betw ?f ?A ?A'"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   785
    apply(rule bij_betw_byWitness[where f' = tl])
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   786
    using assms 
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   787
    by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   788
  have 3: "\<And>xs:: nat list. xs \<noteq> [] \<Longrightarrow> hd xs + (listsum xs - hd xs) = listsum xs"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   789
    by (metis 1 listsum_simps(2) 2)
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   790
  have g: "bij_betw ?g ?B ?B'"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   791
    apply(rule bij_betw_byWitness[where f' = "\<lambda> l. (hd l - 1) # tl l"])
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   792
    using assms
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   793
    by (auto simp: 2 length_0_conv[symmetric] intro!: 3
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   794
      simp del: length_greater_0_conv length_0_conv)
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   795
  { fix M N :: nat have "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   796
    using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto }
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   797
    note fin = this
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   798
  have fin_A: "finite ?A" using fin[of _ "N+1"]
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   799
    by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"], 
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   800
      auto simp: member_le_listsum_nat less_Suc_eq_le)
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   801
  have fin_B: "finite ?B"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   802
    by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"], 
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   803
      auto simp: member_le_listsum_nat less_Suc_eq_le fin)
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   804
  have uni: "?C = ?A' \<union> ?B'" by auto
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   805
  have disj: "?A' \<inter> ?B' = {}" by auto
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   806
  have "card ?C = card(?A' \<union> ?B')" using uni by simp
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   807
  also have "\<dots> = card ?A + card ?B"
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   808
    using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   809
      bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   810
    by presburger
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   811
  finally show ?thesis .
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   812
qed
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   813
58194
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   814
lemma card_length_listsum: --"By Holden Lee, tidied by Tobias Nipkow"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   815
  "card {l::nat list. size l = m \<and> listsum l = N} = (N + m - 1) choose N"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   816
proof (cases m)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   817
  case 0 then show ?thesis
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   818
    by (cases N) (auto simp: cong: conj_cong)
58193
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   819
next
58194
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   820
  case (Suc m')
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   821
    have m: "m\<ge>1" by (simp add: Suc)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   822
    then show ?thesis
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   823
    proof (induct "N + m - 1" arbitrary: N m)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   824
      case 0   -- "In the base case, the only solution is [0]."
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   825
      have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   826
        by (auto simp: length_Suc_conv)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   827
      have "m=1 \<and> N=0" using 0 by linarith
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   828
      then show ?case by simp
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   829
    next
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   830
      case (Suc k)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   831
      
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   832
      have c1: "card {l::nat list. size l = (m - 1) \<and> listsum l =  N} = 
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   833
        (N + (m - 1) - 1) choose N"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   834
      proof cases
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   835
        assume "m = 1"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   836
        with Suc.hyps have "N\<ge>1" by auto
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   837
        with `m = 1` show ?thesis by (simp add: binomial_eq_0)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   838
      next
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   839
        assume "m \<noteq> 1" thus ?thesis using Suc by fastforce
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   840
      qed
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   841
    
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   842
      from Suc have c2: "card {l::nat list. size l = m \<and> listsum l + 1 = N} = 
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   843
        (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   844
      proof -
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   845
        have aux: "\<And>m n. n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" by arith
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   846
        from Suc have "N>0 \<Longrightarrow>
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   847
          card {l::nat list. size l = m \<and> listsum l + 1 = N} = 
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   848
          ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   849
        thus ?thesis by auto
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   850
      qed
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   851
    
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   852
      from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> listsum l = N} + 
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   853
          card {l::nat list. size l = m \<and> listsum l + 1 = N}) = (N + m - 1) choose N"
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   854
        by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   855
      thus ?case using card_length_listsum_rec[OF Suc.prems] by auto
3d90a96fd6a9 Generalised card_length_listsum to all m
paulson <lp15@cam.ac.uk>
parents: 58193
diff changeset
   856
    qed
58193
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   857
qed
ae8a5e111ee1 added lemma
nipkow
parents: 57514
diff changeset
   858
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   859
end