src/HOL/Fact.thy
author paulson <lp15@cam.ac.uk>
Tue, 10 Mar 2015 15:20:40 +0000
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(*  Title       : Fact.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    The integer version of factorial and other additions by Jeremy Avigad.
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*)
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section{*Factorial Function*}
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theory Fact
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imports Main
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begin
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class fact =
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  fixes fact :: "'a \<Rightarrow> 'a"
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instantiation nat :: fact
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begin
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fun
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  fact_nat :: "nat \<Rightarrow> nat"
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where
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  fact_0_nat: "fact_nat 0 = Suc 0"
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| fact_Suc: "fact_nat (Suc x) = Suc x * fact x"
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instance ..
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end
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(* definitions for the integers *)
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instantiation int :: fact
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begin
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definition
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  fact_int :: "int \<Rightarrow> int"
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where
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  "fact_int x = (if x >= 0 then int (fact (nat x)) else 0)"
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instance proof qed
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end
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subsection {* Set up Transfer *}
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lemma transfer_nat_int_factorial:
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  "(x::int) >= 0 \<Longrightarrow> fact (nat x) = nat (fact x)"
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  unfolding fact_int_def
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  by auto
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lemma transfer_nat_int_factorial_closure:
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  "x >= (0::int) \<Longrightarrow> fact x >= 0"
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  by (auto simp add: fact_int_def)
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declare transfer_morphism_nat_int[transfer add return:
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    transfer_nat_int_factorial transfer_nat_int_factorial_closure]
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lemma transfer_int_nat_factorial:
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  "fact (int x) = int (fact x)"
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  unfolding fact_int_def by auto
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lemma transfer_int_nat_factorial_closure:
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  "is_nat x \<Longrightarrow> fact x >= 0"
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  by (auto simp add: fact_int_def)
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declare transfer_morphism_int_nat[transfer add return:
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    transfer_int_nat_factorial transfer_int_nat_factorial_closure]
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subsection {* Factorial *}
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lemma fact_0_int [simp]: "fact (0::int) = 1"
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  by (simp add: fact_int_def)
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lemma fact_1_nat [simp]: "fact (1::nat) = 1"
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  by simp
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lemma fact_Suc_0_nat [simp]: "fact (Suc 0) = Suc 0"
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  by simp
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lemma fact_1_int [simp]: "fact (1::int) = 1"
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  by (simp add: fact_int_def)
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lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n"
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  by simp
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lemma fact_plus_one_int:
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  assumes "n >= 0"
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  shows "fact ((n::int) + 1) = (n + 1) * fact n"
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  using assms unfolding fact_int_def
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  by (simp add: nat_add_distrib algebra_simps int_mult)
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lemma fact_reduce_nat: "(n::nat) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
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  apply (subgoal_tac "n = Suc (n - 1)")
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  apply (erule ssubst)
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  apply (subst fact_Suc)
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  apply simp_all
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  done
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lemma fact_reduce_int: "(n::int) > 0 \<Longrightarrow> fact n = n * fact (n - 1)"
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  apply (subgoal_tac "n = (n - 1) + 1")
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  apply (erule ssubst)
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  apply (subst fact_plus_one_int)
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  apply simp_all
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  done
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lemma fact_nonzero_nat [simp]: "fact (n::nat) \<noteq> 0"
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  apply (induct n)
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  apply (auto simp add: fact_plus_one_nat)
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  done
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lemma fact_nonzero_int [simp]: "n >= 0 \<Longrightarrow> fact (n::int) ~= 0"
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  by (simp add: fact_int_def)
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lemma fact_gt_zero_nat [simp]: "fact (n :: nat) > 0"
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  by (insert fact_nonzero_nat [of n], arith)
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lemma fact_gt_zero_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) > 0"
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  by (auto simp add: fact_int_def)
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lemma fact_ge_one_nat [simp]: "fact (n :: nat) >= 1"
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  by (insert fact_nonzero_nat [of n], arith)
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lemma fact_ge_Suc_0_nat [simp]: "fact (n :: nat) >= Suc 0"
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  by (insert fact_nonzero_nat [of n], arith)
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lemma fact_ge_one_int [simp]: "n >= 0 \<Longrightarrow> fact (n :: int) >= 1"
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  apply (auto simp add: fact_int_def)
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  apply (subgoal_tac "1 = int 1")
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  apply (erule ssubst)
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  apply (subst zle_int)
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  apply auto
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  done
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8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   138
lemma dvd_fact_nat [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::nat)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   139
  apply (induct n)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   140
  apply force
32047
c141f139ce26 Changed fact_Suc_nat back to fact_Suc
avigad
parents: 32039
diff changeset
   141
  apply (auto simp only: fact_Suc)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   142
  apply (subgoal_tac "m = Suc n")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   143
  apply (erule ssubst)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   144
  apply (rule dvd_triv_left)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   145
  apply auto
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40033
diff changeset
   146
  done
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   147
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   148
lemma dvd_fact_int [rule_format]: "1 <= m \<longrightarrow> m <= n \<longrightarrow> m dvd fact (n::int)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   149
  apply (case_tac "1 <= n")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   150
  apply (induct n rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   151
  apply (auto simp add: fact_plus_one_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   152
  apply (subgoal_tac "m = i + 1")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   153
  apply auto
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 40033
diff changeset
   154
  done
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   155
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   156
lemma interval_plus_one_nat: "(i::nat) <= j + 1 \<Longrightarrow>
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   157
  {i..j+1} = {i..j} Un {j+1}"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   158
  by auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   159
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   160
lemma interval_Suc: "i <= Suc j \<Longrightarrow> {i..Suc j} = {i..j} Un {Suc j}"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   161
  by auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   162
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   163
lemma interval_plus_one_int: "(i::int) <= j + 1 \<Longrightarrow> {i..j+1} = {i..j} Un {j+1}"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   164
  by auto
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   165
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   166
lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   167
  apply (induct n)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   168
  apply force
32047
c141f139ce26 Changed fact_Suc_nat back to fact_Suc
avigad
parents: 32039
diff changeset
   169
  apply (subst fact_Suc)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   170
  apply (subst interval_Suc)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   171
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   172
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   173
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   174
lemma fact_altdef_int: "n >= 0 \<Longrightarrow> fact (n::int) = (PROD i:{1..n}. i)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   175
  apply (induct n rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   176
  apply force
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   177
  apply (subst fact_plus_one_int, assumption)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   178
  apply (subst interval_plus_one_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   179
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   180
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   181
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   182
lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd fact (m::nat)"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   183
  by (auto simp add: fact_altdef_nat intro!: setprod_dvd_setprod_subset)
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   184
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   185
lemma fact_mod: "m \<le> (n::nat) \<Longrightarrow> fact n mod fact m = 0"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   186
  by (auto simp add: dvd_imp_mod_0 fact_dvd)
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   187
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   188
lemma fact_div_fact:
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   189
  assumes "m \<ge> (n :: nat)"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   190
  shows "(fact m) div (fact n) = \<Prod>{n + 1..m}"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   191
proof -
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   192
  obtain d where "d = m - n" by auto
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   193
  from assms this have "m = n + d" by auto
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   194
  have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   195
  proof (induct d)
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   196
    case 0
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   197
    show ?case by simp
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   198
  next
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   199
    case (Suc d')
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   200
    have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   201
      by simp
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   202
    also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}"
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   203
      unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   204
    also have "... = \<Prod>{n + 1..n + Suc d'}"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   205
      by (simp add: atLeastAtMostSuc_conv setprod.insert)
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   206
    finally show ?case .
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   207
  qed
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   208
  from this `m = n + d` show ?thesis by simp
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   209
qed
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   210
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   211
lemma fact_mono_nat: "(m::nat) \<le> n \<Longrightarrow> fact m \<le> fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   212
apply (drule le_imp_less_or_eq)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   213
apply (auto dest!: less_imp_Suc_add)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   214
apply (induct_tac k, auto)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   215
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   216
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   217
lemma fact_neg_int [simp]: "m < (0::int) \<Longrightarrow> fact m = 0"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   218
  unfolding fact_int_def by auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   219
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   220
lemma fact_ge_zero_int [simp]: "fact m >= (0::int)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   221
  apply (case_tac "m >= 0")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   222
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   223
  apply (frule fact_gt_zero_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   224
  apply arith
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   225
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   226
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   227
lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \<Longrightarrow>
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   228
    fact (m + k) >= fact m"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   229
  apply (case_tac "m < 0")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   230
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   231
  apply (induct k rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   232
  apply auto
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57418
diff changeset
   233
  apply (subst add.assoc [symmetric])
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   234
  apply (subst fact_plus_one_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   235
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   236
  apply (erule order_trans)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   237
  apply (subst mult_le_cancel_right1)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   238
  apply (subgoal_tac "fact (m + i) >= 0")
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   239
  apply arith
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   240
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   241
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   242
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   243
lemma fact_mono_int: "(m::int) <= n \<Longrightarrow> fact m <= fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   244
  apply (insert fact_mono_int_aux [of "n - m" "m"])
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   245
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   246
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   247
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   248
text{*Note that @{term "fact 0 = fact 1"}*}
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   249
lemma fact_less_mono_nat: "[| (0::nat) < m; m < n |] ==> fact m < fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   250
apply (drule_tac m = m in less_imp_Suc_add, auto)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   251
apply (induct_tac k, auto)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   252
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   253
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   254
lemma fact_less_mono_int_aux: "k >= 0 \<Longrightarrow> (0::int) < m \<Longrightarrow>
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   255
    fact m < fact ((m + 1) + k)"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   256
  apply (induct k rule: int_ge_induct)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   257
  apply (simp add: fact_plus_one_int)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   258
  apply (subst (2) fact_reduce_int)
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   259
  apply (auto simp add: ac_simps)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   260
  apply (erule order_less_le_trans)
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   261
  apply auto
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   262
  done
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   263
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   264
lemma fact_less_mono_int: "(0::int) < m \<Longrightarrow> m < n \<Longrightarrow> fact m < fact n"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   265
  apply (insert fact_less_mono_int_aux [of "n - (m + 1)" "m"])
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   266
  apply auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   267
done
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   268
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   269
lemma fact_num_eq_if_nat: "fact (m::nat) =
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   270
  (if m=0 then 1 else m * fact (m - 1))"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   271
by (cases m) auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   272
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   273
lemma fact_add_num_eq_if_nat:
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   274
  "fact ((m::nat) + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   275
by (cases "m + n") auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   276
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   277
lemma fact_add_num_eq_if2_nat:
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   278
  "fact ((m::nat) + n) =
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   279
    (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))"
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   280
by (cases m) auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   281
45930
2a882ef2cd73 add lemmas
noschinl
parents: 41550
diff changeset
   282
lemma fact_le_power: "fact n \<le> n^n"
2a882ef2cd73 add lemmas
noschinl
parents: 41550
diff changeset
   283
proof (induct n)
2a882ef2cd73 add lemmas
noschinl
parents: 41550
diff changeset
   284
  case (Suc n)
2a882ef2cd73 add lemmas
noschinl
parents: 41550
diff changeset
   285
  then have "fact n \<le> Suc n ^ n" by (rule le_trans) (simp add: power_mono)
2a882ef2cd73 add lemmas
noschinl
parents: 41550
diff changeset
   286
  then show ?case by (simp add: add_le_mono)
2a882ef2cd73 add lemmas
noschinl
parents: 41550
diff changeset
   287
qed simp
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   288
32039
400a519bc888 Use term antiquotation to refer to constant names in subsection title.
berghofe
parents: 32036
diff changeset
   289
subsection {* @{term fact} and @{term of_nat} *}
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   290
29693
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
parents: 28952
diff changeset
   291
lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \<noteq> (0::'a::semiring_char_0)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   292
by auto
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   293
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33319
diff changeset
   294
lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto
29693
708dcf7dec9f moved upwards in thy graph, real related theorems moved to Transcendental.thy
chaieb
parents: 28952
diff changeset
   295
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33319
diff changeset
   296
lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_semidom) \<le> of_nat(fact n)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   297
by simp
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   298
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33319
diff changeset
   299
lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::linordered_field) < inverse (of_nat (fact n))"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   300
by (auto simp add: positive_imp_inverse_positive)
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   301
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 33319
diff changeset
   302
lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::linordered_field) \<le> inverse (of_nat (fact n))"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   303
by (auto intro: order_less_imp_le)
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   304
50224
aacd6da09825 add binomial_ge_n_over_k_pow_k
hoelzl
parents: 46240
diff changeset
   305
lemma fact_eq_rev_setprod_nat: "fact (k::nat) = (\<Prod>i<k. k - i)"
aacd6da09825 add binomial_ge_n_over_k_pow_k
hoelzl
parents: 46240
diff changeset
   306
  unfolding fact_altdef_nat
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57113
diff changeset
   307
  by (rule setprod.reindex_bij_witness[where i="\<lambda>i. k - i" and j="\<lambda>i. k - i"]) auto
50224
aacd6da09825 add binomial_ge_n_over_k_pow_k
hoelzl
parents: 46240
diff changeset
   308
50240
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   309
lemma fact_div_fact_le_pow:
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   310
  assumes "r \<le> n" shows "fact n div fact (n - r) \<le> n ^ r"
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   311
proof -
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   312
  have "\<And>r. r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   313
    by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
50240
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   314
  with assms show ?thesis
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   315
    by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   316
qed
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   317
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 50240
diff changeset
   318
lemma fact_numeral:  --{*Evaluation for specific numerals*}
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 50240
diff changeset
   319
  "fact (numeral k) = (numeral k) * (fact (pred_numeral k))"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 50240
diff changeset
   320
  by (simp add: numeral_eq_Suc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 50240
diff changeset
   321
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   322
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   323
text {* This development is based on the work of Andy Gordon and
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   324
  Florian Kammueller. *}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   325
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   326
subsection {* Basic definitions and lemmas *}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   327
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   328
primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   329
where
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   330
  "0 choose k = (if k = 0 then 1 else 0)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   331
| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   332
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   333
lemma binomial_n_0 [simp]: "(n choose 0) = 1"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   334
  by (cases n) simp_all
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   335
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   336
lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   337
  by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   338
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   339
lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   340
  by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   341
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   342
lemma choose_reduce_nat:
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   343
  "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   344
    (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   345
  by (metis Suc_diff_1 binomial.simps(2) neq0_conv)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   346
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   347
lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   348
  by (induct n arbitrary: k) auto
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   349
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   350
declare binomial.simps [simp del]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   351
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   352
lemma binomial_n_n [simp]: "n choose n = 1"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   353
  by (induct n) (simp_all add: binomial_eq_0)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   354
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   355
lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   356
  by (induct n) simp_all
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   357
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   358
lemma binomial_1 [simp]: "n choose Suc 0 = n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   359
  by (induct n) simp_all
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   360
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   361
lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   362
  by (induct n k rule: diff_induct) simp_all
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   363
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   364
lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   365
  by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   366
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   367
lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   368
  by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   369
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   370
lemma Suc_times_binomial_eq:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   371
  "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   372
  apply (induct n arbitrary: k, simp add: binomial.simps)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   373
  apply (case_tac k)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   374
   apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   375
  done
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   376
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   377
text{*The absorption property*}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   378
lemma Suc_times_binomial:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   379
  "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   380
  using Suc_times_binomial_eq by auto
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   381
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   382
text{*This is the well-known version of absorption, but it's harder to use because of the
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   383
  need to reason about division.*}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   384
lemma binomial_Suc_Suc_eq_times:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   385
    "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   386
  by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   387
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   388
text{*Another version of absorption, with -1 instead of Suc.*}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   389
lemma times_binomial_minus1_eq:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   390
  "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   391
  using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   392
  by (auto split add: nat_diff_split)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   393
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   394
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   395
subsection {* Combinatorial theorems involving @{text "choose"} *}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   396
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   397
text {*By Florian Kamm\"uller, tidied by LCP.*}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   398
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   399
lemma card_s_0_eq_empty: "finite A \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 1"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   400
  by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   401
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   402
lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   403
    {s. s \<subseteq> insert x M \<and> card s = Suc k} =
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   404
    {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}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   405
  apply safe
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   406
     apply (auto intro: finite_subset [THEN card_insert_disjoint])
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   407
  by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   408
     card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   409
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   410
lemma finite_bex_subset [simp]:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   411
  assumes "finite B"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   412
    and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   413
  shows "finite {x. \<exists>A \<subseteq> B. P x A}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   414
  by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   415
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   416
text{*There are as many subsets of @{term A} having cardinality @{term k}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   417
 as there are sets obtained from the former by inserting a fixed element
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   418
 @{term x} into each.*}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   419
lemma constr_bij:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   420
   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow>
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   421
    card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   422
    card {B. B \<subseteq> A & card(B) = k}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   423
  apply (rule card_bij_eq [where f = "\<lambda>s. s - {x}" and g = "insert x"])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   424
  apply (auto elim!: equalityE simp add: inj_on_def)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   425
  apply (metis card_Diff_singleton_if finite_subset in_mono)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   426
  done
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   427
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   428
text {*
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   429
  Main theorem: combinatorial statement about number of subsets of a set.
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   430
*}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   431
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   432
theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A choose k)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   433
proof (induct k arbitrary: A)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   434
  case 0 then show ?case by (simp add: card_s_0_eq_empty)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   435
next
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   436
  case (Suc k)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   437
  show ?case using `finite A`
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   438
  proof (induct A)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   439
    case empty show ?case by (simp add: card_s_0_eq_empty)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   440
  next
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   441
    case (insert x A)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   442
    then show ?case using Suc.hyps
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   443
      apply (simp add: card_s_0_eq_empty choose_deconstruct)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   444
      apply (subst card_Un_disjoint)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   445
         prefer 4 apply (force simp add: constr_bij)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   446
        prefer 3 apply force
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   447
       prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   448
         finite_subset [of _ "Pow (insert x F)" for F])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   449
      apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   450
      done
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   451
  qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   452
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   453
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   454
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   455
subsection {* The binomial theorem (courtesy of Tobias Nipkow): *}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   456
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   457
text{* Avigad's version, generalized to any commutative ring *}
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   458
theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n =
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   459
  (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n")
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   460
proof (induct n)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   461
  case 0 then show "?P 0" by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   462
next
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   463
  case (Suc n)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   464
  have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   465
    by auto
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   466
  have decomp2: "{0..n} = {0} Un {1..n}"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   467
    by auto
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   468
  have "(a+b)^(n+1) =
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   469
      (a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   470
    using Suc.hyps by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   471
  also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k)) +
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   472
                   b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   473
    by (rule distrib_right)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   474
  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   475
                  (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   476
    by (auto simp add: setsum_right_distrib ac_simps)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   477
  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) +
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   478
                  (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-k))"
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   479
    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   480
        del:setsum_cl_ivl_Suc)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   481
  also have "\<dots> = a^(n+1) + b^(n+1) +
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   482
                  (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) +
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   483
                  (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   484
    by (simp add: decomp2)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   485
  also have
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   486
      "\<dots> = a^(n+1) + b^(n+1) +
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   487
            (\<Sum>k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   488
    by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   489
  also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   490
    using decomp by (simp add: field_simps)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   491
  finally show "?P (Suc n)" by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   492
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   493
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   494
text{* Original version for the naturals *}
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   495
corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   496
    using binomial_ring [of "int a" "int b" n]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   497
  by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   498
           of_nat_setsum [symmetric]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   499
           of_nat_eq_iff of_nat_id)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   500
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   501
lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   502
proof (induct n arbitrary: k rule: nat_less_induct)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   503
  fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   504
                      fact m" and kn: "k \<le> n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   505
  let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   506
  { assume "n=0" then have ?ths using kn by simp }
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   507
  moreover
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   508
  { assume "k=0" then have ?ths using kn by simp }
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   509
  moreover
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   510
  { assume nk: "n=k" then have ?ths by simp }
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   511
  moreover
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   512
  { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   513
    from n have mn: "m < n" by arith
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   514
    from hm have hm': "h \<le> m" by arith
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   515
    from hm h n kn have km: "k \<le> m" by arith
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   516
    have "m - h = Suc (m - Suc h)" using  h km hm by arith
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   517
    with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   518
      by simp
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   519
    from n h th0
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   520
    have "fact k * fact (n - k) * (n choose k) =
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   521
        k * (fact h * fact (m - h) * (m choose h)) +
59658
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   522
        (m - h) * (fact k * fact (m - k) * (m choose k))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   523
      by (simp add: field_simps)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   524
    also have "\<dots> = (k + (m - h)) * fact m"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   525
      using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   526
      by (simp add: field_simps)
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   527
    finally have ?ths using h n km by simp }
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   528
  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < m)"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   529
    using kn by presburger
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   530
  ultimately show ?ths by blast
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   531
qed
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   532
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   533
lemma binomial_fact:
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   534
  assumes kn: "k \<le> n"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   535
  shows "(of_nat (n choose k) :: 'a::{field,ring_char_0}) =
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   536
    of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   537
  using binomial_fact_lemma[OF kn]
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   538
  by (simp add: field_simps of_nat_mult [symmetric])
0cc388370041 sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   539
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   540
lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   541
  using binomial [of 1 "1" n]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   542
  by (simp add: numeral_2_eq_2)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   543
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   544
lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   545
  by (induct n) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   546
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   547
lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   548
  by (induct n) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   549
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   550
lemma natsum_reverse_index:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   551
  fixes m::nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   552
  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)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   553
  by (rule setsum.reindex_bij_witness[where i="\<lambda>k. m+n-k" and j="\<lambda>k. m+n-k"]) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   554
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   555
text{*NW diagonal sum property*}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   556
lemma sum_choose_diagonal:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   557
  assumes "m\<le>n" shows "(\<Sum>k=0..m. (n-k) choose (m-k)) = Suc n choose m"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   558
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   559
  have "(\<Sum>k=0..m. (n-k) choose (m-k)) = (\<Sum>k=0..m. (n-m+k) choose k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   560
    by (rule natsum_reverse_index) (simp add: assms)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   561
  also have "... = Suc (n-m+m) choose m"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   562
    by (rule sum_choose_lower)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   563
  also have "... = Suc n choose m" using assms
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   564
    by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   565
  finally show ?thesis .
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   566
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   567
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   568
subsection{* Pochhammer's symbol : generalized rising factorial *}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   569
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   570
text {* See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"} *}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   571
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   572
definition "pochhammer (a::'a::comm_semiring_1) n =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   573
  (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   574
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   575
lemma pochhammer_0 [simp]: "pochhammer a 0 = 1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   576
  by (simp add: pochhammer_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   577
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   578
lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   579
  by (simp add: pochhammer_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   580
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   581
lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   582
  by (simp add: pochhammer_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   583
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   584
lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   585
  by (simp add: pochhammer_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   586
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   587
lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   588
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   589
  have "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   590
  then show ?thesis by (simp add: field_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   591
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   592
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   593
lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   594
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   595
  have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   596
  then show ?thesis by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   597
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   598
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   599
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   600
lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   601
proof (cases n)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   602
  case 0
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   603
  then show ?thesis by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   604
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   605
  case (Suc n)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   606
  show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   607
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   608
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   609
lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   610
proof (cases "n = 0")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   611
  case True
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   612
  then show ?thesis by (simp add: pochhammer_Suc_setprod)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   613
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   614
  case False
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   615
  have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   616
  have eq: "insert 0 {1 .. n} = {0..n}" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   617
  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)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   618
    apply (rule setprod.reindex_cong [where l = Suc])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   619
    using False
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   620
    apply (auto simp add: fun_eq_iff field_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   621
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   622
  show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   623
    apply (simp add: pochhammer_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   624
    unfolding setprod.insert [OF *, unfolded eq]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   625
    using ** apply (simp add: field_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   626
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   627
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   628
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   629
lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   630
  unfolding fact_altdef_nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   631
  apply (cases n)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   632
   apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   633
  apply (rule setprod.reindex_cong [where l = Suc])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   634
    apply (auto simp add: fun_eq_iff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   635
  done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   636
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   637
lemma pochhammer_of_nat_eq_0_lemma:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   638
  assumes "k > n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   639
  shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   640
proof (cases "n = 0")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   641
  case True
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   642
  then show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   643
    using assms by (cases k) (simp_all add: pochhammer_rec)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   644
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   645
  case False
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   646
  from assms obtain h where "k = Suc h" by (cases k) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   647
  then show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   648
    by (simp add: pochhammer_Suc_setprod)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   649
       (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1))
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   650
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   651
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   652
lemma pochhammer_of_nat_eq_0_lemma':
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   653
  assumes kn: "k \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   654
  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   655
proof (cases k)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   656
  case 0
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   657
  then show ?thesis by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   658
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   659
  case (Suc h)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   660
  then show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   661
    apply (simp add: pochhammer_Suc_setprod)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   662
    using Suc kn apply (auto simp add: algebra_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   663
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   664
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   665
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   666
lemma pochhammer_of_nat_eq_0_iff:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   667
  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   668
  (is "?l = ?r")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   669
  using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   670
    pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   671
  by (auto simp add: not_le[symmetric])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   672
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   673
lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   674
  apply (auto simp add: pochhammer_of_nat_eq_0_iff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   675
  apply (cases n)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   676
   apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   677
  apply (metis leD not_less_eq)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   678
  done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   679
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   680
lemma pochhammer_eq_0_mono:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   681
  "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   682
  unfolding pochhammer_eq_0_iff by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   683
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   684
lemma pochhammer_neq_0_mono:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   685
  "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   686
  unfolding pochhammer_eq_0_iff by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   687
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   688
lemma pochhammer_minus:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   689
  assumes kn: "k \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   690
  shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   691
proof (cases k)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   692
  case 0
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   693
  then show ?thesis by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   694
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   695
  case (Suc h)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   696
  have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i=0..h. - 1)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   697
    using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   698
    by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   699
  show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   700
    unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   701
    by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   702
       (auto simp: of_nat_diff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   703
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   704
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   705
lemma pochhammer_minus':
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   706
  assumes kn: "k \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   707
  shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   708
  unfolding pochhammer_minus[OF kn, where b=b]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   709
  unfolding mult.assoc[symmetric]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   710
  unfolding power_add[symmetric]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   711
  by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   712
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   713
lemma pochhammer_same: "pochhammer (- of_nat n) n =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   714
    ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   715
  unfolding pochhammer_minus[OF le_refl[of n]]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   716
  by (simp add: of_nat_diff pochhammer_fact)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   717
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   718
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   719
subsection{* Generalized binomial coefficients *}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   720
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   721
definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   722
  where "a gchoose n =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   723
    (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   724
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   725
lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   726
  apply (simp_all add: gbinomial_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   727
  apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   728
   apply (simp del:setprod_zero_iff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   729
  apply simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   730
  done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   731
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   732
lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   733
proof (cases "n = 0")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   734
  case True
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   735
  then show ?thesis by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   736
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   737
  case False
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   738
  from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   739
  have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   740
    by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   741
  from False show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   742
    by (simp add: pochhammer_def gbinomial_def field_simps
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   743
      eq setprod.distrib[symmetric])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   744
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   745
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   746
lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   747
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   748
  { assume kn: "k > n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   749
    then have ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   750
      by (subst binomial_eq_0[OF kn])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   751
         (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   752
  moreover
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   753
  { assume "k=0" then have ?thesis by simp }
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   754
  moreover
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   755
  { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   756
    from k0 obtain h where h: "k = Suc h" by (cases k) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   757
    from h
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   758
    have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   759
      by (subst setprod_constant) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   760
    have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   761
        using h kn
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   762
      by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   763
         (auto simp: of_nat_diff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   764
    have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   765
        "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   766
        eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   767
      using h kn by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   768
    from eq[symmetric]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   769
    have ?thesis using kn
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   770
      apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   771
        gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   772
      apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   773
        of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   774
      unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   775
      unfolding mult.assoc[symmetric]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   776
      unfolding setprod.distrib[symmetric]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   777
      apply simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   778
      apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   779
      apply (auto simp: of_nat_diff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   780
      done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   781
  }
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   782
  moreover
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   783
  have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   784
  ultimately show ?thesis by blast
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   785
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   786
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   787
lemma gbinomial_1[simp]: "a gchoose 1 = a"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   788
  by (simp add: gbinomial_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   789
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   790
lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   791
  by (simp add: gbinomial_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   792
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   793
lemma gbinomial_mult_1:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   794
  "a * (a gchoose n) =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   795
    of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?r")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   796
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   797
  have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   798
    unfolding gbinomial_pochhammer
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   799
      pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   800
    by (simp add:  field_simps del: of_nat_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   801
  also have "\<dots> = ?l" unfolding gbinomial_pochhammer
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   802
    by (simp add: field_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   803
  finally show ?thesis ..
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   804
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   805
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   806
lemma gbinomial_mult_1':
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   807
    "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   808
  by (simp add: mult.commute gbinomial_mult_1)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   809
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   810
lemma gbinomial_Suc:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   811
    "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   812
  by (simp add: gbinomial_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   813
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   814
lemma gbinomial_mult_fact:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   815
  "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   816
    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   817
  by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   818
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   819
lemma gbinomial_mult_fact':
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   820
  "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   821
    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   822
  using gbinomial_mult_fact[of k a]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   823
  by (subst mult.commute)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   824
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   825
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   826
lemma gbinomial_Suc_Suc:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   827
  "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   828
proof (cases k)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   829
  case 0
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   830
  then show ?thesis by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   831
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   832
  case (Suc h)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   833
  have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   834
    apply (rule setprod.reindex_cong [where l = Suc])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   835
      using Suc
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   836
      apply auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   837
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   838
  have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   839
    ((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)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   840
    apply (simp add: Suc field_simps del: fact_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   841
    unfolding gbinomial_mult_fact'
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   842
    apply (subst fact_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   843
    unfolding of_nat_mult
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   844
    apply (subst mult.commute)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   845
    unfolding mult.assoc
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   846
    unfolding gbinomial_mult_fact
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   847
    apply (simp add: field_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   848
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   849
  also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   850
    unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   851
    by (simp add: field_simps Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   852
  also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   853
    using eq0
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   854
    by (simp add: Suc setprod_nat_ivl_1_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   855
  also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   856
    unfolding gbinomial_mult_fact ..
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   857
  finally show ?thesis by (simp del: fact_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   858
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   859
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   860
lemma gbinomial_reduce_nat:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   861
  "0 < k \<Longrightarrow> (a::'a::field_char_0) gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   862
by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   863
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   864
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   865
lemma binomial_symmetric:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   866
  assumes kn: "k \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   867
  shows "n choose k = n choose (n - k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   868
proof-
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   869
  from kn have kn': "n - k \<le> n" by arith
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   870
  from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   871
  have "fact k * fact (n - k) * (n choose k) =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   872
    fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   873
  then show ?thesis using kn by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   874
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   875
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   876
text{*Contributed by Manuel Eberl, generalised by LCP.
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   877
  Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"} *}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   878
lemma gbinomial_altdef_of_nat:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   879
  fixes k :: nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   880
    and x :: "'a :: {field_char_0,field_inverse_zero}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   881
  shows "x gchoose k = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   882
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   883
  have "(x gchoose k) = (\<Prod>i<k. x - of_nat i) / of_nat (fact k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   884
    unfolding gbinomial_def
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   885
    by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   886
  also have "\<dots> = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   887
    unfolding fact_eq_rev_setprod_nat of_nat_setprod
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   888
    by (auto simp add: setprod_dividef intro!: setprod.cong of_nat_diff[symmetric])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   889
  finally show ?thesis .
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   890
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   891
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   892
lemma gbinomial_ge_n_over_k_pow_k:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   893
  fixes k :: nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   894
    and x :: "'a :: linordered_field_inverse_zero"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   895
  assumes "of_nat k \<le> x"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   896
  shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   897
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   898
  have x: "0 \<le> x"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   899
    using assms of_nat_0_le_iff order_trans by blast
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   900
  have "(x / of_nat k :: 'a) ^ k = (\<Prod>i<k. x / of_nat k :: 'a)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   901
    by (simp add: setprod_constant)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   902
  also have "\<dots> \<le> x gchoose k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   903
    unfolding gbinomial_altdef_of_nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   904
  proof (safe intro!: setprod_mono)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   905
    fix i :: nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   906
    assume ik: "i < k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   907
    from assms have "x * of_nat i \<ge> of_nat (i * k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   908
      by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   909
    then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)" by arith
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   910
    then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   911
      using ik
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   912
      by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   913
    then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   914
      unfolding of_nat_mult[symmetric] of_nat_le_iff .
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   915
    with assms show "x / of_nat k \<le> (x - of_nat i) / (of_nat (k - i) :: 'a)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   916
      using `i < k` by (simp add: field_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   917
  qed (simp add: x zero_le_divide_iff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   918
  finally show ?thesis .
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   919
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   920
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   921
text{*Versions of the theorems above for the natural-number version of "choose"*}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   922
lemma binomial_altdef_of_nat:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   923
  fixes n k :: nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   924
    and x :: "'a :: {field_char_0,field_inverse_zero}"  --{*the point is to constrain @{typ 'a}*}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   925
  assumes "k \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   926
  shows "of_nat (n choose k) = (\<Prod>i<k. of_nat (n - i) / of_nat (k - i) :: 'a)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   927
using assms
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   928
by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   929
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   930
lemma binomial_ge_n_over_k_pow_k:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   931
  fixes k n :: nat
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   932
    and x :: "'a :: linordered_field_inverse_zero"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   933
  assumes "k \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   934
  shows "(of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   935
by (simp add: assms gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   936
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   937
lemma binomial_le_pow:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   938
  assumes "r \<le> n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   939
  shows "n choose r \<le> n ^ r"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   940
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   941
  have "n choose r \<le> fact n div fact (n - r)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   942
    using `r \<le> n` by (subst binomial_fact_lemma[symmetric]) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   943
  with fact_div_fact_le_pow [OF assms] show ?thesis by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   944
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   945
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   946
lemma binomial_altdef_nat: "(k::nat) \<le> n \<Longrightarrow>
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   947
    n choose k = fact n div (fact k * fact (n - k))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   948
 by (subst binomial_fact_lemma [symmetric]) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   949
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   950
lemma choose_dvd_nat: "(k::nat) \<le> n \<Longrightarrow> fact k * fact (n - k) dvd fact n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   951
by (metis binomial_fact_lemma dvd_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   952
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   953
lemma choose_dvd_int:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   954
  assumes "(0::int) <= k" and "k <= n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   955
  shows "fact k * fact (n - k) dvd fact n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   956
  apply (subst tsub_eq [symmetric], rule assms)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   957
  apply (rule choose_dvd_nat [transferred])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   958
  using assms apply auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   959
  done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   960
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   961
lemma fact_fact_dvd_fact: fixes k::nat shows "fact k * fact n dvd fact (n + k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   962
by (metis add.commute add_diff_cancel_left' choose_dvd_nat le_add2)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   963
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   964
lemma choose_mult_lemma:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   965
     "((m+r+k) choose (m+k)) * ((m+k) choose k) = ((m+r+k) choose k) * ((m+r) choose m)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   966
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   967
  have "((m+r+k) choose (m+k)) * ((m+k) choose k) =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   968
        fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   969
    by (simp add: assms binomial_altdef_nat)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   970
  also have "... = fact (m+r+k) div (fact r * (fact k * fact m))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   971
    apply (subst div_mult_div_if_dvd)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   972
    apply (auto simp: fact_fact_dvd_fact)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   973
    apply (metis add.assoc add.commute fact_fact_dvd_fact)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   974
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   975
  also have "... = (fact (m+r+k) * fact (m+r)) div (fact r * (fact k * fact m) * fact (m+r))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   976
    apply (subst div_mult_div_if_dvd [symmetric])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   977
    apply (auto simp: fact_fact_dvd_fact)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   978
    apply (metis dvd_trans dvd.dual_order.refl fact_fact_dvd_fact mult_dvd_mono mult.left_commute)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   979
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   980
  also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   981
    apply (subst div_mult_div_if_dvd)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   982
    apply (auto simp: fact_fact_dvd_fact)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   983
    apply(metis mult.left_commute)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   984
    done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   985
  finally show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   986
    by (simp add: binomial_altdef_nat mult.commute)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   987
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   988
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   989
text{*The "Subset of a Subset" identity*}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   990
lemma choose_mult:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   991
  assumes "k\<le>m" "m\<le>n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   992
    shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   993
using assms choose_mult_lemma [of "m-k" "n-m" k]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   994
by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   995
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   996
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   997
subsection {* Binomial coefficients *}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   998
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
   999
lemma choose_one: "(n::nat) choose 1 = n"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1000
  by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1001
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1002
(*FIXME: messy and apparently unused*)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1003
lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \<longrightarrow>
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1004
    (ALL n. P (Suc n) 0) \<longrightarrow> (ALL n. (ALL k < n. P n k \<longrightarrow> P n (Suc k) \<longrightarrow>
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1005
    P (Suc n) (Suc k))) \<longrightarrow> (ALL k <= n. P n k)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1006
  apply (induct n)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1007
  apply auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1008
  apply (case_tac "k = 0")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1009
  apply auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1010
  apply (case_tac "k = Suc n")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1011
  apply auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1012
  apply (metis Suc_le_eq fact_nat.cases le_Suc_eq le_eq_less_or_eq)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1013
  done
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1014
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1015
lemma card_UNION:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1016
  assumes "finite A" and "\<forall>k \<in> A. finite k"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1017
  shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1018
  (is "?lhs = ?rhs")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1019
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1020
  have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))" by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1021
  also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1022
    by(subst setsum_right_distrib) simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1023
  also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1024
    using assms by(subst setsum.Sigma)(auto)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1025
  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))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1026
    by (rule setsum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1027
  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))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1028
    using assms by(auto intro!: setsum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1029
  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)))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1030
    using assms by(subst setsum.Sigma) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1031
  also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "setsum ?lhs _ = _")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1032
  proof(rule setsum.cong[OF refl])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1033
    fix x
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1034
    assume x: "x \<in> \<Union>A"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1035
    def K \<equiv> "{X \<in> A. x \<in> X}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1036
    with `finite A` have K: "finite K" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1037
    let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1038
    have "inj_on snd (SIGMA i:{1..card A}. ?I i)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1039
      using assms by(auto intro!: inj_onI)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1040
    moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1041
      using assms by(auto intro!: rev_image_eqI[where x="(card a, a)" for a]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1042
        simp add: card_gt_0_iff[folded Suc_le_eq]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1043
        dest: finite_subset intro: card_mono)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1044
    ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1045
      by (rule setsum.reindex_cong [where l = snd]) fastforce
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1046
    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)))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1047
      using assms by(subst setsum.Sigma) auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1048
    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))"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1049
      by(subst setsum_right_distrib) simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1050
    also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))" (is "_ = ?rhs")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1051
    proof(rule setsum.mono_neutral_cong_right[rule_format])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1052
      show "{1..card K} \<subseteq> {1..card A}" using `finite A`
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1053
        by(auto simp add: K_def intro: card_mono)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1054
    next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1055
      fix i
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1056
      assume "i \<in> {1..card A} - {1..card K}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1057
      hence i: "i \<le> card A" "card K < i" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1058
      have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1059
        by(auto simp add: K_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1060
      also have "\<dots> = {}" using `finite A` i
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1061
        by(auto simp add: K_def dest: card_mono[rotated 1])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1062
      finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1063
        by(simp only:) simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1064
    next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1065
      fix i
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1066
      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)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1067
        (is "?lhs = ?rhs")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1068
        by(rule setsum.cong)(auto simp add: K_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1069
      thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1070
    qed simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1071
    also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}" using assms
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1072
      by(auto simp add: card_eq_0_iff K_def dest: finite_subset)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1073
    hence "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1074
      by(subst (2) setsum_head_Suc)(simp_all )
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1075
    also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1076
      using K by(subst n_subsets[symmetric]) simp_all
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1077
    also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1078
      by(subst setsum_right_distrib[symmetric]) simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1079
    also have "\<dots> =  - ((-1 + 1) ^ card K) + 1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1080
      by(subst binomial_ring)(simp add: ac_simps)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1081
    also have "\<dots> = 1" using x K by(auto simp add: K_def card_gt_0_iff)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1082
    finally show "?lhs x = 1" .
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1083
  qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1084
  also have "nat \<dots> = card (\<Union>A)" by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1085
  finally show ?thesis ..
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1086
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1087
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1088
text{* The number of nat lists of length @{text m} summing to @{text N} is
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1089
@{term "(N + m - 1) choose N"}: *}
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1090
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1091
lemma card_length_listsum_rec:
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1092
  assumes "m\<ge>1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1093
  shows "card {l::nat list. length l = m \<and> listsum l = N} =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1094
    (card {l. length l = (m - 1) \<and> listsum l = N} +
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1095
    card {l. length l = m \<and> listsum l + 1 =  N})"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1096
    (is "card ?C = (card ?A + card ?B)")
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1097
proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1098
  let ?A'="{l. length l = m \<and> listsum l = N \<and> hd l = 0}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1099
  let ?B'="{l. length l = m \<and> listsum l = N \<and> hd l \<noteq> 0}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1100
  let ?f ="\<lambda> l. 0#l"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1101
  let ?g ="\<lambda> l. (hd l + 1) # tl l"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1102
  have 1: "\<And>xs x. xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1103
  have 2: "\<And>xs. (xs::nat list) \<noteq> [] \<Longrightarrow> listsum(tl xs) = listsum xs - hd xs"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1104
    by(auto simp add: neq_Nil_conv)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1105
  have f: "bij_betw ?f ?A ?A'"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1106
    apply(rule bij_betw_byWitness[where f' = tl])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1107
    using assms
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1108
    by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1109
  have 3: "\<And>xs:: nat list. xs \<noteq> [] \<Longrightarrow> hd xs + (listsum xs - hd xs) = listsum xs"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1110
    by (metis 1 listsum_simps(2) 2)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1111
  have g: "bij_betw ?g ?B ?B'"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1112
    apply(rule bij_betw_byWitness[where f' = "\<lambda> l. (hd l - 1) # tl l"])
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1113
    using assms
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1114
    by (auto simp: 2 length_0_conv[symmetric] intro!: 3
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1115
      simp del: length_greater_0_conv length_0_conv)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1116
  { fix M N :: nat have "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1117
    using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto }
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1118
    note fin = this
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1119
  have fin_A: "finite ?A" using fin[of _ "N+1"]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1120
    by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"],
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1121
      auto simp: member_le_listsum_nat less_Suc_eq_le)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1122
  have fin_B: "finite ?B"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1123
    by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"],
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1124
      auto simp: member_le_listsum_nat less_Suc_eq_le fin)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1125
  have uni: "?C = ?A' \<union> ?B'" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1126
  have disj: "?A' \<inter> ?B' = {}" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1127
  have "card ?C = card(?A' \<union> ?B')" using uni by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1128
  also have "\<dots> = card ?A + card ?B"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1129
    using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g]
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1130
      bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1131
    by presburger
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1132
  finally show ?thesis .
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1133
qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1134
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1135
lemma card_length_listsum: --"By Holden Lee, tidied by Tobias Nipkow"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1136
  "card {l::nat list. size l = m \<and> listsum l = N} = (N + m - 1) choose N"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1137
proof (cases m)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1138
  case 0 then show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1139
    by (cases N) (auto simp: cong: conj_cong)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1140
next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1141
  case (Suc m')
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1142
    have m: "m\<ge>1" by (simp add: Suc)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1143
    then show ?thesis
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1144
    proof (induct "N + m - 1" arbitrary: N m)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1145
      case 0   -- "In the base case, the only solution is [0]."
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1146
      have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1147
        by (auto simp: length_Suc_conv)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1148
      have "m=1 \<and> N=0" using 0 by linarith
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1149
      then show ?case by simp
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1150
    next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1151
      case (Suc k)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1152
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1153
      have c1: "card {l::nat list. size l = (m - 1) \<and> listsum l =  N} =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1154
        (N + (m - 1) - 1) choose N"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1155
      proof cases
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1156
        assume "m = 1"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1157
        with Suc.hyps have "N\<ge>1" by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1158
        with `m = 1` show ?thesis by (simp add: binomial_eq_0)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1159
      next
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1160
        assume "m \<noteq> 1" thus ?thesis using Suc by fastforce
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1161
      qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1162
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1163
      from Suc have c2: "card {l::nat list. size l = m \<and> listsum l + 1 = N} =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1164
        (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1165
      proof -
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1166
        have aux: "\<And>m n. n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" by arith
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1167
        from Suc have "N>0 \<Longrightarrow>
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1168
          card {l::nat list. size l = m \<and> listsum l + 1 = N} =
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1169
          ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1170
        thus ?thesis by auto
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1171
      qed
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1172
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1173
      from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> listsum l = N} +
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1174
          card {l::nat list. size l = m \<and> listsum l + 1 = N}) = (N + m - 1) choose N"
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1175
        by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def)
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 59658
diff changeset
  1176
      thus ?case using card_length_listsum_rec[OF Suc.prems] by auto